Stochastic Processes and Functional Analysis: In Celebration of M.M. Rao's 65th Birthday 9781003067597, 9780824798017, 0824798015

Table of contents :
Cover......Page 1
Half Title......Page 2
Title Page......Page 8
Copyright Page......Page 9
Preface......Page 10
Table of Contents......Page 12
Contributors......Page 14
Biography of M. M. Rao......Page 16
Published Writings of M. M. Rao......Page 18
Family Tree of Mathematical Ancestors......Page 23
Ph.D. Theses Completed Under the Direction of M. M. Rao and His Students......Page 24
Some Problems of Real and Stochastic Analysis Arising from Applications......Page 28
Quasi-periodic Solutions of Hamiltonian Evolution Equations......Page 44
Scaling Limits for Lattice Gas Models......Page 66
Multivariate Distributions with Gaussian Conditional Structure......Page 72
The Minimal Projection from L1 onto πn......Page 88
“Proofs” and Proofs of the Eckart-Young Theorem......Page 98
An Analytic Semigroup Associated to a Degenerate Evolution Equation......Page 112
Degenerate Nonlinear Parabolic Problems: The Influence of Probability Theory......Page 128
An Application of Measure Theory to Perfect Competition......Page 140
Dilations of Hilbert-Schmidt Class Operator-Valued Measures and Applications......Page 150
Transient Solution of the M/M/1 Queueing System via Randomization......Page 164
A Characterization of Hida Measure......Page 174
New Results in the Simplex Method in Linear Programming......Page 180
An Estimate of the Semi-stable Measure of Small Balls in Banach Spaces......Page 198
Nonsquare Constants of Orlicz Spaces......Page 206
Recursive Multiple Wiener Expansion for Nonlinear Filtering of Diffusion Processes......Page 226
A Berry-Esseen Type Estimate for Hilbert Space Valued U-statistics and On Bootstrapping Von Mises Statistics......Page 236
On the Strong Form of the Faber Theorem......Page 242
Nonlinear Filtering Theory for Stochastic Reaction-Diffusion Equations......Page 246
An Operator Characterization of Oscillatory Harmonizable Processes......Page 262
Operator Algebraic Aspects for Sufficiency......Page 272
Nonlinear Parabolic Equations, Favard Classes, and Regularity......Page 280
Index......Page 292

Citation preview

stochastic processes and functional analysis

PURE AND APPLIED MATHEMATICS A Program of Monographs, Textbooks, and Lecture Notes EXECUTIVE EDITORS Earl J. Taft Rutgers University New Brunswick, New Jersey

Zuhair Nashed University o f Delaware Newark, Delaware

EDITORIAL BOARD M. S. Baouendi University o f California, San Diego Jane Cronin Rutgers University Jack K. Hale Georgia Institute o f Technology

Anil Nerode Cornell University Donald Passman University o f Wisconsin, Madison Fred S. Roberts Rutgers University

S. Kobayashi University o f California, Berkeley

Gian-Carlo Rota Massachusetts Institute o f Technology

Marvin Marcus University o f California, Santa Barbara

David L. Russell Virginia Polytechnic Institute and State University

W. S. Massey Yale University

Walter Schempp Universitat Siegen

Mark Teply University o f Wisconsin, Milwaukee

LECTURE NOTES IN PURE AND APPLIED MATHEMATICS

1. 2.

N. Jacob son, Exceptional Lie Algebras L.-A . Lindahl a n d F. Poulsen, Thin Sets in H arm onic Analysis

3.

/. S ata ke, C lassification Th eo ry of Sem i-S im ple A lgebraic Groups

4.

F. H irzebruch, W. D. N e w m a n n , a n d S. S. Koh, D iffere n tiab le M a n ifo ld s a n d Q uadratic Form s

5. 6.

/. C havel', Riem annian S ym m etric Spaces of Rank One R. B. Burckel, C haracterization of C (X ) A m ong Its Subalgebras

7.

B. R. M c D o n a ld ', A . R. M a g id , a n d K. C. S m ith, Ring Th eo ry:

Proceedings of the O klahom a

C onference 8.

Y .-T. Siu, Techniques of Extension on A nalytic Objects

9.

S. R. Caradus, W. £

P faffen berger, a n d B. Yood, Calkin A lgebras and A lgebras of O perators

10.

on Banach Spaces £ O. Roxin, P .-T. Liu, a n d R. L. S ternberg, D ifferential G am es and Control Th eo ry

11.

M . O rzech a n d C. Sm all, Th e Brauer Group of C o m m u tative Rings

12.

S. Thornier, Topology and Its A pplications

1 3.

J. M . Lopez a n d K. A . Ross, Sidon Sets

14.

W. W. C o m fo rt a n d S. N egrepontis, Continuous P seudom etrics

15. 16.

K. M cK enn on a n d J. M . Robertson, Locally C onvex Spaces M . C arm eli a n d S. M alin, R epresentations of the R otation

and

Lorentz

Groups:

An

Introduction 17.

G. B. S e/igm an, Rational M e th o d s in Lie Algebras

18.

D. G. de Figueiredo, Functional A nalysis:

Proceedings of the Brazilian M a th e m atical S ociety

S ym posium 19.

L.

Cesari, R. Kannan, a n d J. D.

Equations:

Schuur,

Nonlinear Functional

Analysis and

D ifferential

Proceedings of th e M ichig an S tate U niversity C on feren ce

20.

J. J. S ch a ffe r, G eo m etry of Spheres in Norm ed Spaces

21.

K. Yano a n d M . Kon, A n ti-In varian t S ubm anifolds

22.

W. V. Vasconce/os, T h e Rings of Dim ension T w o

23.

R. £

24.

S. P. Franklin a n d B. V. S. Thom as, Topology:

Chandler, H au sdo rff C om pactification s Proceedings of the M e m p his S tate U niversity

C onference 25.

S. K. Jain, Ring Theory:

26.

B. R. M c D o n a ld an d R. A . M orris, Ring Th eo ry II:

Proceedings of the Ohio U niversity C onference Proceedings of the Second O klahom a C on­

feren ce 27.

R. B. M u ra a n d A . R hem tulla, Orderable Groups

28.

J. R. G raef, S tability of D ynam ical S ystem s: Th eo ry and A pplications

29. 30. 31. 32. 33.

H .-C . W ang, H om ogeneous Branch Algebras £ O. Roxin, P .-T . Liu, a n d R. L. S ternberg, D ifferential G am es and Control Th eo ry II R. D. P orter, Introduction to Fibre Bundles M . A ltm a n , C on tracto rs and C on tracto r D irections T h eo ry and A pplications J. S. Golan, D ecom position and Dim ension in M o du le C ategories

34. 35.

G. F a irw eath er, Finite E lem ent Galerkin M e th o d s for D ifferential Equations J. D. Sally, N um bers of G enerators of Ideals in Local Rings

36.

S. 5 . M iller, C om plex Analysis:

37. 38. 39.

R. Gordon, R epresentation Th eo ry of A lgebras: Proceedings of th e Philadelphia C onference M . Goto a n d F. D. Grosshans, S em isim ple Lie A lgebras A . I. A rru d a, N. C. A . da Costa, a n d R. Chuaqui, M a th e m a tic a l Logic: Proceedings of the

Proceedings of the S .U .N .Y . Brockport C on feren ce

40.

£

Van O ystaeyen, Ring Th eo ry:

41.

£

Van O ystaeyen a n d A .

First Brazilian C onference Proceedings of th e 1 9 7 7 A n tw e rp C on feren ce

Verschoren, R eflectors and Localization:

A pplication to S heaf

T h eo ry 42.

M . S aty an a ray an a, Positively Ordered Sem igroups

43.

D. L Russell, M a th e m a tic s of Finite-D im ensional Control S ystem s

44.

P .-T . Liu a n d £

Roxin, D ifferential Gam es and Control T h eo ry III: Proceedings of the Third

Kingston C onferen ce, Part A 45.

A . G eram ita a n d J. Seberry, Orthogonal Designs:

46.

J. Cigler, V. Losert, a n d P. M icho r, Banach M o du les and Functors on C ategories of Banach S paces P .-T. Liu a n d J. G. S utinen, Control Th eo ry in M a th e m a tic a l Econom ics: Proceedings of the Third Kingston C on feren ce, Part B

47.

Q uadratic Forms and H adam ard M a trices

48.

C. Byrnes, Partial D ifferential Equations and G eo m etry

49.

G. K /am bauer, Problems and Propositions in A nalysis

50.

J. K no pfm acher, A nalytic A rithm etic of A lgebraic Function Fields

51.

F. Van O ystaeyen, Ring Theory:

52. 53.

B. Kadem , Binary Tim e Series J. B arros-N eto a n d R. A . A rtino , Hypoelliptic B oundary-V alue Problems

54.

R.

55.

Equations in Engineering and Applied Science B. R. M c D o n a ld , Ring Th eo ry and Algebra III: Proceedings of the Third O klahom a C onference

56.

J. S. Golan, S tructure Sheaves Over a N o n co m m u ta tive Ring

57.

L.

S ternberg,

A.

J.

Proceedings of the 1 9 7 8 A n tw erp C onference

K alinow ski,

a n d J.

S.

Papadakis,

Nonlinear

Partial

D ifferential

T. V. N arayana, J. G. Williams, an d R. M . M a th s en , C om binatorics, R ep resentatio n Theory and S tatistical M e th o d s in Groups: Y O U N G D A Y Proceedings

58.

T. A . Burton, M odeling and D ifferential Equations in Biology

59.

K. H. Kim a n d F. W. Roush, Introduction to M a th e m atical Consensus Th eo ry

60.

J. Banas a n d K. Goebel, M easu res of N oncom pactness in Banach Spaces

61.

0 . A . Nielson, D irect Integral Theory

62.

J. E. S m ith, G. 0 . Kenny, an d R. N. Ball, Ordered Groups:

Proceedings of the Boise S tate

C onference 63.

J. Cronin, M a th e m a tic s of Cell Electrophysiology

64.

J. W. B rew er, P ow er Series Over C o m m u tative Rings

65.

P. K. K am than an d M . Gupta,

66.

T. G. M cLaughlin, Regressive Sets and the Th eo ry of Isols

67.

T. L. H erdm an, Equations

68.

R. D raper, C o m m u tative A lgebra: A nalytic M e tho ds

69.

S.

M.

S equence Spaces and Series

Rankin III, an d H.

W.

S tech,

Integral and

Functional D ifferential

W. G. M c K a y an d J. P atera, Tables of D im ensions, Indices, and Branching Rules for Repre­ sentations of Sim ple Lie A lgebras

70.

R. L. D e v a n e y an d Z. FI. N itecki, Classical M e chanics and D ynam ical System s

71.

J. Van Geel, Places and V alu atio ns in N o n co m m u ta tive Ring Th eo ry

72.

C. Faith, Injective M odules and Injective Q uotient Rings

73.

A . Fiacco, M a th e m a tic a l Program m ing w ith D ata P erturbations I

74.

P. S chultz, C. Praeger, an d R. Sullivan, A lgebraic Structures and A pplications: Proceedings of the First W e s te rn A ustralian C onference on Algebra

75.

L Bican, T. Kepka, a n d P. N em ec, Rings, M o du les, and Preradicals

76.

D. C. K ay a n d M . Breen, C onvexity and Related C om binatorial G eom etry:

Proceedings of the

Second U niversity of O klahom a C onference 77. 78. 79. 80. 81.

P. F letch er a n d W. F. Lindgren, Q uasi-U niform Spaces C.-C. Yang, Factorization Th eo ry of M ero m o rp hic Functions O. Taussky, Te rn ary Q uadratic Forms and Norm s S. P. Singh an d J. H. Burry, Nonlinear Analysis and A pplications K. B. H an nsg en, T. L. H erdm an, H. W. S tech, an d R. L. W heeler, V o lterra D ifferential Equations

82.

N. L. Johnson, M . J. K allaher, an d C. T. Long, Finite G eom etries: Proceedings of a C on fer­

83. 84.

ence in Honor of T . G. Ostrom G. /. Z a p ata, Functional Analysis, H olom orphy, and A pp ro xim atio n Th eo ry S. Greco an d G. Valla, C om m u tative A lgebra: Proceedings of the T ren to C onference

85.

A . V. Fiacco, M a th e m atical Program m ing w ith D ata Perturbations II

and Functional

86.

J.-B . H iriart-U rru ty, W. Oettli, an d J. S toer, O p tim ization: Th eo ry and A lgorithm s

87.

A . Figa Ta/am anca a n d M . A . Picardello, H arm onic Analysis on Free Groups

88. 89. 90.

M . H arada, Factor C ategories w ith A pplications to D irect D ecom position of M odules V. /. Istrcifescu, S trict C onvexity and C om plex S trict C onvexity V. Laksh m ikantham , Trends in T h eo ry and Practice of Nonlinear D ifferential Equations

91.

H. L. M a n o c h a a n d J. B. S rivastava, A lgebra and Its A pplications

92.

D.

V. C hudnovsky a n d G.

V. Chudnovsky, Classical and Q uantum M o dels

Problem s 93.

J. W. Longley, Least Squares C om putations Using O rthogonalization M e th o d s

94. 95.

L. P. de A lca n ta ra, M a th e m atical Logic and Formal System s C. £ A u/I, Rings of C ontinuous Functions

96.

R. Chuaqui, A nalysis, G eo m etry, and Probability

97. 98.

L. Fuchs a n d L. S alce, M odules O ver V alu atio n D om ains P. Fisch er a n d W. R. S m ith, Chaos, Fractals, and D ynam ics

99.

W. B. P o w e ll a n d C. Tsinakis, Ordered A lgebraic S tructures

and A rith m etic

100.

G. M . Rassias an d T. M . Rassias, D ifferential G eo m etry, Calculus of V ariatio ns, and Th eir A pplications

101.

R.-E. H o ffm a n n an d K. H. H o fm an n, C ontinuous Lattices and Their Applications

102.

J. H. Lightbourne III an d S. M .

Rankin III,

Physical M a th e m a tic s

and

Nonlinear Partial

D ifferential Equations 103. 104.

C. A . B aker an d L. M. B atten, Finite G eom etries

J. W. B rew er, J. W. Bunce, an d F. S. Van Vleck, Linear S ystem s Over C o m m u tative Rings

105.

C. M c C ro ry a n d T. Shifrin, G eo m etry and Topology:

106.

D. W. Kueker, E. G. K. Lopez-Escobar, a n d C. H. S m ith, M a th e m a tic a l Logic and T h eo retical

M a nifo lds, V arietie s, and Knots

107.

B.-L. Lin an d S. Sim ons, Nonlinear and C onvex Analysis:

108.

S. J. Lee, O perator M e th o d s for Optim al Control Problems

C om puter Science Proceedings in Honor of Ky Fan

109.

V. Laksh m ikantham , Nonlinear Analysis and A pplications

110.

S. F. M c C o rm ick , M ultigrid M e th o d s: T h eo ry, A pplications, and S upercom puting

111.

M . C. Tangora, C om puters in Algebra

112.

D. V. C hudnovsky an d G. V. Chudnovsky, Search Th eo ry:

113.

D. V. C hudnovsky an d R. D. Jenks, C om pu ter Algebra

114.

M . C. Tangora, C om puters in G eo m etry and Topology

115.

P. Nelson, V. Faber, T. A . M a n te u ffe l, D. L. S eth, an d A . B. W hite, Jr., T ran sp o rt T h eo ry, Invariant Im bedding, and Integral Equations:

Som e R ecent D evelop m ents

Proceedings in Honor of G. M . W in g 's 6 5 th

Birthday 116.

P. Clem ent, S. Invernizzi, £ M itidieri, a n d I. I. Vrabie, Sem igroup Th eo ry and A pplications

117.

J. Vinuesa, Orthogonal Polynom ials and Their A pplications:

Proceedings of th e International

Congress 118.

C. M . D aferm os, G. Ladas, an d G. Papanicolaou, D ifferential Equations:

Proceedings of the

E Q U A D IFF C onference 119.

£

0 . Roxin, M o dern Optim al Control:

A C onference in Honor of Solom on Lefschetz and Jo ­

seph P. Lasalle 120.

J. C. Diaz, M a th e m a tic s for Large Scale C om puting

121.

P. S. M ilo je v it, Nonlinear Functional A nalysis

122.

C. Sadosky, A nalysis and Partial D ifferential Equations:

A C ollection of Papers D edicated to

M ischa Cotlar 123.

R. M . S hortt, General Topology and A pplications:

Proceedings of th e 1 9 8 8 N orth east Con­

feren ce 124.

R. Wong, A sym ptotic and C om putational A nalysis: O lver's 6 5 th Birthday

C on feren ce in Honor of Frank W . J.

125.

D. V. C hudnovsky an d R. D. Jenks, C om puters in M a th e m a tic s

126.

W. D. Wallis, H. Shen, W. Wei, a n d L. Zhu, C om binatorial Designs and A pplications

127.

S. Elaydi, D ifferential Equations:

128.

G. Chen, £ B. Lee, W. Littm an, a n d L. M arku s, D istributed P aram eter Control S ystem s:

129. 130.

Trends and A pplications W. N. Everitt, Inequalities: Fifty Y ears On from H ardy, Littlew ood and Polya H. G. K aper a n d M . Garbey, A sym p to tic Analysis and the N um erical Solution of Partial D iffe r­ ential Equations

131. 132.

S tability and Control

O. A rino, D. £ A xelro d, a n d M . K im m el, M a th e m a tic a l Population D ynam ics: the Second International C onference S. Coen, G eo m etry and C om plex Variables

N ew

Proceedings of

133.

J. A . Goldstein, F. Kappel, a n d W. S chapp ach er, D ifferential Equations w ith A pplications in Biology, Physics, and Engineering

134.

S. J. A nd im a, R. K opperm an, P. R. M isra, J. Z. R eichm an, a n d A . R. Todd, General Topology

135.

P C lem ent, £

and Applications M itidieri, B. de P agter,

Sem igroup Th eo ry and Evolution Equations:

The

Second International C onference 136. 137. 138.

K. Jarosz, Function Spaces J. M . Bayod, N. D e G rand e-D e K im pe, a n d J. M a rtin e z-M a u ric a , p -ad ic Functional Analysis G. A . A n astassio u, A pp ro xim atio n Th eo ry: Proceedings of th e S ixth S ou theastern A pp ro xi­ m ation Theorists A nnual C onference

139.

R. S. Rees, Graphs, M a tric e s , and Designs: Festschrift in Honor of N orm an J. Pullm an

140.

G. A b ram s, J. H aefn er, a n d K. M . R an g as w am y, M e th o d s in M o du le T h eo ry

141.

G. L. M u llen a n d P. J .-S . Shiue, C om m unications and Com puting

142.

M . C. Jo sh i a n d A . V. Balakrishnan, M a th e m atical T h eo ry of Control: Proceedings of the In ternational C onference

Finite

Fields,

Coding

T h eo ry,

and

A d vances

in

143.

G. K om atsu a n d Y. S akane, C om plex G eom etry: Proceedings of th e O saka International Con­ fe ren ce

144.

/. J. B akelm an, G eom etric A nalysis and Nonlinear Partial D ifferential Equations

145.

T. M a b u c h i a n d S. M u k a i, Einstein M e trics and Y a n g -M ills C onnections: Proceedings of the 2 7 th Taniguchi International S ym posium

146.

L. Fuchs a n d Ft. Gobel, Abelian Groups: Proceedings of the 1 9 9 1 C uracao

147.

A . D.

C on feren ce

148.

G. D ore, A . Favini, E. O brecht, a n d A . Venni, D ifferential Equations in Banach S paces

Pollington an d W. M o ra n , N um ber T h eo ry w ith an Emphasis on th e M a rk o ff S pectrum

149.

T. W est, C ontinuum Th eo ry and D ynam ical S ystem s

150.

K. D. B ierstedt, A . Pietsch, W. Ruess, a n d D. Vogt, Functional A nalysis

151.

K. G. Fischer, P. Loustaunau, J. Shapiro, E. L. Green, a n d D. Farkas, C om pu tation al A lgebra

152.

K. D. E /w orth y, W. N. E veritt, a n d E. B. Lee, D ifferential Equations, D ynam ical S ystem s, and

153.

P .-J. Cahen, D. L. Costa, M . Fontana, a n d S.-E. Kabbaj, C o m m u tative Ring T h eo ry

154.

S. C. C ooper a n d W. J. Thron, Continued Fractions and O rthogonal Functions: Th eo ry and

155.

P. C lem en t an d G. Lum er, Evolution Equations, Control T h eo ry, and B iom athem atics

156.

M . G yllenberg a n d L. Persson, A nalysis, A lgebra, and C om puters in M a th e m a tic a l R esearch:

Control Science

A pplications

Proceedings of th e T w e n ty -F irs t Nordic Congress of M a th e m aticia n s 157.

W. 0 . Bray, P. S. MHojevic a n d 6 . V. Stanojevid, Fourier Analysis: A nalytic and G eom etric A sp ects

158. 159.

J. Bergen a n d S. M o n tg o m ery, A dvances in H opf A lgebras A . R. M a g id , Rings, Extensions, and C ohom ology

160.

N. H. P avel, Optim al Control of D ifferential Equations

161.

M.

Ik a w a ,

Spectral

and

S cattering

Th eo ry:

Proceedings

of the

Taniguchi

In ternational

W orksh op 162.

X . Liu a n d D. S iegel, Com parison M e th o d s and Stability Th eo ry

163. 164. 165.

J.-P . Zolesio, Boundary Control and V ariatio n M . Krizek, P. N eittaan m a ki, a n d R. S ten berg, Finite Elem ent M e th o d s: Fifty Y ears of th e Courant E lem ent G. D a P rato a n d L. Tubaro, Control of Partial D ifferential Equations

166.

E. Ballico, Projective G eo m etry w ith A pplications

167.

M . Costabel, M . D auge, a n d S. N icaise, Boundary V alu e Problems and Integral Equations in N onsm ooth D om ains

168.

G. Ferreyra, G. R. Goldstein, a n d F. N eu brand er, Evolution Equations

169. 170. 171.

S. H u g g e tt, T w is to r Th eo ry H. Cook, W. T. Ingram , K. T. Kuperberg, A . Le/ek, a n d P. M in e, C ontinua: W ith th e H ouston Problem Book D. F. A nd erson a n d D. E. D obbs, Zero-D im ensional C o m m u tative Rings

172.

K. Jarosz, Function Spaces: The Second C onference

173.

V. A n c o n a , E. Ballico, a n d A . Silva, C om plex Analysis and G eo m etry

174. 175.

E. C asas, Control of Partial D ifferential Equations and A pplications N. K alton, E. S aab, a n d S. M o n tg o m e ry -S m ith , In teraction B etw een Functional A nalysis, H ar­

176. 177.

m onic A nalysis, and Probability Z. D eng, Z. Liang, G. Lu, a n d S. Ruan, D ifferential Equations and Control Th eo ry P. M arcellin i, G. Talenti, a n d E. Vesentini, Partial D ifferential Equations and A pplications:

178.

A . K artsatos, T h eo ry and A pplications of Nonlinear O perators of A ccretive and M o n o to n e

C ollected Papers in Honor of Carlo Pucci Ty p e 179.

M . M a ru y a m a , Moduli of V e c to r Bundles

180.

A . U rsini a n d P. A g/iand , Logic and A lgebra

181.

X . H. Cao, S. X . Liu, K. P. Shum , a n d C. C. Yang, Rings, Groups, and A lgebras

182.

D. A rn o ld a n d R. M . R an g as w am y, A belian Groups and M odules

183.

S. R. C h a kra varth y a n d A . S. A lfa , M a trix -A n a ly tic M e th o d s in S tochastic M o dels

184.

J. E. A n d ersen , J. D upont, H. Pedersen, a n d A . S w a n n , G eo m etry and Physics

185.

P .-J.

Cahen, M . Fontana, E. H ou sto n, a n d S.-E. Kabbaj, C o m m u tative Ring Th eo ry: Pro­

ceedings of th e II In ternational C onference 186.

J. A .

Goldstein, N. E. G retsky, a n d J. J.

Uhl, J r., S tochastic Processes and Functional

Analysis

Additional Volumes in Preparation

stochastic processes and functional analysis in celebration of M.M. Rao’s 65th birthday

edited by Jerome A. Goldstein University of Memphis Memphis, Tennessee

Neil E. Gnetsky University of California-Riverside Riverside, California

J. J. Uhl, Jr. University of Illinois

Marcel Dekker, Inc.

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Library of Congress Cataloging-in-Publication Data Stochastic processes and functional analysis : in celebration of M. M. Rao’s 65th birthday / edited by Jerome A. Goldstein, Neil E. Gretsky, John Jerry Uhl, Jr. p. cm. — (Lecture notes in pure and applied mathematics : v. 186) Held at the Univ. of Calif. — Riverside, Nov. 18-20, 1994. “Published writings of M. M. Rao”: p. ISBN 13: 978-0-8247-9801-7 1. Stochastic processes— Congresses. 2. Functional analysis-Congresses. I. Rao, M. M. (Malempati Madhusudana), II. Goldstein, Jerome A. III. Gretsky, Neil E. IV. Uhl, J. J. (J. Jerry) V. Series. QA274.A1S7665 1997 515'.7— dc21 96-48137 CIP

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Preface

A conference in Modern Analysis and Probability on the occasion of his 65th birthday was held in honor of M. M. Rao at the University of California, Riverside. Over a hundred friends, colleagues, students, and other mathematicians attended during the three-day meeting. The Mathematics Department provided copious amounts of coffee and doughnuts. An oncampus dinner was held on a Friday night and an off-campus dinner was held Saturday night following a gala reception at the home of M. M. and his wife, Durgamba. Support for the conference was supplied by the College of Natural and Agricultural Sciences at the University of California-Riverside, the Mathematics Department at the University of CaliforniaRiverside, and the National Science Foundation. This festschrift volume contains most of the talks given at the conference as well as several that were contributed later. The beginning portions of the book include a biography of M. M. Rao, a bibliography of his published writings, an ancestral mathematical family tree, and a list of Ph.D. theses written under Rao and his students. The talks at the conference included four keynote addresses by Rao, Jean Bourgain, S. R. S. Varadhan, and Michael Crandall. All but Crandall’s talk are contained here; Crandall’s talk will appear in a paper that will be published elsewhere. Rao’s paper is an account of that portion of his work which originated in problems arising in applications. It is organized by area and features the work of his students as much as his own. The breadth and depth of Rao’s mathematical work and its impact on analysis, probability, and stochastic processes can be seen not only by what is included in this paper but also by the portion of his bibliography which is not in this paper. The editors enjoyed this paper immensely. Even as good as the paper is, it does not capture the charm and the emotion with which the talk was given. Jean Bourgain’s paper is a long, densely written survey (an “expose” in his terminology) of persistency of quasi-periodic solutions of linear or integrable partial differential equations after Hamiltonian perturbation. Much of the original work is due to Bourgain and is not in print elsewhere. The talk given by Varadhan reported on joint work with H. T. Yau concerning scaling limits for lattice gas models. This provides a way to give a simplified description of the state of a large system of interacting particles which is evolving in time. The results typify recent deep research involving hydrodynamic limits, which establish that nonlinear partial differential equations govern many large particle systems in the limit. The remaining eighteen papers are original contributions in probability and statistics, stochastic processes, Banach space theory, measure theory, and differential equations—both deterministic and stochastic. Many other people attended the conference who did not give talks for one reason or another. Although we cannot list all of them (our sincere apologies) we would like to mention two esteemed intellectual colleagues of M. M. Rao, Mannie Parzen and Howard Tucker, as well as two former students, William Kraynek and Marc Mehlman. Jerome A. Goldstein Neil E. Gretsky J . J. Uhl, Jr. iii

Contents

Preface

iii

Contributors

vii

Biography of M. M. Rao

ix

Published Writings of M. M. Rao

xi

Family Tree of Mathematical Ancestors

xvi

Ph.D. Theses Completed Under the Direction of M. M. Rao and His Students

xvii

Some Problems of Real and Stochastic Analysis Arising from Applications Malempati M. Rao

1

Quasi-periodic Solutions of Hamiltonian Evolution Equations Jean Bourgain

17

Scaling Limits for Lattice Gas Models S.R.S. Varadhan and H.T. Yau

39

Multivariate Distributions with Gaussian Conditional Structure Barry Arnold and Jacek Wesolowski

45

The Minimal Projection from L1 onto nn Bruce L. Chalmers and Frederic T. M etcalf

61

“Proofs” and Proofs of the Eckart-Young Theorem John S. Chipman

71

An Analytic Semigroup Associated to a Degenerate Evolution Equation Angelo Favini, Jerome A. Goldstein, and Silvia Romanelli

85

Degenerate Nonlinear Parabolic Problems: The Influence of Probability Theory Jerome A. Goldstein, Chin-Yuan Lin, and Kunyang Wang

101

An Application of Measure Theory to Perfect Competition Neil E. Gretsky, Joseph M. Ostroy, and William R. Zame

113

Dilations of Hilbert-Schmidt Class Operator-Valued Measures and Applications Yuichiro Kakihara

123

Transient Solution of the M l M l I Queueing System via Randomization Alan Krinik, Daniel Marcus, Dan Kalman, and Terry Cheng

137

A Characterization of Hida Measure Hui-Hsiung Kuo

147

New Results in the Simplex Method in Linear Programming Roger Pedersen

153

v

vi

Contents

An Estimate of the Semi-stable Measure of Small Balls in Banach Spaces Balram Rajput

171

Nonsquare Constants of Orlicz Spaces Zhongdau Ren

179

Recursive Multiple Wiener Expansion for Nonlinear Filtering of Diffusion Processes Sergey Lototsky and Boris Rosovskii

199

A Berry-Esseen Type Estimate for Hilbert Space Valued [/-statistics and On Bootstrapping Von Mises Statistics V. V. Sazonov

209

On the Strong Form of the Faber Theorem Boris Shektman

215

Nonlinear Filtering Theory for Stochastic Reaction-Diffusion Equations Stephen L. Hobbs and S. S. Sritharan

219

An Operator Characterization of Oscillatory Harmonizable Processes Randall J. Swift

235

Operator Algebraic Aspects for Sufficiency Makato Tsukada

245

Nonlinear Parabolic Equations, Favard Classes, and Regularity Gisele Ruiz Goldstein

253

Index

265

Contributors

Barry C. Arnold Jean Bourgain Jersey

Department of Statistics, University of California, Riverside, California

School of Mathematics, Institute for Advanced Study, Princeton, New

Bruce L. Chalmers California

Department of Mathematics, University of California, Riverside,

Terry Cheng Department of Mathematics, Irvine Valley Community College, Irvine, California John S. Chipman Minnesota Angelo Favini

Department of Economics, University of Minnesota, Minneapolis,

Dipartimento di Matematica, Universita di Bologna, Bologna, Italy

Gisele Ruiz Goldstein Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana, and CERI and Department of Mathematical Sciences, University of Memphis, Memphis, Tennessee Jerome A. Goldstein Memphis, Tennessee Neil E. Gretsky California S. L. Hobbs California

Department of Mathematics, University of California, Riverside,

Naval Command Control and Ocean Surveillance Center, San Diego,

Yuichiro Kakihara California Dan Kalman DC

Department of Mathematical Sciences, University of Memphis,

Department of Mathematics, University of California, Riverside,

Department of Mathematics and Statistics, American University, Washington,

Alan Krinik Department of Mathematics, California State Polytechnic University, Pomona, California Hui-Hsiung Kuo Louisiana

Department of Mathematics, Louisiana State University, Baton Rouge,

Chin-Yuan Lin Department of Mathematics, University of South Carolina, Columbia, South Carolina, and Department of Mathematics, National Central University, Chang-Li, Republic of China Sergey Lototsky Center for Applied Mathematical Sciences, University of Southern California, Los Angeles, California

viii

Contributors

Daniel Marcus Department of Mathematics, California State Polytechnic University, Pomona, California Frederic T. Metcalf Department of Mathematics, University of California, Riverside, California Joseph M. Ostroy California

Department of Economics, University of California, Los Angeles,

Roger N. Pedersen Pennsylvania Balram S. Rajput Tennessee

Department of Mathematics, The University of Tennessee, Knoxville,

Malempati M. Rao California Zhongdau Ren

Department of Mathematics, Carnegie Mellon University, Pittsburgh,

Department of Mathematics, University of California, Riverside,

Department of Mathematics, University of California, Riverside, California

Silvia Romanelli Dipartimento di Matematica, Universita di Bari, Bari, Italy Boris L. Rozovskii Center for Applied Mathematical Sciences, University of Southern California, Los Angeles, California Y. V. Sazonov Steklov Mathematical Institute, Moscow, Russia, and Department of Mathematics, Hong Kong University of Science and Technology, Hong Kong Boris Shektman S. S. Sritharan California

Department of Mathematics, University of South Florida, Tampa, Florida Naval Command Control and Ocean Surveillance Center, San Diego,

Randall J. Swift Green, Kentucky Makato Tsukada Chiba, Japan

Department of Mathematics, Western Kentucky University, Bowling Department of Information Sciences, Toho University, Funabashi City,

S. R. S. Varadhan Courant Institute of Mathematical Sciences, New York University, New York, New York Kunyang Wang Louisiana

Department of Mathematics, Louisiana State University, Baton Rouge,

Jacek Wesolowski Mathematical Institute, Warsaw University of Technology, Warsaw, Poland H. T. Yau New York

Courant Institute of Mathematical Sciences, New York University, New York,

William R. Zame California

Department of Mathematics, University of California, Los Angeles,

Biography of M. M. Rao

M. M. Rao was born M alempati M adhusudana Rao in the village of Nimmagadda in the state of A ndhra Pradesh in India on June 6 , 1929. After studying at the College of Andhra University and the Presidency College of Madras University, he came to the United States and obtained his Ph.D in 1959 at the University of Minnesota under the supervision of Monroe Donsker (as well as Bernard R. Gelbaum, Leonid Hurwicz, and I. Richard Savage). His first academic appointm ent was at Carnegie Institute of Technology (now called Carnegie Mellon University) in 1959. In 1972, he accepted an offer from University of California, Riverside where he remains even today. Along the way, he has held visiting positions at the Institute for Advanced Study (Princeton), the Indian Statistical Institute, University of Vienna, University of Strassbourg, and the Mathematical Sciences Research Institute (Berkeley). In 1966 he married Durgamba Kolluru in India. They have twin daughters Leela and Uma. M.M. started out in probability and mathem atical statistics but his voracious m athem at­ ical appetite found him branching out into a wide range of m athem atical analysis including stochastic processes, functional analysis, ergodic theory and related asymptotics, differential equations and difference equations. His breadth of interest is mirrored by his students, many of whom are recognized as experts in diverse fields such as measure theory, operator theory, partial differential equations, etc. M.M. has always strived for complete understanding and full generality and has seldom been willing to accept less from others. This attitude has always affected his teaching as well. Many generations of students have found him to be a demanding but truly inspiring teacher and mentor. He has always championed the open door policy for his Ph. D. students and has always put their welfare above his own. I t ’s no accident that, throughout his career, he has had his share of the best available graduate students lined up to work with him. His article in this volume is a remarkable summary of his broad research program and how he is able to view many diverse topics in a naturally unified way. M athematicians rarely get to see in print such an intim ate view of the thought processes of a leading m athem atician and how he involved his students in his work. We are delighted th at M.M. wrote this article for all of us to share. M.M. continues to work on and we hope th at he will be around twenty years from now to write another such article, to enjoy his beloved mathematics, his beloved students, and his beloved family. ix

M. M. Rao

Published Writings of M. M. Rao [1] Note on a remark of Wald, Amer. Math. Monthly 65 (1958), 277-278. [2] Lower bounds for risk functions in estimation, Proc. N at’l Acad, of Sciences 45 (1959), 1168-1171. [3] Estim ation by periodogram, Trabajos de Estadistica 11 (1960), 123-137. [4] Two probability limit theorems and an application, Indagationes M athematicae 23 (1961), 551-559. [5] Theory of lowei; bounds for risk functions in estimation, Mathematische Annalen 143 (1961), 379-398. [6] Consistency and limit distributions of estimators of param eters in explosive stochastic difference equations, Annals of Math. Stat. 32 (1961), 195-218. [7] Some remarks on independence of statistics, Trabajos de Estadistica 12 (1961), 19-26. [8] Remarks on a multivariate gamma distribution, Amer. Math. Monthly 68 (1961), (with P. R. Krishnaiah, 342-346). [9] Theory of order statistics, M athematische Annalen 147 (1962), 298-312. [10] Nonsymmetric projections in Hilbert Space, Pacific J. Math. 12 (1962), 343-357, (with V. J. Mizel). [11] Characterizing normal law and a nonlinear integral equation, J. Math. & Mech. 12 (1963), 869-880. [12] Inference in stochastic processes-I, Teoria Veroyatnastei i ee Primeneniya 8 (1963), 282-298. [13] Some inference theorems in stochastic processes, Bull. Amer. Math. Soc. 68 (1963), 72-77. [14] Discriminant analysis, Annals of Inst, of Stat. Math. 15 (1963), 11-24. [15] Bayes estim ation with convex loss, Annals of Math. Stat. 34 (1963), 839-846, (with M. H. DeGroot). [16] Stochastic give-and-take, J. Math. Anal. & Appl. 7 (1963), 489-498, (with M.H. DeGroot). [17] Projections, generalized inverses, and quadratic forms, J. Math. Anal. & Appl. 9 (1964), 1- 11 , (with J. S. Chipman). [18] Decomposition of vector measures, Proceedings of N at’l. Acad, of Sciences 51 (1964), 771-774. [19] Linear functionals on Orlicz spaces, Nieuw Archief voor Wiskunde 312 (1964), 77-98. [20] The treatm ent of linear restrictions in regression analysis, Econometrica 32 (1964), 198-209, (with J.S. Chipman). [21] Conditional expectations and closed projections, Indagationes M athematicae 27 (1965), 100-112. [22] Smoothness of Orlicz spaces-I and II, Indagationes Mathematicae 27 (1965), 671-680, 681-690. [23] Existence and determ ination of optimal estimators relative to convex loss, Annals of Inst, of Stat. M ath 17 (1965), 113-147. [24] Interpolation, ergodicity, and martingales, J. of Math. & Mech. 16 (1965), 543-567. [25] Inference in stochastic processes-II, Zeitschrift fur Wahrscheinlichkeitstheorie 5 (1966), 317-335. [26] Approximations to some statistical tests, Trabajos de Estadistica 17 (1966), 85-100. [27] Multidimensional information inequalities and prediction, Proceedings of In t’l. Sym­ posium on M ultivariate Anal., Academic Press, (1966) 287-313, (with M.H. DeGroot). xi

xii

Published Writings of M. M. Rao

[28] Convolutions of vector fields and interpolation, Proceedings of N at’l. Acad. Sciences 57 (1967), 222-226. [29] A bstract Lebesgue-Radon-Nikodym theorems, Annali di M atem atica P ura ed Applicata (4) 76 (1967), 107-132. [30] Characterizing Hilbert space by smoothness, Indagationes M athematicae 29 (1967), 132-135. [31] Notes on pointwise convergence of closed martingales, Indagationes M athematicae 29 (1967), 170-176. [32] Inference in stochastic processes-III, Zeitschrift fur Wahrscheinlichkeitstheorie 8 (1967), 49-72. [33] C haracterization and extension of generalized harmonizable random fields, Proceed­ ings N at’l. Acad. Sciences 58 (1967), 1213-1219. [34] Local functionals and generalized random fields, Bull. Amer. Math. Soc. 74 (1968), 288-293. [35] Extensions of the Hausdorff-Young theorem, Israel J. of Math. 6 (1968), 133-149. [36] Linear functionals on Orlicz spaces: General theory, Pacific J. Math. 25 (1968), 553-585. [37] Almost every Orlicz space is isomorphic to a strictly convex Orlicz space, Proceedings Amer. M ath. Soc. 19 (1968), 377-379. [38] Predictions nonlineares et martingales d ’operateurs, Comptes rendus (Academie des Sciences, Paris), Ser. A, 267 (1968) 122-124. [39] Representation theory of multidimensional generalized random fields, Proceedings 2d In t’l. Sympt. M ultivariate Anal., Academic Press (1969), 411-436. [40] O perateurs de moyennes et moyennes conditionnelles, C.R. Acad. Sciences, Paris, Ser. A, 268 (1969), 795-797. [41] Produits tensoriels et espaces de fontions, C.R. Acad. Sci., Paris 268 (1969), 15991601. [42] Stone-Weierstrass theorems for function spaces, J. Math. Anal. 25 (1969), 362-371. [43] Contractive projections and prediction operators, Bull. Amer. Math. Soc. 75 (1969), 1369-1373. [44] Generalized martingales, Proceedings 1st Midwestern Symp. on Ergodic Theory & Prob., Lecture Notes in Math., Springer-Verlag, 160 (1970), 241-261. [45] Linear operations, tensor products and contractive projections in function spaces, Studia Math. 38, 131-186, Addendum 48 (1970), 307-308. [46] Approximately tam e algebras of operators, Bull. Acad. Pol. Sci., Ser. Math. 19 (1971), 43-47. [47] A bstract nonlinear prediction and operator martingales, J. M ultivariate Anal. 1 (1971), 129-157, Erratum , 9. p. 646. [48] Local functionals and generalized random fields with independent values, Teor. Verojatnost., Prem. 16 (1971), 466-483. [49] Projective limits of probability spaces, J. Multivariate Anal. 1 (1971), 28-57. [50] Contractive projections and conditional expectations, J. M ultivariate Anal. 2 (1972), 262-381, (with N. Dinculeanu). [51] Prediction sequences in smooth Banach spaces, Ann. Inst. Henri Poincare, Ser. B, 8 (1972), 319-332. [52] Notes on characterizing Hilbert space by smoothness and smooth Orlicz spaces, J. Math. Anal. & Appl. 37 (1972), 228-234.

Published Writings of M. M. Rao

xiii

[53] A bstract martingales and ergodic theory, Proc. 3rd Symp. on M ultivariate Anal., Academic Press (1973), 100-116. [54] Remarks on a Radon-Nikodym theorem for vector measures, Proc. Symp. on Vector & O perator Valued Measures and Appl., Academic Press (1973), 303-317. [55] Inference in stochastic processes-IV: Predictors and projections, Sankhya, Ser. A 36 (1974), 63-120. [56] Inference in stochastic processes-V: Admissible means, Sankhya, Ser. A. 37 (1974), 538-549. [57] Extensions of stochastic transformations, Trab. Estadistica 26 (1975), 473-485. [58] Conditional measures and operators, J. M ultivariate Anal. 5 (1975), 330-413. [59] Compact operators and tensor products, Bull. Acad. Pol. Sci. Ser. Math. 23 (1975), 1175-1179. [60] Two characterizations of conditional probability, Proc. Amer. Math. Soc. 59 (1976), 75-80. [61] Conjugate series, convergence and martingales, Rev. Roum. Math. Pures et Appl. 22 (1977), 219-254. [62] Inference in stochastic processes-VI: Translates and densities, Proc. 4th Symp. Mul­ tivariate Anal., North Holland, (1977), 311-324. [63] Bistochastic operators, Commentationes Mathematicae, Vol. 21 March, (1978), 301313. [64] Asymptotic distribution of an estimator of the boundary param eter of an unstable process, Ann. Statistics 6 (1978), 185-190. [651 Convariance analysis of nonstationary time series, Developments in Statistics 1 (1978), 171-225. [66 ] Non Ll-bounded martingales, Stochastic Control Theory and Stochastic Differential Systems, Lecture Notes in Control and Information Sciences, 16 (1979), 527-538, Springer Verlag. [67] Processus lineaires sur Coo(G), C. R. Acad. Sci., Paris, 289 (1979), 139-141. [68 ] Convolutions of vector fields-I, Math. Zeitschrift, 174 (1980), 63-79. [69] Local Functionals on Coo(G) and probability, J. Functional Analysis 39 (1980), 23-41. [70] Local functionals, Proceedings of Oberwolfach Conference on Measure Theory, Lec­ ture Notes in Math. 794, Springer-Verlag (1980), 484-496. [71] Structure and convexity of Orlicz spaces of vector fields, Proceedings of the F.B. Jones Conference on General Topology and Modern Analysis, University of California, Riverside (1981), 457-473. [72] Representation of weakly harmonizable processes, Proc. Nat. Acad. Sci., 79, No. 9 (1981), 5288-5289. [73] Stochastic processes and cylindrical probabilities, Sankhya, Ser. A (1981), 149-169. [74] Application and extension of Cram er’s Theorem on distributions of ratios, In Con­ tributions to Statistics and Probability, North Holland (1981), 617-633. [75] Harmonizable processes: structure theory, L’Enseignement M athematique, 28 (1982), 295-351. [76] Domination problem for vector measures and applications to non-stationary pro­ cesses, Oberwolfach Measure Theory Proceedings, Springer Lecture Notes in Math. 945 (1982), 296-313. [77] Bimeasures and sampling theorems for weakly harmonizable processes, Stochastic Anal. Appl. 1 (1983), 21-55, (with D.K. Chang).

xiv

Published Writings of M. M. Rao

[78] Filtering and smoothing of nonstationary processes, Proceedings of the ONR work­ shop on ”Signal Processing” , Marcel-Dekker Publishing (1984), 59-65. [79] The spectral domain of m ultivariate harmonizable processes, Proc. Nat. Acad. Sci., 81 (1984), 4611-4612. [80] Harmonizable, Cramer, and Karhunen classes of processes, Handbook in Statistics, Vol. 5 (1985), 279-310. [81] Bimeasures and nonstationary processes, Real and Stochastic Analysis, Wiley & Sons (1986), 7-118, (with D.K. Chang). [82] A commentary on ” On equivalence of infinite product measures” , in S. K akutani’s selected works, Birkhauser Boston Series (1986), 377-379. [83] Probability, Academic Press, Inc., New York, Encyclopedia of Physical Science and Technology, Vol. 11 (1987), pp. 290-310. [84] Special representations of weakly harmonizable processes, Stochastic Anal. Appl., 6 (1988), 169-189, (with D.K. Chang). [85] Paradoxes in conditional probability, J. M ultivariate Anal., 27, (1988), pp. 434-446. [86 ] Harmonizable signal extraction, filtering and sampling Springer-Verlag, New York, Topics in Non-Guassian Signal Processing, Vol. II (1989), pp. 98-117. [87] A view of harmonizable processes, North-Holland, New York, in Statistical D ata Analysis and Inference (1989), pp. 597-615. [88 ] Bimeasures and harmonizable processes; (analysis, classification, and representation), Springer-Verlag Lecture Notes in M ath., 1379, (1989), pp. 254-298. [89] Sampling and prediction for harmonizable isotropic random fields, J. Combinatorial Analysis, Inform ation & System Sciences, Vol 16 (1991), pp. 207-220. [90] L 2,2 - boundedness, harmonizability and filtering, Stochastic Anal. Appl., Vol 10 (1992), pp. 323-342. [91] Probability (expanded for 2nd ed.), Encyclopedia of Physical Science and Technology, Vol 13 (1992), pp. 491-512. [92] Stochastic integration: a unified approach, C. R. Acad. Sci., Paris, Vol 314 (Series 1), (1992), pp. 629-633. [93] A projective limit theorem for probability spaces and applications, Theor.Prob. and Appl., Vol 38 (1993), (with V. V. Sazonov, in Russian), pp. 345-355. [94] Exact evaluation of conditional expectations in the Kolmogonov model, Indian J. M ath., Vol 35 (1993) pp 57-70. [95] An approach to stochastic integration (a generalized and unified treatm ent), in Mul­ tivariate Analysis: Future Directions, Elsivier Science Publishers, The Netherlands (1993), pp. 347-374. [96] Harmonizable processes and inference: unbiased prediction forstochastic flows, J. Statistic. Planning and Inf., Vol 39 (1994), pp. 187-209. [97] Packing in Orlicz sequence spaces, (1995), (with Z. D. Ren), 18 pages (to appear). [98] C haracterization of isotropic harmonizable covariances and representations, (1995), 14 pages, (to appear). Books Edited [1] General Topology and Modern Analysis. Proceedings of the F.B. Jones Conference, Academic Press, Inc., New York (1981), 514 pages, (Edited jointly with L.F. McCauley). [2] Handbook in Statistics, Volume 5, Time Series in the Time Domain, (Edited jointly with E.J. Hannan, P.R. Krishnaiah), North-Holland Publishing Co., Amsterdam (1985), 484

Published Writings of M. M. Rao

xv

pages. [3] Real and Stochastic Analysis, (Editor), Wiley & Sons, New York (1986), 347 pages. [4] M ultivariate Statistics and Probability, (Edited jointly with C.R. Rao), Academic Press Inc., Boston (1989), 565 pgs. Books Written [5] Stochastic Processes and Integration. Sijthoff & NoordhofF International Publishers, Alpehn aan den Rijn, The Netherlands, (1979), 460 pages. [6] Foundations of Stochastic Analysis, Academic Press, Inc., New York, (1981), 295 pages. [7] Probability Theory with Applications, Academic Press, Inc. New York, (1984), 495 pages. [8 ] Measure Theory and Integration, Wiley-Interscience, New York (1987), 540 pages. [9] Theory of Orlicz Spaces (jointly with Z. D. Ren), Marcel Dekker Inc., New York (1991), 449 pages. [10] Conditional Measures and Applications, Marcel Dekker Inc., New York (1993), 417 pages. [11] Stochastic Processes: General Theory, Kluwer Academic Publishers, The Netherlands (1995), 620 pages.

A Family Tree of Mathematical Ancestors

Godfrey H. Hardy Norbert Wiener Robert H. Cameron Munroe D. Donsker U Malempati M. Rao

xvi

Ph.D. Theses Completed Under the Direction of M. M. Rao and His Students M. M. Rao at Carnegie-Mellon University D ietm ar R. Borchers (1964), “Second order stochastic differential equations and related Ito processes.” J. Jerry Uhl. Jr ( 1966), “Orlicz spaces of additive set functions and set m artingales.” Jerome A. Goldstein (1967), “Stochastic differential equations and nonlinear semi-groups.” Neil E. Gretsky (1967), “Representation theorems on Banach function spaces.” W illiam T. Kraynek (1968), “Interpolation of sub-linear operators on generalized Orlicz and Hardy spaces.” Robert L. Rosenberg (1968), “Compactness in Orlicz spaces based on sets of probability measures.” George Y. H. Chi (1969), “Nonlinear prediction and multiplicity of generalized random processes.” M. M. Rao at University of California, Riverside Vera Darlean Briggs (1973), “Densities for infinitely divisible processes.” Stephen V. Noltie (1975), “Integral representations of chains and vector measures.” Theodore R. Hillmann (1977), “Besicovitch - Orlicz spaces of almost periodic functions.” Michael D. Brennan (1978), “Planar semi-martingales and stochastic integrals.” James P. Kelsh (1978), “Linear analysis of harmonizable time series.” Alan C. Krinik (1978), “Stroock - Varadhan theory of diffusion in a Hilbert space and likelihood ratios.” Derek K. Chang (1983), “Bimeasures, harmonizable process and filtering.” Marc H. Mehlman (1990), “Moving average representation and prediction for multidi­ mensional strongly harmonizable process.” Randall J. Swift (1992), “Structural and sample path analysis of harmonizable random fields.” Michael L. Green (1995), “M ulti-parameter semi-martingale integrals and boundedness principles.” Heroe Soedjak (1996), “Estim ation problems for harmonizable random processes and fields.” Jerome A. Goldstein at Tulane University Charles J. Monlezun (1972), “Temporally inhomogeneous scattering theory.” James T. Sandefur, Jr. (1974), “Higher order abstract Cauchy problems.” Joseph H. Hendrickson (1974), “Temporally inhomogeneous scattering with modified wave operators.” Bruno J. Wichnoski (1974), “Temporally inhomogeneous scattering theory.” B. Clay Burch (1975), “A semigroup treatm ent of the Hamilton-Jacobi equation in several space variables.” (co-advised with E. D. Conway) Janet M. Hughes (Diem) (1976), “Estimation in a non-stationary Markov chain.” (co­ advised with C.B. Bell and A. Levine) Kuong Lin OuYoung (1978), “Periodic solutions of conservation laws.” Michael E. Ballotti (1983), “Modern versions of the theorems of Kneser and W iener.” xvii

xviii

Ph.D. Theses

Gisele Ruiz Rieder (1986), “M athematical contributions to Thomas-Fermi Theory.” Lige Li (1986), “Positive solutions of some predator-prey interacting systems.” (co­ advised with E.D. Conway) Chin-Yuan Lin (1987), “Degenerate non-linear parabolic boundary value problems.” Douglas D. Pickett (1989), “Scattering theory for higher order equations.” Mi Ai Park (1990), “Model equations in fluid dynamics.” Gabriella Segall (1991), “Linear integro-differential equations in Banach spaces.” Denise Kirschner (1991), “Epidemiology and immunology in AIDS.” Chien-an Lung (1992), “The nuclear cusp condition in spin polarized Thomas-Fermi the­ ory.” Jerome A. Goldstein at Louisiana State University Genbao Shi (1995), “M athematical contributions to elasticity and quantum theory.” Kunyang Wang (1995), “The generalized Kompaneets equation.” Andrei Breazna, in progress. Aurora Breazna, in progress. Radu Cascaval, in progress. M atthew Cliff, in progress. Neil Gretsky at University of California, Riverside Joseph J. Sroka (1987), “Factorization of pNA games through Banach spaces.” Bahman Soltani (1989), “A game-theoretic analysis of hostile takeovers.” Zhongdau Ren (1994), “Packing in Orlicz function spaces.” Tim othy W atson (1994), “Potential in infinite games.” Jaime Londono, in progress. J.J. Uhl, Jr. at University of Illinois, Champaign-Urbana Barry T urett (1976) Elias Saab (1979) P aulette Saab (1979) Robert Geitz (1980) Kevin Andrews (1980) Frank Page (1981) Larry Riddle (1982) Minos Petrakis (1985) Russ Gordon (1986) Maria Girardi (1989) Lige Li at Kansas State University Afshin Ghoreishi (1990), “Positive solutions of interacting models in a heterogeneous enviroment under mixed boundary conditions.” Roger Logan (1990), “A study of a two'-species competing interaction model in m athe­ matical biology.” Inkying Ahn (1992), “Elliptic interacting systems with nonlinear diffusions.” Mufid Abudiab (1993), “Dynamic analysis of AIDS infection: Reaction-diffusion model­ ing.”

Ph.D. Theses

Yaping Liu (1993), “Positive solutions to general ellipic interacting systems.” Elias Saab at University of Missouri, Columbia Musbah Abdul-A1 Salam Narcisse Randianantoanina Paulette Saab at University of Missouri, Columbia Brenda Smith

Some Problems of Real and Stochastic Analysis Arising from Applications

M ALEMPATI M. RAO CA 92521-0135

D epartm ent of M athem atics, University of California, Riverside,

In tro d u c tio n The following is a brief account of some problems coming from appli­ cations, chiefly economics, meteorology, physics and statistical practice. They have led to substantial studies in real and stochastic analysis with a nontrivial interaction between these areas leading to further understanding of the subjects. The ensuing questions have helped increase the interests of several of my colleagues and me in those areas of m athe­ matics. More often the problems th a t came to our attention are those for which one has to go beyond the usual limits of the classical areas th a t were considered by m any people in the past, and this appears often enough to keep the curiosity, as seen from the accounts described below.

I. I d e m p o te n t m a tr ic e s a n d n o n - s y m m e tric p r o je c tio n s In some studies of economic analysis, for instance in the work with Cobb-Douglas production functions, linear restrictions play an im portant role after a suitable conversion of the data. The problem here, after a logarithmic transform ation, leads to linear regression with restrictions which involved a use of non-symmetric idem potent matrices. This was treated in detail by John C hipm an and me in the latter p art of 1950’s and the results were first issued as a report and then w ritten up in two papers separating the applications and structural analysis in 1964. A typical result is the following: 1

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T h e o r e m 1 . Let X be an n-vector which is normally distributed with mean p and covari­ ance E. Let A be an n-by-n matrix. Then the quadratic form Q — X 1A X is distributed as a non-central chi-squared variable with r degrees of freedom (=rank of A ) with the non­ centrality parameter A = ^ p lA p, if and only if the (not necessarily symmetric) matrix AH is idempotent of rank r. [Here X 1, p l are transposes of the vectors X, p.] The non-symm etric idem potent (=projection) operators were not treated thoroughly in the literature until then. The problem was analyzed by Victor Mizel and myself (1962, 1963) in the H ilbert space context, first for (non-symmetric) projections which have sev­ eral curious properties (regarding their spectral and convergence aspects), and then for operators satisfying a quadratic equation a x 2 + bx + c = 0. A basic assertion on the spectral analysis is given by the following: T h e o r e m 2 . A continuous (non-symmetric) linear operator T \TL-^TL, a Hilbert space, satisfies the equation T 2 — bT + c l = 0 if and only if there is a selfadjoint positive definite operator S : H —> TL such that (i) S 2 — bS + c l = 0 , (ii) a unitary operator U which is invariant on the closure of the range R g i-b s + c i, and (Hi) its restriction to the latter space satisfies: (a) U2 = - I , (b) S U = U{bl - S), (c) T = S + U ( S 2 - bS + c i y . W hen b = 1 and c = 0, this was first studied in (1962) and then the work was extended to the general case stated in the above theorem. Later these results led to a structural analysis of Reynolds operators and conditional expectations. A detailed account of some of this analysis is given in my paper (1975). 1. J. S. Chipm an and M. M. Rao (1964), “The treatm ent of linear restrictions in re­ gression analysis,” Econometrica, 3 2 , 198-209. 2 . _____________ (1964),“Projections, generalized inverses and quadratic forms,” J. Math. Anal. Appl., 9, 1- 11 . 3 . V. J. Mizel and M. M. Rao (1962), “Non-symmetric projections in H ilbert space,” Pacific J. Math., 12 , 343-357. 4 . ______________ (1963), “Averaging and quadratic equations in operators,” Technical Report No.9, Carnegie-Mellon Univ., P ittsburgh, PA. 5. M. M. Rao (1975), “Conditional measures and operators,” J. Multivar. Anal., 5, 330-413. II. G e n e r a liz e d C r a m e r - R a o in e q u a litie s a n d f u n c tio n sp a c e s Suppose X is a random variable (or vector) having an absolutely continuous distribution F w ith density f{ 'j0 ) depending on a param eter 0 E A, an open (nonempty) interval. If exists and is dom inated by an F-integrable function, and if 0(= @{X)) is a Borel function of X , called an estim ator of 9, let E q{9) — f Q 6 ( X ) d P = 6 + b(6). Then

Problems of Analysis Arising from Applications

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holds, and is called the Cramer-Rao inequality. This depends on a clever use of the CBS (=Cauchy-Buniyakowski-Schwarz) inequality presented in C ram er’s book (cf. H. Cram er (1946), p .479), and was also derived independently at about the same time by C. R. Rao (hence the name C-R inequality) who published it in a paper in C alcutta. The left side of (*) can be replaced by Eg(\6 — 9|p) , l < p < oo, and the corresponding result based on H older’s inequality is valid, and it was given by E. W. Barankin (1949). Unaware of B arankin’s work, I obtained the same result ten years later. (I. R. Savage informed me of B arankin’s results on seeing my m anuscript.) After analyzing the com putations and applications, I realized th a t the convexity of a function p is actually at work, and hence sought a lower bound for Eo(p(9 —6)) where p : R -> R+ is a symmetric Young function, w ith p(x) = 0 if and only if x = 0 , so th a t p(x) — \x\P,p > 1 includes the above results. This led to a set of inequalities, a typical one of which is as follows. Let p (x) — p i ( x ) k for some k > 1 where p \ satisfies the same conditions as p. Let D \ = as above. Then the inequality below obtained in (1959) reduces to B arankin’s when p i ( x ) = |x|, and to (*) if k = 2 in addition. A multidimensional extension was given in (1963), after a series type bound (extension of B hattacharyya bounds) in (1961), and an application to Bayes estim ation was then obtained by Morris DeGroot and me (1963). T h e o r e m 1 . Under the regularity conditions and the integrability of p { D \), as in (*), one has with b' = jji ( aw ^ j l + bf(0)^k fE g (\D i\)-jk' E , W 9 - 9 ) ) > ^ ( ^ i5 n 5 ) ™ P [-jjD ^ r] •

where k' = k / ( k — 1) and if k! — oo, then ||Di||fc/ is the essential supremum norm.

1. 2. 3. 4. 5.

6.

E. W. Barankin (1949), “Locally best unbiased estim ates,” Ann. Math. Statist., 2 0 , 477-501. H. Cram er (1946), Mathematical Methods of Statistics, Princeton Univ. Press, Prince­ ton, NJ. M. H. DeGroot and M. M. Rao (1963), “Bayes estim ation with convex loss,” Ann. Math. Statist., 34, 839-846. M. M. Rao (1959), “Lower bounds for risk functions in estim ation,” Proc. Nat. Acad. Sci., 45, 1168-1171. ________ (1961), “Theory of lower bounds for risk functions in estim ation,” Math. Annalen, 143, 379-398. ________ (1963), “Some inference theorems in stochastic processes,” Bulletin Am. Math. Soc., 69, 72-77.

(i) Dual spaces (L^)* on the scene. The preceding work on convex (loss) functions was not sufficient for constructing the best 0 which gives equality in the above lower bounds, as was possible in B arankin’s case. The corresponding study demands an analysis of L ^-spaces (called Orlicz spaces), and their adjoint spaces (L^)*. The structure of the la tte r space was an open problem at the time, as indicated in the book by Krasnosel’skii and Rutickii

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(1960). This problem was solved after much work (1964, 1968). It can be stated as follows. Let 'ij; : x ip(x) = sup{|x|?/ — ip(y) : y > 0}, the com plementary Young function to ip. An additive set function v : A R, where A is an algebra of subsets of a set 11, has finite ^-bounded variation relative to p if n

u(A)

||i / ||^ = s u p { y ^ ( i =i

%) n ( A i ) : A j *'

disjoint,

e A } < oo.

Then one has on writing E = a (A): T h e o re m

2.

(L^(/i))* = 2i^(^) =

: E -> R |

||z/||^ < oo}.

For the construction problem of optim al estim ators, it is also necessary to analyze 21^, (p) for sm ooth (p. This was done in (1965a) and the desired construction of the estim ator 6 was accomplished in (1965b). However, the structure of the space (21^(p))* = (L ^ (//))** was also needed in its generality for some related applications. If ip(x) = \x\P,p > 1 , the corresponding space was analyzed by S. Leader (1953), but the more general (L ^/i))** has not been studied before. The problem was solved by J. Jerry Uhl (1967) who also considered the vector valued function space (L^(/i))** and analyzed some of its subspaces when A is a Banach space. But here a new Radon-Nikodym theorem for Y-valued additive set functions was needed. He settled these questions and went on to apply the results for (additive) vector set m artingales (1969) and then chose to classify Banach spaces X w ith the Radon-N ikodym property as the next project.

1. 2. 3.

4. 5.

6. 7. 8.

M. A. K rasnosel’skii and Ya. B. Rutickii (1960), Convex Functions and Orlicz Spaces, P. Noordhoff Ltd., Groningen, The Netherlands. S. Leader (1953), “The theory of L p spaces for finitely additive set functions,” Ann. Math., 58, 528-543. M. M. Rao (1964) “Linear functionals on Orlicz spaces,” Nieuw Arch. Wisk., 12, 77-98. ________ (1965a), “Smoothness of Orlicz spaces,” Indag. Math., 27, 671-690. ________ (1965b), “Existence and determ ination of optim al estim ators relative to convex loss,” Ann. Inst. Statist. Math., 17, 133-147. ________ (1968), “Linear functionals on Orlicz spaces: general theory,” Pacific J. Math., 25, 553-585. J. J. Uhl, Jr (1967), “Orlicz spaces of additive set functions,” Studia Math., 29, 19-58. ________ (1969), “M artingales of vector valued set functions,” Pacific J. Math., 3 0 , 533-548.

(ii)Some extensions and applications. In this study a great many argum ents appear to generalize for Banach function spaces L p(ffi), studied by several people at the time, espe­ cially by A. C. Zaanen and his associates in the Netherlands. Here p(-) is a function norm, and L p{p) is also called a Riesz space. The corresponding problem of a characterization of

Problems of Analysis Arising from Applications

5

(L ^(/i))* has been analyzed in detail by Neil Gretsky and his results appear in a memoir (1968). L ater he and Uhl (1972, 1981) used these ideas in obtaining various other results. During this tim e, certain applications to ergodic theory based on the Lebesgue spaces L p, 1 < p < oo, were being studied by R. V. Chacon, and a corresponding analysis in L (pspaces was n atu ral but was dependent on an interpolation theorem. The desired extension (even w ith changes of measures) and applications to ergodic theory and m artingales, es­ pecially for Reynolds operators considered by R ota (1964) in the context of L 2-spaces, were obtained in (1966). A further generalization of some of the latter results to sublinear operators and a detailed analysis of it in Hardy-Orlicz spaces (being initiated by a student of W. Orlicz at th a t time) were undertaken in his thesis by W illiam Kraynek (1970, 1972). 1. N. E. G retsky (1968), “R epresentation theorems for Banach function spaces,” M em ­ oirs Am. Math. Soc., 84, 1-56. 2 . _________ and J. J. Uhl, Jr (1972) “Bounded linear operators on Banach function spaces of vector valued functions,” Trans. Am. Math. Soc., 167, 263-277. 3 . _____________ (1981), “Carlem an and Korotkov operators on Banach spaces,” Acta Sci. Math., 43, 207-218. 4. W. T. Kraynek (1970), “Interpolation of multilinear functionals on L ^ -s p a c e s ,” J. Math. Anal. A p p l, 31, 414-430. 5. ________ (1972), “Interpolation of sublinear operators on generalized Orlicz and Hardy-Orlicz spaces,” Studia Math., 43, 93-123. 6. M. M. Rao (1966), “Interpolation, ergodicity and m artingales,” J. Math. Mach., 16, 543-567.. 7. G.-C. R ota (1964), “Reynolds operators,” Proc. Am. Math. Soc. Symp. Appl. Math., 16, 70-83. (iii) Nonlinear prediction and related problems. During these investigations some applica­ tions to probability theory have been made. The smoothness of L^(^)-space analysis has led to a study of prediction problems on these spaces. I considered the problem on real L^-spaces (1965) and also the L vx -case for 1 < p < oo with A as a sm ooth Banach space (1967). A further analysis has been undertaken to determine the structure of projection operators, and their relation to the conditional expectation operators, by Nicolae Dinculeanu and me (1974). The work led to an abstract formulation of prediction problems in general sm ooth Banach spaces (1972). Thus a familiarity with Orlicz spaces provided a strong m otivation to obtain several results th a t can be studied on the ZT-spaces to the more inclusive I N - versions. Such an extension in most cases is not straightforw ard since unlike the former case w ith 1 < p < oo, the reflexivity, uniform convexity, smoothness and other properties are all distinguished and separated in Orlicz spaces. For this reason, these are useful “test spaces” to try new ideas and methods. In this way, Theodore Hillmann (1977) has studied the Besicovitch-Orlicz spaces of almost periodic functions. In a similar m anner, there is an analysis of Fenchel-Orlicz spaces of vector functions by J. B arry T urett (1980), a thesis w ritten under the supervision of Uhl. Thus m ost of the advances since the publication of Krasnosel’skii-Rutickii volume noted earlier, obtained by my associates, me, and several other researchers abroad, have recently

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been compiled in a m onograph in collaboration with Z. D. Ren who came from P. R. China to spend tim e here at Riverside with us. He brought the work on Orlicz spaces from his side, since much of it was published in the Chinese language, and fortunately he can read Russian to verify the corresponding contributions from th a t side. Also further results on sphere packing in Orlicz spaces were recently completed by him (1994), in a thesis under G retsky’s guidance. Thus at Riverside an im portant phase of this work has been accomplished.

1. 2. 3. 4. 5.

6. 7. 8.

N. Dinculeanu and M. M. Rao (1974) “Contractive projections and conditional ex­ pectations,” J. Multivar. A n a l, 2 , 362-381. T. R. Hillm ann (1977), “Besicovitvh-Orlicz spaces of almost periodic functions,” Ph.D . thesis at UCR, (see also C hapter 2 in Real and Stochastic Analysis, Wiley, New York, (1986), 119-167). M. M. Rao (1965) “Conditional expectations and closed projections,” Indag. Math., 27, 100-112. (1967), “Inference in stochastic processes-III,” Z. Wahrs., 8, 49-72. ________ (1972), “Prediction sequences in smooth Banach spaces,” Ann. Inst. H. Poincare, Ser. B, 8 , 319-332. M. M. Rao and Z. D. Ren (1991), Theory of Orlicz Spaces, Marcel Dekker, Inc., New York. Z. D. Ren (1994), “Packing in Orlicz function spaces,” Ph.D. thesis at UCR, River­ side, CA. J. B. T urett (1980), “Fenchel-Orlicz spaces,” Dissertationes Math., 181, 1-55.

I I I . H ig h e r o r d e r s to c h a s tic d iffe re n tia l e q u a tio n s In an early 1960 visit to Carnegie Tech, Prof. S. K akutani remarked to me th a t there are still some fresh ideas in the classical paper on applications of probability to physics, by S. C handrasekhar (1943), and I should read it. There is a treatm ent of Langevin’s equation which is of first order and then there is also a description of the m otion of a simple harmonic oscillator driven by a random disturbance. The latter is a second order linear equation. Several works exist in the literature extending the Langevin’s linear equation to nonlinear problems using Ito ’s theory of stochastic integration. For a detailed modern treatm ent of the subject, see, for instance, Stroock and V aradhan (1979), and for infinite dimensional versions Yor (1974) and K rinik (1986). It seemed to me th a t the corresponding equation of a simple harm onic oscillator should be similarly generalized and studied, using a suitable definition of (higher order) stochastic derivative. W hen the work on this problem was started, D ietm ar Borchers, a graduate student at Carnegie, was looking for a dissertation topic and I gave this to him when he wanted to work on his thesis with me. The equation is: (+)

d X t + q(t, X t , X t )dt = a ( t , X u X t)dZt ,

w ith X q = A, X q = B as initial conditions when in fact all these are finite vector (or suitable m atrix) functions, and Z t is a m artingale process with independent increments. Under some suitable Lipschitz conditions and using an appropriate mean derivative con­ cept, Borchers (1964) established the existence and uniqueness of the solutions of (+)

Problems of Analysis Arising from Applications

7

and showed th a t the vector process Vt = ( X t , X t) is Markovian. He also calculated the generator of the associated semigroup of linear operators when the Z-process is Brownian m otion and 0 ) moments. The case w ith 5 = 0, as well as the limit distributions of the estim ators in th a t context are still unresolved. If k = 1 and \a\ = 1 , then the limit distribution is already complicated, as shown by me (1978), and the case th a t k > 1 with some roots on the unit circle is yet to be solved.

1.

T. W. Anderson (1959), “On asym ptotic distribution of estim ates of param eters of stochastic difference equations,” Ann. Math. Statist., 30, 676-687. 2 . T. L. Lai and C. Z. Wei (1983), “Asym ptotic properties of general autoregressive models and strong consistency of least squares estim ators of the param eters,” J. Multivar. Anal., 13, 1-23. 3. H. B. M ann and A. Wald (1943), “On the statistical treatm ent of linear stochastic difference equations,” Econometrica, 11 , 173-220. 4. M. M. Rao (1961), “Consistency and limit distributions of estim ators of param eters in explosive stochastic difference equations,” Ann. Math. Statist., 32, 195-218. 5.___________ (1978), “Asym ptotic distribution of an estim ator ofthe boundary param ­ eter of an unstable process,” Ann. Statist., 6 , 185-190. 6 . J. S. W hite (1958), “The limiting distribution of the serial correlation coefficientin the explosive case,” Ann. Math. Statist., 29, 1188-1197. V . H a rm o n iz a b le p ro c e s s e s In the 1940’s Loeve defined a class of second order processes, called harmonizable, whose covariance is the Fourier transform of a continuous function (necessarily positive definite) of finite Vitali variation in the complex plane. This generalization of (K hintchine’s) stationarity prom pted (and he asked for) a characteriza­ tion of the corresponding harmonizable covariance. This is messy and in 1967, I gave a solution which reduces to the classical Bochner-Khintchine case for stationary processes. In 1956 in the third Berkeley symposium paper, quoted in Section III, Bochner introduced a concept called “R-boundedness” and noted th a t it includes the Loeve harmonizability, and then presented some of its properties. Later Rozanov (1959) defined a related notion, also term ed “harm onizable” , and made some applications. It is found th a t Rozanov’s definition is more general th an Loeve’s and in fact it coincides w ith Bochner’s concept of V-boundedness. In his thesis Niemi (1975) studied the Fourier transform s of certain H ilbert space valued measures directly. To understand and classify these classes, I have called the Loeve concept strong harmonizability, and the Bochner-Rozanov concept weak harmonizability, and presented a structure theory of these processes (1982). To clarify the distinctions noted above, it was necessary to use the Morse-Transue theory of integra­ tion of scalar functions relative to a bimeasure of finite Frechet variation. This integral is weaker th an the Lebesgue-Stieltjes concept and coincides with the latter iff the Frechet

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variation is replaced by the V itali variation. A slightly restricted form of the M T-integral was useful for this work. This w ith ap­ plications to linear filtering were then developed, jointly with D. K. Chang (1986), and further extensions to random fields in (1989). The area of harmonizable processes is now suitable for analysis by extending (nontrivially) many results from the stationary case. Some applications of these processes were considered in studies begining w ith J. Kelsh (1978), and continuing the work by Marc M ehlman (1991). It was noted in the book by Yadrenko (1983) th a t only the trivial (i.e., constant) stationary isotropic random fields satisfy the Laplacian. This m otivated an introduction of isotropy for harmonizable classes for which nontrivial solutions of the above equation exist, and the desired concept is pre­ sented and studied in (1991). Further structural analysis and applications of this class are given by R. J. Swift (1994). Much can be (and remains to be) done here, and in fact ho­ mogeneous (= stationary) isotropic random currents introduced by Ito (1956) and recently analyzed in more detail by Wong and Zakai (1992) can be extended to the harmonizable class and this is actively pursued in the present research. The area is an interesting place in which a (not necessarily abelian) harmonic analysis of vector (or H ilbert space valued) measures and probability theory meet, and a fruitful interaction results (1989). It is also possible to consider some of the ideas by generalizing strict stationarity (i.e., invariance of finite dimensional distributions under transtations) with vector harmonic analysis. This is som ewhat more restricted th an the preceding class, as seen from Hosoya (1982), but can be studied for its own interest.

1. 2. 3. 4. 5.

6. 7. 8. 9.

10 . 11 . 12 .

D. K. Chang and M. M. Rao (1986), “Bimeasures and nonstationary processes,” in Real and Stochastic Analysis, Wiley, New York, 7-118. Y. Hosoya (1982), “Harmonizable stable processes,” Z. Wahrs., 60, 517-533. K. Ito (1956), “Isotropic random current,” Proc. Third Berkeley Symp. Math. Statist. Prob., 2 , 125-132. J. P. Kelsh (1978), “Linear analysis of harmonizable tim e series,” Ph.D . thesis at UCR, Riverside, CA. M. H. M ehlman (1991), “Structure and moving average representation for m ultidi­ m ensional strongly harmonizable processes,” Stochastic Anal. A p p l, 9, 323-361. H. Niemi (1975), “Stochastic processes as Fourier transform s of stochastic m easures,” Ann. Acad. Sci. Fenn. AI. Math., 591, 1-47. M. M. Rao (1967), “Characterization and extension of generalized harmonizable random fields,” Proc. Nat. Acad. Sci., 58, 1213-1219. ________ (1982), “Harmonizable processes: structure theory,” L ’Ensign. Math., 28, 295-351. ________ (1989), “Bimeasures and nonstationary process,” in Probability on GroupsIX, Lect. Notes M ath., 1379, 254-298. ________ (1991), “Sampling and prediction for harmonizable isotropic random fields,” J. Comb. Info. Syst. Sci., 16, 207-220. Yu. A. Rozanov (1959), “Spectral analysis of abstract functions,” Theor. Prob. A p p l, 4, 271-287. R. J. Swift (1994), “The structure of harmonizable isotropic random fields,” Stochas­ tic Anal. A p p l, 12 , 583-616.

Problems of Analysis Arising from Applications

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13. M. I. Yadrenko (1983), Spectral Theory of Random Fields, O ptim ization Software Inc., New York. V I. P r e c is e m e a s u r e m e n ts a n d g e n e ra liz e d ra n d o m fields Stochastic models describing a random phenomenon can be interpreted and verified when observations on random variables are measured at specified instants. If the process is continuous, the value o f-X t {u) at the instant t cannot exactly be observed w ithout affecting a nearby instance. Consequently one can only have at best an averaged version, i.e., j T X t(p(t)dt where

L 2(P), called generalized random functions or functionals. K. Ito and I. M. Gel’fand independently introduced these objects in the middle 1950’s. In their seminal work, Gel’fand and Vilenkin (1964) gave various properties of these functionals and presented representation theorems for them . They also considered to have independent values, i.e., if (pj G D ,j = 1, 2, with p>\ • g>2 — 0, then 4>((/?i), $ ( ^ 2) are m utually independent random variables. If L( d i ( x i ) • •• V I A 1 *)}-

(18)

Bourgain

20

In the Dirichlet case, one gets {»»! + ■■■ where {Aj',Pj} and

{ p n '^ n }

+

( 19 )

■' Pi t t ed)}

are as above.

(iv) A = ( - A + p )1/ 2

( 20 )

K = ( M 2 + p )1/2

(21)

w ith eigenvalues

and the exponentials as eigenfunction s. Observe th a t the sequence {(k + p ) 1/ 2 \ k G Z+} consists of rationally independent numbers for typical p. Coming back to ( 1 ), denote { p ni p n } the eigenfunction s and corresponding eigenvalues for A, i.e. A fPn — p n P n •

Fix some specific indices n i, n 0 (because they lead to some technical simplifications later on). There is dependence of the new frequency A' on {aj} and the perturbation e

In fact, assuming

A'1?. . . , \'b rationally independent, a tim e shift perm its to assume aj( 1 < j < 6) real and hence A' depends only on {|flj|}. The solution (25) corresponds by (27) to a perturbed torus

7 ^{|a^|}

of To{| | } in the phase space.

Consider A = (Ai,...,A&) as a param eter taken in a param eter set A.

Persistency of

quasi-periodic solutions (22 ) after the H am iltonian perturbation as described above will occur for A in a large subset of the frequency set. More precisely, the persistency will hold provided A ^ A fe (\aj\ (1 < j < 6), perturbation) where

£—>-0

mes A'e — >0 .

(32)

(33)

The preceding deals w ith perturbations of linear equations. In order to obtain families of quasi-periodic solutions w ith b frequencies, one needs to consider a param eter dependent linear equation w ith 6-param eters p = ( p i , . . . ,Pb), such th a t if {An (p)} are the eigenvalues of A(p), then det

\ dpk J 1 e.

(6 0 )

To perform their actual sum m ation is a very delicate issue (leading to re-norm alization problems) and was even in finite dimensional phase space (i.e. n ranges in a finite set) only recently fully understood^*). The m ethod followed here is to solve (58) by a Newton iteration scheme, converging much faster (double exponentially fast) and therefore less affected by the presence of small divisors. On the other hand, it will require to control the inverse of the linearization of (58) which is a non-diagonal operator. We first perform a Lyapunov-Schmidt type decomposition. Consider the b equations

( - \ ,j + \ j ) aj + £ ^ - ( n j , ej ) = 0

(j = l , . . . , b )

(61)

obtained by taking (n, k) G S of (26), (29). They form the (finite) system of Q-equations. The remaining (infinite) system &H (-(A ', k) + /in) u(n, k) + e — (n, k) = 0

(n, k) £ S

(62)

are called the P-equations. The general procedure is to determ ine u|(n,fc)^«s from (62) (de­ pending on A') and then substitute in (61) to obtain the new frequencies A' = (A'l 5. . . , \'b). Now these frequencies A^, . . . , A^ need to be real. Thus (61) expresses, in fact, 2b con­ ditions w ith b param eters. The formal solvability is a consequence of the H am iltonian nature of ( 1 ) (result due to Poincare). Assume H (u ,u ) is a sum of monomial s u^uk with real coefficients. In proving the persistency result, we may assume aj G R (j = 1 , . . . , 5), considering tim e shifts as observed earlier. Hence, the system u\(n,k)$s produced from solving (62) will be real and so will be

(rij, ej) in (61). Thus (61) yields a real solution

in A' and the formal solvability is clear in this case. The system (62) cannot be treated by a standard implicit function theorem because of the appearance of small divisors. We denote v = u.

(63)

Assuming the eigenfunction basis for A in ( 1) given by exponentials, we have thus f)(n, k) = u ( —n, —k) W

cf. [El]

(64)

26

Bourgain

(if one would consider a real eigenfunction basis {^ n} 5 then clearly ii(n, k ) = fi(n, —A:)). Duplicate the equations (62) considering the system f

+ n n ) u( n, k ) + e ^ - ( n , k )

\ ( ( X , k ) + n - n) v ( n , k ) + £

=

f f ( n,k) = 0

0

{ n , k ) 0. This fact clearly yields a corresponding off-diagonaldecay estimate for each of the 4 matrixes S# appearing in (71) S ^ X i x ' ) < C e - \ x~x'\c.

(74)

The main difficulty consists in controlling the inverse T _1, due to the fact that the diagonal elements may be arbitrarily small. Fixing some e

(89)

Coming back to (38), assume (except for (n, k ) G 5) I — (&> A) + for some constant C. Assuming

| > (1 + |fc|) c

(90)

as in (28), it follows that also

|- (k, A') + Hn\ > t (1 + \k\)~c for |fc| < e ~ 1/2C.

(91)

N < £ - 1/2C

(92)

Thus, for

one may invert T n =

Djy + s S n by a Neumann series, i.e. oo T n 1 = DJf1 + £ ( - 1 )j ej ( S h D J Y

3= 1 since ||S at

II
ety= o \ (-(A ', k) + n„) u{n, k) + e f f (n, k) =

0

(■?= !>•••>&)

(99)

for (n, fc) e 0 is a typical number in the sense of linear independence of the Vn = (n2 + p ) 1/2.

(125) sequence (126)

Fix a sequence 0 < n\
° f Pos^ ve measure, in fact, of asymptotically full measure when \a\ —> 0, such that for a G C the solution (128) of (129) persists fo r (125) b u(x, t ) = ^ qj cos njX • cos X'jt + 0 (|a |3). 3 =1

(130)

34

Bourgain

The persistency problem for higher dimensional wave equations seems more difficult, due n \ ) 1/ 2 when d >

to the behavior of the frequencies \n\ = (n\ +

2.

One may tre at

however, the special case of tim e periodic solutions in any dimension (this is also the case for NLS). TH EO REM 131. Consider the periodic wave equation in dimension d

m u tt — A u + pu + (u 3 + higher order terms) =

0

(132)

where again p > 0 is a typical number. More precisely, we require p to satisfy acondition of the form & ■

^

" fo r all {kj } e Z r+ 1\{0}.

> (E

3= 0 Fix no Po

GC

G Z d\{0}.

(133)

There is a Cantor set C of positive measure in an interval [0,(5] and for

a solution of (132) of the form u(x, t) = po cos ((n 0, x) + Xt) +

0 (pi)

(134)

where >? = K p o ) 2 =

In o l2 + P +

Q

+ o ( l ) ) Po-

(1 3 5 )

The next two results are norm al form reductions, bringing the problem back to p e rtu rb a­ tions of a linear problem w ith param eters, (cf. the discussion (39)-(45)). TH EO REM 136. Consider a ID NLS iut — u xx + m u + f ( \ u\ 2)u = 0

(137)

where f is a polynomial or real analytic and satisfying a non-degeneracy condition /'( 0 ) + 0.

(138)

Consider (137) with periodic boundary conditions say. Fix a sequence of positive integers ni < n 2 < ... < t v Then f o r a =

(139)

in a Cantor family C of positive measure, there is a quasi-periodic

solution

t)= Y, 9j ei(njX+x'jt^ + 0 ( |a |3) b

u{x,

3= 1

(140)

Solutions of Hamiltonian Evolution Equations

with frequencies

...,

35

where A'- = n 2 +

m + 0 ( | a | 2)

( j

=

(141)

l , . . . , b ) .

This result is due to S. Kuksin and J. Poschel (under Dirichlet boundary conditions) [K-P]. TH EO REM 142. Consider the 2D cubic NLS [B3] iut - A u -f cu\u\2 — 0

( c / 0)

(143)

or, more generally, an 2D NLS (144)

iut —A u + f ( \ u \ 2)u = 0 with f as above in Theorem 136, with periodic boundary conditions. For the modes n x , . . .■, n b €e Z 2, we fix 2 lattice points n\,

nj\

=

R

{ j

=

712 on

a same circle

1,2),«i ^ - n 2

(145)

(more complicated structures involving more then 2 points may be treated as well but this is the simplest case). There is a Cantor family C of positive measure such that f or a = { d j } j =i t2 £ C ( U S ) (144) has a quasi-periodic solution 2

(146)

j =1 with frequencies X' = (A^, A2)

x'j

= K f + o(M2) U = 1 , 2).

(147)

F u r t h e r C o m m e n ts . It seems a n atu ral program to extend the classical theory of sm ooth dynam ical systems to the setting of infinite dimensional phase space, in particular in the context of H am iltonian P D E ’s as discussed here. A subject closely related to persistency of invariant tori is th a t of Nekhoroshev stability [N]. This phenomenon may be roughly stated as follows. Consider in 2N-dimensional phase space a perturbed H am iltonian H ( h , ..., In ,

T n ) = H 0{I) + e H i ( I , ( —1)), and thus \/3(v)\ > 0. establishing the independence. Note that a(t)Vi(x (t)) will be Vi(t) of Theorem A, i = 1. . . . ,n. The homogeneity of the system (10) allows one of the rriij to be normalized to be 1, leaving exactly n 2 equations in n 2 unknowns. Finally, by the Haar condition, for each t E ( - 1 , -hi), k(t. s) = u(t) -v(s), as constructed according to the above prescription, changes sign only at s = X{(t) ( i = 1 , . . . , n - 1) and thus A = u(t) • V( x( t) ) = f T \k(t, s)\dv(s) = L(t) > 0, where L(t) is the Lebesgue function for P. Thus Theorem A guarantees that we have PminFinally, by Theorem B, Pmjn is unique. I REMARK. System (3) is equivalent to

(3a)

where 'd>i = u ,/ u n ( i = 1 . . . . . n —1), and

Vi(x)ipi +

b Vn - i ( x ) ^ n- i + Vn {x)

(3b)

Thus, if Vn_ i := [ui ,..., ^n_i] is also Haar, then the (n — 1) x (n — 1) matrix in (3a) above is invertible. In particular, in the algebraic case, Vn_ i := [1, £ , . . . . t n~2) and the (n — 1) x (n - 1) matrix above is in fact the classical (invertible) Vandermonde matrix. 3. A p p lic a tio n s All the applications in this section are directed towards the determination of the minimal projection from L ^ - R l ] onto 7rn_ i (i.e., the action A = / , the measure v is standard Lebesgue measure, and V = [1, t , . . . , £n -1]). The first two applications are repeated from [1] for the sake of completeness and as an aid to the reader to identify the various parts of the Prescription.

66

Chalmers and Metcalf

In Application 1 ( n = 2), for each M , the single function x(t), being the root of a quadratic, is determined explicitly, and then M := d iag(l,m ) and A are determined to meet the (two) remaining (after symmetry) orthonormality conditions (via a numerical method). In Application 2 (n = 3), for each M , the two functions £*(£), i = 1,2, are also deter­ mined explicitly (in terms of the solution of a quartic), and then M and A are determined to meet the (five) remaining (after symmetry) orthonormality conditions (via a numerical method. In Applications n-1 (4 < n < 6), for each M , determining the n — 1 functions £{(£), i = 1,..., n - 1, involves solving (for each t) an (n —1) x (n - 1) system of polynomial equations of degree n and thus must be determined entirely numerically (e.g. by Newton’s method). M and A are then determined to meet the remaining (after symmetry) orthonormality conditions (via a numerical method). In all cases we give the M matrix (up to 4 decimal places) and the projection norm A (up to 5 decimal places). APPLICATION 1. V = [ U] ( - i , i ] (Franchetti-Cheney [3]). Consider the process described by the above prescription. In this case n = 2, V\(x) = 2x and V^(^) = x 2 - 1. Using symmetry considerations, equation (5) becomes 2x(t) 1i

x 2(t) — 1 ’ m tI

from which the admissible solution is x(t) = m t —s g n ( m t ) \ / m 2t 2 + 1. Equations (3a), (3b) and (7) are then ip(t) = ~ x (t ) ..

( = u i ( t ) / u 2( t ) )

Acr(t) 2x(t)ip(t) + x 2(t) — 1

a{t) - sgn x(t),

(1.3a) (1.36) (1.7)

which give

Using the symmetry of u\ and u 2, equations (9) become (1.9) which are identical to equations (8) and (10) of [3], and result in the equation 2-0(i)[i -

+ v»(i)] log IV’(i)! + 1 - V>2( i) = o

( 12)

67

Minimal Projection from Ll onto n,

for ^ (1 ), and hence ra. It then follows th at A > 0 and Theorem P min. In this case

1

1

guarantees we have

0

M - . \ 0

1.3605

and Pmin = X = 1.22040... . APPLICA TIO N 2. V = [1 ,M 2][-U ] (Chalmers-M etcalf [1]). In this case n — 3 and ^ Vx{ x i , x 2) = 2(xi - x 2 + 1),

-s V2( x i , x 2) = x \ - x \ ,

2 V3( x i , x 2) = - ( 1 + x \ - x 2).

Using sym m etry considerations, equations (5) become 2[xi(t) — x 2(t) + 1] _ xj(t ) - x \ { t ) = |[ x f (t ) - x \{t ) + 1] m i l + m.i2t 2 Letting

t

+rni3t2

mu T. =

2t

m 31 + m 33t 2

“ d

7j =

(2.5)

m3i + m33t2 p •

equations (2.5) may be rewritten Xi —X2 + 1 = 7 i(x i - x l ) ,

x\ - x \ + 1 =

Introducing the variables yi = x \ — X2 and

2/i + 1=7i2/i2/2

and

7/2

= xi + y1

73

(x? - x\ ) .

(2.5)'

leads to

+1

3t/o +

= 732/12/2,

which reduce to the single quartic equation for 7/ 2 : (712/2 -

+

4(71 - 73 ) 2/2 - 4 ] + 1

=

0.

This equation is then solved yielding admissible x \ and X 2 ( —1 < x \ (t) < X2 (t) < 1). The function a(t) (in (7)) is - 1 for \t\ < t 0 and +1 for t Q < |£|, where ±£0 are points where the admissible solutions of the quartic equation switch from one pair of roots to another. The values of A, ran, ra13, ra3i, and ra33 are determined from the five non-trivial orthonormality conditions i

1

i

1 = y u i ( t ) d t = J t 2us(t) dt = J t u 2 (t) dt (2-9) 0 = y t 2ui(t) dt = J uz{t)dt. -1

Chalmers and Metcalf

68

The solution of these equations (for example, by the iteration m ethod of §3 in [1]) yields to = 0 .45710... and the values of A and M given below.

Hence, the X{(t) are

specified, and Pmm = u © v, where u (t) is given by V2 {x(t)) xi(t) x 2 (t)

=

and

V3 (x(t))\ / ui(t)\ x\(t) u 2 (t) x\(t) J \ u 3 ( t ) J

3Asgn(^2)

min|[ — A.

M - |

1 0 -.1 5 5 2

0 .9336 0

-.6 6 7 5 ' 0 1.2711

and P min = A = 1.35948.. APPLICATION 3. V = [1,t ,t 2, t 3]E_ lfl]. /

1 0 M = -.0797 V 0

0 1.4760 0 -.2 6 4 8

-.4 7 2 6 0 1.0520 0

0 \ -1.1095 0 1.3017 /

and A = 1.46184... . APPLICATION 4. V = [1 , m W /

1 0 -.1 5 9 0

0

\ - -0.0126

4](_ m ].

0

.2767

0

1.2925

0

2.3605

0

-.1 5 2 3

0

-.1 178

0 -.7046

0

1.1257

0

and Pmin = A = 1.54874... . APPLICATION 5. V = [L M 2,*3,*4,*5] ^ ] .

-.9 2 5 0 \

0

-1.8019

0

1.0642 /

Minimal Projection from L l onto n„

/ M =

1 0 -.0781

0

-.0743

V 0

69

0 .8893

0

-.5 5 9 3

0

-.1 7 6 9

-.0 4 66

0

1.6151

0

-.0 6 7 5

0

0 .9039

0

3.5525

0

.6439

-.4955

0

-.9 9 0 0

0

.9124

0

0

-1.4609

\

0

-2.6180

0

.2685 /

and A = 1.61031... .

R EFER EN C ES [1] Chalmers, B. L. and F. T. Metcalf, The determination of minimal projections and extensions in L 1, Trans. Amer. M ath. Soc. 329(1992), 289-305. [2] Cheney, E. W., Applications of fixed-point theorem,s to approximation theory, Theory of Approx. w ith Appl., Academic Press Inc., New York, (1976), 1- 8 . [31 Franchetti, C. and E. W. Cheney, Minimal projections in L l -space, Duke M ath. 43(1976), 501-510.

J.

[4] Hobby, Charles R. and John R. Rice, A moment problem in L 1 approximation, Proc. Amer. M ath. Soc. 16(1965), 665-670. [5] Morris, P. D. and Cheney, E. W., On the existence and characterization of minimal projections, J. Reine Angcw. M ath., 270(1974), 61-76.

“Proofs” and Proofs of the Eckart-Young Theorem JOHN S. CHIPMAN Minnesota 55455

D epartm ent of Economics, University of Minnesota, Minneapolis,

INTRODUCTION In 1936 Eckart and Young formulated the problem of approximating an n x k m atrix X of rank k by an n x k m atrix of rank r < k. This has come to be known as the Eckart-Young theorem. It has im portant applications to factor analysis in psychometrics (for which it was originally developed by Eckart and Young), to clustering and aggregation in econometrics (cf. Fisher, 1962, 1969), to quantum chemistry (cf. Goldstein and Levy, 1991; Aiken, Erdos and Goldstein, 1980), as well as to the theory of biased estim ation (cf. M arquardt, 1970) in statistics. M arquardt showed th at if in the regression model y = X/3 + e,

£e = 0 ,

See' = cr21,

Work supported by a Humboldt Research Award for Senior U. S. Scientists. I wish to thank John Eagon, Joel Roberts, and Paul Garrett of the University of Minnesota’s School of Mathematics for their help. In particular, Lemma 2 and Theorem 1 were supplied by Roberts and the idea for Theorem 3 by Garrett, with whom I had many valuable discussions. Both of them declined coauthorship, but they deserve most of the credit for the results. Upon presentation of this paper at the Delhi Workshop on Generalized Inverses, 14 December 1992, George Styan drew my attention to an unpublished technical report by Rao and Styan (1976), some of the results of which were reported by Rao (1979, 1980); this raised some of the same issues as the present paper, and presented alternative proofs. I was also privileged to read some unpublished notes by Styan (1976). A relevant unpublished paper by Sondermann (1980) should also be mentioned. Finally, I wish to thank Renate Meyer for bringing by attention to the paper by Mirsky (1960)—see also Schmidt (1907), von Neumann (1937), Stewart & Sun (1990), and Meyer (1993, p. 67)— and to Jerome Goldstein for stimulating conversations. 71

72

Chipman

where rank A = &, the square of the normalized length of /3 (i.e., /T/?/ r, this m atrix X has rank r, by Theorem 1. Let a singular-value decomposition of X be denoted X = Q D P'

(6)

where D is an n x k diagonal m atrix of the form

0 0 0

S

D =

(7 )

and S is an r x r diagonal m atrix d iag jsi, s 2, . . . , sr } with s\ ^ £2 ^ ^ sr > 0 . The main task of the proof is to show th at D = where the latter is obtained from D in the manner described in the statem ent of the theorem. Define D = Q 'X P , (8) and denote Du D 21

D

D\2 D 22

(9 )

where D n is of order r x r. D has rank p. Owing to the orthogonal invariance of the Frobenius norm, it follows from ( 6 ) and ( 8 ) th at (1) is equivalent to \\D -D \\=

min \\D - D\\. rank d 2i 0

D

This has the same rank as 5, which is r; hence the matrix X = Q DP ' has rank r, and by the orthogonal invariance of the Frobenius norm, ||* - 1 || = ||* - Q D P 11| = WQ’X P - D\\ = ||D - D\\. But from (9) and (12), Du —S 0

||5-D ||

D\ 2 D 22

D\\_ — S D 2i


r, and Z)2i and £>i2 have been shown to be zero, we can find a partition D 22,11

D 22 =

d

2

^ 22,21 D 22 22

of D 22 such that £>2 2,11 7^ 0 and the n x k matrix

D =

Dn 0

0

0

I>22,11 0

0

0

0

has rank r. Accordingly,

\\

d

-

d \\

=

0 0 0

£>•22,21

D 22,12 D 2 ^ 22,22

~ Dn - S
22,11 £>22,21

£>22,12 £>22,22

= \\D-D\\.

>11 = r. S tep 3b. We show finally that D u — S. Suppose not; then, defining

D =

Dn

0

0

0

this matrix has rank r, and \\D-D\\=

0n

0

0

£>22


n-5

0

0

£>22

= \\D-D\\,

76

Chipman

leading to a contradiction, as before. Therefore D must be of the form (11). S te p 4. Now let A 22 = Q22R P 22

(13)

be a singular-value decomposition of the (n — r) x ( k — r) m atrix Z)22, where Q22 and P22 are, respectively, (n — r) x (n r) and ( k — r) x ( k — r ) orthogonal matrices, and R is an (n — rj x ( k — r) diagonal m atrix of singular values of D 22. Define further the partitions P = [Pi, P 2] and Q = [Qx, Q2] of P and Q into their first r and last k — r and n — r columns, respectively. Finally, define the rectangular n X k diagonal m atrix S

D

(14)

0

and the k x k and n x n matrices p = [PuPi

0

Ir

0

Q

P22

—

[ Q

i? Q 2

Ir

0

0

(15)

Q 22

which are readily verified to be orthogonal. Then we verify from (15), (14), (13), (11), and (8 ) th at Q D P ' = Q D P ' = X; (16) thus, Q DP ' is a singular-value decomposition of X , in accordance with (5); and from the orthogonal invariance of the Frobenius norm, m

= m

- i mi .

On the other hand, it is clear from (15), (7), and (6 ) that Q D P' = QDP' = X , so th at Q D P ' is a singular-value decomposition of X . It remains to show th at establishing (4). From (16), (14), and (13) we have

(17) D

= Z)(r),

(18) so th at X has the diagonal elements of S as r of its singular values, and the sum of squares of its remaining m — r singular values is equal to ||^ 221|2- From (16), (17), (14), (7), and (13) we have \ \ X - X \ \ = \ \ D - D \ \ = \\D22\\ = \\R\\. (19) Since by hypothesis, (19) is a minimum (satisfying ( 1)), this can only be the case if, in (18), the diagonal elements of S are the r largest singular values of X , and those of R are the m — r smallest (with possible ties). It follows that, if the singular values of X are ordered as Si ^ s 2 ^ ^ sr > s r+1 > . . . > sm, S must contain s x, s 2, . . . , s r , and R must contain sr+1, . . . , s m. (If sr = sr+1,X is not unique.) Applying this requirement to (7) and (14) we have D = D(r) and the main part of the theorem is proved. S te p 5. Finally, let X = Q D P 1 be any other singular-value decomposition of X and let I)(r) be obtained from D by replacing all but a set of its r largest singular values by Os. Define X — Q D ^ P ' . Then by the orthogonal invariance of the Frobenius norm we have \\X - 1 || = ||Z> - A ,)ll = \\D - D(rill = II* - * ||. l

77

Eckart-Young Theorem

This may be compared with the theorem as presented by Golub and Kahan (1965, p. 220), who proceed as follows1 (where I have substituted the notation of the present paper): “TH EO REM ” Let X be an n x k m atrix of rank p ^ k < n and let its singular-value decomposition be given by (5), where D is an n x k diagonal m atrix of the form D =

0 0 0

S

and S is a p x p diagonal m atrix of singular values of X in descending order si > s 2 ^ ^ sp. Let X r be the set of all n x k matrices of rank r < p. Let D be the n x k diagonal m atrix obtained from D by setting all but its r largest singular values equal to zero, and define X = QDP'. Then \ \ X - X \ \ £ \ \ X - X \ \ for all x e a ;. “Proof” :

From the orthogonal invariance of the Frobenius norm, \ \ X - X \ \ = \\D -Q 'XP\\.

Denote D = QfX P . Then ||D - D\\2 = U s , - d , , f + j= 1 i^j

> U s , - d,,f. j=1

Since \\X — X \\2 = ||D - J9 ||2 = Z^j=r+l )L r+1 it follows th at \\X — X\\ is minimized when 0 otherwise. = Sj for j = 1, 2 , . . . , v and As noted above, the set X r is not closed; and no use in the proof is made of the hypothesis th at X , hence Z), has rank r. However, the problem is simpler: the last sentence asserts in effect th at k U j

k * i -

=1

4 ) 2^

E

Sj

for all

= Q ' X P such th at X E X r.

D

j= r+ l

Suppose k = p = 3 and r = 2 , and let Si = 3, s 2 =

D

2 , and

3

3 3

2

2 2

1

1 .5

s3 = 1 . Then the m atrix

provides a counterexample to the statem ent. (By setting all elements in the bottom row of b equal to .5, one would obtain the same result but violate the rank condition.) Thus, nondiagonal b would have to be disposed of by a separate argument (cf. Styan, 1976).

THEOREM 3 Let X be a fixed element of X and have rank p < m = m in(n, fc), and for any r < p let X be any m atrix E X r that is closest to X . Then there exists a singular-value ^ e e also Problem 10 in section lf.3 of Rao (1965, p. 56; 1973, p. 70), and section 21 of Chapter 6 of Ben-Israel k Greville (1974, pp. 246-9), as well as the revision in Ben-Israel k Greville (1980, pp. 246-9), where a few more references will also be found.

78

Chipman

decomposition X — Q D P ' of X , where dx

~

D =

and d\ ^ d2 ^

^ dp > 0, such th at, defining a m atrix D by di

D =

0 where dr > 0, we have X = Q D P ' . Thus, X has rank r. Proof: Let A G X be a m atrix having one of the three patterns (block decompositions) * *

0

O

0

0

*

0

1 0

0

°

5

0

where the northwest block is r X r and the southeast block is (n — r) x [k — r). Let X E X r be a m atrix closest to X (which exists by Lemma 1), and let X = Q D P ' be a singular-value decomposition of X such that ~ Dx 0 D =

0 0

where D\ has diagonal entries di > d2 ^ ^ dr > 0. Define, for real £, and for A of one of the three patterns above, X{ t) = X + t Q A P' . Then X ( t ) is certainly still in X r. Since X £ X r minimizes \\X — X \ \ 2, it follows that d s ( i i ^ - x (i) r ) L = o dt for A of all three patterns above. Multiplying this out and taking the derivative, we obtain ( X - X , Q A P ’) = Q. Rearranging, this is ( Q ' X P - D , A ) = 0. Since this holds for A of all three patterns above, it must be th at Q ' X P — D is of the form Q'XP - D

0 0 0 Z 22

79

Eckart-Young Theorem

where Z 22 is some (n —r) x (k — r) m atrix. It follows then th at Q'XP =

Di

0

0 Z 22

Let A 22 and B 22 be orthogonal matrices of orders n — r and k — r respectively, such th at Z 22 has singular-value decomposition Z 22 = A 22D 2i? 22>where

0 D2 = with dr+1 ^ dr+2 ^

>1 =

^ dm ^ 0. Define /r

0

0 A 22

Ir

0

0 B 22

and D =

D1

0

0 D2

Then A ' Q ' X P B = D. But Q A and P B are orthogonal matrices, so the singular values of X are the diagonal elements of D\ and D 2. But the singular values of X are the diagonal elements of D\ plus adjoining zeros. Thus, defining P = P B and Q = QA, we have Q \X - X)P =

0 0 0 D2

Therefore, the distance between X and X is I I * - * l l = (d2r+l + . . . + d 2m) L >. The perm utations of the singular values which minimize this distance are obviously those for which the m — r singular values dr+1, . . . , dm are the m — r smallest and the r singular values d i , . . . ,dr the r largest. Since X has been defined as a m atrix closest to X in X r, Di must contain the r largest singular values; and the singular-value decomposition X = QD P' was already chosen so th at the singular values of X , which by the above are the r largest singular values of X , are in descending order. Likewise the singular-value decomposition Z 22 = A 22D 2B 22 was chosen so th at the singular values dr+1, . . . , dm are in descending order. Hence, Q X P ' arranges all the singular values of X in descending order. Since r < p and dp > 0, clearly dr > 0 and X has rank exactly r. M We conclude with an extremely simple proof of necessity of the Eckart-Young condition furnished to me by Heinz Neudecker, which is contained in the Appendix following.2

2For the methodology followed see Magnus and Neudecker (1991), pp. 358ff.

80

Chipman

A P P E N D IX : A P roof of the Eckart-Young Theorem HEINZ NEUDECKER, University of Amsterdam, Amsterdam, The Netherlands Let X be closest to the given n x k m atrix X in the Frobenius norm. Its rows may be expressed without loss of generality as linear combinations of r 1 x k orthonorm al vectors, i.e., X = A B ', B ' B = I r where A is n x r and B is k x r. We therefore wish to find A and B th at solve the problem Minimize tr ( X —A B ' ) ' ( X —A B ')

subject to

B ' B = I,

Maximize 0 = 2 tr B A ' X —tr A'A

subject to

B ' B = I.

or equivalently,

Setting up the Lagrangean expression p = 2 tr B A ' X - tr A'A - tr L ( B ' B - 7), we see without difficulty th at since B ' B is symmetric, without loss of generality the La­ grangean multiplier m atrix L may be taken to be symmetric. Using this symmetry we obtain for variations in A and B dp =

2 tr

( B ' X ' - A')dA + 2 tr { A' X - LB' )dB .

Setting dip — 0 for arbitrary dA and dB yields, with the given constraint, (i)

XB = A

(ii)

A ' X = LB'

(iii)

B ' B = I.

From these three equations we obtain A'A = A ' X B = L B ' B = L whence L is also positive definite. From the first two equations and the sym m etry of L we obtain X ' X B = X ' A = B V = BL. From these it follows th at ■0 = 2 tr B A ' X - tr A' A = 2 tr B L B ' - tr L = tr L, which is to be a maximum. W rite L = TAT' where T is orthogonal and A is diagonal, and define A = AT,

B = BT.

81

Eckart-Young Theorem

Then A' A = T ' A !A T = T ' L T = A and B ' B = T ' B ' B T = T ' T = /. Equations (i) to (iii) above then become (i')

XB = A

(ii')

A ' X = T L B ' = T ' L T B ' = AB'

(iii')

B ' B = I.

From these equations it follows th at X ' X B = BA

and

B ' B = I.

Thus, A, whose trace is to be maximized (being equal to the trace of L), is a diagonal m atrix of r eigenvalues of X ' X , and B is the m atrix whose r columns constitute an associated orthonormal set of r eigenvectors of X ' X . A is maximized when these r eigenvalues are a set of r largest eigenvalues of X ' X . ■

REFERENCES Aiken, John G., John A. Erdos and Jerome A. Goldstein (1980). U nitary Approximation of Positive Operators, Illinois Journal of Mathematics, 24 (Spring): 61-72. Ben-Israel, Adi, and Thomas N. E. Greville (1974). Generalized Inverses: Theory and Ap ­ plications. New York: John Wiley fc Sons. Reprint edition with corrections, Huntington, New York: Robert E. Krieger Publishing Company, 1980. Chipman, John S. (1978). Towards the Construction of an Optimal Aggregative Model of In­ ternational Trade: West Germany, 1963-1975, Annals of Economic and Social Measurement, 6 (W inter-Spring): 535-554. Chipman, John S. (1983). Dynamic Adjustment of Internal Prices to External Price Changes, Federal Republic of Germany, 1958-1979: An Application of Rank-Reduced DistributedLag Estim ation by Spline Functions, Quantitative Studies on Production and Prices (Wolf­ gang Eichhorn, Rudolf Henn, Klaus Neumann, and Ronald W. Shephard, eds.), Wurzburg: Physica-Verlag, Rudolf Liebing GmbH, pp. 195-230. Chipman, John S. (1985). Testing for Reduction of Mean-Square Error by Aggregation in Dynamic Econometric Models, Multivariate Analysis - VI. Proceedings of the Sixth Inter­ national Symposium on Multivariate Analysis (Paruchuri R. Krishnaiah, ed.), Amsterdam: North-Holland Publishing Company, pp. 97-119. Eckart, Carl, and Gale Young (1936). The Approximation of One M atrix by Another of Lower Rank, Psychometrika, 1 (September): 211-218. Eckart, Carl, and Gale Young (1939). A Principal Axis Transformation for Non-Hermitian Matrices, Bulletin of the American Mathematical Society, 45 (February): 118-121.

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Fisher, W alter D. (1962). Optim al Aggregation in M ulti-Equation Prediction Models, Econometrica, 30 (October): 774-769. Fisher, W alter D. (1969). Clustering and Aggregation in Economics. Baltimore: The Johns Hopkins Press. Goldstein, Jerome A., and Mel Levy (1991). Linear Algebra and Q uantum Chemistry, American Mathematical Monthly, 98 (October): 710-718. Golub, G., and W. Kahan (1965). Calculating the Singular Values and Pseudo-Inverse of a M atrix, Journal of the Society for Industrial and Applied Mathematics, Series B, Numerical Analysis, 2 (No. 2 ): 205-224. Householder, A. S., and Gale Young (1938). M atrix Approximations and Latent Roots, American Mathematical Monthly, f 5 (March): 165-171. Magnus, Jan R., and Heinz Neudecker (1991). Matrix Differential Calculus with Applications in Statistics and Econometrics. Chichester and New York: John Wiley & Sons. Reprinted 1994. M arquardt, Donald W. (1970). Generalized Inverses, Ridge Regression, Biased Linear Esti­ mation, and Nonlinear Estim ation, Technometrics, 12 (August): 591-612. Meyer, R enate (1993). Matrix-Approximation in der multivariaten Statistik. Aachen: Verlag der Augustinus Buchhandlung. Mirsky, L. (1960). Symmetric Gauge Functions and Unitarily Invariant Norms, Quarterly Journal of Mathematics, Oxford Second Series, 11 (March): 50-59. von Neumann, John (1937). Some M atrix-Inequalities and M etrization of Matric-Space, Tomsk Univ. Rev., 1: 286-300. Rao, C. Radhakrishna (1965). Linear Statistical Inference and Its Applications. New York: John Wiley & Sons. 2 nd edition, 1973. Rao, C. Radhakrishna (1979). Separation Theorems for Singular Values of Matrices and Their Applications in M ultivariate Analysis, Journal of Multivariate Analysis, 9: 362-377. Rao, C. Radhakrishna (1980). M atrix Approximations and Reduction of Dimensionality in M ultivariate Statistical Analysis, Multivariate Analysis - V. Proceedings of the Fifth Inter­ national Symposium on Multivariate Analysis. (Paruchuri R. Krishnaiah, ed.), Amsterdam: North-Holland Publishing Company, pp. 3-22. Rao, C. Radhakrishna, and George P. H. Styan (1976). Notes on a M atrix Approximation Problem and Some Related M atrix Inequalities, Indian Statistical Institute, Delhi Campus, Discussion Paper No. 137, March. Schmidt, Erhard (1907). Zur Theorie der linearen und nichtlinearen Integralgleichungen. I. Theil: Entwicklung willkiirlicher Funktionen nach Systemen vorgeschriebener, Mathema­ tische Annalen, 63: 433-476. Sondermann, Dieter (1980). Best Approximate Solutions to M atrix Equations under Rank Restrictions. Report No. 23/80, Institute for Advanced Studies, The Hebrew University,

Eckart-Young Theorem

83

Mount Scopus, Jerusalem, Israel (August). Stewart, G. W. (1973). Introduction to Matrix Computations. New York: Academic Press. Stewart, G. W. and Ji-guang Sun (1990). Matrix Perturbation Theory. San Diego: Academic Press, Inc. Styan, George P. H. (1976). ‘T h e Berlin Notes” (MS).

An Analytic Semigroup Associated to a Degenerate Evolution Equation ANGELO FAVINI* D ipartim ento di M atem atica, University di Bologna, Piazza di P orta S. Donato, 5, 40126 Bologna (Italy)

JEROME A. GOLDSTEIN** Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152 SILVIA ROM ANELLI * D ipartim ento di M atem atica, U niversita’ di Bari, via E. O rabona, 4, 70125 Bari (Italy)

1 . I n t r o d u c t i o n It is well known th a t an im portant diffusion process is described with the help of the differential operator Au( x) := x ( l —x)u"{x)

x

6 (0 , 1)

whose dom ain D( A) includes the so-called Wentzell boundary conditions, i.e. lim

x->0+ ,x —^1~

A u( x) — 0 .

the corresponding semigroup has been studied by m any authors since Feller’s work [9]. It arises in m any ways in the applications, for instance, in a diffusion approxim ation limit *Supported by M .U.R.S.T. 60% and 40% and by G.N.A.F.A. of C.N.R. **Partially supported by a USNSF grant.

85

86

Favini et al.

for a sequence of Markov chains related to the W right - Fisher model in genetics (see [8 ], C hapter 10). From the point of view of the generation problem, the results of Clement and T im m erm ans in [6], assure th a t A w ith dom ain D ( A ) := {u

6

C [0 , 1] fi C 2 (0 , 1 )|

A u( x) =

lim x -> 0 +

0}

is the generator of a C 0-contraction semigroup on C[ 0 , 1] equipped with the sup-norm H-Hoo and m any interesting consequences are derived in approxim ation theory as shown in the m onograph [1]. A subsequent direct approach to the study of existence and uniqueness results concern­ ing Cauchy problems associated to the p artial differential equation , x d 2u , . du, x a ( x ) - ( x , t ) - — (x,t) = Q, with boundary conditions

0
0 for all x € [0,1], was given in the space f f ^( 0, l ) by Fichera in [10], highlighting also other properties concerning, in particular, the eigenvalues of A. Hence, in a n atural way the question arose if analyticity holds for the semigroup generated by (A , D ( A )) in some of the above considered spaces. Stim ulated by these investigations, in [2 ] A ttalienti and Romanelli examined the more general problem of analyticity for CQ-semigroups generated by differential operators of the type A au = au" on C [0 ,1] with Wentzell boundary conditions, provided th a t a E C [0 ,1] and a(x) > 0, for x E (0,1) and a(0) = 0 = a( l ) . Unfortunately, assum ptions on a leading to analyticity of the semigroup gave rise to some restrictions of A a included the condition th a t /

j (0 ,1)

—r = = dx = -f oo,

which obviously fails when a (x ) := x ( l —x). Recently, analyticity of the semigroups generated by operators of the type A a , w ith or w ithout Wentzell boundary conditions, in L p-weighted spaces (1 < p < oo), was studied in [3]. In particular, for a(x) x ( l —x), it follows th a t if D ( A a ) is defined as the completion of Co°( 0 , 1) in the norm

then ( A a ^ D ( A a )) generates an analytic semigroup on £ i ( 0 , l ) := { u e L?OC(0 , 1)| f J (

o,i) »(£)

< +oo}.

O ur purpose, here, is to give an explicit description of the domain of A in H i (0 , 1 ), which allows us to obtain the analyticity of the associated semigroup. Interesting consequences

87

Analytic Semigroup and Degenerate Evolution Equation

are derived in connection w ith the adjoint problem (see [5]). It is also shown th a t the operator A (with suitable domain) generates a holomorphic semigroup on W 1,p( 0 , 1 ), for 1 < p < oo. This work was completed during the visit of J.A. Goldstein at the Universities of Bari and Bologna, in May 1996. The authors are grateful to G.N.A.F.A. of C.N.R. for having supported this invitation and J.A. Goldstein is m ost grateful for the exceptional hospitality of his two coauthors and Enrico Obrecht during this visit and a previous visit to Bologna and Bari in 1994, when prelim inary insight into this research was initiated. 2. M a in r e su lts in H ilb e r t sp a ces given by D ( A) := {u

e

Let us introduce the operator A on i/* ( 0,1)

Hq ( 0 , 1)| u;/exists (in the sense of distributions) w ith x ( l- x ) ii//

e

H ^ ( 0 , 1)}

and Au

x ( l — x)u",

for u e D ( A ) .

We have T h eo rem 1 . ( A , D ( A ) ) generates a uniformly bounded semigroup analytic in the right half plane on if* ( 0 , 1). Proof First of all, let us consider H q (0, 1) endowed with the inner product < u^v> \— /

u ( x ) v f( x ) d x

u, v

G H i (0,1),

0 , 1)

th a t is equivalent to the usual inner product / u(x)v(x) dx + / u ( x ) v f( x ) d x J{ o,i) ^(M) in view of the Poincare inequality. To m otivate our choice of the space, we observe th a t if Re A > 0 with (1)

Xu - x ( l - x) u" = /

G H l ( 0 , 1),

u

e

D( A) ,

then nn

\

u

u"=

L

____ ___

x ( l — x)

x (l —x)

implies necessarily (2 )

f

u ' ( x ) u ( x ) dx = -

7(0,1)

th a t is, [u' ( x ) u ( x ) ]^ Z q vanishes.

f j { 0,1)

\ u ( x ) \ 2 dx,

Favini et al.

88

Indeed, f u"( x) u( x) dx = [uf ( x ) u ( x ) ] ^ Z q — [ \u'(x)\2 dx. J(Od) J(OA)

(3) Moreover, f A /

\u{x)\2 f r dx —

-7 -

J{ 0 , 1 ) X ( I “ x ) Now /

/

7(0,1)

e

U( v_, v j f u (x ) u ( x ) d x =

A o ,i)

f(x)u(x) ■ \ —■■ / dx.

^(1 — x )

-A m )

Hq(0, 1) gives

^x ( lr ~ ^x ) d x = J( i 0 ± )tX(t1 — - X) Jo

1J(±yl )V r /'w * \ 2d x x (l -- !x )1Jx

^ [

[ t t - t j I 1 \ f ( t ) \ dt)2 J {0 l ) xw( l r- ~x ) ^Jof \ f ( t ) \ dt)2d* + J(±f 1) zX\1 ~ x ) Jx

< f

— -— -(/

J{ 0,-i) X(1 _ x ) Jo

(4)

12 d t ) ( f

Jo

\f{ t)\2dt)dx+

+ J(-k, f ,J_ x )AJxf 12dt){[Jx \f{t)\2dt)dx ' ( i i 1)) < 2 ||/|| h i ,

f (X i U [ X ) and therefore ffn u — ------ dx converges since u 6 H%(0,1). This implies th a t u" u is (u,i; — X) sum m able on (0,1) and (3) is verified. On the other hand, this also implies th a t both limits lim u ( x) u( x ), X-—>-0+

lim u (x)u(x) x-+l~

exist and belong to C. In order to show th a t they vanish, we prove th at, for all u lim u ' ( x ), £ —►0+

exist and are in C. To see this, observe th a t x ( l - x) u" = g

6

lim u'(x) x —>1 ~

G H ^ Q , 1), so th a t

Is0*01 < V ^( [ W( t) \2 dt)i < Cy j x Jo for a suitable constant C depending on u. Analogously,

D( A) , the lim its

g(x) = Jq g'{t) dt yields

89

Analytic Semigroup and Degenerate Evolution Equation

gives

\g(x)\ < C V 1 - x,

x

e

(0 , 1).

Hence, 0 < y < x < i implies

!« '( * ) - « '( y ) | = | ( X u"{t)dt\ = \ [ Jy

Jy

f

‘ #1(1- - ^i ) 1

< /

,y

i)( [

7(o,i)

x ( l - x ) \ u " { x ) \ 2 dx)±.

Notice th a t x ( l - x ) u n E H ^ ( 0 , 1 ) implies [

——-— - \ x ( l - x ) u "( x )\ 2 dx = f

7(o,i) x \l ~ x )

x ( l - x)\u"{x)\2 dx < +oo,

7(o,i)

by the above rem ark. Hence x { \ - a;)|u"(a ;)|2 dx)* < 4 ||/ ||Hi(o,i)-

( f 7(0,1)

Moreover, since f^Q^ f ( x ) u " ( x ) dx is convergent by

f

_ - a /x (1 - x ) u f/(x) d x ,

f ( x ) u " ( x ) dx = f

7(o,i) > /x (l - x)

7(o,i)

and the Cauchy-Schwarz inequality, we deduce th a t [

f ( x W ' ( x ) dx = [f {x ) u{ x )] x z l - [

7ro.n

/'(x)TZ'(x) dx

7ro,i)

and then both limits lim / ( x ) i/( x ) , x->0+

lim f ( x ) u ' ( x ) x— >1—

exist. Since / G 7f*(0,1) and both lim its lima._^0+ u f(x) and l i m ^ i - u'(x) belong to C, we conclude th a t for all u verifying ( 1) we have f

f( x ) u n(x)dx = -

7/ (0 , d) ( o ,1

I

/'(x )i7 (x ) dx.

7 (0 ( o,,1d)

Therefore, from (4), rew ritten as x IK lli2 + [

7(o,i)

x ( l —x )\u"(x)\2 dx = f f'(x)u(x)dx, 7(o,i)

Analytic Semigroup and Degenerate Evolution Equation

91

we deduce the a priori bound |A|||u||ifi
=

(:r(l —x ) u " ) fv f(x) dx 7(0,1)

= — f

7(0,1)

x ( l — x)u'f(x)v"(x) dx

= [ u ( x ) ( x ( 1 - x ) v /rY(x) dx 7(o,i) = < u,Av > for all u , v £ D( A) . Moreover x ( l —x)\u"(x)\2 dx < 0 ,

— — / 7(o,i)

so th a t A is nonpositive. On the other hand, we observe th a t, for all u , d G D( A) , < (I - A)u, v > / / i = / u ( x ) v ' ( x ) ° 7(o,i)

dx + /

x ( l — x) u" (x)v" (x) dx.

7(o,i)

Let us introduce the H ilbert space V defined by V := {u E H i (0 , 1)| [

x (l —x)\ u"(x)\ 2 dx < oo}.

7(o,i)

It coincides w ith the completion of C ^ O , 1) w ith respect to the norm \\u\\y := f \uf (x)\2 d x f 7(o,i)

x ( l — x )\uff(x)\2 dx. 7(o,i)

To see this, note th a t for u G F , /

\u (x)\2 dx —

/

7(o,i)

—u ( x ) u " { x ) d x

7(o,i)

= i?e[ [

7 (0,1)

- (av')' = g = ( f -b c);, v G i i ( 5 ) c c then, necessarily w = (u -I- —)7 = where u + — satisfies for all c, Re A > 0, A

A

A(u + ^ ) - a ( u + j ) " = f + c, a( u(x) + y ) /X A

0.

for x —>0 + , x —> I - . Therefore, taking into account th a t \ f ( x ) \ < y/x\\g\\L2 ,

x € (0 , 1 )

and, hence WfWh 0 on (0,1) and a(0) = 0 = a(l). I f (Ap , D ( A P)) is defined by D ( A p) := {u

n L p( 0 , 1)1 a u 1 €

e

W

lQ ' p {

0 , 1)}

A pu := ( a u ) ' then (A p, D ( A p)) generates a C0 - analytic semigroup on L p (0 , 1), for

1
2 . If / G L p( 0 , 1) and A G C , Re X > 0, there exists a unique u G D(Ap) such th a t (8 )

Xu - (a u ) f = f .

Let us m ultiply (8 ) by u\u\p~ 2 and integrate from 0 to 1. Thus we obtain (9)

X\\u\\p -

f

0,1)

( a u ) '( x ) \ u \ p~2(x) dx = f f (x)u(x)\u\p~2(x) d x . 7(°,l)

Defining (3 := f

f (x)u(x)\u\p~2(x) dx

7(o,i)

7

(a u /)/(x)itI(a:)|u|p“ 2 (a;) dx,

:= — / 7(o,i)

we can rewrite the equality (9) as follows AIMI£ + 7 = /?•

Analytic Semigroup and Degenerate Evolution Equation

95

By H older’s inequality, l/J| < \ \ f \ \ P \ M p - % ' = l l / I W M i r 1

where - +

= 1. Also, integration by parts yields

V

7=

V

( a u ,){x)(u(vm)2^ ~Y (x) dx

/

7( 0, 1) = f a ( x ) u ( x)uf(x)\u\p~2(x) dx-\7(o,i) +

[ 2

a ( x ) u \ x ) u ( x ) \ u \ p~A(x)(u (x)u(x) + u (x ) uf(x)) dx

7(o,i)

= j a (x )\ u (x)\2\u\p~2(x) dx + (p — 2) I a(x)\u\p~A( x ) ( u u ) ( x ) R e ( u u ) ( x ) dx. 7(o ,n 7•'(0,1) (o, d '(0,1) Taking real and im aginary p arts in (9), we deduce th at, respectively, (ReX)\\u\\p + I

a(x)\u'\2(x)\u\p 2( x ) d x + f

7 (o ,i)

=

f

a(x)\u\p 4( x)(Re (ufu)(x))

7 (o,i)

Re (f u\ u\p~2)(x) dx

7' (( 0o ,, 1d)

< u / u M i r 1’ \ I m X\\\u\\p + (p — 2) I a(x)\u\p A( x ) s i g n ( I m X ) I m ( u ,u ) ( x ) R e ( u u ) ( x ) d x 7'(o,i) ( 0, 1) = sign(ImX) f I r n ( f u \ u \ p~ 2) ( x ) d x 7Ao,i) ( o, d < n / i i p h i i r 1Thus, for 0 < c < 1 it follows th a t C|A ||M |£ < (/?.eA + C|/m A |)|M |£ < ( - 1 + c(p - 2)) I a ( x ) |u '|2(:r)M P -2 (:E) dx + (1 + c ) ||/||p ||u||p_1 J( 0,1)
2, p - 2 IMI* < ^ 1 1 /II p where cp = 2(p —2) (hence cp —>oo asp -> oo ) and this holds for all / € L 2(0,1) n L ° ° ( 0 ,1) and all A w ith Re X > 0.

96

Favini et al.

Now, we assume th a t 1 < p < 2 and observe th at, by duality we have < A u , v >=< u , A v > where A u = (a u ') ' with boundary conditions a ( x) u' (x) -* 0 as x -» 0,1 and w, u are in various spaces. Thus, formally, A* = A p>, where A p (resp. A*) acts on L p{0,1) (resp. L p>(0,1)) and (pO-1 + p ~ l = 1, for 1 < p < oo. Since \ \ ( \ - a * ) - 1\\ = \ \ $ - a ) - x this second case on p can be deduced from the first case and the estim ates used in its proof. Moreover the fact th a t R ( X I - A) (for R e A > 0) is dense in L p{0,1) follows from the L 2 case.

□ Let 1 < p < oo and define B pu := au!' for u E D ( B p), where D ( B P)

{u 6 W 1,p( 0 ,1) D

0 , 1)| B pu E W 1,p( 0 ,1) and lim a ( x) u " ( x ) = 0} x—^0,1

i.e. B p is equipped w ith the Wentzell boundary conditions. Thus, we can prove the following

T heorem 2. ( Bp, D ( B p)) (for 1 < p < oo) generates a CQ analytic semigroup on W ^ p{0,1). Proof Let 1 < p < oo and F E W 1,P(0,1) with F ( 0) = F ( 1) = 0 and take / = F f L p (0,1). From for every A E C , R e \ > 0, there is v E D ( B P) such th a t

(10)

E

Xv — ( av')f = /

with a( x) v' (x ) —>0, a sx —>>0,1. Let ifc(x)

J* v(s) ds. Then ?/ = u and integrating (10) from 0 to £ we deduce th a t Xu —au" — F

and (a?/) (a;) = (au")(x) —> 0 as x -+ 0,1. Thus u E D { B p) with (A — B p)u = F and M „ = IM IP
) := {u

e

C l O A l n C ^ O A y au' € C 1^ , 1]}

and A u := (a u ')', generates a CQ differentiable contraction semigroup on C[0,1]. Now, let F E C 1[0,1], A > 0 and consider the equation Xu — ( a u ) ' = F 1.

(11)

Since F ' E C [0 ,1], it has a unique solution u E D(^oo)> w ith IMIc[0,l] < ^ll^'||c[0,l]Hence, integrating (11) from 0 to x, we deduce th a t u(y) dy — a ( x ) u (x) = F ( x ) — F ( 0),

A / J (0 ,x )

namely

Let / x f w(x) := /

, , , F ( 0) u{ y) dy + —— .

J(o ,x )

Then w

E C 1[0, 1].

Moreover, a( x) w"( x) = a ( x ) u f(x) —>0

and a w"

A

E C x[0,1], hence

w

E D( B) .

a s# -» 0+ , 1

We also notice th a t, if

98

Favini et al.

and this implies th a t w' — 0, hence w(x) = const. Consequently w" = 0 and therefore w = 0. Thus the uniqueness holds. Now, we m ust estim ate the norm of w. To this aim, we observe th a t the norm H l i := m a x { |w (0 )|,||u /||C[o,i]} is equivalent to the usual norm IH Ic i := m ax{||w ||C[0,i], ||w'||c[o,i]} because obviously IMIl < IMIc>

and, on the other hand, w(x)~w(0)+ / w' {t)dt Jr ((00,x) ,2 ) implies th a t for every x € [0,1]

\w(x)\ < |w(0)| + f

\w'(t)\dt

J(Q ,x)

< |w(0)| + ||w/||C[o,i] < 2||w ||i. Hence \\w\\ci < m ax{2||w ||i, ||w'||c[o,i]} < 2 N l i Let us come back to our estim ate I H li = m ax{|w(0)|, ||w '||c[ 0,i]} = m a x { | t ^ | ) ||w||c[0,i]}

A

A

= imax{|J=,(0>[,||JF'|lc|o,I]} = > 1 1 ,. Then, (B , D ( B )) generates a C0-contraction semigroup on C^O, 1]. Moreover, for A in a suitable region E as described in [12], Theorem 4.7 p.54, there exists c > 0 such th a t IMIc[0,i] < c l 1 + l-^^D II-F'llcio.i]' On the other hand

Analytic Semigroup and Degenerate Evolution Equation

99

\w(0)\ = ±\F(0)\a._>i- a( x) u"( x) = 0). In particular, all previous results hold for a( x) := x ( l — x ) m( x ) , where x G [0,1] and m G C^O, 1], with m ( x) > 0 in [0,1]. F in a l rem arks. The long standing conjecture in this area concerns Au

x(l —x)uN

with Wentzell boundary conditions. By Clement and Tim m erm ans [6], A generates a CQ- contraction semigroup on C [0 ,1]. Is this semigroup analytic ? After this work was done, but while the final revisions were being made, G. M etafune kindly provided us with a preprint [11], which states th a t on CfO, 1], A generates a semigroup analytic in the right half plane. Thus, despite the boundary degeneracy, the operator u —> x(l —x)u" w ith Wentzell boundary conditions generates an analytic semigroup on many spaces of interest. R eferences 1. F. Altomare - M. Campiti, K orovkin -type A p p ro x im a tio n T heory an d its A p p lica tio n s , de Gruyter Studies in Mathematics, 17 Walter de Gruyter Co., Berlin, New York, 1994. 2. A. Attalienti - S. Romanelli, On som e classes o f an alytic sem igrou ps on C ([a ,b ]) related to R o r T a dm issible m appin gs , Evolution Equations, G. Ferreyra - G.R. Goldstein - F. Neubrander (eds) Lect. Notes in Pure and Applied Math. 168, M.Dekker, New York - Basel - Hong Kong, 1995, pp. 29-34. 3. V. Barbu - A. Favini - S. Romanelli, D egenerate evolu tion equations and regularity o f th e ir associated sem igrou ps , Funkc. Eqvc. (to appear). 4. H. Brezis - W. Rosenkrantz - B. Singer, On a degenerate elliptic-parabolic equation occurring in the theory o f probability, Comm. Pure Appl. Math. 2 4 (1971), 395 - 416. 5. M. Campiti - G. Metafune - D. Pallara, D egenerate self - ad jo in t evolu tion equations on the u n it in terva l , Semigroup Forum (to appear). 6. Ph. Clement - C.A. Timmermans, On C o-sem igrou ps gen erated by differen tial operators satisfyin g VentceVs boundary con dition s, Indag. Math. 89 (1986), 379 -387. 7. R. F. Curtain - II. Zwart, A n In trodu ction to In fin ite - dim en sion al L in ear S yste m s T h eory , Springer, 1995.

100

Favini et al.

8. S. N. Ethier - T. G. Kurtz, M arkov P rocesses , C h a racterization and C onvergence , Wiley Series in Probability and Mathematical Statistics, J.Wiley, 1986. 9. W. Feller, The parabolic differen tial equations and the associated sem i-groups o f tra n sfo rm a tio n s , Ann. of Math. (2) 5 5 (1952), 468-519. 10. G.Fichera, On a degenerate evolu tion problem , Partial Differential Equations with Real Analysis, H. Begehr - A. Jeffrey (eds), Pitman Research Notes in Mathematics Series 263, Longman Scientific and Technical, 1992, pp. 15-42. 11. G. Metafune, A n a ly tic ity fo r som e degenerate evolu tion equations on the u n it in terva l , preprint (1996). 12. A. Pazy, Sem igrou ps o f linear O perators and A p p lica tio n s to P a rtia l D ifferen tial E qu ation s , Springer Verlag, Berlin - Heidelberg -Tokyo, 1986. 13. H. Tanabe, E qu ation s of E volution, Pitman Monographs and Studies in Math., London, San Francisco, Melbourne, 1979.

Degenerate Nonlinear Parabolic Problems: The Influence of Probability Theory

JER O M E A. GOLDSTEIN* D epartm ent of M athem atics, Louisiana S tate University, Ba­ ton Rouge, LA 70803, USA CHIN-YUAN LIN D epartm ent of M athem atics, University of South Carolina, Columbia, S.C. 29208 and D epartm ent of M athem atics, N ational C entral University, Chang-Li 320, Republic of China KUNYANG WANG D epartm ent of M athem atics, Louisiana State University, B aton Rouge, La. 70803, USA

1. INTRODUCTION Of concern are mixed initial-boundary problems for the nonlinear equation ut = (x,u,ux)

(1)

for x £ [0,1] and t > 0. Here ip is continuous and positive on (0 ,1) x IR, but ip(x,£) may approach 0 as a:tends to either 0 or 1. Thus the diffusion coefficient may degenerate on the spatial boundary. Problem s like this w ith nonlinear, degenerate diffusion coefficients arise in a variety of contexts in fluid dynamics and elsewhere. The particular example dv

1 d r d/

du

a = ? f S [!, 0) arises in physics and reduces to (1) when one sets u ( t , x ) = v(t, t a n ( |x ) ) . The theory of the Kom paneets equation (2) is far from complete; in par­ ticular, well-posedness for the Cauchy problem is not yet established. (But see Goldstein [11] for p artial results and related references.) A system atic study of (1) was begun by Goldstein and Lin in [12] in 1987 and continued in [13]-[15], [17], [18]. Among the related articles we cite the interesting work of Dorroh and G. R. Goldstein [7], [5], [6] who allow ip = (p(x, u, ux ) to depend on u as well. But because this case does not adm it a global quasi-dissipative estim ate, only local existence is established in general; as global existence is our main concern, we restrict our attention to (1) here. Suppose (p(x,£) > (po(x) where 0, then I K - U2\loo < I K - M oo(C3) The graph of A is closed in X x X, andT)(A) is dense in X. The last condition is easy and we will not discuss it further. (C2) is the dissipativity of A. (C l) is the range condition. (C l) is the hard part so we begin with (C2).

106

Goldstein et al.

Let (B l), (B2) hold and let hi,X,U{, be as in (C 2 ). Let u — u\ — u 2 (or u 2 — u\ if necessary). Choose x 0 G [0,1] such th a t u ( x 0) = |M|oo- If 0 < x 0 < 1 then u ' ( x o) = 0,u"(xo) < 0 , whence 11^1 “

^ 2 11oo = U ( X 0 ) < U ( X Q) ~ \ ( f { X o , U* ( X 0 ) ) u " ( X 0 )

= (ui - Ai4ixi)(rc0) - {u2 - AA u 2) ( x0) -Xi ipi xo, Ui(x0), u [ ( x 0)) - tl)(x0, u 2(x0) , u 2{x0))} since u \ { x o) = u 2{xo), A > 0, cp > 0 , (ui — U2 )"(xo) < 0 ,




1

X + (j,

•

is a stochastic matrix corresponding to the following embedded Markov chain.

Krinik et al.

140

Diagram 2

x

with

and

q=

x + n

In particular,

_

_-(X.+n)t

I

(A, + ^i)N tN N!

N= 0

(N) PiJ

(N)

for i, j = 0, 1, 2,... where p - j is the probability of going from state i to state j in N steps for theembedded Markov chain pictured in Diagram 2. Therefore, we have an explicit solution of P j j ( t ) for the M/M/1 queueing system once we have an expression for p ^ . accomplished in the following theorem.

This is

TH E O R E M 1 Suppose i and j are any two states: i, j = 0, 1, 2,... Assume N = 1, 2, 3,...

(

|LL

and q = ----------. Then p • • , the X + fi X + \i N-step transition probability from state i to state j for the embedded Markov chain in Diagram 2 is given by is chosen and let c = i + j and d = j - i and p =

f ,( N )

_


\

j

V

2

N-d

q 2

N + c + 2m + 2

N + c + 2m

V

N -d

fq 2 p 2

N

I

when N + d is even and

N N+c+2

V

N-d

m=l

(

j

N+d

+ m ------

p 2

141

Transient Solution of the M/M/1 Queueing System N-d-1

pS}=

r

I

N

^

r -

N + c + 2m + 1

N

A

N-d+l

N+d-1

+ m ----------- m

>

N+c+2m+3

m=0

V

V

J

2

J

2

when N + d is odd. Note, we adopt the convention that ^ j = 0 if

M >

N

or

m

< 0.

PROOF. The number of sample paths in Diagram 2 from state i to state j in N steps having either 2m or 2m+l loops at state zero is provided in the following lemma. This counting result allows (N)

one to determine the contribution to p t j from sample paths having a prescribed number of loops at state zero. For any finite sequence of numbers S = si S2 S3 ... sn we adopt the notation that Sk means

Y,sn.

The following definitions are motivated by classifying sample paths from i to j in

n=l

Diagram 2 according to whether we loop at zero exactly m times, merely touch state zero without looping or never even visit state zero during our journey from i to j. D E FIN IT IO N 2 Suppose i, j are two arbitrary states from 0,1,2,3, ... and assume N=l,2,3,... is given. A sample path of length N from state i to state j in Diagram 2 may be represented by a finite sequence of N elements A = aj ao a3 ... aN such that Ao = 0 by convention, A n = j - i and

0

or

1

if

A n-1 = - i for n=l,2,...N .

an =

-1 or

1

if

A n-1 > - i

The set of all sample paths, A, of length N from i to j having exactly m of its elements equal to zero (that is, a n ^. = 0 for k=l,2,...m for some subsequence of a n ) is denoted by S y (m) where m > 1.

S y (0) will denote those sample paths A of length N from i to j for which a n

^

0

for each n=l,2,...,N and yet A n = -i for some n=l,2,...,N -l. Finally, let R y represent all the remaining sample paths A of length N from i to j; that is, those sample paths A that go from i to j in N steps such that A n ^

-i for each n=l,2,...,N -l.

The next proposition establishes that S y (m) and S y +m (0) have the same number of sample paths for m = 0,l,..., N-i-j. The following graphs illustrate how this one-to-one correspondence may be visualized when N=9, i= l, j=2 and m=0,l or 2. Start with a sample path

142

Krinik et al.

S

C^

lj2 (0) such as shown in Figure 1. If we change the right most "zero to one" segment (dashed in Figure 1) into a horizontal segment and lower the graph to the right of this segment one unit, we obtain the sample path in S f j (1) pictured in Figure 2. Repeat the process again by changing the right most "zero to one" segment of Figure 2 (the dashed segment) into another horizontal segment and once again lower the part of the graph to the right of the new horizontal segment one unit to obtain the sample path in S ^ q (2) of Figure 3. These operations are easy to reverse by successively replacing horizontal segments by "zero to one" segments and raising the graph to the right of each replacement one unit. The reasoning as to why this procedure is a oneto-one correspondence is contained within the proof of the following proposition.

1 } . Let S'c and S p c denote the complexification of S ' and S p, respectively. For any p > 1 and 0 < /? < 1, let A Pip denote the space of all complex-valued function ip on S'c satisfying the following two conditions: * Research supported by the U.S. Army Research Office grant DAAH04-94-G-0249

147

148

Kuo

(a) p is an analytic function on S'p c. (b) There exists some constant C > 0 such th a t \v{x)\

< C ex p [

J

(

l

V x e S p C.

For

1 and

J s, exp

< °°-

This theorem has been proved for the case (3 = 0 by Lee in [4]. However, the proof for the necessity p art in [4] cannot be adapted to the case ( 3 ^ 0 . To prove the sufficiency of the above theorem , let ip G Ap. Then /

\ v (x )\dv (x)

J S'P

= ^(M z)|exp

- ^(1 + /? )M I+ '3] ) exp ^(1 + ( 3) \ x\ iy ] du(x)

< M \ p.P j ex p [ h l + / ? ) |a ; |i p 'J] dv{x). JSP This implies th a t the linear functional = /

\ i A * j3, there exists some q > 1 such th a t $ p e A* p and $A q > 1 we have

|z |_ p =

\A ~

px

\q

= \ A - b - d A - qx\o < 2 ~ {p~q)\ A - qx\o — 2~(p~q^\x\-q.

151

Characterization of Hida Measures

We can choose large p such th a t p > q +

j

s,e xp

+

p

^ x \^ p )

Then

dv- J s,e xp

d v (x )

Thus by equation (4) we have f exp ^ ( 1 + P ) \ x \ t ^ | dv{x) < e3^1+/?^ 2 $ Iy('0) < oo. B ut |x |_ p = oo for any x € S 1 \ S p. Thus the last inequality implies th a t the measure v is supported on S p and

This completes the proof of the theorem. E x a m p le . The probability measure v \ in Section 1 is a Hida measure of order 1 — A. Hence by the above theorem , it is supported in S'p for some p > 1 and we have

REFEREN CES [1] Kondratiev, Yu. G. and Streit, L.: Spaces of white noise distributions: Construc­ tions, Descriptions, Applications. I; Reports on Math. Phys. 33 (1993) 341-366 [2] Kuo, H.-H.: Gaussian Measures in Banach Spaces. Lecture Notes in M ath. 463, Springer-Verlag, 1975 [3] Kuo, H.-H.: White Noise Distribution Theory. CRC Press, 1996 [4] Lee, Y.-J.: Analytic version of test functionals, Fourier transform and a characteri­ zation of measures in white noise calculus; J. Funct. Anal. 100 (1991) 359-380 [5] Reed M. and Simon, B.: Methods of Modern Mathematical Physics I: Functional Analysis. Academic Press, 1972

New Results in the Simplex Method in Linear Programming ROGER N. PEDERSEN Pittsburgh, PA 15213

Departm ent of M athematics, Carnegie Mellon University,

“N otation is im portant. It can even solve problems. But, at some point, you must do some work yourself.” K. 0 . Friedrichs.

1.

IN T R O D U C T IO N A N D STA TEM E N T O F T H E P R O BLEM .

W ithout using any symbols at all, we can give a precise statem ent of the problem by saying th a t it is to find the maximum, if it exists, of a linear function of a finite number of real variables on a convex plane polyhedron of the same variables. The simplex m ethod of solving the problem is then to find a vertex of the polyhedron and then to proceed along edges from one vertex to the next, in a m anner th at the linear function increases, until the maximum is reached. All the d ata needed to state and solve the problem can be stored in an (ra + 1) x (n + 1) m atrix A. The analytical statem ent of the problem then is to find the maximum of the objective function n (l.i)

subject to the constraints n

y ) A i j X j + A itn+1 > 0, i = 1 , m. l=i 153

(1.2)

154

Pedersen

By defining A to be the m atrix comprising the first m rows and first n columns of A and b to be the transpose of A.ijn+i,..,An,n+i, the constraint (1.2) takes the simpler form Ax + 6 > 0,

(1.3)

meaning,of course, th a t each component of the column vector isnon-negative. The vector x issuperfluous for the purpose of applying the simplex algorithm. But, working only with the m atrix A , can lead to misconceptions as we shall see in the next section. But first, let us find another notation for the constraint set by using Ai to denote the rows of A. Then (1.3) can be replaced by Li(x) = ( A i x ) + bi > 0, z = l , m where bi is the

(1.4)

coordinate of b and (,) represents the canonical inner product.

2. BUT, THOSE SLACK VARIABLES ARE UNNECESSARY. Let us re-write (1.3) as {A C) ( C ~ 1X ) + b > 0

(2.1)

where C is any non-singular n x n matrix, noting th at this does not require an equality. Now, assuming A has rank n, we may apply elementary column operations to reduced echelon form. If C is the product of the corresponding elementary column matrices and y = C ~ l X , the first n coordinate of (2.1) are Vi + h > 0. (2.2) Then, by making the translation Z{ = yi + b{, we may assume the constraint set to be in, what is commonly called, canonical form. Furthermore, if for one j, 1 < j < n, we put Xj = Zj —f3j in (1.1),(1.2) we see th a t this corresponds to multiplying the column of the full m atrix A by fa and subtracting it from the (n + 1 ) ^ column; th at is, it is an elementary column operation. I prefer doing elementary row operations on the transpose. Thus the simplex m ethod reduces to transposing the m atrix A and applying elementary row operations until the first n column are in reduced echelon form, with the restriction th a t the pivots are to be picked from the first n rows of A T. The only question th at remains is when to start using the simplex pivoting strategy. After the system is in canonical form, we must use the simplex strategy; before th a t we may use instead the standard Gaussian Elimination Strategy. Note th a t the simplex strategy requires picking the maximum positive element of the current column and hence is a partial pivoting strategy. We shall have more to say about this in Section 5.

3. EMPTY SETS, REDUNDANT CONSTRAINTS AND LOWER DIMENSIONAL SETS. Let us now suppose th a t the normals of the first n constraints form a linearly independent set. Then, for any k > n, n

Ak and hence

^ C^kiAi 2—1

(3.1)

n

L k{x) =Y1 a kiLi{x) + 2=1

(3.2)

155

Simplex Method in Linear Programming

with ^ k — A k yn+l

n 'y i=,1

1.

It follows from (1.4) and (3.2) th at if ( a ^ i,.,..., a^)n, A k) are all non-negative, the kfi1 constraint is redundant and th a t if they are all negative the set is empty. If for some i < n, a.ki > 0, oikj < 0 for j / i and A*, < 0, then the constraint is redundant. In all other cases where none of the numbers ( o ^ i , o ^ , A*,) is zero it is easily shown th a t the set formed for the first n and the /cth , is non-empty. The other im portant special case occurs when A^ = 0 and < 0 for i — 1, ...,n. Then the entire constraint set is contained in the set where Lk(x) = 0. Hence, we may use this constraint to eliminate a variable and obtain a lower dimensional set. This means that, by reducing the number of dimensions, we may assume th a t this case does not occur. We note from (3.2) and (3.3) th at, when the constraint set is in canonical form, Ai,n+\ = 0, i = 1,..., n, so the % ’s and A& are just the coefficients of the constraint equation. From this point on we shall assume th a t the set is in canonical form. The origin will be called the basic vertex, the first n constraints the basic constraints and the rest of the constraints the non-basic constraints. 4.

THE SIMPLEX ALGORITHM WITH A NON-DEGENERATE BASIC VERTEX.

A vertex which is the intersection of more than n-planes is called a degenerate vertex. This means th at, when the basic vertex is non-degenerate, all of the non-basic constraints have non-zero constants. The simplex strategy then is to increase by one the number of positive constants among these until they are all positive and then to increase the constant in the objective function. Let us assume th a t the constraints are ordered so th at Ai n+i > 0 ,

i 0 there is no maximum while if Am+i,k < 0 we may set = 0 and continue in one less dimension. If p < m we simply set t = p, observing th at the simplex method requires only one step. Next, we interchange the non-basic constraint with the k basic constraint and put the constraint set back into canonical form. This requires applying Gaussian elimination to the k ^ 1 column of A T . The new elements of the m atrix then are A f^k ~

(4-4)

156

Pedersen

A j =

M,k

j ^ k

(4-5)

and when i / £,

A ,k = 4 ^ A j = A
0 and th a t when Aik < 0 it is positive as a consequence of the choice (4.3) of t. Hence, in any case, the first p — 1 constants remain positive and if £ = p, *4p,n+i is also positive and we have increased p by one. B ut we also see from (4.7) th at if I < p, Ap jTl+ 1 > A p,n+1

(4-9)

Since the constraint set has only a finite number of vertices, we shall, in a finite number of steps either find the set to be empty, prove th at A'p n+i > 0 or arrive at a degenerate vertex. 5.

THE CASE OF A DEGENERATE VERTEX.

The case of a degenerate vertex occurs when there are zero constants Ai,n+i — 0. Suppose th a t we apply the previous strategy to the basic constraints and the non-basic constraints with non-zero constants. Then we see from (4.7) th at when A i )U+i = 0, -< n +i = - ( ^ ) A t

(5-1)

and since At j < o, Atk > 0, we have A'in+l > 0 whenever Ai $ > 0. There is no reason th a t this should be the case, but, by applying the simplex strategy to the first n columns of A , with the playing the roll of the constants, we can use the simplex strategy to achieve this. Because the algorithm is slightly more complicated when the degeneracy is of higher order, it is convenient to introduce constants a ^ P k satisfying, after reordering the constraints and variables A i tk = 0, n + 1 < i < a k > 0 , a k < i < /3k
k — 1 , are all zero the elementary row operation correspond to adding zero to the rows of A T indexed by j > k. Hence the otj s and /? /s, j > k so they are unchanged. We now redefine the a / s and f3jS for j < k returning to [2].

[9] R eturn to [2]. The program will term inate at [10]. [10] The maximum is *4.m+i,n+iWe have tacitly assumed the maximum to exist, leaving to the reader the task of adding the lines, explained in Section 3, regarding empty sets, redundant con­ straints, lower dimensional problems and problems with no maximum.

6.

SMALL PIVOTS AND DEGENERATE VERTICES.

In running the above algorithm, it is crucial th a t one distinguish between non-zero numbers and zeros represented by round-off errors. The author has studied this problem extensively on the Radio Shack Color Computer and on the Tandy 1000. Computing, respectively, to 9 and 16 places, base 10, the Random Number generator was used to supply the d ata and, computing to p places base 10 , the test for determining whether or not a number is zero was by comparison with 107-p, 2 < 7 < p/2. In order to increase the probability th a t the set is not empty, the probability th a t the origin satisfying a constraint is set at 7r, 0 < it < 1. W ith

158

Pedersen

no other restriction, a degenerate vertex has never been found. By building in the condition of degeneracy, e.g. by applying a similarity transform ation to a known degenerate situation and adding more constraints, the program seems to work as well as in the non-degenerate case. The problem, in each case, is checked by re-running the program on the constraints forming the final basic vertex and by evaluating the objective function at the intersection of their planes. We have also never found an ill-conditioned m atrix with the random number generator. By putting in the Hilbert m atrix [2], prob. 169, p. 337, we find the obvious difficulty. However, by computing to a sufficient number of places, we have always been able to overcome the difficulty.

7. FURTHER METHODS OF SPEEDING UP THE PROGRAM. The Simplest Method of Speeding Up the Program is to remove the redundant constraint using the test of Section 3, noting th at the test requires only sign-tests of quantities th a t are computed anyway. Its disadvantage is th at a constraint th at shows up as redundant in one coordinate system does not necessarily in another. The number of degenerate constraints can be increased by adding the condition th at the objective function be greater than its value at the current basic vertex. Another m ethod of possibly speeding up the program is to use the fact th a t once a vertex has been found we know th at the constraint set is non-empty. Then we can eliminate a variable using any of the constraints. If the constraint used was redundant, the new set will be empty. Otherwise, we obtain the maximum on an (n - l)-dimensional face. The weakness of this m ethod is th at we lose time when we use a redundant constraint to eliminate a variable.

8.

THE STATEMENT OF THE CONDITION THAT THE SET BE EMPTY OR CONTAIN A REDUNDANT CONSTRAINT.

In this section we iterate the formulas (4.4) - (4.8) for the constraint set n

Y -AijXj + .4.,n + i > 0 , i = 1 , m

( 8 .1)

j =1

in canonical form. T hat is, A\j

^iji ^

1, ..., 72 H- 1, J

1, ..., 72.

(8.2)

Specifically, wegeneralize the condition th at the set is empty when Akj < 0 for all j — 1,..., n + 1 and contains aredundant constraint when the set { A k i , A k n , A k ,n+i } consists only or non-negative elements or Ak,n+i < 0 and Akj > 0 for exactly one j < n. In this section we shall use the above stated condition to obtain a result for appropri­ ate union by obtaining explicit formulas for the coefficients in the constraints when the constraints Ay,..., Ay, h i > n + \ , r < n (8.3) have been interchanged with the constraints £u ..., 4 , £i < n,

(8.4)

Simplex Method in Linear Programming

159

in the order ki,£i, i = 1 , ...,r and the constraint set is returned to canonical form at each step. In order to state the formulas, we denote by fa i f l 5

V - jit "'■>]a)

(8-5)

the minor determ inant of Aij, z = ra-F 1 ,. . . , ra, j = 1 ,..., n indexed by the rows i \ , ..., in and the columns j i , . . . , j r. Then, with representing the original m atrix and A\j the m atrix after the constraints indexed by £4 , . . . , kr have replaced those indexed by £, . . . , £ r , ICr =

{ k u ...,kr} ,/CJ. = [l,m ] - Kr, 4 = { 4 ...,4 } , 4 = [ l,n + 1] - 4 ,

Dr =

f r (ku ...,ki : i u ...,er)

(8.6) we have the formulas for i

G /CJ.,

= /r+ i (ki,-..,kr, i : l u ...,lr, j ) / Dr, j € / D r, £, e £ r ,

(8 .8 )

= ( - i r + 1-7 r-(* i,...,fc r: eu ...,ei. 1,ei+1,...,er, j ) / D r, j e 4 ,

(8-9)

- 4 ^ = ( - l ) r- J/r {ki,...,kj - u kj+i ,. . . , k r, i : i u and for

(8.7)

G /Cr , ^

• 4 U = ( - 1)i+,’/ r - i (fci,...,A:i - i . fci + i . - . fcr :

Ij G 4 .

(8 .10 )

Before stating the condition for redundant constraints or empty sets, we shall prove the following theorem.

Theorem 8 . 1 . The formulas (8.7) - (8.10) are invariant under permutation of &i,..., Ay or t \ , . . . , i r in the sense the sign o f either (8.7), (8.8) or (8.9), (8.10) for fixed i and j = l , . . . , n + l are invariant. This makes it possible to state the condition for e mpty sets or redundant constraints using only the pair (8.7), (8.8) in the order r = 1, 2,..., n. Proof.

F irst let us note th a t we may assume th a t the A;’s and Vs are in increasing order. This follows from the fact th a t when k \ , ..., kn are perm utations of the same set, then k \ , ..., £y_i, kj+i, ...kr, j = 1 , ...,n are merely w ritten down in a different order. To prove this by induction, let a = (ki ,. . . , kr) and aj = (Au,..., kj_i, kj+i , ..., kr) and suppose th a t the largest element y of a is indexed by £. Then after interchanging the y with the last elements of a and ir+i can be rewritten

(10 .8 ) where the num erator is the determ inant of the 2 x 2 m atrix indexed by p = 4 + i 4 and v — 4 + i j and is, in fact, just the Lemma 10.1 with k = 2 after a change of indices. More generally, we can use Lemma 10.1 to prove inductively that, for 1 < p < r + 1 ,

(10.9) where the num erator is the determ inant of the (p + 1) x (p + l)m atrix indexed by p — 4 +2-p, •••, 4 + i 4 and v = 4 +i-p, •••, 4 + i,j- In particular, when p = r + 1, (10.9) reduces, in view of (8.5), to ( 10 .10)

for any i e /Cr+i, j G Cr+X. In particular, ( 10.11)

or 'kr+2/r+2'

( 10 .12 )

Since this is true for each r, we have proved (8.7) with i /CJ.+1, j G £ r+i as a consequence of ( 10 .10 ) and ( 10 .11) with r replaced by r — 1 . By eliminating A ^+1 between (10.9), (10.10), setting i = /cr+2, j = 4 +2, and using 8.6 for a t +i- p, we obtain the interesting identity

f r +2 (kl ,

4 + 2 • 4 , •••, 4 + 2 ) ( D r + l - p Y

— det ( / r_(_2-p ( 4 , •••, 4 + i-p , M • 4 , •••, 4 + i—p, ^))p~)-i with p and v ranging over the indices make of this identity is:

4 +2-p, •••, 4+2

and

4 +2-p, •■•,4 +2- The

(10.13)

main use we

T h e o re m 10 . 2 . Consider the identity (10.13) with p = 1 . I f three of the four minors com­ prising the determinant on the right have sign opposite the fourth then D r / 0 and the sign of f r+2 is determined by the identity

162

Pedersen

Weshall also need the following identity / r+

+

1

(£ +

•••5

£V+ 1 • h i

•••5

1

}

•••? ^ r + l ,j ) f r (£ + •••} k r • A ; •••A ')

/ r+ 1 (£+ 5

•■•5 £>r+l • ^1 j•••?^ r >J ) f r

/ r+ 1

• " i £V+1

• ^1? ■•■5 ^ r+ 1^

(£+

?••■5 k r • ^ 1 5 •••? 1 5^ i+ 1 5 •••?^ r + l )

f r (£+ 5 •••5 k r

• ^ 1 5 •••5

1 5 ^ i+ 1 5

(1

0

.1

4

)

*••j ?J ) —0*

If we suppress the dependence on £q,..., fcr_i and £+,..., 4 - i 5 the left side of (10.14) is the 3 x 3 determ inant of the m atrix with rows indexed by (£4 , £4 , &2) and column indexed by l r,£r+i, j. Since the first two rows are equal the determ inant is zero. 11.

COMPLETION OF THE PROOFS OF THE IDENTITIES.

We now have the main tools sufficient for the proofs of (8 .7),...,( 8 .10) by induction. Note th a t we have proved (8.7) for all r and i £ /Cr , j £ £ r , the proofs of the cases (8 .8 ), (8 .9 ), (8.10). Hence, we may use i = £;r+ 1, j = 4 + i in (8.7) to express (9.1) as Ar + 1

_

kr+l/r +l ~

f r (£+5 - - - , k r : £1, . . . , ^ r )

FJ

J J r -1-1

/i i i \

\i i A )

•

This is the prom oted version of (8.10) with i — r + 1 , j — r + 1 . By putting i = kr+1 , j i £ r+i into (8.7) and substituting (11.1) into (8.10), we obtain

/r+1l 1(£+ •••5 £t+1 5 h 5Jj)) m , •••? ^ r + 1 •• h * 4 ,j ••■

_

AT+l

fcr + U

/ 11

M

“

o\

Fi

~Dr

which is the formula (8.9) corresponding to the pair £v+i,j with j £ £ r. It follows from (9.4) th a t for ki £ /Cr , j Cr+\.

Ar£ = Arkit] - + “+ > + + ■ +

(11 .3 )

^r +1+ +1 After substituting (8.7) and (8.9), and setting the result equal to (8.9) with r replaced by r + 1 , we obtain the identity (10.14). This completes the proof of the remaining cases in (8.9). The proof of the promoted version of (8 .8 ) is isomorphic. There remains the case indexed by ki G JCr and £j G Cr. We obtain from (9.4), (8 .8 ), (8.9), (8.10) and (10.11) with r replaced by r — 1 into (11.4), we obtain + X

=

fr (^1)

^

+ +

{fr~ (fcl> 1

kr ■^li

>

k 3~ 1 k i + 1

£ j - 1; lj+1;

/r+1 (fcl,

»

kr ■

+ •••, 4-1++1, •••+

)

^r-l) f r (fej) fcj— li fc*+l, •••> £^i— 1 • ^1,..., ^r+l) £^r+l • ^1) •••5 A’+l)

We now apply (10.13) in the form

/r+1 (£+ —

fr

{ku fr

£^r+l : h i •••?4+l) f r - 1(£+,..., £jj_i, £jj+i,..., kr . ^i,..., £ i —h

•••) kj —h

kj +15 •••) £r+l

£{+

1, ...£f)

: ^1?■••5 — 15^z+1?^r+l) f r (£+ •••?kr . ^j_, ..., £r)

{ku •••? kj —i, /Cj+ l, •••, £^r+l • ^1?•••) ^r) f r {ku •••, kr . ^i, ..., £i—U £{+U •••> ^r) •

(11.5)

After substituting (11.5) into (11.4), we have the promoted version of (8.10) for ki G /Q, G £ j. Since the case of kr+1 C JCr+i,4 + i C £ r+i has already been disposed of, the proof is complete.

163

Simplex Method in Linear Programming

12.

DUPLICATIONS OF CONSTRAINTS.

The formulas (8.7) - (8.10) are derived under the assumption th at the sets (k i, . .. ,k r) and (£i, ...,£r) are distinct. In particular the £’s are a subset of (1, ...,n) so we must have r < n. On the other hand, it follows from the recursion formulas (9.1) - (9.4) th a t we may, at any time, start over with a new m atrix and continue until there is a duplication in either the A:’s or the f s . In this section we resolve the question of such a duplication in the second step. The new m atrix coefficients, after interchanging the non-basic constraint with the £ ^ basic constraint and then returning to reduced echelon form by the use of elementary column operations, are A '£ = 1 / A u ( 12 . 1 ) A lf3 = - A ij/ A li, j ^ £

( 12 .2 )

Aki = A m / A n,

(12.3)

Akj — Akj — Aki A i j / A n , j ^ £.

(12.4)

and for k ^ z,

Now let us interchange the new constraint with the £ ^ basic constraint. By analogy with ( 12 . 1), ( 12 .2 ) the coefficients for the new constraint are A h — l/A'ki

(12.5)

A l j = A k]/A!kb j / £.

( 12 .6 )

and After substituting from ( 12 .1) - (12.4) there becomes A i = A n / A ki

(12.7)

Akj — Aij —An Akj/Aki, j ~t~£•

(12.8)

and

These are ju st the param eters obtained after interchanging the ^ constraint with the £ ^ and returning to reduced echelon form. But they are in the position of the k ^ 1.The new coefficients for the constraint are A'h = A i / A h A j = A 'j -

A i A 'k j/A 'k i,

(12.9) j± L

(12.10)

Again, after substituting from ( 12 . 1) - (12.4) and taking into account cancellations, these become A t = 11 Aki, (12.11) A 3 = - A k3/ A ' k£, j ± L

(12.12)

They are the coefficients for the k ^ constraint after interchanging the k ^ constraint with the A 1, and they are in the position of the For r / /c or i, r > n, the new coefficients for the A constraint are A ”i — A i / A i ,

(12.13)

164

Pedersen

A rj — A rj

A h A kj / A kl, j 7^ I

(12.14)

After substituting from ( 12 .1) - (12.4), these become A!lt = A r t / A ki, A rj — A rj — A ri A rj / A k i 5i ^

(12.15) (12.16)

which are ju st the coefficient obtained after interchanging the constraint with the in the original matrix. This together with the remarks following(12.8) and (12.12) yields a proof of the following theorem. T h e o r e m 12.1. Interchanging the z*^ non-basic constraint with the £ ^ , updating and then interchanging the k ^ with the and updating is equivalent to merely interchanging the k ^ with the in the original matrix, updating and then interchanging the z**1 and k ^ 1. Nowlet usdetermine the effect of interchanging one non-basic constraint with two dif­ ferent basic constraints. If after obtaining the formulas (12.1) -(12.4), weinterchange the zth constraint with the basic constraint, q / z, the new param eter for the z ^ constraint are A'!q = 1/ A ' q, (12.17) A i = - A i ! A q,

(12-18)

4 , = - 4 ; / 4 , , 3 * I,*-

(12-19)

K q = A q/ A q

( 12 -20 )

A'ki = A'ki - A'kq A!it / A'iq

(12.21)

A'kj = A'kj - A ’kq A ’tJ / A ' q, j + q, I

( 12 .22 )

and The formulas (12.17) - (12.19), after substituting from ( 12 .1) -(12.4) are just the formulas obtained after interchanging the z ^ with the q ^1 in the original matrix. For k ^ q,

and

Again, after substituting from ( 12 . 1) - (12.4), these are just the formula for the k ^ constraint after interchanging the k ^ with the q ^ in the original m atrix except th at the q ^ and variables have been interchanged. T h e o re m 12 . 2 . I f we interchange the z ^ non-basic constraint with the basic constraint, update and then interchange the new z ^ constraint with the q ^ , q ^ z, and update, this is equivalent to merely interchanging the z ^ with the q ^ updating and permuting the q ^ 1 and f i 1 variables.

Simplex Method in Linear Programming

165

13 . THE CASE OF (n + 2) CONSTRAINTS. Let us assume the constraint set to be in canonical form. If a is any subset of the non-basic indices, we shall denote by S a the corresponding set of non-basic constraints and by Sa the set S a together with the basic constraints. For a single index z we define 5+ = {j < n : Aij > 0} and S r = {j < n : A iy


0 and

|S+| — n or a;>n+i
0 there exists an index £ < n with a^i > 0. After

interchanging the z ^ and £ ^ constraint and putting the set back into canonical form the set Si has

= n + 1 —cr. It follows th at interchanging two constraints in an (n + 1) - constraint

set cannot change its status relative to being empty, or having a redundant constraint. Hence, if neither Si nor S j has this property, we can find an empty set or redundant constraint in an (n + 1) constraint set only by interchanging Si with a basic constraint and examining S j , j / i or conversely. In particular, after making this interchange, the new constant term is A' 1 “— A A 1 — 'A-il 4 sii,n+l (13.3) _ A ji I A Ail A u \ A t 'n+1

A 3lAhn+l

Hence, th a t constant term in the z ^ constraint, after interchanging the and the has the same or opposite sign as the j ^ 1 constant, after interchanging the z ^ and the £ ^ , according to whether A j i and A n have the opposite or the same signs. Let us now study the constraint after interchanging the z*^ and the £ ^ with A n < 0 and Aj n > 0.This requires analyzing the signs of A * = iAit r

(13-4)

and •+ =

- T ^ -

(13-5)

Since A j i and A u have opposite signs, it follows from (13.4) th at +

0 / 2 ( m : 2,3) - / i (i ■2 ) / i (* : 1) ~ h ( i , 3 : 1.3) V ~ h ( 3 , P ■i, 2 ) / 2 ( i , p : l , 2 ) f 3 { i, j , p : 1,2,3)

/ f i (j '■2 )

i

2

/ 2 ( i,j : 2,4) ~ h { h h 1,4) / 3 (i, j , p : 1, 2 ,4)

Now let the coefficients of the p *!1 constraint be denoted by A

(14.20)

It follows from (14.12) th a t

A'pX < Oand Ap2 0, this configuration does not contribute to the promoted version of Tk+. Of course, this statem ent does not apply if the interchange is made with respect to some other constraint. Let us now examine the other admissible exchanges within the present matrix. From (14.19) it appears th a t the interchange of the and second variables is one such possibility. But this follows the interchange of the and the first. But this is, by Theorem 12 .2 , the interchange of the z ^ and second followed by a permutation. From (14.19) we see th at the only other admissible interchange is the interchange of the and third constraints under the condition (14.22) This interchange gives the m atrix i —

2

/ f i (j : 3) —f i (i : 3) —f i (j : 1) / i (z : 1) \ ~ / 2 (i,P : 1,3) f 2 {i,p: 1,3)

3,2) h { i , j ■3,4) —/ 2 (i , j : 1 , 2 ) ~ h ( i ,j ■1,4) / 3 (i, j , p : 1, 3, 2) F2 { i, j , p : 1,3,4)

(14.23)

169

Simplex Method in Linear Programming

with D 2 — / 2 {i,j : 1,3) which by (14.22) is positive. By (14.18) we have / 2 (z,p : 1,3) < 0. This configuration appears to have insufficient information to resolve the sign of / 2 (j, p : 1,3). However, if the assumption (14.22) leads to a legitimate simplex step it does impose the ad­ ditional sign f 2 {i,j : 3,4) > 0. (14.24) In any case, the previous configuration was sufficient to resolve the case of the constraints in three variables. When there are more constraints the additional condition (14.24) may be helpful in analyzing the interaction of various sets of three non-basic constraints combined with the basic constraints. We remark,also, th a t if the same constraints z and j solve the maximum problem deter­ mining the nextsimplex step for twosteps in a row, the analysis of (14.20) is sufficient to produce either a complete simplex step or to find a redundant constraint. T hat this be the case when both maximums are achieved by the constraint would require the interchange i — 3. By (14.19) this is impossible since both the 1 , 3 and 3,3 elements are positive. Finally, we consider the sign configuration

1 2

4

z © j + © p + +

+

(14-25)

The interchange of the z ^ and first constraints leads to -

-

+ +

+ +

-

+

+

-

(14.26)

instead of (14.19). Some of these signs are determined as before and the others are conse­ quences of Theorem 10.2. Now we notice th a t the interchange of the constraint with the second is the only admissible simplex interchange. Now to apply the preceding analysis to (14.20), we need only (14.21). This is again a consequence of Theorem 10.2. 15.

THE CASE OF SIX CONSTRAINTS IN THREE VARIABLES.

The analysis of the preceding section yields the following Theorem.

Theorem 15.1. Let us consider

a set of Six Constraints in Three Variables which is in Canonical form and with only one constraint not satisfying the basic vertex. I f completing a simplex step or finding a redundant constraint or finding the set to be empty requires more than three steps then up to a permutation of the first three columns we may assume the configuration of the non-basic constraints

z j V

© + ± + © ± + + -

+ +

(15.1)

-

We leave open the question of whether the number of steps can be reduced from three to two by starting with the configuration + + H— for the p ^

constraint.

(15.2)

170

Pedersen

References [1] Dantzig, Linear Programming and Extensions, Princeton Univ. Press. [4] Polya, G., Szego, G., Problems and Theorems in Analysis, Springer-Verlag, New York, Heidelberg, Berlin, Berlin, 1972. [5] Strang, G., Linear Algebra and its Applications (3rd Ed.), Harcourt, Brace, Jovanovich, San Diego. [6] Wu, S. and Coppins, R., Linear Programming and Extension, McGraw Hill. A ck now ledgem ent I would like to thank Jenny Bourne Wahl for criticizing an earlier version of Sections 7 of this manuscript.

1-

An Estimate of the Semi-Stable Measure of Small Balls in Banach Spaces BALRAM S. R A JP U T Knoxville, TN 37923

D epartm ent of M athem atics, The University of Tennessee,

Abstract. Let (E, || • ||) be a separable Banach space. Let fi be a sym m etric r-semistable probability measure of index 0 < a < 2 on E, and let 0 < q < a. It is proven th a t if Je ||x ||gd/i = 1 then /x{| |rc|| < t} < const. £a / 2, for all t > 0, where const, depends only on r,q and a (and not E or fi). This result compliments similar known results for sym m etric Gaussian and a-stable probability measures on E. Two other related results are also proved; these are needed for the proof of the above m ain result.

1.

IN TR O D UC TIO N A N D PRELIM INARIES

Let (E, || • ||) be a separable Banach space. Let /i be a sym m etric Borel probability m easure on E. In a recent paper, M. Lewandowski, M. Ryzner, and T. Zak (1992) showed th a t, if /i is a-stable, satisfying / E ||x || qdfi = 1 w ith 0 < q < a, then /x{||a:|| < t} < const. £, where const, depends only on a and q (and not E or /i). In the case when \i is centered Gaussian, a similar result is proved earlier by S. Szarek (1991) and also by X. Fernique and by J. Sawa; Sawa requires in addition th a t E be a H ilbert space. (For a discussion and references of the Fernique and Sawa contribution, we refer the reader to Lewandowski, Ryzner, and Zak (1992)).

This research is partially supported by the University of Tennessee Science Alliance, a State of Tennessee Center of Excellence.

171

172

Rajput

The m ain effort of this paper is aimed at proving a version of the above result of Lewandowski, Ryzner, and Zak (1992) for the larger class of semi-stable probability mea­ sures. Specifically, we prove the result stated in the abstract. The proof of Lewandowski, Ryzner, and Zak (1992) in the stable case is based on the fact th a t every E-valued sym m et­ ric a-stab le random variable is conditionally Gaussian and on the well known Anderson Inequality for Gaussian measures. Since a semi-stable random variable in general is not conditionally Gaussian (Rosinski (1991), p .32), the m ethods used in Lewandowski, Ryzner, and Zak (1992) do not apply in the more general semi-stable case; a similar situation seems to prevail w ith regard to the m ethods of proof used by Szarek and Sawa. Our proof, like the one due to Fernique in the Gaussian case (see Lewandowski, Ryzner, and Zak (1992)), is based on the well known K antor Inequality. In the Gaussian case (a = 2 ), this approach yields the same upper bound for //{IM l < t} as obtained in Lewandowski, Ryzner, and Zak (1992) in the stable case (namely, const, t ). In the proper semi-stable case, on the other hand, this approach yields the upper bound for q{|M I < as const. t a / 2. which, in the interesting case, i.e., when t is close to 0, is worse th an const, t . (For more on this point see Concluding Rem ark). For our proof of the m ain result, in addition to the K antor Inequality, we also need an estim ate for the lower bound of the tail of symmetric semi-stable probability measures on E; this is obtained in Lemma 1. This lower bound is obtained by using the PaleyZygmund Inequality and another result which provides a comparison between moments of a sem i-stable probability measure and a related F-norm (Proposition 1). T hroughout, r and a will denote real numbers satisfying 0 < r < 1 and 0 < a < 2; and the n otation r — S S ( a ) will mean “ r-sem i-stable index a ” . Further, throughout E will denote a real separable Banach space. By a measure on E, we shall always m ean th a t it is defined on its Borel