Format: Paperback / softback, 125 pages, height x width: 235x155 mm, 15 Illustrations,
color; 3 Illustrations, black and white; XIV, 125 p. 18 illus., 15 illus. in color.,
Series: SpringerBriefs in Mathematical Physics 47
Pub. Date: 01-May-2023
ISBN-13: 9789811995262
Hermite's theorem makes it known that there are three levels of mathematical frames in which a simple addition formula is valid. They are rational, q-analogue, and elliptic-analogue. Based on the addition formula and associated mathematical structures, productive studies have been carried out in the process of q-extension of the rational (classical) formulas in enumerative combinatorics, theory of special functions, representation theory, study of integrable systems, and so on. Originating from the paper by Date, Jimbo, Kuniba, Miwa, and Okado on the exactly solvable statistical mechanics models using the theta function identities (1987), the formulas obtained at the q-level are now extended to the elliptic level in many research fields in mathematics and theoretical physics. In the present monograph, the recent progress of the elliptic extensions in the study of statistical and stochastic models in equilibrium and nonequilibrium statistical mechanics and probability theory is shown. At the elliptic level, many special functions are used, including Jacobi's theta functions, Weierstrass elliptic functions, Jacobi's elliptic functions, and others. This monograph is not intended to be a handbook of mathematical formulas of these elliptic functions, however. Thus, use is made only of the theta function of a complex-valued argument and a real-valued nome, which is a simplified version of the four kinds of Jacobi's theta functions. Then, the seven systems of orthogonal theta functions, written using a polynomial of the argument multiplied by a single theta function, or pairs of such functions, can be defined. They were introduced by Rosengren and Schlosser (2006), in association with the seven irreducible reduced affine root systems. Using Rosengren and Schlosser's theta functions, non-colliding Brownian bridges on a one-dimensional torus and an interval are discussed, along with determinantal point processes on a two-dimensional torus. Their scaling limits are argued, and the infinite particle systems are derived. Such limit transitions will be regarded as the mathematical realizations of the thermodynamic or hydrodynamic limits that are central subjects of statistical mechanics.
Introduction.- Brownian Motion and Theta Functions.- Biorthogonal
Systems of Theta Functions and Macdonald Denominators.- KMLGV Determinants
and Noncolliding Brownian Bridges.- Determinantal Point Processes Associated
with Biorthogonal Systems.- Doubly Periodic Determinantal Point Processes.-
Future Problems.
Format: Paperback / softback, 209 pages, height x width: 235x155 mm,
1 Illustrations, black and white; XII, 209 p. 1 illus.,
Series: La Matematica per il 3+2 144
Pub. Date: 20-Apr-2023
ISBN-13: 9783031215605
This book provides an introduction to information theory, focussing on Shannon’s three foundational theorems of 1948?1949. Shannon’s first two theorems, based on the notion of entropy in probability theory, specify the extent to which a message can be compressed for fast transmission and how to erase errors associated with poor transmission. The third theorem, using Fourier theory, ensures that a signal can be reconstructed from a sufficiently fine sampling of it. These three theorems constitute the roadmap of the book.
The first chapter studies the entropy of a discrete random variable and related notions. The second chapter, on compression and error correcting, introduces the concept of coding, proves the existence of optimal codes and good codes (Shannon's first theorem), and shows how information can be transmitted in the presence of noise (Shannon's second theorem). The third chapter proves the sampling theorem (Shannon's third theorem) and looks at its connections with other results, such as the Poisson summation formula. Finally, there is a discussion of the uncertainty principle in information theory.
Featuring a good supply of exercises (with solutions), and an introductory chapter covering the prerequisites, this text stems out lectures given to mathematics/computer science students at the beginning graduate level.
Elements of Theory of Probability.- Entropy and Mutual Information.-
Coding.- Sampling.- Solutions to Exercises.- Bibliography.- Notation.- Index.
Format: Hardback, 326 pages, height x width: 235x155 mm, 70 Tables, color;
10 Illustrations, color; 198 Illustrations, black and white; IX, 326 p. 208 illus., 10 illus. in color.,
Series: UNITEXT for Physics
Pub. Date: 28-Apr-2023
ISBN-13: 9783031251535
This textbook introduces the language and the techniques of the theory of dynamical systems of finite dimension for an audience of physicists, engineers, and mathematicians at the beginning of graduation. Author addresses geometric, measure, and computational aspects of the theory of dynamical systems. Some freedom is used in the more formal aspects, using only proofs when there is an algorithmic advantage or because a result is simple and powerful.
The first part is an introductory course on dynamical systems theory. It can be taught at the master's level during one semester, not requiring specialized mathematical training. In the second part, the author describes some applications of the theory of dynamical systems. Topics often appear in modern dynamical systems and complexity theories, such as singular perturbation theory, delayed equations, cellular automata, fractal sets, maps of the complex plane, and stochastic iterations of function systems are briefly explored for advanced students. The author also explores applications in mechanics, electromagnetism, celestial mechanics, nonlinear control theory, and macroeconomy. A set of problems consolidating the knowledge of the different subjects, including more elaborated exercises, are provided for all chapte
Differential Equations as Dynamical Systems.- Stability of fixed
points.- Difference equations as dynamical systems.- Classification of fixed
points.- Hamiltonian systems.- Numerical Methods.-Strange Attractors and Maps
of an Interval.- Stable, Unstable and Centre manifolds.-Dynamics in the
Centre Manifold.- Lyapunov Exponents and Oseledets Theorem.- Chaos.- Limit
and Recurrent Sets.-Poincare Maps.- The Poincare-Bendixon Theorem.-
Bifurcations of Differential Equations.-Singular Pertubations and
Ducks.-Strange Attractors in Delay Equations.- Complexity of Strange
Attractors.-Intermittency.- Cellular Automata.- Maps of the Complex Plane.-
Stochastic Iteration of Function Systems.- Linear Maps on the Torus and
Symbolic Dynamics.- Parametric Resonance.- Robot Motion.- Synchronisation of
Pendula.- Synchronisation of Clocks.- Chaos in Stormer Problem.-Introduction
to Celestial mechanics.- Introduction to non-Liner control Theory.-
Appendices.
Format: Hardback, 409 pages, height x width: 235x155 mm, 119 Illustrations, color;
41 Illustrations, black and white; X, 409 p. 160 illus., 119 illus. in color.,
Series: Industrial and Applied Mathematics
Pub. Date: 02-May-2023
ISBN-13: 9789811999086
This book collects select chapters on modern industrial problems related to uncertainties and vagueness in the expert domain of knowledge. The book further provides the knowledge related to application of various mathematical and statistical tools in these areas. The results presented in the book help the researchers and scientists in handling complicated projects in their domains. Useful to industrialists, academicians, researchers and students alike, the book aims to help managers and technical specialists in designing and implementation of reliability and risk programs as below:
Ensure the system safety and risk informed asset management Follow a proper strategy to maintain the mechanical components of the system Schedule the proper actions throughout the product life cycle Understand the structure and cost of a complex system Plan the proper schedule to improve the reliability and life of the system Identify unwanted failures and set up preventive and correction action
Degradation and Failure Mechanisms of Complex Systems: Principles.-
Simplified Approach to Analyse Fuzzy Reliability of a Repairable System.-
Fault-Tolerant and Resilient Neural Control for Discrete-Time Nonlinear
Systems.- Bayesian Reliability Analysis of the Topp-Leone Model under
Different Loss Functions.- Availability Analysis of Non-Markovian Models with
Rejuvenation and Check Pointing.- Reliability Metrics of Textile Confection
Plant Using Copula Linguistic.- An Application of Soft Computing in Oil
Condition Monitoring.- A Multi-Parameter Occupational Safety Risk Assessment
Model for Chemicals in the University Laboratories by an MCDM-Sorting
Method.- Failure Mode and Effect Analysis (FMEA) for Safety-Critical Systems
in the Context of Industry 4.0.- Optimization of Redundancy Allocation
Problem Using QPSO Algorithm under Uncertain Environment.- Resilience:
Enterprise Sustainability Based to Risk Management.- Reliability Analysis of
Process Systems Using Intuitionistic Fuzzy Set Theory.- Smart Systems Risk
Management in IoT-Based Supply Chain.- Risk and Reliability Analysis in the
Era of Digital Transformation.- Distributed System Reliability Analysis with
Two Coverage Factors: A Copula Approach.- Qualitative Analysis Method for
Evaluation of Risk and Failures in Wind Power Plants: A Case Study of
Turkey.- Some Discrete Parametric Markov-Chain System Models to Analyze
Reliability.- Repair and Maintenance Management System of Food Processing
Equipment: A Systematic Literature Review.- Reliability, Availability,
Maintainability and Dependability of a Serial Rice Mill Plant with the
Incorporation of Coverage Factor.
Format: Hardback, 345 pages, height x width: 235x155 mm, XXII, 345 p.,
Pub. Date: 04-May-2023
ISBN-13: 9783031270949
The goal of this monograph is to answer the question, is it possible to solve the dynamics problem inside the configuration space instead of the phase space? By introducing a proper class of vector field ? the Cartesian vector field ? given in a Riemann space, the authors explore the connections between the first order ordinary differential equations (ODEs) associated to the Cartesian vector field in the configuration space of a given mechanical system and its dynamics. The result is a new perspective for studying the dynamics of mechanical systems, which allows the authors to present new cases of integrability for the Suslov and Veselova problem; establish the relation between the Cartesian vector field and the integrability of the geodesic flow in a special class of homogeneous surfaces; discuss the importance of the Nambu bracket in the study of first order ODEs; and offer a solution of the inverse problem in celestial mechanics.
Chapter.
1. Dynamics via the first order ordinary differential
equationsChapter.
2. Constrained Cartesian vector fieldsChapter.
3. Three
dimensional constrained Cartesian vector fieldsChapter.
4.
Cartesian-Synge-Cinsov vector fieldChapter.
5. Generalized Cartesian-Nambu
vector fieldsChapter.
6. Integrability of generalized Cartesian-Nambu vector
fields
Format: Hardback, 417 pages, height x width: 235x155 mm, 115 Illustrations,
color; 14 Illustrations, black and white; XXIV, 417 p. 129 illus., 115 illus. in color.
Pub. Date: 11-May-2023
ISBN-13: 9789819901463
The book is intended to serve as an introductory course in group theory geared towards second-year university students. It aims to provide them with the background needed to pursue more advanced courses in algebra and to provide a rich source of examples and exercises. Studying group theory began in the late eighteenth century and is still gaining importance due to its applications in physics, chemistry, geometry, and many fields in mathematics.
The text is broadly divided into three parts. The first part establishes the prerequisite knowledge required to study group theory. This includes topics in set theory, geometry, and number theory. Each of the chapters ends with solved and unsolved exercises relating to the topic. By doing this, the authors hope to fill the gaps between all the branches in mathematics that are linked to group theory. The second part is the core of the book which discusses topics on semigroups, groups, symmetric groups, subgroups, homomorphisms, isomorphism, and Abelian groups. The last part of the book introduces SAGE, a mathematical software that is used to solve group theory problems. Here, most of the important commands in SAGE are explained, and many examples and exercises are provided.
Background Results in Set Theory.- Algebraic Operations on Integers.-
The Integers Modulo.- Semigroups.- Groups.- The Symmetric Group.- Subgroups.-
Groups Homomorphisms and Isomorphic Groups.- Classification of Finite Abelian
Groups.- Group Theory and SageMath.