Mathematical Analysis and Analytical Modeling 1774073218, 9781774073216

Table of contents :
Cover
Title Page
Copyright
ABOUT THE AUTHOR
TABLE OF CONTENTS
List of Figures
List of Tables
Preface
Chapter 1 Preliminaries
1.1. Metric Spaces
1.2. Complex Variable
1.3. Complex Functions
1.4. Singularities Classes And Classes Of Functions
1.5. Curves, Contours, And Simple-Minded Domains Related
Chapter 2 Normal Families
2.1. Sequence Of Functions
2.2. Compactness And Convergence In The Space Of Analytical Functions
2.3. Arsela-Ascoli Theorem And Montel
Chapter 3 Iteration Functions, Fixed Points And Fatou And Julia Sets
3.1. Iteration Of Functions And Points
3.2. Fatou And Julia Set
3.3. Fatou Assembly Components
3.4. Classification Of Periodic Components For Functions
Chapter 4 Baker Theorem For Completely Invariant Components In The Set Of Fatou
4.1. Definitions And Results
4.2. Demonstration Of Baker’s Theorem
Chapter 5 Mathematical Analysis Of Significant Transformations Of The Transverse Projects
Chapter 6 Theoretical Framework
6.1. Theoretical Background
6.2. Conceptual Framework
6.3. Regulatory Framework
6.4. Developed Method
6.5. Recommendations And Limitations
Chapter 7 Applying Analytical Functions In Educational Research
7.1. Approach And Problem Formulation
7.2. Background To The Problem Under Study
7.3. Vie Model Overview
7.4. Variability Model Of Educational Research Or Model Vie
7.5. The Extra-Framework Component
7.6. Encompassing Theoretical Empirical Result
7.7. Patterns Methodology
7.8. Contrasting Operations And Validation
7.9. Balancing The Components Which Are Logical-Structural
7.10. Base Epistemological Predominant
Chapter 8 Methodological Framework
8.1. Level Or Stage Of Research
8.2. Design Chosen For Data Collection
8.3. Study Units
8.4. Techniques And Instruments For Data Collection
8.5. Technical Data Analysis
8.6. Technical Reliability Of Life And Employee Instrument
8.7. Process
Chapter 9 Conclusions And Recommendations
9.1. Recommendations For The Study
Bibliography
Index
Back Cover

Citation preview

MATHEMATICAL ANALYSIS AND ANALYTICAL MODELING

MATHEMATICAL ANALYSIS AND ANALYTICAL MODELING

Ivan Stanimirović

ARCLER

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www.arclerpress.com

Mathematical Analysis and Analytical Modeling Ivan Stanimirović

Arcler Press 2010 Winston Park Drive, 2nd Floor Oakville, ON L6H 5R7 Canada www.arclerpress.com Tel: 001-289-291-7705 001-905-616-2116 Fax: 001-289-291-7601 Email: [email protected] e-book Edition 2020 ISBN: 978-1-77407-407-7 (e-book) This book contains information obtained from highly regarded resources. Reprinted material sources are indicated and copyright remains with the original owners. Copyright for images and other graphics remains with the original owners as indicated. A Wide variety of references are listed. Reasonable efforts have been made to publish reliable data. Authors or Editors or Publishers are not responsible for the accuracy of the information in the published chapters or consequences of their use. The publisher assumes no responsibility for any damage or grievance to the persons or property arising out of the use of any materials, instructions, methods or thoughts in the book. The authors or editors and the publisher have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission has not been obtained. If any copyright holder has not been acknowledged, please write to us so we may rectify.

Notice: Registered trademark of products or corporate names are used only for explanation and identification without intent of infringement. © 2020 Arcler Press ISBN: 978-1-77407-321-6 (Hardcover) Arcler Press publishes wide variety of books and eBooks. For more information about Arcler Press and its products, visit our website at www.arclerpress.com

ABOUT THE AUTHOR

Ivan Stanimirovic gained his PhD from University of Niš, Serbia in 2013. His work spans from multi-objective optimization methods to applications of generalized matrix inverses in areas such as image processing and computer graphics and visualisations. He is currently working as an Assistant professor at Faculty of Sciences and Mathematics at University of Niš on computing generalized matrix inverses and its applications.

TABLE OF CONTENTS

List of Figures ........................................................................................................ix List of Tables .........................................................................................................xi Preface........................................................................ .......................................xiii Chapter 1

Preliminaries ............................................................................................. 1 1.1. Metric Spaces ..................................................................................... 1 1.2. Complex Variable ............................................................................... 5 1.3. Complex Functions ............................................................................. 9 1.4. Singularities Classes And Classes Of Functions ................................. 13 1.5. Curves, Contours, And Simple-Minded Domains Related.................. 14

Chapter 2

Normal Families ...................................................................................... 17 2.1. Sequence Of Functions ..................................................................... 17 2.2. Compactness And Convergence In The Space Of Analytical Functions....................................................................... 18 2.3. Arsela-Ascoli Theorem And Montel ................................................... 20

Chapter 3

Iteration Functions, Fixed Points And Fatou And Julia Sets ................................................................................ 27 3.1. Iteration Of Functions And Points...................................................... 27 3.2. Fatou And Julia Set ............................................................................ 31 3.3. Fatou Assembly Components ............................................................ 32 3.4. Classification Of Periodic Components For Functions ....................... 33

Chapter 4

Baker Theorem For Completely Invariant Components In The Set Of Fatou ..................................................................................... 37 4.1. Definitions And Results ..................................................................... 37 4.2. Demonstration Of Baker’s Theorem................................................... 38

Chapter 5

Mathematical Analysis Of Significant Transformations Of The Transverse Projects........................................................................... 41

Chapter 6

Theoretical Framework ........................................................................... 55 6.1. Theoretical Background .................................................................... 55 6.2. Conceptual Framework ..................................................................... 64 6.3. Regulatory Framework ...................................................................... 76 6.4. Developed Method ........................................................................... 80 6.5. Recommendations And Limitations ................................................. 110

Chapter 7

Applying Analytical Functions In Educational Research ............................................................................ 113 7.1. Approach And Problem Formulation ............................................... 114 7.2. Background To The Problem Under Study ....................................... 118 7.3. Vie Model Overview ....................................................................... 126 7.4. Variability Model Of Educational Research Or Model Vie ............... 128 7.5. The Extra-Framework Component ................................................... 129 7.6. Encompassing Theoretical Empirical Result ..................................... 136 7.7. Patterns Methodology ..................................................................... 138 7.8. Contrasting Operations And Validation ........................................... 139 7.9. Balancing The Components Which Are Logical-Structural ............... 141 7.10. Base Epistemological Predominant ............................................... 142

Chapter 8

Methodological Framework .................................................................. 143 8.1. Level Or Stage Of Research ............................................................ 143 8.2. Design Chosen For Data Collection ................................................ 143 8.3. Study Units ..................................................................................... 143 8.4. Techniques And Instruments For Data Collection ............................ 144 8.5. Technical Data Analysis .................................................................. 144 8.6. Technical Reliability Of Life And Employee Instrument .................................................................... 144 8.7. Process ........................................................................................... 144

Chapter 9

Conclusions And Recommendations ..................................................... 147 9.1. Recommendations For The Study .................................................... 148 Bibliography .......................................................................................... 151 Index ..................................................................................................... 157

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LIST OF FIGURES Figure 1.1. Stereographic projection Figure 3.1. Periodic component Fatou set Figure 3.2. Figure 3.3 . Component wanderer Figure 3.4. Domain botcher Figure 3.5. Domain leau Figure 3.6. Disk Siegel Figure 3.7. Domain baker Figure 4.1. Figure 4.2. Figure 6.1. Figure 6.2. The arts and music classes help you in your academic process Figure 6.3. I would like to have more space to approach the artistic proposals Figure 6.4. Encuentras relationship between projects and cross your performance Figure 6.5. Know the purpose of cross-school projects Figure 6.6. Meet the objectives of these activities Figure 6.7. You might like extra-class processes Figure 6.8. I like to participate in proposing such activities and proposals Figure 6.9 . Image associated by teachers to transverse projects

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x

LIST OF TABLES Table 6.1 Table 6.2. Table 6.3. Table 6.4. Table 6.5. Table 6.6. Table 6.7. Table 6.8. Table 6.9. Table 6.10.

PREFACE

The study of holomorphic dynamical systems generated by iterating holomorphic functions has its beginning in the late nineteenth century, motivated by the analysis of convergence for Newton’s method. From the work of Pierre Fatou (1878–1929) and Gaston Julia (1893–1978), in the twentieth century, the global theory was seriously studied. These two mathematicians, mainly studied iteration of rational functions of the Riemann sphere. In 1926, Pierre Fatou first studied the transcendental entire functions (functions with a singularity at infinity). The most important innovation introduced by Fatou and Julia was undoubtedly done using the theory of normal families to divide the field into two sets of dynamic behavior totally different. These sets are now known as Fatou and Julia sets, or equivalently stable set and chaotic set of the holomorphic function in question. Although Julia and Fatou are extensively described in detail in Chapter 3 , the set of Fatou and Julia left on the road to many open problems. The fundamental properties of sets Fatou and Julia were first established for rational functions in Baker (1962, 1964), and transcendental entire functions in Baker (1963). Finally in his chapter, Fatou studied in more detail about the iteration of transcendent entire functions by giving examples and pointing to significant differences to the theory that had been developed for rational functions. We can ask the following fundamental questions about an entire function: (I) Are the dense periodic points of the function f denoted as J(f)? (ii) Are there examples such that J (f) = C? In particular, it is satisfied for f (z) = ez? (iii) Can J (f) be totally disconnected? Most of the questions raised by Fatou were resolved by Baker during the decade 1965–1975. In his investigation, Baker worked on many problems

of complex analysis and had a wide range of partners, but the theory of iteration was his obsession. However, when the subject turned to revive around 1980, partly as a result of the advent of computer graphics, it became apparent that the new followers of Baker had been for many years quietly and carefully completing the foundation begun to the beginning of the century by the French mathematicians Pierre Fatou and Gaston Julia. Also indicated that the road to many future developments, both to test new findings as to pose difficult problems. This thesis will study one of many theorems introduced by Baker . The first chapter sets out the preliminaries; these are complex variable overall results of mathematical analysis, topology, that help us better understand the demonstration of propositions, lemmas, and theorems given in subsequent chapters. In the second chapter, general findings related to normal families which help us study the Fatou and Julia sets are set. In the third chapter, the study of fixed points is done along with classification, and examples of them are given. Furthermore, Fatou sets are defined and Julia some basic properties of these sets are stated, and the classification of the components Fatou. In chapter four, the main theorem proves the hypothesis. The planning and formulation of the problem is given, as well as the objectives and rationale of the study. Next, based on the theoretical framework of reference, the context was studied related to this research and theoretical aspects that underlie exposed. A methodological approach is provided, such as the mode of research, study unit, technical, operating, and other procedures performed to achieve the objectives stated elements. The results with their analysis are given in the last chapter , and the conclusions were reached and recommendations considered relevant. Mathematical analysis of significant transformations of the transverse projects along with the theoretical framework are provided in Chapters 5 and 6. Applying analytical functions in educational research is discussed in the seventh chapter and the conclusions are drawn in the final chapter as well as recommendations for future research.

xiv

1 Preliminaries

Let’s assume a certain familiarity with the complex analysis of a variable; however, remember some results that are of particular importance for our purpose. The results shown in this chapter can be found in the following literature (Agarwal et al., 2011; Antimirov et al., 1998; Conway, 1978; Fernandez and de la Torre, 1983; Iribarren, 2008; Noguchi, 1998; Palka, 1990; Shirali and Vasudeva, 2006; Tamariz, 1982).

1.1. METRIC SPACES Definition 1.1. A metric space is a pair (E, d) formed by a con-together E and an application d: E × E → R called distance E (or metric) that meet the following axioms:

• d(X, y) = 0 if and only if x = y; • ∀x, y ∈ E, d(X, y) = d (y, x); and • ∀X and Z ∈ E, d(X, y) ≤ d (x, y) + d (y, z). Definition 1.2. Let M be a metric space, x ∈ M and r> 0, then we define:

B(X, r) = {y∈X: D (x, y) 0 must be B* (X, r) ∩ A ≠∅.

We denote the set of points of accumulation of A by A’.

Theorem 1.1. (Fernandez and Torre, 1983) let M be a metric space, TO ⊂ M and x accumulation point A. For each V ∈ V(X) the set V ∩ A contains infinite points.

Definition 1.7. Let M be a metric space, a set F ⊂ M is closed if it contains all its accumulation points.

Definition 1.8. Let M be a metric space, A⊂ M. de ne inside A, into A, as the set {G: G is open and G ⊂TO}. It is called close of A, A, to set ∩ {F: F is closed and F⊃ TO}.

Definition 1.9. (Property Cantor) Let M be a metric space is said to M, possesses the property of Cantor, if all accounting family conjunctives {A0, TO1,…}, each nonempty or closed, ∀n ∈N: An+1⊂TOn and in f {δ (A)} = 0, has non-empty intersection. Definition 1.10. Let {x1, x2,…} be a sequence in a metric space M and x ∈ M, we say {xn} Converges ax (X = lim xn, where σxn → x) If for each ϵ >0 exists an integer N such that d (x, xn) 0 exists an integer N such that d (xn, xm) 0 corresponds to-one with infinite points x1, x2, x3,…, xn∈ A such that n

A⊂ ∪B ( xk , ε ) . k =1

Theorem 1.4. (Iribarren, 2008) If A is a set of a paracompact is patio metric (M, d), all nonempty subset or A is paracompact. Definition 1.14. Let M be a metric space and A ⊂M vac or not. We say that A possesses the property of Bolzano-Weierstrass, if all T⊂ A subset in supports an accumulation point A, that is T ‘∩ A ≠∅. A call such sets the BW. Definition 1.15. Let M be a metric space and A⊂ M is said to be compact, if any open cover A supports a finite sub coating. Theorem 1.5. (Fernandez and Torre, 1983) Let M be a metric space, and A ⊂ M is infinite, then A is compact. Definition 1.16. Let M be a metric space, a set A⊂ M is said relatively compact if A is compact.

Mathematical Analysis and Analytical Modeling

4

Definition 1.17. Let M be a metric space and A⊂M be its subset. We say that A is sequentially compact if each sequence in convergent subsequence A has in A. Theorem 1.6. (Conway, 1978) (Cover Lebesgue] If (X, d) is a sequentially compact space and if F is an open deck X, then there is an ε> 0 such that if x ∈ X, a set G ∈ F B(X, ε)⊂ G.

Definition 1.18. Sean M of a metric space and a collection F-together to subcontract M, F is said to have the property of intersection infinite if for all {F1, F2,…., Fn} ⊂ F, F1 ∩ F2∩…∩ Fn≠∅. Theorem 1.7. (Shirali and Vasudeva, 2006) All in one compact set is paracompact patio metric is.

Theorem 1.8. (Tamariz, 1982) Let M be a metric space, a set A ⊂ M is compact if and only if, each collection F∩A closed subsets with the property of finite intersection has {F: F∈ F} ≠∅. Corollary 1.1. (Shirali and Vasudeva, 2006) All metric compact space is complete.

Lemma 1.1. (Iribarren, 2008) [Property Bolzano-Weierstrass] A conjunct-to A of a metric space M is relatively compact if and only if, all succession of elements A supports a convergent partial subsequence (not necessarily in A). Theorem 1.9. (Conway, 1978) If M is a metric space, then the NEXT-s statements are equivalent. • • • •

M is compact. Every infinite set has a point for receipt. M is sequentially compact. M is complete and for each ε>0 there is a finite number of points x1, x2,…, xn M such that M =B(xk, ε).

1.1.2. Connectedness

Definition 1.19. Let M be a metric space, a subset of T M, is called arc joining the points x, y ∈ M if a compact interval [A, b], where b > a and a φ application: [a, b] → M continuous on [a, b] such that φ ([a, b]) = M with φ (a) = x, and φ (b) = y.

Preliminaries

5

Definition 1.20. Let M be a metric space, a subset A of M, is called path connected if for each pair x, y ∈ A there is an arc in A connecting points x, y.

Definition 1.21. Let M be a metric space and A ⊂M or not. A is disconnected if there are two sets S and T in the subspace A such that • S, T ≠∅ and S ∩ T =∅. • S, T are open sets in A. • TO = S∪ T. If A is not unconnected, we say that A is connected.

Proposition 1.2. (Iribarren, 2008) Let M be a metric space, M is connected if and only if, the only sets of M that are simultaneously open and closed are the ∅ and M. Definition 1.22. Let M be a metric space and A ⊂ M or not. It is said that A is a domain, if A is open and connected.

Definition 1.23. Let M be a metric space and A ⊂ M or not. It is said that A is a continuous, if A is compact and connected.

Definition 1.24. Let M be a metric space, M is said locally connected if for each point x ∈ S M and x all around, there is a T x environment such that T ⊂ S and T connected.

1.2. COMPLEX VARIABLE

In this section the complex number is studied, the basic operations are defined: addition, subtraction, multiplication, and division of complex numbers. In addition, the definition of module, conjugate complex number is given. Finally, give some other properties of complex numbers and some important results. Definition 1.25. A complex number z is an ordered pair (a, b) with a, b ∈R, where a is called the real part of z and b is called the imaginary part of z. We denote this z and b a = Re = Im z. The set of complex numbers is denoted by C. Two complex numbers, zone = (X1, Y2) and Z2 = (X2, Y2) Are equal if and only if, its real and imaginary parts are equal, i.e., x1 = x2 and y1 = y2.

Definition 1.26. Sean z1 = (X1, Y1) and Z2 = (X2, Y2) Complex numbers, where x1, x2, Y1, Y2∈R, the following operations: •

z1 ± z2= (X1 ± x2, Y1 ±2).

Mathematical Analysis and Analytical Modeling

6

• z1z2= (X1x2 – X1Y2, X1Y2+ X2Y1). Given v, w, and z complex numbers, the following field properties are met: • • • • • •

•

Commutativity of addition: v + w = w + v. Commutativity of multiplication: vw = wv. Associativity of the sum (v + w) = v + z + (w + z). Associativity multiplication: v (wz) = (vw) z. Distributive law: (v + w) =vz + wz. There complex number zero, 0 = (0, 0) such that z + z = 0 for each z∈C and the complex number, 1 = (1, 0) such that 1z = z for each z∈C. For each z = (x, y) with x, y ∈ C x o y not zero inverse supports,

x −y 2 2 which is given the form z = ( x + y , x + y 2 ) Proposition 1.3. Assembly (C, +, ·) with addition and multiplication from Definition 1.26 forms a field. Definition 1.27. A complex number z of the form z = (0, y), where y is a real number, is a pure complex number. The complex number (0, 1) is called the imaginary unit and is denoted by the symbol i: i = (0, 1). The number (0, y) can be considered as the product of the real number y = (y, 0) and the imaginary unit (0, 1), (Y, 0) · (0, 1) = (y · 0 to 0 · 1 and 0 · 1 + 0). We can then be written as: (0, y) = i. Multiplying the imaginary unit itself we have: i · i= (0, 1) (0, 1) = (0 · 0–1 · 1 0 · 1 + 1 · 0) = (–1, 0), That is, i2 = –1. Definition 1.28. Algebraic form of a complex number is z = (x, y) = (x, 0) + (0, y) where x, y ∈ R, where z = x + iy. –1

2

Definition 1.29. Let z= a + bi , with a, b ∈ R , a number resort, conjugate of z as z defined as z= a − bi , and it is denoted z .

Definition 1.30. Let = a + bi , with a, b ∈ R , a number resort, z module

Preliminaries

z is defined by=

= zz

7

x2 + y 2

Theorem 1.10. (Palka, 1990) The absolute value and conjugate or module. A complex number satisfying the following properties. Let z, w∈ C, we have: • | zw | = | z | | w |. • | z + w | ≤ | z | + | w |. • | z + w | ≥ || z | – | w ||. • | z/w | = | z | /|w|, with w ≠ 0. • |Re z | ≤ | z |, |Im z | ≤ | z | • | z | = | Z | z z = z .2 • Re z = (z + z )/2, Im z = (z – z )/2i. Definition 1.31. Trigonometric form of a complex number z = x + iy is given by, z = r (cos θ + isinθ ), where x = r cosθ, y = r sinθ and r = | z |. In the definition 1.31 at θ angle is called the argument of z and is denoted by θ = arg z, this is the measured counterclockwise to clockwise angle, thus a periodicity of 2π taking, so z is represented as: z θ

= R (cos z (θ ± 2nπ) + i sin (θ ± 2nπ)), = arg z, r = | z |, n = 0, 1, 2,… Another way to express the complex number z = r (cos θ + θ i no), is using the exponential form. If θ is a real number, we define the expression eiθ called Euler’s formula as: eiθ= R (cos θ + i sin nθ )

Definition 1.32. Exponential form of a complex number z = r (cos θ + θ i sin) is given by: z = reiθ or

z = | z | eiargz.

A expression reiθ It is known as the polar form of complex number z = x + iy, and θ is called the argument of z. Definition 1.33. Since n ∈ N, the complex number w = z1/n is called z.

Therefore, z is a complex number, if wn= z.

Theorem 1.11. (Antimirov et al., 1998). A different complex number

z = r (cos θ + I sin θ ) of zero, has exactly n different roots, which are given by the formula

8

Mathematical Analysis and Analytical Modeling

Definition 1.34. The exponential function, ez, by the expression ez= ex+ i= ex(cos y + i sin y). Of Euler’s formula, y = nθ doing, we obtain the following formula known as Moivre formula, (cos θ + i sin θ)n = cos + i sin nθ.

1.2.1. Extended Plane and Its Spherical Representation Often in complex analysis functions, they are approaching infinity or take infinite value when the variable approaches a given point are used. To discuss this situation, we introduce the extended plane which is C∪ {∞} ≡ C1. Also, we introduce a distance function C1 to discuss the properties of continuous functions assuming the infinite value. To achieve this and give a specific drawing from C1, representation to C1 as the unit sphere in R3, this is, S = {(x1, x2, x3) ∈ R3| x21 + x22 + x2. 3 = 1}, where S is known as Riemann sphere.

Let N = (0, 0, 1); i.e., N is the North Pole S. In addition, we identified C with {(x1, x2, 0): x1, x2∈R} C then cut along the S Ecuador. For each point z∈ we consider the straight line C in R3 passing z in N. This intersects the sphere at one point, Z ≠ N. If | z | > 1 then Z is in the north of the area. If | z | 0 there is some δ> 0 such that ∀z ∈ TO, 0