Sampling In Combinatorial And Geometric Set Systems

Download or read book Sampling in Combinatorial and Geometric Set Systems written by Nabil H. Mustafa and published by American Mathematical Society. This book was released on 2022-01-14 with total page 251 pages. Available in PDF, EPUB and Kindle. Book excerpt: Understanding the behavior of basic sampling techniques and intrinsic geometric attributes of data is an invaluable skill that is in high demand for both graduate students and researchers in mathematics, machine learning, and theoretical computer science. The last ten years have seen significant progress in this area, with many open problems having been resolved during this time. These include optimal lower bounds for epsilon-nets for many geometric set systems, the use of shallow-cell complexity to unify proofs, simpler and more efficient algorithms, and the use of epsilon-approximations for construction of coresets, to name a few. This book presents a thorough treatment of these probabilistic, combinatorial, and geometric methods, as well as their combinatorial and algorithmic applications. It also revisits classical results, but with new and more elegant proofs. While mathematical maturity will certainly help in appreciating the ideas presented here, only a basic familiarity with discrete mathematics, probability, and combinatorics is required to understand the material.

Download or read book Self-similar and Self-affine Sets and Measures written by Balázs Bárány and published by American Mathematical Society. This book was released on 2023-11-16 with total page 466 pages. Available in PDF, EPUB and Kindle. Book excerpt: Although there is no precise definition of a “fractal”, it is usually understood to be a set whose smaller parts, when magnified, resemble the whole. Self-similar and self-affine sets are those for which this resemblance is precise and given by a contracting similitude or affine transformation. The present book is devoted to this most basic class of fractal objects. The book contains both introductory material for beginners and more advanced topics, which continue to be the focus of active research. Among the latter are self-similar sets and measures with overlaps, including the much-studied infinite Bernoulli convolutions. Self-affine systems pose additional challenges; their study is often based on ergodic theory and dynamical systems methods. In the last twenty years there have been many breakthroughs in these fields, and our aim is to give introduction to some of them, often in the simplest nontrivial cases. The book is intended for a wide audience of mathematicians interested in fractal geometry, including students. Parts of the book can be used for graduate and even advanced undergraduate courses.

Download or read book Approximation and Online Algorithms written by Jarosław Byrka and published by Springer Nature. This book was released on 2023-12-21 with total page 246 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book constitutes the refereed proceedings of the 21st International Workshop on Approximation and Online Algorithms, WAOA 2023, held in Amsterdam, The Netherlands, during September 7–8, 2023 The 16 full papers included in this book are carefully reviewed and selected from 43 submissions. The topics of WAOA 2023 were algorithmic game theory, algorithmic trading, coloring and partitioning, competitive analysis, computational advertising, computational finance, cuts and connectivity, FPT-approximation algorithms, geometric problems, graph algorithms, inapproximability results, mechanism design, network design, packing and covering, paradigms for the design and analysis of approximation and online algorithms, resource augmentation, and scheduling problems

Download or read book Multidimensional Residue Theory and Applications written by Alekos Vidras and published by American Mathematical Society. This book was released on 2023-10-18 with total page 556 pages. Available in PDF, EPUB and Kindle. Book excerpt: Residue theory is an active area of complex analysis with connections and applications to fields as diverse as partial differential and integral equations, computer algebra, arithmetic or diophantine geometry, and mathematical physics. Multidimensional Residue Theory and Applications defines and studies multidimensional residues via analytic continuation for holomorphic bundle-valued current maps. This point of view offers versatility and flexibility to the tools and constructions proposed, allowing these residues to be defined and studied outside the classical case of complete intersection. The book goes on to show how these residues are algebraic in nature, and how they relate and apply to a wide range of situations, most notably to membership problems, such as the Briançon–Skoda theorem and Hilbert's Nullstellensatz, to arithmetic intersection theory and to tropical geometry. This book will supersede the existing literature in this area, which dates back more than three decades. It will be appreciated by mathematicians and graduate students in multivariate complex analysis. But thanks to the gentle treatment of the one-dimensional case in Chapter 1 and the rich background material in the appendices, it may also be read by specialists in arithmetic, diophantine, or tropical geometry, as well as in mathematical physics or computer algebra.

Download or read book Algebras, Lattices, Varieties written by Ralph S. Freese and published by American Mathematical Society. This book was released on 2022-10-28 with total page 496 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is the second of a three-volume set of books on the theory of algebras, a study that provides a consistent framework for understanding algebraic systems, including groups, rings, modules, semigroups and lattices. Volume I, first published in the 1980s, built the foundations of the theory and is considered to be a classic in this field. The long-awaited volumes II and III are now available. Taken together, the three volumes provide a comprehensive picture of the state of art in general algebra today, and serve as a valuable resource for anyone working in the general theory of algebraic systems or in related fields. The two new volumes are arranged around six themes first introduced in Volume I. Volume II covers the Classification of Varieties, Equational Logic, and Rudiments of Model Theory, and Volume III covers Finite Algebras and their Clones, Abstract Clone Theory, and the Commutator. These topics are presented in six chapters with independent expositions, but are linked by themes and motifs that run through all three volumes.

Download or read book Amenability of Discrete Groups by Examples written by Kate Juschenko and published by American Mathematical Society. This book was released on 2022-06-30 with total page 180 pages. Available in PDF, EPUB and Kindle. Book excerpt: The main topic of the book is amenable groups, i.e., groups on which there exist invariant finitely additive measures. It was discovered that the existence or non-existence of amenability is responsible for many interesting phenomena such as, e.g., the Banach-Tarski Paradox about breaking a sphere into two spheres of the same radius. Since then, amenability has been actively studied and a number of different approaches resulted in many examples of amenable and non-amenable groups. In the book, the author puts together main approaches to study amenability. A novel feature of the book is that the exposition of the material starts with examples which introduce a method rather than illustrating it. This allows the reader to quickly move on to meaningful material without learning and remembering a lot of additional definitions and preparatory results; those are presented after analyzing the main examples. The techniques that are used for proving amenability in this book are mainly a combination of analytic and probabilistic tools with geometric group theory.

Download or read book Completion Problems on Operator Matrices written by Dragana S. Cvetković Ilić and published by American Mathematical Society. This book was released on 2022-06-07 with total page 170 pages. Available in PDF, EPUB and Kindle. Book excerpt: Completion problems for operator matrices are concerned with the question of whether a partially specified operator matrix can be completed to form an operator of a desired type. The research devoted to this topic provides an excellent means to investigate the structure of operators. This book provides an overview of completion problems dealing with completions to different types of operators and can be considered as a natural extension of classical results concerned with matrix completions. The book assumes some basic familiarity with functional analysis and operator theory. It will be useful for graduate students and researchers interested in operator theory and the problem of matrix completions.