Series: SpringerBriefs in Mathematics
1st ed. 2017, XI, 125 p. 31 illus., 9 illus. in
color.
Printed book
Softcover
ISBN 978-3-319-50864-1
* Offers an introduction to asymptotic analysis techniques and various
finite element methods for elliptic problems
* Presents numerous case studies on modeling techniques of
multiscale PDEs, in one- and two-dimensional domains
* Explores modeling through both continuous and numerical
approximations
This book is about numerical modeling of multiscale problems, and introduces several
asymptotic analysis and numerical techniques which are necessary for a proper
approximation of equations that depend on different physical scales. Aimed at advanced
undergraduate and graduate students in mathematics, engineering and physics
or researchers seeking a no-nonsense approach , it discusses examples in their
simplest possible settings, removing mathematical hurdles that might hinder a clear
understanding of the methods.
The problems considered are given by singular perturbed reaction advection diffusion
equations in one and two-dimensional domains, partial differential equations in domains
with rough boundaries, and equations with oscillatory coefficients. This work shows how
asymptotic analysis can be used to develop and analyze models and numerical methods
that are robust and work well for a wide range of parameters.
1
Series: Probability Theory and Stochastic Modelling, Vol. 40
2017, XVII, 249 p. 17 illus.
Printed book
Hardcover
ISBN 978-3-319-50928-0
* Promotes original analytic methods to determine the invariant
measure of two-dimensional random walks in domains with
boundaries. These processes appear in several mathematical areas
(Stochastic Networks, Analytic Combinatorics, Quantum Physics).
* The second edition includes additional recent results on the group of
the random walk. It presents also case-studies from queueing theory
and enumerative combinatorics.
This monograph aims to promote original mathematical methods to determine the
invariant measure of two-dimensional random walks in domains with boundaries.
Such processes arise in numerous applications and are of interest in several areas of
mathematical research, such as Stochastic Networks, Analytic Combinatorics, and Quantum
Physics. This second edition consists of two parts.
Part I is a revised upgrade of the first edition (1999), with additional recent results on the
group of a random walk. The theoretical approach given therein has been developed
by the authors since the early 1970s. By using Complex Function Theory, Boundary Value
Problems, Riemann Surfaces, and Galois Theory, completely new methods are proposed
for solving functional equations of two complex variables, which can also be applied to
characterize the Transient Behavior of the walks, as well as to find explicit solutions to
the one-dimensional Quantum Three-Body Problem, or to tackle a new class of Integrable
Systems.
Part II borrows special case-studies from queueing theory (in particular, the famous
problem of Joining the Shorter of Two Queues) and enumerative combinatorics
(Counting, Asymptotics).
Researchers and graduate students should find this book very useful.
1
Series: Interdisciplinary Applied Mathematics, Vol. 46
1st ed. 2017, XVIII, 413 p. 130 illus., 89 illus.
in color.
Printed book
Hardcover
ISBN 978-3-319-50804-7
* Details the current state-of-the-art in modeling epidemics on
networks
* Offers direct comparison of the main network epidemic models and
works out their hierarchy
* Identifies opportunities for further rigorous mathematical
exploration
* Features practical simulation algorithms written in pseudocode
with implemented code available online
This textbook provides an exciting new addition to the area of network science featuring
a stronger and more methodical link of models to their mathematical origin and explains
how these relate to each other with special focus on epidemic spread on networks.
The content of the book is at the interface of graph theory, stochastic processes and
dynamical systems. The authors set out to make a significant contribution to closing the
gap between model development and the supporting mathematics. This is done by:
* Summarising and presenting the state-of-the-art in modeling epidemics on networks
with results and readily usable models signposted throughout the book;
* Presenting different mathematical approaches to formulate exact and solvable models;
* Identifying the concrete links between approximate models and their rigorous
mathematical representation;
* Presenting a model hierarchy and clearly highlighting the links between model
assumptions and model complexity;
* Providing a reference source for advanced undergraduate students, as well as doctoral
students, postdoctoral researchers and academic experts who are engaged in modeling
stochastic processes on networks;
* Providing software that can solve the differential equation models or directly simulate
epidemics in networks.
Replete with numerous diagrams, examples, instructive exercises, and online access to
simulation algorithms and readily usable code, this book will appeal to a wide spectrum of
readers from different backgrounds and academic levels.
Series: Interdisciplinary Applied Mathematics, Vol. 45
1st ed. 2017, XVIII, 660 p. 341 illus.
Printed book
Hardcover
ISBN 978-0-387-87709-9
* Presents a mathematical theory to quantify and model biological
growth processes
* Shows how mechanics and geometry are coupled during growth and
how complex forms and morphological patterns may arise due to
instabilities
* Illustrated with simple examples and detailed biological applications
This monograph presents a general mechanical theory for biological growth. It provides
both a conceptual and a technical foundation for the understanding and analysis of
problems arising in biology and physiology. The theory and methods is illustrated on a
wide range of examples and applications.
A process of extreme complexity, growth plays a fundamental role in many biological
processes and is considered to be the hallmark of life itself. Its description has been one
of the fundamental problems of life sciences, but until recently, it has not attracted much
attention from mathematicians, physicists, and engineers. The author herein presents
the first major technical monograph on the problem of growth since D’Arcy Wentworth
Thompson’s 1917 book On Growth and Form.
The emphasis of the book is on the proper mathematical formulation of growth
kinematics and mechanics. Accordingly, the discussion proceeds in order of complexity
and the book is divided into five parts. First, a general introduction on the problem of
growth from a historical perspective is given. Then, basic concepts are introduced within
the context of growth in filamentary structures. These ideas are then generalized to
surfaces and membranes and eventually to the general case of volumetric growth. The
book concludes with a discussion of open problems and outstanding challenges.
Thoughtfully written and richly illustrated to be accessible to readers of varying interests
and background, the text will appeal to life scientists, biophysicists, biomedical engineers,
and applied mathematicians alike.
336pp Feb 2017
ISBN: 978-981-3200-49-4 (hardcover)
This book is a monograph on harmonic analysis and fractal analysis over local fields. It can also be used as lecture notes/textbook or as recommended reading for courses on modern harmonic and fractal analysis. It is as reliable as Fourier Analysis on Local Fields published in 1975 which is regarded as the first monograph in this research field.
The book is self-contained, with wide scope and deep knowledge, taking modern mathematics (such as modern algebra, point set topology, functional analysis, distribution theory, and so on) as bases. Specially, fractal analysis is studied in the viewpoint of local fields, and fractal calculus is established by pseudo-differential operators over local fields. A frame of fractal PDE is constructed based on fractal calculus instead of classical calculus. On the other hand, the author does his best to make those difficult concepts accessible to readers, illustrate clear comparison between harmonic analysis on Euclidean spaces and that on local fields, and at the same time provide motivations underlying the new concepts and techniques. Overall, it is a high quality, up to date and valuable book for interested readers.
Preliminary
Character Group Γp of Local Field Kp
Harmonic Analysis on Local Fields
Function Spaces on Local Fields
Fractal Analysis on Local Fields
Fractal PDE on Local Fields
Local Field Analysis and Fractal Analysis in Applications to Medicine Science
Readership: Undergraduate students, graduate students and researchers.
348pp Feb 2017
ISBN: 978-981-3202-44-3 (hardcover)
This volume goes beyond the understanding of symmetries and exploits them in the study of the behavior of both classical and quantum physical systems. Thus it is important to study the symmetries described by continuous (Lie) groups of transformations. We then discuss how we get operators that form a Lie algebra. Of particular interest to physics is the representation of the elements of the algebra and the group in terms of matrices and, in particular, the irreducible representations. These representations can be identified with physical observables.
This leads to the study of the classical Lie algebras, associated with unitary, unimodular, orthogonal and symplectic transformations. We also discuss some special algebras in some detail. The discussion proceeds along the lines of the Cartan-Weyl theory via the root vectors and root diagrams and, in particular, the Dynkin representation of the roots. Thus the representations are expressed in terms of weights, which are generated by the application of the elements of the algebra on uniquely specified highest weight states. Alternatively these representations can be described in terms of tensors labeled by the Young tableaux associated with the discrete symmetry Sn. The connection between the Young tableaux and the Dynkin weights is also discussed. It is also shown that in many physical systems the quantum numbers needed to specify the physical states involve not only the highest symmetry but also a number of sub-symmetries contained in them. This leads to the study of the role of subalgebras and in particular the possible maximal subalgebras. In many applications the physical system can be considered as composed of subsystems obeying a given symmetry. In such cases the reduction of the Kronecker product of irreducible representations of classical and special algebras becomes relevant and is discussed in some detail. The method of obtaining the relevant Clebsch-Gordan (C-G) coefficients for such algebras is discussed and some relevant algorithms are provided. In some simple cases suitable numerical tables of C-G are also included.
The above exposition contains many examples, both as illustrations of the main ideas as well as well motivated applications. To this end two appendices of 51 pages ? 11 tables in Appendix A, summarizing the material discussed in the main text and 39 tables in Appendix B containing results of more sophisticated examples are supplied. Reference to the tables is given in the main text and a guide to the appropriate section of the main text is given in the tables.
Elements of Group Theory
Study of the SU(2) and SO(3) Representations and Applications
Elements of Lie Groups
Lie Algebras
The Classical Algebras L, A, B, C
The Dynkin Diagrams ? Another Classification of Classical Lie Algebras
Weights of Irreducible Representations ? Maximal Subalgebras
Construction of Irreducible Representations ? Young Tableaux
Construction of Irreducible Representations, Kronecker Products, Clebsch-Gordan Coefficients
Some Non-Compact Algebras and Applications
Some Symmetries Involved in Particle Physics
Appendices:
Summary of Useful Expressions
Some Useful Tables
Readership: Advanced undergraduate students in particle physics/high energy physics.