Format: Hardback, 105 pages, height x width: 235x155 mm, 4 Illustrations,
black and white; XX, 105 p. 4 illus.
Series: SpringerBriefs in Statistics
Pub. Date: 13-Feb-2023
ISBN-13: 9783031147968
This book presents a critical overview of statistical fiber bundle models, including existing models and potential new ones. The authors focus on both the physical and statistical aspects of a specific load-sharing example: the breakdown for circuits of capacitors and related dielectrics. In addition, they investigate some areas of open research.
This book is designed for graduate students and researchers in statistics, materials science, engineering, physics, and related fields, as well as practitioners and technicians in materials science and mechanical engineering.
1. Introduction and Preliminaries.-
2. Electrical Circuits of Ordinary Capacitors.-
3. Breakdown of Thin-Film Dielectrics.-
4. Cell Models for Dielectrics.-
5. Electrical Breakdown and the Breakdown Formalism.-
6. Statistical Properties of a Load-Sharing Bundle.-
7. Statistical Analysis of Kim and Lee (2004)'s Data.-
8. Circuits of Ordinary Capacitors.-
9. Size Effects Relationships Motivated by the Load-Sharing Cell Model.-
10. Concluding Comments and Future Research Directions.
Format: Hardback, 535 pages, height x width: 235x155 mm, XV, 535 p., 1 Hardback
Series: Birkhauser Advanced Texts / Basler Lehrbucher
Pub. Date: 13-Dec-2022
ISBN-13: 9783031178368
This book, which is the first of two volumes, presents, in a unique way, some of the most relevant research tools of modern analysis. This work empowers young researchers with all the necessary techniques to explore the various subfields of this broad subject, and introduces relevant frameworks where these tools can be immediately deployed.
Volume I starts with the foundations of modern analysis. The first three chapters are devoted to topology, measure theory, and functional analysis. Chapter 4 offers a comprehensive analysis of the main function spaces, while Chapter 5 covers more concrete subjects, like multivariate analysis, which are closely related to applications and more difficult to find in compact form. Chapter 6 deals with smooth and non-smooth calculus of functions; Chapter 7 introduces certain important classes of nonlinear operators; and Chapter 8 complements the previous three chapters with topics of variational analysis.
Each chapter of this volume finishes with a list of problems ? handy for understanding and self-study ? and historical notes that give the reader a more vivid picture of how the theory developed. Volume II consists of various applications using the tools and techniques developed in this volume.
By offering a clear and wide picture of the tools and applications of modern analysis, this work can be of great benefit not only to mature graduate students seeking topics for research, but also to experienced researchers with an interest in this vast and rich field of mathematics.
Volume I - Theory: - Topology.- Measure Theory.- Banach Space Theory.- Function Spaces.- Multivalued Analysis.- Smooth and Nonsmooth Calculus.- Nonlinear Operators.- Variational Analysis.- References.
Format: Hardback, 200 pages, height x width: 235x155 mm, 2 Illustrations, color;
5 Illustrations, black and white; X, 200 p. 7 illus., 2 illus. in color
Series: Trends in Mathematics
Pub. Date: 08-Jan-2023
ISBN-13: 9783031190810
The chapters in this contributed volume explore new results and existing problems in algebra, analysis, and related topics. This broad coverage will help generate new ideas to solve various challenges that face researchers in pure mathematics. Specific topics covered include maximal rotational hypersurfaces, k-Horadam sequences, quantum dynamical semigroups, and more. Additionally, several applications of algebraic number theory and analysis are presented. Algebra, Analysis, and Associated Topics will appeal to researchers, graduate students, and engineers interested in learning more about the impact pure mathematics has on various fields.
Maximal Rotational Hypersurfaces Having Spacelike Axis, Spacelike Profile Curve in Minkowski Geometry.- On the generalized k-Horadam like sequences.- New Results on (p1, p2...,pn, k) -Analogue of Lauricella function with transforms and Fractional calculus operator.- Absolute Linear Method of Summation for Orthogonal Series.- Derivations and Special Functions over Fields.- On Equalities of Central Automorphisms Group With Various Automorphism Groups.- Automorphism Group and Laplacian Spectrum of a Graph over Brandt Semigroups.- Unified iteration scheme in CAT(0) spaces and fixed point approximation of mean nonexpansive mappings.- Semigroups of Completely Positive Maps.- On Sumset Problems And Their Various Types.- Vector-Valued Affine Bi-Frames on Local Fields.- A New Perspective on I2-statistical limit points and I2-statistical cluster points in probabilistic normed spaces.- Evaluation of Integral transforms in terms of Humbert and Lauricella functions and their applications.- Some Spaces in Neutrosophic e-Open Sets.- Generalized Finite Continuous Ridgelet Transform.
Format: Paperback / softback, 178 pages, height x width: 235x155 mm, XIV, 178 p.
Series: Lecture Notes in Mathematics 2323
Pub. Date: 22-Dec-2022
ISBN-13: 9783031194351
This book is about extending index theory to some examples where non-Fredholm operators arise. It focuses on one aspect of the problem of what replaces the notion of spectral flow and the Fredholm index when the operators in question have zero in their essential spectrum. Most work in this topic stems from the so-called Witten index that is discussed at length here. The new direction described in these notes is the introduction of `spectral flow beyond the Fredholm case'.
Creating a coherent picture of numerous investigations and scattered notions of the past 50 years, this work carefully introduces spectral flow, the Witten index and the spectral shift function and describes their relationship. After the introduction, Chapter 2 carefully reviews Double Operator Integrals, Chapter 3 describes the class of so-called p-relative trace class perturbations, followed by the construction of Krein's spectral shift function in Chapter 4. Chapter 5 reviews the analytic approach to spectral flow, culminating in Chapter 6 in the main abstract result of the book, namely the so-called principal trace formula. Chapter 7 completes the work with illustrations of the main results using explicit computations on two examples: the Dirac operator in Rd, and a differential operator on an interval. Throughout, attention is paid to the history of the subject and earlier references are provided accordingly.
The book is aimed at experts in index theory as well as newcomers to the field.
1. Introduction.-
2. Double Operator Integrals.-
3. The Model Operator and its Approximants.-
4. The Spectral Shift Function.-
5. Spectral Flow.-
6. The Principal Trace Formula and its Applications.-
7. Examples.
Format: Hardback, 695 pages, height x width: 235x155 mm, 276 Illustrations, black and white; XIII, 695 p. 276 illus
Series: Applied Mathematical Sciences 112
Pub. Date: 11-Feb-2023
ISBN-13: 9783031220067
Other books in subject:
Providing readers with a solid basis in dynamical systems theory, as well as explicit procedures for application of general mathematical results to particular problems, the focus here is on efficient numerical implementations of the developed techniques. The book is designed for advanced undergraduates or graduates in applied mathematics, as well as for Ph.D. students and researchers in physics, biology, engineering, and economics who use dynamical systems as model tools in their studies. A moderate mathematical background is assumed, and, whenever possible, only elementary mathematical tools are used. This new edition preserves the structure of the first while updating the context to incorporate recent theoretical developments, in particular new and improved numerical methods for bifurcation analysis.
1 Introduction to Dynamical Systems.- 2 Topological Equivalence, Bifurcations, and Structural Stability of Dynamical Systems.- 3 One-Parameter Bifurcations of Equilibria in Continuous-Time Dynamical Systems.- 4 One-Parameter Bifurcations of Fixed Points in Discrete-Time Dynamical Systems.- 5 Bifurcations of Equilibria and Periodic Orbits in n-Dimensional Dynamical Systems.- 6 Bifurcations of Orbits Homoclinic and Heteroclinic to Hyperbolic Equilibria.- 7 Other One-Parameter Bifurcations in Continuous-Time Dynamical Systems.- 8 Two-Parameter Bifurcations of Equilibria in Continuous-Time Dynamical Systems.- 9 Two-Parameter Bifurcations of Fixed Points in Discrete-Time Dynamical Systems.- 10 Numerical Analysis of Bifurcations.- A Basic Notions from Algebra, Analysis, and Geometry.- A.1 Algebra.- A.1.1 Matrices.- A.1.2 Vector spaces and linear transformations.- A.1.3 Eigenvectors and eigenvalues.- A.1.4 Invariant subspaces, generalized eigenvectors, and Jordan normal form.- A.1.5 Fredholm Alternative Theorem.- A.1.6 Groups.- A.2 Analysis.- A.2.1 Implicit and Inverse Function Theorems.- A.2.2 Taylor expansion.- A.2.3 Metric, normed, and other spaces.- A.3 Geometry.- A.3.1 Sets.- A.3.2 Maps.- A.3.3 Manifolds.- References.
Format: Hardback, 940 pages, height x width: 235x155 mm, 2 Tables, color; 2 Illustrations, color;
12 Illustrations, black and white; X, 940 p. 14 illus., 2 illus. in color.
Pub. Date: 01-Feb-2023
ISBN-13: 9783031219115
This monograph provides a rigorous, encyclopedic treatment of the fundamental topics in real analysis, functional analysis, and measure theory. The result of many years of the authorfs careful and extensive work, this text synthesizes and builds upon the existing literature in an effort to develop and solidify the theory of measure-theoretic calculus in abstract spaces. Standard results and proofs are illustrated in general abstract settings under rigorous treatment, and numerous ancillary topics are also covered in detail, such as functional analytic treatment of optimization, probability theory, and the theory of Sobolev spaces. Applied mathematicians and researchers working in control theory, operations research, economics, optimization theory, and many other areas will find this text to be a comprehensive and invaluable resource. It can also serve as an analysis textbook for graduate-level students.
Introduction.- Set Theory.- Topological Spaces.- Metric Spaces.- Compact and Locally Compact Spaces.- Vector Spaces.- Banach Spaces.- Global Theory of Optimization.- Differentiation in Banach Spaces.- Local Theory of Optimization.- General Measure and Integration.- Differentiation and Integration.- Hilbert Spaces.- Probability Theory.
Format: Hardback, 500 pages, height x width: 235x155 mm, 95 Illustrations, black and white; XXVIII, 500 p. 95 illus
Pub. Date: 07-Feb-2023
ISBN-13: 9783031224218
This textbook demonstrates the strong interconnections between linear algebra and group theory by presenting them simultaneously, a pedagogical strategy ideal for an interdisciplinary audience. Being approached together at the same time, these two topics complete one another, allowing students to attain a deeper understanding of both subjects. The opening chapters introduce linear algebra with applications to mechanics and statistics, followed by group theory with applications to projective geometry. Then, high-order finite elements are presented to design a regular mesh and assemble the stiffness and mass matrices in advanced applications in quantum chemistry and general relativity.
This text is ideal for undergraduates majoring in engineering, physics, chemistry, computer science, or applied mathematics. It is mostly self-contained?readers should only be familiar with elementary calculus. There are numerous exercises, with hints or full solutions provided. A series of roadmaps are also provided to help instructors choose the optimal teaching approach for their discipline.
The second edition has been revised and updated throughout and includes new material on the Jordan form, the Hermitian matrix and its eigenbasis, and applications in numerical relativity and electromagnetics.
Part I: Introduction to Linear Algebra.- Vectors and Matrices.- Determinant and Vector Product in Physics.- Markov Matrix and its Spectrum: Towards Search Engines.- Special Relativity: Algebraic Point of View.- Part II: Introduction to Group Theory.- Groups and Isomorphism Theorems.- Projective Geometry in Computer Graphics.- Quantum Mechanics: Algebraic Point of View.- Part III: Polynomials and Basis Functions.- Polynomials and Their Gradient.- Basis Functions: Barycentric Coordinates in 3D.- Part IV: Finite Elements in 3-D. - Automatic Mesh Generation.- Mesh Regularity.- Numerical Integration.- Spline: Variational Model in 3D.- Part V: Permuation Group in Quantum Chemistry.- Determinant and Electronic Structure.- Part VI: The Jordan Form.- The Jordan Form.- Jordan Decomposition.- Algebras and their Derivation.- Part VII: Linearization in Numerical Relativity.- Einstein Equations and their Linearization.