Proceedings of the Sixth International Conference on Mathematics and Computing: ICMC 2020 981158060X, 9789811580604

Table of contents :
Committee
Technical Program Committee
Message from General Co-chairs
Message from Program Co-chairs
Preface
Contents
About the Editors
Preventing Differential Fault Analysis Attack on AEGIS Family of Ciphers
1 Introduction
2 Background
2.1 AEGIS Family of Ciphers
2.2 Attack Description for AEGIS Family
3 Countermeasures
3.1 Countermeasure for AEGIS-128
3.2 Countermeasure for AEGIS-256
3.3 Countermeasure for AEGIS-128L
4 Comparison of the Modified AEGIS with AEGIS
5 Conclusion
References
A Geometric-Based User Authentication Scheme for Multi-server Architecture: Cryptanalysis and Enhancement
1 Introduction
2 Related Works
3 Brief Review of Lin et al.'s Scheme
4 Weaknesses of Lin et al.'s Scheme
4.1 Minor Error
4.2 Security Weaknesses
5 Our Improved Scheme
6 Analysis of Our Improved Scheme
7 Conclusion
References
Solving the Search-LWE Problem by Lattice Reduction over Projected Bases
1 Introduction
2 Mathematical Preliminaries
2.1 Lattices and Their Bases
2.2 Lattice Basis Reduction
2.3 Typical Reduction Algorithms
3 The LWE Problem and Lattice Problems
3.1 Introduction to the LWE Problem
3.2 Reduction of the Search-LWE Problem to Lattice Problems
4 Solving the Search-LWE Problem Using Projected Lattices
4.1 Our Strategy
4.2 Development of Lattice Reduction for a Projected Basis
5 Experiments for the LWE Challenge
5.1 Implementation
5.2 Details of Experiments and Setting of Parameters
5.3 Experimental Results
6 Conclusion
References
Robust Watermarking Scheme for Compressed Image Through DCT Exploiting Superpixel and Arnold Transform
1 Introduction
2 Related Work
3 Superpixel
4 Proposed Scheme
4.1 Embedding Phase
4.2 Extraction Phase
5 Experimental Results and Comparisons
6 Conclusion
References
User Preference Multi-criteria Recommendations Using Neural Collaborative Filtering Methods
1 Introduction
2 Related Background
2.1 Neural Collaborative Filtering
2.2 Multi-Criteria Recommendation Systems
3 Proposed Approach
3.1 Phase I
3.2 Phase II
4 Performance Evaluation
4.1 Dataset and Evaluation Metrics
4.2 Results and Analysis
5 Conclusion and Future Work
References
Bifurcation Analysis of Tsunami Waves for the Modified Geophysical Korteweg–de Vries Equation
1 Introduction
2 Modified Geophysical Korteweg–de Vries Equation
3 Bifurcation Analysis
4 Different Tsunami Waves
5 Conclusions
References
Effect of Heating Location on Mixed Convection of a Nanofluid in a Partially Heated Enclosure with the Presence of Magnetic Field Using Two-Phase Model
1 Introduction
2 Mathematical Model
2.1 Nusselt Number and Entropy Generation
3 Numerical Methods
4 Grid Independence Test and Validation of Code
5 Results and Discussion
5.1 Study of Flow and Thermal Field
5.2 Effect of Position of the Heat Source on Nuav and Sav
5.3 Effect of Nanoparticle Volume Fraction on Nuav and Sav
5.4 Effect of Magnetic Field on Nuav and Sav
6 Conclusion
References
Weighted Matrix-Based Random Data Hiding Scheme Within a Pair of Interpolated Image
1 Introduction
2 Motivation
3 Proposed Scheme
3.1 Data Embedding
3.2 Data Extraction
4 Experimental Results
5 Conclusion
References
Moller Energy for an Exterior Metric of Relativistic Stars
1 Introduction
2 The Exterior Metric of Relativistic Stars in Scalar-Gauss-Bonnet Gravity
3 Result of Komar Mass
4 Moller Energy-Momentum Complex
5 Discussion
References
Modeling and Multistability of Ion-Acoustic Waves in Titan's Atmosphere
1 Introduction
2 Governing Model
3 The Nonlinear Schrödinger Equation (NLSE)
4 Formation of Dynamical System
5 Multistability Properties
6 Conclusion
References
Cryptanalysis of Kalyna Block Cipher Using Impossible Differential Technique
1 Introduction
2 Related Work
3 Preliminaries
3.1 Key Schedule Algorithm
3.2 Challenges in Kalyna Block Cipher
4 Impossible Differential Attack on Different Variants of Kalyna Block Cipher
4.1 Impossible Differential Attack on Kalyna 128/256
4.2 Impossible Differential Attack on Kalyna 256/512
4.3 Impossible Differential Attack on Kalyna-256/256
4.4 Impossible Differential Attack on Kalyna-512/512
4.5 Impossible Differential Attack on Kalyna-128/128
5 Results
6 Conclusion
References
Weighted Slope One with Threshold Filtering
1 Introduction
2 Related Work
3 Background
4 Weighted Slope Slope One with Threshold Filtering: Proposed Approach
5 Experimental Result
5.1 Metrics
6 Conclusion and Future Work
References
An Intelligent Phishing Detection Scheme Using Machine Learning
1 Introduction
1.1 Contributions
2 Related Work
3 Problem Formulation
4 Proposed Work
4.1 Methodology
5 Results
5.1 Dataset
5.2 Impact of Machine Learning Models
5.3 Impact of Ensemble Approach
5.4 Cross-Validation
6 Conclusion
References
Application of Measure of Non-compactness for the Existence of Solutions of an Infinite System of Differential Equations in the Sequence Spaces of Convergent and Bounded Series
1 Introduction and Preliminaries
2 Infinite System of Second-Order Differential Equations
3 Solvability in cs Space
4 Examples
5 Solvability in bs Space
References
Linear Secret Sharing Schemes with Finer Access Structure
1 Introduction
1.1 Our Contribution
2 Preliminaries
2.1 Coding Theory
2.2 Secret Sharing
3 Proposed Secret Sharing Scheme
3.1 Access Structure
3.2 Share Construction
3.3 Secret Reconstruction
4 Analysis of the Proposed Scheme
4.1 Cheating Detection and Cheating Identification
References
A Deep Learning Approach with Line Drawing for Isolated Online Bangla Character Recognition
1 Introduction
2 Dataset
3 Proposed Method
3.1 Reconstruction of Characters
3.2 Deep Learning Framework
4 Result and Discussion
5 Conclusion
References
Inverse Kinematics Based Computational Framework for Robot Manipulation Inspired by Human Movements
1 Introduction
2 Computational Framework for Vision System
3 Data-Acquisition System
3.1 Design of Bands and Singular Detection
3.2 Combining Multiple Colours to Locate Key-Points
3.3 Experimental Results
4 Forward and Inverse Kinematics
5 Validation
6 Conclusion
References
Slope One Meets Neighbourhood: Revisiting Slope One Predictor in Collaborative Filtering
1 Introduction
2 Related Work
3 Background
4 Proposed Methodology
4.1 Revisiting Slope One Prediction for Personalizing
4.2 Dataset
4.3 Metric for Evaluation
5 Experimental Results and Comparison
6 Conclusion and Future Work
References
Multi-Sensor Tracking Simulator Design and Its Challenges
1 Introduction
1.1 Coordinate Systems
2 Requirements Specifications
2.1 Hardware Requirements
2.2 Software Non-Functional Requirements
2.3 Software Functional Requirements
3 Architectural Design
4 Detailed Design
4.1 Edit Settings, Save Settings, and Load Settings on Initialization
4.2 Graphical Representation of Nominal Data
4.3 Visualization of Flight Parameters
4.4 Formulae Used for the Creation of Packets for EOTS, Radar, and Telemetry
4.5 Create an EOTS/Radar/Telemetry Packet
4.6 Send Packets
4.7 Control of Data Transmission
5 Testing
5.1 Socket Creation and Data Transmission
5.2 Rate of Transmission
5.3 Reading Nominal Files
5.4 Transmission of the Packet of Tracking Sensors
5.5 Graphical Representation and Visualization of Data
5.6 GUI Controls
6 Conclusion and Future Scope
References
Discovering Biomarkers in Parkinson's Disease Using Module Correspondence and Pathway Information
1 Introduction
2 Literature Review
3 Materials and Methods
4 Description
4.1 Topological Aspect of Module Correspondence
4.2 Module Correspondence from Pathway Point of View
5 Identification of Interesting Biomarkers
6 Hub Gene Analysis in Modules Based on Centrality
7 Conclusion and Future Work
References
Indian Regional Spoken Language Identification Using Deep Learning Approach
1 Introduction
2 Literature Review
3 Dataset and Pre-processing
3.1 Dataset Used
3.2 Pre-processing
4 Feature Extraction
4.1 Framing
4.2 Windowing
4.3 Fast Fourier Transform (FFT)
4.4 Mel Frequency Wrapping
4.5 Mel Cepstrum Coefficient
5 Result and Discussion
6 Conclusion
References
A Deep Learning Based Android Application to Detect the Leaf Diseases of Maize
1 Introduction
2 Material and Methods
2.1 About Dataset
2.2 About MobileNet Architecture
2.3 Android Application for Maize Leaf Detection
3 Discussion
4 Conclusion
References
Inversion Formula for the Wavelet Transform Associated with Legendre Transform
1 Introduction
2 Preliminaries
3 Continuous Legendre Wavelet Transformation (CLWT)
3.1 Admissible Legendre Wavelet
3.2 Continuous Legendre Wavelet Transform (CLWT)
4 Basic Properties of CLWT
5 Plancherel and Parseval's Relation for CLWT
References
Android Forensics Using Sleuth Kit Autopsy
1 Introduction
1.1 Android File System
1.2 Android SDK
2 Related Work
3 Methodology
3.1 Data Acquisition Stage
3.2 Artifact Finding Stage
4 Conclusion
References
A New Variant of Genetic Algorithm for Solving Gene Selection Problem
1 Introduction
2 Methodologies
2.1 Simple Genetic Algorithm (SGA)
2.2 Twin Removal Genetic Algorithm (TRGA)
2.3 Elitist Twin Removal Genetic Algorithm (ETRGA)
3 Gene Selection by Sample Classification
4 Experimental Results
4.1 Comparison Among Different Variants of GA Based on Benchmark Functions
4.2 Statistical Test
4.3 Convergence Graph for Benchmark Functions
4.4 Comparison Among Different Variants of GA on Gene Selection
5 Conclusion
References
Smartphone Traffic Analysis: A Contemporary Survey of the State-of-the-Art
1 Introduction
2 Smartphone Platforms
3 Smartphone Security Analysis
3.1 Behaviour Analysis
3.2 System Identification
3.3 Malware Detection
4 Conclusion and Future Direction
References
An Approach Towards IoT-Based Healthcare Management System
1 Introduction
1.1 IMS Layers
1.2 SIP
2 Related Work and Research Gaps
3 Proposed Work
4 Results and Discussions
5 Conclusion and Future Scope
References
DPL Model for Hyperthermia Treatment of Cancerous Cells Using Laser Heating Technique: A Numerical Study
1 Introduction
2 Mathematical Model
3 Computational Scheme
4 Results and Discussion
5 Conclusion
References
Unitary Equivalence of Quantum States in a Bipartite System
1 Introduction
2 Main Results
References
Dynamic Self-dual DeepBKZ Lattice Reduction with Free Dimensions
1 Introduction
2 Preliminaries
2.1 Lattices and Their Bases
2.2 Typical Lattice Problems
3 Lattice Basis Reduction
3.1 Typical Reduction Algorithms
3.2 Measurement of the Output Quality
4 Self-dual DeepBKZ with Free Dimensions
4.1 The Basic Idea of Our Algorithm
4.2 Selection of Free Dimensions
4.3 Detailed Algorithm
5 Experiments
5.1 Implementation
5.2 Experimental Results
6 Conclusion
References
Generation of Pseudorandom Sequence Using Regula-Falsi Method
1 Introduction
2 Regula-Falsi Method
3 Proposed Technique of Pseudorandom Sequence Generation
4 Experimental Results
5 Randomness Test and Security Analysis
5.1 NIST Randomness Test
5.2 Security Analysis
5.3 Information Entropy Analysis
6 Conclusion
References
1D-3v PIC-MCC Based Modeling and Simulation of Magnetized Low-Temperature Plasmas
1 Introduction
2 PIC-MCC Based Kinetic Model
2.1 Constraints
2.2 PIC-MCC Modules
2.3 Plasma Chemistry
2.4 Challenges
3 Plasma Transport in ROBIN Negative Ion Source
4 Comparison of Experimental Results with 1D-3V PIC-MCC Simulation Result
5 Conclusion
References
Dynamical Behavior of Ion-Acoustic Periodic and Solitary Structures in Magnetized Solar Wind Plasma
1 Introduction
2 Model Equations
3 Derivation of the Schamel Equation
4 Dynamical System
5 Exact Solution of the Schamel Equation
6 Conclusion
References
Substructuring Waveform Relaxation Methods with Time-Dependent Relaxation Parameter
1 Introduction
2 Classical DNWR and Its Convergence Results
3 Classical NNWR and Its Convergence Result
4 Analysis with Time-Dependent Parameter
4.1 DNWR Algorithm
4.2 NNWR Algorithm
5 Numerical Illustration
6 Concluding Remarks
References
A Generalized Hilbert Operator on Bloch Space and BMOA Spaces
1 Introduction
2 Main Results
References
Determining the Disease Status Using Gene Expression Analysis
1 Introduction
2 Similarity Identification Problem
3 Methodology
3.1 Data-Driven Grouping
3.2 Benjamini-Hochberg
3.3 Euclidian Distance and Cosine Similarity
4 Data and Experimental Results
4.1 Data and Experimental Setting
4.2 Results and Discussion
5 Conclusion
References
Generalized Double Statistical Convergence in Topological Groups
1 Introduction
2 Definitions and Notation
3 λ-Double Statistical Convergence
References
Exact Soliton Solutions to the Nano-Bioscience and Biophysics Equations Through the Modified Simple Equation Method
1 Introduction
2 Description of the Method
3 Formulation of the Solutions
3.1 The Equation of Nano-Ionic Currents Along MTs
3.2 The Equation of MTs as Nonlinear RLC Transmission Line
4 Graphical Representations and Discussion
4.1 Physical Interpretations of the Soliton Solutions of Nano-Ionic Currents Along MTs
4.2 Physical Interpretations of the Solitons of MTs as Nonlinear RLC Transmission Line
5 Conclusion
References
Design of Optimal Bayesian Reliability Test Plans for a Parallel System Based on Type-II Censoring
1 Introduction
2 Optimal Design of the Problem
3 Numerical Results
4 Conclusions
References
l2 Norm Prior-Based Modified Bright Channel for Low-Illumination Images
1 Introduction
2 Background of Bright Channel
3 Proposed Model
4 Advantage of the Proposed Method
5 Discussion and Results
6 Conclusion
References
Existence Results of Mild Solutions for Impulsive Fractional Differential Equations with Almost Sectorial Operators
1 Introduction
2 Preliminaries
3 Existence Results
3.1 Example
References
Effective Algebraic Methods are Widely Applicable
1 Introduction
2 Reductions of Associative Algebras
2.1 Associative Algebra
2.2 Construction of a Basis
2.3 Discriminant of Algebra
2.4 ALS-Criterion
3 Shemesh Criterion and Its Generalisation
4 Root Classifications of Polynomials
4.1 Discrimination Matrix and Discriminant Sequence
4.2 Number of Positive/Negative Roots
5 Applications
5.1 Dynamical Systems
5.2 Stability of Dynamical Systems
5.3 Decoherence-Free Subspaces
5.4 Cellular Networks
References
n-Fractals in Partial Metric Spaces
1 Introduction
2 Preliminaries
3 n-Fractals in Partial Metric Spaces
References
Some Existence Results on Impulsive Differential Equations
1 Introduction
2 Preliminaries
3 Main Results
4 Integro-Differential Equation
5 Application
References
Weighted Norm Inequality for General One-Sided Vector Valued Maximal Function
1 Introduction
2 Preliminaries
3 Main Results
References
Author Index

Citation preview

Advances in Intelligent Systems and Computing 1262

Debasis Giri · Rajkumar Buyya · S. Ponnusamy · Debashis De · Andrew Adamatzky · Jemal H. Abawajy   Editors

Proceedings of the Sixth International Conference on Mathematics and Computing ICMC 2020

Advances in Intelligent Systems and Computing Volume 1262

Series Editor Janusz Kacprzyk, Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland Advisory Editors Nikhil R. Pal, Indian Statistical Institute, Kolkata, India Rafael Bello Perez, Faculty of Mathematics, Physics and Computing, Universidad Central de Las Villas, Santa Clara, Cuba Emilio S. Corchado, University of Salamanca, Salamanca, Spain Hani Hagras, School of Computer Science and Electronic Engineering, University of Essex, Colchester, UK László T. Kóczy, Department of Automation, Széchenyi István University, Gyor, Hungary Vladik Kreinovich, Department of Computer Science, University of Texas at El Paso, El Paso, TX, USA Chin-Teng Lin, Department of Electrical Engineering, National Chiao Tung University, Hsinchu, Taiwan Jie Lu, Faculty of Engineering and Information Technology, University of Technology Sydney, Sydney, NSW, Australia Patricia Melin, Graduate Program of Computer Science, Tijuana Institute of Technology, Tijuana, Mexico Nadia Nedjah, Department of Electronics Engineering, University of Rio de Janeiro, Rio de Janeiro, Brazil Ngoc Thanh Nguyen , Faculty of Computer Science and Management, Wrocław University of Technology, Wrocław, Poland Jun Wang, Department of Mechanical and Automation Engineering, The Chinese University of Hong Kong, Shatin, Hong Kong

The series “Advances in Intelligent Systems and Computing” contains publications on theory, applications, and design methods of Intelligent Systems and Intelligent Computing. Virtually all disciplines such as engineering, natural sciences, computer and information science, ICT, economics, business, e-commerce, environment, healthcare, life science are covered. The list of topics spans all the areas of modern intelligent systems and computing such as: computational intelligence, soft computing including neural networks, fuzzy systems, evolutionary computing and the fusion of these paradigms, social intelligence, ambient intelligence, computational neuroscience, artificial life, virtual worlds and society, cognitive science and systems, Perception and Vision, DNA and immune based systems, self-organizing and adaptive systems, e-Learning and teaching, human-centered and human-centric computing, recommender systems, intelligent control, robotics and mechatronics including human-machine teaming, knowledge-based paradigms, learning paradigms, machine ethics, intelligent data analysis, knowledge management, intelligent agents, intelligent decision making and support, intelligent network security, trust management, interactive entertainment, Web intelligence and multimedia. The publications within “Advances in Intelligent Systems and Computing” are primarily proceedings of important conferences, symposia and congresses. They cover significant recent developments in the field, both of a foundational and applicable character. An important characteristic feature of the series is the short publication time and world-wide distribution. This permits a rapid and broad dissemination of research results. Indexed by SCOPUS, DBLP, EI Compendex, INSPEC, WTI Frankfurt eG, zbMATH, Japanese Science and Technology Agency (JST), SCImago. All books published in the series are submitted for consideration in Web of Science.

More information about this series at http://www.springer.com/series/11156

Debasis Giri Rajkumar Buyya S. Ponnusamy Debashis De Andrew Adamatzky Jemal H. Abawajy •

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Editors

Proceedings of the Sixth International Conference on Mathematics and Computing ICMC 2020

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Editors Debasis Giri Department of Information Technology Maulana Abul Kalam Azad University of Technology Haringhata, West Bengal, India S. Ponnusamy Department of Mathematics Indian Institute of Technology Madras Chennai, India Andrew Adamatzky Unconventional Computing Laboratory Department of Computer Science and Creative Technologies University of the West of England Bristol, UK

Rajkumar Buyya School of Computing and Information Systems University of Melbourne Melbourne, VIC, Australia Debashis De Department of Computer Science and Engineering Maulana Abul Kalam Azad University of Technology Haringhata, West Bengal, India Jemal H. Abawajy Faculty of Science Engineering and Built Environment Deakin University Geelong Geelong, VIC, Australia

ISSN 2194-5357 ISSN 2194-5365 (electronic) Advances in Intelligent Systems and Computing ISBN 978-981-15-8060-4 ISBN 978-981-15-8061-1 (eBook) https://doi.org/10.1007/978-981-15-8061-1 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Committee

Chief Patron Prof. Avinash Khare, Honorable Vice Chancellor, Sikkim University, India Patron Prof. Rajkumar Buyya, University of Melbourne, Australia General Co-chairs Dr. P. K. Saxena, Former Director, DRDO, SAG, Delhi, India Prof. P. D. Srivastava, IIT Bhilai, India Prof. Debashis De, Maulana Abul Kalam Azad University of Technology, WB, India Program Co-chairs Dr. Debasis Giri, Maulana Abul Kalam Azad University of Technology, WB, India Prof. S. Ponnusamy, IIT Madras, India Dr. Satish Narayana Srirama, University of Tartu, Estonia Prof. Jemal Hussien, Deakin University, Australia Prof. Andrew Adamatzky, University of West of England, UK Publicity Chair Dr. Nilanjan Dey, Techno India College of Technology, Kolkata, India Organizing Chair Partha Pratim Ray, Department of CA, Sikkim University, India Organizing Committee Dr. Swarup Roy, Department of CA, Sikkim University, India Dr. Mohan Pratap Pradhan, Department of CA, Sikkim University, India Mrs. Chunnu Khawas, Department of CA, Sikkim University, India

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Committee

Dr. Rebika Rai, Department of CA, Sikkim University, India Mrs. Lekhika Chettri, Department of CA, Sikkim University, India Dr. Thoudam Roshan Singh, Department of Mathematics, Sikkim University, India Dr. Namita Behera, Department of Mathematics, Sikkim University, India Dr. Bipul Pal, Department of Mathematics, Sikkim University, India Ms. Rinkila Bhutia, Department of Mathematics, Sikkim University, India

Technical Program Committee TPC for Computing (Alphabetically) Abderrahmen Mtibaa, New Mexico State University, Mexico Amlan Chakrabarti, Calcutta University, India Anand Kumar M, NIT Karnataka, India Andrew Adamatzky, University of the West of England, UK Andy Adamatzky University of the West of England, UK Anilkumar Devarapu, Albany State University, USA Anirban Mondal, Ashoka University, India Arif Ahmed Sk, University of Tromsø, Norway Ashok Kumar Das, IIIT Hyderabad, India Athanasios V. Vasilakos, Luleå University of Technology, Sweden Bidyut Patra, NIT Rourkela, India Bivas Mitra, IIT Kharagpur, India Chandan Kumar Chanda, IIEST, India Chandrashekhar Y. Meshram, Rani Durgavati University, India Christina Boura, Université de Versailles Saint-Quentin-en-Yvelines, France Christine Fernandez, University of Poitiers, UMR CNRS, France Dhananjoy Dey, SAG, DRDO, India Debashis De, Maulana Abul Kalam Azad University of Technology, WB, India Debasis Giri, Maulana Abul Kalam Azad University of Technology, WB, India Debiao He, Wuhan University, China Debi Prosad Dogra IIT Bhubaneswar, India Dhananjoy Dey, SAG, DRDO, India Dinesh Dash, NIT Patna, India Dipanwita Roy Chowdhury, IIT kharagpur, India Dung Hoang Duong, University of Wollongongm, Australia Fagen Li, Univ. of Electronic Science and Technology of China, China Fahreddin Abdullaye, v Kyrgyz Turkey Manas University, Turkey Fernando Velez, Universidade da Beira Interior, Portugal Gerardo Pelosi, Politecnico di Milano, Italy Goutham Reddy Alavalapati, NIT Andhra Pradesh, India Indivar Gupta, SAG, DRDO, India Janka Chlebikova, University of Portmouth, UK

Committee

Jaydeb Bhaumik, Jadavpur University, India Jemal Hussien, Deakin University, Australia Jiqiang Lu, Institute for Infocomm Research, Singapore Jorge Sa Silva, University of Coimbra, Portugal Junwei Zhou, Wuhan University of Technology, China Kanesaraj Ramasamy, Multimedia University, Malaysia Keshav Dahal, University of the West of Scotland, UK Kolin Paul, IIT Delhi, India Kouichi Sakurai, Kyushu University, Japan Kuheli Sai, University of Pittsburgh, Pennsylvania Marko Hölbl, University of Maribor, Slovenia Michal Choras, ITTI Ltd., Poland Meng Yu, Roosevelt University, USA Mohd Helmy Abd Wahab, Universiti Tun Hussein Onn Malaysia, Malaysia Nai-Wei Lo, NTU of Science and Technology, Taiwan Niladri Puhan, IIT Bhubaneswar, India Nilanjan Dey, Techno India College of Technology, Kolkata, India Noboru Kunihiro, The University of Tokyo, Japan Olivier, Blazy, Université de Limoges, France Oscar Castillo, Tijuana Institute of Technology, Mexico P. K. Saxena, SCCS & Former Director, SAG DRDO, India Philippe Gaborit, University of Limoges, France Prasanna Mishra, DRDO, India Rajendra Prasath, IIIT Sricity, India Rajkumar Buyya, University of Melbourne, Australia Ranbir Sanasam, IIT Guwahati, India Raviraj Pandian, Kalaignar Karunanidhi Institute of Technology, India Saibal Pal, DRDO, India Samiran Chattopadhyay Jadavpur University, India Sandip Karmakar, IIIT Kalyani, India Sarmistha Neogy, Jadavpur University, India Satish Narayana Srirama, University of Tartu, Estonia Seonghan Shin, NIAIST, Japan Sharma Chakravarthy, The University of Texas at Arlington, USA Shehzad Ashraf Chaudhry, International Islamic University, Pakistan Sherali Zeadally, University of Kentucky, USA Siddhartha Bhattacharyya, RCC IIT, India Sk Hafizul Islam, IIIT Kalyani, India Sokratis Katsikas, NTNU, Norway Somitra Sanadhya, IIT Ropar, India Subhankar Joardar, Haldia Institute of Technology, India Subrata Dutta, NIT Jamshedpur, India Sudip Misra, IIT Kharagpur, India Svetla Nikova, KU Leuven, Belgium Tanmoy Maitra, KIIT University, India

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Committee

Thoudam Doren Singh, CDAC, Mumbai, India Valentina Emilia Balas, Aurel Vlaicu University of Arad, Romania Vasudha Bhatnagar, Delhi University, India Yao Zhao, Beijing Jiaotong University, China TPC for Mathematics (Alphabetically) Abdullah M. Rababah, Jordan University of S & T, Jordan Ameeya Nayak, IIT Roorkee, India Anirban Banerjee, IISER, Kolkata, India Arya Kumar Bedabrata Chand, IIT Madras, India Avishek Adhikari, University of Calcutta, India B. N. Mandal, ISI, Kolkata, India Bapan Ghosh, NIT Meghalaya, India Binod Chandra Tripathy, Tripura University, India Diana Mendes, ISCTE-IUL, Portugal Dipak Jana, Haldia Institue of Technology, India Debjani Chakraborty, IIT Kharagpur, India Don Hong, Middle Tennessee State University, USA Duan Li, The Chinese University of Hong Kong, Hong Kong Edgar Martinez-Moro, Universidad de Valladolid, Spain Emel Aşıcı, Karadeniz Technical University, Turkey Geetanjali Panda, Indian Institute of Technology Kharagpur, India Gennadii Demidenko, Sobolev Institute of Mathematics, Russia Heinrich Begehr, Free University Berlin, Germany Inessa Matveeva, Sobolev Institute of Mathematics, Russia Junzo Watada Waseda University, Japan Kinkar Das, Sungkyunkwan University, South Korea Konstantin Volkov, Kingston University, London Lakshmi Kanta Patra, IIIT Ranchi, India Leopoldo Eduardo, Cárdenas-Barrón Tecnológico de Monterrey, Mexico Ljubisa Kocinac, University of Nis, Serbia Madhumangal Pal, Vidyasagar University, India Margareta Heilmann, University of Wuppertal, Germany María A. Navascués, Universidad de Zaragoza, Spain Manoranjan Maiti Vidyasagar University, India Muhammad Noor, COMSATS IIT, Pakistan Mujahid Abbas, University of Pretoria, South Africa Pamini Thangarajah, Mount Royal University, Canada P. D. Srivastava, IIT Bhilai, India Praveen Kumar Gupta, NIT Silchar, India Ravi P. Agarwal, Texas A & M University, USA S. Ponnusamy, IIT Madras, India Santanu Sarkar, IIT Madras, India Saru Kumari, Ch. Charan Singh University, India

Committee

ix

Shanta Laishram Indian Statistical Institute, New Delhi, India Shay Gueron, University of Haifa, Israel Shuang Li, NIAST, China Soumen Maity, IISER Pune, India Srinivas Jangirala, O. P. Jindal Global University, India Subhas Khajanchi, IIT Roorkee, India Subhash Bhalla University of Aizu, Japan Suchandan Kayal, NIT Rourkela, India Susanta Maity, NIT Arunachal Pradesh, India Takeshi Koshiba, Waseda University, Japan Teodor Bulboaca, Babes-Bolyai University, Cluj-Napoca, Romania Tian-Xiao He, Illinois Wesleyan University, USA Vilem Novak, University of Ostrava, Czech Republic Weizhi Meng, Technical Universtiy of Denmark, Denmark Zakia Hammouch, FST Errachidia Moulay Ismail University, Morocco Zhisheng Shuai, University of Central Florida, USA Additional Reviewers Gopal Shit, Amit Kumar Verma, Partha Pratim Ray, Buddhananda Banerjee, Subhabrata Barman, Tanmoy Chakraborty, Sanjay Chatterji, Barun Das, Manju Khan, Dilip Maity, Amit Maji, Mousumi Mandal, Sourav Mandal, Tufan Naiya, Saroj Padhan, Pratima Panigrahi, Soumen Pati, Suparna Saha, Kuheli Sai.

Message from General Co-chairs

It gives me a great pleasure to welcome you to ICMC 2020, the 6th edition of the premier annual conference on Mathematics and Computing. This year, ICMC will be held in the Department of Computer Applications, Sikkim University, Gangtok, Sikkim, India, where advanced technology infrastructure and collection of creative talents are gathered, making it an excellent position to develop advanced Computing. ICMC has been the most impactful conference on all aspects of Mathematics and Computing. Papers published in this impactful conference represent the hard work of many outstanding researchers from around the world. We are very delighted to report that ICMC remains at the forefront of computer networks. This year, the main conference embodies a set of 45 papers out of 172 submissions, selected through careful and rigorous peer review by TPC members as best of the best submissions, and organized around 2 tracks and 18 keynote sessions. It is our honour to have invited most prominent scholars as our conference: Prof. Dipanwita Roy Chowdhury, IIT Kharagpur, India; Prof. P. D. Srivastava, IIT Bhilai, India; Dr. P. K. Saxena, DRDO, India; Prof. S. Ponnusamy, IIT Madras, India; Prof. Samiran Chattopadhyay, Jadavpur University, India; Dr. Ashok Kumar Das, IIIT Hyderabad; Dr. Sanasam Ranbir Singh, IIT Guwahati, India; Dr. Biswapati Jana, Vidyasagar University, India; Prof. Ekrem SAVAS, Usak University, Usak/Turkey; Mr. Aninda Bose, Springer, India; Prof. Duan Li, City University of Hong Kong, Hong Kong; Prof. Bidyut B. Chaudhuri, Techno India University, Kolkata, and ISI Kolkata, India; Prof. Neeraj Kumar, Thapar Institute of Engineering and Technology, India; Mr. Giovanni BluMitolo, Italy; Dr. Bidyut Kr. Patra, NIT Rourkela, Dr. Debasis Giri, MAKAUT, WB, India; Prof. Debashis De, MAKAUT, WB, India; and Dr. Swadesh Kumar Sahoo, Indian Institute of Technology Indore, India. First, we would like to thank to all program members and organizing committee members who have done an outstanding job in carrying out the paper review tasks. In particular, we would like to express our appreciation to Dr. Debasis Giri and Mr. Partha Pratim Ray who spent great effort to develop the review system. We would like to thank Honourable Vice-Chancellor of Sikkim University, Sikkim, for his support for providing Excellent Venue. Last but not least, we thank our patrons xi

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and sponsors for their warm support. We must thank IEEE Fellow Prof. Rajkumar Buyya for his tireless effort and constant support of ICMC 2020 for his guidance and direction. Finally, we thank all the conference participants for making ICMC a success. P. K. Saxena P. D. Srivastava Debashis De

Message from Program Co-chairs

It is a great pleasure for us to organize the Sixth International Conference on Mathematics and Computing 2020 held from March 18–20, 2020, at Gangtok, Sikkim, India. Our main goal is to provide an opportunity to the participants to learn about contemporary research in Mathematics and Computing, and exchange ideas among themselves and with experts present in the conference as tutorial presenters and the plenary as well as invited speakers. Sixteen speakers from India and abroad agreed to deliver their talks and some of them acted as session chairs. After an initial call for papers, 172 papers were submitted at the conference. All submitted papers were sent to external referees and, after refereeing, 45 papers were recommended for publication for the conference proceedings that will be published by Springer series: Advances in Intelligent Systems and Computing. ICMC 2020 has come up as an international platform to deliver and share novel knowledge in various fields on applied mathematics and computing of interest. We are grateful to the chief patron, patron, general co-chairs, program co-chairs, publicity chair, speakers, participants, referees, organizers, sponsors, and funding agencies for their support and help without which it would have been impossible to organize the conference. We owe our gratitude to the volunteers who work behind the scene tirelessly in taking care of the details in making this conference a success. Debasis Giri S. Ponnusamy Satish Narayana Srirama Jemal Hussien Andrew Adamatzky

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Preface

In the last two decades, scientific computing has become an important contributor to all scientific research programs. It is particularly important for the solution of research problems that are unsolvable by traditional theory and experimental approaches, hazardous to study in the laboratory, or time-consuming or expensive to be solved by traditional means. With mathematical modeling and computational algorithms, many more problems from the realm of science, commerce as well as other walks of life can be solved efficiently. The International Conference on Mathematics and Computing (ICMC) is such a premier forum for the presentation of new advances and research results in the fields of Cryptography, Network Security, Cybersecurity, Internet of Things, Due & Edge Computing, Mathematics, Statistics and Scientific Computing. The conference will bring together leading academic scientists, experts from industry, and researchers in their domains of expertise from around the world. Earlier, ICMC was organized in 2013, 2015, 2017, 2018, and 2019. The 6th ICMC 2020 aims to bring together both novice and experienced scientists with developers, to meet new colleagues, collect new ideas, and establish new cooperation between research groups and provide a platform for researchers from academic and industry to present their original work and exchange ideas, information, techniques, and applications in the field of Computational Applied Mathematics, including, but not limited to the broad topics of Operations Research, Soft Computing, Cryptology, Network Security, Cybersecurity, Internet of Things, Due & Edge Computing, Image Processing, Pure and Applied Mathematics, and other emerging areas of research. The 6th ICMC 2020 is organized by the Department of Computer Applications, Sikkim University, Gangtok, Sikkim, India. This conference received 172 papers from different parts of India as well as world. This book of abstracts contains 45 full papers which fall under mathematics and computing tracks. We would like to thank the Vice-Chancellor and officials of Sikkim University, patron, respected general co-chairs, publicity chair, organizing committee members, volunteers, sponsors,

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attendees, and authors who attended and made this conference a success. ICMC 2020 proceedings will be published in the Springer series: Advances in Intelligent Systems and Computing. Haringhata, India Melbourne, Australia Chennai, India Haringhata, India Bristol, UK Geelong, Australia

Debasis Giri Rajkumar Buyya S. Ponnusamy Debashis De Andrew Adamatzky Jemal Hussien

Contents

Preventing Differential Fault Analysis Attack on AEGIS Family of Ciphers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Swapan Maiti and Dipanwita Roy Chowdhury

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A Geometric-Based User Authentication Scheme for Multi-server Architecture: Cryptanalysis and Enhancement . . . . . . . . . . . . . . . . . . . Debasis Giri and Tanmoy Maitra

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Solving the Search-LWE Problem by Lattice Reduction over Projected Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Satoshi Nakamura, Nariaki Tateiwa, Koha Kinjo, Yasuhiko Ikematsu, Masaya Yasuda, and Katsuki Fujisawa Robust Watermarking Scheme for Compressed Image Through DCT Exploiting Superpixel and Arnold Transform . . . . . . . . . . . . . . . . . . . . Prabhash Kumar Singh, Biswapati Jana, and Kakali Datta User Preference Multi-criteria Recommendations Using Neural Collaborative Filtering Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kaira Nithin Goud, Y. V. Ramanjaneyulu, Korra Sathya Babu, and Bidyut Kr. Patra Bifurcation Analysis of Tsunami Waves for the Modified Geophysical Korteweg–de Vries Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Aranya Jha, Manav Tyagi, Harshvardhan Anand, and Asit Saha Effect of Heating Location on Mixed Convection of a Nanofluid in a Partially Heated Enclosure with the Presence of Magnetic Field Using Two-Phase Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Subhasree Dutta and Somnath Bhattacharyya Weighted Matrix-Based Random Data Hiding Scheme Within a Pair of Interpolated Image . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Debkumar Bera, Biswapati Jana, Partha Chowdhuri, and Debasis Giri

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Moller Energy for an Exterior Metric of Relativistic Stars . . . . . . . . . . 103 Wang Liwei Modeling and Multistability of Ion-Acoustic Waves in Titan’s Atmosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 Jharna Tamang and Asit Saha Cryptanalysis of Kalyna Block Cipher Using Impossible Differential Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 Sunny Kumar Gupta, Mohona Ghosh, and Sraban Kumar Mohanty Weighted Slope One with Threshold Filtering . . . . . . . . . . . . . . . . . . . . 143 Subrata Das, Bidyut Kumar Patra, and Jitendra Kumar An Intelligent Phishing Detection Scheme Using Machine Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 Aaisha Makkar, Neeraj Kumar, Lakshit Sama, Satyam Mishra, and Yash Samdani Application of Measure of Non-compactness for the Existence of Solutions of an Infinite System of Differential Equations in the Sequence Spaces of Convergent and Bounded Series . . . . . . . . . . 167 Niraj Sapkota and Rituparna Das Linear Secret Sharing Schemes with Finer Access Structure . . . . . . . . . 179 Sanyam Mehta and Vishal Saraswat A Deep Learning Approach with Line Drawing for Isolated Online Bangla Character Recognition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 Himadri Mukherjee, Chandrima Majumder, Ankita Dhar, Shibaprasad Sen, Sk Md Obaidullah, and Kaushik Roy Inverse Kinematics Based Computational Framework for Robot Manipulation Inspired by Human Movements . . . . . . . . . . . . . . . . . . . . 201 Geet Patel, Roshani, Tanya Garg, Sarangi Patel, Tapas Kumar Maiti, and Bhaskar Chaudhury Slope One Meets Neighbourhood: Revisiting Slope One Predictor in Collaborative Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 Rabi Shaw and Bidyut Kumar Patra Multi-Sensor Tracking Simulator Design and Its Challenges . . . . . . . . . 227 Sourav Kaity, Biswapati Jana, P. K. Das Gupta, Rakesh Barua, and Lalatendu Das Discovering Biomarkers in Parkinson’s Disease Using Module Correspondence and Pathway Information . . . . . . . . . . . . . . . . . . . . . . 249 Pooja Sharma, Anuj K. Pandey, Dhruba K. Bhattacharyya, Jugal K. Kalita, and Subhash C. Dutta

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Indian Regional Spoken Language Identification Using Deep Learning Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 Bachchu Paul, Santanu Phadikar, and Somnath Bera A Deep Learning Based Android Application to Detect the Leaf Diseases of Maize . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 Utpal Barman, Diganto Sahu, and Golap Gunjan Barman Inversion Formula for the Wavelet Transform Associated with Legendre Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 Jyoti Saikia and C. P. Pandey Android Forensics Using Sleuth Kit Autopsy . . . . . . . . . . . . . . . . . . . . . 297 Atonu Ghosh, Koushik Majumder, and Debashis De A New Variant of Genetic Algorithm for Solving Gene Selection Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309 Priya Das, Biswajit Jana, and Sriyankar Acharyya Smartphone Traffic Analysis: A Contemporary Survey of the State-of-the-Art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325 Sumit Kumar, S. Indu, and Gurjit Singh Walia An Approach Towards IoT-Based Healthcare Management System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 Khushboo Singla, Rudra Arora, and Sakshi Kaushal DPL Model for Hyperthermia Treatment of Cancerous Cells Using Laser Heating Technique: A Numerical Study . . . . . . . . . . . . . . . 357 G. C. Shit and Amal Bera Unitary Equivalence of Quantum States in a Bipartite System . . . . . . . 371 Arnab Patra, Amit Shrivastava, Rohit Sharma, and P. D. Srivastava Dynamic Self-dual DeepBKZ Lattice Reduction with Free Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377 Satoshi Nakamura, Yasuhiko Ikematsu, and Masaya Yasuda Generation of Pseudorandom Sequence Using Regula-Falsi Method . . . 393 Aakash Paul, Shyamalendu Kandar, and Bibhas Chandra Dhara 1D-3v PIC-MCC Based Modeling and Simulation of Magnetized Low-Temperature Plasmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407 Miral Shah, Bhaskar Chaudhury, Mainak Bandyopadhyay, and Arun Chakraborty Dynamical Behavior of Ion-Acoustic Periodic and Solitary Structures in Magnetized Solar Wind Plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419 Punam Kumari Prasad and Asit Saha

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Substructuring Waveform Relaxation Methods with Time-Dependent Relaxation Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429 Bankim C. Mandal and Soura Sana A Generalized Hilbert Operator on Bloch Space and BMOA Spaces . . . 441 S. Naik and P. K. Nath Determining the Disease Status Using Gene Expression Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451 Dulal Adak, Suman Mitra, Biswajit Jana, and Sriyankar Acharyya Generalized Double Statistical Convergence in Topological Groups . . . . 461 Ekrem Savas Exact Soliton Solutions to the Nano-Bioscience and Biophysics Equations Through the Modified Simple Equation Method . . . . . . . . . . 469 Md. Abdul Kayum, Hemonta Kumar Barman, and M. Ali Akbar Design of Optimal Bayesian Reliability Test Plans for a Parallel System Based on Type-II Censoring . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483 P. N. Bajeel and M. Kumar l2 Norm Prior-Based Modified Bright Channel for Low-Illumination Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491 Riya, Bhupendra Gupta, and Subir Singh Lamba Existence Results of Mild Solutions for Impulsive Fractional Differential Equations with Almost Sectorial Operators . . . . . . . . . . . . . 501 M. C. Ranjini Effective Algebraic Methods are Widely Applicable . . . . . . . . . . . . . . . . 515 Takeo Kamizawa n-Fractals in Partial Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529 S. Minirani Some Existence Results on Impulsive Differential Equations . . . . . . . . . 535 Rajib Haloi Weighted Norm Inequality for General One-Sided Vector Valued Maximal Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 549 Duranta Chutia and Rajib Haloi Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565

About the Editors

Dr. Debasis Giri is currently working as an Associate Professor in the Department of Information Technology, Maulana Abul Kalam Azad University of Technology (formerly known as West Bengal University of Technology), West Bengal, India. Prior to this, he also held academic positions as the Professor in the Department of Computer Science and Engineering and Dean in the School of Electronics, Computer Science and Informatics, Haldia Institute of Technology, Haldia, India. He did his masters (M.Tech. and M.Sc.) both from IIT Kharagpur, India, and also completed his Ph.D. from IIT Kharagpur, India. He is tenth all India rank holder in Graduate Aptitude Test in Engineering (GATE) in 1999. He received a certificate from All India Science Teachers’ Association in Science Aptitude & Talent Search Test in 1988. His current research interests include cryptography, information security, e-commerce security, and design and analysis of algorithms. He has authored more than 75 research papers in reputed international journals and conference proceedings. He is serving as an Associate Editorial of the Journal of (i) Information Security and Applications (Elsevier), (ii) Security and Privacy Journal (Wiley), (iii) International Journal of Communication Systems (Wiley), (iv) Security and Communication Networks, (v) Electrical and Computer Engineering Innovations, and (vi) Azerbaijan Journal of High Performance Computing. He has been involved as a Technical Program Committee member of several international conferences in repute. He is the Founder of the International Conference on Mathematics and Computing. He is also a program committee member of several international conferences. He is a member of IEEE, and a life member of Cryptology Research Society of India, and the International Society for Analysis, its Applications and Computation (ISAAC). Dr. Rajkumar Buyya is a Redmond Barry Distinguished Professor and Director of the Cloud Computing and Distributed Systems (CLOUDS) Laboratory at the University of Melbourne, Australia. He is also serving as the founding CEO of Manjrasoft, a spin-off company of the university, commercializing its innovations in cloud computing. He served as a Future Fellow of the Australian Research

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About the Editors

Council during 2012–2016. He has authored over 625 publications and 7 textbooks including “Mastering Cloud Computing” published by McGraw Hill, China Machine Press, and Morgan Kaufmann for Indian, Chinese, and international markets, respectively. He also edited several books including “Cloud Computing: Principles and Paradigms” (Wiley Press, USA, February 2011). He is one of the highly cited authors in computer science and software engineering worldwide (h-index = 132, g-index = 294, 92,500+ citations). “A Scientometric Analysis of Cloud Computing Literature” by German scientists ranked him as the World’s Top-Cited (#1) Author and the World's Most-Productive (#1) Author in Cloud Computing. He is recognized as a “Web of Science Highly Cited Researcher” for four consecutive years since 2016, a Fellow of IEEE, and Scopus Researcher of the Year 2017 with Excellence in Innovative Research Award by Elsevier and recently (2019) received “Lifetime Achievement Awards” from two Indian universities for his outstanding contributions to cloud computing and distributed systems. Software technologies for grid and cloud computing developed under his leadership have gained rapid acceptance and are in use at several academic institutions and commercial enterprises in 40 countries around the world. He served as the founding Editor-in-Chief of the IEEE Transactions on Cloud Computing. He is currently serving as the Co-Editor-in-Chief of Journal of Software: Practice and Experience, which was established 50 years ago. Dr. S. Ponnusamy is a Institute Chair Professor in the Department of Mathematics of IIT Madras. He has earned his B.Sc. (1980) and M.Sc. (1992) from the University of Madras. He completed his Ph.D. (1989) at IIT Kanpur. His current research interest includes complex analysis, quasiconformal and harmonic mappings, special functions, and functions spaces. He is the founding “Fellow of the Forum de Analystes, Chennai, India,” 1992. He was elected as a “Fellow of The National Academy of Sciences, India” in the year 2002. He served 5 years as the Head of the Indian Statistical Institute, Chennai Centre (October 2012–October 2017). He is the currently the President of the Ramanujan Mathematical Society, India. He has been associated with a number of organizations for popularization of science and serves on the editorial boards of several peer-reviewed international journals. He is often invited to give plenary talks at international conferences and to give lectures in universities all over the world. He took part as a researcher in many funded projects in India and Abroad. He has been refereeing for more than 100 journals. He has written four books and another book with Herb Sivlerman. One of his books won the Best Selling Author Award in 2002. He has edited several volumes and international conference proceedings. He has published more than 250 technical articles in reputed international journals (such as Advances in Mathematics, Annales Academia Scientiarum Fennica Mathematica, Applied Mathematics and Computation, Archiv der Mathematik, Bulletin des Sciences Mathematiques, Bulletin/Journal of the Australian Mathematical Society, Complex Variables and Elliptic Equations, Complex Analysis and Operator Theory, Computational Methods and Function Theory, Indagationes Mathematicae, Journal

About the Editors

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of London Mathematical Society, Integral Equations and Operator Theory, Journal Computational and Applied Mathematics, Journal of Mathematical Analysis and Applications, Mathematika, Mathematishe Annalen, Mathematische Nachrichten, Mathematische Zeitschrift, Monatshefte fuer Mathematik, Nonlinear Analysis, Potential Analysis, Proceedings of American Mathematical Society, Results in Mathematics, Rocky Mountain Journal of Mathematics The Journal of Geometric Analysis) and has solved several long-standing open problems and conjectures. He has been a Visiting Professor of a number of universities in abroad (e.g., Hengyang Normal University and Hunan Normal University, China; Kazan Federal University and Petrozavodsk State University, Russia; University Sains Malaysia, Malaysia; the University of Aalto, the University of Turku, and the University of Helsinki, Finland; Texas Tech University, Lubbock, USA). Prof. Debashis De earned his M.Tech. from the University of Calcutta in 2002 and his Ph.D. (Engineering) from Jadavpur University in 2005. He is the Professor and Director in the Department of Computer Science and Engineering of the Maulana Abul Kalam Azad University of Technology, West Bengal (Former West Bengal University of Technology), India, and Adjunct Research Fellow at the University of Western Australia, Australia. He is a senior member of the IEEE, life member of CSI, and member of the International Union of Radio science. He was awarded the prestigious Boyscast Fellowship by the Department of Science and Technology, Government of India, to work at the Herriot-Watt University, Scotland, UK. He received the Endeavour Fellowship Award during 2008–2009 by DEST Australia to work at the University of Western Australia. He received the Young Scientist Award both in 2005 at New Delhi and in 2011 at Istanbul, Turkey, from the International Union of Radio Science, Head Quarter, Belgium. His research interests include mobile cloud computing and Green mobile networks. He has published in more than 250 peer-reviewed international journals and 200 international conference papers, 6 research monographs, and 10 textbooks. His h-index is 26 and i10 index is 91. Total citation is 3330. He is an Associate Editor of journal IEEE ACCESS, Editor Hybrid computational intelligence, Journal Array, Elsevier. Dr. Andrew Adamatzky is the Professor of Unconventional Computing and Director of the Unconventional Computing Laboratory, Department of Computer Science, University of the West of England, Bristol, UK. He does research in molecular computing, reaction-diffusion computing, collision-based computing, cellular automata, slime mould computing, massive parallel computation, applied mathematics, complexity, nature-inspired optimization, collective intelligence and robotics, bionics, computational psychology, nonlinear science, novel hardware, and future and emergent computation. He authored 7 books, mostly notable are “Reaction-Diffusion Computing,” “Dynamics of Crow Minds,” and “Physarum Machines” and edited 22 books in computing, most notable are “Collision Based Computing,” “Game of Life Cellular Automata,” and “Memristor Networks”; he also produced a series of influential artworks published in the atlas “Silence of Slime Mould”. He is the founding Editor-in-Chief of “Journal of Cellular

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Automata” and “Journal of Unconventional Computing” and Editor-in-Chief of “Journal of Parallel, Emergent, Distributed Systems” and “Parallel Processing Letters.” Dr. Jemal H. Abawajy (SM’11) is a Full Professor at the Faculty of Science, Engineering and Built Environment, Deakin University, Australia. He has delivered more than 60 keynote sessions and seminars worldwide and has been involved in the organization of more than 300 international conferences in various capacity including chair and general co-chair. He has also served on the editorial board of numerous international journals including IEEE Transaction on Cloud Computing. He is the author/co–author of more than 400 refereed articles and supervised numerous Ph.D. students to completion.

Preventing Differential Fault Analysis Attack on AEGIS Family of Ciphers Swapan Maiti and Dipanwita Roy Chowdhury

Abstract AEGIS, a dedicated authenticated encryption algorithm is one of the winners of the CAESAR portfolio. In literature, there exist fault attacks on AEGIS family of ciphers. Fault attack is one of the most efficient forms of side-channel attacks against implementations of cryptographic algorithms, and the protection against fault attack is vital for security-related devices. In this paper, we propose countermeasures for AEGIS family of ciphers. The proposed countermeasures show that the state of the ciphers can not be recovered faster than exhaustive search because it needs 2128 time to recover a state of each cipher. Keywords Authenticated encryption · Fault attack · Countermeasures · CAESAR · AES

1 Introduction Authenticated encryption (AE) and authenticated encryption with associated data (AEAD) are forms of encryption which simultaneously assure the confidentiality and authenticity of data. CAESAR [1], a competition for authenticated encryption was started in 2013, which targets to identify a portfolio of AEAD. AEGIS [2], a dedicated AES [3]-based authenticated encryption algorithm is a one of the winners of the CAESAR competition. Fault attack is one of the most efficient forms of side-channel attack against implementations of cryptographic algorithms. In this kind of attack, faults are injected during cipher operations. The attacker then analyzes the fault-free and faulty ciphertexts or keystreams to deduce partial or full information of the secret key. S. Maiti (B) · D. Roy Chowdhury Indian Institute of Technology Kharagpur, Kharagpur, India e-mail: [email protected] D. Roy Chowdhury e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 D. Giri et al. (eds.), Proceedings of the Sixth International Conference on Mathematics and Computing, Advances in Intelligent Systems and Computing 1262, https://doi.org/10.1007/978-981-15-8061-1_1

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S. Maiti and D. Roy Chowdhury

In literature, there exist some fault attacks on authenticated encryption stream ciphers and the countermeasures [4–6]. AEGIS is also susceptible to Differential Fault Attack (DFA) [7]. The main contribution of this work can be summarized below: – Proposed countermeasures against differential fault attack for each of the ciphers of AEGIS family. – Furnished a comparison of the modified AEGIS with AEGIS family The organization of the rest of the paper is as follows. Section 2 discusses AEGIS family of ciphers. The differential fault analysis on each of the ciphers is briefly described in this section. Section 3 presents the countermeasures against fault attack on each of the ciphers. A comparison of the modified AEGIS with AEGIS family is shown in Sect. 4. Finally, the paper is concluded in Sect. 5.

2 Background The authenticated encryption algorithms of AEGIS family of ciphers are introduced in [2]. In this section, we briefly describe the ciphers and the attack strategies on them. The AEGIS family of ciphers extensively uses one keyed round function (AE S Round Function) of AES [3] as follows: AE S Round Function(A, B) = τ (A) ⊕ B where τ (.) = MixColumns(ShiftRows(SubBytes(.))), and each of A and B is a 16byte block. The following operators are used in AEGIS: ⊕ : bit-wise exclusive OR & : bit-wise AND || : concatenation

2.1 AEGIS Family of Ciphers In this section, we briefly describe the ciphers AEGIS-128, AEGIS-256, and AEGIS128L. The ciphertext and tag generation for each cipher of AEGIS family are done in four phases: (1) The initialization, (2) Processing the associated data, (3) The encryption, and (4) The finalization. Each of the ciphers takes a 128-bit key and a 128-bit nonce. AEGIS-128 The 80-byte state Si of AEGIS-128 at round i can be defined as Si = Si,0 ||Si,1 ||Si,2 ||Si,3 ||Si,4

Preventing Differential Fault Analysis Attack on AEGIS Family of Ciphers

3

Fig. 1 The state update function of AEGIS-128

where each Si, j , j = 0 to 4, is a 16-byte block. At each round i, a 16-byte data block m i is used to update the state Si . The next state Si+1 is computed as follows: Si+1,0 =AE S Round Function(Si,4 , Si,0 ⊕ m i ) Si+1,1 =AE S Round Function(Si,0 , Si,1 ) Si+1,2 =AE S Round Function(Si,1 , Si,2 ) Si+1,3 =AE S Round Function(Si,2 , Si,3 ) Si+1,4 =AE S Round Function(Si,3 , Si,4 ) The state update function is shown in Fig. 1. In the encryption (i.e., third phase), a 16-byte plaintext Pi at round i is used to update the state, and Pi is encrypted to a 16-byte ciphertext Ci as Ci = Pi ⊕ Z i , where Z i = Si,1 ⊕ Si,4 ⊕ (Si,2 &Si,3 ) is a 16-byte keystream block. AEGIS-256 The 96-byte state Si of AEGIS-256 at round i can be defined as Si = Si,0 ||Si,1 ||Si,2 ||Si,3 ||Si,4 ||Si,5 where each Si, j , j = 0 to 5, is a 16-byte block. At each round i, a 16-byte data block m i is used to update the state Si . The next state Si+1 is computed as follows:

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Si+1,0 = AE S Round Function(Si,5 , Si,0 ⊕ m i ) Si+1,1 = AE S Round Function(Si,0 , Si,1 ) Si+1,2 = AE S Round Function(Si,1 , Si,2 ) Si+1,3 = AE S Round Function(Si,2 , Si,3 ) Si+1,4 = AE S Round Function(Si,3 , Si,4 ) Si+1,5 = AE S Round Function(Si,4 , Si,5 ) In the encryption (i.e., third phase), a 16-byte plaintext Pi at round i is used to update the state, and Pi is encrypted to a 16-byte ciphertext Ci as Ci = Pi ⊕ Z i , where Z i = Si,1 ⊕ Si,4 ⊕ Si,5 ⊕ (Si,2 &Si,3 ) is a 16-byte keystream block. AEGIS-128L The 128-byte state Si of AEGIS-128L at round i can be defined as Si = Si,0 ||Si,1 ||Si,2 ||Si,3 ||Si,4 ||Si,5 ||Si,6 ||Si,7 where each Si, j , j = 0 to 7, is a 16-byte block. At each round i, two 16-byte data blocks m 2i and m 2i+1 are used to update the state Si . The next state Si+1 is computed as follows: Si+1,0 = AE S Round Function(Si,7 , Si,0 ⊕ m 2i ) Si+1,1 = AE S Round Function(Si,0 , Si,1 ) Si+1,2 = AE S Round Function(Si,1 , Si,2 ) Si+1,3 = AE S Round Function(Si,2 , Si,3 ) Si+1,4 = AE S Round Function(Si,3 , Si,4 ⊕ m 2i+1 ) Si+1,5 = AE S Round Function(Si,4 , Si,5 ) Si+1,6 = AE S Round Function(Si,5 , Si,6 ) Si+1,7 = AE S Round Function(Si,6 , Si,7 ) In the encryption phase, two 16-byte plaintext P2i and P2i+1 at round i are used to update the state, and P2i and P2i+1 are encrypted to two 16-byte ciphertext C2i as C2i = P2i ⊕ Z 2i and C2i+1 as C2i+1 as C2i+1 = P2i+1 ⊕ Z 2i+1 , respectively, where Z 2i = Si,1 ⊕ Si,6 ⊕ (Si,2 &Si,3 ) and Z 2i+1 = Si,2 ⊕ Si,5 ⊕ (Si,6 &Si,7 ) are two 16-byte keystream blocks.

2.2 Attack Description for AEGIS Family Differential Fault Analysis of AEGIS family of ciphers is introduced in [7]. One can find a state of the ciphers by DFA, and eventually, mount forgery attack [8] by the change of one ciphertext and the associated authentication tag. Here, we present the attack description in detail.

Preventing Differential Fault Analysis Attack on AEGIS Family of Ciphers

2.2.1

5

Attack Model

The attacker can run the cipher with the same secret key, public parameters, and plaintext several times. The attacker is able to inject single-bit faults. The attacker has control on the timing of fault injection, and on the fault location. The plaintext and the corresponding fault-free/faulty ciphertext are available to the attacker. Attack on AEGIS-128 The states of the cipher at rounds i and i + 1 are as follows: Si = Si,0 ||Si,1 ||Si,2 ||Si,3 ||Si,4 Si+1 = Si+1,0 ||Si+1,1 ||Si+1,2 ||Si+1,3 ||Si+1,4 By the encryption, the generated ciphertext Ci at round i, Ci+1 at round i + 1 are given by Ci = Pi ⊕ Z i and Ci+1 = Pi+1 ⊕ Z i+1 , respectively, where Z i = Si,1 ⊕ Si,4 ⊕ (Si,2 &Si,3 ) Z i+1 = Si+1,1 ⊕ Si+1,4 ⊕ (Si+1,2 &Si+1,3 )

(1) (2)

The state Si = Si,0 ||Si,1 ||Si,2 ||Si,3 ||Si,4 can be recovered by injecting 3 × 128 singlebit faults. The state recovery procedure is illustrated in Table 1. Attack on AEGIS-256 Let the states of the cipher at rounds i, i + 1, and i + 2 be Si , Si+1 , and Si+2 , respectively. By the encryption, the generated ciphertext Ci at round i, Ci+1 at round i + 1, and Ci+2 at round i + 2 are given by Ci = Pi ⊕ Z i , Ci+1 = Pi+1 ⊕ Z i+1 , and Ci+2 = Pi+2 ⊕ Z i+2 , respectively, where

Table 1 Recovering the cipher state of AEGIS-128 Steps Blocks Recovering procedure recovered 1

Si,2

2

Si,3

3

Si+1,2

4

Si,1

5 6 7 8 9

Si,4 Si+1,3 Si+1,4 Si+1,1 Si,0

Injecting 128 single-bit faults into Si,3 block, and using faulty Z i and fault-free Z i by Eq. (1) Injecting 128 single-bit faults into Si,2 block, and using faulty Z i and fault-free Z i by Eq. (1) Injecting 128 single-bit faults into Si+1,3 block, and using faulty Z i+1 and fault-free Z i+1 by Eq. (2) Using Si+1,2 = AE S Round Function(Si,1 , Si,2 ) = τ (Si,1 ) ⊕ Si,2 by the inverse round function Using Eq. (1) Using Si+1,3 = AE S Round Function(Si,2 , Si,3 ) = τ (Si,2 ) ⊕ Si,3 Using Si+1,4 = AE S Round Function(Si,3 , Si,4 ) = τ (Si,3 ) ⊕ Si,4 Using Eq. (2) Using Si+1,1 = AE S Round Function(Si,0 , Si,1 ) = τ (Si,0 ) ⊕ Si,1 by the inverse round function

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Table 2 Recovering the cipher state of AEGIS-256 Steps Blocks Recovering procedure recovered 1 2 3 4 5 6 7 8 9 10 11 12

Si,2

Injecting 128 single-bit faults into Si,3 block, and using faulty Z i and fault-free Z i by Eq. (3) Si,3 Injecting 128 single-bit faults into Si,2 block, and using faulty Z i and fault-free Z i by Eq. (3) Si+1,2 Injecting 128 single-bit faults into Si+1,3 block, and using faulty Z i+1 and fault-free Z i+1 by Eq. (4) Si+1,3 Using Si+1,3 = AE S Round Function(Si,2 , Si,3 ) = τ (Si,2 ) ⊕ Si,3 Si,1 Using Si+1,2 = AE S Round Function(Si,1 , Si,2 ) = τ (Si,1 ) ⊕ Si,2 by the inverse round function Si+2,3 Using Si+2,3 = AE S Round Function(Si+1,2 , Si+1,3 ) = τ (Si+1,2 ) ⊕ Si+1,3 Si+2,2 Injecting 128 single-bit faults into Si+2,3 block, and using faulty Z i+2 and fault-free Z i+2 by Eq. (5) Si+1,1 Using Si+2,2 = AE S Round Function(Si+1,1 , Si+1,2 ) = τ (Si+1,1 ) ⊕ Si+1,2 by the inverse round function Si,0 Using Si+1,1 = AE S Round Function(Si,0 , Si,1 ) = τ (Si,0 ) ⊕ Si,1 by the inverse round function (Si,4 ⊕ Si,5 ) Using Eq. (3) Si,4 Using Eq. (4) defined as Z i+1 = Si+1,1 ⊕ τ (Si,3 ) ⊕ Si,4 ⊕ τ (Si,4 ) ⊕ Si,5 ⊕ (Si+1,2 &Si+1,3 ) Si,5 From known (Si,4 ⊕ Si,5 ) and Si,4

Z i = Si,1 ⊕ Si,4 ⊕ Si,5 ⊕ (Si,2 &Si,3 ) Z i+1 = Si+1,1 ⊕ Si+1,4 ⊕ Si+1,5 ⊕ (Si+1,2 &Si+1,3 ) Z i+2 = Si+2,1 ⊕ Si+2,4 ⊕ Si+2,5 ⊕ (Si+2,2 &Si+2,3 )

(3) (4) (5)

The state Si = Si,0 ||Si,1 ||Si,2 ||Si,3 ||Si,4 ||Si,5 can be recovered by 4 × 128 single-bit faults. The state recovery procedure is illustrated in Table 2. Attack on AEGIS-128L Let the states of the cipher at rounds i and i + 1 be Si and Si+1 , respectively. By the encryption, the generated ciphertext C2i and C2i+1 at round i, C2i+2 and C2i+3 at round i + 1 are given by C2i = P2i ⊕ Z 2i , C2i+1 = P2i+1 ⊕ Z 2i+1 , C2i+2 = P2i+2 ⊕ Z 2i+2 , and C2i+3 = P2i+3 ⊕ Z 2i+3 , respectively, where Z 2i = Si,1 ⊕ Si,6 ⊕ (Si,2 &Si,3 ) Z 2i+1 = Si,2 ⊕ Si,5 ⊕ (Si,6 &Si,7 )

(6) (7)

Z 2i+2 = Si+1,1 ⊕ Si+1,6 ⊕ (Si+1,2 &Si+1,3 ) Z 2i+3 = Si+1,2 ⊕ Si+1,5 ⊕ (Si+1,6 &Si+1,7 )

(8) (9)

Preventing Differential Fault Analysis Attack on AEGIS Family of Ciphers

7

Table 3 Recovering the cipher state of AEGIS-128L Steps Blocks Recovering procedure recovered 1

Si,2

2

Si,3

3

Si,6

4

Si,7

5 6 7 8 9 10 11 12 13

Si,1 Si,5 Si+1,2 Si+1,3 Si+1,6 Si+1,7 Si+1,1 Si+1,5 Si,0

14

Si,4

Injecting 128 single-bit faults into Si,3 block, and using faulty Z 2i and fault-free Z 2i by Eq. (6) Injecting 128 single-bit faults into Si,2 block, and using faulty Z 2i and fault-free Z 2i by Eq. (6) Injecting 128 single-bit faults into Si,7 block, and using faulty Z 2i+1 and fault-free Z 2i+1 by Eq. (7) Injecting 128 single-bit faults into Si,6 block, and using faulty Z 2i+1 and fault-free Z 2i+1 by Eq. (7) Using Eq. (6) Using Eq. (7) Using Si+1,2 = AE S Round Function(Si,1 , Si,2 ) = τ (Si,1 ) ⊕ Si,2 Using Si+1,3 = AE S Round Function(Si,2 , Si,3 ) = τ (Si,2 ) ⊕ Si,3 Using Si+1,6 = AE S Round Function(Si,5 , Si,6 ) = τ (Si,5 ) ⊕ Si,6 Using Si+1,7 = AE S Round Function(Si,6 , Si,7 ) = τ (Si,6 ) ⊕ Si,7 Using Eq. (8) Using Eq. (9) Using Si+1,1 = AE S Round Function(Si,0 , Si,1 ) = τ (Si,0 ) ⊕ Si,1 by the inverse round function Using Si+1,5 = AE S Round Function(Si,4 , Si,5 ) = τ (Si,4 ) ⊕ Si,5 by the inverse round function

The state Si = Si,0 ||Si,1 ||Si,2 ||Si,3 ||Si,4 ||Si,5 ||Si,6 ||Si,7 can be recovered by 4 × 128 single-bit faults. The state recovery procedure is illustrated in Table 3.

3 Countermeasures Here, we propose the countermeasures against differential fault attack on AEGIS family by slight modification of the 16-byte block keystream generation in the encryption function. By the modified encryption functions, one can not recover a state of the ciphers of AEGIS family, and hence can not mount a forgery attack on the tag generation.

3.1 Countermeasure for AEGIS-128 We modify the keystream generation function. Under this modification, the keystreams at rounds i and i + 1 are as follows:

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S. Maiti and D. Roy Chowdhury

Fig. 2 The state recovering of the modified AEGIS-128

Z i = Si,1 ⊕ Si,4 ⊕ (Si,2 &Si,3 ) ⊕ (Si,0 &Si,2 ) ⊕ (Si,0 &Si,3 )

(10)

Z i+1 = Si+1,1 ⊕ Si+1,4 ⊕ (Si+1,2 &Si+1,3 ) ⊕ (Si+1,0 &Si+1,2 ) ⊕ (Si+1,0 &Si+1,3 ) (11)

As per the attack model (ref. Sect. 2.2), the attacker knows the plaintext and the corresponding fault-free/faulty ciphertext. Therefore, fault-free/faulty Z i and faultfree/faulty Z i+1 are available to the attacker. The attacker can recover (Si,2 ⊕ Si,3 ) by injecting 128 single-bit faults into the block Si,0 , and using fault-free and faulty Z i . Similarly, (Si,0 ⊕ Si,2 ) can be recovered by injecting 128 single-bit faults into the block Si,3 , and using fault-free and faulty Z i . The attacker can recover (Si+1,2 ⊕ Si+1,3 ) by injecting 128 single-bit faults into the block Si+1,0 , and using fault-free and faulty Z i+1 . Similarly, (Si+1,0 ⊕ Si+1,3 ) can be recovered by injecting 128 single-bit faults into the block Si+1,2 , and using fault-free and faulty Z i+1 . Suppose, attacker guesses the block Si,2 . Considering the guess value of the block Si,2 , Si,0 , Si,3 , Si+1,3 , Si+1,2 , Si,1 , Si+1,0 , and finally, Si,4 can be recovered. The steps to recover the blocks are illustrated in Fig. 2 as well as in Table 4. The computed value of (Si,1 ⊕ Si,4 ⊕ (Si,2 &Si,3 ) ⊕ (Si,0 &Si,2 ) ⊕ (Si,0 &Si,3 )) is compared with fault-free Z i whether it matches with Z i or not. If it does not match then the attacker guesses another value of the block Si,2 , and repeat the state recovery procedure. Thus, the 80-byte state Si of the cipher can be recovered by 4 × 128 single-bit fault injections, but it needs 2128 time because the block Si,2 is a 16-byte block. Hence, the state recovery is not feasible in the practical scenario.

Preventing Differential Fault Analysis Attack on AEGIS Family of Ciphers

9

Table 4 Recovering the cipher state of the modified AEGIS-128 Steps Blocks Recovering procedure recovered 1 2 3 4 5 6 7 8 9 10 11 12

(Si,2 ⊕ Si,3 ) Injecting 128 single-bit faults into Si,0 block, and using faulty Z i and fault-free Z i by Eq. (10) (Si,0 ⊕ Si,2 ) Injecting 128 single-bit faults into Si,3 block, and using faulty Z i and fault-free Z i by Eq. (10) (Si+1,2 ⊕ Injecting 128 single-bit faults into Si+1,0 block, and using faulty Z i+1 Si+1,3 ) and fault-free Z i+1 by Eq. (11) (Si+1,0 ⊕ Injecting 128 single-bit faults into Si+1,2 block, and using faulty Z i+1 Si+1,3 ) and fault-free Z i+1 by Eq. (11) Si,2 Guessing the block Si,2 Si,0 From known (Si,0 ⊕ Si,2 ) and Si,2 Si,3 From known (Si,2 ⊕ Si,3 ) and Si,2 Si+1,3 Using Si+1,3 = AE S Round Function(Si,2 , Si,3 ) = τ (Si,2 ) ⊕ Si,3 Si+1,2 From known (Si+1,2 ⊕ Si+1,3 ) and Si+1,3 Si,1 Using Si+1,2 = AE S Round Function(Si,1 , Si,2 ) = τ (Si,1 ) ⊕ Si,2 by the inverse round function Si+1,0 From known (Si+1,0 ⊕ Si+1,3 ) and Si+1,3 Si,4 Using Si+1,0 = AE S Round Function(Si,4 , Si,0 ⊕ Pi ) = τ (Si,4 ) ⊕ Si,0 ⊕ Pi by the inverse round function

3.2 Countermeasure for AEGIS-256 We modify the keystream generation function. Under this modification, the keystreams at rounds i, i + 1, and i + 2 are as follows: Z i = Si,1 ⊕ Si,4 ⊕ Si,5 ⊕ (Si,2 &Si,3 ) ⊕ (Si,0 &Si,2 ) ⊕ (Si,0 &Si,3 )

(12)

Z i+1 = Si+1,1 ⊕ Si+1,4 ⊕ Si+1,5 ⊕ (Si+1,2 &Si+1,3 ) ⊕ (Si+1,0 &Si+1,2 ) ⊕ (Si+1,0 &Si+1,3 )

(13) Z i+2 = Si+2,1 ⊕ Si+2,4 ⊕ Si+2,5 ⊕ (Si+2,2 &Si+2,3 ) ⊕ (Si+2,0 &Si+2,2 ) ⊕ (Si+2,0 &Si+2,3 )

(14) As per the attack model (ref. Sect. 2.2), the attacker knows the plaintext and the corresponding fault-free/faulty ciphertext. Therefore, fault-free/faulty Z i and faultfree/faulty Z i+1 are available to the attacker. The attacker can recover (Si,2 ⊕ Si,3 ) by injecting 128 single-bit faults into the block Si,0 , and using fault-free and faulty Z i . Similarly, (Si,0 ⊕ Si,2 ) can be recovered by injecting 128 single-bit faults into the block Si,3 , and using fault-free and faulty Z i . The attacker can recover (Si+1,2 ⊕ Si+1,3 ) by injecting 128 single-bit faults into the block Si+1,0 , and using fault-free and faulty Z i+1 . Similarly, (Si+1,0 ⊕ Si+1,3 ) can be recovered by injecting 128 single-bit

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Table 5 Recovering the cipher state of the modified AEGIS-256 Steps Blocks Recovering procedure recovered 1 2 3 4 5 6 7 8 9 10 11 12

13 14 15 16 17

(Si,2 ⊕ Si,3 ) Injecting 128 single-bit faults into Si,0 block, and using faulty Z i and fault-free Z i by Eq. (12) (Si,0 ⊕ Si,2 ) Injecting 128 single-bit faults into Si,3 block, and using faulty Z i and fault-free Z i by Eq. (12) (Si+1,2 ⊕ Injecting 128 single-bit faults into Si+1,0 block, and using faulty Z i+1 Si+1,3 ) and fault-free Z i+1 by Eq. (13) (Si+1,0 ⊕ Injecting 128 single-bit faults into Si+1,2 block, and using faulty Z i+1 Si+1,3 ) and fault-free Z i+1 by Eq. (13) Si,2 Guessing the block Si,2 Si,0 From known (Si,0 ⊕ Si,2 ) and Si,2 Si,3 From known (Si,2 ⊕ Si,3 ) and Si,2 Si+1,3 Using Si+1,3 = AE S Round Function(Si,2 , Si,3 ) = τ (Si,2 ) ⊕ Si,3 Si+1,2 From known (Si+1,2 ⊕ Si+1,3 ) and Si+1,3 Si,1 Using Si+1,2 = AE S Round Function(Si,1 , Si,2 ) = τ (Si,1 ) ⊕ Si,2 by the inverse round function Si+1,0 From known (Si+1,0 ⊕ Si+1,3 ) and Si+1,3 Si,5 Using Si+1,0 = AE S Round Function(Si,5 , Si,0 ⊕ Pi ) = τ (Si,5 ) ⊕ Si,0 ⊕ Pi by the inverse round function Si+2,3 Using Si+2,3 = AE S Round Function(Si+1,2 , Si+1,3 ) = τ (Si+1,2 ) ⊕ Si+1,3 (Si+2,0 ⊕ Injecting 128 single-bit faults into Si+2,2 block, and using faulty Z i+2 Si+2,3 ) and fault-free Z i+2 by Eq. (14) Si+2,0 From known (Si+2,0 ⊕ Si+2,3 ) and Si+2,3 Si+1,5 Using Si+2,0 = AE S Round Function(Si+1,5 , Si+1,0 ⊕ Pi+1 ) = τ (Si+1,5 ) ⊕ Si+1,0 ⊕ Pi+1 by the inverse round function Si,4 Using Si+1,5 = AE S Round Function(Si,4 , Si,5 ) = τ (Si,4 ) ⊕ Si,5 by the inverse round function

faults into the block Si+1,2 , and using fault-free and faulty Z i+1 . Suppose, attacker guesses the block Si,2 . Considering the guess value of the block Si,2 , Si,0 , Si,3 , Si+1,3 , Si+1,2 , Si,1 , Si+1,0 , Si,5 , and Si+2,3 can be recovered. Then (Si+2,0 ⊕ Si+2,3 ) can be recovered by injecting 128 single-bit faults into the block Si+2,2 . Then Si+2,0 , Si+1,5 , and finally Si,4 can be recovered. The steps to recover the blocks are illustrated in Table 5. The computed value of (Si,1 ⊕ Si,4 ⊕ Si,5 ⊕ (Si,2 &Si,3 ) ⊕ (Si,0 &Si,2 ) ⊕ (Si,0 &Si,3 )) is compared with fault-free Z i , whether it matches with Z i or not. If it does not match then the attacker guesses another value of the block Si,2 , and repeat the state recovery procedure. Thus, the 96-byte state Si of the cipher can be recovered by 5 × 128 single-bit fault injections, but it needs 2128 time because the block Si,2 is a 16-byte block. Hence, the state recovery is not feasible in the practical scenario.

Preventing Differential Fault Analysis Attack on AEGIS Family of Ciphers

11

3.3 Countermeasure for AEGIS-128L We modify the keystream generation function. Under this modification, the keystreams at rounds i and i + 1 are as follows:

Table 6 Recovering the cipher state of the modified AEGIS-128L Steps Blocks Recovering procedure recovered 1 2 3 4 5 6 7 8 9 10 11 12

13 14 15 16 17 18 19 20

(Si,2 ⊕ Si,3 ) Injecting 128 single-bit faults into Si,0 block, and using faulty Z 2i and fault-free Z 2i by Eq. (15) (Si,0 ⊕ Si,2 ) Injecting 128 single-bit faults into Si,3 block, and using faulty Z 2i and fault-free Z 2i by Eq. (15) (Si+1,2 ⊕ Injecting 128 single-bit faults into Si+1,0 block, and using Si+1,3 ) faulty Z 2i+2 and fault-free Z 2i+2 by Eq. (17) (Si+1,0 ⊕ Injecting 128 single-bit faults into Si+1,2 block, and using Si+1,3 ) faulty Z 2i+2 and fault-free Z 2i+2 by Eq. (17) Si,2 Guessing the block Si,2 Si,0 From known (Si,0 ⊕ Si,2 ) and Si,2 Si,3 From known (Si,2 ⊕ Si,3 ) and Si,2 Si+1,3 Using Si+1,3 = AE S Round Function(Si,2 , Si,3 ) = τ (Si,2 ) ⊕ Si,3 Si+1,2 From known (Si+1,2 ⊕ Si+1,3 ) and Si+1,3 Si,1 Using Si+1,2 = AE S Round Function(Si,1 , Si,2 ) = τ (Si,1 ) ⊕ Si,2 by the inverse round function Si+1,0 From known (Si+1,0 ⊕ Si+1,3 ) and Si+1,3 Si,7 Using Si+1,0 = AE S Round Function(Si,7 , Si,0 ⊕ P2i ) = τ (Si,7 ) ⊕ Si,0 ⊕ P2i by the inverse round function (Si,6 ⊕ Si,7 ) Injecting 128 single-bit faults into Si,4 block, and using faulty Z 2i+1 and fault-free Z 2i+1 by Eq. (16) Si,6 From known (Si,6 ⊕ Si,7 ) and Si,7 (Si,4 ⊕ Si,7 ) Injecting 128 single-bit faults into Si,6 block, and using faulty Z 2i+1 and fault-free Z 2i+1 by Eq. (16) Si,4 From known (Si,4 ⊕ Si,7 ) and Si,7 Si+1,7 Using Si+1,7 = AE S Round Function(Si,6 , Si,7 ) = τ (Si,6 ) ⊕ Si,7 (Si+1,6 ⊕ Injecting 128 single-bit faults into Si+1,4 block, and using Si+1,7 ) faulty Z 2i+3 and fault-free Z 2i+3 by Eq. (18) Si+1,6 From known (Si+1,6 ⊕ Si+1,7 ) and Si+1,7 Si,5 Using Si+1,6 = AE S Round Function(Si,5 , Si,6 ) = τ (Si,5 ) ⊕ Si,6 by the inverse round function

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S. Maiti and D. Roy Chowdhury

Z 2i = Si,1 ⊕ Si,6 ⊕ (Si,2 &Si,3 ) ⊕ (Si,0 &Si,2 ) ⊕ (Si,0 &Si,3 )

(15)

Z 2i+1 = Si,2 ⊕ Si,5 ⊕ (Si,6 &Si,7 ) ⊕ (Si,4 &Si,6 ) ⊕ (Si,4 &Si,7 ) (16) Z 2i+2 = Si+1,1 ⊕ Si+1,6 ⊕ (Si+1,2 &Si+1,3 ) ⊕ (Si+1,0 &Si+1,2 ) ⊕ (Si+1,0 &Si+1,3 ) (17) Z 2i+3 = Si+1,2 ⊕ Si+1,5 ⊕ (Si+1,6 &Si+1,7 ) ⊕ (Si+1,4 &Si+1,6 ) ⊕ (Si+1,4 &Si+1,7 ) (18) As per the attack model (ref. Sect. 2.2), the attacker knows the plaintext and the corresponding fault-free/faulty ciphertext. Therefore, fault-free/faulty Z 2i , Z 2i+1 , Z 2i+2 , and Z 2i+3 are available to the attacker. The steps to recover the blocks are illustrated in Table 6. Here also, the attacker guesses the value of Si,2 . The computed value of (Si,1 ⊕ Si,6 ⊕ (Si,2 &Si,3 ) ⊕ (Si,0 &Si,2 ) ⊕ (Si,0 &Si,3 )) is compared with fault-free Z 2i , whether it matches with Z 2i or not. The computed value of (Si,2 ⊕ Si,5 ⊕ (Si,6 &Si,7 ) ⊕ (Si,4 &Si,6 ) ⊕ (Si,4 &Si,7 )) is compared with fault-free Z 2i+1 , whether it matches with Z 2i+1 or not. If any one of these two cases does not match then the attacker guesses another value of the block Si,2 and repeat the state recovery procedure. Thus, the 128-byte state Si of the cipher can be recovered by 7 × 128 single-bit fault injections, but it needs 2128 time because the block Si,2 is a 16-byte block. Hence, the state recovery is not feasible in the practical scenario.

4 Comparison of the Modified AEGIS with AEGIS Two extra 128-bit AND gates and two extra 128-bit XOR gates are needed for each of the keystream generation functions of the modified AEGIS-128 and AEGIS-256. Four extra 128-bit AND gates and four extra 128-bit XOR gates are needed for the keystream generation function of the modified AEGIS-128L. A comparison between the standard AEGIS and the proposed modified AEGIS is shown in Table 7.

5 Conclusion In this paper, we have proposed countermeasures against differential fault attack on AEGIS family of ciphers AEGIS-128, AEGIS-256, and AEGIS-128L. It shows that a state of the ciphers can not be recovered faster than exhaustive search, and hence the forgery attack by changing the authentication tag is not successful.

Preventing Differential Fault Analysis Attack on AEGIS Family of Ciphers Table 7 Comparison with AEGIS Encryption function

AEGIS[2]

Modified AEGIS

Ci = Pi ⊕ Si,1 ⊕ Si,4 ⊕ (Si,2 &Si,3 ) [AEGIS-128] Ci = Pi ⊕ Si,1 ⊕ Si,4 ⊕ Si,5 ⊕ (Si,2 &Si,3 ) [AEGIS-256] C2i = P2i ⊕ Si,1 ⊕ Si,6 ⊕ (Si,2 &Si,3 ) [AEGIS-128L] C2i+1 = P2i+1 ⊕ Si,2 ⊕ Si,5 ⊕ (Si,6 &Si,7 ) [AEGIS-128L] Ci = Pi ⊕ Si,1 ⊕ Si,4 ⊕ (Si,2 &Si,3 ) ⊕(Si,0 &Si,2 ) ⊕ (Si,0 &Si,3 ) [AEGIS-128] Ci = Pi ⊕ Si,1 ⊕ Si,4 ⊕ Si,5 ⊕ (Si,2 &Si,3 ) ⊕(Si,0 &Si,2 ) ⊕ (Si,0 &Si,3 ) [AEGIS-256] C2i = P2i ⊕ Si,1 ⊕ Si,6 ⊕ (Si,2 &Si,3 ) ⊕(Si,0 &Si,2 ) ⊕ (Si,0 &Si,3 ) [AEGIS-128L] C2i+1 = P2i+1 ⊕ Si,2 ⊕ Si,5 ⊕ (Si,6 &Si,7 ) ⊕(Si,4 &Si,6 ) ⊕ (Si,4 &Si,7 ) [AEGIS-128L]

Security

State recovery

13

Tag changing attack

Vulnerable to Yes fault attacks [7] [7]

Yes [8]

Resistant to fault attacks [Our work]

No

No

References 1. CAESAR (Competition for Authenticated Encryption: Security, Applicability, and Robustness) (2013). http://competitions.cr.yp.to/caesar.html 2. Wu H, Preneel B (2013) AEGIS: a fast authenticated encryption algorithm. In: Selected areas in cryptography—SAC 2013—20th international conference. Burnaby, BC, Canada, Revised Selected Papers. pp 185–201. https://doi.org/10.1007/978-3-662-43414-7_10 3. Daemen J, Rijmen V (2002) The design of rijndael: AES—the advanced encryption standard. Information security and cryptography. Springer, Berlin. https://doi.org/10.1007/978-3-66204722-4 4. Salam MI, Simpson L, Bartlett H, Dawson E, Wong KK (2018) Fault attacks on the authenticated encryption stream cipher MORUS. Cryptography 2(1):4. https://doi.org/10.3390/ cryptography2010004 5. Zhang X, Feng X, Lin D (2017) Fault attack on ACORN v3. IACR Cryptol ePrint Arch 855. http://eprint.iacr.org/2017/855 6. Adomnicai A, Fournier JJA, Masson L (2018) Masking the lightweight authenticated ciphers ACORN and Ascon in software. IACR Cryptol ePrint Arch 708 (2018). https://eprint.iacr.org/ 2018/708

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7. Dey P, Rohit RS, Sarkar S, Adhikari A (2016) Differential fault analysis on tiaoxin and AEGIS family of ciphers. In: Security in computing and communications—4th international symposium, SSCC. Jaipur, India, Proceedings, pp 74–86. https://doi.org/10.1007/978-981-10-2738-3_7 8. El-Hadary, A., Megahed, M., ElAzeem, M.: A proposed solutions to two possible attacks over AEGIS authenticated encryption algorithm. International Journal of Computer Theory and Engineering 10, 185–189 (01 2018)

A Geometric-Based User Authentication Scheme for Multi-server Architecture: Cryptanalysis and Enhancement Debasis Giri

and Tanmoy Maitra

Abstract An authentication system provides such an environment by which remote server can easily verify the authorized remote users over a public communication channel like the Internet. In this regard, Lin et al. proposed a timestamp-based remote user authentication scheme based on the geometric properties on the Euclidean plane for multi-server architecture. Unfortunately, we show in this paper that Lin et al.’s scheme has some security weaknesses. Furthermore, we propose an improvement of Lin et al.’s scheme to withstand their issues. Our improved scheme is analyzed and it shows that the proposed scheme can protect various security attacks. This work also compares the proposed scheme with the related schemes to check the efficiency. Keywords Attack · Authentication · Remote server · Users · Timestamp

1 Introduction In a remote user authentication scheme, only authorized remote users can be authenticated by the remote server over a public communication channel in a multi-server network environment and can access the resources and/or the services provided by the server. In such a scheme, the users usually hold their respective passwords and the server stores a secret key. Typically, a user generates a “request message” with the help of his identity, password, and some public parameters stored either in his smart card or other storage devices. The server can check the validity of the request message with the help of its secret key. However, there could be some more additional D. Giri Department of Information Technology, Maulana Abul Kalam Azad University of Technology, Haringhata, Nadia 741249, West Bengal, India e-mail: [email protected] T. Maitra (B) School of Computer Engineering, KIIT, Deemed to be University, Bhubaneswar 751024, Odisha, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 D. Giri et al. (eds.), Proceedings of the Sixth International Conference on Mathematics and Computing, Advances in Intelligent Systems and Computing 1262, https://doi.org/10.1007/978-981-15-8061-1_2

15

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D. Giri and T. Maitra

procedures to authenticate a remote user by the server, for example, verification of the correctness of the identity of a user and/or the correctness of the time interval between the timestamp of the user at the time of generating the request message and the timestamp of the server receiving the request message from the user. The rest of this work is organized as follows. We first briefly review some related works in Sect. 2. We discuss the Lin et al.’s timestamp-based password authentication scheme [1] in Sect. 3. Section 4 shows the security weaknesses of Lin et al.’s scheme. Section 5 shows the improvement of Lin et al.’s scheme. In Sect. 6, we describe the security analysis of the improved scheme. Finally, Sect. 7 concludes the paper.

2 Related Works In 1981, Lamport first proposed a well-known hash-based password authentication scheme [2] to authenticate a remote user by a remote server over an insecure channel. His scheme resists on “replay attack,” but it requires a verification (password) table to verify the legitimacy of a login user. In 2000, Hwang and Li [3] pointed out that Lamport’s scheme [2] has weaknesses such as the risk to modify the password table and the cost of managing and protecting the table. After his scheme, a number of remote user authentication schemes had been proposed in the studies [4–7]. However, these schemes did not agree with the multi-server environment. In 2001, Li et al. proposed a remote user password authentication scheme [8] by using neural networks. It can be applicable in multi-server environment, but it spends too much time on training the neural networks. Latter, there are a lot of works on the authentication system have been proposed in [9–12] which are based on password plus biometric in a different multi-server environment. Lin et al. proposed a remote user authentication scheme in [1] based on the simple geometric properties on the Euclidean plane, which can be applicable to the multi-server environments. In this paper, we show that Lin et al.’s scheme has some security weaknesses.

3 Brief Review of Lin et al.’s Scheme In this section, we briefly review the Lin et al.’s scheme [1]. Their scheme consists of four phases, namely initialization, registration, login and authentication phases. Initialization phase: Let Sm = {Ser 1 , Ser 2 , · · · , Ser m } be a set of m servers in a multi-server environment and there exist a central manager (C M) in the system. The C M chooses two system parameters p, where p is a large prime and g a primitive element in Z ∗p . Then, the C M chooses randomly a secret key di ∈ G F( p) for each server Seri ∈ Sm , where G F( p) represents a Galois field over prime p and computes the corresponding public key ei as

A Geometric-Based User Authentication Scheme for Multi-server Architecture …

ei = g di mod ( p − 1).

17

(1)

Then, the C M sends the key pair (ei , di ) securely to Seri . Registration phase: When a new user U wants to log into the multi-server computer system, he delivers his identity I D and password P W to the C M. If U is granted registration only by a set Sn of servers, where Sn ⊆ Sm , the C M performs the following steps for each Seri ∈ Sn . 1. Compute X i = I D ei mod p, Yi = I D di mod p,

(2) (3)

Di = eiI D mod p, Wi = eiP W mod p,

(4) (5)

where (X i , Yi ) and (Di , Wi ) are two points on the Euclidean plane. 2. Construct a line L i joining the two points (X i , Yi ) and (Di , Wi ). The equation of the line L i is Y = f (X ) = a X + b mod p, where a = (Wi − Yi ) · (Di − X i )−1 mod p and b = Yi − X i · (Wi − Yi ) · (Di − X i )−1 mod p. 3. Choose randomly a line L S representing Y = g(X ) = a  X + b mod p, where both a  and b are chosen randomly and then compute the intersection point (K i , Q i ) between the two lines L i and L S. 4. Compute ri = g ki mod p after choosing a random number ki such that gcd(ki , p − 1) = 1 and then compute si = ki−1 (S Pi − di · ri ) mod ( p − 1), where S Pi represents the service period for the server Seri containing the identity of U and the service expiration date. The signature of the service period with the server Seri is the pair (ri , si ). 5. Load the public parameters (S Pi , ri , si , K i , L S) into the memory of smart card or other storage devices of the user U corresponding to the server Seri . Figure 1 illustrates the concept of the registration phase. Login phase: If a user U wants to login to a server, say Seri , he submits his identity I D and password P W to the multi-server’s authentication system. Then, the authentication system performs the following steps: 1. Retrieve current timestamp T which is like a timestamp from the terminal of the user U . 2. Generate a random number Rani and compute Ai = g Rani mod p, Bi = eiRani ·T mod p. 3. Compute Q i = g(K i ) from the line L S, and the point (Di , Wi ), and then reconstruct the line L i : Y = f (X ) = a X + b joining the two points (K i , Q i ) and (Di , Wi ).

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Y

L2

L1

L3

(X 2 ,Y 2 )

Li

(D 3 ,W 3 )

(D 1 ,W 1 )

LS (K 3 ,Q 3 ) (K 2 ,Q 2 )

(K 1,Q 1) (D 2 ,W 2 ) (X 3 ,Y 3 ) (X 1 ,Y 1 ) K1

K2

K3

X

Fig. 1 The graphical concept of registration phase

4. Compute Z i = f (Bi ) from the line L i and transmit the login request message M = I D, (K i , Q i ), Z i , Ai , T, S Pi , (ri , si ) to Seri . The user can also login into another registered server at the same time. Authentication phase: On receiving M = I D, (K i , Q i ), Z i , Ai , T, S Pi , (ri , si ) at timestamp T  , the server Seri authenticates the user U as follows. 1. If (T  − T ) < ΔT , go to Step 2; otherwise Seri rejects it. ΔT is the expected legal time interval for transmission delay between the login terminal and the server Seri . 2. Check the correctness of I D. In case of incorrectness, Ser i reject the login request; otherwise go to Step 3. 3. Check whether the condition eiri · risi = g S Pi mod p holds. If the condition holds good, go to Step 4; otherwise Seri rejects it. 4. Compute Bi = Aidi ·T mod p (= eRani ·T mod p)

(6)

and construct the line L i joining the two points (K i , Q i ) and (Bi , Z i ). 5. Compute the point (X i , Yi ) from Eqs. (2) and (3), and then verify whether the point (X i , Yi ) lies on the line L i . If the point lies on the line L i , Seri accepts the login request message; otherwise Seri rejects it.

A Geometric-Based User Authentication Scheme for Multi-server Architecture …

19

Password change phase: In order to change the old password of a user U by a new password at the server Seri , the following steps need to be performed. 1. U submits his old password P W and a new password P W  . 2. The system constructs the line L i using Step 3 as same as that in the login phase of Lin et al.’s scheme [1]. 3. The system calculates Yi from f (X i ), where X i can be calculated by Eq. (2). According to the new password P W  , the system can construct new point (Di , Wi ) after computing Di and Wi by Eqs. (4) and (5), respectively. 4. The system constructs a new line L i joining the two points (Di , Wi ) and (X i , Yi ). Then, the system computes the intersection point (K i , Q i ) between the lines L i and L S. 5. Finally, the system updates the old value K i by the new value K i . We note that the drawback of the Lin et al.’s Password change phase is that if a user U submits his I D, the old password and also new a password to change his old password, no verification is required to validate the old password. As a result, after stolen the smart card or other storage devices whatever the user U is using, one can change the old password by a new password and he can update the information on U ’s smart card or other storage devices.

4 Weaknesses of Lin et al.’s Scheme 4.1 Minor Error In this subsection, we point out a minor error in the presentation of Lin et al.’s scheme. In the initialization phase of Lin et al.’s scheme, the C M computes the public key ei for the server Seri as ei = g di mod ( p − 1) shown in Eq. (1). Further, Bi = Aidi ·T mod p shown in Eq. 6. Now, Bi = eiRani ·T mod p

= (g di mod ( p − 1))Rani ·T mod p = (g Rani mod ( p − 1))di ·T mod p = Aidi ·T mod p [As Ai = g Rani mod p ]

Thus, in general, Bi = Aidi ·T mod p if ei = g di mod ( p − 1). The correct form to compute ei for the server Seri by the C M should be ei = g di mod p In the rest of the paper, we use Eq. (1∗ ) instead of Eq. (1) to compute ei .

(1∗ )

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D. Giri and T. Maitra

4.2 Security Weaknesses In this subsection, we point out some security weaknesses of Lin et al.’s scheme. Description of these weaknesses are outlined below. Attack 1: For security consideration, it is a common practice that users change their respective passwords frequently. By getting an old password P W old of a user U , an adversary A can forge a login request message at an arbitrary timestamp T ∗ using a valid login request message M old = I D, (K iold , Q iold ), Z iold , Aiold , T old , S Pi , (ri , si ) corresponding to the old password P W old of the user U , which was trapped by A from the previous transactions between the user U and the server Seri over the public communication channel. With the help of the above information, the adversary A can create a forged login request message by following the procedure given below. 1. As the line L S is a public parameter, the adversary A is aware of the line L S. Another possibility of the adversary A to construct the line L S is that A first retrieves two points (K i,a , Q i,a ) and (K j,b , Q j,b ) by intercepting the message Ma = I Da , (K i,a , Q i,a ), Z i,a , Ai,a , Ta , S Pi,a , (ri,a , si,a ) sent by the user Ua to Seri and the message Mb = I Db , (K j,b , Q j,b ), Z j,b , A j,b , Tb , S P j,b , (r j,b , s j,b ) sent by the user Ub to Ser j , respectively. Then, A easily reconstructs the line L S joining the two points (K i,a , Q i,a ) and (K j,b , Q j,b ). old 2. A then computes Di = eiI D mod p and Wiold = eiP W mod p from Eqs. (4) and (5), respectively, replacing P W in Eq. (5) by the old password P W old of the user U. 3. A constructs the old line L iold joining the two points (K iold , Q iold ) and (Di , Wiold ) corresponding to the old password P W old . 4. A then computes Yi after substituting X i in the line L iold . It is noted that A can compute X i as X i = I D ei mod p from the public parameters I D, ei and p. arb 5. A chooses an arbitrary password P W arb and computes Wi∗ = eiP W mod p corarb responding to the password P W . 6. A constructs a line L i∗ joining the two points (Di , Wi∗ ) and (X i , Yi ). 7. A computes the point (K i∗ , Q i∗ ) of intersection between two lines L i∗ and L S. ∗ Ran∗ ·T ∗ mod p after choosing a ran8. A computes Ai∗ = g Rani mod p and Bi∗ = ei i dom number Rani∗ . 9. A then computes Z i∗ as Y co-ordinate after substituting Bi∗ as X co-ordinate in the line L i∗ . 10. A transmits a login request message M ∗ = I D, (K i∗ , Q i∗ ), Z i∗ , Ai∗ , T ∗ , S Pi , (ri , si ) to the server Seri . The concept of this attack is explained in Fig. 2. Now, in the following, in order to show M ∗ = I D, (K i∗ , Q i∗ ), Z i∗ , Ai∗ , T ∗ , S Pi , (ri , si ) is a forged login request message, it is sufficient to prove that the authentication phase of Lin et al.’s scheme will pass successfully by the server Seri .

A Geometric-Based User Authentication Scheme for Multi-server Architecture …

21

Y old

L*i

Li

(X i ,Yi ) (K *i ,Q *i )

LS

old

old

(K i , Q i )

(D i ,W*i )

(D ,W

old

Ki

K*i

old i )

i

X Di

Fig. 2 The graphical concept of Attack 1

1. Let the server Ser i receive the login request message M ∗ at timestamp T  . The condition T  − T ∗ < ΔT in Step 1 of the authentication phase of Lin et al.’s scheme holds, if the login request message M ∗ = I D, (K i∗ , Q i∗ ), Z i∗ , Ai∗ , T ∗ , S Pi , (ri , si ) can be transmitted to the server Seri before the timestamp T ∗ + ΔT with T  < T ∗ + ΔT . By guessing the value of T  , A can choose T ∗ in such a fashion that T  − T ∗ < T holds. As a result, Step 1 in the authentication phase of Lin et al.’s scheme can be passed successfully. 2. Since I D is the identity of a valid user U , it is obvious that Step 2 in the authentication phase of Lin et al.’s scheme is passed successfully. 3. We note that since M old = I D, (K iold , Q iold ), Z iold , Aiold , T old , S Pi , (ri , si ) is a valid login request message trapped by A, the condition eri · risi = g S Pi mod p holds in the Step 3 of the authentication phase of Lin et al.’s scheme. Since the values of S Pi , ri and si in the message M ∗ remains unchanged as those in M old , obviously the condition eri · risi = g S Pi mod p holds for the values of ri , si , S Pi in the message M old . Hence, Step 3 of the authentication phase of Lin et al.’s scheme is passed successfully for the message M ∗ . ∗ ∗ ∗ ∗ 4. It follows that Bi∗ = (Ai∗ )di ·T mod p, since (Ai∗ )di ·T = (g Rani )di ·T mod p ∗ ∗ ∗ ∗ Ran ·T mod p = Bi∗ mod p). We note that L i∗ is a (= (g di )Rani ·T mod p = ei i ∗ ∗ line joining the two points (K i , Q i ) and (Bi∗ , Z i∗ ). Now, in the following, we show that the point (X i , Yi ) lies on the line L i∗ , where (X i , Yi ) can be calculated by Eqs. (2) and (3). It is noted that L i∗ is the line joining the two points (Di , Wi∗ )

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D. Giri and T. Maitra

and (X i , Yi ). Further, (K i∗ , Q i∗ ) is the point of intersection between two lines L i∗ and L S. Therefore, (K i∗ , Q i∗ ) is also a point lies on the line L i∗ . Again, according to Step 9 of the above attack procedure, (Bi∗ , Z i∗ ) is a point lies on the line L i∗ . So, all the three points (X i , Yi ), (K i∗ , Q i∗ ) and (Bi∗ , Z i∗ ) lie on the line L i∗ . Therefore, the point (X i , Yi ) lies on the line joining the two points (K i∗ , Q i∗ ) and (Bi∗ , Z i∗ ). Further, L i∗ is the line joining the two points (K i∗ , Q i∗ ) and (Bi∗ , Z i∗ ). Hence, the point (X i , Yi ) lies on the line L i∗ . Thus, Step 5 in the authentication phase of Lin et al.’s scheme is passed successfully. Hence, an adversary A can impersonate user U to access the server Seri by sending a forged login request message at an arbitrary timestamp. Attack 2: In this attack, we show that if there exists an integer t such that I Db = I Dat mod p, where I Da and I Db are two identities of two different users Ua and Ub , respectively, then one of the users Ua and Ub can impersonate other by sending a forged login request message to a server. In the following, we claim that Ua being an adversary can impersonate as user Ub by creating a forged login request message after revealing t. 1. Ua intercepts a valid login request message Mb = I Db , (K i,b , Q i,b ), Z i,b , Ai,b , Tb , S Pi,b , (ri,b , si,b ) from the previous transaction records sent by the user Ub to the server Seri over a public channel. 2. Ua creates its own login request message Ma = I Da , (K i,a , Q i,a ), Z i,a , Ai,a , Ta , S Pi,a , (ri,a , si,a ). 3. Ua then computes Di,a = eiI Da mod p and Wi,a = eiP Wa mod p. 4. Ua constructs a line L i,a joining the two points (K i,a , Q i,a ) and (Di,a , Wi,a ). 5. Ua computes X i,a = I Daei mod p and then computes Yi,a after substituting X i,a as X co-ordinate in the line L i,a . t mod p (= I Dat·ei mod p = I Dbei mod p) and Yi,b = 6. Ua computes X i,b = X i,a t mod p (= I Dat·di mod p = I Dbdi mod p). Yi,a 7. Ua then computes Di,b = eiI Db mod p. PW∗ ∗ = ei b mod p. 8. Ua chooses an arbitrary password P Wb∗ and computes Wi,b ∗ ∗ joining between two points (X i,b , Yi,b ) and (Di,b , Wi,b ). 9. Ua constructs a line L i,b ∗ ∗ ∗ 10. Ua finds the intersection point (K i,b , Q i,b ) between the two lines L i,b and L S. It is noted that the line L S is known to A as described in Step 1 of Attack 1. ∗ Ran∗ ·T ∗ ∗ ∗ ∗ 11. Ua computes Ai,b = g Rani,b mod p and Bi,b = ei i,b b mod p, where Rani,b is ∗ a chosen random number, Tb a chosen timestamp in such a manner that the difference between Tb∗ and the approximated time of the forged request message reached at the server Seri is less than ΔT. ∗ ∗ as Y co-ordinate after substituting Bi,b as X co-ordinate in the 12. Ua computes Z i,b ∗ line L i,b . ∗ ∗ ∗ , Q i,b ), Z i,b , 13. Ua finally transmits a login request message Mb∗ = I Db , (K i,b ∗ ∗ Ai,b , Tb , S Pi,b , (ri,b , si,b ) to the Seri .

A Geometric-Based User Authentication Scheme for Multi-server Architecture …

23

∗ ∗ ∗ ∗ It can be shown that the login request message Mb∗ = I Db , (K i,b , Q i,b ), Z i,b , Ai,b , ∗ Tb , S Pi,b , (ri,b , si,b ) can be a forged login request message which can be proved in the similar fashion as described in Attack 1. Therefore, the user Ua being an adversary can impersonate as user Ub by sending a forged login request message to Seri at an arbitrary timestamp. In the similar fashion, the user Ub being an adversary can impersonate as user Ua by sending a forged login request message to Seri at an arbitrary timestamp.

Attack 3: Let Ua and Ub be two users. We now show that if there is an integer t such that I Db = I Dat mod p, where I Da and I Db are two identities of two different users Ua and Ub , respectively, then one of them can reveal the password of others. We describe that Ua being an adversary can guess the password of the user Ub after intercepting a valid login request message Mb = I Db , (K i,b , Q i,b ), Z i,b , Ai,b , Tb , S Pi,b , (ri,b , si,b ) from the previous transaction records sent by the user Ub to the server Seri over a public channel. In order to do so, the user Ua performs the followings. Ua executes Step 1 through Step 6 as that described in Attack 2. After that Ua constructs a line L i,b joining the two points (X i,b , Yi,b ) and (K i,b , Q i,b ). Ua computes Di,b = eiI Db mod p using the public parameters ei , p. Ua then computes Wi,b after substituting Di,b as X co-ordinate in the line L i,b . In general, most passwords are meaningful strings each of less number of bit length, say l. In such a case, Ua can choose all possible pass words P W  ∈ {0, 1}l and for each P W  , Ua can check whether Wi,b = eiP W mod p holds. If Ua finds such a P W  so that the above condition becomes true, it will be the actual password of Ub ; otherwise Ua attempts for another one. In a similar fashion, Ub can also recover the password of Ua .

5 Our Improved Scheme In this section, we propose an improved version of Lin et al.’s scheme [1] to remedy their weaknesses. Our scheme consists of four phases, namely, initialization, registration, login and authentication phases as in Lin et al.’s scheme. Initialization phase: We assume that there are m servers in a multi-server environment. Let Sm = {Ser 1 , Ser 2 , · · · , Ser m } be a set of such m servers. Let H (·) be a secure cryptographic hash function [13]. The C M chooses two system parameters p, where p is a large prime and g a primitive element in Z ∗p . The C M chooses randomly a secret key di ∈ G F( p) for each server Seri ∈ Sm , computes the corresponding public key ei as ei = g di mod p. and then the C M sends the key pair (ei , di ) securely to Seri .

(7)

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D. Giri and T. Maitra

Registration phase: When a new user U wants to log into the multi-server computer system, he submits his identity I D and password P W securely to the C M. Now, if a set Sn (Sn ⊆ Sm ) of servers grant the user U , then only the C M requires to perform the following steps for each Seri ∈ Sn . 1. CM computes X i = H (I D, di ), Yi = X idi mod p,

(8) (9)

Ci = H (Yi ), Di = eiI D mod p,

(10) (11)

Wi = eiP W mod p,

(12)

where (X i , Yi ) and (Di , Wi ) are two points on the Euclidean plane. 2. CM constructs a line L i joining two points (X i , Yi ) and (Di , Wi ) as derived in Step 2 in registration phase of Lin et al.’s scheme. 3. Similar to Step 3 in registration phase of Lin et al.’s scheme, CM chooses randomly a line L S representing Y = g(X ) = a  X + b mod p, where both a  and b are chosen randomly and then compute the intersection point (K i , Q i ) between the two lines L i and L S. 4. Using the secret key di of the server Seri , identity I D of U and the service expiration date E D, CM computes S Pi = H (di ||I D||E D) as a service period for the server Seri . 5. CM loads the public parameters (S Pi , K i , L S, Ci , Vi ) and a secret parameter X i into the memory of smart card or other storage devices of the user U corresponding to the server Seri . Login phase: This phase is similar to that in Lin et al.’s scheme [1]. In this phase, the system will send the massage M in the format M = I D, (K i , Q i ), Z i , Ai , T, S Pi  instead of M = I D, (K i , Q i ), Z i , Ai ,T, S Pi , (ri , si ) in Lin et al.’s scheme. Authentication phase: On receiving M = I D, (K i , Q i ), Z i , Ai , T, S Pi  at timestamp T  , the server Seri authenticates the user U by performing the following steps. 1. If (T  − T ) < ΔT , where ΔT is the expected legal time interval for transmission delay between the login terminal and the server Seri , go to Step 2. Otherwise, Seri rejects it. 2. If I D is not valid, Seri rejects the login request; otherwise, go to Step 3. 3. Compute S Pi∗ = H (di ||I D||E D) and then check whether S Pi∗ = S Pi holds. If the condition holds good, go to Step 4; otherwise Seri rejects it. 4. Compute Bi = Aidi ·T mod p (= eRani ·T mod p)

(13)

A Geometric-Based User Authentication Scheme for Multi-server Architecture …

25

and construct the line L i joining the two points (K i , Q i ) and (Bi , Z i ). 5. Compute the point (X i , Yi ) from Eqs. (8) and (9), and then verify whether the point (X i , Yi ) lies on the line L i . If the point lies on the line L i , Seri accepts the login request message; otherwise Seri rejects it. Password change phase: In order to change the old password of a user U by a new password at the server Seri , the following steps need to be performed. 1. U first submits his old password P W . In order to update his old password P W , he requires to enter a new password. Let this new password be P W  . 2. The system computes Q i from the line L S using the point K i . 3. The system then reconstructs the line L i joining two points (K i , Q i ) and (Di , Wi ), where Di and Wi can be computed by Eqs. (11) and (12). 4. The system calculates Yi∗ from the line L i using the secret parameter X i and verifies the condition Ci = H (Yi∗ ). If it holds, go to Step 5; otherwise, that system rejects it. 5. According to the new password P W  , the system computes Wi by Eq. (12). 6. The system then constructs a new line L i joining two points (Di , Wi ) and (X i , Yi ). The system also computes the intersection point (K i , Q i ) between the lines L i and L S. 7. Finally, the system updates the old value K i by the new value K i .

6 Analysis of Our Improved Scheme In this section, we analyze the security of our proposed scheme. [Forgery attacks] We now describe that our improved scheme is secure against two proposed attacks: Attack-1 and Attack-2 which are already described in Sect. 4.2. The proposed Attack-1 We can show that Attack-1 described in Sect. 4.2 is infeasible on our improved scheme. Let an adversary A proceed Step 1 through Step 5 in similar fashions described as in Attack-1 in Sect. 4.2. In order to proceed step 6 of Attack-1, attacker needs to compute the new line L i∗ . But, it is computationally hard to compute the line L i∗ by A corresponding to the new password P W ∗ , as X i is secret parameter not known to the user A. As a result, it is hard for the adversary A to compute new K i∗ . Hence, the attacker could not mount this attack and our scheme is secure against Attack-1. The proposed Attack-2 In the following, we can show that Attack-2 described in Sect. 4.2 could not be mounted on our improved scheme. Ua being an adversary can proceed from Step 1 to Step 5 in similar fashions as t they are described in in Attack-2 of Sect. 4.2. Since X i,b = X i,a mod p, Ua can not

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compute X i,b and Yi,b with the help of the relation I Db = I Dbt mod p for some ∗ . integer t. As a result, it is computationally hard to construct for Ua a new line L i,b Hence, our scheme is secure against Attack-2. [Password-guessing attack] In our scheme, let an adversary A trap a valid login request message M = I D, (K i , Q i ), Z i , Ai , T, S Pi  of a user U . We note that P W is embedded only in Wi . The adversary A needs to reveal Wi . To guess the password of the user U , the adversary A constructs the line L i . To construct the line L i , A needs to compute Bi . Now, if Bi is known, A can easily construct the line L i by joining two points (K i , Q i ) and (Bi , Z i ). But, since di is a secret key only hold by the server Seri , it is computationally hard to compute Bi from the given knowledge. Hence, our scheme resists the password-guessing attack. [Replay attack] In our scheme, let an adversary A attempt to record the exchanged messages between a user U and the server, say, Seri . The replay of the old request message M = I D, (K i , Q i ), Z i , Ai , T, S Pi  sent by the user U to the Seri fails because the validity of a message can be checked through the timestamp. As a result, our scheme is secure against a replay attack.

7 Conclusion In this paper, we have pointed out some security weaknesses of Lin et al.’s remote user authentication scheme. We have described two types of forgery attacks on their scheme described in Attack 1 and Attack 2. Further, in Attack 3, we have shown that password guessing attack can be mounted on their scheme. In addition, in Attack 4, the password change protocol in the password change phase of their scheme is not secure, because of the fact that, after stolen the smart card or some other storage devices of a user what he uses, one can easily change the old password by a new password for that user. Further, we propose an improvement of their scheme to withstand their weaknesses.

References 1. Lin IC, Hwang MS, Li LH (2003) A new remote user authentication scheme for multi-server architecture. Future Gener Comput Syst 19:13–22 2. Lamport L (1981) Password authentication with insecure communication. Commun ACM 24:770–772 3. Hwang MS, Li L (2000) A new remote user authentication scheme using smart cards. IEEE Trans Consum Electron 46(1):28–30 4. Chien HY, Jan JK, Tseng YM (2002) An efficient and practical solution to remote authentication: smart card. Comput Secur 21(4):372–375

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5. Kim KW, Jeon JC, Yoo KY (2005) An improvement on yang et al.’s password authentication schemes. Appl Math Comput 170:207–215 6. Shen JJ, Lin CW, Hwang MS (2003) Security enhancement for the timestamp-based password authentication scheme using smart cards. Comput Secur 22(7):591–595 7. Yoon EJ, Ryu EK, Yoo KY (2004) Efficient remote user authentication scheme based on generalized elgamal signature scheme. IEEE Trans Consum Electron 50(2):568–570 8. Li LH, Lin IC, Hwang MS (2001) A remote password authentication scheme for multi-server architecture using neural networks. IEEE Trans Neural Netw 12(6):1498–1504 9. Amin R, Maitra T, Giri D (2013) Article: an improved efficient remote user authentication scheme in multi-server environment using smart card. Int J Comput Appl 69(22):1–6 10. Giri D, Sherratt RS, Maitra T (2016) A novel and efficient session spanning biometric and password based three-factor authentication protocol for consumer usb mass storage devices. IEEE Trans Consum Electron 62(3):283–291 11. Maitra T, Giri D (2014) An efficient biometric and password-based remote user authentication using smart card for telecare medical information systems in multi-server environment. J Med Syst 38(12):142 12. Maitra T, Islam SH, Amin R, Giri D, Khan MK, Kumar N (2016) An enhanced multi-server authentication protocol using password and smart-card: cryptanalysis and design. Secur Commun Netw 9(17):4615–4638 13. NIST: Secure hash standard (1995) Federal Information Processing Standard. FIPS-180-1

Solving the Search-LWE Problem by Lattice Reduction over Projected Bases Satoshi Nakamura, Nariaki Tateiwa, Koha Kinjo, Yasuhiko Ikematsu, Masaya Yasuda, and Katsuki Fujisawa

Abstract The learning with errors (LWE) problem assures the security of modern lattice-based cryptosystems. It can be reduced to classical lattice problems such as the shortest vector problem (SVP) and the closest vector problem (CVP). In particular, the search-LWE problem is reduced to a particular case of SVP by Kannan’s embedding technique. Lattice basis reduction is a mandatory tool to solve lattice problems. In this paper, we give a new strategy to solve the search-LWE problem by lattice reduction over projected bases. Compared with a conventional method of reducing a whole lattice basis, our strategy reduces only a part of the basis and, hence, it gives a practical speed-up in solving the problem. We also develop a reduction algorithm for a projected basis, and apply it to solving several instances in the LWE challenge, which has been initiated since the middle of 2016 in order to assess the hardness of the LWE problem. Keywords Learning with errors (LWE) · Lattice reduction · Projected lattices

S. Nakamura · K. Kinjo NTT Secure Platform Laboratories, Musashino, Japan e-mail: [email protected]; [email protected] K. Kinjo e-mail: [email protected] S. Nakamura · N. Tateiwa Graduate School of Mathematics, Kyushu University, Fukuoka, Japan e-mail: [email protected] Y. Ikematsu · K. Fujisawa Institute of Mathematics for Industry, Kyushu University, Fukuoka, Japan e-mail: [email protected] K. Fujisawa e-mail: [email protected] M. Yasuda (B) Department of Mathematics, Rikkyo University, Tokyo, Japan e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 D. Giri et al. (eds.), Proceedings of the Sixth International Conference on Mathematics and Computing, Advances in Intelligent Systems and Computing 1262, https://doi.org/10.1007/978-981-15-8061-1_3

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1 Introduction Since the US National Institute of Standards and Technology (NIST) began a process to develop new standards for post-quantum cryptography (PQC) in 2015 and called for proposals in 2016, lattice-based cryptography has been paid attention as a candidate of PQC. (See [18] for the process.) At the submission deadline of November 30, 2017, for the call, NIST received more than 20 proposals of lattice-based schemes. At the end of January 2019, NIST announced that 26 candidates can move to the second round of the process, and 12 candidates are based on lattices among them. In particular, the security of the 12 lattice-based candidates relies on the hardness of the LWE or NTRU problem, and 9 candidates are based on the LWE problem [13]. Therefore, it is becoming much more important to precisely analyze the hardness of the LWE problem. We here focus on the search-LWE problem, which is reduced by Kannan’s embedding technique [9] to unique SVP, a particular case of SVP. Lattice basis reduction is a strong tool to solve lattice problems. In particular, BKZ [16] and its variants such as BKZ 2.0 [7] are de facto reduction algorithms to estimate the security level of lattice-based cryptography. (See [1] for details.) Given a blocksize β, BKZ repeatedly calls an SVP oracle in a β-dimensional lattice to find short lattice vectors. (Larger β enables us to find shorter lattice vectors.) In security estimation, it is discussed which blocksizes β are required for BKZ to find a lattice vector of target length. In this paper, we give a new strategy of solving the search-LWE problem. While a conventional method reduces a whole basis B, our strategy reduces a projected basis π (B) for some projection position . We also develop a reduction algorithm for a projected basis π (B). Our reduction algorithm is based on DeepBKZ [20], a mathematical improvement of BKZ. In particular, our reduction algorithm handles only a part of the basis B to reduce the projected basis π (B). Hence, our strategy with our reduction algorithm gives a practical speed-up to solve the search-LWE problem, compared with the conventional method reducing the whole basis B. We also report experimental results of solving several instances in the LWE challenge [6] by our reduction algorithm. In particular, we show which blocksizes β our reduction algorithm requires to solve the search-LWE problem. Notation The symbols Z and R denote the ring of integers and the field of real numbers, respectively. We always drepresent all vectors in column format. We let ai bi between two vectors√a = (a1 , . . . , ad ) a, b denote the inner product i=1  and b = (b1 , . . . , bd ) , and let a denote the Euclidean norm a, a.

2 Mathematical Preliminaries In this section, we briefly review several definitions and properties on lattices. We also introduce some notions of lattice basis reduction and algorithms to achieve them.

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2.1 Lattices and Their Bases Let b1 , . . . , bd be d linearly independent vectors in Rd for a positive integer d. The lattice with basis B = [b1 , . . . , bd ] is the set of all integral linear combinations of the bi ’s, defined as  L = L(B) :=

d 

 xi bi : xi ∈ Z (1 ≤ i ≤ d) .

i=1

The lattice L is full-rank, and its dimension is d. We regard every basis B as a d × d matrix. Every lattice has infinitely many bases for d ≥ 2. In particular, if two bases B1 and B2 span the same lattice, there exists a unimodular matrix V satisfying B1 = B2 V. The volume of L is defined as vol(L) = | det(B)|, independent of the choice of bases. The Gram–Schmidt orthogonalization of a basis B is the orthogonal family B =  [b1 , . . . , bd ], recursively defined by b1 := b1 and for 2 ≤ i ≤ d bi := bi −

i−1 

μi, j bj with μi, j :=

j=1

bi , bj  bj 2

.

Remark that the Gram–Schmidt vectors bi depend on the order of the basis vectors bi . Let U = (μi, j ) be the lower triangular matrix, where set μi,i = 1 for all i and μi, j = 0 d for all j > i. Then we clearly have B = B U and hence vol(L) = i=1 bi . We let π denote the orthogonal projection over the orthogonal supplement of the R-vector space b1 , . . . , b−1 R for 2 ≤  ≤ d. Note that the projection π depends on B. (For convenience, set π1 = id as the identity map.) We let π (L) denote the projected lattice spanned by the projected basis π (B) := [π (b ), π (b+1 ), . . . , π (bd )]. Moreover, we denote by B[i, j] the local projected block [πi (bi ), πi (bi+1 ), . . . , πi (b j )], and by L [i, j] the lattice spanned by B[i, j] for i ≤ j. We let λi (L) denote the i-th successive minimum for 1 ≤ i ≤ d. In particular, the first successive minimum λ1 (L) is equal to the length of a shortest non-zero vector in L.

2.2 Lattice Basis Reduction From a basis of a lattice L, lattice basis reduction finds a new basis [b1 , . . . , bd ] of L with short and nearly orthogonal vectors bi ’s. It is mandatory to solve lattice problems such as SVP and CVP. In this section, we introduce typical reduction algorithms.

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Algorithm 1 Size-reduction at position k 1: procedure Size reduce(B, k)

B = [b1 , . . . , bd ]: basis, k: position 2:

Assume that Gram–Schmidt coefficients μi, j are pre-computed 3: for  = k − 1 downto 1 do 4: if |μk, | > η then

Set η = 21 in theory 5: r ← μk, , bk ← bk − r b

Update of the basis vector bk 6: for j = 1 to  do 7: μk, j ← μk, j − r μ, j

Update of Gram–Schmidt coefficients (set μ, = 1) 8: end for 9: end if 10: end for 11: return B

Size-reduced basis at position k 12: end procedure

2.3 Typical Reduction Algorithms Lenstra-Lenstra-Lovász (LLL) A basis B = [b1 , . . . , bd ] is called δ-LLL-reduced for a reduction parameter 41 < δ < 1 if it satisfies the following two conditions: (i) It is size-reduced; the Gram–Schmidt coefficients satisfy |μi, j | ≤ i > j. (ii) It satisfies Lovász’ condition δbk−1 2 ≤ πk−1 (bk )2 for all k.

1 2

for all

Every δ-LLL-reduced basis of a lattice L satisfies both b1  ≤ α

d−1 2

λ1 (L) and b1  ≤ α

d−1 4

1

vol(L) d ,

4 . (See [5] or [12].) The LLL algorithm [10] finds an LLL-reduced where α = 4δ−1 basis by swapping adjacent basis vectors bk−1 and bk when they do not satisfy Lovász’ condition. It also calls the size reduction algorithm (Algorithm 1) as a subroutine in order to achieve the above condition (i). (The size reduction algorithm shall be a key ingredient in Sect. 4 later.) Furthermore, the LLL algorithm is applicable also for linearly dependent vectors to remove its linear dependency.

LLL with deep insertions (DeepLLL) This is a simple generalization of LLL [16], in which non-adjacent basis vectors can be changed; If πi (bk )2 < δbi 2 is satisfied for some i < k, the basis vector bk is inserted between bi−1 and bi as σi,k : B ← [b1 , . . . , bi−1 , bk , bi , . . . , bk−1 , bk+1 , . . . , bd ].

(1)

Every output basis satisfies the following condition: A basis B = [b1 , . . . , bd ] is called δ-DeepLLL-reduced for a reduction parameter 41 < δ < 1 if it is size-reduced and it also satisfies δbi 2 ≤ πi (bk )2 for all i < k. This basis satisfies both b1  ≤

 √  α  (d−1)(d−2) α  d−2 d−1 1 2 4d α 1+ λ1 (L) and b1  ≤ α 2d 1 + vol(L) d , 4 4

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where α is the same as in the LLL case. (See [21] for a proof.) These properties are strictly stronger than LLL. Block Korkine–Zolotarev (BKZ) A basis B = [b1 , . . . , bd ] of a lattice L is called Hermite–Korkine–Zolotarev (HKZ) reduced if it is size-reduced and it satisfies bi  = λ1 (πi (L)) for every 1 ≤ i ≤ d. The BKZ-reduction is a local block of HKZreduction [14–16]; A basis B of a lattice L is called β-BKZ-reduced for a blocksize β ≥ 2 if it is size-reduced and every local block B[ j, j+β−1] is HKZ-reduced for 1 ≤ j ≤ d − β + 1. In particular, the second condition means bj  = λ1 (L [ j,k] ) for every 1 ≤ j ≤ d − 1 with k = min( j + β − 1, d). Every β-BKZ-reduced basis (d−1)/(β−1) λ1 (L). (See [15] for a proof.) This implies that larger satisfies b1  ≤ γβ β finds a shorter lattice vector. The BKZ algorithm [16] finds an almost β-BKZreduced basis, and it calls LLL as a subroutine to reduce every local block before the enumeration of the shortest vector over the block lattice. Improved algorithms have been proposed such as BKZ 2.0 [7] and progressive-BKZ [4]. DeepBKZ This is a combination of DeepLLL and BKZ [20]. Specifically, it calls DeepLLL as a subroutine alternative to LLL in BKZ. Thus, every output basis of DeepBKZ satisfies the following condition of reduction; A basis is called (δ, β)DeepBKZ-reduced for a reduction parameter 41 < δ < 1 and a blocksize β ≥ 2, if it is both δ-DeepLLL-reduced and β-BKZ-reduced. Experiments in [20] show that DeepBKZ often finds short lattice vectors by smaller blocksizes than BKZ in practice.

3 The LWE Problem and Lattice Problems In this section, we give the definition of the LWE problem and also review how to reduce the search-LWE problem to unique-SVP by Kannan’s embedding technique [9], which seems the most efficient in practice.

3.1 Introduction to the LWE Problem The LWE problem was first introduced by Regev [13]. For an odd prime q, let Z

q denote a set of representatives of integers modulo q. (e.g., We set Zq = Z ∩ − q2 , q2 .) We also let [a]q ∈ Zq denote the reduction of an integer a by modulo q. Definition 1 (LWE) Let n be a dimension parameter, q a prime modulus parameter, and χ an error distribution over Z. Let s ∈ Zqn denote a secret vector, whose entries are chosen uniformly at random from Zq . For a positive integer d, given d LWE samples

 (2) ai , [ai , s + ei ]q ∈ Zqn × Zq

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for 1 ≤ i ≤ d, where ai ’s are uniformly chosen at random from the set Zqn and ei ’s are sampled from the distribution χ . Then two questions are asked in the LWE problem: – The decision-LWE problem is to distinguish whether given t = (t1 , . . . , td ) is obtained from (2) with ti = [ai , s + ei ]q for some ai , or uniformly at random. – The search-LWE problem is to recover the secret vector s from LWE samples (2). Here, we take the discrete Gaussian distribution DZ,σ with standard deviation σ for χ . As mentioned in Sect. 1, we focus on the search-LWE problem. From d LWE samples (2), set an error vector e = (e1 , . . . , ed ) and a target vector t = (t1 , . . . , td ) with ti = [ai , s + ei ]q for 1 ≤ i ≤ d. We set A = [a1 , . . . , ad ] ∈ Zqn×d . (Recall that we always write vectors in column format.) Then the LWE samples (2) are simply written as the pair (A, t) satisfying t ≡ A s + e mod q. Namely, the search-LWE problem is to recover the secret vector s (or equivalently the error vector e) from the pair (A, t).

3.2 Reduction of the Search-LWE Problem to Lattice Problems According to a survey work [3], there are several strategies to solve the searchLWE problem. Here, we describe how to reduce the search-LWE problem to lattice problems. Reduction to a particular case of CVP Given an LWE instance (A, t), let

Λq (A) := x ∈ Zd | x ≡ A z mod q for some z ∈ Zqn denote a q-ary lattice of dimension d.

(See [11] for q-ary lattices.) The columns of the  d × (d + n) matrix C = qId | A form a system of generators of the q-ary lattice Λq (A), where Id denotes the d × d identity matrix. A basis B of the q-ary lattice Λq (A) is given by computing LLL (or Hermite normal form) for the columns of C. (Note that B is a d × d matrix.) Namely, the columns of B is linearly independent and we have Λq (A) = L(B). Then we can regard the target vector t as a vector bounded (A). (The minimum distance in distance from the vector A s in the q-ary lattice Λq√ between t and A s over Λq (A) is equal to e ≈ σ d.) In other words, we can regard the LWE instance (A, t) as an instance of bounded distance decoding (BDD), which is a particular case of CVP. Reduction to a particular case of SVP There are several methods such as Kannan’s embedding technique [9] to reduce BDD to unique SVP, a particular case of SVP finding a non-zero shortest vector in a lattice L under λ2 (L) > γ λ1 (L) for some γ . Procedures of Kannan’s embedding for the LWE instance (A, t) are the following steps:

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Step 1. With a basis B of the q-ary lattice Λq (A) (note that B is a d × d matrix), we construct the (d + 1) × (d + 1) matrix B =

 B 0

 t , 1

(3)

where let 0 denote the row vector of length d with all entries 0. Let L  := L(B ) denote the lattice spanned by the columns of B , and its dimension is d + 1. For the error vector e, the lattice L  includes very short vectors ±e , where we set        e t A s  mod q ∈ Zqd+1 . ≡ − (4) e := 0 1 1 Step 2. Reduce B by lattice basis reduction to find short lattice vectors ±e ∈ L  . (Note that ±e are shortest in the lattice L  in most cases for large d such as d ≥ 2n.) Then we can recover the error vector e from e , and hence the secret vector s.

4 Solving the Search-LWE Problem Using Projected Lattices In this section, we describe our strategy of solving the search-LWE problem over projected lattices, and also develop a lattice reduction algorithm for a projected basis.

4.1 Our Strategy As described in the previous subsection, we need to find the shortest non-zero vector e , defined by (4), in the (d + 1)-dimensional lattice L  from the basis B = [b1 , . . . , bd+1 ] of L  . (See Eq. (3) for B .) A basic idea of our strategy is to find the projected vector π (e ) in π (L  ) for some position , and recover the target use the same assumption lattice vector e in L  from the projected vector. We here √ √ d−i+1  as in [2] on lengths of projected vectors; πi (e ) ≈ √d e  ≈ d − i + 1 · σ for every i. In our strategy, we take the maximum position  > 1 satisfying ( − 1) inequalities πi (e ) ≈

√ 1 d − i + 1 · σ < bi  (1 ≤ ∀i < ), 2

(5)

  where b 1 , . . . , bd+1 denotes the Gram–Schmidt vectors of the basis B . Then we   proceed the following two steps to find the target lattice vector e in L :

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(i) We reduce the projected basis π (B ) = [π (b ), . . . , π (bd+1 )] to find a vector v in L  satisfying π (v)√= π (e ). In practice, it is sufficient to find a vector v of length π (v) ≈ d −  + 1 · σ ≈ π (e ). (Such a projected vector is unique up to signs of plus and minus since π (e ) is unusually short in the projected lattice π (L  ).) (ii) We recover the target lattice vector e from the lattice vector v ∈ L  . For step (i), we shall develop a reduction algorithm for the projected basis π (B ) in the next subsection. For step (ii), we apply size reduction at position  (Algorithm 1) to the sub-basis {b1 , . . . , b−1 , v} to recover the target vector e . Let v denote the -th basis vector in L  after size reduction. Then it satisfies v − π (v) ∈ P  (b1 , . . . , b−1 ), where P



(b1 , . . . , b−1 )

=

 −1  i=1

xi bi

 1 : xi ∈ R with |xi | < (1 ≤ ∀i < ) 2

denotes the fundamental parallelepiped spanned by the Gram–Schmidt vectors of ( − 1) vectors b1 , . . . , b−1 . (See [12] for properties of size reduction.) On the other hand, the difference vector e − π (e ) is included in the same parallelepiped P  (b1 , . . . , b−1 ) since it follows from inequalities (5) that |e , bi | = |πi (e ), bi | ≤ πi (e ) · bi 
π j (v) then 13: z ←−1 14: {b1 , . . . , bh } ← LLL({b1 , . . . , b j−1 , v, b j , . . . , bh }, δ) (at stage k) 15:

Insert v at j-th position, and remove the linear dependency by LLL 16: Proj-DeepLLL({b1 , . . . , bh }, δ, ) (at stage j) 17:

DeepLLL-reduce the projected basis π (B) (Algorithm 2) 18: else 19: z ← z+1 20: Proj-DeepLLL({b1 , . . . , bh }, δ, ) (at stage h − 1)

Algorithm 2 21: end if 22: Size reduce(B, )

Size reduction at position  (Algorithm 1) 23: end while 24: return B

π (B) is (δ, β)-DeepBKZ-reduced 25: end procedure

5 Experiments for the LWE Challenge In this section, we apply our lattice reduction (Algorithm 3) to solving several instances in the LWE challenge [6] and report its experimental results. In the LWE challenge, every instance of the search-LWE problem is parametrized by a pair of two parameters (n, α). LWE parameters (n, q, σ, m) is set as follows: (i) m = n 2 is the maximal number of LWE sample pairs (2), (ii) q is the smallest odd prime number exceeding n 2 , and (iii) σ = αq is the standard deviation of the discrete Gaussian distribution χ = DZ,σ .

5.1 Implementation We implemented our DeepBKZ-reduction for a projected basis (Algorithm 3) in C++ programs with the NTL library [17]. We used the g++ complier (version 9.1.0) with -O2 -std=c++17 option. We set a triple of B = [b1 , . . . , bd ], μ = (μi, j )1≤ j 0.

i. If A > 0, B < 0, then Eq. (9) has one fixed point P0 (0, 0). ii. If A < 0, B > 0 then Eq. (9) has one fixed point P0 (0, 0). iii. If  A > 0, B > 0or A < 0,  B < 0 then Eq. (9) has three fixed points P0 (0, 0),   A A P1 , 0 , P2 − B , 0 B If τ be the trace and Δ be the determinant of the Jacobian matrix for system (9), then τ = 0 and Δ = 3Bu 2 − A. The eigenvalues of the dynamical system (9) at any fixed point are given by  (11) λ1,2 = ± A − 3Bu 2 . Next, we consider each of the four cases obtained earlier so as to classify the fixed points. Case I: When A > 0, B < 0, we get the fixed point as P0 (0, 0). On solving the determinant, we get Δ = −A which further implies that Δ < 0 and τ = 0. Therefore, we obtain real and opposite eigenvalues. So P0 (0, 0) is a saddle point and there are no closed orbits. Case II: When A < 0, B > 0, we get only one fixed point P0 (0, 0). Then Δ = −A. But Δ > 0 as A is negative. Also τ 2 − 4Δ < 0. Thus P0 (0, 0) is a center and a set of nonlinear periodic orbits (N P O1,0 ) are available around the equilibrium points. Case III: A > 0, B > 0. i. For the fixed point P0 (0, 0), Δ = −A which implies delta is negative. Therefore, we get real and opposite eigenvalues and so P0 (0, 0) comes out to be a saddle point.    A ii. For P1 , 0 , Δ > 0 since A is positive. Also τ 2 − 4Δ < 0. Thus, B   A P1 , 0 is a center. B    A iii. For P2 − B , 0 , Δ > 0 since A is positive. Also τ 2 − 4Δ < 0. Thus,    A P2 − B , 0 is a center. There are pair of nonlinear homoclinic orbits (N H O1,0 ) and two groups of nonlinear periodic orbits (N P O1,0 ) lying inside the nonlinear homoclinic orbits. A group of superperiodic orbits (S P O3,1 ) is also present surrounding the three fixed points and a separatrix layer.

Bifurcation Analysis of Tsunami Waves for the Modified Geophysical …

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u

Fig. 1 Phase plots of the system (9) for a A > 0, B < 0, b A < 0, B > 0, c A > 0, B > 0, d A < 0, B < 0

Case IV: A < 0, B < 0. i. For the fixed point P0 (0, 0), Δ = −A which implies Δ > 0 as A is positive. Also τ2 − 4Δ < 0.Thus P0 (0,0) is a center. A ii. For the fixed point P1 , 0 , Δ < 0 since Δ = 2 A but A is negative. Thus B the fixed point is a saddle point.    iii. For the fixed point P2 − BA , 0 , Δ < 0 since Δ = 2 A but A is positive. Thus, the fixed point is a saddle point.



 A ,0 B

There is a pair of heteroclinic orbits (H C O1,0 ) joining P1 and    P2 − BA , 0 . Also there is a set of periodic orbits surrounding fixed point P0 (0, 0).

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(b)

(a) a=4 a=8

ω0 = 5

0.4

0.2

−5

0

5

0 −10

10

−5

0 −10

10

5

0

(f) 0

−0.2

−0.2

−0.2

−0.6

a=4 a=8 −5

0

5

10

u

0

u

0

−0.4

10

5

0

ξ

(e)

(d)

−0.8 −10

−5

ξ

ξ

u

0.4

0.2

0.2

v=1 v=3

0.6

u

0.4

0.8

ω0 = 2

0.6

u

u

0.6

0 −10

(c)

0.8

0.8

−0.4 ω0 = 2

−0.6

ω0 = 5

−0.8 −10

−5

0

5

10

ξ

ξ

−0.4 −0.6 −0.8 −10

v=1 v=3 −5

0

5

10

ξ

Fig. 2 Variation of solitary tsunami waves of mGKdV Eq. (3) for different values of a a with b = 3, ω0 = 2, and v = 1, b ω0 with a = 8, b = 3, and v = 1, c v with a = 8, b = 3, and ω0 = 2, d same as Fig. 2a, e same as Fig. 2b, f same as Fig. 2c

4 Different Tsunami Waves The solitary tsunami wave solutions corresponding to the pair of homoclinic orbits at P0 (0, 0) in Fig. 1c of the mGKdV equation are given by  u=±

√ 2A sech( Aξ). B

(12)

In Fig. 2a, d, we show compressive and rarefactive solitary tsunami waves of mGKdV Eq. (3) for various a corresponding to nonlinear homoclinic orbits N H O1,0 of dynamical system (9) at P0 (0, 0) Fig. 1c. It is observed that if a is increased, amplitude and width of the solitary tsunami waves are decreased. As a result solitary tsunami waves are diminished as nonlinear coefficient a increases. In Fig. 2b, e, we show compressive and rarefactive solitary tsunami waves of mGKdV Eq. (3) for various ω0 corresponding to nonlinear homoclinic orbits N H O1,0 of dynamical system (9) at P0 (0, 0) Fig. 1c. It is found that amplitude of the solitary tsunami waves increases, but width decrease when ω0 increases. Thus, the solitary tsunami waves become spiky. In Fig. 2c, e, we depict compressive and rarefactive solitary tsunami waves of mGKdV Eq. (3) for various v corresponding to nonlinear homoclinic orbits N H O1,0 of dynamical system (9) at P0 (0, 0) Fig. 1c. It is found that amplitude of the solitary

Bifurcation Analysis of Tsunami Waves for the Modified Geophysical …

(a)

(b)

(c)

0.7

a=24

0.65

a=26

0.9

ω =1

0.8

0.6

0.7

0.55

u

u

u

0.55 0.5

v=1 v=1.8

0

ω0=1.2

0.65

0.6

71

0.6

0.5

0.45

0.5

0.45

0.4

0.4

0.35 380

0.35 380

385

390

395

400

0.4 385

390

ξ

395

400

385

380

ξ

390

395

400

ξ

Fig. 3 Variation of periodic tsunami waves of the mGKdV Eq. (3) for different values of a a with b = 2, ω0 = 1, and v = 1, b ω0 with a = 24, b = 2, and v = 1, c v with a = 24, b = 2, and ω0 = 1

(a)

(c)

(b) ω =1

a=24 a=28

1 0.5

1

0

0.5

u

u

u

0.5

0

0

0

−0.5

−0.5

−0.5

−1

20

30

40

ξ

50

60

v=1 v=1.8

0

ω =1.4

1

−1

20

30

40

ξ

50

−1

20

30

40

50

ξ

Fig. 4 Variation of superperiodic tsunami waves of the mGKdV Eq. (3) for different values of a a with b = 2, ω0 = 1, and v = 1, b ω0 with a = 24, b = 2, and v = 1, c v with a = 24, b = 2, and ω0 = 1

tsunami waves increases, but width decrease slowly when v increases. Thus, the solitary tsunami waves become slightly spiky as v increases. In Fig. 3, we present periodic tsunami waves of mGKdV Eq. (3) for various values of a, ω0 , and v corresponding to nonlinear period orbits (N P O1,0 ) of dynamical system (9) around P1 in Fig. 1c. It is noted that if a increases, amplitude and width of the periodic tsunami waves decrease. As a result, periodic tsunami waves are diminished when nonlinear coefficient a increases. On the other hand, if ω0 and v increase, then amplitude of the periodic tsunami wave increases and width decreases. So smoothness of the periodic tsunami waves decreases. In Fig. 4, we depict superperiodic tsunami waves of mGKdV Eq. (3) for various values of a, ω0 , and v corresponding to superperiodic orbits (S P O3,1 ) of dynamical system (9). It is perceived that if a increases, amplitude and width of the superperiodic tsunami waves decrease. As a result superperiodic tsunami waves are diminished

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when nonlinear coefficient a increases. On the other hand if ω0 and v increase, then amplitude of the superperiodic tsunami wave increases and width decreases. So smoothness of the superperiodic tsunami waves decreases.

5 Conclusions Bifurcation analysis of tsunami waves for the modified geophysical KdV equation has been addressed through phase plots and time series plots. Using traveling wave transformation, the mGKdV equation has been deduced to a dynamical system. Using analysis of phase plots, existence of the solitary, periodic, superperiodic tsunami waves has been obtained. The Coriolis parameter (ω0 ), nonlinear parameter (a), and velocity (v) of traveling wave have significant effects on these tsunami waves. Tsunami waves are enhanced when values of the nonlinear coefficient (a) is decreased. On the other hand, smoothness of the tsunami waves decreases if Coriolis parameter (ω0 ) and velocity (v) of the tsunami waves are increased. Acknowledgements This research work is funded by TMA Pai University Research Grant (6100/SMIT/R&D/26/2019), SMIT, SMU.

References 1. Lighthill J (2001) Waves in fluids, 2nd edn. Cambridge University Press, Cambridge 2. Geyer A, Quirchmayr R (2018) Shallow water equations for equatorial tsunami waves. Philos Trans R Soc A 376(2111):20170100. https://doi.org/10.1098/rsta.2017.0100 3. Constantin A, Johnson RS (2008) On the non-dimensionalisation, scaling and resulting interpretation of the classical governing equations for water waves. J Nonlinear Math Phys 15:58–73. https://doi.org/10.2991/jnmp.2008.15.s2.5 4. Korteweg DJ, de Vries G (1895) XLI. On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves. https://doi.org/10.1080/ 14786449508620739 5. Wazwaz A-M (2017) A two-mode modified KdV equation with multiple soliton solutions. Appl Math Lett 70:1–6. https://doi.org/10.1016/j.aml.2017.02.015 6. van Wijmgaarden L (1972) On the motion of gas bubbles in a perfect fluid. Ann Rev Fluid Mech 4:369–373. https://doi.org/10.1007/BF00037735 7. Stuhlmeier R (2009) KdV theory and the Chilean tsunami of 1960. Discret Continous Dyn Syst Ser B 12:623–632. https://doi.org/10.3934/dcdsb.2009.12.623 8. Karunakar P, Chakraverty S (2019) Effect of Coriolis constant on geophysical Korteweg-de Vries equation. J Ocean Eng Sci 4(2):113–121. https://doi.org/10.1016/j.joes.2019.02.002 9. Kirby JT, Shi F, Tehranirad B, Harris JC, Grilli ST (2013) Dispersive tsunami waves in the ocean: model equations and sensitivity to dispersion and Coriolis effects. Ocean Model 62:39– 55. https://doi.org/10.1016/j.ocemod.2012.11.009 10. Lakshmanan M, Rajasekar S (2003) Nonlinear Dynamics. Springer, Heidelberg 11. Saha A (2012) Bifurcation of traveling wave solutions for the generalized KP-MEW equations. Commun Nonlinear Sci Numer Simul. 17:3539. https://doi.org/10.1016/j.cnsns.2012.01.005

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12. Saha A (2017) Bifurcation, periodic and chaotic motions of the modified equal width burgers (MEW-Burgers) equation with external periodic perturbation. Nonlinear Dyn 87:2193–2201. https://doi.org/10.1007/s11071-016-3183-5

Effect of Heating Location on Mixed Convection of a Nanofluid in a Partially Heated Enclosure with the Presence of Magnetic Field Using Two-Phase Model Subhasree Dutta and Somnath Bhattacharyya

Abstract Mixed convective heat transfer in a lid-driven square enclosure in presence of discrete heat source at different location of the left wall is considered. The aim of this study is to analyze numerically the effect of external magnetic field inside the enclosure filled with Al2 O3 –water nanofluid, using Buongiorno’s two-phase model. The temperature of the right wall is lower than that of the heater, placed on the left wall. A control volume method over a staggered grid arrangement is used to discretize the governing equations. The discretized equations of two-dimensional continuity, momentum, energy and volume fraction are solved through a pressure correction based SIMPLE algorithm. The effect of several parameters such as Richardson number (0.1 ≤ Ri ≤ 5), Hartman number (0 ≤ H a ≤ 60), nanoparticle volume fraction (0 ≤ ϕb ≤ 0.05) on the mixed convection of the nanofluid in heat transfer and entropy generation is studied by considering the position of the heat source to vary from bottom to top. The work has a remarkable contribution for the improvement of thermal performance with minimal energy consumption in several engineering applications. Keywords Two-phase model · Mixed convection · Nanofluid · Magnetic field · Entropy generation

1 Introduction Heat transfer performance of nanofluid is one of the most essential needs among the researchers. Nanofluids are characterised by suspended nano-sized particles together with the base fluid such as ethyl alcohol and oil. Inclusion of nanoparticles inside the base fluid enhances the thermal conductivity of the fluid. Several researchers investigated different aspects of nanofluids and its industrial applications. Tiwari and Das [1] studied the effect of the inclusion of Cu–water nanofluid due to the movement of heated side walls and found that the heat transfer rate is enhanced S. Dutta (B) · S. Bhattacharyya Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur 721302, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 D. Giri et al. (eds.), Proceedings of the Sixth International Conference on Mathematics and Computing, Advances in Intelligent Systems and Computing 1262, https://doi.org/10.1007/978-981-15-8061-1_7

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in the presence of nanofluid. Xuan and Li [2] investigated experimentally the heat transfer performance due to the inclusion of nanoparticles and concluded that the rate of heat transfer is augmented as the nanofluid is used. Basak and Chamkha [3] used the Galerkin finite element method to study the non-uniform heating due to the existence of several types of nanofluid. They reported that Alumina–water nanofluid and Cu–water nanofluid exhibit higher enhancement in heat transfer. Several studies on the heat transfer using discrete heat sources placed on different locations are being made. Muftuoglu and Bilgen [4] investigated the effect of different positions of heaters in the open cavity so that the maximum heat transfer is achieved. Sivasankaran et al. [5] numerically studied the effect of discrete heating in an inclined cavity in presence of discrete heat source considering the size of the heat source as a parameter. They conclude that the heat transfer rate not only depends on the position, but the size of the heater also. Natural convective heat transfer under localised heating is studied numerically by Öztop et al. [6] considering several positions of heaters. They concluded that the heat and flow behaviour of fluid changes depending upon the configurations. Magnetohydrodynamics is an interesting topic because of its applications in industrial devices, engineering fields, medicine, geothermal energy extractions, etc. The study of the properties of electrically conducting fluid in presence of electromagnetic field is called Magnetohydrodynamics (MHD). Chamkha [7] studied the unsteady, convective flow field in a square enclosure in presence of magnetic field. They concluded that the transverse magnetic field retarded the movement of the flow and diminishes the heat transfer rate. Natural convection in nanofluid under the influence of magnetic field is numerically studied by Ghasemi et al. [8]. They found that the heat transfer rate decreases due to the enhancement in Hartman number. The effect of inclined magnetic field in the double-sided lid-driven square enclosure is numerically analysed by Hussain et al. [9] and found that the heat transfer rate attenuates with the augmentation of both Hartman number and nanoparticle volume fraction. All the previous studies are based on the single-phase model, incorporating constant thermophysical properties of the nanofluid. In the two-phase model, the slip velocity between the nanoparticles and the base fluid is generated due to seven slip mechanisms inertia namely, Brownian diffusion, thermophoresis, diffusiophoresis, Magnus effect, fluid drainage and gravity. Buongiorno’s [10] study shows that the Brownian diffusion and thermophoresis are the most important slip mechanisms to develop the slip velocity. Several studies are made on the two-phase model incorporating the effect of Brownian diffusion and thermophoresis. Motlagh and Soltanipour [11] studied the effect of natural convective heat transfer in the inclined enclosure considering Buongiorno’s two-phase model [10]. They concluded that the increment of Rayleigh number increases the non-uniformity inside the enclosure. Ho et al. [12] experimentally studied the effect of Brownian diffusion and thermophoresis using the Rayleigh–Bénard convection of Al2 O3 –water nanofluids and found that experimental results are in a good agreement with the numerical results for the nonhomogeneous model in comparison with the homogeneous model. Garoosi et al. [13] studied the convective heat transfer on an enclosure using the nonhomogeneous

Effect of Heating Location on Mixed Convection …

77

model and studied the effect of Richardson number on the non-uniformity of the nanoparticle distribution. The effect of several positions of discrete heaters in a square enclosure incorporating the effects of Brownian diffusion and thermophoresis in presence of Al2 O3 –water nanofluid is dealt in our present study. We investigate the effect of several parameters such as Richardson number, bulk volume fraction, Hartman number on the flow field, heat transfer and entropy generation. To the best of our knowledge, mixed convection using the nonhomogeneous model in presence of magnetic field, considering different locations of the heater has not yet been studied in the literature.

2 Mathematical Model A two-dimensional model for mixed convection in a square enclosure filled with Al2 O3 –water nanofluid is shown in Fig. 1. In our mathematical model, the top wall of the enclosure is moving horizontally with a constant velocity U0 . The top and bottom walls are considered to be thermally insulated. The right wall of the enclosure is kept at a lower temperature Tc in comparison with the left wall, which is taken to be partially heated. The positions of the heater are indicated by three different cases (Case 1, Case 2 and Case 3) where the heater of temperature Th is placed on the bottom, centre and top position of the wall, respectively. The length of the heater is kept fixed at one third of the length of the wall. The remaining portion of the left wall is maintained to be thermally insulated. A uniform magnetic field B0 is applied parallel to the x-axis. The nanofluid is assumed to be sufficiently dilute. Based on our model, the flow is considered to be laminar, steady and the fluid is Newtonian. The existence of relative velocity between the base fluid and the nanoparticles due to the presence of Brownian diffusion and thermophoresis is considered in our two-phase model. We assume that the density variation obeys Boussinesq approximation whereas the radiation and chemical reaction are neglected.

Fig. 1 Schematic diagram of the physical system

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The governing equations of continuity, momentum, energy and the equation of continuity for the nanoparticles in dimensional form can be expressed as ∇ · u∗ = 0 ρn f

 ∂u∗ ∂t ∗

(1)

   + u∗ · ∇u∗ = −∇ p ∗ + ∇ · μn f ∇ · u∗ + (ρβ)n f (T − Tc )g + σn f (u∗ × B∗ ) × B∗

(ρC p )n f

 ∂T ∂t ∗ ρp

∗







(2)

+ u · ∇T = ∇ · kn f ∇T − C p J p · ∇T

(3)

  ∂ϕ ∗ ∗ ∗ u = −∇ · J p + ρ · ∇ϕ p ∂t ∗

(4)

where u∗ , T , p ∗ , B ∗ and J p represent the dimensional velocity, temperature, pressure field, magnetic field and nanoparticle mass flux, respectively. The flux of nanoparticle is governed by J p = J P,B + J P,T , where J P,B is the nanoparticle mass flux due to Brownian diffusion, calculated based on EinsteinStokes’ model as J P,B = −ρ p D B ∇ϕ, D B being the Brownian diffusion coeffiKBT , K B being the Boltzmann constant, and J P,T is the cient given by D B = 3πμ f dp mass flux due to thermophoretic effect, which is obtained by using McNab–Meisen approximation for the thermophoretic velocity of particles dispersed in a liquid. Thus, J P,T = −ρ p DT ∇T where DT is the thermal diffusion coefficient given by T Kf μ f ϕb DT = 0.26 K p +2K , K p , K f being the thermal conductivity of the fluid and ρf f nanoparticle, respectively. The non-dimensional form of the governing equations based on the nonhomogeneous model can be expressed as ∇ ·u=0 ρn f ρf

(ρC p )n f (ρC p ) f





      ρn f μn f ∂u 1 ∇p + ∇ ·u + u · ∇u = − ∇· ∂t ρf Re μf     (ρβ)n f σn f θ · eg + + Ri (u × B) × B ρfβf σf

∂θ + u · ∇θ ∂t



(5)

(6)

     kn f 1 1 = ∇· ∇θ · ∇ϕ ∇θ + Re · Pr kf Re · Pr · Le   1 ∇θ · ∇θ (7) + Re · Pr · Le · N BT

∂ϕ 1 1 + u · ∇ϕ = ∇2ϕ + ∇2θ ∂t Re · Sc Re · Sc · N BT

(8)

Effect of Heating Location on Mixed Convection …

79

A boundary conditions for the computational domain in non-dimensional form are u = 1, v = 0, u = 0, v = 0,

∂θ ∂y ∂θ ∂y

= 0, = 0,

u = v = 0, θ = 0, u=v=0

∂ϕ ∂x

∂ϕ ∂y ∂ϕ ∂y

= 0 at y = 1 and 0 ≤ x ≤ 1 = 0 at y = 0 and 0 ≤ x ≤ 1

= − N1BT

∂θ ∂x

at x = 0 and 0 ≤ y ≤ 1.

Case1: θ = 1, ∂ϕ = − N1BT ∂∂θx at x = 0 and 0 ≤ y ≤ ε + 1/6, ∂x ∂ϕ ∂θ = 0, ∂ x = 0 at x = 0 and ε + 1/6 ≤ y ≤ l. ∂x Case2: ∂∂θx = 0, ∂ϕ = 0 at x = 0 and 0 ≤ y ≤ ε − 1/6 ∂x ∂ϕ 1 ∂θ θ = 1, ∂ x = − N BT ∂ x at x = 0 and ε − 1/6 ≤ y ≤ ε + 1/6 ∂θ = 0, ∂ϕ = 0 at x = 0 and ε + 1/6 ≤ y ≤ 1 ∂x ∂x Case3: ∂∂θx = 0, ∂ϕ = 0 at x = 0 and 0 ≤ y ≤ ε − 1/6 ∂x ∂ϕ 1 ∂θ θ = 1, ∂ x = − N BT ∂ x at x = 0 and ε − 1/6 ≤ y ≤ 1 ε = 1/6, 3/6, 5/6 being the dimensionless distance of centre of the heat source from the bottom of the cavity. Heat source of non-dimensional length L H = 1/3 is placed on different position of the left wall. Here the dimensionless variables are defined by x = x ∗ /H , y = y ∗ /H , t = ∗ t U0 /H , h = h ∗ /H , θ = (T − Tc )/(Th − Tc ), u = u ∗ /U0 , v = v ∗ /U0 , p = p ∗ / ρn f U02 . The volume fraction is scaled by the bulk nanoparticle volume fraction ϕb , i.e. ϕ = ϕ ∗ /ϕb . B∗ is the non-dimensional vector for the magnetic field. The dimenρ U H ν sionless parameters are Reynolds number Re = f μ f0 , Prandtl number Pr = α ff , β g(T −T )H 3

f h c Gr Richardson number Ri = Re . The Schmidt 2 and Grashof number Gr = νf 2 μf number Sc = ρ f D B represents the ratio of momentum diffusivity and Brownian dif-

kf measures (ρC p ) f D B ϕb ϕb D B N BT = DT ΔT denotes

fusivity whereas the Lewis number Le =

the ratio of thermal

diffusivity to the Brownian diffusivity. the relative performance between the Brownian diffusion to that of the thermophoretic diffusion. The thermophysical properties of the nanofluid such as the effective density (ρn f ), heat capacity ((C p )n f ), and the thermal expansion coefficient (βn f ) are obtained through the relations (ρ)n f = (1 − ϕ)ρ f + ϕρ p , (ρc p )n f = (1 − ϕ)(ρc p ) f + φ(ρc p ) p , βn f = (1 − ϕ)β f + ϕβ p , The thermal diffusivity of nanofluid is k expressed as αn f = (ρCnpf)n f where the subscripts f , p, n f refers to the fluid, particle and the nanofluid, respectively. Depending upon the thermophysical properties of the base fluid and nanoparticles, in this study the Corcione model is used to determine the thermal conductivity and dynamic viscosity of the nanofluid.  10  0.03 kn f Kp T 0.4 0.66 = 1 + 4.4Reb Pr ϕ 0.66 kf Tfr Kf

(9)

where T is the temperature, Pr is the Prandtl number, T f r is the base fluid freezing point, Reb is the Reynolds number corresponding to the nanoparticle

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as Reb =

ρ f u B dp μf

where u B = −23

2K B T πμ f d p 2

is the Brownian velocity of nanoparticles.

K B = 1.38065 × 10 is the Boltzmann constant, d p and d f are the diameter of the nanoparticles and the base fluid, respectively. The electrical conductivity ratio is represented by   σp 3 − 1 ϕ σf σn f    . (10) =1+  σp σp σf + 2 − − 1 ϕ σf σf The thermophysical properties for water and Al2 O3 (at T = 310 K), used in the present study, is considered same as provided by Garoosi et al. [13].

2.1 Nusselt Number and Entropy Generation The expression   for the local Nusselt number in non-dimensional form is given by k ∂θ , n being the unit outward normal to the left wall. The averaged N u = − knff ∂n value of Nusselt number (N u av ) can be obtained by integrating the local N u along the heater. Entropy generation in non-dimensional form is defined as

Sgen =

μn f  ∂u 2 kn f  ∂θ 2 ∂v 2 ∂θ ∂v ∂u + ) ( ) + ( )2 + χ 2( ) + 2( )2 + ( k f ∂x ∂y μf ∂y ∂x ∂x ∂y

= Sh + S f Sh and S f are, respectively, the entropy generation due to heat transfer irreversibility and fluid friction irreversibility. χ is the irreversibility factor defined by χ = (μ f T0 U0 2 )/k f (ΔT ) where T0 = (Th + Tc )/2 is the reference temperature. The average entropy generation Sav is defined as Sav =

1 A

Sgen d xd y A

3 Numerical Methods The governing non-dimensional equations are solved numerically using finite volume method in which the non-linear equations are integrated over each cell of the control volume. A first-order implicit scheme is incorporated to discretize the time derivatives. A pressure correction based iterative algorithm SIMPLE (Semi-Implicit

Effect of Heating Location on Mixed Convection …

81

Method for Pressure Linked Equations) [14] is used for computation. In the staggered grid arrangement, each of the velocity components (u, v) are stored at the midpoint of the sides on which they are normal and the pressure as well as the temperature and nanoparticle volume fraction are stored at the centre of each cell. The algebraic equation is solved iteratively by a cyclic series of guess-and-correct operations using block elimination algorithm. The convergence criterion of this iteration is expressed − ϕikj | ≤ 10−5 where the symbol ϕ denotes for the non-dimensional as max | ϕik+1 j ij

velocities and temperature and the subscripts i, j indicates the index of cell, superscript k is the index. of the computed iteration.

4 Grid Independence Test and Validation of Code To evaluate the proper grid size, the grid independence test for the present model is performed. For this, we compute the local Nusselt number for grid size 150 × 150, 180 × 180, 210 × 210 at Re = 100, φ = 0.05, Ri = 1, and H a = 0 along the centred heated portion of the left wall (Fig. 2a). It is evident from the result that further improvement from 180 × 180 does not make any noticeable change in the result (Fig. 2a). The grid size 180 × 180 is considered to be optimal as further improvement does not make any effective changes. To validate the accuracy of the present code, the result for the N u av is compared with the numerically obtained results by Sivasankaran et al. [5] in a lid-driven square cavity flow due to the pure fluid (φ = 0) and Re = 100, with the heater at the bottom of the left wall. Analysis indicates a good agreement between the existing result and the present one. Figure 2c presents a comparative study of N u av for the mixed convection in presence of magnetic field at Re = 100

(a) 13

5.5

150X150 180X180 210X210

5 4.5

6 Present Work Sivakumar et al.

10

3

4

9 8

Nux

Nuav

3.5

ε = 5/6

7

2.5

Present results Oztop et al.

5

11

4

Nu

(c)

(b)

12

Ha = 0

3 2

Ha = 30

6

2

1

5

1.5 0.35

0.4

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0.5

X

0.55

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4 -2 10

Ha = 50

0

10

-1

10

0

Ri

10

1

10

2

0.5

0.6

0.7

0.8

0.9

1

X

Fig. 2 Grid independency test and comparison with the published results: a Local Nusselt number N u along the heated portion of the wall at different grid size when ϕb = 0.05, Re = 100, Ri = 10 and the heat source on the centre of the left wall; b Comparison of the present result for N u av with the results of Sivakumar et al. [5] for ϕb = 0 (clear fluid) when, Re = 100 and ε = 5/6; c Comparison of the local Nusselt number along the bottom wall with the numerical results by Oztop et al. [15] for MHD mixed convection in a lid-driven square cavity due to the presence of corner heaters with Gr = 105 , Re = 100 and X H = Y H = 0.5 for homogeneous model

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and Ri = 1, in a square cavity with discrete heat sources at right and bottom wall with the result of Oztop et al. [15]. The figure shows our results are in good agreement with the existing results.

5 Results and Discussion The fluid flow and heat transfer in presence of Magnetic field in a square enclosure filled with Al2 O3 –water nanofluid incorporating the effect of Brownian diffusion and thermophoresis has been investigated numerically. The mixed convective flow is studied considering the Reynolds number fixed at Re = 100, nanoparticle diameter d p = 25 nm, and varying the bulk volume fraction (0 ≤ ϕb ≤ 0.05), Richardson number (0.1 ≤ Ri ≤ 5) and Hartman number (0 ≤ H a ≤ 60) and the parameter ε for the position of the heat source. Subsequently, the average Nusselt number and average entropy generation are evaluated.

5.1 Study of Flow and Thermal Field The impact of several locations of the heat sources (bottom, middle and top) and the effect of magnetic field on the streamline is displayed in Fig. 3a–c where the mixed convection takes place, i.e. Ri = 1, ϕb = 0.05, Re = 100 and d p = 25 nm with and without the presence of the magnetic field. Since the left wall is uniformly heated, the fluid will move in the upward direction and as the right wall is kept cold the fluid will move in downward direction. Since the top wall is insulated and moving, due to the combined contribution of shear and buoyancy, the fluid flows in a clockwise flow pattern and the primary vortex covered almost all part of the cavity. In the mixed convection dominated case. With the increment of Richardson number, the strength of flow become stronger and secondary vortex appears. For the sake of brevity, we here

(a)

(b)

(c)

Fig. 3 Streamline patterns for different positions (bottom (a), middle (b) and top (c)) of the heat source at Richardson number Ri = 1, ϕb = 0.05 and Reynolds number Re = 100. Dashed line represents H a = 0, and Solid line represents H a = 60

Effect of Heating Location on Mixed Convection …

(a)

(b)

83

(c)

Fig. 4 Isotherm for different position (bottom ((a), middle (b) and top (c)) of the heat source at Richardson number Ri = 10, ϕb = 0.05, and Reynolds number Re = 100; Dashed line represents H a = 0, and Solid line represents H a = 60

show the mixed convection dominated case. The convection becomes stronger with the movement of the heat sources from bottom to top. Otherwise, no considerable change in the stream function is observed due to the movement of the heater. With the inclusion magnetic field, the change in the flow field is observed. The stream function is reduced due to the fact that the thickness of the thermal boundary layer increases and the fluid is confined in the lower part of the enclosure. As the heat source is shifted from bottom to top, the top cell gradually becomes smaller and the bottom cell is enlarged. Figure 4a–c represents the variation of thermal field for Al2 O3 –water nanofluid for three different positions of the heater. For the figure, it is seen that the isotherms are clustered in the vicinity of the heater, approximately parallel to each other, which indicates that the heat is transferred due to conduction. In the core region of the enclosure, the isotherms are distorted, implying a higher thermal mixing due to convection. The convection becomes higher with the increment of Ri, implying a well mixing with the hot fluid. As the value of ε increases, i.e. the heater moves from bottom to top, the convection increases. Figure 4a–c shows that the magnetic field has an unfavourable influence on the thermal field. As the magnetic field is imposed, the heat is transferred due to conduction and the convective heat transfer becomes slower. The isotherms become parallel to the lid-driven wall and with the increment of ε, this conduction becomes higher.

5.2 Effect of Position of the Heat Source on N u av and Sav The variation of heat transfer and entropy generation with the change of Richardson number is represented by Fig. 5a–c for different values of bulk volume fraction for the nonhomogeneous mode. It is seen from results of N u av that the rate of heat transfer enhances with the increment of Ri. This is because, with the increment of Ri, the thermal gradient increases, resulting a higher thermophoretic diffusion. The effect of both Brownian diffusion and thermophoresis becomes stronger and the thickness of the thermal boundary layer reduces, which causes a higher heat transfer

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rate. For a higher value of Ri, the buoyancy force dominates the shear force and hence the heat transfer is augmented. The homogeneous model obtains the similar trend of variation with the nonhomogeneous model but the nonhomogeneous model predicts higher heat transfer than the homogeneous model due to the presence of Brownian diffusion and thermophoresis. The trend is almost similar for all values of ϕb . It is seen from the first row of Fig. 5a–c that the value of N u av increases with the increment of ε, indicating that the top position of the left wall predicts the optimum heat transfer for any value of ϕb . The results for Sav , presented in the second row of Fig. 5a–c shows that the variation of the average entropy generation with Richardson number is similar to that of the corresponding variation of the average Nusselt number. It is evident from the figure that the average entropy generation increases with the increment of Ri as the increment in Ri enhances the rate of heat transfer, indicating a higher heat transfer irreversibility and higher entropy generation. The value of Sav is augmented with the increment of ε showing the optimum position for higher entropy generation.

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5.3 Effect of Nanoparticle Volume Fraction on N u av and Sav Figure 6a–c shows the variation of average Nusselt number N u av and entropy generation Sav as a function of bulk volume fraction at different values of Hartman number. Results show that the values of N u av are always increasing with the increment of nanoparticle volume fraction. Inclusion of nanoparticle enhances the effective thermal conductivity of the nanofluid and modifies the thermal boundary layer. As a result, the heat transfer rate increases both in the presence and absence (H a = 0) of the magnetic field. It is also seen that the average entropy generation increases with the increment of ϕb as the inclusion of nanoparticle enhances the fluid friction irreversibility. Due to the strong impact of S f on Sav , the entropy generation rate is enhanced with the addition of nanoparticle, for all values of Hartman number.

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5.4 Effect of Magnetic Field on N u av and Sav The Impact of magnetic field on the heat transfer as well as the entropy generation is presented in Fig. 7a–c. As the Hartman number increases, the decrement in heat transfer rate occurs. This is due to the fact that magnetic field suppresses the convective flows and retards the velocity. So the heat transfer rate reduces. As the magnetic field is imposed, the thickness of the thermal boundary layer increases. As a result, the temperature gradient reduces, having an unfavourable influence in the heat transfer. The fluid is approached to be carried away due to conduction, as a result the N u av reduces as the Hartman number is increased for any position of the heat source. Entropy generation also reduces with the increment of H a. This is due to the fact that with the increment of H a, the fluid movement decreases. As the increment in Lorentz force attenuates the temperature gradient, the thermal performance reduces. Due to the reduction of the heat transfer irreversibility, the total entropy generation reduces with the increment of magnetic force.

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6 Conclusion The study investigates the mixed convection of Al2 O3 –water nanofluid using Buingirno’s two-phase model in presence of Magnetic field. This analysis considers the two-phase model in which a relative velocity between the base fluid and nanoparticles is considered due to the presence of Brownian diffusion and thermophoresis. The effect of different parameters such as Richardson number (0.1 ≤ Ri ≤ 5), bulk volume fraction (0% ≤ ϕb ≤ 5%), Hartman number (0 ≤ H a ≤ 60), and position of the heater (1/6 ≤ ε ≤ 5/6) on fluid flow, heat transfer and entropy generation are investigated. Our study demonstrates that the nonhomogeneous model, which considers a relative velocity between the base fluid and the nanoparticles, predicts higher heat transfer than the homogeneous model. The rate of heat transfer enhances with the increment of Richardson number due to the higher temperature gradient. As a result, the entropy generation increases. Inclusion of nanoparticles increases the heat transfer as the thermal conductivity of nanofluid is augmented due to the increment of bulk volume fraction. The entropy generation is also enhanced due to the fluid friction irreversibility. The magnetic field has an unfavourable effect on the heat transfer performance. The heat transfer and entropy generation reduces with the increment of Hartman number due to the fact that the imposed magnetic field retarded the convection and suppressed the velocity field. The position of the heater has an effective impact on the flow field and augmentation of heat transfer. The favourable position to achieve the maximum heat transfer is obtained in our study.

References 1. Tiwari RK, Das MK (2007) Heat transfer augmentation in a two-sided lid-driven differentially heated square cavity utilizing nanofluids. Int J Heat Mass Transf 50(9–10):2002–2018. https:// doi.org/10.1016/j.ijheatmasstransfer.2006.09.034 2. Xuan Y, Li Q (2003) Investigation on convective heat transfer and flow features of nanofluids. J Heat Transf 125(1):151–155. https://doi.org/10.1115/1.1532008 3. Basak T, Chamkha AJ (2012) Heatline analysis on natural convection for nanofluids confined within square cavities with various thermal boundary conditions. Int J Heat Mass Transf 55(21– 22):5526–5543. https://doi.org/10.1016/j.ijheatmasstransfer.2012.05.025 4. Muftuoglu A, Bilgen E (2008) Conjugate heat transfer in open cavities with a discrete heater at its optimized position. Int J Heat Mass Transf 51(3–4):779–788. https://doi.org/10.1016/j. ijheatmasstransfer.2007.04.017 5. Sivasankaran S, Sivakumar V, Hussein AK (2013) Numerical study on mixed convection in an inclined lid-driven cavity with discrete heating. Int Commun Heat Mass Transf 46:112–125. https://doi.org/10.1016/j.icheatmasstransfer.2013.05.022 6. Öztop HF, Estellé P, Yan WM, Al-Salem K, Orfi J, Mahian O (2015) A brief review of natural convection in enclosures under localized heating with and without nanofluids. Int Commun Heat Mass Transf 60:37–44. https://doi.org/10.1016/j.icheatmasstransfer.2014.11.001 7. Chamkha AJ (2002) Hydromagnetic combined convection flow in a vertical lid-driven cavity with internal heat generation or absorption. Numer. Heat Transf.: Part A: Appl. 41(5):529–546. https://doi.org/10.1080/104077802753570356

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8. Ghasemi B, Aminossadati SM, Raisi A (2011) Magnetic field effect on natural convection in a nanofluid-filled square enclosure. Int J Therm Sci 50(9):1748–1756. https://doi.org/10.1080/ 104077802753570356 9. Hussain S, Öztop HF, Mehmood K, Abu-Hamdeh N (2018) Effects of inclined magnetic field on mixed convection in a nanofluid filled double lid-driven cavity with volumetric heat generation or absorption using finite element method. Chin J Phys 56(2):484–501. https://doi.org/10.1016/ j.cjph.2018.02.002 10. Buongiorno J (2006) Convective transport in nanofluids. J Heat Transf 128(3):240–250. https:// doi.org/10.1115/1.2150834 11. Motlagh SY, Soltanipour H (2017) Natural convection of Al2 O3 -water nanofluid in an inclined cavity using Buongiorno’s two-phase model. Int J Therm Sci 111:310–320. https://doi.org/10. 1016/j.ijthermalsci.2016.08.022 12. Ho CJ, Chen DS, Yan WM, Mahian O (2014) Rayleigh-Bénard convection of Al2O3/water nanofluids in a cavity considering sedimentation, thermophoresis, and Brownian motion. Int Commun Heat Mass Transf 22–26. https://doi.org/10.1016/j.icheatmasstransfer.2014.07.014 13. Garoosi F, Jahanshaloo L, Garoosi S (2015) Numerical simulation of mixed convection of the nanofluid in heat exchangers using a Buongiorno model. Powder Technol 269:296–311. https:// doi.org/10.1016/j.powtec.2014.09.009 14. Patankar S (1980) Numerical heat transfer and fluid flow. CRC Press, Boca Raton. https://doi. org/10.1201/9781482234213 15. Oztop HF, Al-Salem K, Pop I (2011) MHD mixed convection in a lid-driven cavity with corner heater. Int J Heat Mass Transf 54(15–16):3494–3504. https://doi.org/10.1016/j. ijheatmasstransfer.2011.03.036

Weighted Matrix-Based Random Data Hiding Scheme Within a Pair of Interpolated Image Debkumar Bera, Biswapati Jana, Partha Chowdhuri, and Debasis Giri

Abstract Secure, high payload, reversible data hiding scheme with good visual quality is still an important research issue in data hiding and designing such a scheme is a technically challenging problem. Tseng et al. proposed a securely weighted matrix-based data hiding scheme for the binary image which can hide only two bits secret data within a (3 × 3) pixel block; meanwhile, Fan et al. suggested an improved weighted matrix-based data hiding scheme for grayscale image which can hide 4 bits secret data within a (3 × 3) pixel block. Both of these weighted matrix-based data hiding schemes performed only one modular sum of entry-wise multiplication operation between image block and weighted matrix. To improve the capacity along with visual quality, we have proposed an innovative, reversible, secure, high payload data embedding scheme in a random location of a pair of color image. The randomness has been achieved by using SHA-512 for selecting the image and PRNG is used to select the specific block of the desired image. The weighted matrix helps to embed the secret information by performing entry-wise multiplication with the image block. The embedding capacity has been increased by performing 16 repeated entry-wise multiplication operations and security has been enhanced by modification of the weighted matrix for each block. Finally, the experimental results are compared with existing state-of-the-art methods and observed that our scheme is superior in terms of capacity, quality, and security. D. Bera · B. Jana (B) · P. Chowdhuri Department of Computer Science, Vidyasagar University, West Midnapore 721102, India e-mail: [email protected] D. Bera e-mail: [email protected] P. Chowdhuri e-mail: [email protected] D. Giri Department of Information Technology, Maulana Abul Kalam Azad University of Technology, Haringhata, West Bengal, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 D. Giri et al. (eds.), Proceedings of the Sixth International Conference on Mathematics and Computing, Advances in Intelligent Systems and Computing 1262, https://doi.org/10.1007/978-981-15-8061-1_8

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Keywords Weighted matrix · Data hiding · Image interpolation · Cover image · Stego image

1 Introduction Data hiding is the technique of hiding secret information within an image, audio, video, text, etc. Various kinds of multimedia objects can be used as cover media to hide the existence of secret information from an eavesdropper, but digital images are the most commonly used media because of ubiquitous and, moreover, images speak more than words. For the past few decades, digital images are being used as cover media in data hiding applications, due to the higher degree of distortion tolerance with a larger hiding capability. In 2001, the F5 algorithm was proposed by Westfeld [1] which is resistant to visual and statistical attack. The efficiency of embedding was increased due to the use of matrix encoding. Fan et al. [2] developed an algorithm to increase the embedding capacity and efficiency of Westfeld [1] scheme. It was achieved by modifying the original hash function and adding an n-layer extension to the original algorithm. Tseng et al. [3] developed a data hiding scheme based on the weighted matrix for binary images. In this scheme, a (3 × 3) pixel block is able to hide 2 bits of information. Later, Fan et al. [4] proposed an improved data hiding scheme based on the weighted matrix for grayscale images. The modular sum of entry wise multiplication of (3 × 3) pixel block with a weighted matrix is used to hide the information. The scheme managed to hide 4 bits of secret information in a (3 × 3) pixel block in a single embedding operation with the help of the weighted matrix. Chang et al. [5] proposed a dual-image data hiding scheme where secret bits are converted to digits in base-5 number system and a pair of digits are embedded into a single pixel pair by distributing them into dual stego images. Lee and Huang [6] proposed a dual-image data hiding scheme that also uses digits in base-5 number system to increase the capacity of embedding. They managed to keep the excellent visual quality of the stego images by restricting the value of the changed pixel at most plus or minus one. Lu et al. [7] used LSB matching-based method for embedding secret data into dual images by employing seven different rules to camouflage the modification of the pixels. Jafar et al. [8] developed a dual-image data hiding scheme that is performed in three phases. Four rules are used to embed secret data in the first phase. Other two phases are used to extract secret data from the dual stego images. The Centre Folding Strategy (CFS)-based data hiding scheme for dual images was first proposed by Lu et al. [9]. Yao et al. [10] established an improved version of Lu et al.’s scheme [9] by using a strategy that selects the coordinates of shiftable pixels with minimum distortion.

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2 Motivation The key factors of secured hidden data communications are high security, high embedding capacity, and good imperceptibility. Each of these requirements occupies each corner of a triangle in a data hiding system and there is always a tradeoff between these contradictory requirements. Imperceptibility: The first and foremost requirement of any data hiding algorithm is the imperceptibility. The embedded secret data within the cover image should not cause any degradation in visual quality. The secret message should remain invisible, it should not be detectable to the human eyes and there should not be any visual distortion within stego image so that it remains unsusceptible and unsafe. Higher the stego image quality, more invisible the hidden message which can be measured through PSNR. A higher PSNR value means a lower degree of distortion. Payload: The amount of inserted information within stego image is considered as payload. The payload should be higher as much as possible with an acceptable resultant stego quality. It is measured by some absolute value or relative measurement (bits per pixel) or data embedding rate. The importance of data hiding schemes is based on the tradeoff between payload or data hiding capacity and stego image quality. So, a scheme does have its contribution to the research field if it increases the payload while maintaining an acceptable quality of stego image or improves the quality of image with the same hiding capacity or better. Robustness: Robustness is the level of difficulty required by an eavesdropper to decide whether an image contains a hidden message(s) or not. An effective data hiding scheme would be the one where an image can sustain under steganographic attack that may prove inconclusive. Statistical analysis is the practice of detecting hidden information through applying statistical tests on the stego image. Design of any security scheme is not enough, but their security guarantee is of paramount impermanence. If the detection of secret information within a media is made by an eavesdropper then the data hiding scheme will fail.

3 Proposed Scheme 3.1 Data Embedding Consider two original images I1 and I2 of size (N × N ). increase the size in double i.e (2N − 1 × 2N − 1) using interpolation generate stego images S1 and S2 . Partitioned original images into (3 × 3) pixel blocks and stego images (5 × 5) pixel blocks.

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⎧ Imin = min C(i, j), C(i + 2, j), C(i, j + 2), C(i + 2, j + 2) ⎪ ⎪ ⎪ ⎪ Imax = max C(i, j), C(i + 2, j), C(i, j + 2), C(i + 2, j + 2) ⎪ ⎪ ⎪ ⎪ AD = 3 X Imin4 +Imax ⎨ C(i, j) = I (i, j) ⎪ j+2))/2 ⎪ C(i, j + 1) = AD+(C(i, j)+C(i, ⎪ 2 ⎪ ⎪ AD+(C(i, j)+C(i+2, j))/2 ⎪ ⎪ C(i + 1, j) = ⎪ 2 ⎩ (C(i, j)+C(i+1, j)+C(i, j+1) C(i + 1, j + 1) = 3

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where i = 2m, j = 2n, m, n = 0, 1, 2, . . . k. Then CRS calculate the difference value d1 , d2 and d3 between pixels C(i, j + 1), C(i + 1, j), and C(i + 1, j + 1), respectively. Consider the weighted matrix W of size (3 × 3). To select the one between two pictures to retrieve original data blocks (Fi ), SHA-512 will be used and the value of the (Fi ) will be determined by PRNG(K). After selecting the original data block, the operation SUM(Fi ⊗ Wi ) will be performed with the weighted matrix W . The embedding position of data to be calculated by subtraction of the modular sum (V al) from the secret data unit (dec), i.e., pos = (dec − V al). The desired pixel value of interpolated location in the alternate interpolated stego image will be increased/decreased by one unit depending on the

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Fig. 1 Schematic diagram of embedding process

Interpolated Stego Image S 2 (512 x 512) Divide into 5 x 5 Block

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sign of the value of pos. The embedding process will be repeated until all the interpolated location of a (5 × 5) stego image block increased/decreased for different secret data, i.e., 16. After completion of a block, the weighted matrix will be updated using Wi+1 = (Wi × n − 1)%9, where n and 9 are coprime. The Fig. 1 describes the schematic diagram of embedding process and Fig. 2 also describes the numerical example of the embedding process. Algorithm-1: Data embedding Input: Two original Images of size [N × N ] , Weighted Matrix of size [3 × 3] , Secret Data D = {d1 , d2 , d3 , . . . , where di = 4 bits each} , Shared secret key κ , Shared secret value K , i = 1 and l = length[D] ; Output: Two embedded Stego Images of size [2N − 1 × 2N − 1] ; Step-1: Divide two original images i.e I1 and I2 into blocks. Create interpolated stego images S1 and S2 of size [2N − 1 × 2N − 1] from I1 and I2 and divide them into 5 × 5 blocks. No of blocks, m = [N × N ]/9 ; Step-2: Generate SHA-512 secret bits using shared secret key κ . Step-3: Check the bit in SHA-512 stream if the bit is 0, then select the image 1 (I1 ) otherwise select the image 2 (I2 ). Step-4: Generate secret random numbers using PRNG(K) with shared secret value K. Step-5: Check the random number in PRNG(K), then select K i th 3 x 3 block from the original image which is select by Step 3. m − −; Step-6: Fetch the 3 × 3 block(Fi ) selected by Step 5. Fetch 3 × 3 weighted matrix (Wi ). Then calculate SUM(Fi ⊗ Wi ). count = 1, j = 1; Step-7: Take an r bit secret data (Di ) from D. Convert it into BCD, dec = BC D(Di ). Calculate V al = SU M%16 (2r ) and pos = (dec − V al). Step-8: If ( pos > 0) then If ( pos > 8) then pos = (16 − pos) and d = −1 otherwise d = 1. If ( pos < 0) then If ( pos 0 and m > 0) then Update weighted matrix (Wi ) with Wi+1 = (Wi × n − 1)%9, where gcd(n, 9) = 1 i + +; and Go to Step 3. Step-11: End.

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Consider Secret Data for image 1 D = 1000 1010 0011 1011 0111 0100 1101 1111 0001 0101 1001 0110 1100 1110 0000 0010

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Four bit data D i =1000; dec =8; pos =(dec-Val) =7 8; pos =(16-pos) =7; d = -1; pos =pos*d = -7; then decrease at e2 in alternate image block (in fig (b)) by -7.

Four bit data D i =1101; dec =13; pos =(dec-Val) =6 from topological perspective are observed as follows: < 1, 5 >, < 2, 5 >, < 3, 7 >, < 4, 7 >, < 5, 7 >, < 6, 7 >, < 7, 5 >,< 8, 7 >, < 9, 7 >,< 10, 7 >,< 11, 7 >,< 12, 7 >, 1 https://www.webmd.com/parkinsons-disease/guide/parkinsons-faq#1.

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Fig. 1 Conceptual framework of module correspondence between control and disease stage

< 13, 7 >, < 14, 7 >, < 15, 7 >,< 16, 7 >, < 17, 7 >,< 18, 7 >, < 19, 7 >, < 20, 7 >,< 21, 7 >, < 22, 7 >. We observe here that all the 22 control modules either correspond to module D5 or D7 in the disease stage.

4.2 Module Correspondence from Pathway Point of View A pathway in a biological domain represents a series of related biochemical reactions occurring in the living body. Molecules involved in promotion or inhibition of any activity in the living body need to be studied carefully. The number of common pathways found among the members of modules in different stages can be an interesting feature. We, therefore, use the PANTHER tool [16] to find the pathways in each module in the control as well as disease stages. Table 1 represents the number of common pathways as given by PANTHER among the modules in both the stages. In Table 1, we observe that disease modules numbered D7 , D4 , and D5 are the top three modules which showed the maximum overlap with all the control modules in terms of pathway. In this case, D7 emerges as the winner corresponding module for each control module. We further analyze D7 for better understanding the progression of the disease.

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Table 1 Number of common pathways for all the 22 control modules w.r.t. the 7 disease modules Control Number of common pathways modules Diseased modules D1 D2 D3 D4 D5 D6 D7 C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11 C12 C13 C14 C15 C16 C17 C18 C19 C20 C21 C22

22 22 22 22 22 22 22 22 20 22 20 20 20 22 20 22 20 20 22 22 20 22

13 13 13 13 13 13 13 13 12 13 12 12 12 13 12 13 12 12 13 13 12 13

8 9 9 7 9 9 9 8 8 9 8 8 8 8 8 8 8 8 9 8 8 9

27 28 28 26 28 28 28 26 26 28 26 26 26 27 26 27 26 26 28 27 26 28

28 29 29 27 29 29 29 27 28 29 28 28 28 28 28 28 28 28 29 28 28 29

24 25 25 24 25 25 25 23 24 25 24 24 24 24 24 24 24b 24 25 24 24 25

30 31 31 29 31 31 31 29 30 31 30 30 30 30 30 30 30 30 31 30 30 31

5 Identification of Interesting Biomarkers We analyzed initially all the 22 modules for finding interesting features. However, almost in all the modules, there were overlaps among the elements in the range (48–52)%. Therefore, in order to get a subset of modules for our analysis, we took the help of GeneCard [21]. It is a repository which stores the list of causal genes for around 5000 diseases. From this repository, we got 59 genes associated with Parkinson’s Disease. We used this list of causal genes to identify modules with a high number of disease genes. We observed the inclusion of number of causal genes in each control module, i.e., < module no., no o f causalgenes > as follows: < 1, 5 >, < 2, 6 >,< 3, 6 >,< 4, 2 >,< 5, 6 >, < 6, 6 >,< 7, 6 >,< 8, 4 >, < 9, 6 >, < 10, 6 >, < 11, 6 >, < 12, 6 >, < 13, 6 >, < 14, 4 >, < 15, 6 >, < 16, 3 >, < 17, 6 >, < 18, 6 >, < 19, 5 > < 20, 4 >, < 21, 6 >, < 22, 5 >.

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We saw that the maximum number of causal genes found among the modules in the control stage is 6 and {C2 , C3 , C5 , C6 , C7 , C9 , C10 , C11 , C12 , C13 , C15 , C17 , C18 , C21 } module set showed the presence of maximum number of causal genes. Hence, further analysis of these modules was carried out. For each of these 14 modules, we performed an extensive study of the causal genes and their interacting partners among the module members. The interacting partners of the causal genes were found using the STRING tool [5], which obtains the partner genes using informations from coexpression data, other experimental data and text mining. In addition to the causal genes listed in GeneCard, we analyzed each interacting partner with the causal gene in terms of pathways. Table 2 reports the causal genes along with their interacting partners and the number of pathways they share in common with the causal gene for control modules C2 and C3 . The details of the remaining 12 modules are given in Supplementary file which is available upon request. Apart from these information, these tables also higlight the genes which can be possible suspected genes for the disease. We carried out an analysis of the roles of suspected genes which were not yet known to be sharing any pathway with the six causal genes found in the modules from GeneCard. From literature sources, we gathered some information on the association of such genes with the mechanisms involved in the progression of the disease. We now discuss the role of these genes here. (a) ADCY2: Dopamine neurons are rare in the brain and are associated with many day-to-day activities such as movement and learning [6]. The ADCY2 gene, which is an isoform of adenyl cyclase is known to be expressed in the brain. Sources such as [12] have reported mice expressing certain kinds of motor dysfunctionality that affect the striatal dopamine signaling. (b) CNR1: About 40% patients suffering from Parkinson’s Disease show a tendency to undergo depression. Experiments have established the role of cannabinoid receptor gene (CNR1) to be associated with depression brought about by the disruption in the monoamine transmission [1]. (c) GNB5: Ample evidence is available to describe the role of GNB5 in causing attention deficit hyperactivity disorder, which is one of the symptoms of Parkinson’s Disease [22]. GB5, a β subunit of the GTP-binding proteins is present in the Central Nervous System. It is known to form complexes which control the transmission activity of neurons, thus affecting the behavioral consequences. (d) HTR2A: Impulsive behavior is one of the consequences seen in a Parkinson’s patient undergoing treatment [26]. Serotonin pathways associated with serotonin 2 A receptor gene (HTR2A) and dopamine are known to be causing such behavioral changes [14]. (e) GRIN2A, GRIN1: People who are heavily addicted to coffee have shown chances of developing Parkinson’s Disease via mutation in the glutamate receptor gene (GRIN2A) [7]. (f) STX1A: The first symptoms of Parkinson’s Disease is attributed to the loss of dopaminergic neurons [13]. A post-mortem experiment conducted in the brain tissue samples of Parkinson’s Disease patients shows the association of STX1A with

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Table 2 Causal genes along with their interacting partners among each control module Module Causal gene Interacting partners (No. of Suspecting genes No. common pathways with the causal gene) DRD2

2

SLC6A3

TH DDC

SLC18A2

GCH1

DRD2

SLC6A3

3

TH

DDC

SLC18A2

GCH1

ADCY2 (3), CNR1 (1), GNB5 (2), HTR2A (2), ACTL6B (0), SYT1 (0), SLC18A2 (4), SLC6A3 (4), GRIN2A (6) DRD2 (4), TH (5), GRIN1 ADCY2,CNR1,GNB5, (3), DDC (4), PTK2B (0), HTR2A, ACTL6B, SYT1, GCH1 (0) GRIN2A, GRIN1, PTK2B, GCH1, CACNG3, SLC17A7, MOXD1, STX1A, NRXN1 HTR2A (0), SLC6A3 (5), GRIN1 (3), DDC (7) CACNG3 (0), SLC17A7 (0), GCH1 (0), SLC6A3 (4), GRIN2A (4), SLC17A6 (0), MOXD1 (0) SLC6A3 (5), DRD2 (5), STX1A (2), GCH1 (0), SLC17A7 (1), SLC17A6 (1) SYT1 (0), SLC6A3 (0), ADCY2 (0), DDC (1), SLC18A2 (0), NRXN1 (0) ADCY2(3), GNB5 (2), FOS (3), ACTL6B (0), SYT1 (0), SLC18A2 (4), SLC6A3 (4), PTK2B (0), GRIN2A (6) DDC (4), HTR2A (0), DRD2 (4), GRIN2A (3), GRIN1 (1) DDC (7), DDN (0), GRIN1 ADCY2,CNR1,GNB5, (3), SLC17A6 (0), HTR2A HTR2A, ACTL6B, SYT1, (0), FOS (2), SLC17A7 (0) GRIN2A, GRIN1, PTK2B, GCH1, CACNG3, SLC17A7, MOXD1, STX1A, NRXN1, FOS, DDN MOXD1 (0), TH (7), SLC17A6 (0), SLC17A7 (0) SLC6A3 (5), STX1A (2), HTR2A (1), DRD2 (5), SLC17A7 (1), SLC17A6 (1), DDC (5) ADCY2 (0), SYT1 (0), SLC6A3 (0), NRXN1 (0)

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neurotransmitters [18], and hence can be said to be indirectly associated with the disease. (g) SLC17A7, SLC17A6: Glutamate is one of the most important neurotransmitters associated with the brain’s activity. Any mutation in the activity of two proteinsVGLUT1 (SLC17A7) and VGLUT2 (SLC17A6) affects the expression of glutaminergic neurons, which can cause an imbalance to the brain’s functioning leading to depression and ultimately resulting in Parkinson’s Disease [27]. (h) SYT1: Mutations in the Parkin gene may be associated with juvenile Parkinson’s Disease or late Parkinson’s Disease. It monitors the expression of synaptotagmin1 (SYT1) in the brain, which is indirectly associated with synaptic vesicle release. A deficiency in the expression of the Parkin gene can cause an oxidative stress in dopaminergic neurons, thus affecting people’s brain activity, ultimately leading to Parkinson’s Disease.2 (i) FOS: Motor abnormality in Parkinson’s Disease is caused by the degeneration of nigrostriatal pathway, which is often linked to changes in the pain perception capability of the living being. To see its role, certain experiments were done on the rat model of the disease. Rats with nigrostriatal abnormalities showed varying pain perceptions and hyperalgesic responses when they were injected with formalin drug. This kind of response to the injection leads to reduced expression levels of FOS in the hypothalamus, which is directly linked with the sensory stimulus of pain in the brain [25]. (j) ATP1A3: Changes in the ATP1A3 gene are associated with both dystonia Parkinsonism and hemiplegia of childhood [9].

6 Hub Gene Analysis in Modules Based on Centrality In biological networks such as gene-gene or protein networks, the removal of a node may lead to functional changes besides structural changes. Hence, identification of such nodes is important. These nodes are often referred to as hub nodes or essential nodes. Earlier work [28] suggests that a node with a high degree, i.e., more number of inter-connections with other nodes tends to act as one of the central players in the network and removal of such a node would tend to cause structural deformities in the network. However, this measure does not consider the global structure of the network when deciding the significance of each node. To decide upon the essentiality of nodes in the network, a centrality measure can be used to rank nodes based on certain physical characteristics of the network. We used centrality measures [23], viz., betweenness, eigenvector, page rank, closeness, and radiality measure to discover the role of nodes in a human protein-protein interaction network and finally to pick up the best centrality measure(s) for our purpose. Experimental results show that radiality and pagerank measures are more most suitable while analyzing the importance of each node in such a network. We, therefore, use these two measures to determine the 2 grantome.com/grant/NIH/K01-NS047548-01AI.

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essential gene (hub gene) among the Parkinson’s disease gene network. Tables 3 and 4 give the hub genes for each module using the radiality and page rank measures in both the stages. The interacting partners of the hub genes along with their association types are also given. Association type can be either direct or indirect. In direct association, a hub gene is found to be at one hop distance with the other genes as given by the network using the STRING tool, whereas indirect association implies more than one hop distance in the same network.

Table 3 Hub genes based on radiality and pagerank measure for each module in control stage along with their associations with causal and suspected genes Associations with causal genes Measure Hubgene Associated genes Module No Gene name(s) Radiality MAFF

7 1

MYH11

RBFOX

14,20 2, 5 8 3,6,9,10,11,13, 15,17,18, 21 4 16 19, 22

Associations with suspected genes MAFF 7 1 Associations with causal genes Pagerank MOXD1 2,3,5,6,7,9,10, 11,12,13,15,17, 18,21 4 8 14,20 16 19, 22 Associations with suspected genes 3

DRD2, SLC6A3, TH, DDC, SLC18A2, GCH1 DRD2, SLC6A3, TH, DDC, SLC18A2 DRD2, SLC6A3, TH, DDC, DRD2, SLC6A3, TH, DDC, SLC18A2, GCH1 DRD2, SLC6A3, TH, DDC DRD2, SLC6A3, TH, DDC, SLC18A2, GCH1 DRD2, SLC6A3 DRD2, SLC6A3, TH DRD2, SLC6A3, TH, DDC, SLC18A2

Mode of association Indirect Indirect Indirect Indirect Indirect Indirect Indirect Indirect Indirect

FOS, CNR1 FOS, CNR1

Indirect Indirect

DRD2, SLC6A3, TH, DDC, SLC18A2, GCH1

Indirect

DRD2, SLC6A3 DRD2, SLC6A3, TH, DDC DRD2, SLC6A3, TH, DDC DRD2, SLC6A3, TH DRD2, SLC6A3, TH, DDC, SLC18A2

Indirect Indirect Indirect Indirect Indirect

DDC

Direct

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Table 4 Hub genes based on radiality and pagerank measure for each module in disease stage along with their associations with causal and suspected genes Associations with causal genes Measure Hubgene Associated genes

Radiality

Pagerank

Module No

Gene name(s)

ACTR2 IGF1R DDX3Y LY96

1,2,3 4 5 6

ACHE

7

SLC6A3 SLC6A3 SLC6A3, TH SLC6A3, DDC, GCH1 SLC6A3, DDC, GCH1, TH

Mode of association Indirect Indirect Indirect Indirect Indirect

S100A9

MARCKS

2 3 4 5 6 7

SLC6A3 SLC6A3 SLC6A3 SLC6A3, TH SLC6A3, DDC, GCH1 SLC6A3, DDC, GCH1, TH

Indirect Indirect Indirect Indirect Indirect Indirect

In Table 3 and 4, we see that RBFOX1, MAFF, MYH11, ACTR2, IGFIR, LY96, S100A9, and MARCKS were found as the hub genes among modules obtained in the two stages. However, these genes are not found to be associated with the causal genes in the STRING tool’s repository. Taking a closer view of the role of these genes in Parkinson’s Disease, we find that four of them (RBFOX1, MAFF, MYH11, S100A9) are associated with the disease. An experiment was conducted in vitro on mutations of the neurons in a person suffering from PD showed an increase in the expression level of RBFOX1 which is associated with RNA processing activities resulting in phenotypic changes of the patient [15]. Another transcriptomic study conducted on disease and control olfactory neurosphere derived cells of a Parkinson’s patient revealed that MAFF was induced only in PD cells [2]. Dementia is one of the early signs of PD and is diagnosed at least a year before the actual diagnosis of PD.3 A protein called S100A9 has been established as a biomarker of dementia progression and hence can be associated with the disease [10]. Another interesting finding is the association of a hub gene MAFF with FOS and CNR1 gene in the control stage. The associated genes FOS and CNR1 are among our suspecting genes. Hub gene, MOXD1 in module 3 of the control stage is directly found to be associated to the causal gene, which makes it all the way more significant w.r.t. the disease. The other hub genes of the control stage do not form part of modules during the disease stage 3 http://www.alz.org/dementia/parkinsons-disease-symptoms.asp.

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at the given threshold. This may be due to their low subspace overlap value with the seed node. A low subspace overlap value indicates fewer connections of this gene with the rest of the genes in the network. Although we could not find grounded evidence for the association of a few hub genes with the disease, it can be a good starting point for the biologists to conduct experiments and analyze their roles in the genomic structure of patients suffering from the disease.

7 Conclusion and Future Work In this paper, we have used the modules from both control and diseased stage of a Parkinsonian patient. The extracted modules in both the stages are used to find consensus modules that show high correspondence in terms of sharing of common genes and pathways. The concept of consensus modules encourages a detailed analysis of the differentially coexpressed genes across the stages. This analysis helps identify certain new genes such as ADCY2, GNB5, HTR2A, GRIN2A, GRIN1, and SLC17A6 which have been found strongly associated with the causal genes known apriori and, hence may also cause a critical disease like Parkinson. From the view point of centrality analysis, genes such as RBFOX1, MAFF, S100A9 are found to be closely associated with the disease. These suspected biomarkers can then be used as the starting point for the drug designing community to develop drugs which might ease the sufferings of a Parkinson’s Disease patient.

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Indian Regional Spoken Language Identification Using Deep Learning Approach Bachchu Paul, Santanu Phadikar, and Somnath Bera

Abstract In the last decade speech is a thirsty area of research to the researchers. Man–machine interaction through voice is now making us an efficient and effortless mechanism. In our proposed work of language identification, we have taken the International Institute of Information Technology, Hyderabad (IIIT-H) Indic speech corpus where seven languages have been used and each language has 1000 uttered sentence. Thus, a total of 7000 audio samples have been used in our model of language identification. We have done a pre-processing phase, followed by a pitch and Mel Frequency Cepstral Coefficients (MFCC) feature extraction method and finally a Long Short-Term Memory (LSTM) sequence classification has been used for correct identification of the spoken language and obtained a highest training accuracy of 99.8% for the different hyper-parameters discussed in Sect. 5. Keywords DNN · Pitch · Hamming window · MFCC · Deep learning · LSTM · Mini batch

1 Introduction Language identification is the power of a machine used to identify the spoken language. Automatically, language identification determines the spoken language in a segment of speech utterance. Normally the speech is spoken by the speaker which B. Paul (B) · S. Bera Department of Computer Science, Vidyasagar University, Midnapore 721102, West Bengal, India e-mail: [email protected] S. Bera e-mail: [email protected] S. Phadikar Department of Computer Science and Engineering, Maulana Abul Kalam Azad University of Technology, Kolkata 700064, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 D. Giri et al. (eds.), Proceedings of the Sixth International Conference on Mathematics and Computing, Advances in Intelligent Systems and Computing 1262, https://doi.org/10.1007/978-981-15-8061-1_21

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is unknown. Voice command system becomes more popular in today’s world for human–machine interaction. Now it is shown, those who know the spoken language can identify it positively. This process takes very few seconds. But those who are unknown about the language make ’sounds like French’ statements [1–3]. Generally, natural languages work non-randomly and they use characters and sequence of characters regularly. The functions of language include communication, the expression of identity, play, imaginative expression and emotional release. Language identification is necessary for language translation, call forwarding, question answering, etc. [1, 4]. Language identification is an interesting area of research to recognize the spoken language of a given audio segment in speech translation, multilingual speech recognition, document retrieval, emergency call routing, in defence and surveillance applications, etc. [5]. It is noticed that generally phonological traits are provided by prosodic features. For that, it helps to get distinguish tonal from non-tonal languages. MFCC is used to extract features, i.e. too effective to carry information of tone. From this, it is almost say that MFCC is really the most important feature for language identification system which is based on automatic speech recognizer. In modern study, deep neural network and long short-term memory performs a most effective role for language identification system [6]. Deep neural network is a useful method for language identification at acoustic level. It is successfully applied on acoustic modelling, speech recognition, speaker identification and many others. Specifically, deep neural network applied when a huge amount of data is available and their data is used for training purpose [7]. India is a country where many languages are used for communication [11, 12]. The food habits and spoken language change in every 250 kilometres in India [11, 12]. Moreover, the peoples live in the border of the two states, their pronunciation style, accents and some other parameters match with the two states. Actually, the native language matches with the border of the two states. So it is very important for the identification of the language to identify the people who live in the state or region. This is important for so many applications like machine translation, question answering, automatic call forwarding or conversion, etc. Motivated from in this paper we use deep neural network for language identification where feature extraction is done by pitch and MFCC. In our proposed work of language identification, we trained our model using LSTM network with the different hyper-parameters.

2 Literature Review Zazo et al. [3] presented an analysis of the use of LSTM. Here Recurrent Neural Network (RNN) is used for short utterances that identify language automatically. In this paper, they explored many organization of LSTM RNN system and also they compared to i-vector-based system. Using a dataset of NIST CRE 2009, they got the result of better performance of LSTM RNN based system than i-vector-based

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system. They achieved over 70% accuracy in this system and later handled non seen language problem. Padi et al. [5] utilized short-sequence information for developing an end to end neural network framework for language recognition. They presented a hierarchical gated recurrent unit named HGRU based language identification for this language recognition system consists of a mechanism of attention. They performed their research in NIST LRE 2009 using noisy, clean like partially corrupted speech and multi-speaker speech data for language recognition. Here they also compared the computational complexity of i-vector-based system to LSTM baseline system and decided that (condition for 1000sec) HGRU system performance is not better than the baseline system, but better than LSTM based model. Their proposed model runs on a single CPU-based system for the identification of the language. Finally, from their innovation, it can be said that based on the attention mechanism, HGRU model is logically modelling the time series for language classification. Bhanja et al. [6] proposed a two stage Indian language identification system, i.e TS-LID system consists of tonal or non-tonal pre-classification and individual language identification modules. In this system, performance of TS-LID system has analysed which is nearly related to Indian languages. This paper gave a clear idea of the advantages of Mean Hilbert envelope coefficients and Mel frequency cepstral coefficients and both features. Later in this paper provided a discussion of comparison of performance of MHEC + prosodic features and MFCC + prosodic features. In OGI-MLTS database it was seen that performance of MHEC feature is better than MFCC features. But the performance of MFCC features was better for non-tonal languages than tonal languages. However, analysis for utterances level both of them carry complementary information. Here among DNN, LSTM RNN, i-vector-based SVM only DNN models supplied 90.8% highest accuracy for NITS-LD and 89.2% highest accuracy for OGI-MLTS system. From the experimental result of this paper, it is seen that classification of the world’s distinct languages of OGI-MLTS dataset performance is more outstanding than Indian languages of NITS-LD for the effect of pre-classification module. Moreno et al. [7] experimented the use of DNN to identify the language from speech signals using short-term acoustic features. They took state-of-the-art acoustic system based on i-vectors. Finally, they compared their technique using Google 5M LID dataset and NIST LRE 2009 dataset. In Google, 5M LID corpus DNN provided approx. 70% of relative improvement in Cavg terms over baseline systems, whereas in NIST, LRE 2009 DNN provided 15% of relative improvement approx. 15% in Cavg. Geng et al. [8] proposed a recurrent neural network based on attention. This network is used to build an end to end automatic language identification system. In this paper, it was noticed that the attention mechanism is introduced with LSTM encoder for language identification. Here for utterance level, derivation attention vectors are used. Then it is seen that the soft and hard two attention approaches are explored. Here a hybrid test method is used and after testing 8.2% relative equal error rate reduction is compared with the frame level system based on LSTM. Finally,

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Fig. 1 Schematic diagram of the proposed method

comparing the system with the conventional i-vector system, they have seen that 34.33% improvement done in performance. Trong et al. [9] proposed an end to end automatic language identification by using a combination of most advance network architecture including: Convolutional neural network, recurrent neural network and Feedforward neural network. It is the facility to use end to end system, burden of handcrafted features are removed by deep learning in language identification system. From the different length of speech utterances, languages are recognized by end to end deep learning system. This neural network replaced the pipeline of handcrafted features with deep bottleneck features and i-vector. Here also a comparison of network performance is done between BNF i-vector system and deep language. Bartz et al. [10] proposed a language identification system that solves several problems to identify a language in the image domain than the audio domain. In this paper, they found the several language identification problem from perspective vision of computer. Here they have used a hybrid Convolutional neural network in addition with a recurrent neural network to explore a specific language from the audio system. They have done several experiments for showing model applicability globally using this network. They released their huge training set and code for language identification system to the community. But in our proposed research work, the system more accurately can recognize a language by listening to the segment of an uttered speech. The proposed method is given in the schematic diagram in Fig. 1. Our paper is structured as: Sect. 3 explains the dataset used and the pre-processing phase, Sect. 4 focuses the feature extraction phase , Sect. 5 illustrates the LSTM network structure and result of the method and finally Sect. 6 discusses the conclusion.

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3 Dataset and Pre-processing 3.1 Dataset Used In our proposed work of spoken language identification, we used International Institute of Information Technology, Hyderabad (IIIT-H) Indic speech corpus where seven state languages have been used, namely: Bengali, Marathi, Telugu, Tamil, Malayalam, Kannada and Hindi [18]. Each type of language has 1000 audio samples with varying length with uttered by multiple speakers. The audio was recorded with a sampling frequency of 16 KHz with .wav format. The speech samples were recorded in a mono channel with 32 bit representation for each sample. Among 7000 audio data samples, 5600 samples have been used for training and 1400 sample for testing.

3.2 Pre-processing In this phase the voiced activity zone is detected from each of the uttered sentence. This is done by framing the signal of 25ms with 50% overlapping. Then for each of the frame, the average energy and average zero crossing has been computed by the formula given in Eqs. 1 and 2, respectively. The energy of a frame calculates how much information it holds and zero crossing takes decision for a noise or noiseless frame with some threshold [13]. Then we eliminated from both ends of the signal and preserved the information where the speaker starts speaking and stops speaking as shown in Fig. 2.

Fig. 2 Spoken boundary region of a sentence

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En =

[X (m) − W (n − m)]2

(1)

m=−∞

where X(.) is the frame and W(.) is the windowing function. ZC R = where

1 2N

i 

|sgn(x( j) − sgn(x( j − 1))|w(i − j)

(2)

j=i−N +1

 1, i f (x( j) ≥ 0. sgn(x( j)) = 0, i f x( j) < 0.

4 Feature Extraction For voiced activity region of the uttered sentence, we chopped the signal into small segments of frames and then computed the pitch and 14 Mel Frequency Cepstral Coefficients (MFCC) as our feature. The pitch is a perceptual parameter finding the dominant pitch in a sound signal by finding the fundamental frequency. Conversion of the fundamental frequency to pitch is essentially a non-linear quantization of frequency [14]. The pitch is related to the language in the sense of emotional and paralinguistic information of that language. The easiest way of finding the pitch is the autocorrelation based pitch estimation method. For a discrete time signal, the autocorrelation is defined by the Eq. 3. Rx x [l] = lim

N →∞

N 1  x[k]x[k − l] N j=−N

(3)

where l is called lag. The first peak in the autocorrelation function, after the zero lag value, was considered as the inverse of the fundamental frequency [14]. The MFCC is computed in the following steps given in Fig. 3: To find MFCC from the speech signal we used the following steps given in Fig. 2.

4.1 Framing Instead of finding the features of the whole sentence, the signal is truncated into 25ms segment with 50% overlap with the prior segment called a frame. As speech signal is rapidly changes over time and some of the linguistic information carried into following frames, that’s why overlapping is used. Thus, a single frame contains 400 samples i.e. 80 frames per second.

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Fig. 3 Steps taken to find MFCC

4.2 Windowing Since speech is an aperiodic signal, to maintain the continuity at two extreme ends of a frame, the signal is multiplied by a hamming window [13, 14] of same size. The equation of a hamming window is given by Eq. 4.  w(n) = 0.54 − 0.46cos

2π n N −1

 (4)

4.3 Fast Fourier Transform (FFT) The time domain into frequency domain is converted using the FFT [15], to measure the energy distribution over frequencies. The FFT is calculated using the Discrete Fourier Transform (DFT) formula given in Eq. 5. Si (k) =

N  n=1

si (n)e−

j2π kn N

1≤k≤K

(5)

where, K is the DFT length

4.4 Mel Frequency Wrapping In this step, the power spectrum is mapped onto mel scale using 20 number of triangular band pass filter. The relationship between frequency (f) and mel (m) is given in Eq. 6. f ) (6) m = 2595log10 (1 + 700

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4.5 Mel Cepstrum Coefficient The frequency domain into time domain of the signal is converted by Discrete Cosine Transform (DCT) using Eq. 7. Cm =

M 

1 π cos[m(k − ) ]E k 2 M k=1

(7)

here M is the number of filter bank and 20 in our case, 1≤ m ≤ L is the number of MFCC coefficients.

5 Result and Discussion The Schematic diagram of a LSTM network is given in Fig. 3. Where xt is the input, Ct is the cell state, ht is the hidden state, f is forget gate, g is memory cell, I is input gate and o is the output gate. The advantages of using LSTM model are better for classification in sequential data and avoid vanishing gradients with respect to vanilla RNN [16, 17]. From each of the sentence, we obtained a 15 dimensional feature with variable number of frames, which varies from sentence to sentence . All frames for a single sentence are grouped together with their categorical class label are extracted both for training and testing data set. The training set contains a total of 5600 samples and testing set of 1400 samples. Then both training and testing set are mean normalized to convert varying values of feature into close interval for faster convergence. The training dataset is trained with the different hyper-parameters of the proposed method of LSTM model are as (Fig. 4): Number of input units: 15 dimensional features Number of LSTM units: 100 hidden units Number of output units: 7 fully connected layer Maximum epoch: 100 Mini batch size: 512 We obtained the highest accuracy of 99.77% for the training set of 5600 samples. The confusion matrix for highest accuracy obtained is given in Table 1 The prediction accuracy for language identification depends on the data set, the deep learning model used and the hyper-parameters of LSTM network. From the confusion matrix, we observed that a very less misclassification occurs. We have trained our model under three different conditions. Firstly, we didn’t apply any preprocessing phase. We just feed the raw speech sentence into our deep LSTM model and found the accuracy of 90.38% with 200 epoch and it convergences very slowly. In the second case, we did some preprocessing stage whose output is given in Table 1, where we achieved the best prediction. Lastly, we applied a bidirectional LSTM

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Fig. 4 A Long Short-Term Memory Network Table 1 Confusion matrix for class wise prediction on training data set

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Table 2 Confusion matrix using bidirectional LSTM model

model with two hidden layers of 100 units each and found the accuracy in 99.5% for 100 epoch given in Table 2. The testing accuracy obtained for the LSTM model was 97.93% for 1400 samples with our proposed model. The percentage of sensitivity, specificity and F-score for each of the language is given in Fig. 5.

6 Conclusion In our proposed work of language identification, we have discussed the technique in every stage which is developed. The proposed method works very fine for correct identification of the language for very short utterances of audio samples within 5 seconds. Using this architecture, with tuning some of the hyper-parameters of the LSTM model, we performed on different data sets to show the applicability of our proposed system to different situations and to adopt this to a new language. The performance slightly degrades with a large datasets with samples more than 10 seconds

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Fig. 5 Percentage of sensitivity, specificity and F-score for each language

or continuous speech. We need to test our model for out of set class to show how the system gives prediction accuracy. In our future work with will emphasis to train the model with the latest DNN like Light GRU, Parsimonious Memory Unit (PMU), Coordinated gate LSTM etc. to reduce the recognition time and to increase the out of set database performance. It is also necessary to observe whether it is possible to identify the language without listening to the whole utterance, instead of working with tone, syllables and other similar features.

References 1. Papacharissi Z (2012) Without you, I’m nothing: Performances of the self on Twitter. Int J Commun 6:18 2. https://www.cs.cmu.edu/~ref/mlim/chapter7.html 3. Zazo R, Lozano-Diez A, Gonzalez-Dominguez J, Toledano DT, Gonzalez-Rodriguez J (2016) Language identification in short utterances using long short-term memory (LSTM) recurrent neural networks. PloS One 11(1):e0146917 4. Amine A, Elberrichi Z, Simonet M (2010) Automatic language identification: an alternative unsupervised approach using a new hybrid algorithm. IJCSA 7(1):94–107 5. Padi B, Mohan A, Ganapathy S (2019) End-to-end language recognition using attention based hierarchical gated recurrent unit models. In: ICASSP 2019-2019 IEEE international conference on acoustics, speech and signal processing (ICASSP). IEEE, pp 5966–5970 6. Bhanja CC, Laskar MA, Laskar RH, Bandyopadhyay S (2019) Deep neural network based two-stage indian language identification system using glottal closure instants as anchor points. J King Saud Univer Comput Inform Sci

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7. Lopez-Moreno I, Gonzalez-Dominguez J, Plchot O, Martinez D, Gonzalez-Rodriguez J, Moreno P, (2014) Automatic language identification using deep neural networks. In: 2014 IEEE international conference on acoustics, speech and signal processing (ICASSP). IEEE, pp 5337–5341 8. Geng, Wang W, Zhao Y, Cai X, Xu B, Xinyuan C (2016) End-to-end language identification using attention-based recurrent neural networks 9. Trong TN, Hautamäki V, Lee KA (2016) Deep Language: a comprehensive deep learning approach to end-to-end language recognition. In: Odyssey, pp 109–116 10. Bartz C, Herold T, Yang H, Meinel C (2017) Language identification using deep convolutional recurrent neural networks. In: International conference on neural information processing. Springer, Cham, pp 880–889 11. Mohanty AK, Panda M, Pal R (2010) Language policy in education and classroom practices in India. Negotiating language policies in schools: educators as policymakers, 211–231 12. Bhatia TK (2007) Advertising & marketing in rural india: language, culture, and communication. Macmillan 13. Scarr R (1968) Zero crossings as a means of obtaining spectral information in speech analysis. IEEE Trans Audio Electroacoust 16(2):247–255 14. Prasad B, Prasanna SM (eds) (2007) Speech, audio, image and biomedical signal processing using neural networks, vol 83. Springer 15. Dave N (2013) Feature extraction methods LPC, PLP and MFCC in speech recognition. Int J Adv Res Eng Technol 1(6):1–4 16. Palia N, Kant S, Dev A (2019) Performance evaluation of speaker recognition system. J Discrete Math Sci Crypt 22(2):203–218 17. Sak H, Senior A, Beaufays F (2014) Long short-term memory recurrent neural network architectures for large scale acoustic modeling. In: Fifteenth annual conference of the international speech communication association 18. Prahallad K, Kumar EN, Keri V, Rajendran S, Black AW (2012) The IIIT-H Indic speech databases. In: Thirteenth annual conference of the international speech communication association

A Deep Learning Based Android Application to Detect the Leaf Diseases of Maize Utpal Barman, Diganto Sahu, and Golap Gunjan Barman

Abstract Leaf disease is a major problem in plant growth. Farmers should take the utmost care to detect the diseases of the plant. But, early detection of plant disease may not be possible for farmers due to the lack of knowledge of the diseases. Computer automation helps the farmer to detect the diseases of the plant. Here, MobileNet-based convolution neural network is implemented to detect the leaf diseases of maize. The training and validation accuracy of the model is recorded in each epoch. The best training and validation accuracy of the model with the training and validation loss of 0.33343 and 0.3047, respectively, is 93.23% and 93.75%. After the successful training and validation, the model is loaded in android smartphone as an android application for real-time testing of the model. The farmers can detect the diseases of the maize very easily using the app. Keywords Deep learning · MobileNet · Maize · Diseases detection · Android app

1 Introduction Maize is one of the important crops in India. In India, maize can be considered as a traditional and nontraditional crop. Farmers of Bihar, Madhya Pradesh, and Uttar Pradesh often cultivate maize all around the year, but in Karnataka and Andhra Pradesh, farmers used to cultivate the maize in a nontraditional way. During cultivation, farmers notice different diseases of maize in the different plant parts of the maize. Leaf diseases of maize are the major problems from the farmer’s perspective. U. Barman (B) · D. Sahu · G. G. Barman Department of Computer Science and Engineering, Girijananda Chowdhury Institute of Management and Technology, Guwahati, Assam, India e-mail: [email protected] D. Sahu e-mail: [email protected] G. G. Barman e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 D. Giri et al. (eds.), Proceedings of the Sixth International Conference on Mathematics and Computing, Advances in Intelligent Systems and Computing 1262, https://doi.org/10.1007/978-981-15-8061-1_22

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Leaves are the key part of photosynthesis. An unhealthy leaf always degrades the overall growth of the tree and reduces the fruit quality of the tree. So, it is necessary to diagnose the diseases of the leaf at an early stage. Early-stage leaf disease detection is possible with the help of experts and laboratories. These may not be accessible for all the farmers at all times. But, computer technology helps [1] farmer to detect the plant diseases in an easy method. In this paper, computer automation is applied with an android smartphone to overcome the disease classification problem of maize. With the help of the android app, farmers can detect the diseases of the maize leaf by capturing the images of the leaf using a smartphone camera app or uploading the leaf image from the image gallery of the smartphone. In recent decades, computer technology helps humans to solve most of the real-life problems such as farming, health care, etc. Among all these technologies, machine learning is one of the most popular technologies which is widely used to solve the farmer’s problems. In the previous study, researchers often used traditional machine learning methods to classify the different diseases of the plant such as citrus [2], maize [3], tomato [4], potato [5], etc. Among the entire traditional machine learning model, the support vector machine is widely used to classify the diseases of plants and achieved efficient results for citrus [6], banana [7], lemon [7], rose [7], potato [8], etc. But the traditional machine learning model always depends on the humandefined features of the images. It may sometimes ignore the important features of the images. To overcome this problem, the Convolution Neural Network (CNN) based deep learning method is used to detect the leaf diseases of the maize plant. In current times, the deep learning model is widely used to solve different problems of different areas of human society. In agriculture, the deep learning model is also used in plant disease classification and detection. Researchers often used different deep learning models for plant disease classification such as cassava [11], tomato [9], and banana [10]. Probably in all the applications, the accuracy of the deep learning based method is high as compared to the traditional machine learning based model in the bigger dataset. Table 1 shows a basic comparative analysis of machine learning model with its application and accuracy. The most popular deep learning model is the Convolution Neural Network (CNN), and the transfer learning of CNN is the most popular technique to solve the different real problems. In this paper, we developed our system based on the MobileNet architecture of CNN. In past studies, other pre-trained models are also used to classify the different plant diseases such as VGGNet for tomato [9], ResNet for tomato [9], Table 1 Comparative analysis of various machine learning models

Model

Plant

No of class

Accuracy (%)

SVM

Citrus [2]

3

95.56

VGGNet CNN

Tomato [9]

10

83

ResNet CNN

Tomato [9]

10

82.53

GoogLeNet

Cassava [11]

3

84

LeNet

Banana [10]

3

93

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GoogLeNet for cassava [11], and LeNet for banana [10], etc. Among the entire model, MobileNet [12] is widely used for mobile-based deep learning applications. Since our focus is to make an android application, we applied the MobileNet architecture of CNN to classify and detect the leaf diseases of maize. The next part of the paper (Sect. 2) explained the material and methods of the research where the image dataset, MobileNet architecture, classification result, and android application development are explained. Sections 3 and 4 of the paper explain the result discussion and conclusion of the paper, respectively.

2 Material and Methods 2.1 About Dataset A properly labeled dataset is essential to perform the CNN technique. A huge collection of datasets always makes CNN classifier confident to detect the plant diseases. In this paper, we collected four different classes of maize leaf from the Plant Village image dataset [13]. The dataset contains a total of 3852 images out of which 2690 images are infected, and 1162 images are healthy. Among the infected images, 513 images belong to Gray_leaf_spot of maize, 1192 images belong to Common_rust and 985 images belong to Northern_Leaf_Blight. Some of the infected and noninfected leaf images of maize are presented in Fig. 1. All the images are in 256 × 256 dimensions with 96 dpi and 24-bit level. A total of 80% of images of each class are used in training and 20% of images of each class are used in validation (Table 2).

2.2 About MobileNet Architecture MobileNet architecture is introduced by Howard et al. in 2017 [12]. MobileNet architecture introduced a depthwise convolution layer along with a 1 × 1 pointwise convolution to the image pixel. The depthwise convolution applies a single filter to each channel of the input image, and 1 × 1 pointwise convolution combines all the output of the depthwise convolution.

2.2.1

MobileNet for Maize Leaf Classification and Result

In this paper, we have applied MobileNet V2 architecture for the classification of maize leaf [Fig. 2]. The MobileNet V2 block contains three convolution layers instead of two convolution layers such as expansion layer (1 × 1), depthwise convolution layer (3 × 3), and projection layer (1 × 1). The basic algorithmic step of the proposed work is presented in algorithm 1. The model is implemented in the GPU environment

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Fig. 1 Samples of maize dataset Table 2 Training and validation set of the dataset

Disease category

Training

Validation

Gray_leaf_spot

410

103

Common_rust_

953

239

Healthy

929

233

Northern_Leaf_Blight

788

197

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Fig. 2 Architecture of applied model

of the Google Colab research laboratory. For the implementation, TensorFlow-GPU 2.0.0-beta1 is used with TensorFlow-hub = 0.5. The images are normalized into a 224 × 224 dimension with a batch size of 64. The model is implemented using the Keras layer in the sequential model of the Keras library. In the first layer of the Keras, the pre-trained MobileNet V2 architecture model is called from tfhub. Then, a dropout layer is added in the model with a dropout rate of 0.4. Then, we add a dense layer in the model with 512 hidden neurons in the layer. We add the ReLU activation function for the activation. The next layer is again a dropout with a dropout rate of 0.2. In the last layer, we add the final dense layer with four hidden neurons with a softmax activation function. The four activation layers triggered the four different classes of maize leaf diseases. The learning rate of the model is 0.001, and the loss is categorical_crossentropy loss. We compiled the model for 15 epochs. The training accuracy of the applied model is increasing with the increasing of epochs up to the epoch 3 [Table 3] and then it is reduced and again increased in the different epochs with a zigzag path. The training loss of the applied model also follows a zigzag path. The testing accuracy of the model follows a zigzag path with a maximum accuracy of 94.92% and loss of 0.1366 at epoch 11. Algorithm 1: Algorithm of Proposed Work Input: Input Image Output: Model Output Step 1: Apply Transfer Learning with MobileNet V2(). Step 2: Add Dropout in the network with Dropout rate 0.4. Step 3: Add Dense layer in the network with 512 Hidden Neurons. Step 4: Add another Dropout in the network with Dropout rate 0.2. Step 5: Add Dense layer in the network with 4 Hidden Neurons.

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Table 3 Model history of applied model Epoch

Training loss

Training accuracy

Validation loss

Validation accuracy

1

0.5288

0.7104

0.2008

0.9102

2

0.2208

0.9089

0.2242

0.9193

3

0.1893

0.9197

0.1558

0.9362

4

0.1910

0.9249

0.1978

0.9049

5

0.1753

0.9367

0.1530

0.9388

6

0.2132

0.9222

0.1512

0.9375

7

0.1732

0.9353

0.1359

0.9440

8

0.1621

0.9432

0.1294

0.9440

9

0.1674

0.9318

0.1413

0.9453

10

0.1650

0.9390

0.1453

0.9401

11

0.1509

0.9418

0.1366

0.9492

12

0.1567

0.9391

0.1420

0.9492

13

0.1560

0.9351

0.1285

0.9479

14

0.1311

0.9536

0.1351

0.9388

15

0.1579

0.9291

0.1362

0.9453

The accuracy and loss graphs of the model are presented in Fig. 3. From the graph, it comes to know that the model is slightly overfitted. To reduce the small overfitting problem, we upgrade the model with L2 regularization with a regularization rate of 0.01 [Fig. 4]. The regularization is added in the dense layer before the final layer. In the new layer, we add 64 hidden neurons with the regularization and compile the model for 10 epochs. The model history of the new model is presented in Table 4. For the compilation of the model, we keep the same learning rate for the new model as like the older one. The accuracy graph of the model is presented in Fig. 5. The best validation model accuracy is 93.75% at epoch 9 with an error of 0.3047. It indicates that L2 regularization reduces the overfitting at epoch 9, and it takes less time to record the best validation. The training and validation accuracy of the upgraded CNN also follows the zigzag path but the training and validation loss is gradually reduced with the epochs (Fig. 5).

2.3 Android Application for Maize Leaf Detection After successful training and validation, the model is kept ready for android deployment. Initially, the trained model is saved in .pb format and then it is changed into .tflite using the TensorFlow lite converter library. The .tflite model is the lite version of the original version of the TensorFlow. Finally, the lite model is deployed in the

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Fig. 3 Accuracy and loss curves of applied model

Fig. 4 Architecture of upgraded applied model

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Table 4 Model history of upgraded applied model Epoch

Training loss

Training accuracy

Validation loss

Validation accuracy

1

1.5858

0.7019

1.1287

0.9193

2

1.0366

0.8994

0.8657

0.9375

3

0.8265

0.9137

0.7671

0.9076

4

0.6977

0.9074

0.5959

0.9297

5

0.5798

0.9263

0.4959

0.9505

6

0.4889

0.9128

0.4631

0.9089

7

0.4339

0.9233

0.3697

0.9479

8

0.3905

0.9235

0.3409

0.9401

9

0.3343

0.9323

0.3047

0.9375

10

0.3015

0.9290

0.2839

0.9440

Fig. 5 Accuracy and loss curves of upgraded applied model

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android application. In this paper, the android application is developed using Kotlin language. The basic layout of the model is presented in Fig. 6. The first layout is the basic layout of the model which shows the name of the android app. The name of the app is maize. The second layout of the application defines the five activities of the model such as the selection of the maize image from the smartphone image gallery or it can directly access using a smartphone camera. The other activities are a button, leaf panel, and disease name panel. In the last layout, the classified leaf of the maize will appear along with the confidence level of the classifier.

Fig. 6 Smartphone application layout

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3 Discussion Here, it is found that the transfer learning concept is the most valuable and affordable technique to solve the maize disease classification problem. Researchers already proved that deep learning is widely used to classify the different diseases of plants and especially for plant village datasets. Malusi Sibiya [13] already used the plant village maize dataset and achieved 92.85% accuracy for the CNN architecture which is less as compared to this paper. The authors [13] used a simple architecture of CNN with two convolution layers and two pooling layers. The authors [11] recorded more than 80% accuracy for the cassava disease classification using GoogLeNet-based CNN and mobile application. Here, we used MobileNet-based transfer learning concept which effectively gives good results in each epoch. The final validation accuracy of the model is 93.75% at epoch 9. For further validation of the model, we change the learning rate of the model from 0.001 to 0.01. But, the accuracy of the model is very less as compared to the learning rate of 0.001. The analysis of the model is presented in Fig. 7. In Fig. 7, it is shown that the accuracy of the model is very less and is in the

Fig. 7 Accuracy and loss curve of upgraded applied model at Lr = 0.01

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range of 30% to 13%. It means that the model is not learning well at a low learning rate and shows a minimum accuracy for the model. It is recorded that the model is neither overfitted nor underfitted [Fig. 5]. It gives efficient accuracy in each epoch. The accuracy graph follows a zigzag path but it is increasing with epoch and shows the best validation at epoch 9 [Fig. 5]. Again, the loss graph is gradually decreasing with the epochs. The defined maize problem can also be solved using the traditional Artificial Neural Network (ANN), but it takes more time for classification. But ANN depends on the researcher’s defined features of the maize, whereas CNN does not depend on the researcher’s defined features. So, the CNN can be considered as one of the best models for maize leaf disease detection and classification as comapred to ANN. The result of this paper also proves that CNN gives better accuracy in maize leaf disease classification as like other leaf disease classification (Table 1). So, it is a good technique to be considered for any large dataset classification.

4 Conclusion In this paper, MobileNet-based transfer learning concept is applied to classify the diseases of the maize. The selected dataset for the study is plant village. From the experiment, it is found that MobileNet architecture is a suitable architecture to classify the diseases of maize using android smartphones. The models are easily converted into .tflite format, and it shows a high accuracy as compared to other transfer learning architecture. The model is very helpful for maize farmers, and they can easily detect the diseases of the maize using the application. Acknowledgements This research project is supported by Assam Science and Technology University, Guwahati, Assam under TEQIP-III, vide Ref. No.: ASTU/TEQIP-III/Collaborative Research/2019/3598, Dated August 28, 2019.

References 1. Barman U, Choudhury RD (2019) Soil texture classification using multi class support vector machine. Inf Process Agric 2. Barman U, Choudhury RD Bacterial and virus affected citrus leaf disease classification using smartphone and SVM 3. Kai S, Zhikun L, Hang S, Chunhong G (2011) A research of maize disease image recognition of corn based on BP networks. In: 2011 third international conference on measuring technology and mechatronics automation, vol 1, pp 246–249 4. Mokhtar U, Ali MA, Hassanien AE, Hefny H (2015) Identifying two of tomatoes leaf viruses using support vector machine. In: Information systems design and intelligent applications. Springer, pp 771–782 5. Athanikar G, Badar P (2016) Potato leaf diseases detection and classification system. Int J Comput Sci Mob Comput 5(2):76–88

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6. Sharif M, Khan MA, Iqbal Z, Azam MF, Lali MIU, Javed MY (2018) Detection and classification of citrus diseases in agriculture based on optimized weighted segmentation and feature selection. Comput Electron Agric 150:220–234 7. Singh V, Misra AK (2017) Detection of plant leaf diseases using image segmentation and soft computing techniques. Inf Process Agric 4(1):41–49 8. Oppenheim D, Shani G (2017) Potato disease classification using convolution neural networks. Adv Animal Biosci 8(2):244–249 9. Fuentes A, Yoon S, Kim S, Park D (2017) A robust deep-learning-based detector for real-time tomato plant diseases and pests recognition. Sensors 17(9):2022 10. Amara J, Bouaziz B, Algergawy A (2017) A Deep learning-based approach for banana leaf diseases classification. In: BTW (Workshops), pp 79–88 11. Ramcharan A et al (2019) A mobile-based deep learning model for cassava disease diagnosis. Front Plant Sci 10:272 12. Howard AG et al (2017) Mobilenets: efficient convolutional neural networks for mobile vision applications. arXiv preprint. https://arxiv.org/abs/1704.04861 13. https://www.kaggle.com/emmarex/plantdisease

Inversion Formula for the Wavelet Transform Associated with Legendre Transform Jyoti Saikia and C. P. Pandey

Abstract In this paper, we accomplished the concept of convolution of Legendre transform for the study of continuous Legendre wavelet transform. We also presented some discussion on its basic properties such as linearity, shift property, scaling property, symmetry, and parity. Finally, our main goal is to find out the Plancherel and inversion formula for the Continuous Legendre Wavelet Transform (CLWT). Keywords Legendre function · Legendre transforms · Legendre convolution · Wavelet transform

1 Introduction The wavelet transform [1–5], for a function f 1 ∈ L 2 (R), with respect to the wavelet φ ∈ L 2 (R), is defined by ∞ (Wϕ f 1 )(σ2 , σ1 ) =

f 1 (t)ϕσ2 ,σ1 (t)dt, σ2 ∈ R, σ1 > 0,

(1)

−∞

where −1/2

ϕσ2 ,σ1 (t) = σ1

 ϕ

 t − σ2 . σ1

(2)

J. Saikia (B) · C. P. Pandey North Eastern Regional Institute of Science and Technology, Nirjuli 791109, Arunachal Pradesh, India e-mail: [email protected] C. P. Pandey e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 D. Giri et al. (eds.), Proceedings of the Sixth International Conference on Mathematics and Computing, Advances in Intelligent Systems and Computing 1262, https://doi.org/10.1007/978-981-15-8061-1_23

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Translation τσ2 is defined by τσ2 ϕ(t) = ϕ(t − σ2 ), σ2 ∈ R, and dilation Dσ1 is defined by −1/2

Dσ1 ϕ(t) = σ1

 φ

 t , σ1 > 0 σ1

we can write φσ2 ,σ1 (t) = τσ2 Dσ1 φ(t)

(3)

From (1), (2), and (3) it is obvious that wavelet transform of the function f 1 on R is an integral transform for which the kernel is the dilated translate of φ. We can also express (1), as the convolution [6] 

   Wϕ f 1 (σ2 , σ1 ) = f 1 ∗ f 2o,σ1 (σ2 ),

(4)

where f 2 (t) = ϕ(−t). As for every integral transform, there exists a particular type of convolution, one can define wavelet transform with respect to an integral transform using associated convolution. Integral transform including special functions as kernel play a significant role in the theory of partial differential equations. Pathak and Dixit [7] have defined Bessel wavelet using Bessel functions on semiinfinite interval (0, ∞). Then in [8], Upadhyay and Thripathi construct continuous wavelet transform corresponding to Watson transform. In 2009, Pandey and Pathak [6], studied Laguerre wavelet transform and derived the many properties related to Laguerre wavelet transform. Motivating from above ideas we are interested to define the wavelet transform corresponding to Legendre transform and to establish inversion and Plancherel formula for Legendre wavelet transform.

2 Preliminaries Let X denote the space L p (−1, 1), 1 ≤ p < ∞, orC[−1, 1] with the norms

Inversion Formula for the Wavelet Transform Associated …

⎛

∞

1  f1 p = ⎝ 2

⎞1/

289

p

| f1 | d x ⎠

, 1 ≤ p < ∞,

p

0

 f 1 ∞ = ess sup | f 1 (x)|. −170% of the malware and malware detection is significantly affected by ad traffic. Further, using HTTP request and DNS Query, the scheme identified malware and achieved detection results of 40.89% and 69.55%, respectively. Li et al. [49] presented a system that monitors network traffic for android malware detection. There are four components in the system, i.e. traffic monitoring (feature extraction), traffic anomaly recognition (identify malicious software), response processing (responsible for malicious software) and cloud storage (store security policy and data). It mines the feature data and parses data packet’s protocol. It is designed to perk up Android terminal defence ability against mal-attacks, and APT (Advanced Persistent Threat) attacks by using SVM algorithm classification of data and determining whether network traffic is abnormal and uses correlation analysis to locate the app that produces abnormal traffic.

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Table 1 Comparison of smartphone traffic analysis techniques Author’s Name

Year Methodology Dataset

Result

Remark

Esmaeili et al. [33]

2019 Based on app Drebin, Google features and Play, Café Bazar network traffic analysis KNN, Decision tree, Naive Bayesian as classifier

Precision: Estimation for high risk-87% detection as Very high risk- 96% botnet is done using 3 level of risk i.e. average or high or very high

Zulkifli et al. [34]

2018 Dynamic detection technique Records app behaviour at runtime Based on network traffic using Decision Tree

Drebin, Drebin- 98.4% Contagiodumpset, accuracy Google playstore Contagiodumpset97.6%

Drebin achieved higher accuracy in comparison to Contagiodumpset dataset

Genome malware dataset [36]

Detction accuracy94.25%

Combining permissions and traffic features enhanced the rate of detection

Arora et al. [35]

Combined network traffic & system permissions Hybrid model

Wei et al. [37]

Modelling Official Android and learning Market method for network behaviour detection Used WGAN and Bi-LSTM

App classification accuracy- 96.89%

WGAN model helped in improved accuracy of Bi-LSTM by 9%

Wang et al. [38]

Based on text semantics of network flows Used N-gram method from NLP SVM classifier

Accuracy achieved99.15%

Requires few samples for good detection results Able to detect new discovered malware as well

VirusShare [39], Baidu mobile assistant [40], Google play

(continued)

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Table 1 (continued) Author’s Name

Year Methodology Dataset

Result

Remark

Kandukuru 2017 Two level et al. [41] hybrid analysis approach based on permission vector and network traffic

Malgenome project [36], Google playstore [42]

Detection accuracy–95.56%

Uses less time and limited computational resources

Cam et al. [43]

Hybrid model based on sensitive resource accessing &network traffic

Google play, Genome malware project, Droidbench

Accurately detected all 3 of app groups of Droidbench dataset i.e. Access Internet, Neverclick and sensitive resource

Lower FPR viz. false positive detection rate

Lashkari et al. [44]

Five classifiers namely RF, KNN, DT, RT, Regression Focuses on dynamic behaviour of malware

1900 benign and mal-apps of 12 different families

Avg Accuracy (91.41%), Precision (91.24%), FPR (0.085)

Network traffic captured via limited user interaction with installed apps

Cheng et al. [45]

Analysis of relationship b/wnetwork flows &behaviour patterns Random Forest ML algo

Google play [42]

Achieved higher than 95% precision and accuracy

Lacked comparison with other ML methods

Detection Rate TCP Flow –98.16% HTTP Model –99.65% FPR 5.14% and 1.84%

Provides details about detection results

Wang et al. 2016 Combines Drebin [46] [14] network traffic analysis with ML (C4.5 DT) Perform multi-level network traffic analysis

(continued)

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Table 1 (continued) Author’s Name Nancy et al. [15]

Year Methodology Dataset

Result

Compared Android Malware Accuracy –90.32% traffic of Genome Project malware with that of normal-apps Decision tree as classifier

Remark Fails when obfuscation techniques are employed like encrypting the traffic used by malware

Chen et al. 2015 Involves Drebin, Android Detection Rate Analyzed [8] traffic Malware Genome DNS Query: 69.55% malware traffic generation, Project HTTP Request: only capturing and 40.89% behaviour monitoring Arora et al. 2014 Based on Android Malware Accuracy –93.75% [47] network Genome Project traffic features such as ratio of incoming to outgoing bytes, Avg packet size,etc Uses rule-based classifier

The approach is specific to those malwares which in the background connect to any remote server

Feizollah et al. [48]

Mini batch K-means clustering performed better than the other approach i.e. k-means

Based on MalGenome, network Google play traffic generated by Android apps Uses 2 clustering algorithms i.e. k-means and mini batch k-means

K-means & Mini Batch K-means respectively Accuracy: 0.48, 0.62 Homogeneity: 0.008, 0.13 Completeness: 0.16, 0.18 V-Measure: 0.11, 0.15

In [41], authors presented a hybrid model based on network traffic and permission bit-vector for detection of android malware. This 2-level approach used Decision Tree as a classifier to detect malicious behaviour. This hybrid model is quicker akin to static methods and is able to perceive unfamiliar malwares similar to any dynamic method. In NTPDroid [35], authors suggested a hybrid framework that amalgamates the network traffic and system permissions attributes for detection of malware patterns in applications. The model was trained and tested using the FP-Growth algorithm. It

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has two phases, i.e. analysis and detection phase. Analysis phase aimed to generate the recurrent patterns present in legitimate and mal-apps. While in the detection phase, the two sets of recurrent patterns generated are used to identify mal-apps. NeSeDroid [43] is another hybrid analysis method which is analyzed based on n/w traffic analysis and sensitive resource accessing. It used a static approach for the detection of sensitive resource access and dynamic approach for detection of leakage of info via internet connection. In [34], authors presented a dynamic approach analyzing network traffic and capturing app behaviour at runtime. Few n/w traffic features were extracted and tested on Decision tree algorithm using WEKA tool. The performance is analyzed using accuracy, TPR & FPR, ROC and mean absolute error. In [44], researchers introduced n/w-based system which analyzed meaningful deviation of its behaviour in android app to detect and label malware. It used 9 traffic features and it is not only capable of detecting mal-apps, but also identified them whether it is general or some specific type (adware) of malware. It used flow, packet and time-based features as a classification method to characterize malware families. Taking imbalanced learning problem into consideration, authors [45], developed a management and control scheme for the collection of android network traffic and established that using n/w traffic to train ML model is a problematic imbalanced learning via analysis of using the collected n/w traffic. In addition, android malwares are detected by applying four imbalanced algorithms on imbalanced n/w traffic dataset. It showed that the combination of SVM and SMOTE performed best in all combinations. Taking into consideration different network flow generation by different apps using different operations and, as well as different patterns of both benign and malicious flows, Cheng et al. [50] developed a model after studying the relationship involving behaviour patterns & network flows to detect the app that leaks private info of users. It used RF machine learning algorithm for the classification of network flows. Further to improve controllability, authors designed an app called Moledroid, to put into practice network flow detection with an ML algorithm. It achieved accuracy and precision > 95%. Watkins et al. [51], demonstrates network-based IDS which detects malware that either generates no network traffic or traffic that is impossible to differentiate from from genuine network traffic. This network-based tool does not engross dependability upon other traffic sources from an app and also do not rely on interactions based on the host-based dwelling. It does not require its installation on the device and is competent to detect a set of malwares which is unable to produce Wi-Fi network traffic. The study [52], focused on the identification of mal-apps by using URLs visited by apps. This method is responsible for vectorization operations and URL segmentation and does not involve complicated feature engineering. It used a neural network having multiple views and laid emphasis on width and depth of neural network and addressed the challenge of feature selection. Multiple views of input are automatically created using this proposed neural network and hence preserved rich semantic info from the input and then distributed soft attention weights, and its main emphasis is on diverse input features. Wei et al. [37] demonstrated learning and modelling method to analyze android mobile apps network behaviour. To trigger different categories of app behaviours, various app system and environmental factors are simulated.

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The retrieved series of network event behaviour is classified according to behaviour sequence combination via a Bi-LSTM. Additionally, WGAN viz. Wasserstein Generative Adversarial network was used to solve the trouble of time overhead, limited data samples and hence it increased the diversity of data. The authors [38], presented a detection method based on text semantics of network flows. It took HTTP flow generated via android apps as documents which are further processed using N-gram method from NLP, to extract text-level features. These features are used to develop a malware detection model. The method uses SVM classifier to find out whether the traffic is malicious or legitimate. It detects unknown samples only when it possesses some characteristics similar to mal-samples in the training phase. It constructs the malware detection model by using N-gram sequence generation, chi-square feature selection algo and SVM algo. Zaman et al. [15] demonstrated a behaviour detection technique for detecting mobile malware that can commune with blacklisted domains and bypass confidential info. For this, AppURL table was created that records all efforts by apps for interacting with remote servers. Each entry in this record takes care of the app id and the URL that the app makes contact with. Further, authors used domain blacklist and flagged the apps that make contact with any of the domains as malware. Nancy et al. [53] presented a technique based on network traffic for detection of malware. The authors compared network traffic of malwares with that of benign apps and found the distinguishing features. A decision tree classifier was built based on the features to detect mal and benign apps. This method achieved an accuracy of more than 90% and network traffic of malware was captured using actual smartphones rather than using emulators. In [47], authors analyzed the n/w traffic features and built rule-based classifier in order to reveal android malware. Remotely server controlled android malware is detected by it or confidential info of the remote server is disclosed. In the first phase, the method includes analyzing network traffic and identifying the distinguishing features among malware and normal traffic. In the second phase, rule-based classifier was built over found distinguishing features and the accuracy was calculated by running the classifier on test data and hence it achieved an accuracy of 93.75%. Mobile Guard [54], a network-based malware detection method which detects malicious activities within a network and also protects end users from malware attacks which are transmitted via mobile operator’s network. It is capable of detecting the suspicions patterns even if those are not included in the database via machine learning. Shabtai et al. [32] presented an anomaly detection method based on behaviour to identify significant variations in device app n/w behaviour. It safeguards cellular infrastructure companies and mobile devices user from mal-apps via detecting malattacks and repackaging apps. It attempted to detect malware having self-updating capabilities as this type of malware is not detectable using regular dynamic or static analysis approach. The method uses app network traffic patterns only to perform detection. Arora and Peddoju in [16] focused on minimizing the number of features by proposing an algorithm that prioritizes n/w traffic features. Statistical tests are used in this approach to rank the attributes. The end result demonstrated that it reduced training and testing time, as well as gave the high detection accuracy rather using features collectively, and hence achieved F-Measure of 0.9636 with 9 features out

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of a total of 22 features. Further, the training time of 300 apps was reduced to 5.8 s from 11.7 s and testing time was reduced to 17.3 s from 25.1 s for 230 apps. The traffic was captured on an actual smartphone rather than an emulator. It is off-device detection and moreover, few samples that use obfuscation technique escape from getting detected by employing encryption remote server interaction. The authors in [48], evaluated the performance of two clustering algorithm, i.e. mini batch K-Means and K-means to detect android malware via n/w traffic analysis. Further, n/w traffic produced by Android apps, i.e. by 800 malware samples and numerous normal-apps are analyzed, and hence the result depicted a better malware detection by mini batch K-means algorithm achieved in comparison to K-means algorithm. Wei et al. [55] presented a method based on monitoring network behaviour and interaction of apps to characterize app behaviour. From app network traffic, statistical features are derived which are supplied to a ML classifier to derive a common model for each distinctive class of mobile apps. PODBot [33] is a tool based in cooperation with network traffic analysis and app features. It was assessed over a set of botnets of famous types and gives accurate detection of 87% in high threat and 96% in very high threat. By detection via host, bots including which are communicated via Bluetooth or SMS channels are detected but for internet as communication channel, network flow and traffic analysis is effective in detection because of high FPR of static analysis. In [56], the authors proposed a malware detection arrangement built on TCP traffic that can rapidly and aptly detect malware. Here, n/w traffic produced by several apps has been collected and an enormous number of TCP flows resulted after preprocessing. After that initial packets sizes were extracted from TCP flow as features to be fed to detection model. In this method, the feature extraction time from 53,108 network flows is abridged from 39321 to 18041 s (drop of 54%). This method also accomplishes a detection rate of 97%. The authors in [57], suggested an approach grounded on n/w packets fuzzing for Android apps. This system acquires the communication data directed by servers to apps, implements diverse mutation schemes to mutate the different types of novel data, return the mutated response to apps, monitor crash information using log monitoring tools to determine the impending security threats. Four types of problems were exposed by using the above approach. The problems comprise unresponsiveness, crashes originated via JSON data exception, URL redirection and HTML content replacement. The outcomes showed that the suggested technique aptly exposed malign behaviour of mobile applications using network information interaction. Authors in [58], proposed a method for identifying malware based on URLs frequented by the apps. Each URL is divided into various segments using particular characters. The skip-gram algorithm is then used for training. The generated URL vector is then fed into a multi-view neural network-based malware detection model.

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4 Conclusion and Future Direction In this paper, surveying of the different approaches for investigating the network traffic generated by smartphones has been done. We examined state-of-the-art tactics according to the different goals of the network traffic analysis targeting smartphones. The pre-eminent goal encompasses user behaviour analysis, system identification and malware detection. In particular, we observed the approaches used to capture mobile traffic for malware detection/identification. Moreover, we examined that Android is the most sought-after smartphones OS platform targeted by malicious writers and hence by the researchers. To conclude, we account that the most of the reviewed work rely on ML techniques. We also gave details about the methodology, dataset used, as well as the achieved performance of the frameworks. In future, we would like to devise a state-of-the-art machine learning-based traffic model to segregate the malicious traffic from the benign one after applying efficient feature selection algorithm on the extracted traffic features.

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58. Shanshan W et al (2019) Deep and broad URL feature mining for android malware detection. Information Sciences, Elsevier, November

An Approach Towards IoT-Based Healthcare Management System Khushboo Singla, Rudra Arora, and Sakshi Kaushal

Abstract Internet of Things (IoT) can be employed in various applications like agriculture, disaster relief operations, smart and connected health care, this paper presents a healthcare system using IP Multimedia Subsystem (IMS) which can help doctors to monitor patient’s health from anywhere. IMS is the key idea of Next Generation Networks (NGNs) and integrated network of telecommunications that utilizes Internet Protocol (IP) for packet communications. Event-based Session Initiation Protocol (SIP) for Instant Messaging and Presence Leveraging Extensions (SIMPLE) architecture is the best solution for IMS-based systems. SIP protocol is used to communicate with IMS core and data is transmitted inside SIP messages. In this manner, there is no requirement for extra transport protocol. The proposed system can generate automatic alerts by sending calls and messages to various stakeholders like healthcare organization, family members, etc. in emergency situations. IMS client has been implemented using Nubomedia IMS connector which is customized to read and interpret data. The existing IMS clients like UCT and Boghe can’t read different files thereby prohibiting various forms of analysis and data manipulation. A rule-based interpretation of data is then done to trigger messages to the doctors if the values observed are above certain threshold. The developed system can be useful for monitoring patient’s health by collecting health data through smart watches, apps and sensors. The designed IMS platform can be extended to contribute in numerous fields like disaster management, healthcare system, emergency ambulance call setup by reading live data streams, monitoring and issuing notifications. Keywords IP multimedia subsystem · Internet of things · Body sensor networks · Session initiation protocol

K. Singla (B) · R. Arora · S. Kaushal UIET, Panjab University, Chandigarh, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 D. Giri et al. (eds.), Proceedings of the Sixth International Conference on Mathematics and Computing, Advances in Intelligent Systems and Computing 1262, https://doi.org/10.1007/978-981-15-8061-1_27

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1 Introduction To improve the quality of healthcare there is a need to monitor the vitals of the patients. The monitoring in this case must be uninterrupted and work regardless of location of the patients. To address this interest, various commercial products have been delivered such as body sensors, smartwatches which aim to provide constant information about one’s well-being condition [1, 2]. Telemedicine service allows constant observing of patient’s well-being even when they are not present in the hospitals. Those who are living in rural areas can consult their health provider without travel to such long distance using telemedicines. Wearable Sensors Networks (WSN) for well-being monitoring consists of various miniature sensors. These sensors are capable of measuring various vital signs like heart rate, blood pressure, respiration rate, etc. [3]. The vital signs data are communicated either wireless or a wired link to a central node like Personal Digital Assistant (PDA) [4]. With continuous observing of the condition by mean of brilliant medicinal gadget associated with a cell phone application, associated gadgets can gather medical and other required well-being information and utilize the data connection of the cell phone to move gathered data to a doctor [5]. These information are stored in the cloud and shared with doctors using transport protocol of network [6]. IP Multimedia Subsystem (IMS) is a key component of telecommunications [7]. It comprises horizontal control and service layer that is deployed over IP-based versatile and fixed network. One of the significant application that can make use of WSN and IMS is remote healthcare. IMS is an innovative structure that permits conveying IP multimedia services in the telecom region. Call Session Control Function (CSCF) is at the core of IMS. It manages all the signalling from end-user to services and other networks, and it builds a horizontal layer that allows the convergence of different access networks. IMS uses Session Initiation Protocol (SIP) for communication. CSCFs can be divided into three categories: Proxy-CSCF (P-CSCF), Interrogating CSCF (I-CSCF) and Serving CSCF (S-CSCF). P-CSFC is the very first contact point with IMS centre system. All SIP signalling must be steered through this part. After getting a SIP demand, it guarantees that it is sent to the best possible goal. The reactions from IMS are then sent back to the client. This element might be situated in the home system or in the visited system/networks. The central node of IMS network is S-CSCF, and it is situated in the home network. It manages the association and enrolment services. During IMS enrolments, I-CSCF inquiries the Home Subscriber Server (HSS) to choose the suitable S-CSCF which can serve the User Equipment (UE). In the proposed work, IMS is used to send the real-time data of the patients to the doctors. Body sensor systems are the key piece of architecture by empowering the detecting of physiological information [8]. The main advantage of using IMS in healthcare applications is that it facilitates the patient’s mobility so that their health can be monitored anytime and anywhere [9].

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Fig. 1 The architecture of IP multimedia subsystem

1.1 IMS Layers IMS supports a wide scope of services dependent on SIP. Figure 1 shows IMS architecture separated into four layers, i.e. device layer, transport layer, service layer and control layer. Device layer is the very first layer where various devices like laptops, phones connected to IMS. Transport layer is answerable for joining all the systems (GSM, GPRS, 3G, fixed communication) dependent on IP. The main responsibility of the control layer is to provide communication between gadgets and services. Service layer focuses on the services that are accessible to the clients like calls, messages [7].

1.2 SIP IMS uses SIP for communication. IETF organization develops SIP protocol. SIP is used for starting, keeping up and ending ongoing sessions that includes calling and text applications [10]. It is utilized for signalling and controlling interactive media. For secure transmissions of SIP messages over uncertain system interfaces, the protocol might be encoded with Transport Layer Security (TLS).

2 Related Work and Research Gaps This section discusses the work done in the field of health using IMS and IoT. Various technologies used by different authors are discussed here. Some gaps resulting from this study are also presented. Gelogo et al. [11] described how IoT will be helpful to develop u-health care system. IoT is useful for both patients and caregivers because they can monitor health without visiting the hospitals. In this article, architecture is presented in three tiers. Tier 1 in which the patient health is monitored by various monitoring devices. In tier 2, multipurpose gateways are used to receive the data and compute the patient

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health. In tier 3, mobile phones transfer the data to the monitoring centre. This article shows the architecture of u-healthcare w.r.t IoT. Hassanalieragh et al. [12] proposed that multimedia and security activities can be performed in the cloud, permitting portable healthcare organizations to broaden the capacities of their versatile healthcare applications beyond the current cell phone applications. Cell phones are being considered as service platform for portable wellbeing data convey. Anyway, mobiles face difficulties as to conveying secure interactive media-based health services because of limitations in calculation and power supply. The framework depicted in this paper utilizes cloud computing to decrease the weight of organising and improving the elements of existing mobile healthcare frameworks. Paul Fergus et al. [13] shows how the wireless sensor networks help a lot in health improvement. The proposed method contains a body sensor network, the gaming environment and data acquisition manager. Body sensor attached to the patient’s body and any movement in patient are captured and data is transmitted. Authors build gaming environment to improve the health of the patient. The gaming environment used by the doctors to give therapy to the patient. Microsoft Access database for the storage of data and Java Database Connectivity (JDBC) to connect the database and the middleware. This paper uses a neck therapy case study. Mari Carmen Domingo et al. [10] proposed the architecture to integrate the body sensor networks and social media using IMS. The proposed architecture consists of four layers that are device layer, access layer, control layer and service layer. In the first layer, the information which is sensed send to the gateway and then forward to the monitoring station. Access layer role is to give access to the monitoring station to the radio channel. Monitoring station consists of a connectivity layer and abstraction layer. Connectivity of BSN and IMS is done at the connectivity layer. Whereas at the abstraction layer the data is formatted and send to the application servers. Next layer is the control layer which is responsible for the SIP signalling. At service layer, many multimedia services are offered. This article shows the benefits of integrated BSN and social media using IMS. Garcia et al. [14] presented NUBOMEDIA, an open-source platform enabling developers to create and deploy RTC applications with advanced media processing capabilities. For this, NUBOMEDIA introduces the concept of Media Pipeline: chains of interconnected media processing elements. At deployment time, NUBOMEDIA follows a Platform as a Service (PaaS) scheme, which helps developers in most of the complex infrastructure-related tasks such as provisioning, scaling or QoS and network management. Padilla et al. [15] demonstrate the executions of a horizontal M2M network services platform over IMS. The point of research is to associate any M2M devices with M2M application server through IMS network core utilizing an M2M Gateway to build up an M2M Horizontal Services Platform over IMS. The proposed architecture is divided into M2M domain, network domain and application domain and also covers the data flow. M2M devices send the data like temperature and water tank level, and that data is transmitted to end-user using IMS. A stage for M2M innovation has been proposed.

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Barachi et al. [16] proposed the 3GPP IMS emergency service architecture for the supply of context-aware emergency services. The UE recognizes that a crisis session is being set up and register with IMS, using emergency public user id and forward a crisis session foundation solicitation to P-CSCF with the required data. P-CSCF performs verification of clients and sends request to emergency-CSCF (ECSCF). E-CSCF decides the location of suitable PSAP. PSAP calltaker evaluates the circumstance and dispatches fireman, police or ambulance to the individual’s area. Prathima Agrawal et al. [17] discussed the challenges or limitation of 3G networks like roaming problem, limited bandwidth and tunnelling. In this way, to start, end and keep up a session in various versatile systems is more confounded and troublesome than on the fixed Internet. This paper describes method to enhance the signalling and the IP connectivity. IMS is designed to guide real-time Voice over IP (VoIP) and complicated IP multimedia services for a larger quantity of customers. Roch Glitho et al. [18] discussed the scalability and elasticity of the existing IMS could be increased using cloud computing. Cloud computing is the technology which can increase scalability, elasticity, can support more application, good resource usage. To cloudify IMS various approaches have been applied, some target only specific IMS entity like HSS and other whole IMS. In a specific Entity approach, they divide the HSS into resource and management layers. This research paper shows how cloud computing, Software Defined Network and Network Functional Virtualization can use to enhance the capability of IMS. Although IMS Client is based on 3GPP and IETF specifications and recommendations, but is still in developing phase. As a result, most of the available open IMS clients are unstable and not fully functional as per the 3GPP, IETF and ETSI specifications. So, there is an urgent need to create customized IMS client. The existing IMS clients like UCT and Boghe can’t read different files thereby prohibiting various forms of analysis and data manipulation, so to overcome this problem, there is a need to design a client which can be modified according to the analysis performed on interpreted data. Scalability is an issue for an enormous number of users joining an IMS. In this paper, Healthcare Management System is designed using Open IMS Core and various IMS clients like Boghe. For reading and interpretation data, customized IMS client is also built.

3 Proposed Work The proposed model consists of four layers: device layer, cloud and management layer, control layer and service layer as shown in Fig. 2. In the device layer, Body Sensors Networks (BSN) and smartwatches are used to monitor the vital signs of the patients like heart rate, blood pressure, etc. Next layer is cloud and management layer where data is received from the monitoring station and stored in the cloud. At the control layer, IMS server is installed using Open IMS Core. The stored data from the cloud is retrieved through NUBOMEDIA IMS connector and send to IMS server

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Fig. 2 Layered architecture of the proposed model

which is working as the transport platform. In service layer, various services are provided to various stakeholders like messages, calls and other multimedia services. The details are discussed below. (i) Device Layer Body sensors or smartwatches are attached to the patient’s body. Various vital signs such as blood pressure, heart rate are monitored through BSN, smartwatches and sent to the laptop, mobile phones via Bluetooth or WiFi. The sensed data from laptops and phones are stored in the cloud through the 3G/4G connectivity or using WiFi [19–21]. (ii) Cloud and Management Layer Cloud storage is a cloud computing model that stores information on the Internet through a cloud computing provider who manages and operates information storage as a service. The sensor information ought to be prepared, designed and sent to the application servers of IMS centre system. The sensed data is stored in the cloud database. From the cloud storage, the data is transferred to IMS server using IMS clients. For this purpose, Nubomedia IMS connector is used, a java code file is written in IMS connector which reads the data from the cloud and transfers it to IMS server [9].

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Fig. 3 The 3GPP IMS emergency control layer service

(iii) Control Layer At this layer, IMS server is installed because in the proposed work IMS is used as transport protocol. CSCF is answerable for preparing SIP signalling in IMS because it uses SIP for communication. CSCFs can be separated into three classes: P-CSCF, I-CSCF and S-CSCF. The first point of contact is P-CSCF. It additionally validates the client and confirms the correctness of the SIP demands. I-CSCF is used to receive user data from the HSS and SLF databases and routes the SIP request to the destination. S-CSCF is a central node of the control layer. IMS emergency scenario depends upon four substances: Proxy-CSCF, Emergency-CSCF (E-CSCF), Location Retrieval Function (LRF) as shown in Fig. 3. The purpose of P-CSCF is to contact between the user equipment and IMS server. The E-CSCF is in charge of obtaining the location of UE and directing the call to Public Safety Answering Point (PSAP). LRF is responsible for retrieving the area from where the session has started. S-CSCF is the middle node of IMS and situated in the home network. It sits on the way of all signalling messages and can review each message. The UE makes the emergency calls. Many new users were created and added in the FHoSS. P-CSCF performs verification of the session and sends a request to the E-CSCF. Than E-CSCF decides the location of suitable PSAP and calls them. In this way, IMS server can be used in emergency situations to call ambulance, fireman, etc. (iv) Service Layer In this layer, various services are provided to the users like messages, calls and other multimedia services. Stakeholders who are registered on IMS server receives notification of any change in patient’s health through messages. This data is very helpful to translate changes in the physiological condition of an individual. If the value of the vital sign is reached above the threshold value then automatic call will be generated and the necessary action can be taken thereof. For instance, the rapid increase in a patient’s heart rate can be symptomatic of a heart attack [22].

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Algorithm for Auto Alerts Generation 1. Setup profile for patient setImsDomain: 192.168.49.214 setImsLocalInterface: 192.168.49.214 setImsLocalPort: 5060 setImsSecretKey: bob setImsPCSCFAddress: 192.168.54.149 setImsPCSCFPort: 4060 setImsPrivateID: [email protected] setImsPublicID: sip\:[email protected] setImsSourceID: [email protected] setInitialConditions: respective of patient

2. Get the critical conditions of the patient 3. Connector to the patient for dynamic Data from the sensors 4. Get the Source ID to send the alerts 5. Instance of IMS Connector 6. Runs the Connector after every 5 seconds 7. Check if the conditions are critical to the patient 8. Alert to the SourceID if something wrong with the patient 9. Gets the IMSconnector and Alerts the Source 10. Doctors, family members are getting alerts if patient’s condition is critical. function run(){ criticalCondition := getInitialConditions(). patient := patient() sourceID := getImsSourceID cui := ConsoleUEx() while(cui.sleep(5000)){ if ( critical(Patitent.Heartrate,criticalCondition) ){ alert(cui, Patient.Name, Patient.Heartrate, SourceID) }}} function alert(ConsoleUEx, Name, Rate, Source){ ConsoleUEx.start("msg "+Name+"\tCRITICAL: "+rate) }

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4 Results and Discussions The main purpose of this research work is to send the real-time data of patients to the doctors and family members, so that there is no delay in patient’s treatment. For that purpose, IMS is used as transport platform to exchange the data between doctors and patients. IMS clients play a great job to retrieve the patient’s data but the existing open-source platforms like UCT IMS client or Boghe cannot read the data. So to overcome this problem, Nubomedia IMS connector is used to retrieve the patient’s data and to get registered on the IMS server as shown in algorithm. Wire shark is open-source software. It checks registration of user through NUBOMEDIA IMS Connector. In Fig. 4, Wireshark analyzes and captures the network packets and shows the packet information in detail by using SIP filter. It shows the IP address which was register on IMS server. The main motive of this work is to show the real-time data and to analyze the condition of the patients. Various vital signs of the patients were monitored. Nubomedia IMS connector which is working as IMS client reads vital signs data of the patients through java code file in every 5 s and compares the value of data with the threshold value. If vital signs value reaches above threshold value then automatic alerts are generated through calls and messages. Family members and doctors who are registered on IMS server get the information of the patients. Figure 5 shows that pulse rate value of various patients read by the Nubomedia IMS connector and if the value is above or equal to threshold, i.e. α, then it will generate alerts to the doctors and family members (the value of α in this case is set to 100).

Fig. 4 User’s registration captured through Wire shark

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Fig. 5 Pulse rate sent through messages

The results depict clear demonstration of how alerts are successfully generated when a critical situation is seen. The developed system can be useful for monitoring patients’ health by collecting health data through smartwatches, apps and sensors. IMS client has been implemented as Nubomedia IMS connector which is customized to read and interpret data. A rule-based interpretation of data is then done to trigger messages to the doctors if the values observed are above certain threshold. The use of mobile as gateways between sensors and IMS has led to the development of integrated solutions. IMS enables the creation of new services, which were not possible or might be costly and complex to implement, such as video sharing. The creation of new multimedia services can be delivered in short span of time which dramatically reduces the cost of the applications. IMS has such attributes which make it appropriate for choosing as a transport platform. The designed IMS platform can be extended to contribute in numerous fields like disaster management, healthcare system, emergency ambulance call setup by reading live data streams, monitoring and issuing notifications also.

5 Conclusion and Future Scope Internet of Things can be utilized in clinical consideration where patient’s physiological status requires close consideration. IMS is the de-facto standard for IPbased multimedia services. IMS can extraordinarily upgrade its services provisioning capacities and open the entryway to a wide scope of novel multimedia services. The proposed work focuses to contribute in the field of health, by using IMS. The main aim is to generate automatic alerts and to send the real-time data of the patient to the doctors automatically, so that there is no delay in their treatment. The automatic calls to ambulance will also be possible by using IMS and IoT. In disaster relief camp, the health of the patient can be measured and send to the doctors by the emergency team management. Interoperability is going to be a big issue in the coming years over the heterogeneous environment of service providers. To address this, machine

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to machine communication protocols can be implemented for the intercommunication of all the involved devices including sensors at IMS platform in future. IMS has some security issues from various communication protocols such as SIP and RTP. Therefore, in future, progressively strong and secure models are also required for these protocols to shield IMS services and resources from these threats.

References 1. Pantelopoulos A, Bourbakis NG (2010) A survey on wearable sensor-based systems for health monitoring and prognosis. In: IEEE transactions on systems, man and cybernetics Part C: applications and reviews 2. Doukas C, Pliakas T, Maglogiannis I (2010) Mobile healthcare information management utilizing Cloud Computing and Android OS. In: 2010 annual international conference of the IEEE engineering in medicine and biology society, EMBC’10 3. Majumder S, Mondal T, Deen MJ (2017) Wearable sensors for remote health monitoring. Sensors (Switzerland) 4. Mosenia A, Sur-Kolay S, Raghunathan A, Jha NK (2017) Wearable medical sensor-based system design: a survey. IEEE Trans. Multi-Scale Comput Syst 5. Tsakalakis M, Bourbakis NG (2014) Health care sensor - Based systems for point of care monitoring and diagnostic applications: a brief survey. In: 2014 36th annual international conference of the IEEE engineering in medicine and biology society, EMBC 2014 6. Hanen J, Kechaou Z, Ben Ayed M (2016) An enhanced healthcare system in mobile cloud computing environment. Vietnam J Comput Sci 7. Christophe Gourraud SA (2007) Using IMS as a service framework. IEEE Veh Technol Mag 4–11 8. Sood SK, Mahajan I (2017) Wearable IoT sensor based healthcare system for identifying and controlling chikungunya virus. Comput Ind 9. Jaiswal K, Sobhanayak S, Mohanta BK, Jena D (2017) IoT-cloud based framework for patient’s data collection in smart healthcare system using raspberry-pi. In: 2017 international conference on electrical and computing technologies and applications, ICECTA 2017 10. El Barachi M, Alfandi O (2013) The design and implementation of a wireless healthcare application for WSN-enabled IMS environments. In: 2013 IEEE 10th consumer communications and networking conference, CCNC 2013 11. Islam SMR, Kwak D, Kabir MH, Hossain M, Kwak KS (2015) The internet of things for health care: a comprehensive survey. IEEE Access 12. Hassanalieragh M et al (2015) Health monitoring and management using Internet-of-Things (IoT) Sensing with cloud-based processing: opportunities and challenges. In: Proceedings 2015 IEEE international conference on services computing, SCC 2015 13. Fergus P, Kifayat K, Cooper S, Merabti M, El Rhalibi A (2009) A framework for physical health improvement using wireless sensor networks and gaming. In: 2009 3rd international conference on pervasive computing technologies for healthcare - Pervasive Health 2009, PCTHealth 2009, vol 10, no 1, pp 1–6 14. Garcia B, Gallego M, Lopez L, Carella GA, Cheambe A (2016) NUBOMEDIA: an elastic PaaS enabling the convergence of real-time and big data multimedia. In: Proceedings - 2016 IEEE International Conference Smart Cloud, SmartCloud 2016, pp 45–56 15. Padilla JEV, Lee JO, Kim JH (2013) A M2M horizontal services platform implementation over IP multimedia subsystem (IMS). In: 15th Asia-Pacific network operations and management symposium: “Integrated Management of Network Virtualization”, APNOMS 2013, pp 1–3 16. El Barachi M, Kadiwal A, Glitho R, Khendek F, Dssouli R (2008) An architecture for the provision of context-aware emergency services in the IP multimedia subsystem. In: IEEE vehicular technology conference

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17. Agrawal P, Yeh JH, Chen JC, Zhang T (2008) IP multimedia subsystems in 3GPP and 3GPP2: overview and scalability issues. IEEE Commun Mag 2(1):4–11 18. Abu-Lebdeh M, Sahoo J, Glitho R, Tchouati CW (2016) Cloudifying the 3GPP IP multimedia subsystem for 4G and beyond: a survey. IEEE Commun Mag 54(1):91–97 19. Magalotti D et al (2015) Design and implementation of a wireless intelligent personal sensor node for the dosimetry of interventional radiology operators. In: Conference record - IEEE instrumentation and measurement technology conference 20. Petrakis EGM, Sotiriadis S, Soultanopoulos T, Renta PT, Buyya R, Bessis N (2018) Internet of Things as a Service (iTaaS): challenges and solutions for management of sensor data on the cloud and the fog. Internet of Things 21. Basatneh R, Najafi B, Armstrong DG (2018) health sensors, smart home devices, and the internet of medical things: an opportunity for dramatic improvement in care for the lower extremity complications of diabetes. J Diabetes Sci Technol 22. Milenkovi´c A, Otto C, Jovanov E (2006) Wireless sensor networks for personal health monitoring: issues and an implementation. Comput Commun 23. Lomotey RK, Deters R (2014) Mobile-based medical data accessibility in mHealth. In: Proceedings - 2nd IEEE international conference on mobile cloud computing, services, and engineering, MobileCloud 2014 24. De D, Mukherjee A, Sau A, Bhakta I (2017) Design of smart neonatal health monitoring system using SMCC. Healthc Technol Lett 25. Nkosi MT, Mekuria F (2010) Cloud computing for enhanced mobile health applications. In: Proceedings - 2nd IEEE international conference on cloud computing technology and science, CloudCom 2010 26. Qureshi B, Tounsi M (2006) Abluetooth enabled mobile intelligent remote healthcare monitoring system in Saudi Arabia: analysis and design issues. In: 18th national computer conference

DPL Model for Hyperthermia Treatment of Cancerous Cells Using Laser Heating Technique: A Numerical Study G. C. Shit

and Amal Bera

Abstract We have numerically simulated the temperature distribution in the living biological tissues for more effective thermal treatment of tumor cells based on the DPL (Dual Phase Lag) model of bioheat transfer equation having a laser heat source. The temperature distribution from the center of the spherical tissue towards the outer surface (taken as skin surface) is simulated. The numerical solutions of the DPL model for one-dimensional bioheat transfer equation along with laser heating conditions are obtained by using the finite difference method. The effects of the phase lag of the heat flux and the temperature gradient, the blood perfusion rate, thermal conductivity, and the laser heating parameter have been presented. The study shows that the phase lags and thermal conductivity have a greater influence on the temperature distribution for hyperthermia treatment in cancer therapy. Keywords Bioheat transfer · Dual phase lag · Laser heating · Finite difference method

1 Introduction Laser radiation has most widely been used in the field of medical sciences for diagnostic and therapeutic applications. Most of the clinical treatments with laser radiation such as surgery, angioplasty, hyperthermia of tumors [1, 2], burn injury evaluation [3, 4], cryopreservation, and laser tissue soldering are concerned with the thermal effects. Various effects of laser interaction with biological tissues and thermal effects [5] are of special importance because these effects are very complex and result in three distinct phenomena, namely, conversion of light to heat, transfer of heat and the tissue reaction, which are ultimately related to the temperature distribution in time. Pennes in 1948 [6], proposed the bioheat transfer model. He developed this model for predicting heat transfer in the human biological tissues. Later on, Klinger [7], G. C. Shit (B) · A. Bera Department of Mathematics, Jadavpur University, Kolkata 700032, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 D. Giri et al. (eds.), Proceedings of the Sixth International Conference on Mathematics and Computing, Advances in Intelligent Systems and Computing 1262, https://doi.org/10.1007/978-981-15-8061-1_28

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Wulff [8], Chen and Holmes [9], proposed different bioheat transfer models. Several researchers [10–14], have investigated the thermal behavior of biological tissues and most of them were modeled by using the Pennes bioheat transfer equation based on the Classical Fourier’s law q(r, t) = −k∇T (r, t),

(1)

where q is heat flux, k is thermal conductivity , and T is the temperature of the tissue. But an infinite speed of wave propagation occurred in the Fourier’s law which is physically ill-disposed. Due to this impossible fact, Cattaneo [15], Vernotte [16], Weymann [17], have suggested a modified heat flux model to associate a finite speed of wave propagation by introducing a phase lag. According to them, the thermal wave model of bioheat transfer has been proposed based on single phase lagging (SPL) with constitutive relation q(r, t + τq ) = −k∇T (r, t).

(2)

Tzou [18] conjectured the same type of equation by adding two time constants, namely the phase lags of temperature gradient and heat flux, in the Fourier heat flux model. This model is termed as DPL (dual phase lag) model, lumps the microstructural effects into the delayed response in time. To consider the effect of microstructural interactions on the fast-transient process of heat transfer, the DPL model accounts for both the temporal and the spatial heat transfer in one formulation that takes in the form [18, 19]: q(r, t + τq ) = −k∇T (r, t + τT ),

(3)

where T is the temperature, k the thermal conductivity, q the heat flux, t the time, and r the space variable. τq is the phase lag of the heat flux, and τT is the phase lag of the temperature gradient. The heat flux precedes the temperature gradient for τq < τT , on the other hand, the temperature gradient precedes the heat flux for τq > τT . With the DPL model, not only the temperature gradient can precede the heat flux, but also the heat flux may precede the temperature gradient. Many researchers have investigated several heat transfer problems using DPL model. Antaki [20] studied transient heat conduction in a semi-finite thin slab with surface heat flux and in microscale during a short time. Zhang [21] obtained the bioheat equations in the living tissue based on a nonequilibrium heat transfer model. In their work, the phase lag times are expressed in terms of the properties of blood and tissue, the interphase convective heat transfer coefficient, and the blood perfusion rate. Xu et al. [22] constructed three kinds of DPL model. They showed that there exists a very larger discrepancy among the predictions of the Pennes model, thermal wave model, and dual phase lag model, while different dual phase lag models give similar results. Poor and Moradi [23] analytically studied the dual phase lag bioheat transfer equation with a constant and periodic train of heat flux conditions at the skin surface. They have discussed the transient temperature responses for constant and

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time-dependent boundary conditions. Liu and Chen [24] treated dual phase lag (DPL) model to predict the temperature rise behavior in a two-layer concentric spherical region for hyperthermia treatment of tumor by applying the magnetic field. They determined the temperature response in the spherical tissue region by employing the Gaussian elimination algorithm and the numerical inversion of the Laplace transform. Afrin et al. [25] examined the thermal damage induced by laser irradiation on the living biological tissues based on the generalized dual phase lag (DPL) bioheat model with imposing zero heat flux at a depth of the tissue. Singh and Kumar [26] analyzed the effect of phase lags of heat flux and temperature gradient using DPL model in the three-layer skin tissue for freezing procedure during cryosurgery by using constant temperature at the outer surface of the tissue and with keeping the adiabatic temperature at a depth of the tissue. Kumar et al. [27] derived a nonlinear dual phase lag (DPL) bioheat transfer model based on the metabolic heat generation rate depending on the temperature to analyze the heat transfer phenomena in living tissues during thermal ablation treatment. They obtained a numerical solution of the nonlinear problem by using a method that combines the essence of the Runge–Kutta method with the finite difference scheme. Peng and Chen [28] have studied a laser heating problem considering the Hybrid differential transformation method and the finite difference method. Moreover, Lin et al. [29] have presented a new approach for characterizing the thermal behavior of hyperthermia treatment. They performed an optic- fiber laser irradiation system measurement on the bio-heat transfer of finite and infinite heat propagation in the living tissues, together with constant and exponential heating in the spherical coordinate system by the Differential Transformation Method. In this work, we have to obtain a numerical solution of the DPL bioheat transfer equation with the effects of thermal phase lags of heat flux and temperature gradient subject to the laser heating conditions at the center of the biological tissues containing a tumor. The finite difference method has been employed to compute the numerical solution of the DPL bioheat equation of spherical one-dimensional with laser heating boundary conditions at a depth of the tumor, while the skin surface temperature kept at an adiabatic temperature. Our study aims to investigate the temperature distribution in living biological tissues when laser irradiation is used for the treatment. Finally, we determined the optimal choice of the parameters so that no damage will occur for healthy tissues during hyperthermia treatment of cancer therapy.

2 Mathematical Model We consider the living biological tissue as a nonhomogeneous material in spherical shape which contains a tumor and a laser heat source is applied at the center of the tumor. The sphere is considered of finite radius L of which r = 0 is the center of the sphere and r = L is the position of the tissue adjacent to the skin surface. In order to obtain the temperature distribution in this finite domain, we formulate our problem using 1-D Pennes’ bioheat model given by

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ρC

∂q 2 ∂T =− − q + ωb ρb Cb (Tb − T ) + qm + qr , ∂t ∂r r

(4)

where ρ, C, and T, respectively denote the density, specific heat, and temperature of the tissue, ωb , ρb , Cb , and Tb are the perfusion rate, density, specific heat, and temperature of blood, qm the metabolic heat generation, qr the heat source for surface heating. The first order Taylor series expansions of both q and T of the Eq. (3), gives rise to a linear version of DPL model as   ∂T ∂q ∂2 T = −k + τT . (5) q + τq ∂t ∂r ∂t∂r Invoking the assumption of constant blood temperature and τT = τq = 0, Eq. (5), reduces to a classical Fourier’s law that requires immediate response between the temperature gradient and heat flux. In the absence of phase lag time (τT ) for temperature gradient , Eq. (5), reduces to the thermal wave model. Substituting Eq. (5), into the energy conservation equation (4), leads to the heat conduction equation  ρC

∂2 T ∂T + τq 2 ∂t ∂t

 =k

   1 ∂ ∂2 T 2 ∂T r + Wb Cb (Tb − T ) + qm + qr + τ T r 2 ∂r ∂r ∂t∂r   ∂T ∂qm ∂qr +τq −Wb Cb + + , ∂t ∂t ∂t

(6)

where Wb = ωb ρb . From this equation Eq. (6), it is evident that τT = τq = 0 corresponds to the simply Pennes’ bioheat transfer equation. In our model, we assumed that the metabolic heat generation qm = constant and the external heat source qr = 0 . Under this assumption, the Eq. (6), reduces to      1 ∂ ∂2 T ∂2 T ∂T ∂T 2 ∂T = k r + τ + Wb Cb (Tb − T ) + qm − τq Wb Cb . ρC τq + T ∂t 2 ∂t r 2 ∂r ∂r ∂t∂r ∂t

as

(7)

For initial steady state temperature Ti (r, 0) in the tissue, the Eq. (7), can be written    1 ∂ 2 ∂Ti r + Wb Cb (Tb − Ti ) + qm = 0. (8) k 2 r ∂r ∂r

Now subtraction of Eq. (8), from Eq. (7), gives    ∂2θ ∂2θ ∂θ 1 ∂ 2 ∂θ + W r + τ , + (ρC + τ W C ) C θ = k q b b b b T ∂t 2 ∂t r 2 ∂r ∂r ∂t∂r (9) where the temperature difference θ(r, t) = T (r, t) − Ti (r, 0) is denoted as the elevation of temperature in dimensional form (Fig. 1). The initial conditions for the present problem are taken as ρCτq

DPL Model for Hyperthermia Treatment of Cancerous Cells …

361 −βt

Fig. 1 A physical sketch of the spherical tumor with a laser heat source at the center

θ(0,t)=100e

,(β>0)

r

θ(r, 0) = 0, ∂θ(r, 0) =0 ∂t

(10)

θ(0, t) = 100e−βt , ∂θ(L , t) =0 ∂r

(12)

(11)

and the boundary conditions are

(13)

where β > 0 is called a laser heat source parameter or exponential decaying parameter.

3 Computational Scheme In order to solve the partial differential equation (9), subject to the boundary conditions (12) and (13), and the initial conditions (10) and (11) we have developed a finite difference numerical scheme. The partial derivatives involved in the above mentioned equations are discretized using the forward difference scheme in time directions and central difference scheme in space directions (FTCS). To discretize the domain, we first divide the intervals 0 ≤ r ≤ L into m number of subintervals L and Δr is the spacing gap in the space direction. such that m = Δr The numerical values of θ at the mesh point ri (i = 0, 1, 2, ...., m) at any time tn = n ∗ Δt (n = 0, 1, 2, .....) are denoted by θi n . As an illustrative example, the first and second order derivatives are replaced by the following finite differences: u n −u n ( ∂u )n ∼ i+12Δr i−1 + O(Δr 2 ); ∂r i = n n n 2 i+1 ∼ u i−1 −2u i +u + O(Δr 2 ); ( ∂∂ru2 )in = (Δr )2 n+1 u −u n and ( ∂u )n ∼ i Δt i + O(Δt); where i = 1, 2, ...m − 1 and n = 1, 2, · · · ∂t i = By neglecting the truncation error terms the discretized form of Eq. (9), can be expressed in the following system of equations as

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G. C. Shit and A. Bera

   θin+1 − θin θin+1 − 2θin + θin−1 ρCτq + (ρC + τq Wb Cb ) + Wb Cb θin Δt 2 Δt    n n−1 n−1 n  n n θi+1 − 2θin + θi−1 − 2θin + θi−1 − 2θin−1 + θi−1 θi+1 kτT θi+1 =k − + Δr 2 Δt Δr 2 Δr 2    n  n−1 n−1 n n n θi+1 − θi−1 − θi−1 2kτT θi+1 2k θi+1 − θi−1 + − + ri 2Δr ri Δt 2Δr 2Δr 

(14) multiplying by Δt 2 and rearranging the terms we obtain 

ρCτq + Δt (ρC + τq Wb Cb ) θin+1 = 2ρCτq + Δt (ρC + τq Wb Cb ) − Wb Cb Δt 2 θin    n n−1 n−1 n  n − 2θ n + θ n θi+1 θi+1 − 2θin + θi−1 − 2θin−1 + θi−1 θi+1 i i−1 2 + kτT Δt +kΔt − Δr 2 Δr 2 Δr 2     n−1 n − θn n − θn θn−1 − θi−1 kΔt 2 θi+1 kτT Δt θi+1 i−1 i−1 + + − ρCτq θin−1 , − i+1 ri Δr ri Δr Δr 

(15) The initial and boundary conditions in finite difference scheme are formulated as

θin+1

=

θin θ0n

θi0 = 0 (i = 0, 1, 2, ...., m, n = 0, 1, 2, .....)

= 100e−βtn n θmn = θm−1 .

(16) (17) (18) (19)

4 Results and Discussion The DPL model of bioheat transfer equation with the effect of two types of phase lag times has been modeled along with laser heating boundary conditions at the center of the spherical tissue containing a tumor. The numerical solution scheme of the present problem is expressed in the previous section along with the boundary conditions. To compute the numerical solutions, we have assigned the numerical values to the physical variables appearing in the present model as (cf- [29, 30]), given in Table 1. Figure 2 gives the temperature distribution along the radius of the tissue at different time span t = 5s to t = 20s with unequal (τq = τT ) phase lags. We observed from this figure that the heat wave increases along the radius of the spherical tumor when time progresses. The temperature gradually decreases as we have applied exponential decaying exponential function of the laser heating condition. The temperature

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Table 1 Typical values of the physical variables Physical properties Value (unit) Density of tissue (ρ) Density of blood (ρb ) Thermal conductivity of tissue (k) Blood perfusion rate (Wb ) Specific heat of tissue (C) Specific heat of blood (Cb ) Phase lag of heat flux (τq ) Phase lag of temperature gradient (τT ) Radius of the tumor tissue (L) Laser heat source parameter (β)

1000 (Kg/m3 ) 1060 (Kg/m3 ) 0.1–1.0 (W/m ◦ C) 0.5–3.0 Kg/m3 s 4200 (J/kg ◦ C) 4200 (J/kg ◦ C) 5–30 (s) 0.1–10 (s) 0.01208 (m) 0.01–0.1

is greater within a short period of time applying a laser heat source and releases temperature within a small distance of the tumor. More heating duration causes the release of heat from the depth of the tumor slowly and keeps at a normal temperature without damaging the normal tissues. Figure 3 represents the temperature distribution along the radius of the spherical tissue containing a tumor for different phase lags τT due to temperature gradient. For a greater laser heating duration, the heat wave arrived near the length of 0.0015 meters in the radial direction of tumor tissue. The lower phase lag duration due to temperature gradient has an increasing temperature at a depth of the tissue and release temperature at a distance of 0.0025 meters. Therefore, the tumor region can be affected by the heating procedure without damaging the surrounding healthy tissue. For higher values of τT has more flattening of the temperature distribution. Figure 4 indicates that the reduction in temperature is observed at a certain distance from the depth of the tissue with increasing values of the phase lag of heat flux τq , while the trend is reversed at the center of the tissue. This fact lies within the absorption of heat due to laser heating of living tissues. Thus, the controlled temperature due to the application of laser heating is helpful for therapeutic treatment in tumor tissue. Figure 5 represents the variation of a laser heat source parameter β on the tissue temperature in function of radial distance r . We notice that the amplitude of the tissue temperature is lower for higher values of β. Also, for any value of β, the heat wave terminates near the end of the radius of spherical tissue. Therefore, the values of β affect mainly the central region of the living tissue where a tumor is present, up to a certain time duration of laser heating when τq >τT . The hyperthermic temperature can be achieved by controlling the laser heat source parameter, and it can optimize the damage of healthy tissue surrounded by the tumor. The temperature distribution along the radial direction for different thermal conductivity of tissue is shown in Fig. 6. The rate of decrease of the tissue temperature varies proportionally to the rate of decreasing the values of thermal conductivity. Hence, the thermal conductivity of tissue also plays a crucial role in the enhance-

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G. C. Shit and A. Bera 70

t=5 s t=10 s t=15 s t=20 s

Temperature ( θ o C)

60

50

40

30

20

10

0

0

0.5

1

1.5

2

2.5

3 × 10-3

Distance r (m)

Fig. 2 Temperature distribution along the radius of the tissue at different time for ρ = 1000 kg/m3 , C = 4200 J/kg ◦ C, Cb = 4200 J/kg ◦ C, Wb = 0.5 kg/m3 .s, k = 0.5 W/m ◦ C, τq = 20 s, τT = 0.1 s, β = 0.1 18

τ T =0.1

16

τ T =1

Temperature ( θ o C)

14

τ T =5

12 10 8 6 4 2 0

0

0.5

1

1.5

Distance r(m)

2

2.5

3 × 10-3

Fig. 3 Temperature distribution along the radius of the tissue for different phase lags due to temperature gradient with ρ = 1000 kg/m3 , C = 4200 J/kg ◦ C, Cb = 4200 J/kg ◦ C, Wb = 0.5 kg/m3 .s, k = 0.5 W/m ◦ C, τq = 30 s, t = 20 s, β = 0.1

DPL Model for Hyperthermia Treatment of Cancerous Cells …

365

40

τ q =5

35

τ q =10

Temperature ( θ o C)

30

τ q =20 τ q =30

25 20 15 10 5 0

0

0.5

1

1.5

Distance r (m)

2

2.5

3 × 10-3

Fig. 4 Temperature distribution along the radius of the tissue for different phase lags due to heat flux with ρ = 1000 kg/m3 , C = 4200 J/kg ◦ C, Cb = 4200 J/kg ◦ C, Wb = 0.5 kg/m3 .s, k = 0.5 W/m ◦ C, τ = 1 s, t = 20 s, β = 0.05 T 100

β=0.0 β=0.025 β=0.05 β=0.1

90

Temperature ( θ o C)

80 70 60 50 40 30 20 10 0

0

0.5

1

1.5

Distance r (m)

2

2.5

3 × 10-3

Fig. 5 Temperature distribution along the radius of the tissue for different values of β when ρ = 1000 kg/m3 , C = 4200 J/kg ◦ C, Cb = 4200 J/kg ◦ C, Wb = 0.5 kg/m3 .s, k = 0.5 W/m ◦ C, τq = 20 s, τT = 1 s, t = 20 s

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G. C. Shit and A. Bera 40

κ=0.1 κ=0.2 κ=0.5 κ=1

35

Temperature ( θ o C)

30 25 20 15 10 5 0

0

0.5

1

1.5

2

2.5

3

Distance r (m)

× 10-3

Fig. 6 Temperature distribution along the radius of the tissue for different values of thermal conductivity k when ρ = 1000 kg/m3 , C = 4200 J/kg ◦ C, Cb = 4200 J/kg ◦ C, Wb = 0.5 kg/m3 .s, τq = 20 s, τT = 1 s, t = 20 s, β = 0.05 35

τ T =0.1

30

τ T =5 τ T =10

Temperature ( θ o C)

25

20

15

10

5

0

0

10

20

30

40

50

60

70

80

90

100

Time t(s)

Fig. 7 Temperature distribution at r = 0.000025 m for ρ = 1000 kg/m3 , C = 4200 J/kg ◦ C, Cb = 4200 J/kg ◦ C, Wb = 0.5 kg/m3 .s, τq = 20 s, β = 0.025

ment of the bioheat transfer on the living tissues. The required value of thermal conductivity for arriving heat wave near half of the radius of the living tissue is 0.2 2 W/m ◦ C.

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367

14

τ q =5

Temperature ( θ o C)

12

τ q =10 τ q =15

10

τ q =20

8

6

4

2

0

0

10

20

30

40

50

60

70

80

90

100

Time t (s)

Fig. 8 Temperature distribution at r = 0.000025 m for ρ = 1000 kg/m3 , C = 4200 J/kg ◦ C, Cb = 4200 J/kg ◦ C, Wb = 0.5 kg/m3 .s, τT = 1 s, β = 0.025 50 45

β=0 β=0.025 β=0.05 β=0.1

Temperature ( θ o C)

40 35 30 25 20 15 10 5 0

0

10

20

30

40

50

60

70

80

90

100

Time t(s)

Fig. 9 Temperature distribution at r = 0.000025 m for ρ = 1000 kg/m3 , C = 4200 J/kg ◦ C, Cb = 4200 J/kg ◦ C, Wb = 0.5 kg/m3 .s, τq = 20 s, τT = 1 s, t = 20 s

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Figures 7 , 8, 9 illustrate the temperature distribution at the center of the tumor with the heating duration t for different values of τT , τq , and β. Figure 7 shows that for a fixed value phase lag τq = 20s. The temperature is higher for the lower value of phase lag τT and decreases its peak value with a shorter heating duration when the values of τT increased. However, for the longer heating duration, the temperature increases with a rise in the phase lag τT due to temperature gradient, as shown in Fig. 8. Similar observations can also be drawn in the case of phase lag time due to the heat flux of the tissue temperature. The maximum temperature is over the therapeutic value for hyperthermia treatment, i.e. 43 ◦ C and appears around the center of the tumor. Some of the regions of the tumor have the temperature below 43 ◦ C, which characterizes that these heating processes were unable to destroy the tumor completely. Although the change of phase lag arising out of temperature gradient does not increase the temperature of the tumor, but can lead to a decrease of surrounding tissue temperature. It is clear from these figures that when τq >τT , the rate of temperature increase is fast in early heating time near about 5 s, and then it gradually decreases. The difference of the characteristics of the laser heating parameter β is presented in Fig. 9. The temperature increases gradually with heating duration t when β = 0. But for the values of β > 0, the temperature initially increases up to heating duration 10s to 20s and then gradually decreases to maintain a normal temperature of healthy surrounding tissues. The decrease is greater for higher values of β = 0.1. Using laser heating technique, the targeted tumor tissue can be heated by minimizing the damage of normal healthy tissues. For laser heating with lower values of β causes more heating of entire spherical tissues with large heating duration. This model can predict/optimize the use of a laser heat source. Thus, the laser heating technique may throw some light toward the implementation of the hyperthermic protocol.

5 Conclusion We have studied numerically the behavior of dual phase lag (DPL) on the bioheat transfer in the living biological tissues for hyperthermic treatment using a laser heat source. We have derived the DPL model based on the 1-D Pennes bioheat equation in the spherical coordinates and then presented its numerical scheme to solve the temperature distribution. We observed that the phase lag due to heat flux is more important to increase the temperature of the targeted tissue. The decaying parameter also has an essential controlling effect on the heating of tumor cells while using a laser heat source technique. Thus, our study may have important applications in biomedical physics and provide useful information for further improvements of the model by taking into account precisely the blood perfusion rate.

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References 1. Roemer RB (1999) Engineering aspects of hyperthermia therapy. Annu Rev Biomed Eng 1:347–376 2. Minkowycz WJ, Sprrow EM, Abraham JP (2009) Advances in numerical heat transfer, vol 3. CRC Press, Boca Raton, USA 3. Ng EYK, Chua LT (2001) Quick numerical assessment of skin burn injury with spreadsheet in PC. J Mech Med Biol 1:1–10 4. Ng EYK, Chua LT (2002) Comparison of one and two dimensional programmes for predicting the state of skin burns. Burns 28:27–34 5. Dua R, Chakraborty S (2004) A novel modeling and simulation technique of photo-thermal interactions between lasers and living biological tissues undergoing multiple changes in phase. Comput Biol Med 35:447–462 6. Pennes HH (1948) Analysis of tissue and arterial blood temperature in the resting forearm. J Appl Physiol 1:93–122 7. Klinger HG (1974) Heat transfer in perfused biological tissue. I. General theory. Bull Mathe Biol 36:403–415 8. Wulff W (1974) The energy conservation equation for living tissues. IEEE Trans-Biomed Eng 21:494–495 9. Chen MM, Holmes KR (1980) microvascular contributions in tissue heat transfer. Ann New York Acad Sci 335:137–150 10. Shih TC, Yuan P, Lin WL (2007) Hong Sen Kou, Analytical analysis of the Pennes bioheat transfer equation with sinusoidal heat flux condition on skin surface. Med Eng Phys 29:946–953 11. Deng ZS, Liu J (2002) Analytical study on bioheat transfer problems with spatial or transient heat on skin surface or inside biological bodies. J Biomed Eng 124:638–649 12. Diller KR (1992) Modeling of bioheat transfer processes at high and low temperatures. Adv Heat Trans 22:157–357 13. Bhowmik A, Singh R, Repaka R, Mishra SC (2013) Conventional and newly developed bioheat transport models in vascularized tissues: A review. J Therm Biol 38:107–125 14. Nakayama A, Kuwahara F (2008) A general bioheat transfer model based on the theory of porous media. Int J Heat Mass Transf 51:3190–3199 15. Cattaneo C (1958) A form of heat conduction equation which eliminates the paradox of intantaneous propagation. Compte Rendus 247:431–433 16. Vernotte P (1961) Some possible complications in the phenomena of thermal conduction. Compte Rendus 252:2190–2191 17. Weymann HD (1967) Finite speed of propagation in heat conduction, diffusion, and viscous shear motion. Am J Phys 35:488–96 18. Tzou DY (1996) Macro- to microscale heat transfer: the lagging behavior. Taylor and Francis Washington, DC 19. Tzou DY (1995) The generalized lagging response in small-scale and high-rate heating. Int J Heat Mass Transf 38:3231–3240 20. Antaki PJ (2000) Effect of dual-phase-lag heat conduction on ignition of a solid. J Thermophys Heat Transfer 14:276–78 21. Zhang Y (2009) Generalized dual-phase lag bioheat equations based on nonequilibrium heat transfer in living biological tissues. Int J Heat Mass Transf 52:4829–834 22. Xu F, Lu T, Seffen K (2008) Dual-phase-lag model of skin bioheat transfer. International Conference on BioMedical Engineering and Informatics 1 23. Poor HZ, Moosavi H, Moradi A (2016) Analysis of the dual phase lag bio-heat transfer equation with constant and dime dependent heat flux conditions on skin surface. J Therm Sci 20:1457– 1472 24. Liu KC, Chen HT (2009) Analysis for the dual-phase-lag bio-heat transfer during magnetic hyperthermia treatment. Int J Heat Mass Transf 52:1185–1192

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25. Afrin N, Zhou J, Zhang Y, Tzou DY, Chen JK (2012) Numerical simulation of thermal damage to living biological tissues induced by laser irradiation based on a generalised dual phase lag model. Numerical Heat Transfer 61:483–501 26. Singh S, Kumar S (2014) Numerical study on triple layer skin tissue freezing using dual phase lag bio-heat model. Int J Therm Sci 86:12–20 27. Kumar P, Kumar D, Rai KN (2016) Non-linear dual-phase-lag model for analyzing heat transfer phenomena in living tissues during thermal ablation. J Therm Biol 60:204–212 28. Peng HS, Chen CL (2011) Application of hybrid differential transformation and finite difference method on the laser heating problem. Numer Heat Transfer, Part A 59:28–42 29. Lin SY, Lai HY, Chen CK (2011) Hyperthermia treatment for living tissue with laser heating problems by the differential transformation method. Numer Heat Transfer, Part A 60:499–518 30. Liu KC, Chen HT (2009) Analysis for the dual-phase-lag bio-heat transfer during magnetic hyperthermia treatment. Int J Heat Mass Trans 52:1185–1192

Unitary Equivalence of Quantum States in a Bipartite System Arnab Patra, Amit Shrivastava, Rohit Sharma, and P. D. Srivastava

Abstract If two quantum states are unitarily equivalent then their von Neumann entropies are same. Converse statement also holds, which was proved by Kan He et al. [Applied Mathematics Letters, 2012;25(8):1391–1393 [1]]. In this paper, we extend it to bipartite quantum system and prove a sufficient and necessary condition of unitary equivalence of quantum states associated with the von Neumann entropy of the composite states. A sufficient condition is also provided which involves the von Neumann entropy of the quantum state of the component subsystems. Bipartite quantum system is of interest in quantum cryptography and quantum information processing. Quantum key distribution also uses a bipartite quantum system shared between two parties namely Alice and Bob.

Keywords Quantum states · Unitary equivalence · Von Neumann entropy

1 Introduction Any quantum mechanical system associates with a Hilbert space which is known as the state space of the system. A quantum state is completely described by a state vector which is a unit vector in a Hilbert space. An alternative mathematical formulation for this is the density matrix approach. It is well-known that an operator defined over a Hilbert space is a density matrix if and only if it is a positive operator and trace is one [2, Theorem 2.5]. In the quantum mechanical description of physics, systems A. Patra (B) · A. Shrivastava · R. Sharma Advanced Numerical Research and Analysis Group, DRDO, Hyderabad 500058, India e-mail: [email protected] A. Shrivastava e-mail: [email protected] R. Sharma e-mail: [email protected] P. D. Srivastava Department of Mathematics, Indian Institute of Technology Bhilai, GEC Campus, Chhattisgarh 492015, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 D. Giri et al. (eds.), Proceedings of the Sixth International Conference on Mathematics and Computing, Advances in Intelligent Systems and Computing 1262, https://doi.org/10.1007/978-981-15-8061-1_29

371

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are composed of two or more subsystems. If a quantum system is composed of two subsystems namely A and B then the state space of a composite physical system is the tensor product of the state spaces of the component subsystems. Also, if we have systems A and B, prepared in the states ρ A and ρ B , respectively, then the state of the composite system is given by ρ A ⊗ ρ B . Throughout this paper, let H A and H B are the complex Hilbert Spaces with dim(H A ) = m and dim(H B ) = n associated with the systems A and B, respectively. I A and I B denote the identity operator defined over H A and H B . The Von Neumann entropy of a quantum state ρ is denoted by S(ρ) and defined by S(ρ) = − tr (ρ log2 ρ). where tr (ρ) denotes the trace of a square matrix. Since ρ is a positive operator so it has the spectral decomposition. Therefore, if λ1 , λ2 , · · · , λk are the eigenvalues of ρ then k  λi log2 λi S(ρ) = − i=1

where we take 0 log2 0 = 0 by convention. For more details on entropy we refer to [3] and the references therein. Evolution of a closed quantum system is characterized by unitary transformation. A state ρ1 of the system at time t1 is related to the state ρ2 of the system at time t2 by a unitary transformation U such that ρ2 = U ρ1 U ∗ . In this case, two quantum states ρ1 and ρ2 are said to be unitarily equivalent. If two quantum states ρ1 and ρ2 are unitarily equivalent, then they have the same entropy. The converse of this statement holds true and proved by Kan He et al. [1]. A simpler proof is also given in [4]. In this paper, we prove some necessary and sufficient conditions on the unitary equivalence of two quantum states in a bipartite system.

2 Main Results The following theorem is our main result. Theorem 1 Let ρ A ⊗ ρ B and σ A ⊗ σ B are quantum states on H A ⊗ H B . A necessary and sufficient condition that there exists a unitary operator U such that ρ A ⊗ ρ B = U (σ A ⊗ σ B )U ∗ is     1−λ 1−λ I A ⊗ I B = S λσ A ⊗ σ B + IA ⊗ IB S λρ A ⊗ ρ B + nm nm

(1)

Unitary Equivalence of Quantum States in a Bipartite System

373

holds for all λ ∈ (0, 1). Proof If ρ A ⊗ ρ B and σ A ⊗ σ B are unitary equivalence, then it is easy to deduce the Eq. 1. Hence, the necessary part is proved. To prove sufficient part let     1−λ 1−λ I A ⊗ I B and g(λ) = S λσ A ⊗ σ B + IA ⊗ IB f (λ) = S λρ A ⊗ ρ B + nm nm

for λ ∈ (0, 1). Let the eigenvalues of ρ A and ρ B are x1(A) , x2(A) , · · · , xm(A) , and x1(B) , x2(B) , · · · , xn(B) respectively. Then the eigenvalues of ρ A ⊗ ρ B are xi(A) x (B) j , i = 1, 2, · · · , m, j = 1, 2, · · · , n. as a nm-tuple (x1 , x2 , · · · , xnm ) such that Let us enumerate the eigenvalues xi(A) x (B) j x1 ≥ x2 ≥ · · · , ≥ xnm . Also, we have the relations m 

xi(A) = 1 and

n 

i=1

This implies

j=1

m  n 

xi(A) x (B) j

i=1 j=1

Set ηi = xi − (0, 1)

1 nm

x (B) = 1. j

=

mn 

xk = 1.

k=1

for i = 1, 2, · · · , nm. Then we have

i=1

ηi = 0. Then for λ ∈

   1−λ 1−λ log2 λxi + nm nm k=1     nm  1 1 ληi + log2 ληi + . =− nm nm k=1

f (λ) = −

nm  

mn

λxi +

Also d dλ



f (λ) λ



       ηi ηi 1 1 1 log2 ληi + + + ληi + log2 ληi + nm nm λ nm λ ln2 k=1    nm nm  1 1 ηi − log2 ληi + = 2 nmλ nm λ ln2 k=1 k=1  nm  

1 1 = log2 ληi + . nmλ2 nm

=−

nm  

−

1 λ2

i=1

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Let us introduce the polynomial p1 (λ) =

nm 

ληi +

i=1

d nmλ dλ



f (λ) λ

2

1 nm

of degree at most nm. Hence

 = log2 p1 (λ),

(2)

for all λ ∈ (0, 1). In a similar way, Let the eigenvalues of σ A and σ B are y1(A) , y2(A) , · · · , ym(A) , and y1(B) , y2(B) , · · · , yn(B) respectively, and the eigenvalues of σ A ⊗ σ B are yi(A) y (B) j , i = 1, 2, · · · , m, j = 1, 2, · · · , n. Also, let (y1 , y2 , · · · , ynm ) be the enumeration of the eigenvalues of σ A ⊗ σ B such that y1 ≥ y2 ≥ · · · , ≥ ynm . Then

m  i=1

yi(A) =

n 

y (B) = j

j=1

mn 

yk =

k=1

n m  

yi(A) y (B) = 1. j

i=1 j=1

1 Construct ζi such that ζi = yi − nm for i = 1, 2, · · · , nm. Then similar calculation shows that for all λ ∈ (0, 1)

nmλ2

d dλ



g(λ) λ

mn

i=1 ζi

= 0. A

 = log2 p2 (λ),

(3)

where p2 (λ) is a polynomial of degree at most nm defined by p2 (λ) =

nm  i=1

1 λζi + nm

 .

Since Eq. 1, holds, therefore, Eqs. 2 and 3, imply that p1 (λ) = p2 (λ) for all λ ∈ (0, 1). Hence, the sets of the mn-tuples {x1 , x2 , · · · , xnm } and {y1 , y2 , · · · , ynm } are same. Consequently, the quantum states ρ A ⊗ ρ B and σ A ⊗ σ B have the same spectrum, and therefore, they are unitarily equivalent. In the next result, we provide a sufficient condition for unitary equivalence of two states ρ A ⊗ ρ B and σ A ⊗ σ B . Theorem 2 Two states ρ A ⊗ ρ B and σ A ⊗ σ B are unitarily equivalent if     1−λ 1−λ S λρ A + I A = S λσ A + I A and n n

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    1−λ 1−λ I B = S λσ B + IB S λρ B + n n holds for all λ ∈ (0, 1).



Proof Since S λρ A + 1−λ I A = S λσ A + 1−λ I A holds for λ ∈ (0, 1), then by [1], n n there is a unitary matrix U1 such that ρ A = U1 σ A U1∗ . Similarly, there exists a unitary matrix U2 such that ρ B = U2 σ B U2∗ holds. Consider the matrix U = U1 ⊗ U2 . Then, U is unitary since U1 and U2 are unitary (see [5, Property 2.5]). Now U (σ A ⊗ σ B )U ∗ = (U1 ⊗ U2 )(σ A ⊗ σ B )(U1∗ ⊗ U2∗ ) = [(U1 σ A ) ⊗ (U2 σ B )](U1∗ ⊗ U2∗ ) = (U1 σ A U1∗ ) ⊗ (U2 σ B U2∗ ) = ρ A ⊗ ρB . Hence, the result is proved. Remark 1 The sufficient condition in Theorem 2, on the unitary equivalence of two quantum state in a composite system, involves Von Neumann entropy of single quantum states.

References 1. He K, Hou J, Li M (2012) A von neumann entropy condition of unitary equivalence of quantum states. Appl Math Lett 25(8):1153–1156 2. Nielsen MA, Chuang I (2002) Quantum computation and quantum information 3. Bhatia R (2003) Partial traces and entropy inequalities. Linear Algebra Appl 370:125–132 4. Drnovšek R (2013) The von neumann entropy and unitary equivalence of quantum states. Linear Multilinear Algebra 61(10):1391–1393 5. Regalia PA, Sanjit MK (1989) Kronecker products, unitary matrices and signal processing applications. SIAM Rev 31(4):586–613

Dynamic Self-dual DeepBKZ Lattice Reduction with Free Dimensions Satoshi Nakamura, Yasuhiko Ikematsu, and Masaya Yasuda

Abstract Lattice basis reduction is a mandatory tool to solve lattice problems such as the shortest vector problem (SVP), whose hardness assures the security of lattice-based cryptography. The most famous reduction is the celebrated algorithm by Lenstra-Lenstra–Lovász (LLL), and the block Korkine–Zolotarev (BKZ) is its blockwise generalization. At present, BKZ and its variants such as BKZ 2.0 are a de facto standard reduction algorithm to estimate the security level of lattice-based cryptosystems. Recently, DeepBKZ was proposed as a mathematical improvement of BKZ, in which LLL with deep insertions (DeepLLL) is called as a subroutine alternative to LLL. In this paper, we develop a new self-dual variant of DeepBKZ to obtain a reduced basis. Different from conventional self-dual algorithms, we select suitable free dimensions to reduce primal and dual lattice bases in our variant. We also report experimental results to compare our self-dual DeepBKZ with primal BKZ and DeepBKZ for several random lattice bases. Keywords Lattice problems · Primal and dual lattices · Lattice basis reduction · DeepBKZ

S. Nakamura NTT Secure Platform Laboratories, Musashino, Japan e-mail: [email protected]; [email protected] Graduate School of Mathematics, Kyushu University, Fukuoka, Japan Y. Ikematsu Institute of Mathematics for Industry, Kyushu University, Fukuoka, Japan e-mail: [email protected] M. Yasuda (B) Department of Mathematics, Rikkyo University, Tokyo, Japan e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 D. Giri et al. (eds.), Proceedings of the Sixth International Conference on Mathematics and Computing, Advances in Intelligent Systems and Computing 1262, https://doi.org/10.1007/978-981-15-8061-1_30

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1 Introduction With the development of large-scale quantum computers, it has accelerated to research lattice-based cryptography as a candidate of post-quantum cryptography (PQC) (see [23] for the NIST PQC standardization.) The security of lattice-based cryptography is based on lattice problems such as SVP. Lattice basis reduction is a strong tool in cryptanalysis, and it has been used to estimate the security of latticebased cryptosystems. LLL is the most famous reduction algorithm, and its typical improvements are BKZ and DeepLLL, both proposed by Schnorr and Euchner [19]. At present, BKZ [19] and its variants such as BKZ 2.0 [6] are de facto standard to estimate the security level (e.g., see [2] for security estimation.) Given a blocksize β, BKZ repeatedly calls an SVP oracle in a β-dimensional lattice to find a short lattice vector. In security estimation, it is discussed which blocksizes β are required for BKZ to find a short lattice vector of target norm. In [24], DeepBKZ was proposed as a new improvement, which calls DeepLLL as a subroutine alternative to LLL before every SVP oracle to find a short lattice vector by smaller blocksizes than BKZ. In fact, DeepBKZ with around β = 40 had found new solutions for the SVP challenge [7] in most dimensions from 102 to 127. In this paper, we develop a new self-dual variant of DeepBKZ. The output quality of self-dual algorithms such as slide-reduction [9] and self-dual BKZ [14] is proven to be slightly better than primal algorithms in theory. However, experiments in [10, 14, 25] show that the output quality of conventional self-dual variants of BKZ or DeepBKZ is rarely better than that of the original algorithm. To obtain a practical advantage, we take free dimensions for our self-dual variant; For a parameter of free dimensions f and a lattice basis [b1 , . . . , bd ], we repeatedly reduce [b1 , . . . , bd− f +1 ] by DeepBKZ and [π f (b f ), . . . , π f (bd )] by dual DeepBKZ [27] to reduce a whole basis, where π f denotes the projection map over the orthogonal supplement of the space b1 , . . . , b f −1 R . While conventional algorithms set f = 2 inspired by a classical proof of Mordell’s inequality, we select suitable f based on the Gaussian Heuristic (see [15] for both the inequality and the heuristic.) We also report experimental results to compare our self-dual DeepBKZ with BKZ and DeepBKZ for several random bases. In particular, we use the slope metric of fpylll [22] to measure the quality of output bases by reduction algorithms. Notation The symbols Z and R denote the ring of integers and the field of real numbers, respectively. Werepresent all vectors in column format. We let a, b d ai bi between two vectors√a = (a1 , . . . , ad ) and denote the inner product i=1  b = (b1 , . . . , bd ) , and let a denote the Euclidean norm a, a.

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2 Preliminaries In this section, we review basic definitions and properties on lattices. We also present lattice problems whose hardness ensures the security of lattice-based cryptography.

2.1 Lattices and Their Bases (Primal) Lattices For a positive integer d, let b1 , . . . , bd be d linearly independent vectors in Rd . The set of all integral linear combinations of the bi ’s is a (full-rank) lattice  d   L = L(b1 , . . . , bd ) := xi bi : xi ∈ Z (1 ≤ i ≤ d) i=1

of dimension d with basis B = [b1 , . . . , bd ]. (We regard B as a d × d matrix, and write L = L(B) simply.) Every lattice has infinitely many bases if d ≥ 2; If two bases B1 and B2 span the same lattice, there exists a unimodular matrix V ∈ GLd (Z) with B1 = B2 V. The volume of L is defined as vol(L) = | det(B)|, independent of the choice of bases. The Gram–Schmidt orthogonalization for a basis B is the orthogonal family B = [b1 , . . . , bd ], recursively defined by b1 := b1 and for 2 ≤ i ≤ d bi

:= bi −

i−1 

μi, j bj with μi, j :=

j=1

bi , bj  bj 2

.

d Then we have vol(L) = i=1 bi . For 2 ≤  ≤ d, let π denote the orthogonal projection over the orthogonal supplement of the R-vector space b1 , . . . , b−1 R (note that π depends on B, and set π1 = id). For i ≤ j, we denote by B[i, j] the local projected block [πi (bi ), πi (bi+1 ), . . . , πi (b j )], and by L [i, j] the lattice spanned by B[i, j] . The first successive minimum is the length of a shortest non-zero vector in λ1 (L)2 L, denoted by λ1 (L). The supremum of vol(L) 2/d over all d-dimensional lattices L is called Hermite’s constant of dimension d, denoted by γd (see [15] for Hermite’s constants.) 



Dual Lattices The dual of a lattice L is defined as x ∈ spanR (L) : x, y ∈ Z for ∀y ∈ L , denoted by L, where spanR (L) denotes the R-vector space spanned by the lattice vectors in L. The dual lattice L of a full-rank lattice L = L(B) with basis B

−1  has a basis C = B , called the dual basis of B. Write B = [b1 , . . . , bd ] and C = [c1 , . . . , cd ]. Then C · B = Id is maintained, where Id denotes the identity matrix of size d. Now we define the Gram–Schmidt orthogonalization for the dual basis C as in case of the primal basis, but going through the basis vectors in reverse order; cd† := cd and for 1 ≤ j ≤ d − 1

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c†j := c j −

n 

μ j,i ci† with μ j,i :=

i= j+1

c j , ci†  ci† 2

.

For 1 ≤  ≤ d − 1, let τ denote the orthogonal projection over the orthogonal supplement of the space c+1 , . . . , cd R . Set τd = id. The dual basis of [πi (bi ), πi (bi+1 ), . . . , πi (b j )] is given by [τ j (ci ), . . . , τ j (c j )]. The particular case i = j shows bi  · ci†  = 1 for all i.

2.2 Typical Lattice Problems Question 1 [The Shortest Vector Problem, SVP] Given a basis of a lattice L, find a shortest non-zero vector in L, that is, a vector s ∈ L such that s = λ1 (L). SVP can be relaxed by an approximate factor; Given a basis of a lattice L and an approximation factor f ≥ 1, find a non-zero vector v in L with v ≤ f λ1 (L). (Approximate-SVP is exactly SVP when f = 1.) Given a lattice L of dimension d and a measurable set S in Rd , the Gaussian Heuristic predicts that the number of vectors in L ∩ S is roughly equal to vol(S)/vol(L). By applying to the ball of radius λ1 (L) centered at origin in Rd , this heuristic gives the prediction of the length of a shortest non-zero vector in L; The expectation of λ1 (L) according to the Gaussian Heuristic is given by GH(L) :=

1 −1 ωd d vol(L) d

∼

d 1 vol(L) d , 2πe

(1)

where ωd is the volume of the unit ball in Rd . (This is only a heuristic.) Question 2 [The Closest Vector Problem, CVP] Given a basis of a lattice L and a target vector t, find a vector in L closest to t. CVP is at least as hard as SVP. As in the SVP case, we can define an approximate variant of CVP by an approximate factor. Approximate-CVP is also at least as hard as approximate-SVP with the same factor. From a practical point of view, both are considered equally hard, due to Kannan’s embedding technique [12] which can transform approximate-CVP into approximate-SVP. (See [8] for the embedding.) The security of modern lattice-based cryptosystems is based on the hardness of cryptographic lattice problems, such as the LWE and the NTRU problems. (See [23]). Such lattice problems are reduced to approximate-SVP or approximate-CVP (e.g., see [2]).

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3 Lattice Basis Reduction Given any basis of a lattice, lattice basis reduction finds a new basis [b1 , . . . , bd ] of the same lattice with short and nearly orthogonal vectors bi ’s. It is mandatory to solve lattice problems. (For example, it enables to solve approximate-SVP since the length of b1 is guaranteed to be within some exponential factor from the shortest non-zero vector.) In this section, we introduce some notions of reduction and algorithms to achieve them.

3.1 Typical Reduction Algorithms Lenstra-Lenstra–Lovász (LLL) For a parameter 14 < δ < 1, a basis B = [b1 , . . . , bd ] is called δ-LLL-reduced if it satisfies the following two conditions; (i) It is sizereduced, namely, the Gram–Schmidt coefficients satisfy |μi, j | ≤ 21 for all i > j. (ii) It satisfies Lovász’ condition, namely, δbk−1 2 ≤ πk−1 (bk )2 for all k. Every δ-LLL1 d−1 d−1 reduced basis of a lattice L satisfies b1  ≤ α 2 λ1 (L) and b1  ≤ α 4 vol(L) d , 4 (see [4] or [15]). An LLL-reduced basis can be found by the LLL where α = 4δ−1 algorithm [13], in which adjacent basis vectors bk−1 and bk are swapped if Lovász’ condition does not hold. LLL with Deep Insertions (DeepLLL) It is a straightforward generalization of LLL, in which non-adjacent basis vectors can be changed; If the deep exchange condition πi (bk )2 < δbi 2 is satisfied for some i < k, the basis vector bk is inserted between bi−1 and bi as B ← [b1 , . . . , bi−1 , bk , bi , . . . , bk−1 , bk+1 , . . . , bd ]. This is called a deep insertion. Every output basis of DeepLLL satisfies the following condition; For 41 < δ < 1, a basis B = [b1 , . . . , bd ] is called δ-DeepLLLreduced if it is size-reduced and it also satisfies δbi 2 ≤ πi (bk )2 for all i < k.  d−2 √

Every δ-DeepLLL-reduced basis satisfies both b1  ≤ α 1 + α4 2 λ1 (L) and  (d−1)(d−2) d−1

1 4d b1  ≤ α 2d 1 + α4 vol(L) d [26], where α is the same as in LLL. These properties are strictly stronger than LLL. Block Korkine-Zolotarev (BKZ) A basis B = [b1 , . . . , bd ] of a lattice L is called HKZ-reduced if it is size-reduced and it satisfies bi  = λ1 (πi (L)) for every 1 ≤ i ≤ d. The notion of BKZ-reduction is a local block version of HKZ-reduction, defined as follows [16, 17, 19]; For β ≥ 2, a basis B = [b1 , . . . , bd ] of a lattice L is called βBKZ-reduced if it is size-reduced and every local block B[ j, j+β−1] is HKZ-reduced for 1 ≤ j ≤ d − β + 1. The second condition means bj  = λ1 (L [ j,k] ) for every 1 ≤ j ≤ d − 1 with k = min( j + β − 1, d). Every β-BKZ-reduced basis satisfies (d−1)/(β−1) λ1 (L) [17]. The BKZ algorithm [19] finds an almost β-BKZb1  ≤ γβ reduced basis, and it calls LLL to reduce every local block before enumeration of a shortest vector over the block lattice.

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Improved BKZ algorithms have been proposed such as BKZ 2.0 [6] and progressive-BKZ [3]. Variants of BKZ have also been proposed such as slidereduction [10] and self-dual BKZ [14]. Some of them have been implemented in software (e.g., [21]). DeepBKZ It is a combination of DeepLLL and BKZ, proposed in [24]. Specifically, DeepLLL is called as a subroutine alternative to LLL in BKZ. Thus every output basis of DeepBKZ satisfies the following condition of reduction; For 41 < δ < 1 and β ≥ 2, a basis is called (δ, β)-DeepBKZ-reduced if it is both δ-DeepLLL-reduced and β-BKZ-reduced. Experiments in [24] show that short lattice vectors can be often found by DeepBKZ with smaller blocksizes than BKZ in practice. Dual and self-dual variants have been proposed in [27] and [25], respectively.

3.2 Measurement of the Output Quality The Hermite factor is known as a good index to measure the practical output qualb1  ity of a reduction algorithm. It is defined by γ = vol(L) 1/d , where b1 is a shortest basis vector output by a reduction algorithm for a basis of a lattice L of dimension d. Smaller γ means that it finds a shorter lattice vector. It is shown in [10] by exhaustive experiments that for a practical algorithm such as LLL and BKZ, its root factor γ 1/d converges to a constant for high dimensions d ≥ 100. For a reduced basis   ≈ η for some constant η. This is called the geo[b1 , . . . , bd ], we have bi /bi+1 metric series assumption (GSA), first introduced in [18]. This states that the plots of Gram–Schmidt log-norms log bi  are on a straight line (e.g., see [18, Fig. 1].) The constant η depends on the reduction algorithm. While the Hermite factor depends only on the length of a shortest basis vector b1 , the slope metric of fpylll [22] measures an averaged quality of a whole basis [b1 , . . . , bd ]. The slope ρ is a least squares fit of the log bi 2 calculated as d ρ=

i=1

i log bi 2 − 21 (d + 1) d(d 2 − 1)/12

d i=1

log bi 2

.

This slope relates to the root Hermite factor by γ 1/d = exp(−ρ/4) in the GSA model. Under the Gaussian Heuristic and GSA, a limiting value of the root Hermite factor of BKZ with blocksize β is predicted in [5] as 1 1  2(β−1)

1  β−1

1 −β β β (πβ) ∼ . lim γ = ωβ d→∞ 2πe 1 d

(2)

There are experimental evidences supporting this prediction for high blocksizes β > 50. (Note that the Gaussian Heuristic holds in practice for random lattices in high dimensions, but unfortunately it is violated in low dimensions.) In a simple form

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based on the Gaussian Heuristic, the Gram–Schmidt lengths of a β-BKZ-reduced  1/β d−1 −i β . basis [b1 , . . . , bd ] of volume 1 is predicted as bi  ≈ αβ 2 , where αβ = 2πe This is reasonably accurate in practice for β > 50 and β  d. (See [5, 6, 28].) With this prediction, the slope of a BKZ-reduced basis can be estimated. In particular, we have −ρ > 0 and it decreases as β increases (see [1, Fig. 4] for −ρ ranging from 0.048 to 0.040.)

4 Self-dual DeepBKZ with Free Dimensions Experiments in [14, 25] show that the output quality of conventional self-dual variants of BKZ or DeepBKZ is rarely better than that of the original algorithm. In this section, we develop a new algorithm of self-dual DeepBKZ so that it can decrease the slope −ρ of an input basis B = [b1 , . . . , bd ] more efficiently than the original DeepBKZ. (See the previous subsection for the importance of decreasing the slope −ρ > 0.)

4.1 The Basic Idea of Our Algorithm Here, let us describe the basic idea of our algorithm. Given a basis B = [b1 , . . . , bd ] of a lattice L, we consider two types of bases b1 b2

··· ··· · · · bd− f +1 π f (b f ) π f (b f +1 ) · · · · · · · · · π f (bd )

(3)

for a parameter of free dimensions f with 2 ≤ f ≤ d − 1. (The projection map π f depends on the sub-basis [b1 , . . . , b f −1 ].) Like conventional algorithms [10, 14, 25], our self-dual DeepBKZ reduces the sub-basis [b1 , . . . , bd− f +1 ] of L by DeepBKZ [24], and also reduces the basis [π f (b f ), . . . , π f (bd )] of the projected lattice π f (L) by dual DeepBKZ [27]. In particular, forward basis vectors b1 , . . . , b f −1 are reduced only by DeepBKZ (i.e., dual DeepBKZ does not change them), and backward projected basis vectors π f (bd− f +2 ), . . . , π f (bd ) are reduced only by dual DeepBKZ (i.e., primal DeepBKZ does not change them). We expect that forward Gram–Schmidt lengths would decrease and backward lengths would increase as a blocksize β of self-dual DeepBKZ increases. (i.e., The slope −ρ of the whole basis B would decrease by repeating these procedures). Inspired by a classical proof of Mordell’s inequality, conventional algorithms use f = 2 (e.g., see [15] for Mordell’s inequality and its proof.) The output quality of several self-dual algorithms (in the worst case) is proven to be slightly better than primal algorithms in theory, but it is rarely better in practice (e.g., see [14, 25] for experiments.) For our algorithm, we shall select f suitable for practical use in the next subsection.

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4.2 Selection of Free Dimensions Here, we describe how to select a parameter of free dimensions f for our algorithm. We shall consider ideal and practical cases. Ideal Case: Using HKZ and Its Dual HKZ-reduction (resp., dual HKZ-reduction) is an ideal reduction of DeepBZ (resp., dual DeepBKZ). Using the Gaussian Heuristic, we can predict the sequence of Gram–Schmidt lengths i = bi  of an HKZ-reduced basis [b1 , . . . , bd ]. For a lattice of volume 1, the expected sequence is inductively defined as 1 =

d and i = 2πe



⎛

i−1 

1 ⎞− d−i+1

d −i +1 ⎝ j⎠ · 2πe j=1

(2 ≤ i ≤ d).

Recall that the Gaussian Heuristic is known to be violated in small dimensions (e.g., dimensions smaller 40). Similarly, the expected sequence of Gram–Schmidt lengths ˆi = bi  of a dual HKZ-reduced basis [b1 , . . . , bd ] is given by ˆi = −1 d−i+1 for 1 ≤ i ≤ d, mentioned in Sect. 2.1. In Fig. 1, we show a simulation of GSA slopes of HKZ and dual HKZ bases of a lattice of dimension d = 100. (The last 40 Gram– Schmidt lengths of an HKZ-reduced basis are roughly approximated from the forward lengths under the GSA assumption.) Our self-dual algorithm aims to decrease forward Gram–Schmidt lengths and increase backward lengths in order to decrease the slope −ρ of a whole basis. Our simulation in Fig. 1 implies that around f = 19 is suitable as a parameter of free dimensions of our algorithm in dimension d = 100 (see the best model in Fig. 1.) With such simulation, we can obtain a suitable free dimension f for every lattice dimension d (see Fig. 2 for ideal free dimensions f in 90 ≤ d ≤ 200). Practical Case: Using DeepBKZ and Its Dual For high dimensions d such as d ≥ 100, it is infeasible to run HKZ and its dual since it is costly to call an exact-SVP algorithm in such dimensions. Like conventional algorithms, our self-dual DeepBKZ calls an exact-SVP algorithm in dimension β for a fixed blocksize β. For our algorithm, we take a free dimension, approaching the ideal f in Fig. 2 as β increases. (Note that the output basis of DeepBKZ is HKZ-reduced when β = d.) Specifically, we simply take   β f (4) fβ = d − f +1 as an input parameter of free dimension of our self-dual DeepBKZ with blocksize β, where we can select the ideal case parameter f suitably from Fig. 2.

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Fig. 1 A simulation of GSA slopes of HKZ and dual HKZ bases of a lattice of dimension d = 100, based on Gaussian Heuristic (x-axis: index i, y-axis: log2 bi  for 1 ≤ i ≤ d). The last 40 Gram– Schmidt lengths of an HKZ-reduced basis are roughly approximated from the GSA assumption.

Fig. 2 A simulation of ideal free dimensions f of our self-dual algorithm for lattice dimensions d with 90 ≤ d ≤ 200 (The case of dimension d = 100 is given by Fig. 1.)

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4.3 Detailed Algorithm Algorithm 1 is our self-dual DeepBKZ with free dimensions. Like conventional selfdual algorithms [14, 25], it consists of two parts: forward tours and backward tours. Let B = [b1 , . . . , bd ] be an input basis of our self-dual DeepBKZ with blocksize β and free dimension parameter f β . In this subsection, we shall describe each part and terminating condition of our self-dual DeepBKZ with free dimensions. Algorithm 1 Self-dual DeepBKZ with free dimensions 1: procedure Self- dual DeepBKZ(B, β, f β )  B: basis, β: blocksize, f β : free dimension (4) 2: Compute the slope −ρ of the input basis B = [b1 , . . . , bd ] 3: do 4: β-DeepBKZ-reduce [b1 , . . . , bd− fβ +1 ]  forward tours 5: β-Dual-DeepBKZ-reduce [π fβ (b fβ ), . . . , π fβ (bd )]  backward tours 6: Compute the slope −ρ of the current basis B 7: while the slope −ρ decreases 8: return B  The output basis B is reduced 9: end procedure

Forward Tours In this part, we call SVP oracles in dimension β to reduce every local block B[ j, j+β−1] from j = 1 to d − β − f β + 2. (This series of processing is a forward tour.) In addition, we call DeepLLL for the sub-basis [b1 , . . . , bd− fβ +1 ] before enumeration to find a shortest vector over every block lattice L [ j, j+β−1] . Note that this part does not change the backward basis vectors bd− fβ , . . . , bd . In our self-dual algorithm, we terminate this part when the slope −ρ of sub-basis [b1 , . . . , bd− fβ +1 ] does not decrease during 5 forward tours, like the auto-abort termination of BKZ implemented in the fplll library [21]. Backward Tours Similarly to the above part, we call dual SVP oracles in dimension β to reduce the dual basis of every local block B[ j−β+1, j] from j = d down to β + f β − 1. (This series of processing is a backward tour.) We also call dual DeepLLL [27, Algorithm 1], a dual variant of DeepLLL, before the dual enumeration [14, Algorithm 2]. In particular, we perform dual deep insertions so that they does not change the forward sub-basis [b1 , . . . , b fβ −1 ]. This means that we reduce the local projected block [π fβ (b fβ ) . . . , π fβ (bd )] by dual DeepBKZ. As in the above part, we terminate this part when the slope −ρ of the local block does not decrease during 5 backward tours. Terminating Condition We count a pair of forward and backward tours as an overall tour of our self-dual DeepBKZ. We terminate our algorithm when the slope −ρ of the whole basis B does not decrease during 2 overall tours. (cf., Different from our self-dual algorithm, conventional algorithms of [14, 25] repeat a forward tour and a backward tour alternatively until the slope of −ρ of the whole basis B does not decrease enough).

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5 Experiments In this section, we report experimental results of our self-dual DeepBKZ algorithm with free dimensions about how efficiently it decreases the slope −ρ of an input basis.

5.1 Implementation We implemented our self-dual DeepBKZ (Algorithm 1) in C++ programs with the NTL library [20]. We used the g++ complier with -O3 -std=c++11 option. We set a triple of B = [b1 , . . . , bd ], μ = (μi, j )1≤ j 0. Critical point C1 is a saddle point and C2 is a center. There are two distinct type of trajectories in the phase diagram, namely, homoclinic trajectory (H T ) and periodic trajectory (P T ). H T and P T correspond to solitary and periodic wave solutions, respectively. In Fig. 2, we exhibit periodic wave corresponding to periodic trajectory around a critical point C2 (ψ1 , 0) of phase diagram (Fig. 1). Here, amplitude of periodic wave

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0.25

PT

HT

0.2 0.15 0.1 0.05 C

z

1

C2

0 −0.05 −0.1 −0.15 −0.2 −0.25 0

0.05

0.1

0.15

ψ

0.2

Fig. 1 Phase diagram of dynamical system (15) with κ = 4, δ = 0.1, V = 0.9

0.25 Ti Tp

0.3

= 4 , β = −0.1, and

is directly influenced by the parameters, like efficiency of electron trapping (β), ratio of temperature of ion to proton ( TTpi ) of SW, and traveling wave velocity (V ); whereas it is inversely influenced by κ. It is important to observe that κ, β, TTpi , and V do not show significant effect on the width of periodic wave of dynamical system (15).

5 Exact Solution of the Schamel Equation Considering the Galilean transformation χ = ξ − v0 τ and using initial condition 1 φ1 → 0 and dφ → 0, we obtain exact solution of the Schamel Eq. (13) as dχ  φ1 = φ0 sech

4

χ , Δ

(17)

 0 2 where, v0 is wave speed, φ0 = ( 15v ) and Δ = 4 vB0 are amplitude and width of 8A IASW solution of the Schamel equation, respectively.

Dynamical Behavior of Ion-Acoustic Periodic and Solitary Structures … β = 0.1

0.45

Ti/Tp = 4

0.25

0.35 0.3

0.2

ψ

ψ

Ti/Tp = 4.4

0.3

β = −0.1

0.4

425

0.25

0.15

0.2 0.15

0.1

0.1 0.05

0.05 0

120

140

160

χ

180

200

110 120 130 140 150 160 170 180 190 200

220

χ

(b)

(a) 0.3

κ= 4 κ= 5

V = 0.9 V = 0.95

0.3

0.25 0.25 0.2

ψ

ψ

0.2

0.15

0.15 0.1

0.1

0.05

0.05 110 120 130 140 150 160 170 180 190

110 120 130 140 150 160 170 180 190 200

χ

χ

(c)

(d)

Fig. 2 Variation of periodic wave solution of dynamical system (15) for a β with κ = 4, δ = 0.1, TTpi = 4 and V = 0.9, b TTpi with κ = 4, δ = 0.1, β = −0.1 and V = 0.9, c V with κ = 4, δ = 0.1,

Ti Tp

= 4 and β = −0.1, and d κ with δ = 0.1,

Ti Tp

= 4 , β = −0.1 and V = 0.9

In Fig. 3, IASW solution given by Eq. (17) and corresponding to H T of the phase diagram (Fig. 1) is presented. Here, the parameters β, TTpi , and V show a similar effect on the wave characteristics of IASW, i.e., as magnitude of β , TTpi , and v0 increases IASW is amplified, whereas, as κ increases, IASW slowly diminish.

6 Conclusion Dynamical behavior of IA periodic and solitary wave solutions corresponding to Schamel equation has been analyzed in a collisionless and magnetized SW plasma with suprathermal electrons, protons, and heavier ions. We have taken typical SW parameters revealed by various SW satellite observations, such as, T p /Te ≥ 1 (for fast SW) and T p /Te < 1 (for slow SW), n i0 /n e0 ≈ 0.0 − 0.05, Ti /T p ≥ 1 [14, 19,

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Fig. 3 Variation of the IASW solution of the Schamel equation for the different values of a β with κ = 4, δ = 0.1, TTpi = 4 and v0 = 1.3, b TTpi with κ = 4, δ = 0.1, β = −0.1 and v0 = 1.3, c v0 with κ = 4, δ = 0.1, v0 = 1.3

Ti Tp

= 4 and β = −0.1 , and d κ with , δ = 0.1,

Ti Tp

= 4 , β = −0.1 , and

20], to examine the influence of SW parameters on the width and amplitude of IA periodic and solitary wave solutions of the Schamel equation. We observed the following by increasing the parameters β, TTpi , v0 , and κ: 1. β, TTpi and v0 amplify both IA periodic and solitary wave solutions corresponding to Schamel equation. 2. κ reduces the amplitude of IA periodic and solitary waves. 3. There is no significant change in the width of IA periodic and solitary waves. The result of this study can be important to discern various wave profiles of SW plasma.

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References 1. Steinberg JT, Lazaras AJ, Ogilvie KW, Lepping R, Byrnes J (1996) Differential flow between solar wind protons and alpha particles: first wind observations. Geophys Res Lett 23:1183. https://doi.org/10.1029/96GL00628 2. Marsch E, Muhlhauser K-H, Rosenbauer H, Schwenn R, Neubauer FM (1982) Solar wind helium ions: observations of the Helios solar probes between 0.3 and 1 AU. J Geophys Res 87:35. https://doi.org/10.1029/JA087iA01p00035 3. Gurnett DA, Marsch E, Pilipp W, Schwenn R, Rosenbauer H (1979) Ion acoustic waves and related plasma observations in the solar wind. J Geophys Res 84(A5):2029. https://doi.org/10. 1029/JA084iA05p02029 4. Arshad K, Mirza AM, Rehman A (2014) Ion-acoustic waves in non-Maxwellian magnetospheric electron-positron-ion plasma. Astrophys Space Sci 350(2):585–590. https://doi.org/ 10.1007/s10509-014-1788-z 5. Sreeraj T, Singh SV, Lakhina GS (2016) Coupling of electrostatic ion cyclotron and ion acoustic waves in the solar wind. Phys Plasmas 23(8):082901. https://doi.org/10.1063/1.4960657 6. Bacha M, Gougam LA, Tribeche M (2017) Ion-acoustic rogue waves in magnetized solar wind plasma with nonextensive electrons. Phys Stat Mech Appl 466:199–210. https://doi.org/10. 1016/j.physa.2016.09.013 7. Samanta UK, Saha A, Chatterjee P (2013) Bifurcations of dust ion acoustic traveling waves in a magnetized dusty plasma with a q-nonextensive electron velocity distribution. Phys Plasmas 20:022111. https://doi.org/10.1063/1.4791660 8. Selim MM, El-Depsy A, El-Shamy EF (2015) Bifurcations of nonlinear ion-acoustic traveling waves in a multicomponent magnetoplasma with superthermal electrons. Astrophys Space Sci 360:66. https://doi.org/10.1007/s10509-015-2574-2 9. Saha A (2016) Bifurcation, periodic and chaotic motions of the modified equal width-Burgers (MEW-Burgers) equation with external periodic perturbation. Nonlinear Dyn 87(4):2193– 2201. https://doi.org/10.1007/s11071-016-3183-5 10. Saha A (2017) Dynamics of the generalized KP-MEW-Burgers equation with external periodic perturbation. Comput Math Appl 73(9):1879–1885. https://doi.org/10.1016/j.camwa.2017.02. 017 11. Saha A, Ali R, Chatterjee P (2017) Nonlinear excitations for the positron acoustic waves in auroral acceleration regions. Adv Space Res 60:1220. https://doi.org/10.1016/j.asr.2017.06. 012 12. Saha A, Prasad PK, Banerjee S (2019) Bifurcations of ion-acoustic superperiodic waves in auroral zone of Earth’s magnetosphere. Astrophys Space Sci 364:180. https://doi.org/10.1007/ s10509-019-3671-4 13. El-Monier SY, Atteya A (2018) Bifurcation analysis for dust-acoustic waves in a fourcomponent plasma including warm ions. IEEE Trans Plasma Sci 46:815–824. https://doi.org/ 10.1109/TPS.2017.2766097 14. Lakhina GS, Singh SV (2015) Generation of Weak Double Layers and Low-Frequency Electrostatic Waves in the Solar Wind. Solar Phys 290(10):3033–3049. https://doi.org/10.1007/ s11207-015-0773-1 15. Williams G, Verheest F, Hellberg MA, Anowar MGM, Kourakis I (2014) A Schamel equation for ion acoustic waves in superthermal plasmas. Phys Plasmas 21(9):092103. https://doi.org/ 10.1063/1.4894115 16. Schamel H (1973) A modified Korteweg-de Vries equation for ion acoustic wavess due to resonant electrons. J Plasma Phys 9(03):377. https://doi.org/10.1017/S002237780000756X 17. Baluku TK, Hellberg MA, Kourakis I, Saini NS (2010) Dust ion acoustic solitons in a plasma with kappa-distributed electrons. Phys Plasmas 17(5):053702. https://doi.org/10.1063/ 1.3400229 18. El-Kalaawy OH (2011) Exact solitary solution of Schamel equation in plasmas with negative ions. Phys Plasmas 18(11):112302. https://doi.org/10.1063/1.3657422

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19. Mangeney A, Salem C, Lacombe JC, Perche C, Manning R, Kellogg PJ, Geoz K, Monson SJ, Bosqued JM (1999) WIND observations of coherent electrostatic waves in the solar wind. Ann Geophys 17:307–320. https://doi.org/10.1007/s00585-999-0307-y 20. Borovsky JE, Gary SP (2014) How important are the alpha-proton relative drift and the electron heat flux for the proton heating of the solar wind in the inner heliosphere. J Geophys Res—Space Phys 119:5210–5219. https://doi.org/10.1002/2014JA019758 21. Livadiotis G (2015) Introduction to special section on Origins and Properties of Kappa Distributions: Statistical Background and Properties of Kappa Distributions in Space Plasmas. J Geophys Res Space Phys 120(3):1607–1619. https://doi.org/10.1002/2014JA020825

Substructuring Waveform Relaxation Methods with Time-Dependent Relaxation Parameter Bankim C. Mandal and Soura Sana

Abstract We present in this paper a modified variant of Dirichlet–Neumann (DNWR) and Neumann–Neumann Waveform Relaxation (NNWR) methods for solving time-dependent Partial Differential Equations (PDEs). These two domain decomposition methods are based on non-overlapping splitting of domains. Unlike in the case of the classical version, we introduce time-dependent relaxation parameter in the update stage of the iterative process before moving to the next iteration. This makes the analysis complex, but leads to an understanding of the effect of the relaxation parameter on the iteration. We present the details of the modified algorithms for two non-overlapping subdomains, and show conditional convergence properties in a few special cases. We illustrate our findings with numerical results. Keywords Dirichlet–Neumann · Neumann–Neumann · Waveform relaxation · Domain decomposition methods · Relaxation parameter · Quasi-nilpotent operator · Spectrum of operators

1 Introduction There are many numerical algorithms developed over the years to solve PDEs and ODEs; Domain Decomposition Method (DDM) is one of them. It is suitable for computationally complex problems as it is inherently parallel. Starting from Schwarz [1], through Picard [2] and Lindelöf [3], Bjørstad and Widlund [4], the development of DDM has come a long way. In this article, we are particularly interested in DD methods for solving space–time problems. There are mainly two approaches: firstly the method of lines, i.e. the classical approach where the problem is discretized in B. C. Mandal · S. Sana (B) School of Basic Sciences, Indian Institute of Technology Bhubaneswar, Bhubaneswar, India e-mail: [email protected] B. C. Mandal e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 D. Giri et al. (eds.), Proceedings of the Sixth International Conference on Mathematics and Computing, Advances in Intelligent Systems and Computing 1262, https://doi.org/10.1007/978-981-15-8061-1_34

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time to obtain a sequence of elliptic problems, and then DDM [5, 6] are applied to solve them. The disadvantage of this approach is that one is forced to include uniform time steps across the whole domain, which is restrictive for multi-scale problems. The second approach is the Waveform Relaxation (WR) methods: here, one solves the space–time subproblems for the whole time window in one go. There are multiple variants of the WR type DDMs, such as Schwarz waveform relaxation (SWR) [7, 8], Optimized SWR [9, 10], and more recently the DNWR [11, 12], and NNWR [13–15]. In these algorithms, one first decomposes the spatial domain into two or several overlapping or non-overlapping subdomains, followed by subdomain solves by adding suitable artificial boundary condition(s). These boundary conditions are usually called transmission conditions (TCs), which transmit information between neighbouring subdomains via iterations. Depending on the nature of the TCs, one observes difference in convergence behaviour. In this article, we will focus on a few modifications on the classical DNWR and NNWR methods. Usually, a constant relaxation parameter is taken to update the interface values before starting a new iteration. The algorithms are infact analyzed for particular values of the parameter both for parabolic and hyperbolic problems, and the corresponding convergence results are shown: superlinear convergence rate for the heat equation and finite step convergence for the wave equation; for more details see [11, 15]. Here we have considered time-dependent relaxation parameter and analyzed the algorithms for a few particular cases. For constant relaxation parameter, it is shown that θ = 1/2 for the DNWR and θ = 1/4 for the NNWR produce the best convergence result. Few numerical examples for heterogeneous medium were shown in [11, 13, 16], but no concrete analytical study is known for such cases. This article is focussed to enhance the convergence study in heterogeneous medium and it is to establish the effect of time-dependent relaxation parameter in DNWR and NNWR. The study is motivated by various reasons: for elliptic problems, these algorithms are closely related to Finite Element Tearing and Interconnect (FETI) methods [17, 18], which are few of the most commonly used and extensively studied DD methods in existence. Additionally, NN and FETI methods with a coarse grid correction for elliptic problems scale very well in the mesh size and in the number of subdomains. An advantage of the WR variant is the possibility for parallelism in time: see [19] for the coupling of DNWR and NNWR with parareal methods; for more on parallel-in-time algorithms, see [20] and the references therein. We consider the following one-dimensional heat equation as our guiding example ∂t u − ∇ · (κ(x)∇u) = f (x, t), u(x, 0) = u 0 (x), u(x, t) = g(x, t), where Ω ⊂ Rd and κ(x) ≥ κ > 0.

x ∈ Ω, t ∈ (0, T ], x ∈ Ω, x ∈ ∂Ω, t ∈ (0, T ],

(1)

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431

2 Classical DNWR and Its Convergence Results To define the DNWR algorithm for the model problem (1), the domain Ω = (−b, a) is decomposed into two non-overlapping subdomains Ω1 = (−b, 0) and Ω2 = (0, a), and we assume κ(x) = κ1 in Ω1 and κ(x) = κ2 in Ω2 . On the interface  = {0}, we choose an initial guess h [0] (t), t ∈ (0, T ]. Then the DNWR algorithm is given by the following iteration for k = 1, 2, . . . Dirichlet part: ⎧ [k] ∂u ⎪ ⎪ ⎪ t[k]1 ⎨ u 1 (x, 0) [k] ⎪ u (−b, t) ⎪ ⎪ ⎩ 1[k] u 1 (0, t)

Neumann part: [k]

= κ1 u 1 + f, Ω1 × (0, T ], = u 0 (x), Ω1 × {0}, = g(−b, t), ∂Ω1 × (0, T ], = h [k−1] (t),  × (0, T ],

⎧ [k] ∂u ⎪ ⎪ ⎪ t[k]2 ⎨ u 2 (x, 0) [k] ⎪ u (a, t) ⎪ ⎪ ⎩ 2 [k] κ2 ∂ x u 2

[k]

= κ2 u 2 + f, Ω2 × (0, T ], = u 0 (x), Ω2 × {0}, = g(a, t), ∂Ω2 × (0, T ], [k] = κ1 ∂x u 1 (0, t),  × (0, T ],

Update part: [k−1] (t). h [k] (t) = θ u [k] 2 (0, t) + (1 − θ )h

The relaxation parameter θ ∈ (0, 1] is chosen to obtain fast convergence. For convergence analysis one can use Laplace transforms in t to analyze this method; the convergence results [11] are enlisted below for 1D heat equation with homogeneous medium, κ1 = κ2 . Result 1 (Convergence of DNWR for equal subdomain) If the subdomains are of same length, i.e. a = b then the DNWR algorithm converges linearly for 0 < θ < 1. Moreover, for θ = 1/2, it converges to the exact solution in two iterations, independent of the size of time window T . Result 2 (Superlinear convergence for bounded time-window) If θ = 1/2 and the Dirichlet and Neumann subdomain are not of same length, then DNWR algorithm converges superlinearly on bounded time interval t ∈ (0, T ] with the estimate 

[k]

h  L ∞ (0,T ) ≤

b−a b

k

  ka h [0]  L ∞ (0,T ) , erfc √ 2 T

when Dirichlet subdomain is larger than Neumann subdomain, and h

[2k]

√ 2

  L ∞ (0,T ) ≤

1−

e−(2k+1)b2 /T

2k

e−k

b /T

2 2

h [0]  L ∞ (0,T ) ,

when Neumann subdomain is larger than Dirichlet subdomain. Result 3 (Linear convergence for unbounded time window) If θ = 1/2 and the Dirichlet and Neumann subdomain are not of same length, then DNWR algorithm converges linearly on unbounded time interval for |a − b| < 2b with the estimate

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[k]

h  L ∞ (0,∞) ≤

|a − b| 2b

k

h [0]  L ∞ (0,∞) .

3 Classical NNWR and Its Convergence Result In the same way, we define the NNWR algorithm for the model problem (1). The domain of interest Ω = (−b, a) is decomposed into two non-overlapping subdomains Ω1 = (−b, 0) and Ω2 = (0, a), where κ(x) = κ1 in Ω1 and κ(x) = κ2 in Ω2 . On the interface  = {0}, we choose an initial guess w[0] (t), t ∈ (0, T ]. Then the NNWR algorithm is given by the following iteration for k = 1, 2, . . . Dirichlet part: ⎧ [k] ∂t u 1 ⎪ ⎪ ⎪ ⎨ [k] u 1 (x, 0) ⎪ u [k] (−b, t) ⎪ ⎪ 1 ⎩ [k] u 1 (0, t)

[k]

= κ1 u 1 + f, Ω1 × (0, T ), = u 0 (x), Ω1 × {0}, = g(−b, t), ∂Ω1 × (0, T ], = w[k−1] (t),  × (0, T ],

⎧ [k] ∂t u 2 ⎪ ⎪ ⎪ ⎨ [k] u 2 (x, 0) ⎪ u [k] (a, t) ⎪ ⎪ 2 ⎩ [k] u 2 (0, t)

= κ2 u 2 + f, Ω2 × (0, T ), = u 0 (x), Ω2 × {0}, = g(a, t), ∂Ω2 × (0, T ], = w[k−1] (t),  × (0, T ],

⎧ ⎪ ∂t ψ2[k] ⎪ ⎪ ⎨ [k] ψ2 (x, 0) ⎪ ψ2[k] (a, t) ⎪ ⎪ ⎩ κ2 ∂x ψ2[k]

= κ2 ψ2[k] , Ω2 × (0, T ], = 0, Ω2 × {0}, = 0, ∂Ω2 × (0, T ], [k] = κ2 ∂x u [k] 2 − κ1 ∂x u 1 ,  × (0, T ],

[k]

Neumann part: ⎧ ⎪ ∂t ψ1[k] ⎪ ⎪ ⎨ [k] ψ1 (x, 0) ⎪ ψ [k] (−b, t) ⎪ ⎪ ⎩ 1 κ1 ∂x ψ1[k]

= κ1 ψ1[k] , Ω1 × (0, T ], = 0, Ω1 × {0}, = 0, ∂Ω1 × (0, T ], [k] = κ1 ∂x u [k] 1 − κ2 ∂x u 2 ,  × (0, T ],

Update part: w[k] (t) = w[k−1] (t) − θ [ψ1[k] (0, t) + ψ2[k] (0, t)], where the relaxation parameter θ is in (0, 1]. The convergence results from [11] are enlisted bellow for the case κ1 = κ2 . Result 4 (Convergence of NNWR for equal subdomain) If the subdomains are of same length, i.e. a = b then the NNWR algorithm converges linearly for 0 < θ < 1. Moreover, for θ = 1/4, it converges to the exact solution in two iterations, independent of the size of time window T . Result 5 (Superlinear convergence for bounded time window) If θ = 1/4 and the two subdomain are not of same length, then NNWR algorithm converges superlinearly on bounded time interval t ∈ (0, T ] with the estimate w[k]  L ∞ (0,T ) ≤ where m = min{a, b}.



(a − b)2 ab

k

e−k

2

m 2 /T

w[0]  L ∞ (0,T ) ,

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Result 6 (Linear convergence for unbounded time window) If θ = 1/4 and the subdomain are not of same length, then NNWR algorithm converges linearly on unbounded time interval for (a − b)2 < 4ab with the estimate 

[k]

w  L ∞ (0,∞) ≤

(a − b)2 4ab

k w0  L ∞ (0,∞) .

4 Analysis with Time-Dependent Parameter We now extend the analysis of the DNWR and NNWR algorithms from Sect. 2 and Sect. 3, in case the parameter θ varying with time t. We consider few particular cases below.

4.1 DNWR Algorithm Suppose θ is a linear function of time t. i.e. θ (t) = θ0 + θ1 t. After a Laplace transform, the solution of the DNWR algorithm for the error equations in the onedimensional heat equation becomes √ sinh{(x + b) s/κ1 } [k−1] ˆ , =h (s) √ sinh(b s/κ1 ) √  ˆ [k−1] (s) coth(b √s/κ1 ) sinh{(x − a) s/κ2 }. uˆ [k] 2 (x, s) = h cosh(a s/κ2 ) uˆ [k] 1 (x, s)

√ √ From the update part at x = 0 and Gˆ 1 (s) = (tanh(a s/κ2 ) coth(b s/κ1 ) − 1), we get

  d  { κ1 /κ2 Gˆ 1 hˆ [k] (s) = (1 − (1 + κ1 /κ2 )θ0 ) − θ0 κ1 /κ2 Gˆ 1 hˆ [k−1] (s) + θ1 ds

 + (1 + κ1 /κ2 )}hˆ [k−1] (s)  

= (1 − (1 + κ1 /κ2 )θ0 )hˆ [k−1] (s) + κ1 /κ2 {−θ0 Gˆ 1 + θ1 Gˆ 1 }hˆ [k−1] (s)   + θ1 (1 + κ1 /κ2 )hˆ [k−1] (s) + θ1 κ1 /κ2 Gˆ 1 hˆ [k−1] (s). After inverse Laplace transform, we obtain h [k] (t) = {1 − (1 +



 t  κ1 /κ2 )(θ0 + θ1 t)}h [k−1] (t) − (θ0 + θ1 t) κ1 /κ2 G 1 (τ )h [k−1] (t − τ )dτ 0

= {1 − (1 +



κ1 /κ2 )θ(t)}h

[k−1]

 t  (t) − θ(t) κ1 /κ2 G 1 (τ )h [k−1] (t − τ )dτ. 0

(2)

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If we choose θ (t) = θ0 + θ1 t + θ2 t 2 then the update part is modified as d  { κ1 /κ2 Gˆ1 κ1 /κ2 θ0 Gˆ1 }hˆ [k−1] (s) + θ1 ds



  d2  + (1 + κ1 /κ2 )}hˆ [k−1] − θ2 2 { κ1 /κ2 Gˆ1 + (1 + κ1 /κ2 )}hˆ [k−1] ds  

= hˆ [k−1] (s) − (1 + κ1 /κ2 )(θ0 hˆ [k−1] − θ1 hˆ [k−1] + θ2 hˆ

[k−1] ) + κ1 /κ2 [−θ0 Gˆ1 + θ1 Gˆ1   



− θ2 Gˆ1 ]hˆ [k−1] + [θ1 κ1 /κ2 Gˆ1 − (1 + κ1 /κ2 )θ2 Gˆ1 ]hˆ [k−1] − θ2 κ1 /κ2 Gˆ1 hˆ

[k−1] ,

hˆ [k] (s) = {(1 − (1 +



κ1 /κ2 )θ0 ) −



from which an inverse Laplace transform leads to h [k] (t) = [1 − (1 + 

t

−θ1 t



κ1 /κ2 )(θ0 + θ1 t + θ2 t 2 )]h [k−1] +

G 1 (t − τ )h

[k−1]

0

= {1 − (1 +

 t κ1 /κ2 −θ0 G 1 (t − τ )h [k−1] (τ )dτ 0

 (τ )dτ − θ2 t



t

2

τ G 1 (t − τ )h 2

[k−1]

(τ )dτ

0



κ1 /κ2 )θ(t)}h

[k−1]

 t  (t) − θ(t) κ1 /κ2 G 1 (t − τ )h [k−1] (τ )dτ.

(3)

0

Therefore, by combining the Eqs. (2) and (3), we may write the update condition for any degree of polynomial θ (t) as h [k] (t) = φ1 (t)h [k−1] (t) − θ (t)



t

K 1 (t − τ )h [k−1] (τ )dτ,

(4)

0

where φ1 (t) := {1 − (1 +

√

κ1 /κ2 )θ (t)} and K 1 (t) :=

√ κ1 /κ2 G 1 (t).

4.2 NNWR Algorithm We now deduce the expression of update condition for the NNWR method with timevariable parameter. For a linear or quadratic function of time t, i.e. θ (t) = θ0 + θ1 t or θ (t) = θ0 + θ1 t + θ2 t 2 , we obtain the solution of the Dirichlet part of NNWR algorithm for the error equations in Laplace space,  wˆ [k−1] (s) sinh{(x + b) s/κ1 }, √ sinh(b s/κ1 )  wˆ [k−1] (s) sinh{(x + b) s/κ2 }, uˆ [k] √ 2 (x, s) = sinh(b s/κ2 )

uˆ [k] 1 (x, s) =

and the solution of Neumann part becomes

Substructuring Waveform Relaxation Methods …

435

√ sinh{(x + b) s/κ1 } [k] [k] ˆ ψ1 (x, s) = rˆ (s) √ , √ sκ1 cosh(b s/κ1 ) √ sinh{(a − x) s/κ2 } , ψˆ 2[k] (x, s) = rˆ [k] (s) √ √ sκ2 cosh(a s/κ2 ) where rˆ [k] (s) = κ1 ∂x uˆ [k] ˆ [k] 1 (0, s) − κ2 ∂x u 2 (0, s). Thus the update part becomes wˆ [k] (s) = [1 − θˆ (s)(2 +



κ1 /κ2 +



  ˆ κ2 /κ1 )]wˆ [k−1] (s) − θ(s)( κ1 /κ2 Gˆ1 + κ2 /κ1 Gˆ2 )wˆ [k−1] ,

√ √ √ where √Gˆ 1 (s) = (tanh(a s/κ2 ) coth(b s/κ1 ) − 1) and Gˆ 2 (s) = (tanh(b s/κ1 ) coth(a s/κ2 ) − 1). Likewise DNWR analysis, we can show that for any polynomial θ (t), the inverse Laplace transform of the update part becomes   κ1 /κ2 + κ2 /κ1 )θ (t)}w[k−1] (t)  t   − θ (t) ( κ1 /κ2 G 1 + κ2 /κ1 G 2 )(t − τ )w[k−1] (τ )dτ 0  t [k−1] (t) − θ (t) K 2 (t − τ )w[k−1] (τ )dτ, (5) = φ2 (t)w

w[k] (t) = {1 − (2 +

0

√ √ √ with φ2 (t) := {1 − (2 + κ1 /κ2 + κ2 /κ1 )θ (t)} and K 2 (t) = κ1 /κ2 G 1 (t) + √ κ2 /κ1 G 2 (t). Convergence Study: Note that, any continuous function on a compact domain can be approximated arbitrarily close by polynomials, so for continuous θ (t) on compact domain, we may use above deduction (4)–(5). For convergence we need |φi (t)| < 1, for i = 1, 2. Our next goal is to find θ (t) for faster convergence. In operator notion, the update part in both DNWR and NNWR method (4)–(5) can be rewritten as Ph = Sh − V h, 

where Sh(t) = φi (t)h(t) and

V h(t) = θ (t)

t

K i (t − τ )h(τ )dτ.

0

The functions G(t) and θ (t) are continuous on [0, T ]. Therefore, the operatorsP : C([0.T ]) → C([0, T ]) and V are compact on C([0, T ]). However, the operator P can also be defined in other spaces. To do this we note the following standard result. Result 7 Let X and Y be Banach spaces and T is a linear operator from X 1 to Y where X 1 is a dense subspace of X . Suppose that there is a constant c such that ||T x|| ≤ c||x||, ∀x ∈ X 1 . Then T can be uniquely extended to a bounded linear operator from X to Y whereas the above estimate holds on X .

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The operator P can, therefore, be extended to become a bounded linear operator on L 2 ([0, T ]). We have the operator V quasi-nilpotent and compact on L 2 ([0, T ]) which means the spectrum σ (V ) = {0}, i.e. spectral radius ρ(V ) = 0. We are now in a position to prove our main results. We will use φ and K for φi and K i , respectively, in the remaining part. Theorem 1 (θ (t) is constant) If θ (t) is a constant function on a bounded time interval [0, T ], then φi (t) = 0 leads to the optimal convergence rate. Proof  t Let θ (t) = θ is a constant function. Then the operator S = φ and V = θ 0 K dτ commute with each other. So from the properties of spectral radius for commutative operator we have ρ(S − V ) ≤ ρ(S) + ρ(−V ) ≤ ρ(S) + ρ(V ) ≤ ρ(S), which means ρ(P) ≤ ρ(S) and also ρ(S − V ) ≥ |ρ(S) − ρ(V )| ≥ ρ(S), which yields ρ(P) ≥ ρ(S). Combining these inequalities, we get ρ(P) = ρ(S). The convergence rate will be optimal when ρ(P) = 0, i.e. when ρ(S) = 0. This establishes φ = 0. Theorem 2 (θ (t) is continuous) If θ (t) is a continuous function on a bounded time interval [0, T ], then φ(t) = 0 leads to the optimal convergence rate. Proof We have V : L 2 ([0, T ]) → L 2 ([0, T ]) to be compact and φ(t) ∈ C([0, T ]). So by the remark made after Theorem 3.3 in [21], we have that σ (P) = range of φ(t). Therefore, P gives the optimal convergence when φ(t) = 0.

5 Numerical Illustration We now verify our results numerically with the following model problem: ∂t u − ∂x (κ(x)∂x u) = −e−t−x , 2

u(x, 0) = e

−2x

,

u(−3, t) = e

−2t

= u(5, t),

x ∈ (−3, 5), t > 0, x ∈ (−3, 5), t > 0,

where κ(x) = 3, ∀x ∈ (−3, 0), and κ(x) = 5, ∀x ∈ (0, 5). To apply DNWR and NNWR algorithms, the domain Ω = (−3, 5) is decomposed into two non-overlapping subdomains (−3, 0) and (0, 5) and on each subdomain we take

Substructuring Waveform Relaxation Methods …

437

Error Profile for various = 0.164 = 0.264 = 0.364 = 0.464 = 0.564 = 0.664 = 0.764 = 0.864 = 0.964

10-5

10-10

10-15

Error Profile for various

100

ERROR

ERROR

100

= 0.164 = 0.264 = 0.364 = 0.464 = 0.564 = 0.664 = 0.764 = 0.864 = 0.964

10-5

10-10

10-15 0

5

10

15

20

0

5

ITERATION

10

15

20

ITERATION

(b) Long time window

(a) Short time window

Fig. 1 Convergence rate of DNWR algorithm with two subdomains for various values of the relaxation parameter θ; left: T = 2 and right: T = 20 Error Profile for various = 0.096 = 0.146 = 0.196 = 0.246 = 0.296 = 0.346 = 0.396 = 0.446 = 0.496

10-5

10-10

10-15

0

5

10

15

ITERATION

(a) Short time window

Error Profile for various

100

10

ERROR

ERROR

100

= 0.096 = 0.146 = 0.196 = 0.246 = 0.296 = 0.346 = 0.396 = 0.446 = 0.496

-5

10-10 10-15

20

10-20

0

5

10

15

20

ITERATION

(b) Long time window

Fig. 2 Convergence rate of NNWR algorithm with two subdomains for different values of the relaxation parameter θ; left: T = 2 and right: T = 20

81 mesh points in space and 261 time steps. we discretize using centred finite differences in space and backward Euler in time. On the interface x = 0, we choose the initial guess h [ 0](t) = 0, t > 0. In Fig. 1, we plot the error curves for different values of the parameter θ√ for (a) T = 2 and (b) T = 20 for DNWR method. We see that except θ ∗ = 1/(1 + 3/5), convergence rates are linear when θ remain in the neighbourhood of θ ∗ . In Fig. 2, we plot the error curves for different values of the parameter θ for (a) T = (b) T = 20 for NNWR method. We observe that except √ 2 and √ θ # = 1/(2 + 3/5 + 5/3) convergence rate are linear when θ remain in the neighbourhood of θ # . We now consider few continuous parameter θ (t). We assume θ (t) = C0 + C1 t, where C0 and C1 are taken from [0, 1] and [−1, 1] for DNWR and [0, 0.5] and [−1, 1] for NNWR, respectively. We have taken 10 and 200 different values of C0 and C1 respectively. For each value of C0 we check 200 combimantion values of C1

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Table 1 Time-dependent relaxation parameter results in DNWR Error C0 C1 2.3347e − 07 3.7584e − 07 2.1446e − 08 1.4311e − 08 1.7796e − 09 1.8437e − 11 1.8437e − 11 6.1058e − 11 8.6476e − 09 1.7113e − 07 3.3547e − 07

0.66351 0.66351 0.56351 0.56351 0.56351 0.56351 0.56351 0.56351 0.56351 0.56351 0.46351

−0.06 −0.05 −0.04 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.04

Table 2 Time-dependent relaxation parameter results in NNWR Error C0 C1 1.0525e − 07 1.2255e − 07 3.9445e − 07 8.748e − 09 8.748e − 09 2.508e − 09 3.5875e − 07 1.4716e − 07 1.946e − 07

0.29597 0.29597 0.29597 0.24597 0.24597 0.24597 0.19597 0.19597 0.19597

−0.04 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.04

and find the value of C1 for which the error is minimum. This process is continued for 10 different values of C0 . In Table 1, we have shown the errors for DNWR algorithm for θ (t) = C0 + C1 t and for T = 2, after 10th iteration in each combination of (C0 , C1 ). Observe that the error is minimum when C0 = 0.5635, C1 = 0. Therefore, we can verify that, even for linear time-dependent relaxation parameter, θ (t) = 0.5635 gives the best convergence result. In Table 2, we have shown the similar results for NNWR algorithm for θ (t) = C0 + C1 t for time T = 2. The error is minimum for NNWR when C0 = 0.24597, C1 = 0. It illustrates that even for linear time-dependent relaxation parameter θ (t) = 0.24597 yields the best convergence rate.

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6 Concluding Remarks We have analyzed the DNWR and NNWR algorithm using time-dependent relaxation parameter for 1D heat equation. We have √ showed using an operator theory√argument κ1 /κ2 ) for DNWR and θ = 1/(2 + κ1 /κ2 + that for two subdomains θ = 1/(1 + √ κ2 /κ1 ) for NNWR produce the best convergence rate. For a model problem, we have shown the numerical illustrations of the results in different possible cases. The analysis indicates that the time-dependent parameter would have no significant contribution in the convergence behaviour. This study can also be naturally extended for multiple subdomains as well as in a higher dimensional setting.

References 1. Schwarz HA (1870) Über einen Grenzübergang durch alternierendes Verfahren. Vierteljahrsschrift der Naturforschenden Gesellschaft Zürich 15:272–286 2. Picard E (1893) Sur l’application des méthodes d’approximations successives à l’étude de certaines équations différentielles ordinaires. Journal de Mathématiques Pures et Appliquées 217–272 3. Lindelöf E (1894) Sur l’application des méthodes d’approximations successives à l’étude des intégrales réelles des équations différentielles ordinaires. J de Mathématiques Pures et Appliquées 117–128 4. Bjørstad PE, Widlund OB (1986) Iterative methods for the solution of elliptic problems on regions partitioned into substructures. SIAM J Numer Anal 23(6):1097–1120. https://doi.org/ 10.1137/0723075, http://dx.doi.org/10.1137/0723075 5. Cai XC (1991) Additive Schwarz algorithms for parabolic convection-diffusion equations. Numer Math 60:41–61 6. Cai XC (1994) Multiplicative Schwarz methods for parabolic problems. SIAM J Sci Comput 15(3):587–603 7. Gander, MJ (1996) Overlapping schwarz for linear and nonlinear parabolic problems 8. Gander MJ, Stuart AM (1998) Space-time continuous analysis of waveform relaxation for the heat equation. SIAM J Sci Comput 19(6):2014–2031. https://doi.org/10.1137/ S1064827596305337, http://dx.doi.org/10.1137/S1064827596305337 9. Gander MJ, Halpern L (2007) Optimized Schwarz waveform relaxation methods for advection reaction diffusion problems. SIAM J Numer Anal 45(2):666–697 10. Gander MJ, Halpern L, Nataf F (2003) Optimal Schwarz waveform relaxation for the one dimensional wave equation. SIAM J Numer Anal 41(5):1643–1681. https://doi.org/10.1137/ S003614290139559X, http://dx.doi.org/10.1137/S003614290139559X 11. Gander MJ, Kwok F, Mandal BC (2016) Dirichlet-Neumann and Neumann-Neumann waveform relaxation algorithms for parabolic problems. Electron Trans Numer Anal 45:424–456 12. Ong BW, Mandal BC, Pipeline Implementations of Neumann–Neumann and Dirichlet– Neumann waveform relaxation methods. to appear arXiv:1605.08503 13. Hoang TTP (2013) Space-time domain decomposition methods for mixed formulations of flow and transport problems in porous media. PhD thesis, University Paris 6, France 14. Kwok F (2014) Neumann-Neumann waveform relaxation for the time-dependent heat equation. In: Erhel J, Gander MJ, Halpern L, Pichot G, Sassi T, Widlund OB (eds) Domain decomposition in science and engineering XXI, vol 98. Springer, Berlin, pp 189–198 15. Mandal BC (2016) Neumann-Neumann waveform relaxation algorithm in multiple subdomains for hyperbolic problems in 1d and 2d. Numer Methods Partial Differ Equ. https://doi.org/10. 1002/num.22112

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16. Hoang TTP, Jaffré J, Japhet C, Kern M, Roberts JE (2013) Space-time domain decomposition methods for diffusion problems in mixed formulations. SIAM J Numer Anal 51:3532–3559 17. Farhat C, Roux FX (1991) A method of finite element tearing and interconnecting and its parallel solution algorithm. Int J Numer Methods Eng 32:1205–1227 18. Le Tallec P, De Roeck Y, Vidrascu M (1991) Domain decomposition methods for large linearly elliptic three-dimensional problems. J Comput App Math 34:93–117 19. Jiang YL, Song B (2018) Coupling parareal and dirichlet-neumann/neumann-neumann waveform relaxation methods for the heat equation. In: International conference on domain decomposition methods XXIV. Springer, Berlin, pp 405–413 20. Gander MJ (2015) 50 years of time parallel time integration. In: Multiple shooting and time domain decomposition methods. Springer, Berlin, pp 69–113 21. Schep A (1980) Positive diagonal and triangular operators. J Oper Theory 165–178

A Generalized Hilbert Operator on Bloch Space and BMOA Spaces S. Naik and P. K. Nath

Abstract We consider the generalized Hilbert operators Ha,b , Ha,b ( f )(z) =

∞ ∞   (b − a)n μn,k n=0

k=0

n!

 ak z n ,

 k where a, b ∈ C, f (z) = ∞ k=0 ak z analytic on the unit disc D, μ be a positive Borel measure on the interval [0, 1) and μn denote the moment of order n of μ, that is, μn = [0,1) t n dμ(t) with μn,k = μn+k . This is one of the generalization of the classical Hilbert operator. In this paper, we characterize the measures μ and find the conditions on a, b such that Ha,b is bounded and compact on Bloch space and B M O A spaces. Keywords Hilbert operators · Bloch space · B M O A spaces · Boundedness

1 Introduction Let D = {z ∈ C : |z| < 1} denote the open unit disc in the complex plane C and H (D) be the space of all analytic functions in D endowed with topology of uniform convergence in compact subsets. We also let H p (0 < p ≤ ∞) be the classical Hardy spaces. For notation  and kmore results regarding Hardy Spaces see [7]. For f (z) = ∞ k=0 ak z ∈ H (D), the generalized Hilbert operator is defined by

S. Naik (B) Department of Applied Sciences, Gauhati University, Guwahati 781 014, Assam, India e-mail: [email protected] P. K. Nath Department of Mathematics, Pandit Deendayal Upadhyaya Adarsha Mahavidyalaya, Dalgaon, Darrang 784 116, Assam, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 D. Giri et al. (eds.), Proceedings of the Sixth International Conference on Mathematics and Computing, Advances in Intelligent Systems and Computing 1262, https://doi.org/10.1007/978-981-15-8061-1_35

441

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Hμ ( f )(z) =

∞ ∞   n=0

 μn,k ak z n ,

k=0

whenever the right-hand side makes sense and defines an analytic function in D. Here, μ is a finite positive Borel measure on [0,  1) and n = 0, 1, 2, . . . , we let μn denote the moment of order n of μ, that is, μn = [0,1) t n dμ(t), and we let Hμ be the Hankel matrix (μn,k )n,k≥0 with entries μn,k = μn+k . The matrix Hμ induces formally an operator, which will also be called H μ , on the spaces of analytic functions by its action on the Taylor coefficients: an → ∞ k=0 μn,k ak , n = 0, 1, 2, . . . If μ is the Lebesgue measure on [0, 1), the matrix Hμ reduces to the classical Hilbert matrix H = ((n + k + 1)−1 )n,k≥0 , which induces the classical Hilbert operator H which have been extensively studied recently (see [1, 4–6, 11]). Many authors have studied the operator Hμ for its actions on Hardy spaces H p , 0 < p < ∞ by characterizing the measures μ so that Hμ is well defined and bounded on H p . For details see [3, 8]. The integral representation of Hμ is given by  Iμ ( f )(z) = [0,1)

f (t) dμ(t), 1 − tz

where μ is as above, and whatever the right-hand side makes sense and defines an analytic function in D, see [10]. For a, b ∈ C, we define an operator Ha,b as follows: ∞  ∞   (b − a)n μn,k ak z n , Ha,b ( f )(z) = n! n=0 k=0 where (a, n) is the shifted factorial defined by Appel’s symbol (a, n) = a(a + 1)...(a + n − 1) =

Γ (a + n) , n ∈ N = {1, 2, 3, . . .} Γ (a)

and (a, 0) = 1 for a = 0. A simple calculation gives an integral representation of the operator Ha,b which is given as follows:  Ia,b ( f )(z) = [0,1)

f (t) dμ(t), (1 − t z)b−a

whenever right-hand side is well defined and analytic in D. In particular, if we take b − a = 1 then we get

A Generalized Hilbert Operator on Bloch Space and BMOA Spaces

443

Ha,a+1 ( f )(z) = Hμ ( f )(z). Thus operator Ha,b generalize Hμ . If I ⊂ ∂D is an arc, |I | will denote the length of I. The Carleson square S(I ) is |I | ≤ r < 1}. defined as S(I ) = {r eit : eit ∈ I, 1 − 2π If s > 0 and μ is a positive Borel measure on D, we shall say that μ is an s− Carleson measure if there exists a positive constant C such that μ(S(I )) ≤ C|I |s , for any interval I ⊂ ∂D. )) = 0, then we say that μ is a vanishing s-Carleson If μ satisfies lim|I |→0 μ(S(I |I |s measure. A 1-Carleson measure, respectively, a vanishing 1−Carleson measure, will be simply called a Carleson measure, respectively, a vanishing Carleson measure. Let us start considering the Bloch space and B M O A. The Bloch space B consists of all analytic functions f in D with bounded invariant derivative:

f ∈ B ⇔ f B = | f (0)| + sup(1 − |z|2 )| f  (z)| < ∞. z∈D

A classical source for the Bloch space is [2]; see also [12]. The space B M O A is defined to be the class of all analytic functions in H 1 whose boundary values have bounded mean oscillation on the circle ∂D. There are many characterizations of B M O A functions. Let us mention the following: If f is an analytic function in D, then f ∈ B M O A if and only if

f B M O A = | f (0)| + f  , where

f  = sup f oϕa − f (a) H 2 , a∈D

a−z where ϕa (z) = 1− . We mention [3] as general references for the spaces B M O A. az ¯ Let us recall that  H p and B M O A  B. H∞  BM O A  0< p 0 we have | f (z)| < C f B log Let A = have



2 1 − |z|

(1)

2 log 1−t dμ(t) and suppose (i) is true. Using (1) for every f ∈ B we   2 dμ(t) = AC f B | f (t)|dμ(t) ≤ C f B log (2) 1−t

[0,1)

[0,1)

which gives

[0,1)

 [0,1)

| f (t)| AC f B dμ(t) ≤ . |1 − t z|b−a (|1 − |z|)b−a

(3)

Using (2),  (3) and Fubini’s theorem we see that if f ∈ B then: for every n ∈ N, the integral [0,1) t n f (t)dμ(t) converges absolutely and  n t f (t)dμ(t) < ∞. sup n≥0 [0,1) The integral



 [0,1)

f (t) [0,1) (1−t z)b−a dμ(t)

converges absolutely, and

⎛ ⎞  ∞  f (t) (a − b)n n ⎜ ⎟ dμ(t) = t f (t)dμ(t)⎠ z n . ⎝ (1 − t z)b−a n! n=0 [0,1)

A Generalized Hilbert Operator on Bloch Space and BMOA Spaces

445

Thus, if f ∈ B then Ia,b ( f ) is a well-defined analytic function in D and ⎛ Ia,b ( f )(z) =

∞  n=0

⎜ ⎝

⎞



[0,1)

(a − b)n n ⎟ t f (t)dμ(t)⎠ z n . n!

Since B M O A ⊂ B, the implication (ii) ⇒ (iii) is obvious. Now (iii) ⇒ (i). 2 Suppose (iii) hold. Since the function F(z) = log 1−z ∈ B M O A, Ia,b (F)(z) is well defined for every z ∈ D. In particular  Ia,b (F)(0) =

log [0,1)

2 dμ(t) 1−t

2 is a complex number. Since μ is a positive measure and log 1−t > 0 for all t ∈ [0, 1), (i) follows.

Remark 1 If we take b = a + 1 in Theorem 1, we obtain the Theorem 2.1 of [10]. In the next result, we find the conditions on a, b so that Ia,b is bounded in B M O A and B.  2 dμ(t) < ∞ Theorem 2 Let μ be a positive Borel measure on [0, 1) with [0,1) log 1−t and a, b ∈ C. Then the following conditions are equivalent: 2 dμ(t) is a Carleson measure. (i)The measure ν defined by dν(t) = log 1−t (ii)For a − b is an integer the operator Ia,b ( f ) is bounded from B into B M O A. (iii)For a − b is an integer the operator Ia,b ( f ) is bounded from B M O A into itself.  2 Proof (i) ⇒ (ii). Since [0,1) log 1−t dμ(t) < ∞, for all f ∈ B. From (2), we have  | f (t)|dμ(t) < ∞

(4)

[0,1)

For 0 ≤ r < 1, f ∈ B and g ∈ H 1 , from (4) we get, 2π  0 [0,1)

| f (t)||g(eiθ )| dμ(t)dθ < ∞. |1 − r eiθ |b−a

(5)

Since a, b are integers, using Fubini’s theorem and Cauchy’s integral representation of H 1 -functions ([7], Theorem 3.6) in (5), we deduce that whenever f ∈ B and g ∈ H 1 we have

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2π

2π Ia,b ( f )(r eiθ )g(eiθ )dθ =

0

0

⎛ 2π  f (t) ⎝

 = [0,1)

0

⎛ ⎜ ⎝



[0,1)

⎞  g(eiθ )dθ ⎠ dμ(t) = (1 − r eiθ t)b−a

⎞ f (t)dμ(t) ⎟ ⎠ g(eiθ )dθ (1 − r eiθ t)b−a

(6)

f (t)g b−a−1 (r t)dμ(t), 0 ≤ r < 1.

[0,1)

Assume that ν is a Carleson measure and take f ∈ B and g ∈ H 1 . Using (6) and (1), we find 2π   Ia,b ( f )(r eiθ )g(eiθ )dθ =

[0,1)

0

  f B

|g b−a−1 (r t)| log [0,1)

f (t)g b−a−1 (r t)dμ(t)

2 dμ(t) = f B 1−t

 |g b−a−1 (r t)|dν(t) [0,1)

Since ν is a Carleson measure,  |g b−a−1 (r t)|dν(t)  grb−a−1 H 1 ≤ g b−a−1 H 1 . [0,1)

Here, as usual, gr is the function defined by grb−a−1 (z) = g b−a−1 (r z), (z ∈ D). Thus, we have proved that 2π  Ia,b ( f )(r eiθ )g b−a−1 (eiθ )dθ  f B g b−a−1 H 1 ,

f ∈ B, g ∈ H 1 .

0

Using Fefferman’s duality Theorem (see ([9], Theorem 7.1)) we deduce that if f ∈ B then Ia,b ( f ) ∈ B M O A and

Ia,b ( f ) B M O A  f B . The implication (ii) ⇒ (iii) is trivial because B M O A ⊂ B. (iii) ⇒ (i). Assume (iii). Then there exists a positive constant A such that Ia,b ( f ) B M O A ≤ A f B M O A , for all f ∈ B M O A. Let

A Generalized Hilbert Operator on Bloch Space and BMOA Spaces

F(z) = log

447

2 , z ∈ D. 1−z

It is well known that F ∈ B M O A. Then Ia,b (F) ∈ B M O A and

Ia,b (F) B M O A ≤ A F B M O A . Now, using Fefferman’s duality theorem we obtain that 2π  Ia,b (F)(r eiθ )g b−a−1 (eiθ )dθ  g b−a−1 H 1 , g b−a−1 ∈ H 1 . 0

Using (6) and definition of F, this implies  2 dμ(t)  g b−a−1 H 1 , g b−a−1 (r t) log 1−t [0,1)

f ∈ B, g b−a−1 ∈ H 1 .

(7)

Take g ∈ H 1 . Using Proposition 2 of [3], we know that there exists a function G ∈ H 1 with G H 1 = g H 1 and such that |g(s)| ≤ G(s), f or all s ∈ [0, 1). Using these properties and (7) for G, we obtain  |g b−a−1 (r t)| log [0,1)

2 dμ(t) ≤ 1−t

 G b−a−1 (r t) log [0,1)

2 dμ(t) 1−t

≤ C G rb−a−1 H 1 ≤ C G b−a−1 H 1 = C g b−a−1 H 1

for a certain constant C > 0, independent of g. By taking r tend to 1, it follows that  |g b−a−1 (t)| log [0,1)

2 dμ(t)  g b−a−1 H 1 , g ∈ H 1 . 1−t

This is equivalent to saying that ν is a Carleson measure. Remark 2 If we take b = a + 1 in Theorem 2, we obtain the Theorem 2.2 of [10]. We have the following result for compactness. Theorem 3 Let a, b ∈ C. Suppose μ be a positive Borel measure on [0, 1) with  log [0,1)

2 dμ(t) < ∞. 1−t

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2 a If the measure ν defined by dν(t) = log 1−t dμ(t) is a vanishing Carleson measure then (i)For a − b is an integer the operator Ia,b is compact operator from B into B M O A. (ii)For a − b is an integer the operator Ia,b is a compact operator from B M O A into itself.

We recall some facts about Carleson measures. If μ is a Carleson measure on D, we define the Carleson-norm of μ, denoted N (μ), as N (μ) =

μ(S(I )) . |I | I subar co f ∂D sup

We let also ε(μ) denote the norm of the inclusion operator i : H 1 → L 1 (dμ). It turns out that these quantities are equivalent: There exist two positive constants C1 , C2 such that C1 N (μ) ≤ ε(μ) ≤ C2 N (μ), for every Carleson measure μ on D. For a Carleson measure μ on D and 0 < r < 1, we let μr be the measure on D defined by dμr (z) = χ {r < |z| < 1}dμ(z). We have that μ is a vanishing Carleson measure if and only if N (μr ) → 0, as r → 1. Proof Since B M O A ⊂ B, it is sufficient to prove (i). Suppose that ν is a vanishing Carleson measure, let { f n }∞ n=1 be a sequence of Bloch functions with supn≥1 f n B < ∞ and such that f n → 0, uniformly on compact subsets of D. We have to prove that Ia,b ( f n ) → 0 in B M O A. The condition supn≥1 f n B < ∞ implies that there exists a positive constant M such that 2 , z ∈ D, n ≥ 1. (8) | f n (z)| ≤ M log 1 − |z| By taking g ∈ H 1 and r ∈ [0, 1) and using (8), (6) and proceeding similar to the proof of the Theorem 2.8 of [10], we have ⎞ 2π  iθ iθ lim ⎝ lim Ia,b ( f n )(r e )g(e )dθ ⎠ = 0, n→∞ r →1 ⎛

f or all g ∈ H 1 .

0

By the duality relation (H 1 ) = B M O A, this is equivalent to Ia,b ( f n ) → 0 in B M O A. Remark 3 If we take b = a + 1 in Theorem 3, we obtain the Theorem 2.8 of [10]. Conflicts of Interests The authors declare that there is no conflict of interests regarding the publication of this paper.

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Determining the Disease Status Using Gene Expression Analysis Dulal Adak, Suman Mitra, Biswajit Jana, and Sriyankar Acharyya

Abstract Colon Cancer is quite prevalent in the worldwide scenario. A prognostic model of colon cancer has been developed using both microarray gene expression data and patients clinical information. A small subset of informative genes is required to design the prognostic model. Here, in this paper, a two-step methodology has been designed to select a subgroup of genes. In the first step, the Cox Proportional Regression Model has been applied to choose a subgroup of genes. Afterwards, the Benjamini-Hochberg method has been applied to reduce the false discovery rate. Cosine similarity method has been applied to measure the similarity between reference patient and query patient. 5-fold cross validation has been performed to compute the accuracy. A comparative analysis has been conducted between Euclidian distance and Cosine similarity. The experimental results have shown that the performance using Cosine similarity is better than the Euclidean distance. Keywords Gene selection · Cox proportional regression model · Benjamini-Hochberg · Euclidean distance · Cosine similarity

D. Adak · S. Mitra · B. Jana · S. Acharyya (B) Maulana Abul Kalam Azad University of Technology, Kolkata West Bengal,, India e-mail: [email protected] D. Adak e-mail: [email protected] S. Mitra e-mail: [email protected] B. Jana e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 D. Giri et al. (eds.), Proceedings of the Sixth International Conference on Mathematics and Computing, Advances in Intelligent Systems and Computing 1262, https://doi.org/10.1007/978-981-15-8061-1_36

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1 Introduction Colon cancer is found to be one of the most deadly cancers, prevailing in men and women [1]. Gene portrays a significant role in the growth and spread of tumors. There are various other risk factors like age, family history, personal lifestyle, and personal history. The goal of this work is to predict the reference patient from an existing database to stand for (most similar to) the new patient. In other words, it is a mapping from new patients to reference patients. It can help doctors provide proper treatment to the new patient based on the treatment had been provided to the most similar reference patient. This prediction has been performed through selecting disease critical genes using miRNA gene expression dataset and patient clinical information. The experiment has been conducted on microarray gene expression dataset and patient clinical data. The data is collected from NCBI GEO, an open-source biological database. In this work, we have used two datasets: GSE29623 (for reference patient dataset containing 145 patients (samples) and 22618 genes) and GSE17536 (for query patient dataset consisting of 65 patients (samples) and 17500 genes). Here, a brief review of these works has been presented. Eddy et al. [2] have conducted a survey on different types of computational methods to study transcriptomic microarray data and also addressed its limitations. They have also discussed about the Relative Expression analysis, specifically, TSP and K-TSP classifier for high throughput data analysis in cancer diagnosis and prognosis. Tarca et al. [3] have described the data processing, experimental design, and gene selection methods for gene expression profiling. They have also discussed different microarray experiments like class prediction, class comparison, and class discovery. Bao et al. [4] have discussed early stage breast cancer diagnosis and treatment based on the clinical-pathological analysis. Dobbin et al. [5] have developed a nonparametric algorithm to split the dataset into training and testing sets optimally for a specific classifier. Tang et al. [6] have conducted a study to understand the role, clinicpathological functions, and prognostic values of let-7 miRNA family in High-Grade serous ovarian carcinoma. Chen et al. [7] proposed a large number of conditions for the existence of Maximum Partial Likelihood Estimate (MPLE) for fully observed data and Maximum Likelihood Estimate (MLE) with missing data. Mishra et al. [8] have conducted a review on the available treatment plans for patients affected with colorectal cancer. They have discussed on the development of basic research, microRNA-technology, gene delivery, and miRNA. Winkles et al. [9] have discussed the interrelation of lifestyle, diets, and quality of life in causing colon cancer. Cannon and Buechler [10] have predicted the location of the tumor for colon cancer using gene expression data. This insight has been used in the treatment of colon tumors. Kuznetsov et al. [11] have introduced a multi-gene prognostic signature method to divide the patients into different groups based on disease development risk and patient clinical data. Motakis et al. [12] have proposed One-Dimensional Data-Driven grouping (1D-DDg) to rank the sample. 1D-DDg based on Cox Proportional hazard regression model.

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In this paper, a prognostic model has been developed for colon cancer. Onedimensional data-driven grouping has been used to choose a small section of informative genes for colon cancer. Benjamini-Hochberg [13] method has been applied to reduce the false discovery rate in gene selection. A novel distance measurement method, namely, Cosine Similarity [14], has been proposed and implemented. Fivefold cross validation methods has been used to compute the accuracy. The experiment has been conducted on microarray gene expression dataset and clinical reports obtained from the patients affected by colon cancer. Performance comparison has been made between cosine similarity and Euclidian distance. It is observed that the cosine similarity method outperformed the Euclidian distance-based model. This paper is structured into 5 sections. Section 1 contains introduction. Sect. 2 defines the problem. The methodology used in the experiment is described in Sect. 3. The experimental results have been specified in Sect. 4. Section 5 concludes the research work and also provides some futuredirections.

2 Similarity Identification Problem Each patient (sample) is represented by a vector of size N, where N denotes the number of predictor variables or genes. Each patient has been classified into two risk groups (high and low) using Kaplan-Meier curve. High-risk patient is represented as 1 and low-risk patient is represented as –1. This sample vector contains bipolar formatted value (1 and –1) and called as Prognostic Binary Variable Vectors (PBVV). By comparing two PBVV of size N × N , cannot be found with exact similarity. For getting similarity, PBVV has been transformed to Prognostic Signature Vector (PSV). Then, similarity has been computed between a new patient and reference patient using Euclidian distance and Cosine similarity.

3 Methodology 1D-DDg [12] method is used for selecting a small subset of disease critical genes. This method is based on the Cox proportional regression model. The input of this model is the normalized gene expression data and the clinical data collected from the patients. Data-driven grouping, Benjamini-Hochberg and Euclidian distance, and Cosine similarity have been discussed in Sects. 3.1, 3.2 and 3.3, respectively.

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3.1 Data-Driven Grouping Data-Driven grouping [12], has been derived from the semi-parametric Cox Proportional Hazard Regression Model (See Eq. 1). The Cox Proportional Hazard Model has taken several variables (factors) as input simultaneously for computing the survival time of a gene for specific events at a particular instant. Here, Overall Survival (OS) and event (clinical information) are incorporated with the microarray gene expression data of each patient. Using this method, the particular gene obtained is identified as high-risk and low-risk. 



h z i, t = h 0 (t) exp

 m 

 z i, j β j

(1)

i=1

• h (zi , t) is the hazard rate at time t for zi • Where zij = (zi1 , zi2 , ……. zin ) is a predictor variable of individual gene j of n number of patients. • β is going to be calculated using Partial Likelihood Function. • h0 (t) is a baseline hazard rate function where all the covariates denote to zero. Hazard Ratio (HR): The Cox Proportional Hazard model relates the hazard rate of each individual (gene) Xi to baseline hazard value of that genes (See Eq. 2). ⎡ ⎤ n  h(X i , t) H R(X i ) = = exp⎣ Xi j β j ⎦ h 0 (t) j=1 If, HR 1: then-No Effect in Hazard. HR < 1 then Deduction in Hazard. HR > 1 then Escalate in Hazard, as a result, length of survival decrease.

(2)

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3.2 Benjamini-Hochberg 1D-DDg method generated a small subset of informative genes along with P-value. Benjamini-Hochberg (B-H) method has been applied to identify the type-1 error which may occur during gene selection using 1D-DDg. B-H method helps to remove falsely identified genes. The final subset of informative genes has been selected using B-H. The formula is given below   P= i m ∗Q

(3)

Here, ‘i’ represents the rank of individual p-values, ‘m’ denotes a total number of tests and ‘Q’ stands for false discovery rate (predefined constant).

3.3 Euclidian Distance and Cosine Similarity Euclidian distance and Cosine similarity are used to compute the similarity between reference patients and query patient using Prognostic Significance Vector. Cosine similarity (see Eq. 5) is applied here to compute the similarity between a new patient and Reference patients. Outcome of a Cosine similarity is bounded in the range, zero

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to one. The maximum value of Cosine Similarity implies that there is a maximum similarity between query patient and reference patient. Euclidean distance between reference and query patient is computed using Eq. 4 minimum distance implies that there is a maximum similarity between reference patient and query patient. Here query patients are denoted as Xi and reference patients as Xj . The formula of Euclidean distance is

n

    2 X k, j − X k,i (4) d X j, X i = |X i − X i | =  k=1

And the formula of Cosine similarity is n X ·Y j=1 X j Y j   similarit y = cos(θ ) = n n X Y  X2 j=1

j

(5)

2 j=1 Y j

4 Data and Experimental Results Experimental setup, gene expression dataset have been discussed in Sect. 4.1, and results and discussion have been carried upon in Sect. 4.2.

4.1 Data and Experimental Setting The experiment has been performed on two different computers. The first computer has Intel Xeon(R) processor with a processing speed of 2.27 GHz × 8. It has 64-bit linux and 32 GB RAM. The second computer has Intel core i3 with a processing speed 1.70 GHz, 64-bit operating system, and 4.00 GB RAM. Programs are written in R 3.5.0 and Matlab R2018. Two gene expression datasets have been used in this work. Detail of gene expression data is given in Table 1. Each dataset is a two-dimensional matrix, where, row represents genes and column represents sample Id (patients). Each element of a matrix is a gene expression value and it is a floating point number. Initially, gene expression dataset may contain some Table 1 Microarray gene expression datasets collected for experiments Sl. no.

Dataset accession no.

Collected from (site)

Total number of samples (m)

Number of genes (n)

1

GSE17536

GEO, NCBI

145

22,618

2

GSE29623

GEO, NCBI

65

17,500

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missing values. Therefore, it is required to perform normalization on this data. Zscore has been applied here for this purpose. After normalization of this dataset, experiments have been conducted.

4.2 Results and Discussion

PSV values at the ith variable

The algorithm has been executed after initializing all parameters. During the experiment, 36 responsible genes have been selected for colon cancer out of 22,315 genes. The selected genes are:- CACFD1, FTH1P16, KCNK1, LAMC2, DOK5, CDR2L, DGKD, CALB2, ENO2, PTPRR, EPYC, AC026362.1, OGFOD2, OGFOD2, FTH1, CEP170, CEP170, AKT3, TIMP1, NRP1, PLAUR GAS1, KIAA1549L, LY6E, AL049629.2, CD59, SLC35E2B, SLC7A7. In Fig. 1, the horizontal line represents the PSV (prognostic Significant Vector) values of three reference patients and one query patient. The vertical line represents the variable or gene names. This figure showed that the third reference patient is most similar to the query patient. But accurately does not identify the similar reference patient. To solve this problem, Euclidean distance and Cosine similarity method has been applied to find out exact similar patient. Table 2 represents the Euclidian distance and Cosine Similarity of three reference patients versus one query patient. Using Euclidean distance most of the similar reference is GSM437094 with respect to query patient GSM734111. But the cosine similarity identifies the most similar reference patient GSM437099 with respect to query patient GSM734111. 60 40 20 0 -20

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35

reference 1

Ordered variable (ith) axis reference 2

reference 3

query patient

Fig. 1 Two dimensional representations of PSV values

Table 2 Euclidean distance and Cosine similarity between three reference patients with one query patient

Reference patients

Euclidean distance with GSM734111

Cosine similarity with GSM734111

GSM437094

114.46

0.724

GSM437097

244.34

0.477

GSM437099

195.21

0.791

458 5-fold cross validation 150

Accuracy

Fig. 2 Performance of Euclidean distance and Cosine similarity

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100 50 0 1 Euclidean distance

Table 3 Accuracy of Euclidean distance and Cosine similarity

Cosine similarity

Methods name

Accuracy

Euclidean distance

70.46

Cosine similarity

96.0094

Figure 2 shows the performance of two distance calculation methods, namely, Euclidean distance and Cosine similarity. They are used for finding similar patients. The accuracy of each of the distance calculation method is represented in Table 3. From the above table, it is evident that the accuracy of cosine similarity is better than Euclidean distance. Cosine similarity-based prognostic model can identify reference patient with 96.0094% accuracy.

5 Conclusion In this research work, a prognostic model has been developed for colon cancer based on 1D-DDg model. A new distance measurement technique, namely, Cosine Similarity, has been applied here to compute the similarity between query patient and reference patients. This experiment has required patients clinical information and microarray gene expression data. Comparative analysis has been performed between Euclidean distance and cosine similarity. Cosine similarity outperforms the Euclidean distance-based model.In future, this model can be applied to other cancer to make a prognostic model.

References 1. Patil PS, Saklani A, Gambhire P, Mehta S, Engineer R, De’Souza A, Bal M (2017) Colorectal cancer in India: an audit from a tertiary center in a low prevalence area. Indian J Surgical Oncology 8(4):484–490 2. Eddy JA, Sung J, Geman D, Price ND (2010) Relative expression analysis for molecular cancer diagnosis and prognosis. Technol Cancer Res Treat 9(2):149–159 3. Tarca AL, Romero R, Draghici S (2006) Analysis of microarray experiments of gene expression profiling. Am J Obstet Gynecol 195(2):373–388

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4. Bao T, Davidson NE (2008) Gene expression profiling of breast cancer. Adv Surg 42:249–260 5. Dobbin KK, Simon RM (2011) Optimally splitting cases for training and testing high dimensional classifiers. BMC Med Genomics 4(1):31 6. Tang Z, Ow GS, Thiery JP, Ivshina AV, Kuznetsov VA (2014) Meta-analysis of transcriptome reveals let-7b as an unfavorable prognostic biomarker and predicts molecular and clinical subclasses in high-grade serous ovarian carcinoma. Int J Cancer 134(2):306–318 7. Chen MH, Ibrahim JG, Shao QM (2009) Maximum likelihood inference for the Cox regression model with applications to missing covariates. J Multivariate Anal 100(9):2018–2030 8. Mishra J, Drummond J, Quazi SH, Karanki SS, Shaw JJ, Chen B, Kumar N (2013) Prospective of colon cancer treatments and scope for combinatorial approach to enhanced cancer cell apoptosis. Critical Rev Oncology/Hematology 86(3):232–250 9. Winkels RM, Heine-Bröring RC, Zutphen M, Harten-Gerritsen S, Kok DE, Duijnhoven Van FJ, Kampman E (2014) The COLON study: Co lorectal cancer: longitudinal, observational study on nutritional and lifestyle factors that may influence colorectal tumour recurrence, survival and quality of life. BMC Cancer 14(1):374 10. Cannon E, Buechler S (2019) colon cancer tumor location defined by gene expression may disagree with anatomic tumor location. Clin Colorectal Cancer 11. Kuznetsov VA, Ivshina AV, Sen’Ko OV, Kuznetsova AV (1996) Syndrome approach for computer recognition of fuzzy systems and its application to immunological diagnostics and prognosis of human cancer. Math Comput Model 23(6):95–119 12. Motakis E, Ivshina AV, Kuznetsov VA (2009) Data-driven approach to predict survival of cancer patients. IEEE Eng Med Biol Mag 28(4):58–66 13. Benjamini Y, Yekutieli D (2001) The control of the false discovery rate in multiple testing under dependency. Ann Stat 29(4):1165–1188 14. Garcia E (2018) a tutorial on distance and similarity. Accessed from http://www.minerazzi. com/tutorials/distance-similarity-tutorial.pdf

Generalized Double Statistical Convergence in Topological Groups Ekrem Savas

Abstract The notion of statistical convergence of double sequences in topological groups is introduced by Çakalli and Sava¸s. The goal of this paper is to introduce a class of summability methods that can be applied to λ- double statistical convergence in topological groups and is to prove some theorems. Keywords λ- double statistical convergence · Topological groups · Pringsheim limit

1 Introduction Looking through historically at statistical convergence of single sequences, we recall that the concept of statistical convergence of sequences was first studied by Fast [4] (see also Schoenberg [16]). The notion of statistical convergence of a sequence (xk ) in a locally convex Hausdorff topological linear space X was introduced recently by Maddox [8], where it was shown that the slow oscillation of (sk ) was a Tauberian condition for the statistical convergence of (sk ). In [6], statistical convergence to normed spaces was extended by Kolk. Further in [1] and [2], Çakalli extended this notation to topological Hausdorff groups. Savas [15] introduced lacunary statistical convergence of double sequences in topological groups. Also double ideal lacunary statistical convergence in topological groups was studied by Sava¸s (see, [13]). Finally Savas [14] studied Iλ -double statistical convergence of order α in topological groups. More results on double statistical convergence can be seen from [3, 11, 12]. Recall that for a subset F of N the asymptotic density of F, denoted by δ(F), is defined by 1 δ(F) = lim |{k ≤ i : k ∈ F}| i→∞ i

E. Savas (B) Department of Mathematics, U¸sak University, U¸sak, Turkey e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 D. Giri et al. (eds.), Proceedings of the Sixth International Conference on Mathematics and Computing, Advances in Intelligent Systems and Computing 1262, https://doi.org/10.1007/978-981-15-8061-1_37

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if this limit exists, where |{k ≤ i : k ∈ F}| denotes the cardinality of the set |{k ≤ i : k ∈ F}| . A sequence x = (xi ) is statistically convergent to ξ if δ({i : |xi − ξ| ≥ ε}) = 0 for every ε > 0, (see [5]). In this case ξ is called the statistical limit of x. Mursaleen and Edeley [9] defined and studied statistical analogue of convergence and Cauchy for double sequences and also established the relation between statistical convergence and strongly Cesàro summable double sequences. Throughout this paper, we follow some notion used in [3]. By the convergence of a double sequence, we mean the convergence in Pringsheims in the Pringsheims sense [10]. A double sequence x = (xi j ) is said to be   convergent sense if for every ε > 0 there exists N ∈ N such that xi j − ξ  < ε whenever i, j ≥ N . ξ is called the Pringsheim limit of x. A double sequence x = (xi j ) is saidto be Cauchy sequence if for every ε > 0 there exists N ∈ N such that x pq − xi j  < ε for all p ≥ i ≥ N and q ≥ j ≥ N . In a topological group E, the above definitions become as in the following: a double sequence x = (xi j ) in E is said to be convergent to ξ in E in the Pringsheims sense if for every neighbourhood U of 0 there exists N ∈ N such that xi j − ξ ∈ U whenever i, j ≥ N . ξ is called the Pringsheim limit of x. A double sequence x = (xi j ) is said to be a Cauchy sequence if for every neighbourhood U of 0 there exists N ∈ N such that x pq − xi j ∈ U for all p ≥ i ≥ N and q ≥ j ≥ N . The goal of this paper is to introduce the λ- double statistical convergence of double sequences in topological groups and to prove some useful theorems.

2 Definitions and Notation By E, we will denote an abelian topological Hausdorff group, written additively, which satisfies the first axiom of countability. For a subset B of E, s(B) will denote the set of all sequences (xi ) such that xi is in B for i = 1, 2, ..., c(E) will denote the set of all convergent sequences. A sequence (xi ) in E is called to be statistically convergent to an element ξ of E if for each neighbourhood U of 0, ( see, [2]) 1 |{k ≤ i : xi − ξ ∈ / U }| = 0 i→∞ i lim

and is called statistically Cauchy in E if for each neighbourhood U of 0 there exists a positive integer i 0 (U ), depending on the neighbourhood U , such that  1  {k ≤ i : xi − xi0 (U ) ∈ / U } = 0 i→∞ i lim

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where the vertical bars indicate the number of elements in the enclosed set. The set of all statistically convergent sequences in E is denoted by S(E) and the set of all statistically Cauchy sequences in E is denoted by SC(E). It is known that SC(E) = S(E) if E is complete. Let λ = (λ p ) and μ = (μq ) be two non-decreasing sequences of positive real numbers both of which tends to ∞ as p and q approach ∞, respectively. Also let λ p+1 ≤ λ p + 1, λ1 = 1 and μq+1 ≤ μq + 1, μ1 = 1. The collection of such sequence will be denoted by Δ. We write the generalized double de la Valee-Poussin mean by t p,q (x) =

 1 xi j , (λ p μq ) i∈I , j∈J p

q

where I p = [ p − λ p + 1, p] and Jq = [q − μq + 1, q]. Throughout this paper we shall denote λ p μq by λ¯ pq and (i ∈ I p , j ∈ Iq ) by (i, j) ∈ I pq .

3 λ-Double Statistical Convergence Let R ⊂ N × N be a two-dimensional set of positive integers and let R(n, m) be the numbers of (i, j) in R such that i ≤ n and j ≤ m. Then the two-dimensional analogue of natural density can be defined as follows. The lower asymptotic density of a set R ⊂ N × N is defined as δ2 (R) = lim inf n,m

R(n, m) nm

  In case the sequence R(n,m) has a limit in Pringsheims sense, then we say that R nm has a double natural density and is defined as δ2 (R) = lim n,m

R(n, m) nm

Mursaleen and Edely [9] called a real double sequence x = (xi j ) statistically   convergent to the number ξ if for each ε > 0, the set {(i, j), i ≤ n and j ≤ m : xi j − ξ  ≥ ε} has double natural density zero. In this case, we write S2 − limi, j xi j = ξ and we denote the set of all statistically convergent double sequences by S2 . We now define statistical convergence of double sequences x = (xi j ) in a topological group in the following. A double sequence x = (xi j ) is called statistically convergent to a point ξ of E if for each neighbourhood U of 0 the set / U} {(i, k), i ≤ n; and; j ≤ m : xi j − ξ ∈

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has double natural density zero. In this case, we write S2 (E) − limi, j xi j = ξ and we write the set of all statistically convergent double sequences by S2 (E). Definition 1 A double sequence x = (xi j ) is said to be Sλ2 −convergent to ξ of E (or λ- double statistically convergent to ξ of E ) if for each neighbourhood U of 0, the set   /U (i, j) ∈ I pq : xi j − ξ ∈ has double natural density zero. In this case, we write Sλ2 − lim xi j = ξ or i, j→∞

xi j → ξ(Sλ2 )

and we write the set of all λ-statistically convergent double sequences by Sλ2 (E). A λ-statistically convergent double sequence has a unique limit, i.e. if x is λstatistically convergent to elements ξ1 and ξ2 of E, then ξ1 = ξ2 . Theorem 1 A double sequence x = (xi j ) in E is λ- double statistically convergent to ξ if and only if there exists a subset R ⊂ N × N such that δλ2 (R) = 1 and limi, j→∞ xi j = ξ where limit is being taken over the set E, i.e. (i, j) ∈ E. Proof Necessity. Suppose that x be λ- double statistically convergent to ξ, and (Ur ) be a base of nested closed neighbourhoods of 0. Write / Ur } Rr = {(i, j) ∈ I pq : xi j − ξ ∈ Fr = {(i, j) ∈ I pq : xi j − ξ ∈ Ur }

(r = 1, 2, ...)

Then δλ2 (Rr ) = 0 and F1 ⊃ F2 ⊃ ... ⊃ Fi ⊃ Fi+1 ⊃ ...

(1)

and δλ2 (Fr ) = 1, r = 1, 2, ... (2)  Now we have toshow that for (i, j) ∈ Fr , xi j is λ- double statistically convergent to ξ. Suppose that xi j is not λ- double convergent to ξ so that there is a neighbourhood U of 0 such that /U xi j − ξ ∈ for in finitely many terms. Let Ur ⊂ U (r = 1, 2, ...) and FU = {(i, j) : xi j − ξ ∈ U }. Then δλ2 (FU ) = 0  and by (1) , Fr ⊂ FU . Hence δλ2 (Fr ) = 0 which is a contradiction to (2) . Thus, xi j is λ- double statistically convergent to ξ.

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Sufficiency: Suppose that there exists a subset R = {(i, j)} ⊆ N × N} such that δλ2 (R) = 1 and limi j xi j = ξ, i.e. there exists an ro ∈ N such that for each neighbourhood U of 0, xi j − ξ ∈ U for every i, j ≥ ro Now / U } ⊆ N × N\{(iro +1 , jro +1 ), (iro +2 , jro +2 ), ...} RU = {(i, j) : xi j − ξ ∈ Therefore δλ2 (RU ) ≤ 1 − 1 = 0 It follows that x is λ- double statistically convergent to ξ.  Corollary 1 If a doublesequence xi j is λ- double statistically convergent to ξ, then there exists a sequence yi j such that limi, j yi j = ξ and δλ2 {(i, j) : xi j = yi j } = 1, i.e. xi j = yi j for almost all i, j. In a topological group, double sequence x = (xi j ) is called λ- double statistically Cauchy if for each neighbourhood U of 0 there exists G = G(U ) and H = H (U ) / U } has double such that for all i, k ≥ G , j, l ≥ H the set {(i, j) ∈ I pq : xi j − xkl ∈ natural density zero. In this case, we denote the set of all statistically Cauchy double sequences by Sλ2 C(E). Theorem 2 Let E be complete. A double sequence x = (xi j ) in E is λ- double statistically convergent if and only if x is λ- double statistically Cauchy. Proof Let x = (xi j ) be λ- double statistically convergent to ξ. Let U be any neighbourhood of 0. Then we may choose a symmetric neighbourhood W of 0 such that W + W ⊂ U. Then for this neighbourhood W of 0, the set {(i, j) ∈ I pq : xi j − ξ ∈ W } has double natural λ- density 0. For each neighbourhood U of 0, the set {(i, j) ∈ / U } has double natural λ-density zero. Then we may choose numbers I pq : xi j − ξ ∈ / U . Now write G and H such that x G H − ξ ∈ / U} TU = {(i, j) ∈ I pq :: xi j − x G H ∈ / W} L W = {(i, j) ∈ I pq :: xi j − ξ ∈ / W} K W = {(N , M) ∈ I pq :: x G H − ξ ∈

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Then TU ⊂ L W ∪ K W and hence δλ2 (TU ) ≤ δλ2 (L W ) + δλ2 (K W ) = 0. Therefore, we get that x is λ- double statistically Cauchy. To prove the converse suppose that there is a λ- double statistically Cauchy sequence x but it is not λ-double statistically convergent. Then we may find natural numbers G and H such that the set TU has double natural λ-density zero. It follows from this that the set Z U = {(i, j) ∈ I pq : xi j − x G H ∈ U } has double natural density 1. We may choose a neighbourhood W of 0 such that W + W ⊂ U . Now take any fixed non-zero element ξ of E. Let xi j − x G H = xi j − ξ + ξ − x G H . It follows from this equality that xi j − x G H ∈ U if xi j − ξ ∈ W. Since x is not λ- double statistically convergent to ξ, the set L W has double natural density / W } has double natural density 0. Hence 1, i.e. the set {(i, j) i ≤ n, j ≤ m : xi j − ξ ∈ the set {(i, j) i ≤ n, j ≤ m : xi j − x G H ∈ U } has double natural density 0, i.e. the set TU has double natural density 1 which is a contradiction. Finally, from theorems 1 and 2 we can state the following theorem and the proof is easy and omitted. Theorem 3 If E is complete, then the following conditions are equivalent: (a) x is λ- double statistically convergent to ξ; (b) x is λ- double statistically Cauchy; (c) there exists a subsequence y of x such that limi, j yi j = ξ.

References 1. Çakalli H, Lacunary statistical convergence in topological groups, Indian J. Pure Appl. Math. 26 (2),(1995), 113-119. MR 95m:40016 2. Çakalli H, On Statistical Convergence in topological groups, Pure and Appl. Math. Sci. 43, No.1-2, 1996, 27-31. MR 99b:40006 3. Çakalli H, Sava¸s E, Statistical Convergence of Double Sequences in Topological Groups, Journal of Computational Analysis and Applications, Volume: 12 Issues: 2 Pages: 421-426 Published: APR 2010 4. Fast H, Sur la convergence statistique, Colloq. Math. 2, (1951) 241–244. MR 14:29c 5. Fridy JA, On statistical convergence, Analysis, 5, 1985, 301-313. MR 87b:40001 6. Kolk E, The statistical convergence in Banach spaces, Tartu Ul. Toime. 928, 1991, 41-52. MR 93c:40003 7. Maddox IJ, Sequence spaces de ned by a modulus, Math. Cambridge Phil. Soc. 100, 1986, 161-166. MR 87h:46024 8. I. J. Maddox, Statistical convergence in locally convex spaces, Math. Cambr. Phil. Soc. 104,1988, 141-145. MR 89k:40012 9. Mursaleen and Osama H. H. Edely, Statistical convergence of double sequences, Journal of Mathematical Anal. Appl. 288, (2003), 223-231 10. Pringsheim A (1900) Zur Ttheorie der zweifach unendlichen Zahlenfolgen. Math. Ann. 53:289– 321

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11. Rahmet Sava¸s Eren and Ekrem Sava¸s, Double lacunary statistical convergence of order α, Indian J. Math. 57(2015), no. 1, 1-15 12. Sava¸s E (2018) On the lacunary (A, ϕ)-statistical convergence of double sequences. Ukrainian Math. J. 70(6):980–989 13. E. Sava¸s, Double ideal lacunary statistical convergence in topological groups, Advanced Studies in Contemporary Mathematics (Kyungshang) ,28(4),(2018)pp. 635-642 14. E. Sava¸s, Iλ -double statistical convergence of order α in topological groups. Ukran. Mat. Zh. 68(9) (2016), 1251-1258 15. E.Sava¸s, Lacunary statistical convergence of double sequences in topological groups, J. Inequal. Appl., Article Number: 480 Published: DEC 2 2014 16. Schoenberg IJ (1959) The integrability of certain functions and related summability methods. Amer. Math. Monthly 66:361–375

Exact Soliton Solutions to the Nano-Bioscience and Biophysics Equations Through the Modified Simple Equation Method Md. Abdul Kayum, Hemonta Kumar Barman, and M. Ali Akbar

Abstract The nano-ionic currents along microtubules (MTs) and the equation of microtubules in nano-biosciences as nonlinear RLC transmission line are very significant nonlinear evolution equations (NLEEs) in biological physics and applied mathematics. The modified simple equation (MSE) method is useful, functional, and efficacious to extract exact soliton solutions. But, when the balance number is greater than one, it is challenging to find out the solutions. In this article, we have put in use the MSE method to ascertain some solutions accessible in the literature and establish some new soliton solutions to the equations described earlier each of which has a balance number two. In the first instance, we have established a general solution comprising some subjective parameters. We analyze the solitary wave properties of the solutions by depicting 3D graphs. Keywords Microtubules · Nonlinear RLC transmission line · Solitary wave solutions · MSE method · Exact solutions

1 Introduction The mathematical modeling of complex phenomena in cytoskeletal structured materials should be able to account for various scales of MTs, which implicated in different cellular activities, i.e., cell motility, cell division, and intracellular trafficking closely on the study of a variety of systems of ordinary and partial differential equations. Similar models are developed in diverse fields of study, ranging from the natural and Md. A. Kayum · H. K. Barman · M. A. Akbar (B) Department of Applied Mathematics, University of Rajshahi, Rajshahi, Bangladesh e-mail: [email protected] URL: http://103.79.117.242/ru_profile/public/teacher/24305717/profile Md. A. Kayum e-mail: [email protected] H. K. Barman e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 D. Giri et al. (eds.), Proceedings of the Sixth International Conference on Mathematics and Computing, Advances in Intelligent Systems and Computing 1262, https://doi.org/10.1007/978-981-15-8061-1_38

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physical sciences, population ecology to economics, infectious disease epidemiology, neural networks, biology, mechanics, etc. Therefore, mathematical modeling is very important to analyze and comprehend intricate physical systems. In spite of the eclectic nature of the fields wherein these models are formulated, different groups of them contribute adequate common attributes that make it possible to examine them within a unified theoretical structure. Therefore, the investigation of exact solutions to NLEEs plays a very important role to uncover the obscurity of many phenomena and processes throughout the natural sciences. Therefore, in order to find out exact solutions to NLEEs, different groups of researchers have been working tirelessly. Accordingly, in the recent years, they established several methods to search exact solutions, for instance, the Jacobi elliptic function method [1], the He’s homotopy method [2], the Riccati equation method [3], the tanh function method [4], the Lie group symmetry method [5], the inverse scattering method [6], the Hirota’s method [7], the (G  /G)-expansion method [8, 9], the Sine-Gordon method [10], the Adomian decomposition method [11], the first integral method [12], the F-expansion method [13], the ansatz method [14], the Exp-function method [15], the homogeneous balance method [16], the exp(−φ(η))-expansion method [17], the Miura transformation method [18], the modified extended tanh-function (METF) method [19], the MSE method [20–23], and so on. The MSE [20–23] is a lately developed expanding method. Its computations are straightforward, systematic and it is not essential to use the symbolic computation software to manipulate the algebraic and differential equations. But, the method has some drawbacks, when the balance number is greater than one; it is challenging to find out the solutions. The balance numbers of the equation of nano-ionic currents along MTs and the equation of MTs as nonlinear RLC transmission line are two. In this article, through putting in use the MSE method, we have established some solutions accessible in the literature and ascertain some new soliton solutions to the equations described earlier with computerized symbolic computation. The attained solutions might be useful to comprehend the phenomena relating to biological physics and applied mathematics. The article is organized as follows: In Sect. 2, we summarize the method. In Sect. 3, we implement the method to NLEEs with balance number two. In Sect. 4, physical explanations of the determined solutions are discussed and, in Sect. 5, conclusions are given.

2 Description of the Method Let us consider the nonlinear evolution equation of the form H (u, u t , u x , u y , u z , u tt , u x x , ...) = 0

(1)

where u = u(x, y, z, t) is an unidentified function, is a polynomial in u(x, y, z, t) and its partial derivatives. In order to examine (1) by means of the MSE method [20–23], we have to execute the following steps:

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Step 1: The traveling wave variable, u(x, y, z, t) = u(ξ), ξ = k(x + y + z ± ωt)

(2)

allows us to change the Eq. (1) into the subsequent ordinary differential equation (ODE): (3) G(u, u  , u  , ...) = 0 where G is a polynomial in u(ξ) and its derivatives, wherein u  = du/dξ. Step 2: In accordance with the MSE method, we suppose that Eq. (3) has the solution in the form,   i N  s (ξ) ai , (4) u(ξ) = s(ξ) i=0 where ai , i = 0, 1, 2, .., N are unknown constants to be calculated, such that a N = 0, and s(ξ) is an unidentified function to be identified. Accordingly, it is improbable to guess from prior what sort of solutions one might attain through this strategy. This is the uniqueness and qualification of this strategy. Consequently, some distinct solutions may be found by this method. Step 3: The positive integer N appearing in Eq. (4) can be determined by taking into account the homogeneous balance between the highest order nonlinear terms and the derivatives of the highest order occurring in Eq. (3). Step 4: We substitute (4) into (3) and then we account the function s(ξ). As a result of this substitution, we get a polynomial of (1/s(ξ)) and its derivatives. In the resultant polynomial, we equate all the coefficients of (s(ξ))−i , (i = 1, 2, 3, ...) to zero. This procedure gives a system of algebraic and differential equations which can be solved for getting ai (i = 1, 2, 3, ...), s(ξ) and the value of the other needful parameters.

3 Formulation of the Solutions In this section, we will execute the MSE method to extract solitary wave solutions to the equation of nano-ionic currents along MTs and the equation of MTs as nonlinear RLC transmission line, which are very important in the engineering fields.

3.1 The Equation of Nano-Ionic Currents Along MTs In this subsection, we will attempt to achieve some new solitary wave solutions to the equation of nano-ionic currents along MTs by the MSE method. Let the following

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NLEEs, namely the equation of nano-ionic currents along MTs in nano-biosciences (see [19, 24, 25] for details):    l2 1  −1 G0 R Z − G 0 Z u = 0, uu t + u x x x + 2δ − χ C0 3 l (5) 9 where the resistance of the elementary ring (ER) R = 0.34 × 10  with length l = 8 × 10−9 m, the total maximal capacitance of the ER C0 = 1.8 × 10−15 F, the conductance of pertaining nano-pores (NPs) G 0 = 1.1 × 10−13 Si and the characteristic impedance of this system Z = 5.56 × 1010 . Also δ and χ represents nonlinearity of ER capacitor and conductance of NPs in ER, respectively. Now, we can be proceed by imposing the condition, R Z −1 − G 0 Z = 0 enables that the ionic pumps in suitable voltage regime and the Ohmic loss could be balanced by fresh ions injected from NPs. So, the Eq. (5) becomes zC0 z 3/2 C0 u t + 2u x − l l

z 3/2 C0 zC0 u t + 2u x − l l

  l2 G0 uu t + u x x x = 0 2δ − χ C0 3

(6)

To construct solitary wave solutions to the equation (6) by means of the MSE method, we utilize the traveling wave transformation with dimensionless wave variable u(x, t) = U (ξ), ξ =

1 c x− t l τ

(7)

where c be the dimensionless wave velocity and τ = RC0 = 0.61 × 10−6 s be the characteristic time of charging ER capacitor. The traveling wave transformation (7) reduces Eq. (6) to the following ODE in the form: U  − βcUU  + (6 − γc)U  = 0,

(8)

  3/2 where β = 3Z τ C0 2δ − χ GC00 , γ = 3ZτC0 and prime denotes the derivatives with respect to ξ. Integrating Eq. (8) with respect to ξ, we get a new ODE in the form: U  − βc

U2 + (6 − γc)U = 0 2

(9)

Balancing the highest order derivative term U  and the nonlinear highest order term U 2 appearing in the ODE (9), we get N = 2. Thus, the solution (4) takes the form U (ξ) = a0 + a1

    2 s s + a2 , s s

(10)

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where a0 , a1 and a2 are constants to be determined such that a2 = 0, and s(ξ) is an unknown function to be evaluated. Substituting solution (10) into Eq. (9) and then equating the coefficients of s 0 , s −1 , s −2 , s −3 , s −4 to zero, we, respectively, obtain 1 − a0 (−12 + 2cγ + cβa0 ) = 0 2

(11)

a1 (−(−6 + cγ + cβa0 )s  + s  ) = 0

(12)

1 − cβa12 (s  )2 − 3a1 s  s  + a2 (−6 + cγ + cβa0 )(s  )2 + 2(s  )2 + 2s  s  = 0 (13) 2 − (s  )2 (a1 (−2 + cβa2 )s  ) + 10a2 s  = 0

(14)

1 − a2 (−12 + cβa2 )s  + 10a2 s  = 0 2

(15)

From Eqs. (11) and (15), we obtain 12 and a2 = cβ , since a2 = 0. a0 = 0, −2(−6+cγ) cβ Case 1: When a0 = 0, then from Eqs. (12)–(14), we attain √ 12 −6 + cγ , a1 = ∓ cβ and s(ξ) =

√ m1 e±ξ −6+cγ + m 2 −6 + cγ

where m 1 and m 2 are integrating constants. Now, inserting the values of a0 , a1 , a2 and s(ξ) into solution (10), we derive the exponential exact solution of (9) as follows: √

12(−6 + cγ)2 m 1 m 2 e±ξ −6+cγ √ U (ξ) = − cβ(m 1 e±ξ −6+cγ + (−6 + cγ)m 2 )2

(16)

Since m 1 and m 2 are arbitrary constants, one may randomly choose their values. Therefore, if we choose m 1 = (cγ − 6) and m 2 = 1, and simplify then from solution (16), we obtain 3(−6 + cγ) sech2 u 1 (x, t) = − cβ

√

cγ − 6(xτ − clt) 2lτ

(17)

Again, if we choose m 1 = −(cγ − 6) and m 2 = 1, then from solution (16), we attain the following solitary wave solutions in the form

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3(−6 + cγ) cosech2 u 2 (x, t) = cβ

√

cγ − 6(xτ − clt) 2lτ

(18)

The further choices of the values of the integral parameters deliver many other wave solutions to the nano-ionic currents along MTs equation, but for conciseness these solutions have not been noted here. Case 2: When a0 = − 2(−6+cγ) , then from Eqs. (12)–(14), we get cβ √ 12 6 − cγ a1 = ∓ cβ , and s(ξ) = −

√ m1 e±ξ −6+cγ + m 2 −6 + cγ

where m 1 and m 2 are integrating constants. Now, embedding the values of a0 , a1 , a2 and s(ξ) into solution (10), we obtain the subsequent general solution

U (ξ) =

  √ √ 2 m 1 2 e±2ξ 6−cγ + 4(cγ − 6)m 1 m 2 e±ξ 6−cγ − (cγ − 6)2 m 2 2 cβ(6 − cγ)−1 (m 1 e±ξ

√

6−cγ

+ (6 − cγ)m 2 )2

,

(19)

√

−clt) where θ = (6−cγ)(xτ . lτ Since m 1 and m 2 are integral constants, we might set arbitrarily their values. Thus, if we set m 1 = (cγ − 6) and m 2 = 1, from solution (19), we obtain

u 1 (x, t) = −

 √  √ )) 2 ( (6−cγ)(−clt+xτ )) (cγ − 6) 2 + cosh ( (6−cγ)(−clt+xτ cosech lτ 2lτ cβ (20)

On the other hand, if we set m 1 = −(cγ − 6) and m 2 = 1, from solution (19), we extract the solitary wave solution in the form:

u 2 (x, t) = −

 √  √ )) 2 ( (6−cγ)(−clt+xτ )) (cγ − 6) −2 + cosh ( (6−cγ)(−clt+xτ sech lτ 2lτ cβ (21)

The major advantage of the MSE method is that the calculations are not sophisticated and easy to control. It is not essential to use computer algebra system to facilitate the calculations as it take to the Exp-function method, the (G  /G)-expansion method, the tanh function method, the METF method, the homotopy analysis method etc. But the solutions obtained by the MSE method are as realistic as the solutions obtained by the above mentioned method.

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3.2 The Equation of MTs as Nonlinear RLC Transmission Line In this subsection, we will bring to bear the MSE method to find the exact solutions and then the solitary wave solutions to the equation of MTs as nonlinear RLC transmission line [19]: R2 C0 l 2 u x xt + l 2 u x x + 2R1 C0 δuu t − R1 C0 u t = 0,

(22)

where the longitudinal and transversal component of resistance of the elementary ring (ER) R1 = 109  and R2 = 7 × 106 , respectively, with length l = 8 × 10−9 m, the total maximal capacitance of the ER C0 = 1.32 × 10−15 F and the parameter δ when δ < 1, represents nonlinearity of ER capacitor in MT. In order examine the solitary wave solutions of the equation of MTs as nonlinear RLC transmission line by making use of the MSE method. For the equation (22), we will use the same transformation (7) and thus the nonlinear RLC transmission line equation (22) turns into: c c c − R2 C0 U  + U  − 2R1 C0 δ UU  + R1 C0 U  = 0, τ τ τ

(23)

where prime denotes the derivatives with respect to ξ. Now, integrating Eq. (23) with respect to ξ, we get a new ODE in the form: U  − α



U − δU 2 + U c

 = 0,

(24)

where, α = RR21 . Balancing the highest order derivative term U  and the nonlinear highest order term U 2 of the come out in (24), we get N = 2. Therefore, the form of the solution of (24) is identical to the form of the solution (10) and hence it has not been written here. Substituting the solution (10) into (24) and then equating the coefficients of s 0 , s −1 , s −2 , s −3 , s −4 to zero, we, respectively, obtain αa0 (−1 + δa0 ) = 0.

(25)

 a1  cα(−1 + 2δa0 )s  − αs  + cs  = 0. c

(26)

1 2 2 (cαδa12 s  + a1 s  (αs  − 3cs  ) + a2 (cα(−1 + 2δa0 )s  + 2c(s  )2 + s  (−2αs  + 2cs  ))) = 0 c

(27)

1 2 (2s (ca1 (1 + αδa2 )s  + a2 (αs  − 5cs  ))) = 0. c

(28)

a2 (6 + αδa2 )s  = 0.

(29)

4

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From Eqs. (25) and (29), we obtain a0 = 0, and a2 = −

a δ

6 αδ

since a2 = 0. Therefore, we obtain the following two cases for the values of a0 . Case 1: When a0 , from Eqs. (26)–(28), we get

√ √ √ √  √ √  6 α 6 α 2 6 2 6 a1 = √ , c = − , a1 = − √ , c = 5 5 αδ αδ √ 6m 1 √α s(ξ) = ± √ e± 6 ξ + m 2 . α

and

where m 1 and m 2 are integrating constants. Now, putting the values of a0 , a1 , a2 and s(ξ) into solution (10), we obtain the subsequent exponential solution √

2m 1 e U (ξ) =

± √α6 ξ

  √ √ √ ± √α ξ 3m 1 e 6 ± 6 αm 2

√ √ √ ± √α ξ δ( 6m 1 e 6 ± αm 2 )2

(30)

Since m 1 and √ constants, one may randomly opt their values. If we √m 2 are arbitrary opt m 1 = ± α and m 2 = 6, then from solution (30), we obtain u 1 (x, t) =

1 4δ

 √

 √ x α x α tα tα 3 ± 2 tanh √ ± − tanh2 √ ± 10τ 10τ 2 6l 2 6l

(31)

√ √ Furthermore, if we opt m 1 = ∓ α and m 2 = 6, then from solution (30), we accomplish 1 u 2 (x, t) = 4δ



√ √

 x α tα tα 2 x α 3 ± 2 coth √ ± − coth √ ± 10τ 10τ 2 6l 2 6l

(32)

√ √ Alternatively, if we opt m 1 = ± 6 α and m 2 = 3, from solution (30), we achieve the subsequent solitary wave solution:

Exact Soliton Solutions to the Nano-Bioscience and Biophysics Equations …

  √ tα 8 1 ± tanh[ 2x√6lα ± 10τ ] u 3 (x, t) =  2 √ tα δ 3 ± tanh[ 2x√6lα ± 10τ ]

477

(33)

√ √ Moreover, when m 1 = ∓ 6 α and m 2 = 3, then from solution (30), we determine the following solitary wave solutions:   √ tα 8 1 ± coth[ 2x√6lα ± 10τ ] u 4 (x, t) =  2 √ tα δ 3 ± coth[ 2x√6lα ± 10τ ]

(34)

Case 2: When a0 = 1δ , then from Eqs. (26)–(28), we get √ √ 6 α a1 = 0, c = ± 5 √

and

√

6m 1 ± √α ξ s(ξ) = ± √ e 6 + m 2 . α where m 1 and m 2 are constants of integration. Now, setting the values of a0 , a1 , a2 and s(ξ) into solution (10), we obtain the next general solution ⎡ U (ξ) =

1⎢ ⎣1 − √ δ

√ 2α

6m 21 e± 6m 1 e

√ 2α

±

3

ξ

⎤ ξ

⎥ 2 ⎦ √ ± αm 2 3

(35)

√ Also here √ m 1 and m 2 are arbitrary constants. Thus, if we accept m 1 = ± α and m 2 = 6. then from solution (35), we obtain the following solution: 1 u 1 (x, t) = 4δ

√  √

 x α tα tα 2 x α 3 ∓ 2 tanh √ ± − tanh √ ± 10τ 10τ 2 6 2 6

(36)

√ √ Again, if we accept m 1 = ∓ α and m 2 = 6, then from solution (35), we acquired the under mentioned wave solutions: √  √

 x α x α 1 tα tα 3 ∓ 2 coth √ ± − coth2 √ ± (37) u 2 (x, t) = 4δ 10τ 10τ 2 6 2 6 √ √ On the other hand, if we accept m 1 = ± α and m 2 = 2 6, from solution (35), we derive the soliton solutions in the form:

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  √  tα 8 1 ∓ tanh x2√α6 ± 10τ u 3 (x, t) =  2  √ tα δ 3 ∓ tanh x2√α6 ± 10τ

(38)

√ √ Also, if we accept m 1 = ∓ α and m 2 = 2 6, then from solution (35),we attain the following wave solution:   √  tα 8 1 ∓ coth x2√α6 ± 10τ u 4 (x, t) =  2  √ tα δ 3 ∓ coth x2√α6 ± 10τ

(39)

Since m 1 and m 2 are integral constants for other choices of m 1 and m 2 , we might obtain much new and more general exact solutions (30) and (35) of the Eq. (22) by making use of the MSE method without any aid of symbolic computation software.

4 Graphical Representations and Discussion In this section, the graphical representations and physical explanations of the attained solutions have been given to the nano-ionic currents along microtubules (MTs) and the equation of microtubules in nano-biosciences as nonlinear RLC transmission line.

4.1 Physical Interpretations of the Soliton Solutions of Nano-Ionic Currents Along MTs In this subsection, we will depict the graph and signify the obtained solutions to the equation of nano-ionic currents along MTs in a biological nonlinear transmission line for ionic currents. The solitons (17)–(18) and (20)–(21) are plotted for the conditions, cγ > 6 and cγ < 6. The solitons are assuming small region of nonlinearities as follows 2δ − χ GC00 = 0.1. For cγ > 6, the soliton solution (17) represents step bellshaped soliton, the soliton solution (18) represents step anti-bell-shaped soliton, shown in Fig. 1. And for cγ < 6, the soliton solutions (17) and (18) represent periodic soliton shown in Fig. 2. And for cγ < 6, solutions (20)–(21) represent the ionic cloud solitons shown in Fig. 3.

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Fig. 1 Sketch of bell-shaped wave solutions (17) and anti-bell-shaped solitons (18) to the equation of nano-ionic currents along MTs (6) for cγ > 6

Fig. 2 Sketch of the periodic shape ionic cloud solitons (17) and (18) to the equation of nano-ionic currents along MTs (6) for cγ < 6

Fig. 3 Sketch of the ionic cloud soliton (20) and (21) to the equation of nano-ionic currents along MTs (6) for cγ < 6

4.2 Physical Interpretations of the Solitons of MTs as Nonlinear RLC Transmission Line In this sub-section, we will portray the graph and signify the obtained solutions to the equation of MTs as nonlinear RLC transmission line in nano-biosciences. The solutions (31)–(32) represent the kink-shaped solitons and (33)–(34) represent the bell-shaped solitons. Figures 4 and 5 show the shape of the solitons found from (34) to (32), respectively, for α = RR21 = 142.86 and δ < 1. Also, the solutions (36)–(37)

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Fig. 4 Plot of the kink-shaped solitons of (31) to the equation of MTs as nonlinear RLC transmission line (22)

Fig. 5 Plot of the bell-shaped solitons of (33) to the equation of MTs as nonlinear RLC transmission line (22)

represent the kink-shaped solitons and (38)–(39) represent the bell-shaped solitons. The shapes of the solutions (36)–(39) are identical to the shapes of the solutions (31)–(32) for α = RR21 = 142.86 and δ < 1. Therefore, these figures have not been plotted.

5 Conclusion In this article, we have considered a couple of NLEEs each of which the balance number of each is two. If the balance number is greater than one, in general, it is difficult to establish soliton solutions through the MSE method. For this case, if the solution of s(ξ) consists of polynomial of the wave variable ξ, then each coefficient has to be set zero; otherwise, it does not meet the condition |u| → 0 as ξ → ±∞. This constraint is crucial to extract wave solutions to NLEEs for higher balance number. We have analyzed the solitary wave properties of the solutions via the 3D

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graphs. It is seen that the MSE method is concise, straightforward, and powerful mathematical tool to ascertain closed-form soliton solutions. Acknowledgment The authors acknowledge the research grant No. A-1220/5/52/RU/Science37/2019–2020.

References 1. Tala-Tebue E, Zayed EME (2018) New Jacobi elliptic function solutions solitons and other solutions for the (2+1)-dimensional nonlinear electrical transmission line equation. European Phys. J. Plus 133:314 2. Ganji DD, Afrouzi GA, Talarposhti RA (2007) Application of variational iteration method and homotopy perturbation method for nonlinear heat diffusion and heat transfer equations. Phys. Lett. A 368:450–457 3. Zayed EME, Amer YA, Shohib RMA (2014) The improved generalized Riccati equation mapping method and its application for solving a nonlinear partial differential equation (PDE) describing the dynamics of ionic currents along microtubules. Academic J. 9(8):238–248 4. Jawad AJM, Petkovic MD, Laketa P, Biswas A (2013) Dynamics of shallow water waves with Boussinesq equation, Scientia Iranica. Trans. B: Mech. Engr. 20(1):179–184 5. Guo AL, Lin J (2010) Exact solutions of (2+1)-dimensional HNLS equation. Commun. Theor. Phys. 54:401–406 6. Ablowitz MJ, Clarkson PA (1991) Soliton, nonlinear evolution equations and inverse scattering. Cambridge University Press, New York 7. Hirota R (2004) The direct method in soliton theory. Cambridge University Press, Cambridge 8. Akbar MA, Ali NHM, Zayed EME (2012) Abundant exact traveling wave solutions of the generalized Bretherton equation via (G  /G)-expansion method. Commun. Theor. Phys. 57:173–178 9. Alam MN, Akbar MA, Mohyud-Din ST (2014) General traveling wave solutions of the strain wave equation in microstructured solids via the new approach of generalized (G  /G)-expansion method. Alexandria Engg. J. 53:233–241 10. K.K. Ali, A.M. Wazwaz and M.S. Osman, Optical soliton solutions to the generalized nonautonomous nonlinear Schrodinger equations in optical fibers via the Sine-Gordon expansion method, Optik, https://doi.org/10.1016/j.ijleo.2019.164132(in press-available online 26 December 2019) 11. Helal MA, Mehana MS (2006) A comparison between two different methods for solving Boussinesq-Burgers equation. Chaos, Solitons Fract. 28:320–326 12. Akbar MA, Ali NHM, Hussain J (2019) Optical soliton solutions to the (2+1) Chaffee-Infante equation and dimensionless form of the Zakharov equation. Adv. Differ. Equ. 2019:446 13. Islam MS, Akbar MA, Khan K (2018) Analytical Solutions of nonlinear Klein-Gordon equation using the F-expansion method. Opt. Quant. Electron. 50:224 14. Triki H, Kara AH, Biswas A (2014) Domain walls to Boussinesq type equations in (2+1)dimensions. Indian J. Phys. 88(7):751–755 15. H. Naher, A.F. Abdullah and M.A. Akbar, New traveling wave solutions of the higher dimensional nonlinear partial differential equation by the Exp-function method, J. Appl. Math., 2012 (2012) Article ID 575387, 14 pages 16. Wang M (1995) Solitary wave solutions for variant Boussinesq equations. Phy. Lett. A 199:169– 172 17. M.G. Hafez, M.N. Alam and M.A. Akbar, Traveling wave solutions for some important coupled nonlinear physical models via the coupled Higgs equation and the Maccari system, J. King Saud Univ.-Sci., 27 (2015), 105-112 18. Bock TL, Kruskal MD (1979) A two-parameter Miura transformation of the Benjamin-One equation. Phys. Lett. A 74:173–176

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19. Sekulic DL, Sataric MV, Zivanov MB (2011) Symbolic computation of some new nonlinear partial differential equations of nano-biosciences using modified extended tanh-function method. Appl. Math. Comput. 218:3499–3506 20. Khan K, Akbar MA, Alam MN (2013) Traveling wave solutions of the nonlinear Drinfel’dSokolov-Wilson equation and modified Benjamin-Bona-Mahony equations. J. Egyptian Math. Soc. 21:233–240 21. Kamruzzaman Khan and M. Ali Akbar, Exact and solitary wave solutions for the TzitzeicaDodd-Bullough and the modified Boussinesq-Zakharov-Kuznetsov equations using the modified simple equation method, Ain Shams Engr. J., 4 (2013) 903-909 22. K. Khan and M.A. Akbar, Traveling wave solutions of some coupled nonlinear evolution equations, ISRN Math. Phys., 2013 (2013) Article ID 685736, 8 pages 23. Islam MS, Roshid MM, Rahman AKML, Akbar MA (2019) Solitary wave solutions in plasma physics and acoustic gravity waves of some nonlinear evolution equations through enhanced MSE method. J. Phys. Commun. 3(12):125011 24. Sataric MV, Sekulic D, Zivanov M (2010) Solitonic ionic currents along microtubules. J. Comp. Th. Nanosc. 7(11):1–10 25. Sekulic DL, Sataric BM, Tuszynski JA, Sataric MV (2011) Nonlinear ionic pulses along microtubules. Eur. Phys. J. E. 34(5):1–11

Design of Optimal Bayesian Reliability Test Plans for a Parallel System Based on Type-II Censoring P. N. Bajeel and M. Kumar

Abstract Consider a parallel system with n independent components. Assume that the lifetime of ith component follows an exponential distribution with unknown parameter. We assume that each failure rate is distinct and the priori information can be modeled by quasi-density function. Using squared error loss function, a Bayesian estimator for failure rate based on type-II censoring is used to get an estimate of system reliability. An optimal reliability test plan is designed and a non-linear integer optimization problem is formulated satisfying usual probability requirements (type-I and type-II error constraints). Several numerical examples are considered to illustrate the Bayesian approach of obtaining optimal reliability test plan for a parallel system. Keywords Reliability · Life testing · Type-II censoring · Type-I error · Type-II error

1 Introduction The basic problem of component reliability test plan was first introduced by [2], with an unknown constant failure rate and considering consumer’s requirement. This work is extended by [4], with constant failure rate by considering both Producer’s and Consumer’s requirement. Later, [3] developed a test plan for a series system by considering that the failure rate of the component follows Uniform distribution over an interval. [5] constructed a reliability test plan for a parallel system with n independent components under type-II censoring with the assumption that the lifetime of each component follows an exponential distribution with a constant but unknown failure rate parameter. [1] considered a parallel system with n different components P. N. Bajeel (B) Sir Syed College, Taliparamba, Kerala, India e-mail: [email protected] M. Kumar National Institute of Technology Calicut, Calicut, Kerala, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 D. Giri et al. (eds.), Proceedings of the Sixth International Conference on Mathematics and Computing, Advances in Intelligent Systems and Computing 1262, https://doi.org/10.1007/978-981-15-8061-1_39

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by assuming that lifetime of each component follows an exponential distribution with failure rate as a parameter and depends upon k covariates such as temperature, pressure, humidity, etc., through exponential relationships. All problems addressed so far in literature are not a part of pure Bayesian approach. In this paper, we adopt a pure Bayesian approach to design optimal component reliability test plans for parallel systems under type-II censoring. We consider a problem of testing the reliability of a parallel system with n independent components under type-II censoring, where the i − th component has exponential lifetime X i with unknown parameter λi , ∀ i = 1, 2, ..., n. Then the parallel system reliability for unit time period is given by n    1 − e−λi . R =1− i=1

Since the system is highly reliable, the system reliability can be approximated as R ≈1−

n 

λi .

i=1

We consider the quasi-prior g(λi ) =

1 , k ≥ 0, λik

a simple prior as the non-informative quasi-prior for λi . We test ri components of type i for failure. As soon as the component fails, it will be replaced by an identical component, so that the testing continue till the ri − th failure occurs and observe the failure times X i j , 1 ≤ i ≤ n; 1 ≤ j ≤ ri . Since each X i j follows exponential random variable with probability density function f (xi j ) = λi e−λi xi j , λi > 0, xi j > 0, i = 1, 2, ..., n, the likelihood function is given by L=

λri i e

−λi

ri  j=1

xi j

.

Then the posterior distribution of λi is given by g(λi | xi j ) =

∞ λi =0

That is,

L ∗ g(λi ) L ∗ g(λi )dλi

.

Design of Optimal Bayesian Reliability Test Plans …

λri i e g(λi | xi j ) =

∞

λri i e

485 ri 

−λi

−λi

ri 

xi j

j=1

0

 =

ri 

xi j

j=1

1 λik

1 dλi λik

ri −k+1 xi j

j=1

Γ (ri − k + 1)

λri i −k e

−λi

ri 

xi j

j=1

Now the Baye’s estimator of λi under the squared error loss function is given by ri − k + 1 . λˆi = ri  xi j j=1

Then the reliability estimate of the system reliability is obtained using the Bayesian estimator of failure rates is Rˆ = 1 −

n 

λˆi

i=1 n  ri − k + 1 =1− . ri  i=1 xi j j=1

Now, we will accept the system if the estimate of the system reliability based on Bayesian estimator of λi given by Rˆ = 1 −

n  ri − k + 1 ri  i=1 xi j j=1

is greater than or equal to some number d ∈ (0, 1). That is n  ri − k + 1 ≥d 1− ri  i=1 xi j j=1

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⎛ ⇒

⎞

⎜r − k + 1⎟ ⎜ i ⎟ ln ⎜ ri ⎟ ≤ ln(1 − d). ⎝  ⎠ i=1 xi j

n

j=1

i Since X i j follows exponential distribution, rj=1 xi j follows Gamma distribution ri r i and variance σi2 = 2 . Define with mean μi = λi λi ⎞ ⎛ ⎞ ⎛ ri ⎜r − k + 1⎟ ⎟ ⎜ i xi j ⎠ = ln ⎜ ri g⎝ ⎟,  ⎠ ⎝ j=1 xi j j=1

⎛

then

⎞

⎜r − k + 1⎟ −1 ⎜ i ⎟ g ⎜ ri , ⎟ = ri  ⎝  ⎠ xi j xi j j=1

j=1

 g(μi ) = ln

λi (ri − k + 1) ri



and g (μi ) = Then by Delta method g



ri j=1

 xi j follows Normal distribution with mean 

g(μi ) = ln and variance 

−λi . ri

2

λi (ri − k + 1) ri

⎛

g (μi ) σi2 ⎝

ri



⎞ xi j ⎠ =

j=1



1 . ri

 ri − k + 1 Then by Lindeberg central limit theorem i=1 ln ri follows Normal j=1 x i j   n 1 n λi (ri − k + 1) and variance i=1 ln . distribution with mean i=1 ri ri n

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487

A system is said to be satisfactory for unit time if R, the survival probability, is greater than or equal to R1 , the acceptable reliability level (ARL) and, it is said to be unsatisfactory if R is less than or equal to R0 , the unacceptable reliability level (URL), where R0 and R1 are constants such that 0 < R0 < R1 < 1. Then we have the following relations: R ≥ R1 ⇒ 1 −

n 

λi ≥ R1 ⇒

i=1

and R ≤ R0 ⇒ 1 −

n 

n

ln λi ≤ ln(1 − R1 )

i=1

λi ≤ R0 ⇒

i=1

n

ln λi ≥ ln(1 − R0 ).

i=1

2 Optimal Design of the Problem Let ci denote the cost of testing one component of type i for failure. Then the aim is to find the number of failures ri , 1 ≤ i ≤ n that minimize the total testing cost subjected to type-I and type-II error constraints. That is, the problem is to determine the optimum values of by formulating the following optimization problem: n  ci ri Minimize C(r ) = i=1

such that P(Accept the system | System is good) ≥ 1 − α,

(1)

P(Accept the system | System is bad) ≤ β,

(2)

where 0 < β, 1 − α < 1 and C(r ) is the total testing cost. Here, α in the first constraint is usually referred to as producers risk, while β in the second constraint is the consumer’s risk. Using the acceptance rule defined in the previous section, the above two constraints can be written as ⎛

⎞

⎛

⎞

n n  ⎜ ⎜r − k + 1⎟ ⎟  ⎟ ⎜ ⎜ i ⎟ ≤ ln(1 − d) P⎜ ln ⎜ r λi ≤ ln(1 − R1 )⎟ ≥ 1 − α, ⎟ i ⎠ ⎝ ⎝  ⎠ i=1 i=1 xi j

⎛

⎛

j=1

⎜ P⎜ ⎝

⎞

⎞ n 

⎜r − k + 1⎟ ⎟  ⎟ ⎜ i ⎟ ≤ ln(1 − d) ln ⎜ r λi ≥ ln(1 − R0 )⎟ ≤ β, ⎟ i ⎠ ⎝  ⎠ i=1 i=1 xi j

n ⎜

j=1

(3)

(4)

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 ri − k + 1 The exact distribution of i=1 ln ri is not easy to obtain, and in order j=1 x i j to obtain the we therefore need to approximate the distribution  tractable problem,  n ri − k + 1 of i=1 ln ri with Normal distribution, then the constraints (3) and (4) j=1 x i j can be written as n

⎛

⎞  λi (ri − k + 1) ln(1 − d) − ln ⎜ ⎟ n  ⎜ ⎟ ri  i=1 ⎜ Z ≤ min P ⎜ λi ≤ ln(1 − R1 )⎟   ⎟ ≥ 1 − α, n λi  1 ⎝ ⎠ i=1 i=1 ri n 



⎛

⎞





n  λi (ri − k + 1) ln(1 − d) − ln ⎜ ⎟ n  ⎜ ⎟ ri  i=1 ⎟ Z ≤ max P ⎜ λ ≥ ln(1 − R )   0 ⎟ ≤ β, i ⎜ n 1 λi  ⎝ ⎠ i=1 i=1 ri

(5)

(6)

where Z is the standard normal random variable. Since the cumulative distribution function of standard normal random variable is strictly increasing function in its arguments, the constraints (5) and (6) can be written as ⎞  ri − k + 1 lnλi − ln ⎟ ⎜ ln(1 − d) − n  ⎟ ⎜ ri  i=1 i=1 ⎟ min ⎜ λ ≤ ln(1 − R )   1 ⎟ ≥ Z 1−α , i ⎜ n 1 λi ⎝  ⎠ i=1 i=1 ri ⎛

⎛

n 

n 

n 

n 

lnλi − ln ⎜ ln(1 − d) − ⎜ i=1 i=1 max ⎜  n 1 λi ⎜  ⎝ i=1 ri





⎞



(7)

ri − k + 1 ⎟ n  ⎟ ri  λi ≥ ln(1 − R0 )⎟  ⎟ ≤ Z β . (8) ⎠ i=1

Z 1−α and Z β are strictly positive and negative, respectively, for all values of α, β < 0.5, hence clearly the optimum lies in the boundary. Therefore, the constraints (7) and (8) can be written as n 



ri − k + 1 ln(1 − d) − ln(1 − R1 ) − ln ri i=1  n  1 i=1 ri

 ≥ Z 1−α ,

(9)

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489

  n  ri − k + 1 ln(1 − d) − ln(1 − R0 ) − ln ri i=1 ≤ Zβ .  n 1  i=1 ri Now the optimal design n can be written as ci ri Minimize C(r ) = i=1 such that     n  ri − k + 1 1−d   n − ln ln  1 1 − R1 ri i=1  ≤ , r Z 1−α i=1 i     n  ri − k + 1 1−d   n − ln ln  1 1 − R0 ri i=1  ≥ . r Z β i=1 i

(10)

(11)

(12)

This optimization problem can be solved using LINGO11 software.

3 Numerical Results The method presented in the previous section is illustrated below with the help of numerical computation. Let the number of components in a series system be 2 and the cost vector c = (1, 2). Then the optimum values of r1 , r2 , C(r ) and d corresponding to different values of inputs α, β, R0 , R1 and k are presented in the following Table 1.

Table 1 Numerical examples for six component system and comparison of results Parameters Example 1 Example 2 Example 3 α β R0 R1 k r1 r2 C d

0.05 0.05 0.8 0.9 1 3 2 7 0.8618394

0.05 0.05 0.8 0.9 0.5 2 2 6 0.5

0.05 0.05 0.85 0.9 0 1 1 3 0.9999999

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4 Conclusions In this paper, the designing of an optimal Bayesian reliability test plan for a parallel system with a failure rate as a random variable having quasi-density is discussed in detail. The data are obtained through a type-II censoring scheme, and the reliability estimator is obtained by estimating a Bayesian estimator of failure rate. Some numerical examples are also computed to illustrate the Bayesian approach of estimating system reliability and thereby to test the system reliability.

References 1. Bajeel PN, Kumar M (2015) A component reliability test plan for a parallel system with failure rate as the exponential function of covariates. Math Eng Sci Aerosp 6(2):177–190 2. Gal S (1974) Optimal test design for reliability demonstration. Oper Res 22(6):1236–1242 3. Nair JH, Sabnis SV (2002) A reliability test-plan for series systems with components having stochastic failure rates. IEEE T Reliab 51(1):17–22 4. Rajgopal J, Mazumdar M (1988) A type-II censored, log test time based, component-testing procedure for a parallel system. IEEE T Reliab 37(4):406–412 5. Vellaisamy P, Kumar M (2008) Optimal component test plans for a parallel system based on type-II censoring. Stat Methodol 5(5):454–461

l 2 Norm Prior-Based Modified Bright Channel for Low-Illumination Images Riya, Bhupendra Gupta, and Subir Singh Lamba

Abstract Low-illumination image enhancement problem is a very challenging problem in many computer vision applications and, when it comes to nighttime lowillumination images, it becomes more challenging because the depth information of the low-illumination image is not known. Recently, bright channel prior-based methods are used to enhance the overall illumination of the image. The bright channel prior is based on statistical observation on the low-illumination image containing some regions with bright intensity pixels. In this paper, we propose an improved l2 norm-based prior bright channel to enhance the overall illumination of the image by maintaining the image contrast. This new generated bright channel is free from the block effect, which makes our method more robust than other methods. The experimental results show the effectiveness of the proposed method on the low-illumination images as well as on the nighttime low-illumination images. Keywords Low-illumination images · Bright channel prior · Airlight · Image enhancement

1 Introduction Low-illumination images suffer from poor visibility and affect many outdoor computer vision applications [1], hence, the enhancement of the low-illumination images is very important. The enhancement of low-illumination images becomes a very challenging task because of the unknown depth information. The traditional image Riya (B) · B. Gupta · S. S. Lamba Design & Manufacturing, PDPM Indian Institute of Information Technology, Jabalpur 482005, MP, India e-mail: [email protected] B. Gupta e-mail: [email protected] S. S. Lamba e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 D. Giri et al. (eds.), Proceedings of the Sixth International Conference on Mathematics and Computing, Advances in Intelligent Systems and Computing 1262, https://doi.org/10.1007/978-981-15-8061-1_40

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enhancement methods are histogram equalization (HE) [2–4] and contrast limited adapted histogram equalization (CLAHE) [5, 6]. However, HE and CLAHE both over enhance the image and amplify the noisy effect in the images. For very lowilluminated images, Jiang et al. and Zhao et al. introduced a color space conversion method [5, 7], which does not yield much effective results. Piao et al. and Wang et al. proposed retinex transform method for uneven illumination images [8, 9]. The retinex transform method produced good results but not effectively work for nighttime images. A high-speed quantile-based histogram equalization (HSQHE) method [10] is introduced for image enhancement. This method better preserves the brightness of the image and effectively works for low-illuminated images. Recently, methods based on bright channel prior (BCP) have been introduced in the literature [11–13]. BCP-based methods yield good results for low-illuminated images. The bright channel is used for image enhancement because it reflects more target profiles and more grayscale features are shown on the bright channel. The idea of the bright channel is inspired by He’s dark channel prior-based method [14], used for single image haze removal. The dark channel, airlight, and medium transmission play a dominant role to find a haze-free image. A similar concept is used for bright channels prior. In this, the grayscale characteristics of the dark image are used for the bright channel prior-based method. After extracting the bright channel from the low-illuminated image (dark image) to evaluate the airlight component and medium transmission, a well-illuminated image is achieved. However, the bright channel and transmission map obtained from these channels contains block effect and to remove these block effect, the authors used the guided filter, bilateral filter, etc. In this proposed algorithm we introduce a simple and easy l2 norm-based prior to producing a bright channel to enhance the brightness of the image by maintaining the image contrast. In the proposed method, no block effect is visible in the bright channel and transmission map hence produces good results.

2 Background of Bright Channel The idea of a bright channel is based on the statistical analysis of the well-illuminated images. As we know that He’s dark channel prior (DCP)-based method [14] is based on the statistical analysis of outdoor haze-free images. From the statistical analysis, the author found that the dark channel of the haze-free image tends to zero. A similar concept is used in the bright channel prior-based method [11–13] that in most of the local patches, very often that some pixels have very high intensity in at least one color channel in an RGB well-illuminated image. Hence, the bright channel of the well-illuminated image tends to grayish white (near to 255) as shown in Fig. 1. By the statistical analysis on the numerous dark images, it is found that the bright channel of the dark image is grayish black as shown in Fig. 2. In [11], it is found that the dark images are due to the environment with poor light. Hence, the color of the bright channel of the dark image is grayish black. A well-illuminated image

l2 Norm Prior-Based Modified Bright Channel …

493

Fig. 1 Well-illuminated image: original image and bright channel

Fig. 2 Low-illuminated image: original image and bright channel

has sufficient light and, due to the environment with poor light, it becomes dark. Similar to the haze removal algorithm, a dark image is processed to get an image with sufficient light.

3 Proposed Model Based on the He’s dark channel prior model [14] for haze removal, the model described for the low-illuminated image [11] is I (x) = J (x)t (x) + A(1 − t (x)),

(1)

where I is the low-illuminated image, J is the scene radiance, t is the medium transmission describing that portion of light which is not scattered and reaches the camera and A is the airlight. In the proposed method, we find the well-illuminated image J by solving the Eq. 1

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J (x) =

I (x) − A + A. t (x)

(2)

For bright channel, firstly, we find the maximum intensity channel by selecting the maximum intensity from each of the RGB color channel at every pixel. Then we divide the maximum intensity channel into overlapping patches ω(x) (patch centered at pixel x) and calculate the value of l2 norm over each of the patch of the maximum intensity channel. After that, these values are normalized by the number of pixels in the patch and then assigned to that pixel around which the patch is centered. The modified bright channel is shown by J bright and defined by   J bright (x) = maxc∈{RG B} I c (x)x∈ω(x) /N ,

(3)

where N is number of pixels in a patch and . be l2 norm. The bright channel of the dark image is grayish black as shown in Fig. 2. Earlier works on bright channel prior [11–13] suggested that the bright channel of the well-illuminated image should be grayish white (as shown in Fig. 1). The bright channel of the well-illuminated image is   (4) J bright (x) = maxc∈{RG B} I c (x)x∈ω(x) /N + M, where M is chosen as 155. Because we have used l2 norm over each intensity in the local patch ω(x), hence, J bright equally depends on each pixel and least affected by any particular pixel in the local patch. Thus, J bright is more robust than the bright channel proposed in [11–13]. For large values of N, J bright tends to 255. The transmission map is obtained by substituting the Eq. 4 into Eq. 1. We can define t (x) as t (x) =

max c∈{RG B} (I c (x))x∈ω(x) − A × N max c∈{RG B} (I c (x))x∈ω(x) + M − A × N

.

(5)

To obtain the transmission map, we need to calculate the airlight component. For airlight component, we calculate the mean in the bright channel of every patch centered on each pixel and select 1% pixels having minimum mean. Then the mean of the pixels with the mean intensity of these pixels in the input image I is considered as the airlight. In Eq. 2, we see that when the transmission t (x) tends to zero then J (x) will be a very large value. To avoid such problem, we bound t (x) by a smaller but non-zero value t0 . We have taken t0 as 0.11. Thus, the recovered image is obtained from J (x) =

I (x) − A + A. max(t0 , t (x))

In Fig. 3, we have shown the result of the proposed method.

(6)

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Fig. 3 Result of the proposed method on dark image: original image, bright channel, transmission map, and output image

4 Advantage of the Proposed Method In [11–13], bright channel prior-based methods are introduced for low-illuminated images. However, the bright channel and transmission map obtained through these methods contains block effects. Figures 4 and 5 shows the visual comparison between the bright channel and transmission map obtained from these methods and the proposed method. A pixel contributes to the bright channel if it is the brightest pixel in the patch ω(x) in any one of the RG B color channel. Consider, x ∈ ω(x) be the brightest pixel in at least one RG B color channel. Due to the overlapping nature of the patches, a particular pixel x belongs to the n × n neighboring patches. So, a brightest pixel affects the n × n neighboring pixels and these neighboring pixels creates the block effects in the bright channel and so in transmission map. To overcome these block effects, authors used guided filter [11–13], which increases the computational time of the entire process. In the proposed modified bright channel prior, each pixel in the patch equally contributes to obtain the bright channel, which reduces the block effect from the bright channel and so on transmission map.

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Fig. 4 Bright channel obtained from different methods: original image, bright channel obtained from Sun et al. [11] method, bright channel obtained from Shi et al. [13] method and bright channel obtained from proposed method

Fig. 5 Transmission map using Sun et al. [11], refined transmission map using Sun et al. [11], transmission map using Shi et al. [13], refined transmission map using Shi et al. [13] and transmission map using proposed method

5 Discussion and Results We have tested the proposed method on Exclusively Dark (ExDark) Image Dataset (Official Site) [15] and some real-life low-illuminated images. We found that our method performed well on both types of images. In Fig. 6, we represented the results of the proposed method using different patch size. To show the efficiency of the proposed method, we compare the results of the proposed method with bright channel prior-based methods [11, 13]. In Fig. 7, visual comparison of the proposed method with the bright channel prior-based methods [11, 13] has been shown. The comparative methods over enhance the image and amplify the effect of noise (if noise present in the image) and these methods only work well on night-time low-illumination images. While the results obtained by the proposed method appears more natural as

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Fig. 6 Results of proposed method on dark images: original image, result using patch size 15 × 15, result using patch size 9 × 9, and result using patch size 3 × 3

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Fig. 7 Comparison of the proposed method with earlier discussed BCP-based methods: original image, result of Sun et al. [11], result of Shi et al. [13], and result of the proposed method

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Table 1 Results of PSNR and IQE values of the Sun et al. [11] and Shi et al. [13] methods Image Sun et al. [11] Shi et al. [13] Proposed PSNR IQE PSNR IQE PSNR IQE 1. 2. 3. 4. 5. 6. Average

9.98 10.18 11.28 12.20 11.07 10.38 10.85

0.48 0.20 0.42 0.73 0.78 0.56 0.53

5.93 7.84 7.94 7.42 8.60 6.92 7.44

0.32 0.14 0.37 0.41 0.60 0.33 0.36

17.56 17.50 17.81 18.65 17.56 17.48 17.76

0.82 0.53 0.75 0.87 0.90 0.81 0.78

compared to methods presented in [11, 13] and effectively works well for all types of low-illuminated images. For quantitative evaluation, we used two widely used measurements peak signal to noise ratio (PSNR) and image quality index (IQE) [16]. The PSNR and image quality index are used to measure contrast and visibility improvement in all type of images, respectively. Table 1 shows the comparison of the proposed method with the aforementioned methods [11, 13]. The best values of the PSNR and image quality index shows the efficiency of the proposed method.

6 Conclusion In this work, we have introduced an improved l2 norm-based bright channel prior to enhance the overall illumination of the image by maintaining the image contrast. The proposed method effectively work for low-illuminated images and reduce the computational time by reducing the block effect from the bright channel and transmission map of the earlier bright channel prior-based methods. Earlier methods work better for low-illuminated night-time images while these methods also enhance the effect of noise in the images. In this work, Exclusively Dark (ExDark) Image Dataset (Official Site) and some real-life low-illuminated images are used to check the effectiveness of the proposed method. From the experimental results, we clearly observe that the proposed modified bright channel prior method recover the low-illuminated images effectively without affecting its natural appearance.

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References 1. Vishalakshi GR, Gopalakrishna MT, Hanumantharaju MC (2016) In Comprehensive review of video enhancement algorithms for low lighting conditions. In: Satapathy S, Mandal J, Udgata S, Bhateja V (eds), Advances in intelligent systems and computing. Springer, New York, pp. 475–485 2. Wang ZY, Hang MW, Hu P (2006) Image enhancement based on histograms and its realization with MATLAB. Comput Eng Sci 28(2):54–56 3. Jiang DQ, Li MD, Mao JL (2013) The research of the luminance dark color image enhancement technology. Artif Intell Identif 20:81–82 4. Jinag JL, Zhang YS, Xue F (2006) Local histogram equalization with brightness preservation. ACTA ELECTRONICA SINICA 34(5):861–866 5. Pizer SM, Amburn EP, Austin JD, Cromartie R, Geselowitz A, Greer T, Romeny BH, Zimmerman JB, Zuiderveld K (1987) Adaptive histogram equalization and its variations, Comput Vis Graph Image Process 39 355–228 368 6. Zuiderveld K (1994) Contrast limited adaptive histogram equalization. In: Graphics Gems IV; Academic Press Professional Inc: 230 Cambridge. MA, USA, pp 474–485 7. Zhao F, Luo HY, Geng H (2014) An RSSI gradient-based AP localization algorithm. China Commun 11:100–108 8. Piao Y, Liu L, Liu XY (2014) Enhancement technology of video under low illumination. Infrared Laser Eng 43(6):2021–2026 9. Wang WB, Mu XY, Tang N (2014) Algorithm of low illumination image enhancement, computer and modernization, pp. 27–31 10. Tiwari M, Gupta B, Shrivastava M (2015) High-speed quantile-based histogram equalisation for brightness preservation and contrast enhancement. IET Image Process 9:80–89 11. Sun S, Guo X, Image enhancement using bright channel prior. In: 2016 International conference on industrial informatics–computing technology, intelligent technology, Industrial Information Integration. Wuhan, China, pp 83–86 (2016) 12. Singh D, Kumar V (2018) Single image haze removal using integrated dark and bright channel prior. Modern Phys Lett B 32:1–9 13. Shi Z, Zhu MM, Guo B, Zhao M, Zhang C (2018) Nighttime low illumination image enhancement with single image using bright/dark channel prior. EURASIP J Image Video Process 13:1–15 14. He K, Sun J, Tang X (2011) Single image haze removal using dark channel prior. IEEE Trans Pattern Anal Mach Intell 33(12):2341–2353 15. Loh YP, Chan CS (2019) Getting to know low-light images with the exclusively dark dataset. Comput Vision Image Understand 178 16. Wang Z, Bovik AC (2002) A universal image quality index. IEEE Signal Process Lett 9:81–84

Existence Results of Mild Solutions for Impulsive Fractional Differential Equations with Almost Sectorial Operators M. C. Ranjini

Abstract This paper is concerned with the existence and uniqueness of mild solutions for a class of fractional impulsive semilinear differential equations using the concepts of almost sectorial operators. The results are established by using Banach contraction principle and Schauder’s fixed point theorem. Keywords Fractional differential equations · Impulsive condition · Almost sectorial operator · Semigroup of growth α · Mild solution

1 Introduction Sectorial operators, that is, linear operators A defined in Banach spaces, whose spectrum lies in a sector   π Sw = λ ∈ C/{0} | |argλ| ≤ w ∪ {0} f or some 0 ≤ w ≤ 2 and whose resolvent satisfies an estimate ||(λ − A)−1 || ≤ C|λ|−1 ,

∀ λ ∈ C\Sw ,

(1)

have been studied extensively during the last 40 years, both in abstract settings and for their applications to partial differential equations. Many important elliptic differential operators belong to the class of sectorial operators, especially when they are considered in the Lebesgue spaces or in spaces of continuous functions (see [1] and [[2], Chap. 3]). However, if we look at spaces of more regular functions such as the spaces of Holder continuous functions, we find that these elliptic operators do no longer satisfy the estimate (1) and therefore are not sectorial as was pointed out by Von Wahl (see [3], Ex.3.1.33, see [4]). M. C. Ranjini (B) Research Department of Mathematics, MES Mampad College, Malappuram, Kerala, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 D. Giri et al. (eds.), Proceedings of the Sixth International Conference on Mathematics and Computing, Advances in Intelligent Systems and Computing 1262, https://doi.org/10.1007/978-981-15-8061-1_41

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Neverthless, for these operators estimates such as ||(λ − A)−1 || ≤

C , |λ|1−α

λ∈



  = λ ∈ C : |arg(λ − w)| < v

(2)

w,v

where α ∈ (0, 1), w ∈ R and v ∈ ( π2 , π), can be obtained, (see [4]) which allows to define an associated “analytic semigroup” by means of the Dunford Integral T (t) =

1 2πi

 Γθ

eλt (λ − A)−1 dλ,

t >0

(3)

  where Γθ = teiθ : t ∈ R\{0} , θ ∈ (v, π2 ). In the literature, a linear operator A : D(A) ⊂ X →  X which satisfy the condition  (2) is called almost sectorial and the operator family T (t), T (0) = I, t ≥ 0 is said to be the “semigroup of growth α” generated by A. The operator family T (t)t≥0 has properties similar at those of analytic semigroup which allow to study some classes of partial differential equations via the usual methods of semigroup theory. Concerning almost sectorial operators, semigroups of growth α, and applications to partial differential equations, we refer the reader to [4–7, 10] and the references there in. Let us now give a short summary, which is organized in a way that in the first section, we state some results about the analytic semigroups of growth order γ, the definition of almost sectorial operator and introducing two special functions and their properties. Also, we construct a pair of families of operators and present some usual properties for these families which in turn is used to analyze the existence of mild solutions to the cauchy problem. In the second section, we explained the existence of mild solutions to the semilinear cauchy problem and, in the final section, we present an example to illustrate our results.

2 Preliminaries Here, we consider the semilinear impulsive fractional evolution equations in the following form. ⎧ c α ⎪ ⎨ D x(t) + Ax(t) = f (t, x(t)), t ∈ I = [0, T ], t = tk Δx|t=tk = Ik (x(tk− )), t = tk , k = 1, 2, ..., m. ⎪ ⎩ x(0) = x0

(4)

where c D α is the standard Caputo’s fractional derivative of order α, 0 < α < 1 and A : D(A) ⊂ X → X is an almost sectorial operator on a Banach space X. Here, 0 < t1 < t2 < ... < tm = T , Ik ∈ C(X, X), k = 1, 2, ..., m. Let Δx|t=tk = x(tk+ ) − x(tk− ), x(tk+ ) and x(tk− ) represent the right and left limits of x(t) at t = tk respectively.

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In this section, we recall some notations about almost sectorial operators, solution operators, and then present the definition of a mild solution of (3). Definition 1 Let 0 < γ < 1 and 0 < ω < π2 . By γω (X), we denote the family of all linear closed operators A : D(A) ⊂ X → X, which satisfy   (1) σ(A) ⊂ Sω = λ ∈ C\{0} : |argλ| ≤ ω ∪ {0} and (2) for every ω < μ < π, there exists a constant Cμ such that ||(λ − A)−1 || ≤

Cμ |λ|1−γ

, λ ∈ C\Sμ .

(5)

A linear operator A will be called an almost sectorial operator on X if A ∈ γω (X). The above definition allows to define an “associated analytic semigroup” by means of the Dunford Integral, T (t) = e−t z (A) =

1 2πi



e−t z R(z; A)dz,

Γθ

t >0

(6)

    where Γθ = R+ eiθ ∪ R+ e−iθ , is oriented counter-clockwise and ω < θ < μ < π forms an analytic semigroup of growth γ. The operator family T (t)t≥0 has properties similar at those of analytic semigroup which allow to study some classes of partial differential equations via the usual methods of semigroup theory. For more properties on T (t), please see the following proposition. Proposition 1 ([9]) Let A ∈ γω (X) with 0 < γ < 1. Then the following properties are satisfied. (i) The operator A is closed, T(t+s) = T(t)T(s) and AT(t)x = T(t)Ax, ∀t, s ∈ [0, ∞) and each x ∈ D(A). (ii) dtd T (t) = AT (t). (iii) There exists a constant C0 > 0 such that ||An T (t)|| ≤ Cn t −(n+γ) (t > 0). Also, the relation between the resolvent operators of A and the semigroup T(t) is characterized by, Proposition 2 Let A ∈ γω (X) with 0 < γ < 1. Then for every λ ∈ C with Re λ > 0,

∞ one has R(λ, −A) = 0 e−λt T (t)dt. We present some properties of two special functions. Denote by E α,β the generalized Mittag-Leffler function defined by E α,β (z) =

∞  k=0

1 zk = Γ (αk + β) 2πi

 ℘

λα−β eλ dλ, α, β > 0, z ∈ C λα − z

where ℘ is a contour which starts and ends at −∞.

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Consider also the function of Wright-type, Ψα (z) =

∞  n=0

∞ 1  (−z)n (−z)n = Γ (nα)sin(nπα), z ∈ C n!Γ (−αn + 1 − α) π n=1 (n − 1)!

with 0 < α < 1. For −1 < r < ∞, λ > 0, the following results hold: (w1 ) (w2 ) (w3 ) (w4 ) (w5 )

Ψ α (t) ≥ 0, t > 0;

∞ α α Ψ ( 1 )e−λt dt = e−λ ; 0

∞ t α+1 αr t α (1+r ) Ψ (t)t dt = ΓΓ(1+αr ; )

0∞ α −zt dt = E α (−z), z ∈ C;

0∞ Ψα (t)e −zt αtΨ (t)e dt = eα (−z), z ∈ C. α 0

Throughout this section, we will define the two families of operators based on the generalized Mittag-Leffler type functions denoted by,  1 E α (−zt α )R(z; A)dz 2πi Γθ  1 Pα (t) = eα (−zt α )(A) = eα (−zt α )R(z; A)dz 2πi Γθ Sα (t) = E α (−zt α )(A) =

    where Γθ = R+ eiθ ∪ R+ e−iθ , is oriented counterclockwise. We need some basic properties of these families which are used further in this paper. Lemma 1 ([9]) For each fixed t ∈ S 0π −ω , Sα (t) and Pα (t) are linear and bounded 2 operators on X. Moreover, there exist constants Cs , C p > 0 such that for all t > 0, ||Sα (t)|| ≤ Cs t −αγ ,

||Pα (t)|| = C p t −αγ ,

wher e0 < γ < 1.

From the definition defined by (5) and by (w4 ), we will frequently use the representations for Sα (t) and Pα (t) as follows:  Sα (t) =

∞

Ψα (s)T (st α )ds.

(7)

αsΨα (s)T (st α )ds.

(8)

0

 Pα (t) =

∞

0

Lemma 2 ([9]) For t > 0, Sα (t) and Pα (t) are continuous in the uniform operator topology. Moreover, for every r > 0, the continuity is uniform on [r, ∞). Lemma 3 ([9]) If R(λ, −A) is compact for every λ > 0, then T (t) is compact for every t > 0.

Existence Results of Mild Solutions for Impulsive …

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Lemma 4 ([9]) If R(λ, −A) is compact for every λ > 0, then Sα (t), Pα (t) are compact for every t > 0. In order to define the concept of mild solution, we need to define the following space,  + PC(I, X) = x : I → X : x ∈ C((tk , tk+1 ], X),  k = 1, 2, ..., m and there exist x(tk ) − − and x(tk ), k = 1, 2, ...m with x(tk ) = x(tk ) endowed with the norm ||x|| PC = supt∈I ||x(t)||. Now, we define the mild solution of the system (4) as follows: Definition 2 A function x : I → X is called a mild solution of a system (4), if x ∈ PC(I, X) and satisfies the following equation,

x(t) =

⎧

⎪ t ∈ [0, t1 ]; S (t)x0 + 0t (t − s)α−1 Pα (t − s) f (s, x(s))ds, ⎪ ⎪ α 

⎪ ⎪ ⎪ Sα (t − t1 ) x(t1− ) + I1 (x(t1− )) + tt (t − s)α−1 Pα (t − s) f (s, x(s))ds, ⎪ ⎪ 1 ⎪ ⎨.

t ∈ (t1 , t];

⎪ . ⎪ ⎪ ⎪ ⎪ ⎪ . ⎪ ⎪ ⎪ ⎩ S (t − t )x(t − ) + I (x(t − )) + t (t − s)α−1 P (t − s) f (s, x(s))ds, α m m α m m tm

t ∈ (tm , T ].

3 Existence Results In this section, we give the main results on the existence of mild solutions of the system (4). To establish our results, we introduce the following hypotheses. (H1 )

f : I × X → X is continuous and there exists a constant M > 0 such that || f (t, x) − f (t, y)|| ≤ M||x − y||,

∀ t ∈ I, x, y ∈ X

|| f (t, 0)|| ≤ k1 , where k1 is a constant. (H2 ) for each k = 1, 2, ..., m, there exists ρk > 0 such that ||Ik (x) − Ik (y)|| ≤ ρk ||x − y||, ||Ik (0)|| ≤ k2 ,

∀x, y ∈ X

where k2 is a constant.

   T α(1−γ) (H3 ) max Cs T −αγ r + (ρi r + k2 ) + C p (Mr + k1 ) α(1−γ) ≤r 1≤i≤m

Theorem 1 Under the assumptions (H1 ) − (H3 ), the system (4) has a unique mild solution x ∈ PC(I, X) if

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M. C. Ranjini

  T α(1−γ)  0. (H2 ) The function f : I × X → X is continuous and there exists a function μ1 ∈ L ∞ (I, R+ ) such that || f (t, x)|| ≤ μ1 (t), for all t ∈ I, x ∈ X. (H3 ) The functions Ik : X → X, k = 1, 2, ..., m are completely continuous and uniformly bounded. Then the system (4) has atleast one mild solution defined on I. Proof Define operator Γ : PC(I, X) → PC(I, X), as in Theorem [1] by

Γ x(t) =

⎧

⎪ t ∈ [0, t1 ]; S (t)x0 + 0t (t − s)α−1 Pα (t − s) f (s, x(s))ds, ⎪ ⎪ α 

⎪ ⎪ ⎪ Sα (t − t1 ) x(t1− ) + I1 (x(t1− )) + tt (t − s)α−1 Pα (t − s) f (s, x(s))ds, ⎪ ⎪ 1 ⎪ ⎨. ⎪ . ⎪ ⎪ ⎪ ⎪ ⎪ ⎪. ⎪ ⎪ ⎩ S (t − t )x(t − ) + I (x(t − )) + t (t − s)α−1 P (t − s) f (s, x(s))ds, α m m α m m tm

t ∈ (t1 , t];

t ∈ (tm , T ].

We have already proved that Γ is well defined. Now, we prove the theorem in the following three steps. Step 1: To prove Γ is continuous. Let xn be a sequence such that xn → x in PC(I, X). Then, for each t ∈ I , f (s, xn (s)) → f (s, x(s))

as

n→∞

(10)

because the function f is continuous on I × X. Now, for every t ∈ [0, t1 ], we have 

t

||Γ xn (t) − Γ x(t)|| ≤

(t − s)α−1 ||Pα (t − s)|| || f (s, xn (s)) − f (s, x(s))||ds

0

≤ Cp 

T α(1−γ) , α(1 − γ)

 > 0,

 → 0 (n → ∞)

(11)

Existence Results of Mild Solutions for Impulsive …

509

for t ∈ (t1 , t2 ], we have   ||Γ xn (t) − Γ x(t)|| ≤ ||Sα (t − t1 )|| ||xn (t1− ) − x(t1− )|| + ||I1 (xn (t1− )) − I1 (x(t1− ))||  t (t − s)α−1 ||Pα (t − s)|| || f (s, xn (s)) − f (s, x(s))||ds t1

  ≤ Cs (t − t1 )−αγ ||xn (t1− ) − x(t1− )|| + ||I1 (xn (t1− )) − I1 (x(t1− ))|| +C p 

T α(1−γ) , α(1 − γ)

 > 0,

 → 0 as n → ∞

(12)

Moreover, we have ∀ t ∈ (ti , ti+1 ]   ||Γ xn (t) − Γ x(t)|| ≤ Cs T −αγ ||xn (ti− ) − x(ti− )|| + ||Ii (xn (ti− )) − Ii (x(ti− ))|| +C p 

T α(1−γ) , α(1 − γ)

 > 0,

 → 0 as n → ∞

(13)

and ∀ t ∈ (tm , T ],   ||Γ xn (t) − Γ x(t)|| ≤ Cs T −αγ ||xn (tm− ) − x(tm− )|| + ||Ii (xn (tm− )) − Ii (x(tm− ))|| +C p 

T α(1−γ) , α(1 − γ)

 > 0,

 → 0 as n → ∞

(14)

By the continuity of Ik (k = 1, 2, ..., m), as well as from (11) to (14), we have limn→∞ ||Γ xn (t) − Γ x(t)|| PC = 0 which means that Γ is continuous. Step 2: To prove Γ (Br ) is equicontinuous, where Br is defined as in Theorem [1]. Denote Λ = max1≤k≤m {||Ik (x)||}. Let, s1 , s2 ∈ [0, t1 ], with s1 < s2 , then ∀s1 , s2 , we have   s2 ||(Γ x)(s2 ) − (Γ x)(s1 )|| ≤ ||Sα (s2 ) − Sα (s1 )|| ||x0 || + (s2 − s)α−1 ||Pα (s2 − s)|| 0  s1  || f (s, x(s))||ds − (s1 − s)α−1 ||Pα (s1 − s)|| || f (s, x(s))||ds 0

≤ ||Sα (s2 ) − Sα (s1 )|| r  s α(1−γ) − s α(1−γ)  1 +C p ||μ1 || L ∞ (I,R+ ) 2 α(1 − γ)

Similarly, ∀s1 , s2 ∈ (ti , ti+1 ], with s1 < s2 , i = 1, 2, ..., m, we have

510

M. C. Ranjini

||(Γ x)(s2 ) − (Γ x)(s1 )|| ≤ ||Sα (s2 − t1 ) − Sα (s1 − t1 )|| [r + Λ]  s α(1−γ) − s α(1−γ)  1 +C p ||μ1 || L ∞ (I,R+ ) 2 α(1 − γ) Thus, from the above inequalities, we have lims2 →s1 ||(Γ x)(s2 ) − (Γ x)(s1 )|| = 0. So, Γ (Br ) is equicontinuous. Step 3: To prove {Γ x(t) : x ∈ Br } is precompact in X. We decompose Γ by Γ = Γ1 + Γ2 , where ⎧ t α−1 Pα (t − s) f (s, x(s))ds, ⎪ ⎪ 0 (t − s) ⎪

⎪ t α−1 ⎪ Pα (t − s) f (s, x(s))ds, ⎪ t1 (t − s) ⎪ ⎪ ⎨. Γ1 x(t) = ⎪ . ⎪ ⎪ ⎪ ⎪ ⎪. ⎪ ⎪ ⎩ t α−1 Pα (t − s) f (s, x(s))ds, tm (t − s)

t ∈ [0, t1 ]; t ∈ (t1 , t];

t ∈ (tm , T ].

and ⎧ Sα (t)x0 t ∈ [0, t1 ]; ⎪ ⎪ ⎪  − ⎪ − ⎪ t ∈ (t1 , t]; ⎪ ⎪ Sα (t − t1 ) x(t1 ) + I1 (x(t1 )) ⎪ ⎨. Γ2 x(t) = ⎪. ⎪ ⎪ ⎪ ⎪ ⎪ . ⎪ ⎪  ⎩ t ∈ (tm , T ]. Sα (t − tm ) x(tm− ) + Im (x(tm− )) Now, we prove that Γ1 is compact in X. For that let t ∈ (0, T ] be fixed and , δ > 0. For x ∈ Br , using the equation (8) define the map Γ1δ by, ⎧ t− ∞ ατ (t − s)α−1 Ψα (τ )T ((t − s)α τ ) f (s, x(s))dτ ds, ⎪ ⎪

0t− δ∞ ⎪ ⎪ α−1 Ψ (τ )T ((t − s)α τ ) f (s, x(s))dτ ds, ⎪ ⎪ α t1 δ ατ (t − s) ⎪ ⎪ ⎨. Γ1,δ x(t) = ⎪ . ⎪ ⎪ ⎪ ⎪ ⎪ . ⎪ ⎪ ⎩ t− ∞ α−1 Ψ (τ )T ((t − s)α τ ) f (s, x(s))dτ ds, , α tm δ ατ (t − s)

t ∈ [0, t1 ]; t ∈ (t1 , t];

t ∈ (tm , T ].

Existence Results of Mild Solutions for Impulsive …

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For t ∈ [0, t1 ],  t  δ    ||Γ1 x(t) − Γ1,δ x(t)|| ≤  ατ (t − s)α−1 Ψα (τ )T ((t − s)α τ ) f (s, x(s))dτ ds  0 0   t  ∞   ατ (t − s)α−1 Ψα (τ )T ((t − s)α τ ) f (s, x(s))dτ ds  + ≤

 t 0

+ ≤

t− δ

α(t − s)α(1−γ)−1 || f (s, x(s))||ds

 t

 δ 0

α(t − s)α(1−γ)−1 || f (s, x(s))||ds

t−

τ 1−γ Ψα (τ )dτ  ∞ δ

τ 1−γ Ψα (τ )dτ

 α||μ1 || L ∞ (I,R+ )  α(1−γ)  δ 1−γ Γ (2 − γ) t τ Ψα (τ )dτ + α(1−γ) α(1 − γ) Γ (1 + α(1 − γ)) 0

Similarly for t ∈ (ti , ti+1 ], we have  t   ||Γ1 x(t) − Γ1,δ x(t)|| ≤  t1

  +

δ

0

t

t−

≤

  ατ (t − s)α−1 Ψα (τ )T ((t − s)α τ ) f (s, x(s))dτ ds 

 δ

∞

  ατ (t − s)α−1 Ψα (τ )T ((t − s)α τ ) f (s, x(s))dτ ds 

α||μ1 || L ∞ (I,R+ )  (t − ti )α(1−γ) α(1 − γ)



δ 0

τ 1−γ Ψα (τ )dτ + α(1−γ)

 Γ (2 − γ) Γ (1 + α(1 − γ))

Using the total boundedness we have that for each t ∈ (0, T ], {Γ1 x(t) : x ∈ Br } is precompact in X. Now, we show that {Γ2 x(t) : x ∈ Br } is precompact in X. For all t ∈ [0, t1 ], since Γ2 x(t) = Sα (t)x0 , by lemma [4], it follows that {Γ2 x(t) : t ∈ [0, t1 ], x ∈ Br } is precompact in X. On the other hand, for t ∈ (ti , ti+1 ], i ≥ 1 and x ∈ Br , there exists r ∗ > 0 such that ⎧  − − ⎪ ⎨ Sα (t − ti ) y(t i ) + Ii (y(ti )) t ∈ (ti , ti+1 ), y ∈ Br ∗ [Γ2∗ x]i (t) ∈ Sα (ti+1 − ti ) y(ti− ) + Ii (y(ti− )) t = ti+1 , y ∈ Br ∗ ⎪ ⎩ − − y(ti ) + Ii (y(ti )), t = ti , y ∈ Br ∗ where Br ∗ is an open ball of radius r ∗ . From (H3 ) and lemma [4], it follows that Γ2∗ x]i (t) is relatively compact in X, for all t ∈ (ti , ti+1 ], i ≥ 1. Moreover, by the compactness of Ik (k = 1,2,...,m) and the continuity of the evolution operator Sα (t), for all t ∈ [0, T ], we conclude that operator Γ2 is also compact. Therefore, finally, by steps 1–3 and by Arzela–Ascoli theorem, Γ = Γ1 + Γ2 is precompact in X. Thus, by Schauder’s fixed point theorem, Γ has a fixed point, which gives a mild solution. This completes the proof.

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3.1 Example 1 ˆ = W 1,3 (R2 ) (a sobolev space) Let Aˆ = (−iΔ + σ) 2 , D( A) be as in example 6.3([9]), in which the authors demonstrate that Aˆ is an almost sectorial operator for some 0 < w < π2 and γ = − 16 . We denote the semigroup associated 1 with Aˆ by T(t) and ||T (t)|| ≤ C0 t − 6 , where C0 is a constant. Let X = L 3 (R2 ), we consider the following problem.

⎧ 1 e−t c ˆ ⎪ + 2+e t ar ctan(2x(t)), t ∈ I = [0, 1], t = ⎨ D 2 x(t) = Ax(t) Δx( 21 ) = 19 sinx(t), t = 21 ⎪ ⎩ x(0) = 0

1 2

(15)

−t

e Here, f (t, x) = 2+e t ar ctan(2x), for (t, x) ∈ [0, 1] × X, 1 I1 (x) = 9 sin(x) for x ∈ X. By direct computations, we see that || f (t, x) − f (t, y)|| ≤ 23 ||x − y||, and ||I1 (x) − I1 (y)|| ≤ 19 ||x − y|| and m = 1. Then from the equation (9),

L = Cs

10 8 + Cp < 1 9 7

for the suitable values of the constants Cs and C p . Thus, all the assumptions of Theorem [1] are satisfied and hence by the conclusion of the Theorem [1], the impulsive fractional problem (15) has a unique solution on [0,1].

References 1. Da Prato G, Sinestrari E (1987) Differential operators with non-dense domain. Ann. Scuola Norm. Sup. Pisa cl. sci. 14:285–344 2. Lunardi A (1995) Analytic semigroups and optimal regularity in parabolic problems. Birkhauser Verlag, Basel 3. Von Wahl, W (1972) Gebrochene potenzen eines elliptischen operators und parabolische Differentialgleichungen in Raumen holderstetiger Funktionen. Nachr. Akad. Wiss. Gottingen, Math.-Phys. Klasse 11, 231–258 4. Periago F, Straub B (2002) A functional calculus for almost sectorial operators and applications to abstract evolution equations. J. Evol. Equ. 2:41–68 5. Dlotko T (2007) Semilinear Cauchy problems with almost sectorial operaors. Bull. Pol. Acad. Sci. Math. 55:333–346 6. Okazawa N (1974) A generation theorem for semigroups of growth order α. Tohoku Math. L. 26:39–51 7. Periago F (2003) Global existence, uniqueness and continuous dependence for a semilinear initial value problem. J. Math. Anal. Appl. 280:413–423 8. Taira K (1989) The theory of semigroups with weak singularity and its applications to partial differential equations. Tsukuba J. Math. 13:513–562

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9. Wang Rong-Nian, Chen De-Han, Xiaon Ti-jun (2012) Abstract fractional cauchy problems with almost sectorial operators. Journal of Differential Equations. 252:202–235 10. Lu Zhang., Yong Zhou.: Fractional cauchy problems wuth almost sectorial operators. 257, 145-157 (2015). https://doi.org/10.1016/j.amc.2014.07.024 11. Xiao-LiDing., Bashir Ahmad.: Analytical solutions to fractional evolution equations with almost sectorial operators. Advances in Difference Equations, 2016:203 (2016). https://doi. org/10.1186/s13662-016-0927-y 12. DiZhang., Yue Liang.:Existence and controllability of fractional evolution equation with sectorial operator and impulse. Advances in Difference Equations. 2018:219 (2018). https://doi. org/10.1186/s13662-018-1664-1 13. Podlubny I (1993) Fractional Differential Equations. Academic Press, New York 14. Ahmed B, Sivasundaram S (2010) Existence of solutions for Integral boundary value problems of fractional order. Nonlinear Analysis: Hybrid Systems. 4:134–141 15. Lakshmikantham V, Bainov DD, Simenov PS (1989) Theory of Impulsive Differential Equations. World Scientific, Singapore 16. Mophou G (2010) Existence and uniqueness of mild solutions to impulsive fractional differential equations. Nonlinear Analysis: Theory, Methods and Applications. 72:1604–1615 17. Wang RN, Chen DH, Xiao TJ (2012) Abstraft fractional cauchy problems with almost sectorial operators. Journal of Differential Equations. 252:202–235

Effective Algebraic Methods are Widely Applicable Takeo Kamizawa

Abstract This presentation deals with some algebraic methods for mathematical analysis. The main point of using these tools is the ’effectiveness’ of them, where an effective method means that it is a step-by-step procedure (i.e. an algorithm) and the result will be obtained in a finite number of steps. We will study several effective algebraic procedures and explore the fact that they can be used in many applications. Keywords Effective procedures · Reducibility · Root classifications · Differential equations

1 Introduction In order to discuss natural phenomena scientifically, it is important (and necessary) to describe them in mathematical ways. Today, mathematical models of natural phenomena are often described by dynamical systems, where the dynamics are represented by differential or difference equations. However, the models of natural phenomena are usually complicated, and even with the help of latest supercomputers the complete analysis of nature is hardly possible. Even in the linear case, it is extremely difficult to compute the eigenvalues of linear maps (matrices) because, in general, there is no algebraically closed formula for the eigenvalues of a matrix if the dimension is greater or equal to 5 due to the well-known Abel–Ruffini theorem. One approach for this difficulty is the ‘algebraic reduction’ of models. The idea is that, despite the complexity of those model analysis on the whole space, in some occasions we are able to ‘extract’ a part on which the dynamics is relatively simple. If such a part is found, we can analyse the dynamics on this part in detail. Since such a part does not always exist, we are interested in criteria of the existence of it. The important point is that they should be ‘effective’, i.e. we can test the reducibility of T. Kamizawa (B) Department of Information Sciences, Tokyo University of Science, Noda, Japan e-mail: [email protected] URL: http://www.rs.tus.ac.jp/ kamizawa/ © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 D. Giri et al. (eds.), Proceedings of the Sixth International Conference on Mathematics and Computing, Advances in Intelligent Systems and Computing 1262, https://doi.org/10.1007/978-981-15-8061-1_42

515

516

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our model step by step and the result will be obtained in a finite number of steps. The effective algebraic criteria we introduce are quite general and they can be applied in many branches of science. In addition, we introduce effective algebraic procedures which mention the ‘number’ of certain eigenvalues (e.g. positive ones) but not exact values. These methods allow us to discuss the stability of dynamical systems.

2 Reductions of Associative Algebras 2.1 Associative Algebra Let us review some algebraic notions. Let V be an n-dimensional linear space over a field F and A : V → V be a linear map (operator). An A-invariant subspace is a subspace S ⊂ V such that AS ⊂ S . In addition, for a set of linear maps A ⊂ L(V ), a subspace S ⊂ V is said to be A-invariant if S is A-invariant for all A ∈ A. A subset A ⊂ L (V ) is said to be reducible if there is a non-trivial A-invariant subspace S ⊂ V (here, the trivial invariant subspaces are ∅ and V ) and A is said to be irreducible, otherwise. An associative algebra A over a field F is a linear space over F which is closed under a bilinear map, called the multiplication, satisfying the associative law, and A is unital if it is equipped with the multiplicative identity I . For simplicity, we call such A an algebra, and throughout this paper, we assume that dim A = m < +∞. An element A ∈ A is said to be nilpotent if there is some k ∈ N0 such that Ak = O, and P ∈ A is said to be properly nilpotent if P A is nilpotent for all A ∈ A. The set of all properly nilpotent elements is called the radical of A and denoted by radA. An algebra A with radA = {O} is called semi-simple and A is said to be simple if A is nonzero and it has no non-trivial ideal. Semi-simple and simple algebras are strongly related; according to Wedderburn, a semi-simple algebra can be decomposed into a direct sum of simple algebras (cf. [3]): A=

s 

As ,

(1)

k=1

Indeed, a simple algebra is known to be isomorphic to a full matrix algebra M p (F) for some p, so the Wedderburn decomposition (1) means that the elements in A can be simultaneously block-diagonalised. Concerning the reducibility of algebras, Burnside proved a characterisation of reducible algebras (cf. [3]): If V is a finite-dimensional linear space over an algebraically closed field, then the only irreducible algebra A ⊂ L (V ) is L (V ) itself. 2 Hence, if we have a basis B = {Bk }m k=1 of the algebra A to find that m < n , then one can immediately say that A is reducible. If so, in the matrix representation, there is a basis such that all operators in A have the same block-triangular form.

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If an algebra A is semi-simple and reducible, we say that A is completely reducible. If A is known to be completely reducible, there exist at least two simple algebras in the Wedderburn decomposition (1).

2.2 Construction of a Basis How can we check the reducibility of the algebra A generated by our system? Due to the Burnside’s theorem the algebra A is reducible iff dim A = m < n 2 . If  = {A1 , . . . , As } ⊂ Mn (F) is a generating set of matrices (we can assume the matrices are linearly independent w.l.o.g.), a direct method is to compute all possible combinations of products of matrices from  and choose a maximal number of matrices so that they are linearly independent. Note that the ‘length’ of the multiplications is bounded above, which simply follows from the well-known Cayley– Hamilton theorem, but some better bounds have been discovered, for example, by Paz and Pappacena (cf. [11]):    2 1 n 2n 2 n +2 , n + + − 2. (2) 3 (n − 1) 4 2 Thus, our procedure for calculating a basis B = {Bk }m k=1 of A (and then test if A is reducible or not) is effective, which is summarised as follows: Algorithm 1 Let  = {A1 , . . . , As } ⊂ Mn (F) be a generating set of matrices which are linearly independent. 1. Set B1 = A1 , . . . , Bs = As and set B = {Bk }sk=1 .   2. For some j, k ∈ {1, . . . , |B|}, compute B j Bk . If B ∪ B j Bk is linearly independent, set B|B|+1 = B j Bk , add it into B and try with other pairs. 3. Conduct similar process for Bi1 · · · Bi (i k ∈ {1, . . . , |B|}) until the length  meets either of the bounds (2). 4. The obtained set B forms a basis of A. – If m = |B| < n 2 , then the algebra A is reducible. – If m = |B| = n 2 , then the algebra A is irreducible. Example 1 Let us consider an associative algebra A generated by ⎤ ⎡ ⎤ 5 −3 3 −3 2 −2 D1 = ⎣ 0 21 21 ⎦ , D2 = ⎣ − 27 2 −4 ⎦ . 1 −2 23 − 21 −1 −1 2 ⎡

Set  = {I, D1 , D2 } (here I, D1 , D2 are linearly independent). 1. According to (2), the length of the products is bounded above by Set B1 = I , B2 = D1 and B3 = D2 .



32 +2 3

= 4.

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2. Compute Bi1 · · · Bi4 for all patterns (i 1 , . . . , i 4 ) ∈ {1, 2, 3}4 and choose maximal number of linearly independent matrices. Then, one can observe that   B = {B1 , . . . , B5 } = I, D1 , D2 , D1 D2 , D12 is a basis for A. 3. Since dim A = 5 < 32 , the algebra A is reducible.

2.3 Discriminant of Algebra For testing the complete reducibility of our algebra A, once it is known to be reducible, we are left to check the semi-simplicity of A. In order for this, we introduce the so-called ‘discriminant’ of A, which is an effective criterion. Let B = {Bk }m k=1 be a basis of an algebra A. The discriminant matrix DB is given by ⎤ ⎡ tr B1 B1 · · · tr B1 Bm m  ⎥ ⎢ .. .. .. DB = tr Bi B j i, j=1 = ⎣ ⎦, . . . tr Bm B1 · · · tr Bm Bm

and the discriminant of A is the quantity: discB A = det DB . Since the construction of the discriminant matrix DB and the computation of the determinant of a matrix can be done effectively, the total procedure can be completed effectively. Theorem 2 (cf. [9, 14]). A is semi-simple if and only if discB A = 0. Therefore, the discriminant is an effective tool to check the semi-simplicity of A. However, it may take time to calculate the determinant of DB . Since we are interested only in the invertibility of DB , some sufficient (but not necessary) conditions have been discovered (cf. [4, 5, 13]). Example 2 Consider the algebra A generated by D1 and D2 as in Example 1 and let B = {Bk }k be the basis. The discriminant of A with respect to B becomes ⎡

3 ⎢ 5 ⎢ discB A = det ⎢ ⎢ −2 ⎣ −4 15

5 15 −4 −10 53

−2 −4 6 8 −10

−4 −10 8 10 −32

⎤ 15 53 ⎥ ⎥ −10 ⎥ ⎥ = 256 = 0, −32 ⎦ 195

so the algebra A is semi-simple. Since A is reducible, we conclude that it is completely reducible.

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2.4 ALS-Criterion Another effective method for checking the complete reducibility of algebras is the ‘ALS-criterion’. The standard polynomial with  arguments is a polynomial: S (X 1 , . . . , X  ) =



sgn(σ )X σ (1) . . . X σ () ,

σ

where σ is a permutation and the summation is taken over all possible permutations. The standard polynomial S satisfies the following conditions (cf. [15]): 1. The standard polynomial S is multilinear. 2. S (. . . , X i , . . . , X j , . . .) = −S (. . . , X j , . . . , X i , . . .) for any pair of i, j. Amitsur and Levitzki [1], Shapiro [17] studied the relations between algebras and polynomial identities. Combining their results, we obtain the following criterion, which we call the ‘ALS-criterion’ (cf. [11, 15]). Theorem 3 (ALS-criterion, cf. [15]). Let  = {A1 , . . . , As } be a given set of n × n matrices over C , A = A(A1 , . . . , As ) be the algebra generated by A1 , . . . , As and suppose A = A∗ . Then, the followings are equivalent: – S2 p = 0 on A for some p, – A is completely reducible and the dimension of each (decomposed) subalgebra is no greater than p. Example 3 Let us consider the algebra A generated by D1 and D2 . By computing S4 (X 1 , X 2 , X 3 , X 4 ) for all X j ∈ B, we observe that S4 = 0 for all X j ∈ B, which means that S4 = 0 on A and the algebra A is completely reducible. Moreover, the size of each block on the main diagonal does not exceed 2 × 2.

3 Shemesh Criterion and Its Generalisation For given two matrices A, B ∈ Mn (F), where F is an algebraically closed field, it is interesting to determine if they share an eigenvector. Shemesh [18] showed a necessary and sufficient condition for this problem. An important point is that, if F = C, the problem can be solved ’effectively’. The Shemesh criterion was generalised for an arbitrary number of matrices by Jamiołkowski and Pastuszak [7]. Note that the generalised Shemesh criterion can be applied to matrices over not only the complex field but an algebraically closed field due to Laffey [12]. Theorem 4 Let A1 , . . . , As ∈ Mn (F), where F is an algebraically closed field, N (A1 , . . . , As ) =

n−1  ki ,li ≥0

  ker Ak11 , . . . , Aks s , Al11 , . . . , Alss ,

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  where the intersection is taken so that i ki = 0, i li = 0. Then, N (A1 , . . . , As ) is the maximal common invariant subspace containing all common eigenvectors, i.e. A1 , . . . , As have a common eigenvector if and only if N (A1 , . . . , As ) = {0}. Moreover, if F = C, we have N (A1 , . . . , As ) = ker

n−1  

Ak11 , . . . , Aks s , Al11 , . . . , Alss

∗ 

 Ak11 , . . . , Aks s , Al11 , . . . , Alss , (3)

ki ,li ≥0

where the summation is taken so that

 i

ki = 0,



i li

= 0.

Example 4 Let us consider D1 , D2 in Example 1. Then, ⎡

⎤ 17440 −21360 21360 N (D1 , D2 ) = ker ⎣ −21360 35712 −35712 ⎦ 21360 −35712 35712 = span [0, 1, 1]T = {0} , so we can conclude that D1 and D2 have a common eigenvector (in fact [0, 1, 1]T is a common eigenvector) and they can be simultaneously block-triangularised.

4 Root Classifications of Polynomials Another type of effective algebraic methods we introduce is one for counting the number of certain classes of roots of a polynomial, e.g. real roots, positive roots (cf. [6, 20]). This can be applied to the study of eigenvalues of matrices.

4.1 Discrimination Matrix and Discriminant Sequence Definition 1 Let  p ∈ R [x] be a polynomial whose coefficients are real numbers and write p (x) = nk=0 ak x k , where an = 0. The 2n × 2n matrix defined as follows is called the discrimination matrix of p: ⎤ an an−1 an−2 · · · a0 ⎥ ⎢ 0 nan (n − 1) an · · · a1 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ Discr ( p) = ⎢ ⎥. . . .. .. ⎥ ⎢ ⎥ ⎢ ⎣ an an−1 · · · a0 ⎦ 0 nan · · · a1 ⎡

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521

Definition 2 Let p ∈ R [x] with deg p = n, and for each k ∈ {1, . . . , 2n} let Mk be the k-th leading principal minor of Discr ( p). Set Dk = M2k , and the n-tuple D ( p) = (D1 , D2 , . . . , Dn ) is called the discriminant sequence of p. Example 5 For p (x) = a2 x 2 + a1 x + a0 , the discrimination matrix is ⎡

⎤ a2 a1 a0 ⎢ 2a2 a1 ⎥ ⎥. Discr ( p) = ⎢ ⎣ a2 a1 a0 ⎦ 2a2 a1    and the discriminant sequence becomes D ( p) = 2a22 , a22 a12 − 4a0 a2 . Definition 3 Let p ∈ R [x] with deg p = n and D ( p) be the discriminant sequence. The list S ( p) = (sgnD1 , . . . sgnDn ) = (s1 , . . . , sn ) is called the sign list of p. Moreover, if there is some subsequence in S ( p) such that   . . . , sk , sk+1 , . . . , sk+ j−1 , sk+ j , . . . sk = 0, sk+ j = 0, sk+1 = · · · = sk+ j−1 = 0, then let ek+ = (−1)(+1)/2 , and let e = s , otherwise. The list R ( p) = (e1 , . . . , en ) is called the revised sign list of p. Theorem 5 (c f.[20]) Let p ∈ R [x] with deg p = n and R ( p) be the revised sign list. Then, the number of sign changes, say ν, is equal to the number of pairs of distinct complex conjugate roots of p, and s − 2ν, where s is the number of non-zero digits in the revised sign sequence, is equal to the number of distinct real roots of p. Example 6 If p (x) = 5x 3 − 7x 2 + 3, the revised sign list is R( p) = (1, 1, −1), so ν = 1 is the number of pairs of complex conjugate roots. In fact, p has complex roots approximately 0.98 ± 0.36i. Since the non-vanishing elements are s = 3, there is a s − 2ν = 1 distinct real root, which is approximately −0.55. In applications, it is often beneficial if the real parts of the roots of a real polynomial are negative, especially in the stability theory. We give the following criterion for this. Let p (x) = nk=0 αk x k with α0 = 0 (we assume that αn > 0 w.l.o.g.). Construct the so-called the Hurwitz matrix [20]: ⎡

αn−1 ⎢ αn ⎢ ⎢ 0 ⎢ ⎢ H ( p) = ⎢ 0 ⎢ 0 ⎢ ⎢ 0 ⎣ .. .

αn−3 αn−2 αn−1 αn 0 0

αn−5 αn−4 αn−3 αn−2 αn−1 αn .. .

⎤ ··· ···⎥ ⎥ ···⎥ ⎥ ···⎥ ⎥ ∈ Mn (R) . ···⎥ ⎥ ···⎥ ⎦

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Theorem 6 (Routh–Hurwitz criterion, cf. [20]). The real parts of the real polyno mial p (x) = nk=0 αk x k are all negative (i.e. p is a Hurwitz polynomial) if and only if the sign changes do not occur in the sequences (αn , det H1 , det H3 . . . , ) and (1, det H2 , det H4 , . . .) , where det Hk means the k-th leading principal minor of H ( p). Example 7 Consider the real polynomial p (x) = x 3 + 2x 2 + 4x + 1. In this case, ⎡

⎤ 210 H ( p) = ⎣ 1 4 0 ⎦ , 021 so there is no sign change in the lists: (α3 , det H1 , det H3 ) = (1, 2, 7)

(1, det H2 ) = (1, 7) ,

and we conclude that the real parts of the roots are all negative. In fact, p has solutions approximately −0.86 ± 1.67i and −0.28.

4.2 Number of Positive/Negative Roots In some occasions, we may wish to determine the number of positive and negative roots of p ∈ R[x]. Let p (x) = nk=0 αk x k ∈ R [x] with αn = 0, α0 = 0 and n    π± (y) = p ±y 2 = (±1)k αk x 2k . k=0

Then, it can be easily observed that the number of positive (resp. negative) roots of p is equal to half the number of real roots of π+ (resp. π− ). Thus, by constructing the revised sign lists of π± , we are able to count the number of positive and negative roots of p. For constructing the discriminant sequences, in [21] an effective procedure was considered.  Theorem 7 Let p (x) = nk=0 αk x k ∈ R [x] with an = 0 and a0 = 0. Let (d1 , d2 , . . . , d2n+1 ) be the list of principal minors of Discr ( p). Then, the discriminant sequence of π− is equal to (d1 d2 , d2 d3 , . . . , d2n d2n+1 ). In addition, the discriminant sequence of π+ is (D1 (π+ ) , . . . , D2n (π+ )) , up to a factor of the same sign as a0 .

Dk (π+ ) = (−1)k/2 dk ( p) dk+1 ( p)

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5 Applications Let us study some applications of the algebraic methods we introduced above. It should be stressed that the following considerations are only possible examples and the effective tools can be applied in many other branches of science.

5.1 Dynamical Systems The effective algebraic methods can be applied to the analysis of dynamical systems. Consider a linear differential equation on Kn (K = R or C): d x (t) = A (t) x (t) , dt

(4)

where A (t) ∈ Mn (K) is bounded. In general, the closed form of the solution to (4) is not known, so alternative approaches are required for further analysis. It can be  observed that the coefficient matrix A (t) can be written as A (t) = sk=1 αk (t) Ak , where αk : I → K and Ak ∈ Mn (K) are constant matrices. Then, we are able to study the dynamics (4) algebraically by studying the algebra A generated by A (t), equivalently the algebra generated by the constant matrices  = {A1 , . . . , As }. If A is reducible, then all matrices in A can be simultaneously block-triangularised, i.e. A (t) can be block-triangularised. In addition, the ALS-criterion or the discriminant of A can determine if A is semi-simple or not. If A is completely reducible, then the system (4) can be decomposed into several independent linear differential equations, where the dimension of each decomposed system is strictly less than n. Example 8 Consider the system on C3 : d dt

⎡

⎤ 5 sin t − 3 cos t 2 cos t − 3 sin t 3 sin t − 2 cos t 1 − 27 cos t (4 cos t + sin t) 21 (sin t − 8 cos t) ⎦ x (t) boldsymbolx (t) = ⎣ 2 1 t (cos t − 4 sin t) 3 sin − cos t − cos t − sin2 t 2 2 = {(sin t) D1 + (cos t) D2 } x(t). By constructing the algebra A generated by D1 , D2 , we can check that the A is completely reducible as we did above. Thus, the linear system is decomposable. In addition, there is a common eigenvector (which is [0, 1, 1]T ) for D1 , D2 due to the Shemesh criterion, so the linear system is reduced to a scalar equation, which is solvable, on this invariant subspace. A similar idea applies to the linear difference equation in the same analogue.

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5.2 Stability of Dynamical Systems Effective algebraic methods can be applied to the study of stability of dynamical systems. The solution x (t) to the differential equation on Rn : d x (t) = f (t, x (t)) dt is said to be Lyapunov stable if for any ε > 0 there is δ > 0 such that any solution y (t) with x (t0 ) − y (t0 ) < δ satisfy x (t) − y (t) < ε (∀t ≥ t0 ). Moreover, the solution x (t) is said to be asymptotically stable if it is Lyapunov stable and there is η > 0 such that any solution y (t) with x (t0 ) − y (t0 ) < η satisfies limt→∞ x (t) − y (t) = 0. In the case of the linear constant system on Rn : d x(t) = Ax(t), dt

(5)

it has the trivial solution x (t) = 0 and in some applications the stability of 0 is interesting. It is known that the stability of x (t) = 0 depends on the eigenvalues λ1 , . . . , λn ∈ C of A ∈ Mn (R) as follows [19]: – x (t) = 0 is asymptotically stable if  (λk ) < 0 for all k = 1, . . . , n; – x (t) = 0 is Lyapunov unstable if there is some k such that  (λk ) > 0. Thus, by studying the number of eigenvalues having the negative real parts, we are able to mention the stability of 0 in (5). We are able to use the Routh–Hurwitz criterion as an effective criterion. Example 9 Consider the linear system (5) with ⎤ 1 1 0 A = ⎣ −1 −4 5 ⎦ . − 85 −2 1 ⎡

The characteristic polynomial of A is p A (x) = −x 3 − 2x 2 − 4x − 1, so by Example 7 the real parts of the eigenvalues are all negative. Thus, we can conclude that x(t) = 0 is asymptotically stable. As a special case, when the linear constant system (5) on Cn is generated by a Hermitian (or symmetric on Rn ) matrix, it is known that the characteristic polynomial p A of A is a real polynomial having only real roots. Thus, we are able to study the stability by counting the number of negative roots of p A . This idea can be also useful in the study of quantum systems [6]. Example 10 Let

⎡

1 ⎢ −i A=⎢ ⎣2 + i −1

i 2 −1 i

⎤ 2 − i −1 −1 −i ⎥ ⎥ −1 1 + i ⎦ 1−i 1

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525

and consider the linear constant system on C4 . The characteristic polynomial of A is p A (x) = x 4 − 3x 3 − 10x 2 + 31x − 13, so we set   π+ (y) = p A y 2 = y 8 − 3y 6 − 10y 4 + 31y 2 − 13   π− (y) = p A −y 2 = y 8 + 3y 6 − 10y 4 − 31y 2 − 13. The revised sign lists of π± are (1, 1, 1, −1, −1, −1, −1, −1) and (1, −1, −1, −1, + −1, 1, 1, −1), respectively, so we can say that there are s+ −2ν = 8−2 = 3 positive 2 2 s− −2ν− 8−6 roots and 2 = 2 = 1 negative root, which means the system is unstable. More generally, one may be able to study the stability of a time-varying linear system (4). If the system (4) is so-called Lyapunov reducible, there exists a bounded matrix L (t) such that L −1 , dtd L exist and the change of variables x (t) = L (t) y (t) reduces (4) into a constant system (5). Then, the time-varying system (4) and the constant system (5) are in the same equivalence class, so they share the same asymptotic stability property. Hence, by studying the stability of the constant system (5), one can also study that of the time-varying system (4). Lyapunov reducible systems include, for instance, the periodic systems, functionally commutative systems, Wu– Horowitz–Denninson systems and Shifner–Erugin–Salakhova–Chebotarev systems (cf. [11]). Example 11 Consider the time-varying system (4) with  A(t) =

(t−4)t 10t 3 +2t 2 −3t−2 5t 4 +t 3 −t 2 −4t+4 5t 4 +t 3 −t 2 −4t+4 5(t−4)t 10t 3 +t 2 +t−2 − 5t 4 +t 3 −t 2 −4t+4 5t 4 +t 3 −t 2 −4t+4

 .

  One can check that this system is functionally commutative (i.e. A(t), dtd A(t) = O), and in fact this can be transformed to a constant system (5) with the coefficient matrix   2 −8 . 1 −3 The Hurwitz matrix of the characteristic polynomial p(x) = x 2 + x + 2 of this matrix is   10 H ( p) = , 12 so we conclude that the real parts of the eigenvalues are negative, i.e. the system is asymptotically stable.

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5.3 Decoherence-Free Subspaces A quantum system is described by some dynamics on the state space S (H) = {ρ ∈ B (H) | ρ ∗ = ρ, ρ ≥ 0, trρ = 1}, where H is a Hilbert space. If our system has no influence from the environmental noise, then the dynamics is often given by the von Neumann equation: i dtd ρt = [H, ρt ], where  > 0, H is a Hamiltonian. However, usually the evolutions on quantum systems observe noise from the surroundings, so the von Neumann equation is no longer satisfied. In general, the evolution on S (H) is represented as d ρt = L (t) ρt dt

(6)

(master equation) and we would like to study the structure of the system. As one approach, since the master equation (6) is a linear equation, the algebraic decomposition in Sect. 5.1 is valid. If the algebra A generated by L (t) is completely reducible, then we are able to simplify the structure of the master equation (6). As another approach, one can study the existence of so-called decoherence-free subspaces (DFSs). Because of the physical requirement, the evolutions in quantum systems are usually assumed to be completely positive and trace-preserving (CP and TP). It is well known that a CP map can be represented as

(X ) =

s 

Mk X Mk∗

(7)

k=1

 and it is TP iff k Mk∗ Mk = I , so often the quantum evolutions are given by CPTP maps so-called channels instead of master equations. Consider a quantum channel (7). The quantum channel can also observe influence from the environment, so is not unitary on S (H) in general. However, in some cases there is a subspace H D ⊂ H such that is unitary on S (H D ), which means that is free from the noise. Such a subspace is called a decoherence-free subspace (DFS), and the existence of a DFS is useful in applications (e.g. quantum information transmission). In order to test the existence of a DFS, one can apply the generalised Shemesh criterion and the author showed the following result: Theorem 8 (cf. [10]). The quantum channel (7) has a DFS if and only if   N M1 , . . . , Ms , M1∗ , . . . , Ms∗ = {0} . Example 12 Let H S = C3 and consider a quantum channel: Φ(ρ) = M1 ρ M1∗ + M2 ρ M2∗ ,

Effective Algebraic Methods are Widely Applicable

where

⎡

√1 3

0

⎡

⎤

i 2

527

√1 6

i 2

0 

⎤

⎥ ⎢ ⎥ ⎢ 2 M1 = ⎣ 0 √13 0 ⎦ , M2 = ⎢ 0 ⎥ ⎦. ⎣ 0 3 i 1 − 2i 0 − √13 − 2 0 − √6 Notice that N (M1 , M2 ) = N (M1 , M2 , M1∗ , M2∗ ) because M1 , M2 are Hermitian matrices. By computing

Θ=

2 



j

M1 , M2k

∗ 



⎡

⎢ j M1 , M2k = ⎣

j,k=1

−

√ 58393(−3+2 2) 62208

0 0

⎤ 0 ⎥ 0 √ ⎦, 58393(−3+2 2) 0− 62208 0 0

then we can see that span{[0, 1, 0]T } is the kernel of Θ. Thus, we conclude that N (K 1 , K 2 ) = {0} and there is a DFS.

5.4 Cellular Networks The reduction methods can also be applied to some models in biology, e.g. cellular networks, where these systems are often modelled using piecewise linear differential equations. There have been several approaches for the model decompositions in biological systems (see, e.g. [2, 16]). Piecewise linear systems deal with, in general, the nonhomogeneous linear equation: d x (t) = A (t) x (t) + b (t) , dt

(8)

where A (t) = Aσ (t) , b (t) = bσ (t) with some piecewise constant function σ : R → {1, . . . , N } called the switching signal. The approach in Sect. 5.1 can also be applied to this system, and one can study (8) on some ‘good’ subspace. One can also find techniques and applications in biology in [8, 20].

References 1. Amitsur A, Levitzki J (1950) Proc Amer Math Soc 1(4):449–463 2. Eriksso O, Tegnér J (2016) Modeling and model simplification to facilitate biological insights and predictions. In: Geri L, Gomez-Cabrero D (eds.), Uncertainty in biology: a computational modeling approach. Springer, pp. 301–325 3. Farenick D (2012) Algebras of linear transformations. Springer 4. Gil M (2001) Linear Algebra Appl 327:95–104 5. Gil M (2003) Operator functions and localization of spectra. Springer

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6. Jamiołkowski A (2019) On classification of roots of polynomial parametric systems. Lecture at Tokyo University of Science 7. Jamiołkowski A, Pastuszak G (2015) Linear Multilinear Algebra 63:314–325 8. Jones DS, Plank M, Sleeman BD (2009) Differential equations and mathematical biology. CRC 9. Kamizawa T (2017) Open Syst Inf Dyn 24(01):1750002 10. Kamizawa T (2018) Int J Theor Phys 57(5):1272–1284 11. Kamizawa T (2019) Algebraic and Lyapunov reducibilities and the analysis of linear dynamical systems and quantum systems. In: Proceedings of mathematical society japan 58th joint symposium of real analysis and functional analysis sections 12. Laffey TJ (1986) Linear Algebra Appl 84:123–138 13. Marcus M, Minc H (1992) A survey of matrix theory and matrix inequalities. Courier Corporation 14. Mitropolsky YA, Lopatin A (2013) Nonlinear mechanics, groups and symmetry. Springer 15. Pastuszak G, Kamizawa T, Jamiołkowski A (2016) Open Syst Inf Dyn 23:1650003 16. Saez-Rodriguez J, Kremling A, Conzelmann H, Bettenbrock K, Gilles E (2004) IEEE Control Syst Mag 24(4):35–52 17. Shapiro J (1979) Linear Algebra Appl 25:129–137 18. Shemesh D (1984) Linear Algebra Appl 62:11–18 19. Walter W (1998) Ordinary differential equations. Springer 20. Xia B, Yang L (2016) Automated inequality proving and discovering. World Scientific 21. Yang L, Xia B (1997) Explicit criterion to determine the number of positive roots of a polynomial. MM Res Preprints 15:134–145

n-Fractals in Partial Metric Spaces S. Minirani

Abstract A metric space with nonzero self distance gives us a generalization of metric spaces which is coined as a partial metric space. In this paper we discuss the construction of an n-fractal which is the attractor of a collection of n-IFSs in a partial metric space. Keywords Partial metric space · n-fractals · Iterated function system

1 Introduction As a part of the study of denotational semantics of dataflow networks, Steve G Mathews introduced the notion of partial metric space in 1992 [1], which is a generalization of metric space with nonzero self distances. It coincides with the metric space if the self distance becomes zero. S G Mathews proved a partial metric generalization of Banach’s contraction mapping theorem and also established the relationship between partial metric space and weightable quasi-metric spaces. A significant change to Mathews’ definition of partial metrics was proposed by S. J. O’ Neill which extended their range from IR+ to IR [2]. Fractals are defined as sets with Hausdorff dimension greater than the topological dimension by Mandelbrot. Exact, quasi and statistical self-similarity are some of the additional properties of fractals that has been extensively used to model various physical phenomena. An iterated function system (IFS) is a mathematical tool to generate a fractal using a finite collection of contraction mappings on a complete metric space. The fixed point of such a collection is called the attractor of the IFS which is generally a fractal. Various generalisations of IFSs are available in the literature. Some of them are recurrent IFS, random IFS, countable IFS, infinite IFS, and directed graph IFS [3]. S. Minirani (B) MPSTME, NMIMS Deemed to be University, Mumbai, India e-mail: [email protected] URL: http://www.engineering.nmims.edu/faculty-and-research/faculty-profile/minirani-s/ © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 D. Giri et al. (eds.), Proceedings of the Sixth International Conference on Mathematics and Computing, Advances in Intelligent Systems and Computing 1262, https://doi.org/10.1007/978-981-15-8061-1_43

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The fixed point of an iteration function system in partial metric spaces was discussed by Minirani and Mathew in 2014 [4]. A collection of (n, m)-IFSs and its attractor was introduced by Balu and Mathew in 2013. Here we discuss a collection of n-IFSs and their attractor in partial metric spaces.

2 Preliminaries Here we discuss some of the basic definitions and results from fractal theory and partial metric spaces which can be found in [3–7]. A mapping f on a metric space (X, d) is said to be a contraction if there exist a real number c such that 0 < c < 1 and d( f (x), f (y)) ≤ cd(x, y) for all x, y ∈ X . If it becomes an equality then f is called a similarity transformation. Clearly, every contraction mapping is continuous. We call A ⊆ X a self-similar set if A =  n f i (A) where f i s are contraction mappings defined on X . Some of the examples i=1 of self-similar sets which are fractals are Cantor set, Sierpinski triangle, and Koch curve. Definition 1 Let (X,d) be a complete metric space. Let H(X) denote the set of all non-empty compact subsets of X. The Hausdorff metric h on H (X ) is defined by h(A, B) = max{sup{dist(x, B)}, sup{dist(y, A)}} x∈A

y∈B

Barnsley established the space (H(X), h) as a complete metric space and referred to it as the “space of fractals”. Definition 2 An iterated function system (IFS) consist of a complete metric space (X, d) together with a finite set of contraction mappings f i on X with the respective contractivity factor ci for i = 1 . . . n. The notation for the IFS is {X ; f i , i = 1, 2, . . . n} and its contractivity factor is c = max ci . 1≤i≤n

Using this collection of contractive functions f i on a complete metric space (X, d), a new contractive mapping is defined on the space (H (X ), h) whose fixed point is the attractor of the IFS, the existence of which is established in the following theorem by Barnsley. Theorem 1 Let {X ; f i , i = 1, 2, . . . n} be an IFS with contractivity n factor c. Then the transformation W : H (X ) → H (X ) defined by W (B) = f i (B) for all i=1 B ∈ H (X ), is a contraction mapping on the complete metric space (H(X), h) with contractivity factor c. Its unique fixed point A ∈ H (X ) exist and is given by A = lim W ◦n (B) for any B ∈ H (X ). n→∞

In this paper, we are considering a collection of n-IFSs defined on n complete metric spaces. The following definition of an (n, m)-I F S and the theorem on the existence of its attractor is given in [3].

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Definition 3 Let X 1 , X 2 , . . . , X n be a finite sequence of complete metric spaces. Let f i j be a contraction mapping on X i with contractivity factor ci j for i = 1, 2, . . . , n and j = 1, 2, . . . , k. Then the IFS {X i : f i j , j = 1, 2, . . . , k} for each i = 1, 2, . . . , n. This collection is said to be an (n,m)-IFS and its contractivity factor is given by c = max max {ci j }. The (n, m)-IFS is denoted by (X; f) where 1≤i≤n 1≤ j≤m

X = {X 1 , X 2 , . . . , X n } and f is the collection of contraction mappings { f i j }. Here n stands for the number of IFSs and the second coordinate m stands for the total number of contraction mappings involved in the collection of IFSs. The number k may vary for each IFS. Definition 4 Let (X; f) be an (n, m)-IFS with contractivity factor c. Let  h be the n  = metric on H H (X i ) is defined as i=1

 h((A1 , A2 , . . . , An ), (B1 , B2 , . . . , Bn )) = max h i (Ai , Bi ) 1≤i≤n

. Define W :H → H  by for all (A1 , A2 , . . . , An ), (B1 , B2 , . . . , Bn ) ∈ H ⎛  (A1 , A2 , . . . , An ) = ⎝ W



f 1 j (A1 ),

j



f 2 j (A2 ), . . . ,

j



⎞ f n j (An )⎠

j

. Then W  is a contraction mapping on the complete for all (A1 , A2 , . . . , An ) ∈ H ,   obeys A = W  (A) is called the metric space ( H h) and its unique fixed point A ∈ H attractor of the (n, m)-IFS. We define it as an n-fractal.

3 n-Fractals in Partial Metric Spaces We work in a partial metric space (X, p) and H p (X ) where H p (X ) denote the set of all non-empty compact subsets of the partial metric space X . The compact subsets are the closed and bounded subsets of X induced by the partial metric p. Definition 5 Let X 1 , X 2 , . . . , X n be a sequence of partial metric spaces and let n  p p (X ) = H p (X i ). For A = (A1 , A2 , . . . , An ) and B = (B1 , B2 , . . . , Bn ) ∈ H H i=1

p (X ) as (X ), we define the Hausdorff partial metric hp on H hp (A, B) = max h pi (Ai , Bi ) 1≤i≤n

where each h pi is a Hausdorff partial metric on H p (X i ).

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p (X ). Proposition 1 hp is a partial metric on H Proof To prove that hp is a partial metric , we need to prove the following axioms: h p (A, A) ≤  h p (A, B) P0 :   h p (B, A) P1 : h p (A, B) =  h p (A, A) =  h p (A, B) =  h p (B, B) then A = B P2 : If    h p (B, A) − inf p(b, b) P3 : h p (A, C) ≤ h p (A, B) +  b∈B

Here A = (A1 , A2 , . . . , An ), B = (B1 , B2 , . . . , Bn ) and C = (C1 , C2 , . . . , Cn ) P0 : hp (A, A) = hp ((A1 , A2 , . . . , An ), (A1 , A2 , . . . , An )) = max h pi (Ai , Ai ) 1≤i≤n

≤ max h pi (Ai , Bi ), since each h pi is a partial metric 1≤i≤n

= hp ((A1 , A2 , . . . , An ), (B1 , B2 , . . . , Bn )) = hp (A, B) P1 : hp (A, B) is symmetric since hp (A, B) = hp ((A1 , A2 , . . . , An ), (A1 , A2 , . . . , An )) = max h pi (Ai , Bi ) 1≤i≤n

= max h pi (Bi , Ai ) 1≤i≤n

= hp ((B1 , B2 , . . . , Bn ), (A1 , A2 , . . . , An )) P2 : hp (A, B) = hp ((A1 , A2 , . . . , An ), (B1 , B2 , . . . , Bn )) = max h pi (Ai , Bi ) 1≤i≤n

= max h pi (Ai , Ai ) 1≤i≤n

= max h pi (Bi , Bi ) 1≤i≤n

⇒ A=B P3 : hp (A, B) = hp ((A1 , A2 , . . . , An ), (B1 , B2 , . . . , Bn )) = max h pi (Ai , Bi ) 1≤i≤n

≤ max h pi (Ai , Ci ) + h pi (Ci , Bi ) − inf p(c, c) 1≤i≤n

c∈Ci

≤ max h pi (Ai , Ci ) + max h pi (Ci , Bi ) − inf p(c, c) 1≤i≤n

1≤i≤n

c∈Ci

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= hp (A, C) + hp (C, B) − inf p(c, c) c∈Ci

p (X ). Since all the four axioms are satisfied, hp is a partial metric on H n p (X ) = Theorem 2 The space H H p (X i ) is complete in the partial metric i=1  h p. Proof Let {A1 j }, {A2 j }, . . . {An j } be Cauchy sequences in H p (X 1 ), H p (X 2 ), . . . , H p (X n ) respectively converging to A1 , A2 , . . . , An . Given  > 0, ∃N : ∀l, m > N , we have max(h p1 (A1l , A1m ), h p2 (A2l , A2m ), . . . , h pn (Anl , Anm )) <  p (X ). Since each of the Let (A1 j , A2 j , . . . , An j ) be a Cauchy sequence in H sequence {Ai j } is Cauchy and converges to Ai and since each H (X i ) is complete, the n p (X ) = H p (X i ) sequence converges to (A1 , A2 , . . . , An ). This proves that H i=1 is complete in the partial metric hp . Thus, the generalization of Banach’s fixed point theorem to the partial metric spaces p (X ), hp ). This will ensure the existence of can be applied to the complete space ( H p (X ). This unique fixed point a unique fixed point of every contraction mapping in H will be the deterministic n-fractal which is defined as the attractor of the (n, m) − I F S p defined in the theorem given below. Theorem 3 A partial (n,m)-Iterated Function System denoted as (n, m) − I F S p is a collection of a finite sequence of partial metric spaces X 1 , X 2 , . . . , X n together with a collection of contraction mappings { f i j } on X i with contractivity factor ci j . The contractivity factor of this (n, m) − I F S p is defined as c = max ci j . Now we define a ⎛ i, j   p on H p (X ) as W p (A1 , A2 , . . . , An ) = ⎝ transformation W f 1 j (A1 ), f 2 j (A2 ), j





j

p (X ). This mapping W p is a well defined f n j (An ) ∀(A1 , A2 , . . . , An ) ∈ H ..., j p (X ), hp ). Since this partial contraction mapping on the partial metric space ( H p has a unique fixed point given by (A1 , A2 , . . . , An ) = metric space is complete, W p (X ) and obeys (A1 , A2 , . . . , p ◦n (B1 , B2 , . . . , Bn ) for any (B1 , B2 , . . . , Bn ) ∈ H W p (A1 , A2 , . . . , An ) An ) = W

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References 1. Mathews SG (1992) Partial metric topology. In: Proceedings 8th summer conference on topology and its applications. New York 2. S. J. O’Neill (1995) Partial metrics, valuations and domain theory. In: Proceedings of the 11th Summer Conference on General Topology and Applications, vol 806, pp 304–315. The New York Academy of Sciences, Gorham, Me, USA 3. Balu R, Mathew S (2013) On (n, m)-iterated function systems. Asian-Eur J Math 6(4) 4. Minirani S, Mathew S (2014) Fractals in partial metric spaces: fractals, wavelets, and their applications, springer proceedings in mathematics. Springer, Heidelberg, pp 203–215 5. Barnsley MF (1997) Fractals everywhere. Academic Press. https://doi.org/10.1109/HPDC.2001. 945188 6. Falconer KJ (1990) Fractal geometry-foundations and applications. Wiley, New York 7. Aydi et al (2012) Partial Hausdorff metric and Nadler’s fixed point theorem on partial metric spaces. Topol Appl 159:3234–3242

Some Existence Results on Impulsive Differential Equations Rajib Haloi

Abstract We prove the existence and uniqueness of piecewise continuous (PC) mild solutions to fractional impulsive differential equations in a Banach space are established. We use the theory of semigroup of almost sectorial operators and the fixed point theorem. Keywords Impulsive differential equation · Semigroup with growth · Neutral equations · Fixed point theorem

1 Introduction We prove the existence and uniqueness of the solutions to in a complex Banach space (X,  · ): α c Dt [u(t) +

⎫ g(t, u(t)] + Au(t) = f (t, u(t)), t ∈ J = [0, t0 ], t  = tk , k = 1, 2, 3, . . . , m; ⎬ Ik (u(tk− )) = u(tk+ ) − u(tk− ), k = 1, 2, 3, . . . , m; ⎭ u(0) = u 0 ,

(1)

where u : J → X and u 0 ∈ X . The functions f : J × X → X , g : J × X → X are non-linear functions and satisfy some appropriate conditions. Here, A : D(A) ⊂ X → X is linear operator with resolvent satisfies a growth −γ, −1 < γ < 0 in a sector of the complex plane. Throughout the article we denote u(tk+ ) − u(tk− ) = xk . The fractional differential equations arise as models of many fields in engineering and science. This includes electrochemistry, electromagnetics, electrical circuits control theory, visco-elasticity, porous media, neuron modelling etc. [19]. The plentiful occurrence and applications of fractional differential equations motivate the rapid developments and gained much attention in the recent years. For a nice introduction to the theory and its applications, we refer to [1, 12–14, 17, 19]. Impulsive differential equations model various problems in population dynamics, control theory, physics, R. Haloi (B) Tezpur University, Tezpur 784028, Assam, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 D. Giri et al. (eds.), Proceedings of the Sixth International Conference on Mathematics and Computing, Advances in Intelligent Systems and Computing 1262, https://doi.org/10.1007/978-981-15-8061-1_44

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biology and medicine, etc. where the dynamics of the process have some abrupt changes at certain moment like earthquake, harvesting, shock, etc. The investigation of existence and uniqueness of mild solutions for differential and integro-differential equations with impulse effects have been discussed by many authors [8, 15, 16, 21, 24]. The concept of almost sectorial operator has introduced by Wahl [26]. Followed by Wahl, many authors have developed the abstract theory for Cauchy problems with almost sectorial operators. We refer [2–4, 6, 7, 9, 11, 18, 20, 25] and reference therein for more details. Further, Wang et al. [25] proved the existence and uniqueness in a Banach space X , c

η

Dt u(t) = Au(t) + f (t, u(t)), t ∈ [0, T ], 0 ≤ η ≤ 1, u(0) = u 0 ,

 (2)

η

where c Dt is the Caputo fractional derivative of order η, A : D(A) ⊂ X → X is a almost sectorial operator and f : [0, T ] × X → X. However, Problem (1) has not studied when A is almost sectorial operator. With the strong motivation, we study the existence and uniqueness of PC mild solution to Problem (1) using the new notion of solutions introduced in [22], when A is almost sectorial. The results are new and complement to the existing ones that generalize some results in [25]. The paper is organized as follows. In Sect. 2, we recall the definition of the Caputo fractional derivative, Riemann-Liouville integral, the theory of semigroup of bounded linear operators and some lemmas that are used in the remaining part of the article. In Sect. 3, we study the existence and the uniqueness of PC mild solutions Eq. (1). The results obtained in Sect. 3 have extended to integro-differential equation in Sect. 4. Finally, examples are provided to illustrate the analytical results.

2 Preliminaries In this section, the basic definitions, lemmas which are used in the rest part of the article are recalled. Let (X,  · ) be a complex Banach space. We use D(A) for the domain of a operator, σ(A) for its spectrum, ρ(A) : C \ σ(A) for its resolvent, R(λ; A), λ ∈ ρ(A) for the resolvent operator, L(X ) for the space of all bounded linear operators on X . Definition 1 For −1 < γ < 0 and 0 < ω < π/2, a closed linear operator A : D(A) ⊂ X → X is called an almost sectorial operator on X if 1. σ(A) ⊂ ω = {z ∈ C \ {0} : | arg z| ≤ w} ∪ {0} 2. for every ω < μ < π, there exists a constant Cμ such that R(z, A) ≤ Cμ |z|γ for all

z ∈ C \ μ .

We denote the family of all almost sectorial operators by Fωγ (X ).

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Example 1 [25] Let  be the union of two bounded domain in Rn , n ≥ 2 which have smooth boundaries. Consider the following operator 1 B(u, v) = (−Δu + u, − (gv ) ) for (u, v) ∈ D(B), g

(4)

p

where D(B) the domain of B is a dense subset of L p () ⊕ L g (0, 1)(1 ≤ p < ∞) for a smooth function g : [0, 1] → (0, ∞) and Δ is the Laplacian with Neumann boundary condition. The domain D(B) is endowed with the norm  |u| +

(u, v) =

1/ p

1 p



g|v| p

.

0

Then for p > n/2, B is a closed linear operator with compact resolvent. But B is not sectorial and the resolvent operator R(z; −B) satisfies R(z; −B) ≤

C for |z|γ

z ∈ μ \ {0},

where μ = {z ∈ C \ {0} : | arg z| ≤ π − μ} ∪ {0}} ⊂ ρ(−B) for μ ∈ (0, π/2), 0 < γ < 1 − n/2 p and some positive constant C. Thus the operator B is almost sectorial. We refer the reader to [2–4, 25] for more details. Throughout the article we assume the following hypothesis. (B1)

The operator A : D(A) ⊂ X → X is almost sectorial and A ∈ Fωγ (X ) for −1 < γ < 0. Further, we assume that R(λ; −A) is compact for each λ > 0.

Then A generates a semigroup {T (t) : t ≥ 0} of bounded linear operators on X with growth 1 + γ which is analytic in an open sector of the complex plane C, but T (t) is discontinuous at t = 0 in the strong operator topology [25]. Let μ0 = z ∈

0 C \ {0} : | arg z| < μ , where 0 < μ < π be the open sector. For given t ∈ π/2−ω π and ω < φ < μ < 2 − | arg t|, the family T (t) = e−t z (A) =

1 2πi



e−t z R(z; A)dz,

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γφ

where γφ = {R+ eiφ } ∪ {R+ e−iφ }, forms a family of analytic semigroup in L(X ). We note that 0 ∈ ρ(A) does not imply that the negative power of A is well defined. However, A−β is a bounded linear operator on X if β > 1 + γ. We define X β = D(Aβ ) for β > 1 + γ, endowed with the norm xβ = Aβ x for x ∈ X β .

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Then X β is a Banach space endowed with the norm  · β . Before stating more properties, we now recall the generalized Mittag-Leffler and Wright-type functions. The generalized Mittag-Leffler function E α,β is defined as E α,β :=

∞  k=0

1 zk = Γ (αk + β) 2πi

 χ

λα−β eλ dλ for α, β > 0, z ∈ C, λα − z

where χ is a contour starts and ends at −∞ and encircles the disc |λ| ≤ |z|1/α counterclockwise. We denote E α (z) := E α,1 (z), eα (z) := E α,α (z). For 0 < α < 1, the function of Wright-type is defined as Ψα (z) :=

∞  n=0

∞ 1  (−z)n (−z)n = Γ (nα) sin(nπα) for z ∈ C. n!Γ (−αn + 1 − α) π n=1 (n − 1)!

0 Let t ∈ π/2−ω and ω < φ < μ < π/2 − | arg t|. We define the following families of operators:

1 Sα (t) = 2πi



1 E α (−zt )R(z; A)dz, Pα (t) = 2πi α

γφ



eα (−zt α )R(z; A)dz,

γφ

0 , it where γφ = {R+ eiφ } ∪ {R+ e−iφ } is oriented counterclockwise. For t ∈ π/2−ω can be shown that

∞ Sα (t) =

α

∞

Ψα (θ)T (θt )xdθ, Pα (t) = 0

αθΨα (θ)T (θt α )xdθ.

0

0 Lemma 1 [25, Theorem 3.1] Let t ∈ π/2−ω and ω < π/2 − | arg t|. Then Sα (t) and Pα (t) are bounded linear operators on X such that

Γ (−γ) t −α(1+γ) x, ∀t > 0, ∀x ∈ X, Γ (1 − α(1 + γ)) Γ (1 − γ) −α(1+γ) t x, ∀t > 0, ∀x ∈ X Pα (t)x ≤ αC0 Γ (1 − αγ) Sα (t)x ≤ C0

for some positive constant C0 .

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Lemma 2 [25, Theorem 3.2] For t > 0, Sα (t) and Pα (t) are continuous in the uniform operator topology. The continuity is uniform on [r, ∞) for each r > 0. Lemma 3 [25, Theorem 3.5] If R(λ; −A) is compact for every λ > 0, then Sα (t) and Pα (t) are compact for every t > 0. We recall the definition of fractional integral and derivative of a function. Definition 2 The Riemann-Liouville fractional integral of order η of h ∈ L 1 (I ; X ) with the lower limit zero is defined as η Jt h(t)

1 = Γ (η)

t

h(s) ds, t > 0, η > 0 (t − s)1−η

0

provided that the right hand side is defined pointwise on [0, ∞) , where Γ (·) is the Gamma function. η

Definition 3 Let h ∈ C m−1 (I ; X ) and (Jt h)(m) ∈ L 1 (I ; X ). The Caputo derivative of order η of h is defined as 

η

m−η

m c Dt h(t) = Dt Jt

h(t) −

k=0

where Dtm =

 t (k) h (0) , t > 0, m − 1 < η < m, k!

m−1  k

dm . dt m

Lemma 4 [25, Theorem 3.4] The following properties hold. (i) Let β > 1 + γ. For all x ∈ D(Aβ ), lim Sα (t)x = x; t→0+

(ii) For all x ∈ D(A), t > 0, Dtα Sα (t)x = −ASα (t)x; Let u 0 ∈ X and f : [0, T ] → X, with α c Dt u(t)

+ Au(t) = f (t) t ∈ (0, T ], u(0) = u 0 .

 (8)

Definition 4 A continuous function u : (0, T ] → X is called a mild solution of problem (8) if u satisfies t u(t) = Sα (t)u 0 + 0

(t − s)η−1 Pα (t − s) f (s)ds.

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Theorem 1 Let A ∈ Fωγ (X ), where 0 < ω < π2 . Suppose that f ∈ D(A) and A f (t) ∈ L ∞ ((0, T ]; X ). Then for each u 0 ∈ X, Problem (8) has a unique mild solution on (0, t0 ] for some 0 < t0 ≤ T. For a proof of the theorem, we refer to Wang et al. [25, Theorem 4.1]. We use the following lemma. Lemma 5 If 0 < η ≤ 1 and 0 < x ≤ y, then we have |x η − y η | ≤ (y − x)η . Let f and g be three continuous functions. Let V be open set in X . For each v ∈ V , there is a ball such that B(v, r1 ) ⊂ V for r1 > 0 such that (B2)

there exists constant C f > 0 such that the continuous map f : J × V → X with  f (t, x1 ) − f (t, x2 ) ≤ C f (x1 − x2 )

(B3)

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for all x1 , x2 ∈ B, and t ∈ J . let g : [0, T ] × V → X 1 be a continuous map with g(t, x1 ) − g(s, x2 )1 ≤ C g x1 − x2 

(10)

for all x1 , x2 ∈ B and s, t ∈ J and for some constant C g > 0.

3 Main Results The section is devoted to prove the existence of the solution to Problem (1). We define the following space

PC(X ) = u : J → X | u ∈ C((tk , tk+1 ], X ), k = 0, 1, . . . , m, u(tk− ), u(tk+ ) exist , where J = [0, t0 ]. Then PC(X ) is a Banach space endowed with the supremum norm uPC := max{sup u(t + 0), sup u(t − 0)}. t∈J

t∈J

We use the following definition that is introduced by Wang et al. [22].

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Definition 5 A function u : J → X is called a PC-mild solution to problem (1) if (i) u is continuous on J \ {t1 , t2 , . . . , tn }; (ii) the function Pα (t − s)Ag(s, u(s)) is integrable for each s ∈ [0, t); (iii) u satisfies the following integral equation ⎧ Sα (t)[u 0 + g(0, u 0 )] − g(t, u(t)) ⎪ ⎪ ⎪ ⎪ t ⎪ ⎪ ⎪ + (t − s)η−1 Pα (t − s)Ag(s, u(s))ds ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ t ⎪ ⎪ + (t − s)η−1 Pα (t − s) f (s, u(s))ds, t ∈ [0, t1 ], ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Sα (t)[u 0 + g(0, u 0 )] − g(t, u(t)) + Sα (t − t1 )x1 ⎪ ⎪ ⎪ t ⎪ ⎪ ⎪ + (t − s)η−1 Pα (t − s)Ag(s, u(s))ds ⎪ ⎪ ⎪ 0 ⎨ t u(t) = + (t − s)η−1 Pα (t − s) f (s, u(s))ds, t ∈ (t1 , t2 ], ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ .. ⎪ ⎪ ⎪ . ⎪ n ⎪ ⎪ ⎪ Sα (t)[u 0 + g(0, u 0 )] − g(t, u(t)) + i=1 Sα (t − ti )xi ⎪ ⎪ ⎪ ⎪ t ⎪ ⎪ + (t − s)η−1 Pα (t − s)Ag(s, u(s))ds ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ t ⎪ ⎪ ⎪ + (t − s)η−1 Pα (t − s) f (s, u(s))ds, t ∈ (tn , t0 ]. ⎩

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0

−αγ

Let Mg = sup Ag(t, u(t)) and M f = sup  f (t, u(t)). Let t ∗ = t0 t∈J

.

t∈J

Theorem 2 Let the assumptions (B1)–(B3) hold. Then Problem (1) has unique mild solution on [0, t0 ] for each u 0 ∈ X β , β > 1 + γ if C g A−1  + (C g + C f )(C0

Γ (1 − γ) )t ∗ < 1. (−γ)Γ (1 − αγ)

Proof For fixed u 0 ∈ X β , β > 1 + γ, we define a map Q on PC(X ) as

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R. Haloi

⎧ Sα (t)[u 0 + g(0, u 0 )] − g(t, u(t)) ⎪ ⎪ ⎪ ⎪ t ⎪ ⎪ ⎪ + (t − s)α−1 Pα (t − s)Ag(s, u(s))ds ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ t ⎪ ⎪ + (t − s)α−1 Pα (t − s) f (s, u(s))ds, t ∈ [0, t1 ], ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Sα (t)[u 0 + g(0, u 0 )] − g(t, u(t)) + Sα (t − t1 )x1 ⎪ ⎪ ⎪ t ⎪ ⎪ ⎪ + (t − s)α−1 Pα (t − s)Ag(s, u(s))ds ⎪ ⎪ ⎪ 0 ⎨ t Qu(t) = + (t − s)α−1 Pα (t − s) f (s, u(s))ds, t ∈ (t1 , t2 ], ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ .. ⎪ ⎪ ⎪ . ⎪ n ⎪ ⎪ ⎪ Sα (t)[u 0 + g(0, u 0 )] − g(t, u(t)) + i=1 Sα (t − ti )xi ⎪ ⎪ ⎪ t ⎪  ⎪ ⎪ + (t − s)α−1 Pα (t − s)Ag(s, u(s))ds ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ t ⎪ ⎪ ⎪ + (t − s)α−1 Pα (t − s) f (s, u(s))ds, t ∈ (tn , t0 ], ⎩

(13)

0

for u ∈ PC(X ). We show that Q : PC(X ) → PC(X ). Let 0 ≤ τ < t ≤ t1 . Then Qu(t) − Qu(τ ) ≤ [Sα (t) − Sα (τ )][u 0 + g(0, u 0 )] + g(t, u(t)) − g(τ , u(τ )) t +

(t − s)α−1 Pα (t − s)Ag(s, u(s))ds

τ

τ +

(t − s)α−1 [Pα (t − s) − Pα (τ − s)]Ag(s, u(s))ds

0

 + 0 t

+

τ

|(t − s)α−1 − (τ − s)η−1 |Pα (t − s)Ag(s, u(s))ds

(t − s)α−1 Pα (t − s) f (s, u(s))ds

τ

τ +

(t − s)α−1 [Pα (t − s) − Pα (τ − s)] f (s, u(s))ds

0

 + 0

τ

|(t − s)α−1 − (τ − s)α−1 |Pα (t − s) f (s, u(s))ds

≤ Sα (t) − Sα (τ )[u 0 + g(0, u 0 )] + g(t, u(t)) − g(τ , u(τ ))

Some Existence Results on Impulsive Differential Equations

+ (Mg + M f )(αC0

Γ (1 − γ) ) Γ (1 − αγ)

t

543

(t − s)α−1−α(1+γ) ds

τ

τ

+ (Mg + M f ) sup Pα (t − s) − Pα (τ − s) s∈[0,τ ]

(t − s)α−1 ds

0

  τ  Γ (1 − γ)  τ ) (τ − s)α−1 (t − s)−α(1+γ) − (t − s)α−1−α(1+γ) ds . + (Mg + M f )(αC0 Γ (1 − αγ) 0

0

Using Lemma 5 and the fact that τ − s ≤ t − s (0 ≤ s ≤ τ ), we have Qu(t) − Qu(τ ) ≤ Sα (t) − Sα (τ )[u 0 + g(0, u 0 )] + g(t, u(t)) − g(τ , u(τ )) αγ

t1 sup Pα (t − s) − Pα (τ − s) −αγ s∈[0,τ ] Γ (1 − γ) )(t − τ )−αγ . + (Mg + M f )(3C0 (−γ)Γ (1 − αγ) + (Mg + M f )

(14)

The first and third term in (14) tends to zero as t → τ by Lemma 3. The second term in (14) tends to zero as t → τ by the assumption (B3) and the last term in (14) tends to zero as t → τ obviously. Thus, we have Qx ∈ C([0, t1 ]; X ). Similarly, for t1 < τ < t ≤ t2 , we have Qu(t) − Qu(τ ) ≤ Sα (t) − Sα (τ )[u 0 + g(0, u 0 )] + g(t, u(t)) − g(τ , u(τ )) αγ

t sup Pα (t − s) − Pα (τ − s) + Sα (t − t1 ) − Sα (τ − t1 )y1  + (Mg + M f ) 2 −αγ s∈[0,τ ] Γ (1 − γ) )(t − τ )−αγ . + (Mg + M f )(3C0 (15) (−γ)Γ (1 − αγ)

The right hand side of the inequality (15) tends to zero as t → τ by the assumption (B3) and Lemma 3. Hence Qx ∈ C((t1 , t2 ]; X ). Proceeding in the same way, we can show that Qx ∈ C((t2 , t3 ]; X ),...,Qx ∈ C((tn , t0 ]; X ). Q : PC(X ) → PC(X ). Finally, we will claim that Q is a contraction map. For t ∈ [0, t1 ], we have Qu(t) − Qv(t) t ≤ g(t, u(t)) − g(t, v(t)) +

(t − s)α−1 Pα (t − s)A[g(s, u(s)) − g(s, v(s))]ds

0

t + 0

(t − s)α−1 Pα (t − s)[ f (s, u(s)) − f (s, v(s))]ds

544

R. Haloi

t Γ (1 − γ) (t − s)α−1−α(1+γ) ds Γ (1 − αγ) 0  −αγ u − vPC . )t1

≤ C g A−1 u − vPC + u − vPC (C g + C f )(C0 α  ≤ C g A−1  + (C g + C f )(C0

Γ (1 − γ) (−γ)Γ (1 − αγ)

Similarly, for each t ∈ (tk , tk+1 ], we can show that Qu(t) − Qv(t)   Γ (1 − γ) −αγ −1 u − vPC )t ≤ C g A  + (C g + C f )(C0 (−γ)Γ (1 − αγ) k+1   Γ (1 − γ) ≤ C g A−1  + (C g + C f )(C0 )t ∗ u − vPC . (−γ)Γ (1 − αγ) It follows from (12) that Q is a contraction on PC(X ). By the Banach contraction mapping principle, the map Q has fixed point in PC(X ). That Problem (1) has a unique PC mild solution in PC(X )

4 Integro-Differential Equation We prove the existence of PC mild solution to the following problem in (X,  · ): α c Dt [u(t) +

g(t, u(t)] + Au(t) = f (t, u(t)) +

t

a(t − τ )h(τ , u(τ ))dτ ,

0

⎫ ⎪ ⎪ ⎪ ⎪ ⎬

t ∈ J = [0, t0 ], t = tk , k = 1, 2, 3, ..., m; ⎪ ⎪ ⎪ = u(tk+ ) − u(tk− ), k = 1, 2, 3, ..., m; ⎪ ⎭ u(0) = u 0 ,

Ik (u(tk− ))

(16)

where (B4) h : J × V → X satisfies Lipschitz condition in the second variable, that is, there exists a positive constant C h such that h(t, v) − h(s, v ) ≤ C h v − v  for all v, v ∈ B and s, t ∈ J. (B5) a : J → C is continuous on J.

(17)

Some Existence Results on Impulsive Differential Equations

545

Lemma 6 Let h 1 : J → X be a continuous function on J. Suppose that the assumption (B1) holds. Then G ∈ C((tk , tk+1 ], X ), where G is defined as t (t − s)

H (t) =

α−1

s Pα (t − s)

0

a(s − τ )h 1 (τ )dτ . 0

Proof We define a map t G(t) =

(t − s)

α−1

s Pα (t − s)

0

a(s − τ )h 1 (τ )dτ ds. 0

Let tk < η < t ≤ tk+1 . Using Lemma 5, we obtain t H (t) − H (η) ≤

(t − s)α−1 Pα (t − s)

η

s a(s − τ )h 1 (τ )dτ ds 0

τ +

(t − s)α−1 [Pα (t − s) − Pα (τ − s)]

0

+

s a(s − τ )h 1 (τ )dτ ds 0

 τ 0

|(t − s)α−1 − (τ − s)η−1 |Pα (t − s) αγ

s a(s − τ )h 1 (τ )dτ ds 0

t sup Pα (t − s) − Pα (τ − s) ≤ a J Mh 1 1 −αγ s∈[0,τ ] Γ (1 − γ) + a J Mh 1 (3C0 )(t − τ )−αγ , (−γ)Γ (1 − αγ)

where a J = supt∈J

t

(18) (19)

|a(s)|ds and Mh 1 = supt∈J h 1 (t). The right hand side of the

0

inequality (18) tends to zero as t → τ by Lemma 3. Hence H ∈ C((tk , tk+1 ], X ). Theorem 3 If the assumptions (B1)–(B5) hold, then Problem (16) has unique solution for each u 0 ∈ X β , β > 1 + γ if C g A−1  + (C g + C f + C h a J )(C0 −αγ

where t ∗ = t0

.

Γ (1 − γ) )t ∗ < 1, (−γ)Γ (1 − αγ)

(20)

546

R. Haloi

Proof For fixed u 0 ∈ X β , β > 1 + γ, we define a map G as ⎧ Sα (t)[u 0 + g(0, u 0 )] − g(t, u(t)) ⎪ ⎪ ⎪ t ⎪ ⎪ ⎪ + (t − s)α−1 Pα (t − s)Ag(s, u(s))ds ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ t ⎪ ⎪ + (t − s)α−1 Pα (t − s) f (s, u(s))ds ⎪ ⎪ ⎪ ⎪ 0 ⎪ ⎪ t s ⎪ ⎪ α−1 ⎪ + (t − s) P (t − s) a(s − τ )h(τ , u(τ ))dτ , t ∈ [0, t1 ], ⎪ α ⎪ ⎪ 0 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Sα (t)[u 0 + g(0, u 0 )] − g(t, u(t)) + Sα (t − t1 )x1 ⎪ ⎪ ⎪ ⎪ t ⎪ ⎪ ⎪ + (t − s)α−1 Pα (t − s)Ag(s, u(s))ds ⎪ ⎪ ⎪ 0 ⎪ ⎪ t ⎪ ⎪ ⎨ + (t − s)α−1 Pα (t − s) f (s, u(s))ds 0 Gu(t) = (21) ⎪ t s ⎪ ⎪ α−1 ⎪ + (t − s) Pα (t − s) a(s − τ )h(τ , u(τ ))dτ , t ∈ (t1 , t2 ] ⎪ ⎪ ⎪ 0 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ .. ⎪ ⎪ . ⎪ ⎪ n ⎪ ⎪ Sα (t)[u 0 + g(0, u 0 )] − g(t, u(t)) + i=1 Sα (t − ti )xi ⎪ ⎪ ⎪ t ⎪  ⎪ ⎪ ⎪ + (t − s)α−1 Pα (t − s)Ag(s, u(s))ds ⎪ ⎪ ⎪ 0 ⎪ ⎪ t ⎪ ⎪ ⎪ + (t − s)α−1 Pα (t − s) f (s, u(s))ds ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ t s ⎪ ⎪ ⎩ + (t − s)α−1 Pα (t − s) a(s − τ )h(τ , u(τ ))dτ t ∈ (tn , t0 ], 0

0

for u ∈ PC(X ). By Lemma 6, we have G : PC(X ) → PC(X ). To show that the map G is contraction on PC(X ), let u, v ∈ PC(X ). Then for t ∈ (tk , tk+1 ], we have  Gu(t) − Gv(t) ≤ C g A−1  + (C g + C f + C h a J )(C0

 Γ (1 − γ) )t ∗ u − vPC . (−γ)Γ (1 − αγ)

It follows from the hypothesis (20), G is a contraction on PC(X ). By the Banach fixed point theorem, G has a unique fixed point on PC(X ).

5 Application Example 2 Let X = C l ([0, 1]) denote space of all complex valued Hölder continuous functions on [0, 1], 0 < l < 1. For 0 < T < 1 and (x, t) ∈ (0, 1) × (0, T ), we consider the following problem in X :

Some Existence Results on Impulsive Differential Equations

547

⎫ ∂α ∂2u ⎪ 2 [u(t, x) + tu (t, x)] − 2 = f (t, u(t, x)), ⎪ ⎪ ⎪ ⎪ ∂t α ∂x ⎬ u(t, 0) = u(t, 1) = 0, 1− − 2u( 2 ,x) ⎪ ⎪ I1 (u( 21 , x) = , − ⎪ ⎪ 2+u( 21 ,x) ⎪ ⎭ u(0, x) = u 0 (x), where

∂α ∂t α

(22)

denotes the Caputo fractional derivative of order α, 0 < α < 1 and x f (t, u(t, x)) =

b(x, s)u(t, s)ds 0

for a continuous complex valued function b on [0, 1] × [0, 1]. We define u(t, ·) = v(t), Av = −

d2 u , u ∈ D(A), dx 2

(23)

where D(A) = {u ∈ C 2+l [0, 1] : u(0) = u(1) = 0}. Then Problem (22) can be put in the following form α c Dt [v(t)

⎫ + g(t, v(t))] + Av(t) = f (t, v(t)), ⎪ ⎬ −

I1 (v( 21 ) =

−

2v( 21 )

− 2+v( 21 )

,

v(0) = v0 ,

⎪ ⎭

(24)

where g(t, v(t)) = tv 2 (t) and f (t, v(t)) = f (t, u(x, t)). We note that [25] (i) the domain of A is not dense X ; (ii) there exist θ, δ > 0 such that σ(A + θ) ⊂  π2 −δ = {λ ∈ C \ {0} : | arg λ| ≤ R(λ; A + θ)L(C l [0,1]) ≤

C λ1−l/2

π − δ} ∪ {0} 2

, λ ∈ C \  π2 −δ

for some positive constant C. −1+l/2

Thus the operator A ∈ F π −δ (X ). Also the function f and g satisfy the assumptions 2 (B2) and (B3). By Theorem 2, Problem (24) has a unique PC mild solution for each v0 ∈ D(Aβ ) for β > l/2. Acknowledgements The work is supported by the Grant No. SERB/F/12082/2018 − 2019 and NBHM Grant No.02011/9/2019 NBHM(R. P.)/R. and D. I I /1324.

548

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References 1. Agarwal RP, Benchohra M, Hamani S (2010) A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions. Acta Appl Math 109:973–1033 2. Arrieta JM, Carvalho A, Lozada-Cruz G (2006) Dynamics in dumbbell domains I. Continuity of the set of equilibria. J Differ Equ 231:551–597 3. Arrieta JM, Carvalho A, Lozada-Cruz G (2009) Dynamics in dumbbell domains III. Continuity of attractors. J Differ Equ 247:225–259 4. Arrieta JM, Carvalho A, Lozada-Cruz G (2009) Dynamics in dumbbell domains II. The limiting problem. J Differ. Equ 247:174–202 5. Carvalho AN, Dlotko T, Nescimento MJD (2008) Non-autonomous semilinear evolution equations with almost sectorial operators. J Evol Equ 8:631–659 6. Ducrot A, Magal P, Prevost K (2010) Integrated semigroups and parabolic equations. Part I: linear perturbation of almost sectorial operators. J Evol Equ 10:263–291 7. El-Borai MM (2004) Some probability densities and fundamental solutions of fractional evolution equations. Chaos Solitons Fractals 149:823–831 8. Fe˘ckan M, Zhou Y, Wang JR (2012) On the concept and existence of solution for impulsive fractional differential equations. Commun Nonlinear Sci Numer Simul 17:3050–3060 9. Haloi R (2018) On Solutions to fractional neutral differential equations with infinite delay. J Fractional Calculus Appl 9(2):1–16 10. Hernndez E, Pierri M, Prokopczyk A (2011) On a class of abstract neutral functional differential equations. Nonlinear Anal 74(11):3633–3643 11. Hilfer H (2000) Applications of Fractional Calculus in Physics. World Scientific Publ. Co., Singapore 12. Jiang H (2012) Existence results for fractional order functional differential equations with impulse. Comput Math Appl 64(10):3477–3483 13. Kilbas A, Srivastava HM, Trujillo JJ (2006) Theory and Applications of Fractional Differ- ential Equations, North-Holland Mathematics Studies, vol 204. Elsevier Science B.V, Amsterdam 14. Miller KS, Ross B (1993) An introduction to the fractional calculus and differential equations. Wiley, New York 15. Milman VD, Myshkis AD (1960) On the stability of motion in presence of impulses. Siberial Math J 1:233–237 16. Mophou GM (2010) Existence and uniqueness of mild solutions to impulsive fractional differential equations. Nonlinear Anal 72(3–4):1604–1615 17. Oldham KB, Spanier J (1974) The Fractional Calculus. Academic Press, New York 18. Periago F, Straub B (2002) A functional calculus for almost sectorial operators and applications to abstract evolution equations. J Evol Equ 2:41–68 19. Podlubny I (1999) fractional differential equations, math science and engineering, vol 198. Academic Press, San Diego 20. Ranjini MC, Anguraj A (2013) Nonlocal impulsive fractional semilinear differential equations with almost sectorial operators. Malaya J Matematik 2(1):43–53 21. Shu XB, Lai Y, Chen Y (2011) The existence of mild solutions for impulsive fractional partial differential equations. Nonlinear Anal 74(5):2003–2011 22. von Wahl W (1972) Gebrochene Potenzen eines elliptischen Operators und parabolische Differentialgleichungen in Rumen hlderstetiger Funktionen. Nachr Akad Wiss Gttingen Math-Phys Kl 11:231–258 23. Wang RN, Chen DH, Xiao TJ (2012) Abstract fractional Cauchy problems with almost sectorial operators. J Differ Equ 252(1):202–235 24. Wang J, Fe˘ckan M, Zhou Y (2011) On the new concept of solutions and existence results for impulsive fractional evolution equation’s. Dyn Partial Differ Equ 8(4):345–361 25. Zhang X, Zhu C, Wu Z (2012) The Cauchy problem for a class of fractional impulsive differential equations with delay. Electron J Qual Theory Differ Equ 37:1–13 26. Zhu Y, Jiao F (2010) Nonlocal Cauchy problem for fractional evolution equations. Nonlinear Anal Real World Appl 11(5):4465–4475

Weighted Norm Inequality for General One-Sided Vector Valued Maximal Function Duranta Chutia and Rajib Haloi

Abstract In this article, we study the one weight vector valued norm inequality for the general one-sided maximal function Mw+ . We prove a sufficient, as well as a necessary condition for the weighted boundedness of the one-sided maximal function Mw+ in the vector valued setting. We establish an inequality for the operator Mw+ in the scalar setting similar to the Fefferman-Stein’s weighted lemma. Keywords Hardy-Littlewood maximal function · One-sided maximal function · One-sided weight · Weighted norm inequality

1 Introduction 1 Let w be a positive locally integrable function on IR. For f ∈ L loc,w (IR), we define the general one-sided maximal function as

Mw+ (

 f )(x) = sup  x+h h>0

x

1 w(y)dy

 x+h  | f (y)|w(y)dy . x

1 (IR), we want to express that f w is locally integrable. If we By writing f ∈ L loc,w take w = 1, then Mw+ is the classical one-sided Hardy-Littlewood maximal function M +. 1 (IRn ) is defined as The Hardy-Littlewood maximal function M for f ∈ L loc

D. Chutia · R. Haloi (B) Tezpur University, Tezpur 784028, India e-mail: [email protected] D. Chutia e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 D. Giri et al. (eds.), Proceedings of the Sixth International Conference on Mathematics and Computing, Advances in Intelligent Systems and Computing 1262, https://doi.org/10.1007/978-981-15-8061-1_45

549

550

D. Chutia and R. Haloi

M( f )(x) = sup x∈Q

1 |Q|

 | f (y)|dy, Q

where Q is any arbitrary cube in IRn with the sides parallel to the coordinate axes and the supremum is taken over all those cubes containing x. The expression |Q| denotes the Lebesgue measure of Q in IRn . The Hardy-Littlewood maximal function satisfies the following two estimates. 

 M( f )(x) dx ≤ C p

| f (x)| p dx, 1 < p < ∞

p

IRn

IRn

and for each λ > 0 |{x ∈ IRn : M( f )(x) > λ}| ≤

C λ

 | f (x)|dx. IRn

In [1], Muckenhoupt obtained a beautiful class, known as A p weights in the context of generalization of the above two inequalities with weight. It was proved that A p condition is necessary and sufficient for the boundedness of the maximal function with weights. Later, John and Anderson [2], extended the result of [1], to the vector valued set up and also found that A p condition is necessary, as well as sufficient for their result. We note that though a lot of progress has been made in the context of maximal function (see [3]), in the case of one-sided maximal function there are still some open problems which are yet to be addressed. In this context, Sawyer[4], introduced a new class of weights, A+p and establish boundedness for M + in IR. In the case of onesided maximal function, the problem is yet to be solved for IRn , n ≥ 3. Shrivastava [5] extended the result of [2] for the one-sided maximal function M + with the help of A+p weights. Throughout this article, we plan to characterize the weight for the maximal operator Mw+ in the vector valued set up. This work is an extension of the work [5], done by S. Shrivastava where the author obtained the one weight vector valued norm inequality for the one-sided maximal function M + . The result of [5], can be deduced from our result by taking w as a constant function.  In this paper, we write g(E) to denote the integral E g(x)dx, where E ⊂ IRn is any measurable set and g is a positive function. Throughout this paper, C denotes an independent positive constant not necessarily same in all cases. Also, for a sequence of measurable functions f = { f j }, we define the lq norm of f at a point for 1 ≤ q < ∞ as follows,  q1  q  f (x)q = | f j (x)| . j

Weighted Norm Inequality for General One-Sided …

551

Similarly, for the sequence f = { f j } we define Mw+ ( f ) = {Mw+ ( f j )} and the lq norm of Mw+ ( f ) as  q1   +  +  M ( f )(x) = |Mw ( f j )(x)|q . w q j

We organize the article as follows. In Sect. 2, we state some known and existing results. We state and prove the main results in Sect. 3.

2 Preliminaries In this section, we recall some of the basic definitions, lemmas, and theorems that are used in the remaining part of the article. In [6], Fefferman and Stein proved a weighted inequality for Hardy-Littlewood maximal function similar to that of Muckenhoupt [1] but with some new weight M(ψ) on the right hand side. Lemma 1 ([6]) Let q > 1 and f, ψ be two nonnegative locally integrable function IRn . Then the following holds 

 M( f )(x)q ψ(x)dx ≤ Cq

IRn

f (x)q M(ψ)(x)dx, IRn

where Cq is a positive constant. This lemma has many applications in the theory of A p weights. Using this lemma, Fefferman and Stein [6], proved the L p boundedness of the maximal function M in the vector valued setting. It also plays a vital role in the work [2], of Anderson and John where the authors established the one weight vector valued norm inequality for the maximal function. Now, we discuss about generalized one-sided A p weights corresponding to the general maximal function Mw+ . We are using the term generalized because the result of M + can be obtained from the result of Mw+ by taking w = 1. In [7], Martin-Reyes introduced two new classes of weights A+p (w) and S + p (w) for the boundedness of the operator Mw+ and also established an equivalence relation between these two weights. Also the authors [7] defined the A+p (w) weights as similar to the A+p weights. Definition 1 Let (ω, φ) be a nonnegative locally integrable pair of functions on IR. Then (i) the pair (ω, φ) is said to follow A+p (w) condition for 1 < p < ∞ if for every α, β and γ with α ≤ β ≤ γ we have,

552

D. Chutia and R. Haloi

 β

 γ ω(x)dx

 p−1

p

w(x) ρ(x)dx

α

 γ ≤C

p w(x)dx

,

α

β

where ρ(x) = φ(x)− p−1 , (ii) we say (ω, φ) satisfies A+ 1 (w) condition if 1

Mw−



   ω φ (x) ≤ C (x) a.e. w w

Mw− corresponds to the one-sided maximal function defined in the interval [x − h, x). The A+p (w) condition is necessary and sufficient for the weak ( p, p) of the operator Mw+ , but this class is not sufficient to prove the strong ( p, p) of the operator Mw+ with two different weights. Hence, to establish the strong ( p, p) for Mw+ , we need an another class called S + p (w) which ensures the sufficient and necessary conditions for the operator Mw+ . Theorem 1 ([7]) Mw+ is of weak type ( p, p), 1 ≤ p < ∞ with respect to the measures φd x and ωd x if and only if the pair (ω, φ) satisfies the A+p (w) condition. Theorem 2 ([7]) Mw+ is of strong type ( p, p), p > 1 from L p (φ) to L p (φ) if and only if φ satisfies the A+p (w) condition. Thus, it is observed that A+p (w) condition is necessary and sufficient for the one weight problem for Mw+ in the scalar setting. Hence, A+p (w) is necessary in the vector valued set up for the one weight problem of Mw+ . Next, we will state some elementary properties satisfied by A+p (w) weights which are required in proving our main result. The properties are quite similar to the case of A p and A+p weights. Theorem 3 ([7]) If φ ∈ A+ 1 (w), then there exists δ > 0 such that 1 w(α, β)

β φ

1+δ

w

−δ

α

 ≤ Cδ

1 (w(α, β))δ

for each α and β a.e. and for this particular δ,

φ1+δ wδ

β  δ φ φ (β) w α

∈ A+ 1 (w).

Corollary 1 ([7]) The followings are true. 

(i) ω ∈ A+p (w) if and only if w p ρ ∈ A−p (w). (ii) If ω ∈ A+p (w), 1 < p < ∞, then there exists δ > 0 such that ω ∈ A+p−δ (w) with p − δ > 1.

Weighted Norm Inequality for General One-Sided …

553

3 Main Results We begin with the following containment relation for generalized one-sided weights. Lemma 2 If r > p, then A+p (w) ⊂ Ar+ (w). Proof Let ω ∈ A+p (w). Thus, for any α, β, δ with α < β < δ we have,  β  δ  p−1  δ  p 1 p − p−1 ω w ω < w < ∞. α

α

β

We need to show  β  δ r −1  δ r 1 r  − r −1 ω w ω < w < ∞, α

α

β

for each α, β, δ with α < β < δ. Now, for each α < β < δ, r −1  β  δ  p−1  δ r − p  β  δ 1 1 r  − r −1 p − p−1 ω w ω ≤ ω w ω w α

α

β

β

 δ ≤

β

r w

< ∞.

α

We obtain the first inequality by using Hölder inequality with the exponent and then we use the fact that ω ∈ A+p (w).

r −1 p−1

Next, we prove an inequality similar to that of Fefferman-Stein weighted lemma in the scalar setting. The idea of the proof is based on techniques used in [5, 8]. Lemma 3 Let g be a nonnegative locally integrable function on IR. Then for p > 1 1 and f (≥ 0) ∈ L loc,w (IR) there exists a constant C p > 0 such that  IR

Mw+ (

 f )(x) g(x)dx ≤ C p

f (x)

p

IR

p

Mw−



 g (x)w(x)dx. w

Proof We establish the result with the help of Marcinkiewicz interpolation theorem. The above inequality can be viewed as boundedness of the operator f → Mw+ ( f )

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D. Chutia and R. Haloi

in the L p norm with respect to the weights Mw−

  g w and g. It is enough for us w

to prove the weak (1, 1) and weak (∞, ∞) of the operator f → Mw+ ( f ) to get the above inequality. We assume that f is bounded with compact support. Let λ > 0 and consider the set Υλ = {x ∈ IR : Mw+ ( f )(x) > λ}. Then there exists a sequence of disjoint and bounded intervals [8] {(α j , β j )} such that Υλ = ∪(α j , β j ) and 1 λ≤ w(x, β j )

β j f (y)w(y)dy. x

Due to the disjointness property of the sequence (α j , β j ), it is sufficient to prove the following inequality β j αj

C g(x)dx ≤ λ

β j

f (x)Mw− (

αj

g )(x)w(x)dx. w

For a fixed j, we break the interval (α j , β j ) as follows: We put γ 0j = α j and for each k ∈ IN, we define γ kj as β j

1 f (x)w(x)dx = k 2

γ kj

β j f (x)w(x)dx. αj

Thus, we get an increasing sequence {γ kj } with (α j , β j ) =

∞ 

k (γ k−1 j , γ j ] and

k=1

γ j

k+2

λ≤

4 w(γ kj , γ k+2 j )

f (y)w(y)dy. γ k+1 j

Now γ j

k+1

β j f (x)w(x)dx =

γ kj

β j f (x)w(x)dx −

γ kj

=

f (x)w(x)dx

γ k+1 j

1 2k+1

β j αj

1 f (x)w(x)dx = 4

β j f (x)w(x)dx. γ k−1 j

Weighted Norm Inequality for General One-Sided …

555

By changing the order of the integration, we obtain γ j

k

 g(x)dx ≤

λ

w(γ k−1 j , βj)

γ k−1 j

k

f (y)w(y)dy γ k−1 j

γ j

4

 γ j

γ j

k+1

γ kj

γ j γ kj

g(x)dx

 g(x)dx w(y)dy

y

w(γ k−1 j , βj)

f (y)Mw−



γ k−1 j

1

k+1

≤4

γ kj

 f (y)

=4

k

f (y)w(y)dy

w(γ k−1 j , βj)

 g(x)dx

γ k−1 j

k+1

 =

 γ j

β j

1

γ k−1 j

g

(y)w(y)dy. w

Summing the integral on the both sides over k, we get β j αj

Thus

 Υλ

C g(y)dy ≤ λ

C g(y)dy ≤ λ

β j

f (y)Mw−

g

αj

β j

f (y)Mw−



αj

w

(y)w(y)dy.

 g (y)w(y)dy. w

This proves the weak (1, 1) condition. Theorem 4 Let 1 < q < ∞ and f = { f j } be a sequence of measurable functions 1 such that f j ∈ L loc,w (IR), for each j. Then the following inequalities hold. (i) If ω ∈ A+p (w), 1 < p < ∞, then we have the following strong ( p, p) condition, 

||Mw+ (

f )(x)||qp ω(x)dx

 1p

 ≤C

IR

||

f (x)||qp ω(x)dx

 1p

.

IR

(ii) If ω ∈ A+ 1 (w) with p = 1 and for λ > 0, we have the weak (1, 1) inequality, ω



x ∈ IR :

||Mw+ (

f )(x)||q > λ



≤

C λ

 || f (x)||q ω(x)dx. IR

556

D. Chutia and R. Haloi

Proof We break the proof of the theorem into three different parts depending on p and q. Let 1 < p, q < ∞ and ω ∈ A+p (w). Case I : When p = q. Now 

  +  M f (x) p ω(x)dx w

 1p

q



  +  M f (x) p ω(x)dx

=

IR

w

 1p

p

IR

=

   j

≤C

IR

IR

 

 1p

f j (x) ω(x)dx

 1p

 1p

p

 1p

| f j (x)| p ω(x)dx

j

IR



 f (x) ω(x)dx

=

ω(x)dx

Mw+ f j (x) p ω(x)dx

 j

=



j

IR

=

Mw+ f j (x) p

p

 1p

.

IR

This completes the proof for p = q. Case II : When p > q. Since ω ∈ A+p (w), thus from the Corollary 1, there exists δ > 0 such that ω ∈ A+p−δ (w) with p − δ > 1. This is same as saying that there exists p r0 , 1 < r0 < p such that ω ∈ A+ s0 (w) with s0 = r0 . Thus, again applying Corollary 1, 

w s ω − s−1 ∈ A− we get ω ∈ A+ s (w) and s  (w), for all s ≥ s0 . s Now, for ψ ∈ L ω (IR), 

  +  M ( f )(x)q (ψω)(x)dx w

IR

=

q

  IR

=

1

 j

≤C

Mw+ ( f j )(x)q (ψω)(x)dx

j

Mw+ ( f j )(x)q (ψω)(x)dx

IR

 j

IR

| f j (x)|q Mw−

ψω

(x)w(x)dx w

Weighted Norm Inequality for General One-Sided …

=C

 

| f j (x)|q Mw−

ψω

j

IR



 f (x)qq Mw−

=C IR

w

 f (x)qsq ω(x)dx

 1s   

IR

 IR

 ≤C

IR

f (x)qsq

ω(x)dx

s   s1 ψω

s (x)w(x) ω(x)− s dx w

 1s   

Mw−

 s1 ψω s  1 s − s−1 (x) w(x) ω(x) dx w

IR

s   1s     s1 ψω

1 sq s − s−1  f (x)q ω(x)dx (x) w(x) ω(x) dx w IR

  f (x)qsq ω(x)dx

=C

Mw−

IR

 =C

(x)w(x)dx

ψω

(x)w(x)dx w

 ≤C

557

 1s  

IR

s

ψ(x) ω(x)

 s1

.

IR

We choose s = qp and then taking supremum over all ψ in the set B = {ψ ≥ 0 :   ψ(x)s ω(x)dx = 1}, we get, IR



  +  M ( f )(x) p ω(x)dx w

 1p

q

 

≤C

IR

f (x)qp

ω(x)dx

 1p

, 1 < q ≤ r0 < p.

IR

From the first case, we have got the desired inequality for q = p. Thus, by interpolation result the above inequality is true for all q with r0 < q < p. Hence, we can conclude that 

  +  M ( f )(x) p ω(x)dx w

IR

q

 1p

 

≤C

f (x)qp

ω(x)dx

 1p

, 1 < q < p.

IR

Case III: When p < q. We have already established our result for p = q and p > q. Now, we plan to establish the weak ( p, p) version of the result. Thus, by using the result for p = q, weak ( p, p) and interpolation result we get the result for p < q. Hence, it is sufficient for us to prove the following estimate 

 +  C    f (x)qp ω(x)dx, where λ > 0. ω {x ∈ IR : Mw ( f )(x) q > λ} ≤ p λ IR

558

D. Chutia and R. Haloi

We prove this estimate via breaking f as f = g + b and then proving the same estimate for g and b, and finally we use sublinear property of the maximal operator Mw+ . The details of the proof are given below. For λ > 0 consider the set Υ = {x ∈ IR : Mw+ ( f q )(x) > λ}. Then as earlier we get a sequence of disjoint interval {(α j , β j )} j satisfying following three conditions  (i) Υ = (α j , β j ), j

(ii) λ =

1 w(α j ,β j )

β j αj

 f (x)q w(x)dx, ∀ j,

(iii) for x ∈ (α j , β j ), we have λ ≤

1 w(x,β j )

β j

 f (y)q w(y)dy, ∀ j.

x

Thus,  f (x)q ≤ λ, for a.e. x ∈ Υ c . We decompose f as f = g + b, where g = {gk } and b = {bk } are two sequences of measurable functions defined for each k as, gk = f k χΥ c and bk = f k − gk . Now, Mw+ ( f )(x) = Mw+ (g + b)(x) ≤ Mw+ (g)(x) + Mw+ (b)(x), this gives us the following       +  M ( f )(x) ≤  M + (g)(x) +  M + (b)(x) . w w w q q q

(1)

We complete the proof of the theorem by using the following estimates ω



    C  f (x)qp ω(x)dx, x ∈ IR :  Mw+ (g)(x)q > λ ≤ p λ

(2)

IR

ω



    C  f (x)qp ω(x)dx x ∈ IR :  Mw+ (b)(x)q > λ ≤ p λ

(3)

IR

We see the proof of (2) as follows: From the fact ω ∈ A+p (w), we have ω ∈ Aq+ (w). And from the first case, we obtain ω



    C g(x)qq ω(x)dx x ∈ IR :  Mw+ (g)(x)q > λ ≤ q λ IR  C p− p g(x)q+ = q ω(x)dx q λ Υc  C  f (x)qp ω(x)dx ≤ p λ Υc  C  f (x)qp ω(x)dx, ≤ p λ IR

Weighted Norm Inequality for General One-Sided …

559

where the first inequality comes from the Case I and in the second inequality we have used the fact that g(x)q ≤  f (x)q ≤ λ for a.e. x ∈ Υ c . This proves the estimate (2). Next, we prove the estimate (3). We construct a new function f  = { f k } as f k (x)

=

⎧ ⎪ ⎨ ⎪ ⎩

1 w(α j ,β j )

β j αj

f k (y)w(y)dy, if x ∈ (α j , β j ),

0,

otherwise.

For an interval J = (α, β), set J − = (2α − β, α) and define Υ ∗ = Υ ∪ Υ − , where  − Υ = (2α j − β j , α j ). Let us first assume that the following inequalities hold. j

    +  M (b)(x) ≤ C  M + ( f  )(x) , x ∈ / Υ ∗, w w q q

(4)

    ω 2α j − β j , α j ≤ Cω α j , β j ,

(5)

  C ω αj, βj ≤ p λ

β j  f (x)qp ω(x)dx. αj

Using the estimates (6), (4) and (5), we obtain 

   x ∈ IR :  Mw+ (b)(x)q > λ     c = ω x ∈ Υ ∗ ∪ Υ ∗ :  Mw+ (b)(x)q > λ     / Υ ∗ :  Mw+ (b)(x)q > λ = ω(Υ ∗ ) + ω x ∈     ≤ Cω(Υ ) + ω x ∈ / Υ ∗ :  Mw+ ( f  )(x)q > λ    q C  f (x) ω(x)dx ≤ Cω(Υ ) + q q λ IR      q 1  f (x) ω(x)dx ≤ C ω(Υ ) + q q λ Υ     1 | f k (x)|q ω(x)dx = C ω(Υ ) + q λ k ω

Υ

(6)

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D. Chutia and R. Haloi



1 ≤ C ω(Υ ) + q λ 

1 ≤ C ω(Υ ) + q λ

  k

Υ

  k

Υ

 β j w(y)

1 w(α j , β j ) 1 w(α j , β j )

(1− q1 )q 

 1 q q

dy

q

β j | f k (y)|w(y)dy

 ω(x)dx

αj

 β j | f k (y)|q w(y)dy

 q1

αj

 ω(x)dx

αj



1 ≤ C ω(Υ ) + q λ

 Υ

1 w(α j , β j )

 β j  αj





| f k (y)| w(y)dy ω(x)dx q

k

    β j 1 1 q λ w(y)dy ω(x)dx ≤ C ω(Υ ) + q λ w(α j , β j ) α Υ j  C  f (x)qp ω(x)ddx. = Cω(Υ ) ≤ p λ 

IR

Next, we shall prove the three assumptions. / Υ and x ∈ / Υ −. We begin with the proof of the inequality (4). Let x ∈ / Υ ∗ , thus x ∈ For each k > 0 consider the set Yk = {y : x < y < x + k}. We have, 1 w(Yk )

 bk (y)w(y)dy =



1  w(Yk ) j∈J

bk (y)w(y)dy,

Yk ∩(α j ,β j )

Yk

where, J = { j : Yk ∩ (α j , β j ) = φ}. Clearly x ∈ / (α j , β j ) and x ∈ / (2α j − β j , α j ). Now, if j ∈ J then (α j , β j ) ⊂ Y2k . Thus 1 w(Yk )

 bk (y)w(y)dy = Yk



1  w(Yk ) j∈J

bk (y)w(y)dy

Yk ∩(α j ,β j )

1  ≤ w(Yk ) j∈J 1  = w(Yk ) j∈J

β j bk (y)w(y)dy αj

β j ( f k − gk )(y)w(y)dy αj

Weighted Norm Inequality for General One-Sided …

561

1  = w(Yk ) j∈J

β j

f k (y)w(y)dy

αj

≤

w(Y2k ) 1 w(Yk ) w(Y2k )

≤

C Mw+ ( f k )(x).



f k (y)w(y)dy

Y2k

This proves the estimate (4). Next, we proof the inequality (5). Let J = (α, β) be any interval and x ∈ J − = (2α − β, α), then as similar to [5] we obtain 1 1= w(α, β)

β w(y)dy α

w(x, β) 1 = w(α, β) w(x, β)

β χ J (y)w(y)dy x

1 w(2α − β, β) ≤ w(α, β) w(x, β)

β χ J (y)w(y)dy x

≤ C Mw+ (χ J )(x). Thus, x ∈ J − implies that Mw+ (χ J )(x) ≥

1 C

i.e. J − ⊂ {x : Mw+ (χ J )(x) ≥

1 }. C

Now

 1 ω(J − ) ≤ ω x : Mw+ (χ J )(x) ≥ C  p p ≤C χ J (x) ω(x)dx IR

 =C

ω(x)d x = Cω(J ). J

    This establishes that ω 2α j − β j , α j ≤ Cω α j , β j for each j. To prove the third inequality (6) we use the earlier technique to break the interval 0 (α j , β j ) into a disjoint increasing sequence {γk }∞ k=0 . Set γ j = α j and for each k ∈ IN we define γ kj as β j γ kj

1  f (x)q w(x)d x = k 2

β j  f (x)q w(x)dx. αj

562

D. Chutia and R. Haloi

Thus, we get an increasing sequence {γ kj } with (α j , β j ) =

∞ 

k (γ k−1 j , γ j ].

k=1

Now

k ω γ k−1 j , γj  ≤

1 λw(γ k−1 j , βj)

4 λw(γ k−1 j , βj)

 f (x)q w(x)dx

 ω(x)dx

γ k−1 j

γ k−1 j

4 λw(γ k−1 j , βj)

C ≤ p λ

 p  γ j

k



 f (x)q ω(x)

1 1 p− p

w(x)dx

j

 ω(x)dx

γ k−1 j

γ k−1 j

γ k+1

 ≤

k

γ k+1 j

 =

 p  γ j

β j

p  f (x)q

γ k+1

ω(x)dx

1 j p

γ kj

p

w(x) ω(x)

γk



− pp

dx

γ kj

1  p   j p

 ω(x)dx

γ k−1 j

γ k+1 j



p  f (x)q ω(x)dx .

γ kj

Summing over all k, we get C ω(α j , β j ) ≤ p λ

β j

 f (x)qp ω(x)dx .

αj

This concludes the proof of the estimate (6). Hence, the CaseI I I follows. The weak (1, 1) version for the operator Mw+ in the vector valued set up is immediate from the CaseI I I. Hence, it completes the proof. Acknowledgements D. Chutia was supported by the DST INSPIRE (Grant No. DST/INSPIRE Fellowship/2017/IF170509). R. Haloi was supported by the DST MATRICS (Grant No. SERB/F/ 12082/2018-2019).

References 1. Muckenhoupt B (1972) Weighted norm inequalities for the Hardy maximal functions. Trans Am Math Soc 165:207–226 2. Anderson KF, John RT (1980/1981) Weighted inequalities for vector-valued maximal functions and singular integrals. Stud Math 69:19–31 3. Grafakos L (2008) Classical fourier analysis.: graduate texts in mathematics, vol 249, 2nd edn. Springer, Berlin

Weighted Norm Inequality for General One-Sided …

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4. Sawyer E (1986) Weighted inequalities for the one-sided Hardy-Littlewood maximal functions. Trans Am Math Soc 297:53–61 5. Shrivastava S (2016) Weighted and vector-valued inequalities for one-sided maximal functions. Proc Indian Acad Sci (Math Sci) 126:359–380 6. Fefferman C, Stein EM (1971) Some maximal inequalities. Am J Math 93:107–115 7. Martin-Reyes FJ, Salvador PO, de La Torre A (1990) Weighted inequalities for one- sided maximal functions. Trans Am Math Soc 319:517–534 8. Qinsheng L (1996) A note on the weighted norm inequality for the one-sided maximal operator. Proc Am Math Soc 124:527–537

Author Index

A Acharyya, Sriyankar, 309, 451 Adak, Dulal, 451 Akbar, M. Ali, 469 Anand, Harshvardhan, 65 Arora, Rudra, 345

B Bajeel, P. N., 483 Bandyopadhyay, Mainak, 407 Barman, Golap Gunjan, 275 Barman, Hemonta Kumar, 469 Barman, Utpal, 275 Barua, Rakesh, 227 Bera, Amal, 357 Bera, Debkumar, 89 Bera, Somnath, 263 Bhattacharyya, Dhruba K., 249 Bhattacharyya, Somnath, 75

C Chakraborty, Arun, 407 Chaudhury, Bhaskar, 201, 407 Chowdhuri, Partha, 89 Chutia, Duranta, 549

D Das Gupta, P. K., 227 Das, Lalatendu, 227 Das, Priya, 309 Das, Rituparna, 167 Das, Subrata, 143

Datta, Kakali, 43 De, Debashis, 297 Dhara, Bibhas Chandra, 393 Dhar, Ankita, 193 Dutta, Subhash C., 249 Dutta, Subhasree, 75

F Fujisawa, Katsuki, 29

G Garg, Tanya, 201 Ghosh, Atonu, 297 Ghosh, Mohona, 125 Giri, Debasis, 15, 89 Gupta, Bhupendra, 491

H Haloi, Rajib, 535, 549

I Ikematsu, Yasuhiko, 29, 377 Indu, S., 325

J Jana, Biswajit, 309, 451 Jana, Biswapati, 43, 89, 227 Jha, Aranya, 65

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 D. Giri et al. (eds.), Proceedings of the Sixth International Conference on Mathematics and Computing, Advances in Intelligent Systems and Computing 1262, https://doi.org/10.1007/978-981-15-8061-1

565

566 K Kaity, Sourav, 227 Kalita, Jugal K., 249 Kamizawa, Takeo, 515 Kandar, Shyamalendu, 393 Kaushal, Sakshi, 345 Kayum, Md. Abdul, 469 Kinjo, Koha, 29 Kumar Gupta, Sunny, 125 Kumar, Jitendra, 143 Kumar, M., 483 Kumar, Neeraj, 151 Kumar, Sumit, 325

L Lamba, Subir Singh, 491 Liwei, Wang, 103

M Maiti, Swapan, 1 Maiti, Tapas Kumar, 201 Maitra, Tanmoy, 15 Majumder, Chandrima, 193 Majumder, Koushik, 297 Makkar, Aaisha, 151 Mandal, Bankim C., 429 Mehta, Sanyam, 179 Minirani, S., 529 Mishra, Satyam, 151 Mitra, Suman, 451 Mohanty, Sraban Kumar, 125 Mukherjee, Himadri, 193

N Naik, S., 441 Nakamura, Satoshi, 29, 377 Nath, P. K., 441 Nithin Goud, Kaira, 55

O Obaidullah, Sk Md, 193

P Pandey, Anuj K., 249 Pandey, C. P., 287 Patel, Geet, 201 Patel, Sarangi, 201

Author Index Patra, Arnab, 371 Patra, Bidyut Kumar, 55, 143, 217 Paul, Aakash, 393 Paul, Bachchu, 263 Phadikar, Santanu, 263 Prasad, Punam Kumari, 419

R Ramanjaneyulu, Y. V., 55 Ranjini, M. C., 501 Riya, 491 Roshani, 201 Roy Chowdhury, Dipanwita, 1 Roy, Kaushik, 193

S Saha, Asit, 65, 113, 419 Sahu, Diganto, 275 Saikia, Jyoti, 287 Sama, Lakshit, 151 Samdani, Yash, 151 Sana, Soura, 429 Sapkota, Niraj, 167 Saraswat, Vishal, 179 Sathya Babu, Korra, 55 Savas, Ekrem, 461 Sen, Shibaprasad, 193 Shah, Miral, 407 Sharma, Pooja, 249 Sharma, Rohit, 371 Shaw, Rabi, 217 Shit, G. C., 357 Shrivastava, Amit, 371 Singh, Prabhash Kumar, 43 Singla, Khushboo, 345 Srivastava, P. D., 371

T Tamang, Jharna, 113 Tateiwa, Nariaki, 29 Tyagi, Manav, 65

W Walia, Gurjit Singh, 325

Y Yasuda, Masaya, 29, 377