We review several properties of integrals of the Wigner distribution on subsets of the phase space. Along our way, we provide a theoretical proof of the invalidity of Flandrin's conjecture, a fact already proven via numerical arguments in our joint paper [J. Fourier Anal. Appl. 26 (2020), no. 1, article no. 6] with B. Delourme and T. Duyckaerts. We use also the J.G. Wood & A.J. Bracken paper [J. Math. Phys. 46 (2005), no. 4, article no. 042103], for which we offer a mathematical perspective. We review thoroughly the case of subsets of the plane whose boundary is a conic curve and show that Mehler's formula can be helpful in the analysis of these cases, including for the higher dimensional case investigated in the paper [J. Math. Phys. 51 (2010), no. 10, article no. 102101] by E. Lieb and Y. Ostrover. Using the Feichtinger algebra, we show that, generically in the Baire sense, the Wigner distribution of a pulse in L
2
(R
n
) does not belong to L
1
(R
2n
), providing as a byproduct a large class of examples of subsets of the phase space R
2n
on which the integral of the Wigner distribution is infinite. We study as well the case of convex polygons of the plane, with a rather weak estimate depending on the number of vertices, but independent of the area of the polygon.
Frontmatter
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Abstract
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Contents
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Foreword
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1 Preliminaries and definitions
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2 Quantization of radial functions and Mehler’s formula
Download pp. 41?44
3 Conics with eccentricity smaller than 1
Download pp. 45?76
4 Parabolas
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5 Conics with eccentricity greater than 1
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6 Unboundedness is Baire generic
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7 Convex polygons of the plane
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8 Open questions and conjectures
Download pp. 149?151
A Appendix
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References
Download pp. 211?213
Index
Download pp. 215?216
ISBN: 978-1-394-23594-0
October 2024
544 pages
Hardcover
Detailed review of optimization from first principles, supported by rigorous math and computer science explanations and various learning aids
Supported by rigorous math and computer science foundations, Combinatorial and Algorithmic Mathematics: From Foundation to Optimization provides a from-scratch understanding to the field of optimization, discussing 70 algorithms with roughly 220 illustrative examples, 160 nontrivial end-of-chapter exercises with complete solutions to ensure readers can apply appropriate theories, principles, and concepts when required, and Matlab codes that solve some specific problems. This book helps readers to develop mathematical maturity, including skills such as handling increasingly abstract ideas, recognizing mathematical patterns, and generalizing from specific examples to broad concepts.
Starting from first principles of mathematical logic, set-theoretic structures, and analytic and algebraic structures, this book covers both combinatorics and algorithms in separate sections, then brings the material together in a final section on optimization. This book focuses on topics essential for anyone wanting to develop and apply their understanding of optimization to areas such as data structures, algorithms, artificial intelligence, machine learning, data science, computer systems, networks, and computer security.
Combinatorial and Algorithmic Mathematics includes discussion on:
Propositional logic and predicate logic, set-theoretic structures such as sets, relations, and functions, and basic analytic and algebraic structures such as sequences, series, subspaces, convex structures, and polyhedra
Recurrence-solving techniques, counting methods, permutations, combinations, arrangements of objects and sets, and graph basics and properties
Asymptotic notations, techniques for analyzing algorithms, and computational complexity of various algorithms
Linear optimization and its geometry and duality, simplex and non-simplex algorithms for linear optimization, second-order cone programming, and semidefinite programming
Combinatorial and Algorithmic Mathematics is an ideal textbook resource on the subject for students studying discrete structures, combinatorics, algorithms, and optimization. It also caters to scientists across diverse disciplines that incorporate algorithms and academics and researchers who wish to better understand some modern optimization methodologies.
Preface
Acknowledgements
About the Companion Website
PART I FOUNDATIONS
1. Mathematical Logic
2. Set-Theoretic Structures
3. Analytic and Algebraic Structures
PART II COMBINATORICS
4. Graphs
5. Recurrences
6. Counting
PART III ALGORITHMS
7. Analysis of Algorithms
8. Array and Numeric Algorithms
9. Elementary Combinatorial Algorithms
PART IV OPTIMIZATION
10. Linear Programming
11. Second-Order Cone Programming
12. Semidefinite Programming and Combinatorial Optimization
Appendix. Solutions to Chapter Exercises
Bibliography
Index
Format: Paperback / softback, 475 pages, height x width: 235x155 mm, 1 Illustrations,
black and white; XV, 475 p. 1 illus., 1
Series: Probability Theory and Stochastic Modelling 103
Pub. Date: 15-Mar-2024
ISBN-13: 9783662669129
This book provides a compact introduction to the theory of measure-valued branching processes, immigration processes and Ornstein?Uhlenbeck type processes.
Measure-valued branching processes arise as high density limits of branching particle systems. The first part of the book gives an analytic construction of a special class of such processes, the Dawson?Watanabe superprocesses, which includes the finite-dimensional continuous-state branching process as an example. Under natural assumptions, it is shown that the superprocesses have Borel right realizations. Transformations are then used to derive the existence and regularity of several different forms of the superprocesses. This technique simplifies the constructions and gives useful new perspectives. Martingale problems of superprocesses are discussed under Feller type assumptions. The second part investigates immigration structures associated with the measure-valued branching processes. The structures are formulated by skewconvolution semigroups, which are characterized in terms of infinitely divisible probability entrance laws. A theory of stochastic equations for one-dimensional continuous-state branching processes with or without immigration is developed, which plays a key role in the construction of measure flows of those processes. The third part of the book studies a class of Ornstein-Uhlenbeck type processes in Hilbert spaces defined by generalized Mehler semigroups, which arise naturally in fluctuation limit theorems of the immigration superprocesses.
This volume is aimed at researchers in measure-valued processes, branching processes, stochastic analysis, biological and genetic models, and graduate students in probability theory and stochastic processes.
Preface to the Second Edition.- Preface to the First Edition.-
Conventions and Notations.-
1. Random Measures on Metric Spaces.-
2. Measure-Valued Branching Processes.-
3. One-Dimensional Branching Processes.-
4. Branching Particle Systems.-
5. Basic Regularities of Superprocesses.-
6. Constructions by Transformations.-
7. Martingale Problems of Superprocesses.-
8. Entrance Laws and Kuznetsov Measures.-
9. Structures of Independent Immigration.-
10. One-Dimensional Stochastic Equations.-
11. Path-Valued Processes and Stochastic Flows.-
12. State-Dependent Immigration Structures.-
13. Generalized Ornstein-Uhlenbeck Processes.-
14. Small-Branching Fluctuation Limits.-
A. Markov Processes.- References.- Subject Index.-Symbol Index.
Format: Hardback, 218 pages, height x width: 235x155 mm, 72 Illustrations, color;
5 Illustrations, black and white; Approx. 200 p.,
Series: Springer Proceedings in Complexity
Pub. Date: 24-May-2024
ISBN-13: 9783031575143
The International Conference on Complex Networks (CompleNet) brings together researchers and practitioners from diverse disciplines working on areas related to complex networks. CompleNet has been an active conference since 2009. Over the past two decades, we have witnessed an exponential increase in the number of publications and research centres dedicated to this field of Complex Networks (aka Network Science). From biological systems to computer science, from technical to informational networks, and from economic to social systems, complex networks are becoming pervasive for dozens of applications. It is the interdisciplinary nature of complex networks that CompleNet aims to capture and celebrate. The CompleNet conference is one of the most cherished events by scientists in our field. Maybe it is because of its motivating format, consisting of plenary sessions (no parallel sessions); or perhaps the reason is that it finds the perfect balance between young and senior participation, a balance in the demographics of the presenters, or perhaps it is just the quality of the work presented.
Chapter 1: Mapping low-resolution edges to high-resolution paths: the case of traffic measurements in cities.
Chapter 2: From Low Resource Information Extraction to Identifying Influential Nodes in Knowledge Graphs.
Chapter 3: Inhomogenous Marketing Mix Diffusion.
Chapter 4: Modelling both pairwise interactions and group effects in polarization on interaction networks.
Chapter 5: Computing Motifs in Hypergraphs.
Chapter 6: Extending network tools to explore trends in temporal granular trade networks.
Chapter 7: Expressivity of Geometric Inhomogeneous Random Graphs-Metric and Non-Metric.
Chapter 8: Social Interactions Matter: Is Grey Wolf Optimizer a Particle Swarm Optimization Variation?.
Chapter 9: Exploring Ingredient Variability in Classic Russian Cuisine Dishes through Complex Network Analysis.
Chapter 10: Unraveling the Structure of Knowledge: Consistency in Everyday Networks, Diversity in Scientific.
Chapter 11: Kinetic-based force-directed graph embedding.
Chapter12: Deep Graph Machine Learning Models for Epidemic Spread Prediction and Prevention.
Chapter 13: EleMi: A robust method to infer soil ecological networks with better community structure.
Chapter 14: Interpreting Node Embedding Distances Through n-order Proximity Neighbourhoods.
Chapter 15: Edge Dismantling with Geometric Reinforcement Learning.
Chapter 16: Public Transit Inequality in the Context of the Built Environment.
Format: Hardback, 453 pages, height x width: 235x155 mm, XI, 442 p. 10 illus., 1 Hardback
Series: Springer Series in Operations Research and Financial Engineering
Pub. Date: 12-Jun-2024
ISBN-13: 9783031579875
This book offers a comprehensive presentation of the theory and methods of risk-averse optimization and control. Problems of this type arise in finance, energy production and distribution, supply chain management, medicine, and many other areas, where not only the average performance of a stochastic system is essential, but also high-impact and low-probability events must be taken into account. The book is a self-contained presentation of the utility theory, the theory of measures of risk, including systemic and dynamic measures of risk, and their use in optimization and control models. It also covers stochastic dominance relations and their application as constraints in optimization models. Optimality conditions for problems with nondifferentiable and nonconvex functions and operators involving risk measures and stochastic dominance relations are discussed. Much attention is paid to multi-stage risk-averse optimization problems and to risk-averse Markov decision problems.
Elements of the Utility Theory.- Measures of Risk.- Optimization of
Measures of Risk.- Dynamic Risk Optimization.- Optimization with Stochastic
Dominance Constraints.- Multivariate and Sequential Stochastic
Orders.- Numerical Methods for Problems with Stochastic
Dominance Constraints.- Risk-Averse Control of Markov Systems.
Format: Paperback / softback, 442 pages, height x width: 235x155 mm,
127 Illustrations, black and white; XXIV, 442 p. 127 illus.
Series: Grundlehren der mathematischen Wissenschaften 360
Pub. Date: 29-Mar-2024
ISBN-13: 9783031218026
The publication of the first edition of Lagerungen in der Ebene, auf der Kugel und im Raum in 1953 marked the birth of discrete geometry. Since then, the book has had a profound and lasting influence on the development of the field. It included many open problems and conjectures, often accompanied by suggestions for their resolution. A good number of new results were surveyed by L?szlo Fejes Toth in his Notes to the 2nd edition.
The present version of Lagerungen makes this classic monograph available in English for the first time, with updated Notes, completed by extensive surveys of the state of the art. More precisely, this book consists of:
a corrected English translation of the original Lagerungen, the revised and updated Notes on the original text, eight self-contained chapters surveying additional topics in detail.
The English edition provides a comprehensive update to an enduring classic. Combining the lucid exposition of the original text with extensive new material, it will be a valuable resource for researchers in discrete geometry for decades to come.
Part I. Lagerungen - Arrangements in the Plane, on the Sphere, and in Space.-
1. Some Theorems from Elementary Geometry.-
2. Theorems from the Theory of Convex Bodies.-
3. Problems on Packing and Covering in the Plane.-
4. Efficiency of Packings and Coverings with a Sequence of Convex Disks.-
5. Extremal Properties of Regular Polyhedra.-
6. Irregular Packing on the Sphere.-
5. Packing in Space.- Part II. Notes and Additional
Chapters to the English Edition.-
8. Notes.-
9. Finite Variations on the Isoperimetric Problem.-
10. Higher Dimensions.-
11. Ball Packings in Hyperbolic Space.-
12. Mutliple Arrangements.-
13. Neighbors.-
14. Packing and Covering Properties fo Sequences of Convex Bodies.-
15. Four Classic Problems.-
16. Miscellaneous Problems about Packing and Covering.- References for Part I.- References for Part II.- Name Index.- Subject Index.