Inequalities: Theory of Majorization and Its Applications (eBook)

Albert W. Marshall, Ingram Olkin, Barry C. Arnold (Autoren)

eBook Download: PDF

2010 | 2nd ed. 2011
XXVII, 909 Seiten
Springer New York (Verlag)
978-0-387-68276-1 (ISBN)

Infos

This book's first edition has been widely cited by researchers in diverse fields. The following are excerpts from reviews. 'Inequalities: Theory of Majorization and its Applications' merits strong praise. It is innovative, coherent, well written and, most importantly, a pleasure to read. ... This work is a valuable resource!' (Mathematical Reviews). 'The authors ... present an extremely rich collection of inequalities in a remarkably coherent and unified approach. The book is a major work on inequalities, rich in content and original in organization.' (Siam Review). 'The appearance of ... Inequalities in 1979 had a great impact on the mathematical sciences. By showing how a single concept unified a staggering amount of material from widely diverse disciplines-probability, geometry, statistics, operations research, etc.-this work was a revelation to those of us who had been trying to make sense of his own corner of this material.' (Linear Algebra and its Applications). This greatly expanded new edition includes recent research on stochastic, multivariate and group majorization, Lorenz order, and applications in physics and chemistry, in economics and political science, in matrix inequalities, and in probability and statistics. The reference list has almost doubled.

Albert W. Marshall is Professor Emeritus of Statistics at the University of British Columbia. His fundamental contributions to reliability theory have had a profound effect in furthering its development.

Ingram Olkin is Professor Emeritus of Statistics at Stanford University. He has made fundamental contributions in multivariate analysis, and in the development of statistical methods in meta-analysis, which have resulted in its use in many applications.

Barry C. Arnold is Distinguished Professor of Statistics at the University of California, Riverside. His previous books deal with Pareto Distributions, Order Statistics, Record Values, Conditionally Specified Distributions, and the Lorenz Order.

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This book's first edition has been widely cited by researchers in diverse fields. The following are excerpts from reviews. "e;Inequalities: Theory of Majorization and its Applications"e; merits strong praise. It is innovative, coherent, well written and, most importantly, a pleasure to read. ... This work is a valuable resource!"e; (Mathematical Reviews). "e;The authors ... present an extremely rich collection of inequalities in a remarkably coherent and unified approach. The book is a major work on inequalities, rich in content and original in organization."e; (Siam Review). "e;The appearance of ... Inequalities in 1979 had a great impact on the mathematical sciences. By showing how a single concept unified a staggering amount of material from widely diverse disciplines-probability, geometry, statistics, operations research, etc.-this work was a revelation to those of us who had been trying to make sense of his own corner of this material."e; (Linear Algebra and its Applications). This greatly expanded new edition includes recent research on stochastic, multivariate and group majorization, Lorenz order, and applications in physics and chemistry, in economics and political science, in matrix inequalities, and in probability and statistics. The reference list has almost doubled.

Inhalt

Preface and Acknowledgments 
 
6 
History and Preface of the 
 
10 
Overview and Roadmap 
13 
Contents 
15 
Basic Notation and Terminology 
22 
Part I: Theory of Majorization 
26 
1 Introduction 27
A Motivation and Basic Definitions 27
B Majorization as a Partial Ordering 42
C Order-Preserving Functions 43
D Various Generalizations of Majorization 45
2 Doubly Stochastic Matrices 53
A Doubly Stochastic Matrices and Permutation Matrices 
53 
B Characterization of Majorization Using Doubly 
56 
C Doubly Substochastic Matrices and Weak 
60 
D Doubly Superstochastic Matrices and Weak 
66 
E Orderings on D 69
F Proofs of Birkhoff's Theorem and Refinements 71
G Classes of Doubly Stochastic Matrices 76
H More Examples of Doubly Stochastic and Doubly Substochastic Matrices 85
I Properties of Doubly Stochastic Matrices 91
J Diagonal Equivalence of Nonnegative Matrices and Doubly Stochastic Matrices 
100 
3 Schur-Convex Functions 102
A Characterization of Schur-Convex Functions 103
B Compositions Involving Schur-Convex Functions 111
C Some General Classes of Schur-Convex Functions 114
D Examples I. Sums of Convex Functions 124
E Examples II. Products of LogarithmicallyConcave (Convex) Functions 128
F Examples III. Elementary Symmetric Functions 137
G Symmetrization of Convex and Schur-Convex Functions: Muirhead’s Theorem 
143 
H Schur-Convex Functions on D and TheirExtension to R n 155
I Miscellaneous Specific Examples 161
J Integral Transformations Preserving 
168 
K Physical Interpretations of Inequalities 176
4 Equivalent Conditions for Majorization 178
A Characterization by Linear Transformations 178
B Characterization in Terms of Order-Preserving 
179 
C A Geometric Characterization 185
D A Characterization Involving Top Wage Earners 186
5 Preservation and Generation of Majorization 187
A Operations Preserving Majorization 187
B Generation of Majorization 207
C Maximal and Minimal Vectors Under Constraints 214
D Majorization in Integers 216
E Partitions 221
F Linear Transformations That Preserve Majorization 224
6 Rearrangements and Majorization 225
A Majorizations from Additions of Vectors 226
B Majorizations from Functions of Vectors 232
C Weak Majorizations from Rearrangements 235
D L-Superadditive Functions---Properties 
239 
E Inequalities Without Majorization 247
F A Relative Arrangement Partial Order 250
Part II: Mathematical Applications 
262 
7 Combinatorial Analysis 263
A Some Preliminaries on Graphs, Incidence 
263 
B Conjugate Sequences 265
C The Theorem of Gale and Ryser 269
D Some Applications of the Gale--Ryser Theorem 274
E s-Graphs and a Generalization of the Gale--Ryser Theorem 
278 
F Tournaments 280
G Edge Coloring in Graphs 285
H Some Graph Theory Settings in Which 
287 
8 Geometric Inequalities 288
A Inequalities for the Angles of a Triangle 290
B Inequalities for the Sides of a Triangle 295
C Inequalities for the Exradii and Altitudes 301
D Inequalities for the Sides, Exradii, and Medians 303
E Isoperimetric-Type Inequalities for Plane Figures 306
F Duality Between Triangle Inequalities and 
313 
G Inequalities for Polygons and Simplexes 314
9 Matrix Theory 316
A Notation and Preliminaries 317
B Diagonal Elements and Eigenvalues of a Hermitian Matrix 319
C Eigenvalues of a Hermitian Matrix and Its 
327 
D Diagonal Elements and Singular Values 332
E Absolute Value of Eigenvalues and Singular Values 336
F Eigenvalues and Singular Values 343
G Eigenvalues and Singular Values of A, B,and A + B 348
H Eigenvalues and Singular Values of A, B, and AB 357
I Absolute Values of Eigenvalues and Row Sums 366
J Schur or Hadamard Products of Matrices 371
K Diagonal Elements and Eigenvalues of a Totally Positive Matrix and of an M-Matrix 
376 
L Loewner Ordering and Majorization 379
M Nonnegative Matrix-Valued Functions 380
N Zeros of Polynomials 381
O Other Settings in Matrix Theory Where 
382 
10 Numerical Analysis 385
A Unitarily Invariant Norms and Symmetric 
385 
B Matrices Closest to a Given Matrix 388
C Condition Numbers and Linear Equations 394
D Condition Numbers of Submatrices 
398 
E Condition Numbers and Norms 398
Part III: Stochastic Applications 
402 
11 Stochastic Majorizations 403
A Introduction 403
B Convex Functions and Exchangeable 
408 
C Families of Distributions Parameterized to Preserve Symmetry and Convexity 
413 
D Some Consequences of the Stochastic 
417 
E Parameterization to Preserve Schur-Convexity 419
F Additional Stochastic Majorizations and Properties 436
G Weak Stochastic Majorizations 443
H Additional Stochastic Weak Majorizations and Properties 451
I Stochastic Schur-Convexity 456
12 Probabilistic, Statistical, and Other Applications 457
A Sampling from a Finite Population 458
B Majorization Using Jensen's Inequality 472
C Probabilities of Realizing at Least k of n Events 473
D Expected Values of Ordered Random Variables 477
E Eigenvalues of a Random Matrix 485
F Special Results for Bernoulli and Geometric 
490 
G Weighted Sums of Symmetric Random Variables 492
H Stochastic Ordering from Ordered Random 
497 
I Another Stochastic Majorization Based on Stochastic Ordering 503
J Peakedness of Distributions of Linear Combinations 506
K Tail Probabilities for Linear Combinations 510
L Schur-Concave Distribution Functions and Survival 
516 
M Bivariate Probability Distributions with Fixed 
521 
N Combining Random Variables 523
O Concentration Inequalities for Multivariate 
526 
P Miscellaneous Cameo Appearances of Majorization 527
Q Some Other Settings in Which Majorization 
541 
13 Additional Statistical Applications 543
A Unbiasedness of Tests and Monotonicity 
544 
B Linear Combinations of Observations 551
C Ranking and Selection 557
D Majorization in Reliability Theory 565
E Entropy 572
F Measuring Inequality and Diversity 575
G Schur-Convex Likelihood Functions 582
H Probability Content of Geometric Regions 
583 
I Optimal Experimental Design 584
J Comparison of Experiments 586
Part IV: Generalizations 
591 
14 Orderings Extending Majorization 592
A Majorization with Weights 593
B Majorization Relative to d 600
C Semigroup and Group Majorization 602
D Partial Orderings Induced by Convex Cones 610
E Orderings Derived from Function Sets 613
F Other Relatives of Majorization 618
G Majorization with Respect to a Partial Order 620
H Rearrangements and Majorizations for Functions 621
15 Multivariate Majorization 625
A Some Basic Orders 625
B The Order-Preserving Functions 635
C Majorization for Matrices of Differing Dimensions 637
D Additional Extensions 642
E Probability Inequalities 644
Part V: Complementary Topics 
648 
16 Convex Functions and Some Classical Inequalities 649
A Monotone Functions 649
B Convex Functions 653
C Jensen's Inequality 666
D Some Additional Fundamental Inequalities 669
E Matrix-Monotone and Matrix-Convex Functions 682
F Real-Valued Functions of Matrices 696
17 Stochastic Ordering 705
A Some Basic Stochastic Orders 706
B Stochastic Orders from Convex Cones 712
C The Lorenz Order 724
D Lorenz Order: Applications and Related Results 746
E An Uncertainty Order 760
18 Total Positivity 769
A Totally Positive Functions 769
B Pólya Frequency Functions 774
C Pólya Frequency Sequences 779
D Total Positivity of Matrices 779
19 Matrix Factorizations, Compounds,Direct Products, and M-Matrices 781
A Eigenvalue Decompositions 781
B Singular Value Decomposition 783
C Square Roots and the Polar Decomposition 784
D A Duality Between Positive Semidefinite Hermitian Matrices and Complex Matrices 
786 
E Simultaneous Reduction of Two Hermitian Matrices 787
F Compound Matrices 787
G Kronecker Product and Sum 792
H M-Matrices 794
20 Extremal Representations of Matrix Functions 795
A Eigenvalues of a Hermitian Matrix 795
B Singular Values 801
C Other Extremal Representations 806
Biographies 808
References 823
Author Index 889
Subject Index 903

Produkt-Details

• Erscheint lt. Verlag: 25.11.2010
• Reihe/Serie: Springer Series in Statistics
• Reihe/Serie: Springer Series in Statistics
• Zusatzinfo: XXVII, 909 p.
• Verlagsort: New York
• Sprache: englisch
• Original-Titel: Inequalities: Theory of Majorization and Its Applications
• Themenwelt: Mathematik / Informatik ► Mathematik ► Statistik
• Themenwelt: Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik
• Themenwelt: Technik
• ISBN-10: 0-387-68276-7 / 0387682767
• ISBN-13: 978-0-387-68276-1 / 9780387682761