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Numerical Methods for General and Structured Eigenvalue Problems
The purpose of this book is to describe recent developments in solving eig- value problems, in particular with respect to the QR and QZ algorithms as well as structured matrices. Outline Mathematically speaking, the eigenvalues of a square matrix A are the roots of its characteristic polynomial det(A??I). An invariant subspace is a linear subspace that stays invariant under the action of A. In realistic applications, it usually takes a long process of simpli?cations, linearizations and discreti- tions before one comes up with the problem of computing the eigenvalues of a matrix. In some cases, the eigenvalues have an intrinsic meaning, e.g., for the expected long-time behavior of a dynamical system; in others they are just meaningless intermediate values of a computational method. The same applies to invariant subspaces, which for example can describe sets of initial states for which a dynamical system produces exponentially decaying states. Computing eigenvalues has a long history, dating back to at least 1846 when Jacobi [172] wrote his famous paper on solving symmetric eigenvalue problems. Detailed historical accounts of this subject can be found in two papers by Golub and van der Vorst [140, 327].
Nonsmooth Vector Functions and Continuous Optimization
A recent significant innovation in mathematical sciences has been the progressive use of nonsmooth calculus, an extension of the differential calculus, as a key tool of modern analysis in many areas of mathematics, operations research, and engineering. Focusing on the study of nonsmooth vector functions, this book presents a comprehensive account of the calculus of generalized Jacobian matrices and their applications to continuous nonsmooth optimization problems and variational inequalities in finite dimensions.
Non-negative Matrices and Markov Chains
This book is a photographic reproduction of the book of the same title published in 1981, for which there has been continuing demand on account of its accessible technical level. Its appearance also helped generate considerable subsequent work on inhomogeneous products of matrices. This printing adds an additional bibliography on coefficients of ergodicity and a list of corrigenda.
Naive Lie Theory
In this new textbook, acclaimed author John Stillwell presents a lucid introduction to Lie theory suitable for junior and senior level undergraduates. In order to achieve this, he focuses on the so-called "classical groups'' that capture the symmetries of real, complex, and quaternion spaces. These symmetry groups may be represented by matrices, which allows them to be studied by elementary methods from calculus and linear algebra.
MSCs and Innovative Biomaterials in Dentistry
Presents the modern concepts of mesenchymal stem cells (MSCs) and biomaterials as they pertain to the dental field. The book is organized around three main topics: MSCs biology, advanced biomaterials, and clinical applications. The chapters present basic information on stem cell biology and physiology, modern biomaterials that improve bone tissue regeneration, the biomatrices like platelet-rich fibrin (PRF) used to functionalize the biomaterials surface, the strategic and safe intraoral seats of harvesting, the new sources for MSCs, as well as the future perspectives and new challenges in these exciting fields.
Molecules in Superfluid Helium Nanodroplets : Spectroscopy, Structure, and Dynamics
This book covers recent advances in experiments using the ultra-cold, very weakly perturbing superfluid environment provided by helium nanodroplets for high resolution spectroscopic, structural and dynamic studies of molecules and synthetic clusters. The recent infra-red, UV-Vis studies of radicals, molecules, clusters, ions and biomolecules, as well as laser dynamical and laser orientational studies, are reviewed. The Coulomb explosion studies of the uniquely quantum structures of small helium clusters, X-ray imaging of large droplets and electron diffraction of embedded molecules are also described. Particular emphasis is given to the synthesis and detection of new species by mass spectrometry and deposition electron microscopy.
Modern parallel programming with C++ and assembly language : X86 SIMD development using AVX, AVX2, and AVX-512
Understand the essential details about x86 SIMD architectures and instruction sets including AVX, AVX2, and AVX-512. / Master x86 SIMD data types, arithmetic instructions, and data management operations using both integer and floating-point operands. / Code performance-enhancing functions and algorithms that fully exploit the SIMD capabilities of a modern x86 processor. Employ C++ intrinsic functions and x86-64 assembly language code to carry out arithmetic calculations using common programming constructs including arrays, matrices, and user-defined data structures. Harness the x86 SIMD instruction sets to significantly accelerate the performance of computationally intense algorithms in applications such as machine learning, image processing, computer graphics, statistics, and matrix arithmetic. / Apply leading-edge coding strategies and techniques to optimally exploit the x86 SIMD instruction sets for maximum possible performance.
Metallopolymer Nanocomposites
Highly dispersed nanoscale particles in polymer matrices are currently attracting great interest in many fields of chemistry, physics and materials science. This book presents and analyzes the essential data on nanoscale metal clusters dispersed in, or chemically bonded with polymers. Special attention is paid to the in situ synthesis of the nanocomposites, their chemical interactions, and the size and distribution of the particles in the polymer matrix. Numerous novel nanocomposites are described with regard to their mechanical, electrophysical, optical, magnetic, catalytic and biological properties. Their applications, present and future, are outlined. The book is addressed both to researchers who actively use these materials and to students entering this multidisciplinary field.
Matrix Algebra From a Statistician`s Perspective
This book presents matrix algebra in a way that is well-suited for those with an interest in statistics or a related discipline. It provides thorough and unified coverage of the fundamental concepts along with the specialized topics encountered in areas of statistics such as linear statistical models and multivariate analysis. It includes a number of very useful results that have only been available from relatively obscure sources. Detailed proofs are provided for all results. The style and level of presentation are designed to make the contents accessible to a broad audience. The book is essentially self-contained, though it is best-suited for a reader who has had some previous exposure to matrices (of the kind that might be acquired in a beginning course on linear or matrix algebra).
Matrix Algebra : Theory, Computations, and Applications in Statistics
Matrix algebra is one of the most important areas of mathematics for data analysis and for statistical theory. The first part of this book presents the relevant aspects of the theory of matrix algebra for applications in statistics. This part begins with the fundamental concepts of vectors and vector spaces, next covers the basic algebraic properties of matrices, then describes the analytic properties of vectors and matrices in the multivariate calculus, and finally discusses operations on matrices in solutions of linear systems and in eigenanalysis. This part is essentially self-contained.
Infinite matrices and their finite sections : An introduction to the limit operator method
In this book we are concerned with the study of a certain class of infinite matrices and two important properties of them: their Fredholmness and the stability of the approximation by their finite truncations. Let us take these two properties as a starting point for the big picture that shall be presented in what follows. Stability Fredholmness We think of our infinite matrices as bounded linear operators on a Banach space E of two-sided infinite sequences.The class of operators we are interested in consists of those bounded and linear operatorson E which can be approximated in the operator norm by b and matrices. We refer to them as band-dominated operators. Of course, these considerations 2 are not limited to the space E = . We will widen the selection of the underlying space E in three directions: p
Idempotent matrices over complex group algebras
The study of idempotent elements in the group algebras originates from geometric and analytic considerations. … This book provides an introduction to the study of these problems.It collects and presents in a systematic way basic techniques and important results that have been obtained during the past few decades.The book is suitable for independent study. Moreover, all chapters and appendices finish with a sufficient number of exercises that also increase the quality of the book.
Hierarchical Matrices : A Means to Efficiently Solve Elliptic Boundary Value Problems
Hierarchical matrices are an efficient framework for large-scale fully populated matrices arising, e.g., from the finite element discretization of solution operators of elliptic boundary value problems. In addition to storing such matrices, approximations of the usual matrix operations can be computed with logarithmic-linear complexity, which can be exploited to setup approximate preconditioners in an efficient and convenient way. Besides the algorithmic aspects of hierarchical matrices, the main aim of this book is to present their theoretical background. The book contains the existing approximation theory for elliptic problems including partial differential operators with nonsmooth coefficients.
Hamiltonian Reduction by Stages
In this volume readers will find for the first time a detailed account of the theory of symplectic reduction by stages, along with numerous illustrations of the theory. Special emphasis is given to group extensions, including a detailed discussion of the Euclidean group, the oscillator group, the Bott-Virasoro group and other groups of matrices. Ample background theory on symplectic reduction and cotangent bundle reduction in particular is provided. Novel features of the book are the inclusion of a systematic treatment of the cotangent bundle case, including the identification of cocycles with magnetic terms, as well as the general theory of singular reduction by stages.
Grey information : Theory and practical applications
he book covers the latest advances in grey information and systems research, providing a state-of-the-art overview of this important field. Covering the theoretical foundation, fundamental methods and main topics in grey information and systems research, this book includes all the elementary concepts: basic principles, grey numbers and their operations, grey equations and matrices, operators of sequences and generations of grey sequences, grey incidence analysis, grey clusters and grey statistical evaluations, grey systems modeling, grey combined models, grey prediction, grey decisions, grey programming, grey input and output and grey controls, etc.
Geometric Aspects of Functional Analysis : Israel Seminar 2004-2005
Most of the papers deal with different aspects of the Asymptotic Geometric Analysis, ranging from classical topics in the geometry of convex bodies, to inequalities involving volumes of such bodies or, more generally, log-concave measures, to the study of sections or projections of convex bodies. In many of the papers Probability Theory plays an important role; in some limit laws for measures associated with convex bodies, resembling Central Limit Theorems, are derive and in others probabilistic tools are used extensively. There are also papers on related subjects, including a survey on the behavior of the largest eigenvalue of random matrices and some topics in Number Theory.
Frontiers in Number Theory, Physics, and Geometry II : On Conformal Field Theories, Discrete Groups and Renormalization
The present book collects most of the courses and seminars delivered at the meetingentitled"FrontiersinNumberTheory, PhysicsandGeometry", which took place at the Centrede PhysiquedesHouches in theFrenchAlps, March9- 21,2003. Itisdividedintotwovolumes. VolumeIcontainsthecontributionson three broad topics: Random matrices, Zeta functions and Dynamical systems. The present volume contains sixteen contribution sonthreethemes:Conformal?eld theories for strings and branes, Discrete groups and automorphic forms and?nally, Hopf algebras and renormalization. The relation between Mathematics and Physics has a long history.
Frontiers in Number Theory, Physics, and Geometry I : On Random Matrices, Zeta Functions, and Dynamical Systems
This book presents pedagogical contributions on selected topics relating Number Theory, Theoretical Physics and Geometry. The parts are composed of long self-contained pedagogical lectures followed by shorter contributions on specific subjects organized by theme. Most courses and short contributions go up to the recent developments in the fields; some of them follow their author?s original viewpoints. There are contributions on Random Matrix Theory, Quantum Chaos, Non-commutative Geometry, Zeta functions, and Dynamical Systems. The chapters of this book are extended versions of lectures given at a meeting entitled Number Theory, Physics and Geometry, held at Les Houches in March 2003, which gathered mathematicians and physicists.
Foundation Mathematics for Computer Science : A Visual Approach
In this second edition of Foundation Mathematics for Computer Science, John Vince has reviewed and edited the original book and written new chapters on combinatorics, probability, modular arithmetic and complex numbers. These subjects complement the existing chapters on number systems, algebra, logic, trigonometry, coordinate systems, determinants, vectors, matrices, geometric matrix transforms, differential and integral calculus. During this journey, the author touches upon more esoteric topics such as quaternions, octonions, Grassmann algebra, Barrycentric coordinates, transfinite sets and prime numbers.
Essays in Constructive Mathematics
This book aims to promote constructive mathematics, not by defining it or formalizing it, but by practicing it, by basing all definitions and proofs on finite algorithms. The topics covered derive from classic works of nineteenth century mathematics---among them Galois' theory of algebraic equations, Gauss's theory of binary quadratic forms and Abel's theorem about integrals of rational differentials on algebraic curves. It is not surprising that the first two topics can be treated constructively---although the constructive treatments shed a surprising amount of light on them---but the last topic, involving integrals and differentials as it does, might seem to call for infinite processes. In this case too, however, finite algorithms suffice to define the genus of an algebraic curve, to prove that birationally equivalent curves have the same genus, and to prove the Riemann-Roch theorem. The main algorithm in this case is Newton's polygon, which is given a full treatment. Other topics covered include the fundamental theorem of algebra, the factorization of polynomials over an algebraic number field, and the spectral theorem for symmetric matrices.
