Series: Lecture Notes in Mathematics, Vol. 2198
1st ed. 2017, X, 207 p. 12 illus.
Softcover
ISBN 978-3-319-66525-2
* 1. The first monograph on Painleve equations to treat both classical
local aspects and modern global aspects simultaneously
* 2. Introduces a new method in the study of Painleve equations,
combining local analysis and global topology
* 3. Gives a new classification of real solutions of the Third Painleve
equation in terms of their zeros and poles
The purpose of this monograph is two-fold: it introduces a conceptual language for
the geometrical objects underlying Painleve equations, and it offers new results on
a particular Painleve III equation of type PIII (D6), called PIII (0, 0, 4, *4), describing
its relation to isomonodromic families of vector bundles on P1 with meromorphic
connections. This equation is equivalent to the radial sine (or sinh) Gordon equation and,
as such, it appears widely in geometry and physics. It is used here as a very concrete
and classical illustration of the modern theory of vector bundles with meromorphic
connections.
Complex multi-valued solutions on C* are the natural context for most of the monograph,
but in the last four chapters real solutions on R>0 (with or without singularities) are
addressed. These provide examples of variations of TERP structures, which are related
to tt* geometry and harmonic bundles.
As an application, a new global picture of0 is given.
Series: Lecture Notes in Mathematics, Vol. 2199
1st ed. 2017, XIV, 136 p. 12 illus. in color.
Softcover
ISBN 978-3-319-65809-4
* A self-contained exposition of the concept of "mother body" in potential theory
* Intriguing numerical experiments lacking theoretical explanation
* A new class of complex polynomials orthogonal with respect to a non
Lebesgue space type norm
* Optimal storage and exact reconstruction of moments of a class of
planar algebraic domains
This book exploits the classification of a class of linear bounded operators with rank-one
self-commutators in terms of their spectral parameter, known as the principal function.
The resulting dictionary between two dimensional planar shapes with a degree of shade
and Hilbert space operators turns out to be illuminating and beneficial for both sides. An
exponential transform, essentially a Riesz potential at critical exponent, is at the heart
of this novel framework; its best rational approximants unveil a new class of complex
orthogonal polynomials whose asymptotic distribution of zeros is thoroughly studied in
the text. Connections with areas of potential theory, approximation theory in the complex
domain and fluid mechanics are established. The text is addressed, with specific aims,
at experts and beginners in a wide range of areas of current interest: potential theory,
numerical linear algebra, operator theory, inverse problems, image and signal processing,
approximation theory, mathematical physics.
Series: Lecture Notes in Mathematics, Vol. 2203
1st ed. 2017, VIII, 134 p. 19 illus., 2 illus. in color.
Softcover
ISBN 978-3-319-67058-4
* Demonstrates a general approach for studying telegraph processes
and alternating renewal processes
* Considered to be the first comprehensive book on the topic
* Includes numerous examples that illustrate applications of theories
* All theorems and lemmas are proven
This monograph is focused on the derivations of exact distributions of first boundary
crossing times of Poisson processes, compound Poisson processes, and more general
renewal processes. The content is limited to the distributions of first boundary crossing
times and their applications to various stochastic models. This book provides the theory
and techniques for exact computations of distributions and moments of level crossing
times. In addition, these techniques could replace simulations in many cases, thus
providing more insight about the phenomenona studied.
This book takes a general approach for studying telegraph processes and is based on
nearly thirty published papers by the author and collaborators over the past twenty five
years. No prior knowledge of advanced probability is required, making the book widely
available to students and researchers in applied probability, operations research, applied
physics, and applied mathematics.
196pp Sep 2017
ISBN: 978-981-3222-92-2 (hardcover)
This book is more than a mathematics textbook. It discusses various kinds of numbers and curious interconnections between them. Without getting into hardcore and difficult mathematical technicalities, the book lucidly introduces all kinds of numbers that mathematicians have created. Interesting anecdotes involving great mathematicians and their marvelous creations are included. The reader will get a glimpse of the thought process behind the invention of new mathematics. Starting from natural numbers, the book discusses integers, real numbers, imaginary and complex numbers and some special numbers like quaternions, dual numbers and p-adic numbers.
Real numbers include rational, irrational and transcendental numbers. Iterations on real numbers are shown to throw up some unexpected behavior, which has given rise to the new science of "Chaos". Special numbers like e, pi, golden ratio, Euler's constant, Gauss's constant, amongst others, are discussed in great detail.
The origin of imaginary numbers and the use of complex numbers constitute the next topic. It is shown why modern mathematics cannot even be imagined without imaginary numbers. Iterations on complex numbers are shown to generate a new mathematical object called 'Fractal', which is ubiquitous in nature. Finally, some very special numbers, not mentioned in the usual textbooks, and their applications, are introduced at an elementary level.
The level of mathematics discussed in this book is easily accessible to young adults interested in mathematics, high school students, and adults having some interest in basic mathematics. The book concentrates more on the story than on rigorous mathematics.
Introduction
Integers
Real Numbers
Imaginary and Complex Numbers
Special Numbers
Appendix A:
Solution of Equation (1.2)
Brahmagupta's Equation and Its Solution
Solution to Equation (2.26)
Appendix B:
Sum of Integral Powers of Natural Numbers
Appendix C:
Origin of Curious Patterns (Section 3.7.6)
Elementary and secondary education students as well as high school students and non-mathematicians.