This volume presents recent advances in the field of matrix analysis based on contributions at the MAT-TRIAD 2015 conference. Topics covered include interval linear algebra and computational complexity, Birkhoff polynomial basis, tensors, graphs, linear pencils, K-theory and statistic inference, showing the ubiquity of matrices in different mathematical areas. With a particular focus on matrix and operator theory, statistical models and computation, the International Conference on Matrix Analysis and its Applications 2015, held in Coimbra, Portugal, was the sixth in a series of conferences. Applied and Computational Matrix Analysis will appeal to graduate students and researchers in theoretical and applied mathematics, physics and engineering who are seeking an overview of recent problems and methods in matrix analysis.

This volume offers contributions reflecting a selection of the lectures presented at the international conference BAIL 2014, which was held from 15th to 19th September 2014 at the Charles University in Prague, Czech Republic. These are devoted to the theoretical and/or numerical analysis of problems involving boundary and interior layers and methods for solving these problems numerically. The authors are both mathematicians (pure and applied) and engineers, and bring together a large number of interesting ideas. The wide variety of topics treated in the contributions provides an excellent overview of current research into the theory and numerical solution of problems involving boundary and interior layers.

This book offers a thorough and self-contained exposition of the mathematics of time-domain boundary integral equations associated to the wave equation, including applications to scattering of acoustic and elastic waves. The book offers two different approaches for the analysis of these integral equations, including a systematic treatment of their numerical discretization using Galerkin (Boundary Element) methods in the space variables and Convolution Quadrature in the time variable. The first approach follows classical work started in the late eighties, based on Laplace transforms estimates. This approach has been refined and made more accessible by tailoring the necessary mathematical tools, avoiding an excess of generality. A second approach contains a novel point of view that the author and some of his collaborators have been developing in recent years, using the semigroup theory of evolution equations to obtain improved results. The extension to electromagnetic waves is explained in one of the appendices.

Different facets of interplay between harmonic analysis and approximation theory are covered in this volume. The topics included are Fourier analysis, function spaces, optimization theory, partial differential equations, and their links to modern developments in the approximation theory. The articles of this collection were originated from two events. The first event took place during the 9th ISAAC Congress in Krakow, Poland, 5th-9th August 2013, at the section "Approximation Theory and Fourier Analysis". The second event was the conference on Fourier Analysis and Approximation Theory in the Centre de Recerca Matematica (CRM), Barcelona, during 4th-8th November 2013, organized by the editors of this volume. All articles selected to be part of this collection were carefully reviewed.

This volume of LNCSE is a collection of the papers from the proceedings of the third workshop on sparse grids and applications. Sparse grids are a popular approach for the numerical treatment of high-dimensional problems. Where classical numerical discretization schemes fail in more than three or four dimensions, sparse grids, in their different guises, are frequently the method of choice, be it spatially adaptive in the hierarchical basis or via the dimensionally adaptive combination technique. Demonstrating once again the importance of this numerical discretization scheme, the selected articles present recent advances on the numerical analysis of sparse grids as well as efficient data structures. The book also discusses a range of applications, including uncertainty quantification and plasma physics.

This monograph presents some theoretical and computational aspects of the parameterization method for invariant manifolds, focusing on the following contexts: invariant manifolds associated with fixed points, invariant tori in quasi-periodically forced systems, invariant tori in Hamiltonian systems and normally hyperbolic invariant manifolds. This book provides algorithms of computation and some practical details of their implementation. The methodology is illustrated with 12 detailed examples, many of them well known in the literature of numerical computation in dynamical systems. A public version of the software used for some of the examples is available online. The book is aimed at mathematicians, scientists and engineers interested in the theory and applications of computational dynamical systems.

This text is an introduction to harmonic analysis on symmetric spaces, focusing on advanced topics such as higher rank spaces, positive definite matrix space and generalizations. It is intended for beginning graduate students in mathematics or researchers in physics or engineering. As with the introductory book entitled "Harmonic Analysis on Symmetric Spaces - Euclidean Space, the Sphere, and the Poincare Upper Half Plane, the style is informal with an emphasis on motivation, concrete examples, history, and applications. The symmetric spaces considered here are quotients X=G/K, where G is a non-compact real Lie group, such as the general linear group GL(n,P) of all n x n non-singular real matrices, and K=O(n), the maximal compact subgroup of orthogonal matrices. Other examples are Siegel's upper half "plane" and the quaternionic upper half "plane". In the case of the general linear group, one can identify X with the space Pn of n x n positive definite symmetric matrices. Many corrections and updates have been incorporated in this new edition. Updates include discussions of random matrix theory and quantum chaos, as well as recent research on modular forms and their corresponding L-functions in higher rank. Many applications have been added, such as the solution of the heat equation on Pn, the central limit theorem of Donald St. P. Richards for Pn, results on densest lattice packing of spheres in Euclidean space, and GL(n)-analogs of the Weyl law for eigenvalues of the Laplacian in plane domains. Topics featured throughout the text include inversion formulas for Fourier transforms, central limit theorems, fundamental domains in X for discrete groups (such as the modular group GL(n,Z) of n x n matrices with integer entries and determinant +/-1), connections with the problem of finding densest lattice packings of spheres in Euclidean space, automorphic forms, Hecke operators, L-functions, and the Selberg trace formula and its applications in spectral theory as well as number theory.

This volume presents an elaborated version of lecture notes for two advanced courses: (Re)Emerging methods in Commutative Algebra and Representation Theory and Building Bridges Between Algebra and Topology, held at the CRM in the spring of 2015. Homological algebra is a rich and ubiquitous area; it is both an active field of research and a widespread toolbox for many mathematicians. Together, these notes introduce recent applications and interactions of homological methods in commutative algebra, representation theory and topology, narrowing the gap between specialists from different areas wishing to acquaint themselves with a rapidly growing field. The covered topics range from a fresh introduction to the growing area of support theory for triangulated categories to the striking consequences of the formulation in the homotopy theory of classical concepts in commutative algebra. Moreover, they also include a higher categories view of Hall algebras and an introduction to the use of idempotent functors in algebra and topology.

Written by a team of leading experts in the field, this volume presents a self-contained account of the theory, techniques and results in metric type spaces (in particular in G-metric spaces); that is, the text approaches this important area of fixed point analysis beginning from the basic ideas of metric space topology. The text is structured so that it leads the reader from preliminaries and historical notes on metric spaces (in particular G-metric spaces) and on mappings, to Banach type contraction theorems in metric type spaces, fixed point theory in partially ordered G-metric spaces, fixed point theory for expansive mappings in metric type spaces, generalizations, present results and techniques in a very general abstract setting and framework. Fixed point theory is one of the major research areas in nonlinear analysis. This is partly due to the fact that in many real world problems fixed point theory is the basic mathematical tool used to establish the existence of solutions to problems which arise naturally in applications. As a result, fixed point theory is an important area of study in pure and applied mathematics and it is a flourishing area of research.

This volume presents original research contributed to the 3rd Annual International Conference on Computational Mathematics and Computational Geometry (CMCGS 2014), organized and administered by Global Science and Technology Forum (GSTF). Computational Mathematics and Computational Geometry are closely related subjects, but are often studied by separate communities and published in different venues. This volume is unique in its combination of these topics. After the conference, which took place in Singapore, selected contributions chosen for this volume and peer-reviewed. The section on Computational Mathematics contains papers that are concerned with developing new and efficient numerical algorithms for mathematical sciences or scientific computing. They also cover analysis of such algorithms to assess accuracy and reliability. The parts of this project that are related to Computational Geometry aim to develop effective and efficient algorithms for geometrical applications such as representation and computation of surfaces. Other sections in the volume cover Pure Mathematics and Statistics ranging from partial differential equations to matrix analysis, finite difference or finite element methods and function approximation. This volume will appeal to advanced students and researchers in these areas.

This book features research contributions from The Abel Symposium on Statistical Analysis for High Dimensional Data, held in Nyvagar, Lofoten, Norway, in May 2014. The focus of the symposium was on statistical and machine learning methodologies specifically developed for inference in "big data" situations, with particular reference to genomic applications. The contributors, who are among the most prominent researchers on the theory of statistics for high dimensional inference, present new theories and methods, as well as challenging applications and computational solutions. Specific themes include, among others, variable selection and screening, penalised regression, sparsity, thresholding, low dimensional structures, computational challenges, non-convex situations, learning graphical models, sparse covariance and precision matrices, semi- and non-parametric formulations, multiple testing, classification, factor models, clustering, and preselection. Highlighting cutting-edge research and casting light on future research directions, the contributions will benefit graduate students and researchers in computational biology, statistics and the machine learning community.

This book constitutes the thoroughly refereed post-conference proceedings of the 23rd IFIP WG 1.3 International Workshop on Algebraic Development Techniques, WADT 2016, held in September 2016 in Gregynog, UK. The 9 revised papers presented together with two invited talks, one invited paper and two survey papers were carefully reviewed and selected from numerous submissions and focus on foundations of algebraic specification; other approaches to formal specification, including process calculi and models of concurrent, distributed and mobile computing; specification languages, methods, and environments; semantics of conceptual modeling methods and techniques; model-driven development; graph transformations, term rewriting and proof systems; integration of formal specification techniques; formal testing and quality assurance, validation, and verification areas, broadly falling into three categories: multimedia content analysis; multimedia signal processing and communications; and multimedia applications and services.

"Proofs and Fundamentals: A First Course in Abstract Mathematics" 2nd edition is designed as a "transition" course to introduce undergraduates to the writing of rigorous mathematical proofs, and to such fundamental mathematical ideas as sets, functions, relations, and cardinality. The text serves as a bridge between computational courses such as calculus, and more theoretical, proofs-oriented courses such as linear algebra, abstract algebra and real analysis. This 3-part work carefully balances Proofs, Fundamentals, and Extras. Part 1 presents logic and basic proof techniques; Part 2 thoroughly covers fundamental material such as sets, functions and relations; and Part 3 introduces a variety of extra topics such as groups, combinatorics and sequences. A gentle, friendly style is used, in which motivation and informal discussion play a key role, and yet high standards in rigor and in writing are never compromised. New to the second edition: 1) A new section about the foundations of set theory has been added at the end of the chapter about sets. This section includes a very informal discussion of the Zermelo- Fraenkel Axioms for set theory. We do not make use of these axioms subsequently in the text, but it is valuable for any mathematician to be aware that an axiomatic basis for set theory exists. Also included in this new section is a slightly expanded discussion of the Axiom of Choice, and new discussion of Zorn's Lemma, which is used later in the text. 2) The chapter about the cardinality of sets has been rearranged and expanded. There is a new section at the start of the chapter that summarizes various properties of the set of natural numbers; these properties play important roles subsequently in the chapter. The sections on induction and recursion have been slightly expanded, and have been relocated to an earlier place in the chapter (following the new section), both because they are more concrete than the material found in the other sections of the chapter, and because ideas from the sections on induction and recursion are used in the other sections. Next comes the section on the cardinality of sets (which was originally the first section of the chapter); this section gained proofs of the Schroeder-Bernstein theorem and the Trichotomy Law for Sets, and lost most of the material about finite and countable sets, which has now been moved to a new section devoted to those two types of sets. The chapter concludes with the section on the cardinality of the number systems. 3) The chapter on the construction of the natural numbers, integers and rational numbers from the Peano Postulates was removed entirely. That material was originally included to provide the needed background about the number systems, particularly for the discussion of the cardinality of sets, but it was always somewhat out of place given the level and scope of this text. The background material about the natural numbers needed for the cardinality of sets has now been summarized in a new section at the start of that chapter, making the chapter both self-contained and more accessible than it previously was. 4) The section on families of sets has been thoroughly revised, with the focus being on families of sets in general, not necessarily thought of as indexed. 5) A new section about the convergence of sequences has been added to the chapter on selected topics. This new section, which treats a topic from real analysis, adds some diversity to the chapter, which had hitherto contained selected topics of only an algebraic or combinatorial nature. 6) A new section called ``You Are the Professor'' has been added to the end of the last chapter. This new section, which includes a number of attempted proofs taken from actual homework exercises submitted by students, offers the reader the opportunity to solidify her facility for writing proofs by critiquing these submissions as if she were the instructor for the course. 7) All known errors have been corrected. 8) Many minor adjustments of wording have been made throughout the text, with the hope of improving the exposition.

Held during algebraic topology special sessions at the Vietnam Institute for Advanced Studies in Mathematics (VIASM, Hanoi), this set of notes consists of expanded versions of three courses given by G. Ginot, H.-W. Henn and G. Powell. They are all introductory texts and can be used by PhD students and experts in the field. Among the three contributions, two concern stable homotopy of spheres: Henn focusses on the chromatic point of view, the Morava K(n)-localization and the cohomology of the Morava stabilizer groups. Powell's chapter is concerned with the derived functors of the destabilization and iterated loop functors and provides a small complex to compute them. Indications are given for the odd prime case. Providing an introduction to some aspects of string and brane topology, Ginot's contribution focusses on Hochschild homology and its generalizations. It contains a number of new results and fills a gap in the literature.

This book constitutes the thoroughly refereed post-conference proceedings of the 22nd International Workshop on Algebraic Development Techniques, WADT 2014, held in September 2014 in Sinaia, Romania. The 8 revised papers presented were carefully reviewed and selected from 13 presentations and focus together with one invited paper on foundations of algebraic specification, approaches to formal specification including process calculi and models of concurrent, distributed and mobile computing, specification languages, methods, and environments, semantics of conceptual modeling methods and techniques, model-driven development, graph transformations, term rewriting and proof systems, integration of formal specification techniques, formal testing and quality assurance, validation, and verification.

This book gives a thorough and entirely self-contained, in-depth introduction to a specific approach to group theory, in a large sense of that word. The focus lie on the relationships which a group may have with other groups, via "universal properties", a view on that group "from the outside". This method of categorical algebra, is actually not limited to the study of groups alone, but applies equally well to other similar categories of algebraic objects. By introducing protomodular categories and Mal'tsev categories, which form a larger class, the structural properties of the category Gp of groups, show how they emerge from four very basic observations about the algebraic litteral calculus and how, studied for themselves at the conceptual categorical level, they lead to the main striking features of the category Gp of groups. Hardly any previous knowledge of category theory is assumed, and just a little experience with standard algebraic structures such as groups and monoids. Examples and exercises help understanding the basic definitions and results throughout the text.

This book is dedicated to Professor Ernst--Rudiger Olderog on the occasion of his 60th birthday. This volume is a reflection on Professor Olderog's contributions to the scientific community. It provides a sample of research ideas that have been influenced directly by Ernst- Rudiger Olderog's work. After a laudatio section that provides a brief overview of Ernst- Rudiger Olderog's research, the book is comprised of five parts with scientific papers written by colleagues and collaborators of Professor Olderog. The papers address semantics, process algebras, logics for verification, program analysis, and synthesis approaches.

We humans are collectively driven by a powerful - yet not fully explained - instinct to understand. We would like to see everything established, proven, laid bare. The more important an issue, the more we desire to see it clarified, stripped of all secrets, all shades of gray. What could be more important than to understand the Universe and ourselves as a part of it? To find a window onto our origin and our destiny? This book examines how far our modern cosmological theories - with their sometimes audacious models, such as inflation, cyclic histories, quantum creation, parallel universes - can take us towards answering these questions. Can such theories lead us to ultimate truths, leaving nothing unexplained? Last, but not least, Heller addresses the thorny problem of why and whether we should expect to find theories with all-encompassing explicative power.

An ontology is a formal description of concepts and relationships that can exist for a community of human and/or machine agents. The notion of ontologies is crucial for the purpose of enabling knowledge sharing and reuse. The Handbook on Ontologies provides a comprehensive overview of the current status and future prospectives of the field of ontologies considering ontology languages, ontology engineering methods, example ontologies, infrastructures and technologies for ontologies, and how to bring this all into ontology-based infrastructures and applications that are among the best of their kind. The field of ontologies has tremendously developed and grown in the five years since the first edition of the "Handbook on Ontologies". Therefore, its revision includes 21 completely new chapters as well as a major re-working of 15 chapters transferred to this second edition.

This is a monograph that details the use of Siegel’s method and the classical results of homotopy groups of spheres and Lie groups to determine some Gottlieb groups of projective spaces or to give the lower bounds of their orders. Making use of the properties of Whitehead products, the authors also determine some Whitehead center groups of projective spaces that are relevant and new within this monograph.

Karl Menger, one of the founders of dimension theory, is among the most original mathematicians and thinkers of the twentieth century. He was a member of the Vienna Circle and the founder of its mathematical equivalent, the Viennese Mathematical Colloquium. Both during his early years in Vienna and, after his emigration, in the United States, Karl Menger made significant contributions to a wide variety of mathematical fields, and greatly influenced many of his colleagues. These two volumes contain Menger's major mathematical papers, based on his own selection from his extensive writings. They deal with topics as diverse as topology, geometry, analysis and algebra, and also include material on economics, sociology, logic and philosophy. The Selecta Mathematica is a monument to the diversity and originality of Menger's ideas.

Karl Menger, one of the founders of dimension theory, is among the most original mathematicians and thinkers of the twentieth century. He was a member of the Vienna Circle and the founder of its mathematical equivalent, the Viennese Mathematical Colloquium. Both during his early years in Vienna and, after his emigration, in the United States, Karl Menger made significant contributions to a wide variety of mathematical fields, and greatly influenced many of his colleagues. These two volumes contain Menger's major mathematical papers, based on his own selection from his extensive writings. They deal with topics as diverse as topology, geometry, analysis and algebra, and also include material on economics, sociology, logic and philosophy. The Selecta Mathematica is a monument to the diversity and originality of Menger's ideas.

This textbook addresses the mathematical description of sets, categories, topologies and measures, as part of the basis for advanced areas in theoretical computer science like semantics, programming languages, probabilistic process algebras, modal and dynamic logics and Markov transition systems. Using motivations, rigorous definitions, proofs and various examples, the author systematically introduces the Axiom of Choice, explains Banach-Mazur games and the Axiom of Determinacy, discusses the basic constructions of sets and the interplay of coalgebras and Kripke models for modal logics with an emphasis on Kleisli categories, monads and probabilistic systems. The text further shows various ways of defining topologies, building on selected topics like uniform spaces, Goedel's Completeness Theorem and topological systems. Finally, measurability, general integration, Borel sets and measures on Polish spaces, as well as the coalgebraic side of Markov transition kernels along with applications to probabilistic interpretations of modal logics are presented. Special emphasis is given to the integration of (co-)algebraic and measure-theoretic structures, a fairly new and exciting field, which is demonstrated through the interpretation of game logics. Readers familiar with basic mathematical structures like groups, Boolean algebras and elementary calculus including mathematical induction will discover a wealth of useful research tools. Throughout the book, exercises offer additional information, and case studies give examples of how the techniques can be applied in diverse areas of theoretical computer science and logics. References to the relevant mathematical literature enable the reader to find the original works and classical treatises, while the bibliographic notes at the end of each chapter provide further insights and discussions of alternative approaches.

This is a text in methods of applied statistics for researchers who design and conduct experiments, perform statistical inference, and write technical reports. These research activities rely on an adequate knowledge of applied statistics. The reader both builds on basic statistics skills and learns to apply it to applicable scenarios without over-emphasis on the technical aspects. Demonstrations are a very important part of this text. Mathematical expressions are exhibited only if they are defined or intuitively comprehensible. This text may be used as a self review guidebook for applied researchers or as an introductory statistical methods textbook for students not majoring in statistics.​ Discussion includes essential probability models, inference of means, proportions, correlations and regressions, methods for censored survival time data analysis, and sample size determination. The author has over twenty years of experience on applying statistical methods to study design and data analysis in collaborative medical research setting as well as on teaching. He received his PhD from University of Southern California Department of Preventive Medicine, received a post-doctoral training at Harvard Department of Biostatistics, has held faculty appointments at UCLA School of Medicine and Harvard Medical School, and currently a biostatistics faculty member at Massachusetts General Hospital and Harvard Medical School in Boston, Massachusetts, USA. 

This book is a comprehensive, systematic survey of the synthesis problem, and of region theory which underlies its solution, covering the related theory, algorithms, and applications. The authors focus on safe Petri nets and place/transition nets (P/T-nets), treating synthesis as an automated process which, given behavioural specifications or partial specifications of a system to be realized, decides whether the specifications are feasible, and then produces a Petri net realizing them exactly, or if this is not possible produces a Petri net realizing an optimal approximation of the specifications. In Part I the authors introduce elementary net synthesis. In Part II they explain variations of elementary net synthesis and the unified theory of net synthesis. The first three chapters of Part III address the linear algebraic structure of regions, synthesis of P/T-nets from finite initialized transition systems, and the synthesis of unbounded P/T-nets. Finally, the last chapter in Part III and the chapters in Part IV cover more advanced topics and applications: P/T-net with the step firing rule, extracting concurrency from transition systems, process discovery, supervisory control, and the design of speed-independent circuits. Most chapters conclude with exercises, and the book is a valuable reference for both graduate students of computer science and electrical engineering and researchers and engineers in this domain.