Ebook: Total Positivity and Its Applications

This volume contains both invited lectures and contributed talks presented at the meeting on Total Positivity and its Applications held at the guest house of the University of Zaragoza in Jaca, Spain, during the week of September 26-30, 1994. There were present at the meeting almost fifty researchers from fourteen countries. Their interest in thesubject of Total Positivity made for a stimulating and fruitful exchange of scientific information. Interest to participate in the meeting exceeded our expectations. Regrettably, budgetary constraints forced us to restriet the number of attendees. Professor S. Karlin, of Stanford University, who planned to attend the meeting had to cancel his participation at the last moment. Nonetheless, his almost universal spiritual presence energized and inspired all of us in Jaca. More than anyone, he influenced the content, style and quality of the presentations given at the meeting. Every article in these Proceedings (except some by Karlin hirnself) references his influential treatise Total Positivity, Volume I, Stanford University Press, 1968. Since its appearance, this book has intrigued and inspired the minds of many researchers (one of us, in his formative years, read the galley proofs and the other of us first doubted its value but then later became its totally committed disciple). All of us present at the meeting encourage Professor Karlin to return to the task of completing the anxiously awaited Volume 11 of Total Positivity.

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This volume contains articles that document the advances in the subject of Total Positivity during the last two decades. The material is divided into ten chapters. While some of the articles are of a survey nature, others present new results appearing here for the first time. Also, some papers contain introductory material and are therefore accessible to non-experts interested in becoming familiar with the important ideas and techniques of Total Positivity.
Audience: This book will be of value to mathematicians, engineers and computer scientists whose work involves applications of Total Positivity to problems in the theory of spline functions, numerical quadrature, nonlinear analysis, entire functions, probability, mathematical biology, statistics, approximation theory, combinatorics, geometric modelling, matrix theory and integral equations.

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This volume contains articles that document the advances in the subject of Total Positivity during the last two decades. The material is divided into ten chapters. While some of the articles are of a survey nature, others present new results appearing here for the first time. Also, some papers contain introductory material and are therefore accessible to non-experts interested in becoming familiar with the important ideas and techniques of Total Positivity.
Audience: This book will be of value to mathematicians, engineers and computer scientists whose work involves applications of Total Positivity to problems in the theory of spline functions, numerical quadrature, nonlinear analysis, entire functions, probability, mathematical biology, statistics, approximation theory, combinatorics, geometric modelling, matrix theory and integral equations.
Content:
Front Matter....Pages i-x
Front Matter....Pages 1-1
Total Positivity of the Spline Kernel and its Applications....Pages 3-34
Total Positivity Properties of LB-splines....Pages 35-46
Total Positivity and Splines....Pages 47-84
Totally Positive Systems and Interpolation by Subspaces of Generalized Splines....Pages 85-94
Front Matter....Pages 95-95
Preservation of Total Nonnegativity under the Hadamard Product and Related Topics....Pages 97-102
Vertex Implications for Totally Nonnegative Matrices....Pages 103-107
On Factorizations of Totally Positive Matrices....Pages 109-130
Front Matter....Pages 131-131
Total positivity and optimal bases....Pages 133-155
Total Positivity and the Shape of Curves....Pages 157-186
Tchebycheff Curves....Pages 187-218
Front Matter....Pages 219-219
Analytical and Probabilistic Methods for Totally Positive Matrices....Pages 221-246
Some Inequalities of Total Positivity in Pure and Applied Mathematics....Pages 247-261
New Directions and Problems of Multivariate Total Positivity Motivated by Genetic Recombination Probabilities....Pages 263-274
Front Matter....Pages 275-275
Tchebycheff spaces and total positivity....Pages 277-300
?eby?ev and Weak ?eby?ev Systems....Pages 301-332
Front Matter....Pages 333-333
Zero Sets of Entire Fourier Transforms of Multiply Positive Density Functions....Pages 335-356
Front Matter....Pages 357-357
Total Nonnegativity of Moment Matrices and its Application to E-Optimal Designs in Polynomial Regression....Pages 359-383
Analysis of Regression Dependence in 2 ? n Bivariate Distributions And Some Applications in Contingency Tables....Pages 385-400
Front Matter....Pages 401-401
Total Positivity and Nonlinear Analysis....Pages 403-427
Solutions of certain functional equations....Pages 429-432
Front Matter....Pages 401-401
Alternance and Snakes in the Nonlinear Finite Moment Problem....Pages 433-447
Front Matter....Pages 449-449
The Applications of Total Positivity to Combinatorics, and Conversely....Pages 451-473
Front Matter....Pages 475-475
Spectral Properties of Totally Positive Kernels and Matrices....Pages 477-511
Back Matter....Pages 513-518

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This volume contains articles that document the advances in the subject of Total Positivity during the last two decades. The material is divided into ten chapters. While some of the articles are of a survey nature, others present new results appearing here for the first time. Also, some papers contain introductory material and are therefore accessible to non-experts interested in becoming familiar with the important ideas and techniques of Total Positivity.
Audience: This book will be of value to mathematicians, engineers and computer scientists whose work involves applications of Total Positivity to problems in the theory of spline functions, numerical quadrature, nonlinear analysis, entire functions, probability, mathematical biology, statistics, approximation theory, combinatorics, geometric modelling, matrix theory and integral equations.
Content:
Front Matter....Pages i-x
Front Matter....Pages 1-1
Total Positivity of the Spline Kernel and its Applications....Pages 3-34
Total Positivity Properties of LB-splines....Pages 35-46
Total Positivity and Splines....Pages 47-84
Totally Positive Systems and Interpolation by Subspaces of Generalized Splines....Pages 85-94
Front Matter....Pages 95-95
Preservation of Total Nonnegativity under the Hadamard Product and Related Topics....Pages 97-102
Vertex Implications for Totally Nonnegative Matrices....Pages 103-107
On Factorizations of Totally Positive Matrices....Pages 109-130
Front Matter....Pages 131-131
Total positivity and optimal bases....Pages 133-155
Total Positivity and the Shape of Curves....Pages 157-186
Tchebycheff Curves....Pages 187-218
Front Matter....Pages 219-219
Analytical and Probabilistic Methods for Totally Positive Matrices....Pages 221-246
Some Inequalities of Total Positivity in Pure and Applied Mathematics....Pages 247-261
New Directions and Problems of Multivariate Total Positivity Motivated by Genetic Recombination Probabilities....Pages 263-274
Front Matter....Pages 275-275
Tchebycheff spaces and total positivity....Pages 277-300
?eby?ev and Weak ?eby?ev Systems....Pages 301-332
Front Matter....Pages 333-333
Zero Sets of Entire Fourier Transforms of Multiply Positive Density Functions....Pages 335-356
Front Matter....Pages 357-357
Total Nonnegativity of Moment Matrices and its Application to E-Optimal Designs in Polynomial Regression....Pages 359-383
Analysis of Regression Dependence in 2 ? n Bivariate Distributions And Some Applications in Contingency Tables....Pages 385-400
Front Matter....Pages 401-401
Total Positivity and Nonlinear Analysis....Pages 403-427
Solutions of certain functional equations....Pages 429-432
Front Matter....Pages 401-401
Alternance and Snakes in the Nonlinear Finite Moment Problem....Pages 433-447
Front Matter....Pages 449-449
The Applications of Total Positivity to Combinatorics, and Conversely....Pages 451-473
Front Matter....Pages 475-475
Spectral Properties of Totally Positive Kernels and Matrices....Pages 477-511
Back Matter....Pages 513-518
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