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Author: David J. Grynkiewicz
Publisher: Springer Science & Business Media
ISBN: 3319004166
Category : Mathematics
Languages : en
Pages : 425
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Book Description
Nestled between number theory, combinatorics, algebra and analysis lies a rapidly developing subject in mathematics variously known as additive combinatorics, additive number theory, additive group theory, and combinatorial number theory. Its main objects of study are not abelian groups themselves, but rather the additive structure of subsets and subsequences of an abelian group, i.e., sumsets and subsequence sums. This text is a hybrid of a research monograph and an introductory graduate textbook. With few exceptions, all results presented are self-contained, written in great detail, and only reliant upon material covered in an advanced undergraduate curriculum supplemented with some additional Algebra, rendering this book usable as an entry-level text. However, it will perhaps be of even more interest to researchers already in the field. The majority of material is not found in book form and includes many new results as well. Even classical results, when included, are given in greater generality or using new proof variations. The text has a particular focus on results of a more exact and precise nature, results with strong hypotheses and yet stronger conclusions, and on fundamental aspects of the theory. Also included are intricate results often neglected in other texts owing to their complexity. Highlights include an extensive treatment of Freiman Homomorphisms and the Universal Ambient Group of sumsets A+B, an entire chapter devoted to Hamidoune’s Isoperimetric Method, a novel generalization allowing infinite summands in finite sumset questions, weighted zero-sum problems treated in the general context of viewing homomorphisms as weights, and simplified proofs of the Kemperman Structure Theorem and the Partition Theorem for setpartitions.
Author: David J. Grynkiewicz
Publisher: Springer Science & Business Media
ISBN: 3319004166
Category : Mathematics
Languages : en
Pages : 425
Get Book Here
Book Description
Nestled between number theory, combinatorics, algebra and analysis lies a rapidly developing subject in mathematics variously known as additive combinatorics, additive number theory, additive group theory, and combinatorial number theory. Its main objects of study are not abelian groups themselves, but rather the additive structure of subsets and subsequences of an abelian group, i.e., sumsets and subsequence sums. This text is a hybrid of a research monograph and an introductory graduate textbook. With few exceptions, all results presented are self-contained, written in great detail, and only reliant upon material covered in an advanced undergraduate curriculum supplemented with some additional Algebra, rendering this book usable as an entry-level text. However, it will perhaps be of even more interest to researchers already in the field. The majority of material is not found in book form and includes many new results as well. Even classical results, when included, are given in greater generality or using new proof variations. The text has a particular focus on results of a more exact and precise nature, results with strong hypotheses and yet stronger conclusions, and on fundamental aspects of the theory. Also included are intricate results often neglected in other texts owing to their complexity. Highlights include an extensive treatment of Freiman Homomorphisms and the Universal Ambient Group of sumsets A+B, an entire chapter devoted to Hamidoune’s Isoperimetric Method, a novel generalization allowing infinite summands in finite sumset questions, weighted zero-sum problems treated in the general context of viewing homomorphisms as weights, and simplified proofs of the Kemperman Structure Theorem and the Partition Theorem for setpartitions.
Author: Terence Tao
Publisher: Cambridge University Press
ISBN: 1139458345
Category : Mathematics
Languages : en
Pages : 18
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Book Description
Additive combinatorics is the theory of counting additive structures in sets. This theory has seen exciting developments and dramatic changes in direction in recent years thanks to its connections with areas such as number theory, ergodic theory and graph theory. This graduate-level 2006 text will allow students and researchers easy entry into this fascinating field. Here, the authors bring together in a self-contained and systematic manner the many different tools and ideas that are used in the modern theory, presenting them in an accessible, coherent, and intuitively clear manner, and providing immediate applications to problems in additive combinatorics. The power of these tools is well demonstrated in the presentation of recent advances such as Szemerédi's theorem on arithmetic progressions, the Kakeya conjecture and Erdos distance problems, and the developing field of sum-product estimates. The text is supplemented by a large number of exercises and new results.
Author: Alfred Geroldinger
Publisher: Springer Science & Business Media
ISBN: 3764389613
Category : Mathematics
Languages : en
Pages : 324
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Book Description
Additive combinatorics is a relatively recent term coined to comprehend the developments of the more classical additive number theory, mainly focussed on problems related to the addition of integers. Some classical problems like the Waring problem on the sum of k-th powers or the Goldbach conjecture are genuine examples of the original questions addressed in the area. One of the features of contemporary additive combinatorics is the interplay of a great variety of mathematical techniques, including combinatorics, harmonic analysis, convex geometry, graph theory, probability theory, algebraic geometry or ergodic theory. This book gathers the contributions of many of the leading researchers in the area and is divided into three parts. The two first parts correspond to the material of the main courses delivered, Additive combinatorics and non-unique factorizations, by Alfred Geroldinger, and Sumsets and structure, by Imre Z. Ruzsa. The third part collects the notes of most of the seminars which accompanied the main courses, and which cover a reasonably large part of the methods, techniques and problems of contemporary additive combinatorics.
Author: Elisabeth A. Lloyd
Publisher: Princeton University Press
ISBN: 0691223831
Category : Science
Languages : en
Pages : 254
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Book Description
Traditionally a scientific theory is viewed as based on universal laws of nature that serve as axioms for logical deduction. In analyzing the logical structure of evolutionary biology, Elisabeth Lloyd argues that the semantic account is more appropriate and powerful. This book will be of interest to biologists and philosophers alike.
Author: Ted Chinburg
Publisher: American Mathematical Soc.
ISBN: 0821824589
Category : Algebraic number theory
Languages : en
Pages : 167
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Book Description
The main object of this memoir is to describe and, in some cases, to establish, new systems of congruences for the algebraic parts of the leading terms of the expansions of [italic]L-series at [italic lowercase]s = 0. If these congruences hold, together with a conjecture of Stark which states (roughly) that the ratio of the leading term to the regulator is an algebraic integer, then the main conjecture is true. The greater part of the memoir is devoted to the study of these systems of congruences for certain infinite families of quaternion extensions [italic]N/[italic]K (that is, [capital Greek]Gamma quaternion order 8). It is shown that such extensions can be constructed with specified ramification, and that various unit and class groups are calculable. This permits the verification of the congruences, and the main conjecture can be established for one such family of extensions.
Author: K Heiner Kamps
Publisher: World Scientific
ISBN: 9814502553
Category : Mathematics
Languages : en
Pages : 476
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Book Description
The abstract homotopy theory is based on the observation that analogues of much of the topological homotopy theory and simple homotopy theory exist in many other categories (e.g. spaces over a fixed base, groupoids, chain complexes, module categories). Studying categorical versions of homotopy structure, such as cylinders and path space constructions, enables not only a unified development of many examples of known homotopy theories but also reveals the inner working of the classical spatial theory. This demonstrates the logical interdependence of properties (in particular the existence of certain Kan fillers in associated cubical sets) and results (Puppe sequences, Vogt's Iemma, Dold's theorem on fibre homotopy equivalences, and homotopy coherence theory).
Author: Matthias Dehmer
Publisher: Springer Science & Business Media
ISBN: 0817649042
Category : Mathematics
Languages : en
Pages : 409
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Book Description
For over a decade, complex networks have steadily grown as an important tool across a broad array of academic disciplines, with applications ranging from physics to social media. A tightly organized collection of carefully-selected papers on the subject, Towards an Information Theory of Complex Networks: Statistical Methods and Applications presents theoretical and practical results about information-theoretic and statistical models of complex networks in the natural sciences and humanities. The book's major goal is to advocate and promote a combination of graph-theoretic, information-theoretic, and statistical methods as a way to better understand and characterize real-world networks. This volume is the first to present a self-contained, comprehensive overview of information-theoretic models of complex networks with an emphasis on applications. As such, it marks a first step toward establishing advanced statistical information theory as a unified theoretical basis of complex networks for all scientific disciplines and can serve as a valuable resource for a diverse audience of advanced students and professional scientists. While it is primarily intended as a reference for research, the book could also be a useful supplemental graduate text in courses related to information science, graph theory, machine learning, and computational biology, among others.
Author: Roman Garnett
Publisher: Cambridge University Press
ISBN: 110842578X
Category : Computers
Languages : en
Pages : 375
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Book Description
A comprehensive introduction to Bayesian optimization that starts from scratch and carefully develops all the key ideas along the way.
Author: Hasse Ekstedt
Publisher: Routledge
ISBN: 1351984942
Category : Business & Economics
Languages : en
Pages : 288
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Book Description
Economic theory in its neoclassical form is sometimes regarded as free from values; it is simply the theory of economic exchange. This can only hold true if we accept the idea of "Homo Economicus" and the equilibrium economy. But in the real world, away from neoclassical models, there is no intrinsic stability as such. Instead, stability is created by the surrounding social, cultural and political structures. Clearly, it is imperative that ethics features in the analysis of these economic and socio-political structures. Drawing on Aristotle, Kant, Hume and others, this book conceptualizes the analysis of ethics and economic and social structures. It first considers the key philosophical underpinnings and categories which frame the discussion of ethics in economic theory and then considers individual ethics, social action, financial structures and war. Throughout, ethics are examined in a multicultural context with structural complexities, and the difficulties in finding a coherent set of ethics which provides social cohesion and an open society are considered. A key part of this is the comparison of two ethical principles which can be adopted by societies: ius soli or loyalty to constitution, and ius sanguinis or loyalty to "Blood and Soil". The latter is argued to lead to problems of Us and the Other. Introducing the possibility of integrating microscopic ethics into socio-political structures and proposing the eventual existence of a global ethics, this volume is a significant contribution to the emerging literature on economics, social structures and ethics. It will be of particular interest to those working in business and public administration and who have an education in socio-economic areas, but it also has a broad appeal to students and academics in the social sciences.
Author: Lindsay N. Childs
Publisher: American Mathematical Soc.
ISBN: 1470465167
Category : Education
Languages : en
Pages : 311
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Book Description
Hopf algebras have been shown to play a natural role in studying questions of integral module structure in extensions of local or global fields. This book surveys the state of the art in Hopf-Galois theory and Hopf-Galois module theory and can be viewed as a sequel to the first author's book, Taming Wild Extensions: Hopf Algebras and Local Galois Module Theory, which was published in 2000. The book is divided into two parts. Part I is more algebraic and focuses on Hopf-Galois structures on Galois field extensions, as well as the connection between this topic and the theory of skew braces. Part II is more number theoretical and studies the application of Hopf algebras to questions of integral module structure in extensions of local or global fields. Graduate students and researchers with a general background in graduate-level algebra, algebraic number theory, and some familiarity with Hopf algebras will appreciate the overview of the current state of this exciting area and the suggestions for numerous avenues for further research and investigation.
