This book extends classical Hermite-Hadamard type inequalities to the fractional case via establishing fractional integral identities, and discusses Riemann-Liouville and Hadamard integrals, respectively, by various convex functions. Illustrating theoretical results via applications in special means of real numbers, it is an essential reference for applied mathematicians and engineers working with fractional calculus. Contents Introduction Preliminaries Fractional integral identities Hermite-Hadamard inequalities involving Riemann-Liouville fractional integrals Hermite-Hadamard inequalities involving Hadamard fractional integrals

The transparency and power of geometric constructions has been a source of inspiration to generations of mathematicians. The beauty and persuasion of pictures, communicated in words or drawings, continues to provide the intuition and arguments for working with complicated concepts and structures of modern mathematics. This volume contains a selection of articles exploring geometric approaches to problems in algebra, algebraic geometry and number theory.

Key topics include:

* Curves and their Jacobians

* Algebraic surfaces

* Moduli spaces, Shimura varieties

* Motives and motivic integration

* Number-theoretic applications, rational points

* Combinatorial aspects of algebraic geometry

* Quantum cohomology

* Arithmetic dynamical systems

The collection gives a representative sample of problems and most recent results in algebraic and arithmetic geometry; the text can serve as an intense introduction for graduate students and those wishing to pursue research in these areas.

Contributors: I. Bauer, F. Bogomolov, N. Budur, F. Catanese, C.-L. Chai, R. Cluckers, C. De Concini, J.S. Ellenberg, F. Grunewald, B. Hassett, T. Hausel, F. Loeser, J. Pineiro, R. Pink, C. Procesi, M. Spitzweck, P. Swinnerton-Dyer, L. Szpiro, H. Tamvakis, Y. Tschinkel, T.J. Tucker, A. Venkatesh, and Y.G. Zarhin.

From the 28th of February through the 3rd of March, 2001, the Department of Math ematics of the University of Florida hosted a conference on the many aspects of the field of Ordered Algebraic Structures. Officially, the title was "Conference on Lattice Ordered Groups and I-Rings," but its subject matter evolved beyond the limitations one might associate with such a label. This volume is officially the proceedings of that conference, although, likewise, it is more accurate to view it as a complement to that event. The conference was the fourth in wh at has turned into aseries of similar conferences, on Ordered Algebraic Structures, held in consecutive years. The first, held at the University of Florida in Spring, 1998, was a modest and informal affair. The fifth is in the final planning stages at this writing, for March 7-9, 2002, at Vanderbilt University. And although these events remain modest and reasonably informal, their scope has broadened, as they have succeeded in attracting mathematicians from other, related fields, as weIl as from more distant lands."

This book is concerned with cardinal number valued functions defined for any Boolean algebra. Examples of such functions are independence, which assigns to each Boolean algebra the supremum of the cardinalities of its free subalgebras, and cellularity, which gives the supremum of cardinalities of sets of pairwise disjoint elements. Twenty-one such functions are studied in detail, and many more in passing. The questions considered are the behaviour of these functions under algebraic operations such as products, free products, ultraproducts, and their relationships to one another. Assuming familiarity with only the basics of Boolean algebras and set theory, through simple infinite combinatorics and forcing, the book reviews current knowledge about these functions, giving complete proofs for most facts. A special feature of the book is the attention given to open problems, of which 185 are formulated. Based on Cardinal Functions on Boolean Algebras (1990) and Cardinal Invariants on Boolean Algebras (1996) by the same author, the present work is much larger than either of these. It contains solutions to many of the open problems of the earlier volumes. Among the new topics are continuum cardinals on Boolean algebras, with a lengthy treatment of the reaping number. Diagrams at the end of the book summarize the relationships between the functions for many important classes of Boolean algebras, including interval algebras, tree algebras and superatomic algebras.

This is the first of two volumes of a state-of-the-art survey article collection which originates from three commutative algebra sessions at the 2009 Fall Southeastern American Mathematical Society Meeting at Florida Atlantic University. The articles reach into diverse areas of commutative algebra and build a bridge between Noetherian and non-Noetherian commutative algebra. These volumes present current trends in two of the most active areas of commutative algebra: non-noetherian rings (factorization, ideal theory, integrality), and noetherian rings (the local theory, graded situation, and interactions with combinatorics and geometry). This volume contains combinatorial and homological surveys. The combinatorial papers document some of the increasing focus in commutative algebra recently on the interaction between algebra and combinatorics. Specifically, one can use combinatorial techniques to investigate resolutions and other algebraic structures as with the papers of Floystad on Boij-Soederburg theory, of Geramita, Harbourne and Migliore, and of Cooper on Hilbert functions, of Clark on minimal poset resolutions and of Mermin on simplicial resolutions. One can also utilize algebraic invariants to understand combinatorial structures like graphs, hypergraphs, and simplicial complexes such as in the paper of Morey and Villarreal on edge ideals. Homological techniques have become indispensable tools for the study of noetherian rings. These ideas have yielded amazing levels of interaction with other fields like algebraic topology (via differential graded techniques as well as the foundations of homological algebra), analysis (via the study of D-modules), and combinatorics (as described in the previous paragraph). The homological articles the editors have included in this volume relate mostly to how homological techniques help us better understand rings and singularities both noetherian and non-noetherian such as in the papers by Roberts, Yao, Hummel and Leuschke.

In 2008, November 23-28, the workshop of "Classical Problems on Planar Polynomial Vector Fields " was held in the Banff International Research Station, Canada. Called "classical problems", it was concerned with the following: (1) Problems on integrability of planar polynomial vector fields. (2) The problem of the center stated by Poincare for real polynomial differential systems, which asks us to recognize when a planar vector field defined by polynomials of degree at most n possesses a singularity which is a center. (3) Global geometry of specific classes of planar polynomial vector fields. (4) Hilbert's 16th problem. These problems had been posed more than 110 years ago. Therefore, they are called "classical problems" in the studies of the theory of dynamical systems. The qualitative theory and stability theory of differential equations, created by Poincare and Lyapunov at the end of the 19th century, had major developments as two branches of the theory of dynamical systems during the 20th century. As a part of the basic theory of nonlinear science, it is one of the very active areas in the new millennium. This book presents in an elementary way the recent significant developments in the qualitative theory of planar dynamical systems. The subjects are covered as follows: the studies of center and isochronous center problems, multiple Hopf bifurcations and local and global bifurcations of the equivariant planar vector fields which concern with Hilbert's 16th problem. The book is intended for graduate students, post-doctors and researchers in dynamical systems. For all engineers who are interested in the theory of dynamical systems, it is also a reasonable reference. It requires a minimum background of a one-year course on nonlinear differential equations.

This book describes the latest Russian research covering the structure and algorithmic properties of Boolean algebras from the algebraic and model-theoretic points of view. A significantly revised version of the author's Countable Boolean Algebras (Nauka, Novosibirsk, 1989), the text presents new results as well as a selection of open questions on Boolean algebras. Other current features include discussions of the Kottonen algebras in enrichments by ideals and automorphisms, and the properties of the automorphism groups.

The term Noncommutative Dynamics can be interpreted in several ways. It is used in this book to refer to a set of phenomena associated with the dynamics of quantum systems of the simplest kind that involve rigorous mathematical structures associated with infinitely many degrees of freedom. The dynamics of such a system is represented by a one-parameter group of automorphisms of a noncommutative algebra of observables, and the author focuses primarily on the most concrete case in which that algebra consists of all bounded operators on a Hilbert space. This subject overlaps with several mathematical areas of current interest, including quantum field theory, the dynamics of open quantum systems, noncommutative geometry, and both classical and noncommutative probability theory. This is the first book to give a systematic presentation of progress during the past fifteen years on the classification of E-semigroups up to cocycle conjugacy. There are many new results that cannot be found in the existing literature, as well as significant reformulations and simplifications of the theory as it exists today. William Arveson is Professor of Mathematics at the University of California, Berkeley. He has published two previous books with Springer-Verlag, An Invitation to C*-algebras (1976) and A Short Course on Spectral Theory (2001).

This book presents the general theory of categorical closure operators together with examples and applications to the most common categories, such as topological spaces, fuzzy topological spaces, groups, and abelian groups. The main aim of the theory is to develop a categorical characterization of the classical basic concepts in topology via the newly introduced concept of categorical closure operators. This permits many topological ideas to be introduced in a topology-free environment and imported afterwards into a new category, which often yields interesting new insights into its structure. The first part of the book deals with the general theory, starting with basic definitions and gradually moving to more advanced properties. The second part includes applications to the classical concepts of epimorphisms, separation, compactness and connectedness. Every chapter ends with exercises. A comprehensive list of references for the reader who wants to consult original works and a good index complete the book. "Categorical Closure Operators" is self-contained and can be considered as a graduate level text for topics courses in category theory, algebra, and topology. The book appeals mainly to graduate students and researchers in category theory and categorical topology, and to those interested in categorical methods applied to the most common concrete categories. The reader is expected to have some basic knowledge of algebra, topology and category theory; however, all recurrent categorical concepts are included in a preliminary chapter.

This volume is devoted to the development of an algebraic model of databases. The first chapter presents a general introduction. The following sixteen chapters are divided into three main parts. Part I deals with various aspects of universal algebra. The chapters of Part I discuss topics such as sets, algebras and models, fundamental structures, categories, the category of sets, topoi, fuzzy sets, varieties of algebras, axiomatic classes, category algebra and algebraic theories. Part II deals with different approaches to the algebraization of predicate calculus. This material is intended to be applied chiefly to databases, although some discussion of pure algebraic applications is also given. Discussed here are topics such as Boolean algebras and propositional calculus, Halmos algebras and predicate calculus, connections with model theory, and the categorial approach to algebraic logic. Part III is concerned specifically with the algebraic model of databases, which considers the database as an algebraic structure. Topics dealt with in this part are the algebraic aspects of databases, their equivalence and restructuring, symmetries and the Galois theory of databases, and constructions in database theory. The volume closes with a discussion and conclusions, and an extensive bibliography. For mathematicians, computer scientists and database engineers, with an interest in applications of algebra and logic.

This is the first monograph on rings closed to von Neumann regular rings. The following classes of rings are considered: exchange rings, pi-regular rings, weakly regular rings, rings with comparability, V-rings, and max rings. Every Artinian or von Neumann regular ring A is an exchange ring (this means that for every one of its elements a, there exists an idempotent e of A such that aA contains eA and (1-a)A contains (1-e)A). Exchange rings are very useful in the study of direct decompositions of modules, and have many applications to theory of Banach algebras, ring theory, and K-theory. In particular, exchange rings and rings with comparability provide a key to a number of outstanding cancellation problems for finitely generated projective modules. Every von Neumann regular ring is a weakly regular pi-regular ring (a ring A is pi-regular if for every one of its elements a, there is a positive integer n such that a is contained in aAa) and every Artinian ring is a pi-regular max ring (a ring is a max ring if every one of its nonzero modules has a maximal submodule). Thus many results on finite-dimensional algebras and regular rings are extended to essentially larger classes of rings. Starting from a basic understanding of ring theory, the theory of rings close to regular is presented and accompanied with complete proofs.

The book will appeal to readers from beginners to researchers and specialists in algebra; it concludes with an extensive bibliography.

This book contains nineteen papers from among the twenty-five papers presented at the Second International Conference on Fibonacci Numbers and Their Applications. These papers have been selected after a careful review by well known referee's in the field, and they range from elementary number theory to probability and statistics. The Fibonacci numbers are their unifying bond. It is anticipated that this book will be useful to research workers and graduate students interested in the Fibonacci numbers and their applications. October 1987 The Editors Gerald E. Bergum South Dakota State University Brookings, South Dakota, U.S.A. Andreas N. Philippou University of Patras Patras, Greece Alwyn F. Horadam University of New England Armidale, N.S.W., Australia xiii THE ORGANIZING COMMITTEES LOCAL COMMITTEE INTERN A TIONAL COMMITTEE Bergum, G., Chairman Philippou, A. (Greece), Chairman Edgar, H., Co-chalrman Horadam, A. (Australia), Co-chalrman Bergum, G. (U.s.A.) Thoro, D. Kiss, P. (Hungary) Johnson, M. Long, C. (U.S.A.) Lange, L.

The contributions in this book focus on a variety of topics related to discrepancy theory, comprising Fourier techniques to analyze discrepancy, low discrepancy point sets for quasi-Monte Carlo integration, probabilistic discrepancy bounds, dispersion of point sets, pair correlation of sequences, integer points in convex bodies, discrepancy with respect to geometric shapes other than rectangular boxes, and also open problems in discrepany theory.

Kummer's work on cyclotomic fields paved the way for the development of algebraic number theory in general by Dedekind, Weber, Hensel, Hilbert, Takagi, Artin and others. However, the success of this general theory has tended to obscure special facts proved by Kummer about cyclotomic fields which lie deeper than the general theory. For a long period in the 20th century this aspect of Kummer's work seems to have been largely forgotten, except for a few papers, among which are those by Pollaczek [Po], Artin-Hasse [A-H] and Vandiver [Va]. In the mid 1950's, the theory of cyclotomic fields was taken up again by Iwasawa and Leopoldt. Iwasawa viewed cyclotomic fields as being analogues for number fields of the constant field extensions of algebraic geometry, and wrote a great sequence of papers investigating towers of cyclotomic fields, and more generally, Galois extensions of number fields whose Galois group is isomorphic to the additive group of p-adic integers. Leopoldt concentrated on a fixed cyclotomic field, and established various p-adic analogues of the classical complex analytic class number formulas. In particular, this led him to introduce, with Kubota, p-adic analogues of the complex L-functions attached to cyclotomic extensions of the rationals. Finally, in the late 1960's, Iwasawa [Iw 11] made the fundamental discovery that there was a close connection between his work on towers of cyclotomic fields and these p-adic L-functions of Leopoldt - Kubota.

This book focuses on problems at the interplay between the theory of partitions and optimal transport with a view toward applications. Topics covered include problems related to stable marriages and stable partitions, multipartitions, optimal transport for measures and optimal partitions, and finally cooperative and noncooperative partitions. All concepts presented are illustrated by examples from game theory, economics, and learning.

From the reviews of the first edition:

"It is certainly no exaggeration to say that A Singular Introduction to Commutative Algebra aims to lead a further stage in the computational revolution in commutative algebra . Among the great strengths and most distinctive features is a new, completely unified treatment of the global and local theories. making it one of the most flexible and most efficient systems of its type....another strength of Greuel and Pfister's book is its breadth of coverage of theoretical topics in the portions of commutative algebra closest to algebraic geometry, with algorithmic treatments of almost every topic....Greuel and Pfister have written a distinctive and highly useful book that should be in the library of every commutative algebraist and algebraic geometer, expert and novice alike."

J.B. Little, MAA, March 2004

The second edition is substantially enlarged by a chapter on Groebner bases in non-commtative rings, a chapter on characteristic and triangular sets with applications to primary decomposition and polynomial solving and an appendix on polynomial factorization including factorization over algebraic field extensions and absolute factorization, in the uni- and multivariate case."

ThesubjectofthisbookisSemi-In?niteAlgebra,ormorespeci?cally,Semi-In?nite Homological Algebra. The term "semi-in?nite" is loosely associated with objects that can be viewed as extending in both a "positive" and a "negative" direction, withsomenaturalpositioninbetween,perhapsde?nedupto a"?nite"movement. Geometrically, this would mean an in?nite-dimensional variety with a natural class of "semi-in?nite" cycles or subvarieties, having always a ?nite codimension in each other, but in?nite dimension and codimension in the whole variety [37]. (For further instances of semi-in?nite mathematics see, e. g. , [38] and [57], and references below. ) Examples of algebraic objects of the semi-in?nite type range from certain in?nite-dimensional Lie algebras to locally compact totally disconnected topolo- cal groups to ind-schemes of ind-in?nite type to discrete valuation ?elds. From an abstract point of view, these are ind-pro-objects in various categories, often - dowed with additional structures. One contribution we make in this monograph is the demonstration of another class of algebraic objects that should be thought of as "semi-in?nite", even though they do not at ?rst glance look quite similar to the ones in the above list. These are semialgebras over coalgebras, or more generally over corings - the associative algebraic structures of semi-in?nite nature. The subject lies on the border of Homological Algebra with Representation Theory, and the introduction of semialgebras into it provides an additional link with the theory of corings [23], as the semialgebrasare the natural objects dual to corings.

The book describes the history of Jordan algebras, and describes in full mathematical detail the recent structure theory for Jordan algebras of arbitrary dimension due to Efim Zel'manov. To keep the exposition elementary, the structure theory is developed for linear Jordan algebras (where the scalar ring contains 1/2, avoiding the nuisancy distractions of characteristic 2), though the modern quadratic methods are used throughout. Both the quadratic methods and the Zelmanov results go beyond the previous textbooks on Jordan Theory, written in the 1960's and early 1980's before the theory reached its final form. The book is written to serve either as a text for a 2nd year graduate course, or for independent reading, for students who need or wish to know a bit about Jordan algebras. It is not primaily aimed at experts or students going on to do research in the area, and no knowledge is required beyond standard first-year graduate algebra courses. General students of algebra can profit from exposure to nonassociative algebras, and students or professional mathematicians working in areas such as Lie algebras, differential geometry Äsymmetric spaces or bounded symmetric domainsÜ, functional analysis ÄJB algebras and triplesÜ, or exceptional groups and geometry Ärelated to the 27- dimensional Albert algebraÜ can also profit from acquaintance with the material. Jordan algebras crop up in many surprising settings, and find application to a variety of mathematical areas.

The book presents integral formulations for partial differential equations, with the focus on spherical and plane integral operators. The integral relations are obtained for different elliptic and parabolic equations, and both direct and inverse mean value relations are studied. The derived integral equations are used to construct new numerical methods for solving relevant boundary value problems, both deterministic and stochastic based on probabilistic interpretation of the spherical and plane integral operators.

The fundamental theorem of algebra states that any complex polynomial must have a complex root. This book examines three pairs of proofs of the theorem from three different areas of mathematics: abstract algebra, complex analysis and topology. The first proof in each pair is fairly straightforward and depends only on what could be considered elementary mathematics. However, each of these first proofs leads to more general results from which the fundamental theorem can be deduced as a direct consequence. These general results constitute the second proof in each pair. To arrive at each of the proofs, enough of the general theory of each relevant area is developed to understand the proof. In addition to the proofs and techniques themselves, many applications such as the insolvability of the quintic and the transcendence of e and pi are presented. Finally, a series of appendices give six additional proofs including a version of Gauss'original first proof. The book is intended for junior/senior level undergraduate mathematics students or first year graduate students, and would make an ideal "capstone" course in mathematics.

This is a textbook for a course (or self-instruction) in cryptography with emphasis on algebraic methods. The first half of the book is a self-contained informal introduction to areas of algebra, number theory, and computer science that are used in cryptography. Most of the material in the second half - "hidden monomial" systems, combinatorial-algebraic systems, and hyperelliptic systems - has not previously appeared in monograph form. The Appendix by Menezes, Wu, and Zuccherato gives an elementary treatment of hyperelliptic curves. This book is intended for graduate students, advanced undergraduates, and scientists working in various fields of data security.

The book offers an original view on channel coding, based on a unitary approach to block and convolutional codes for error correction. It presents both new concepts and new families of codes. For example, lengthened and modified lengthened cyclic codes are introduced as a bridge towards time-invariant convolutional codes and their extension to time-varying versions. The novel families of codes include turbo codes and low-density parity check (LDPC) codes, the features of which are justified from the structural properties of the component codes. Design procedures for regular LDPC codes are proposed, supported by the presented theory. Quasi-cyclic LDPC codes, in block or convolutional form, represent one of the most original contributions of the book. The use of more than 100 examples allows the reader gradually to gain an understanding of the theory, and the provision of a list of more than 150 definitions, indexed at the end of the book, permits rapid location of sought information.

This book presents advances in matrix and tensor data processing in the domain of signal, image and information processing. The theoretical mathematical approaches are discusses in the context of potential applications in sensor and cognitive systems engineering.
The topics and application include Information Geometry, Differential Geometry of structured Matrix, Positive Definite Matrix, Covariance Matrix, Sensors (Electromagnetic Fields, Acoustic sensors) and Applications in Cognitive systems, in particular Data Mining."