Pages: 272
ISBN: 978-981-12-7891-4 (hardcover)
This book offers the comprehensive study of one of the foundational topics in Mathematics, known as Metric Spaces. The book delivers the concepts in an appropriate and concise manner, at the same time rich in illustrations and exercise problems. Special focus has been laid on important theorems like Baire's Category theorem, Heine?Borel theorem, Ascoli?Arzela Theorem, etc, which play a crucial role in the study of metric spaces.
The additional chapter on Cofinal completeness, UC spaces and finite chainability makes the text unique of its kind. This helps the students in:
taking the secondary step towards analysis on metric spaces,
realizing the connection between the two most important classes of functions, continuous functions and uniformly continuous functions,
understanding the gap between compact metric spaces and complete metric spaces.
Readers will also find brief discussions on various subtleties of continuity like subcontinuity, upper semi-continuity, lower semi-continuity, etc. The interested readers will be motivated to explore the special classes of functions between metric spaces to further extent.
Consequently, the book becomes a complete package: it makes the foundational pillars strong and develops the interest of students to pursue research in metric spaces. The book is useful for third and fourth year undergraduate students and it is also helpful for graduate students and researchers.
Fundamentals of Analysis
Continuity and Some Stronger Notions
Complete Metric Spaces
Compactness
Weaker Notions of Compactness
Real-Valued Functions on Metric Spaces
Connectedness
Undergraduate and graduate students, researchers in the areas of real analysis, analysis on metric spaces, real functions, topology, functional
Pages: 556
ISBN: 978-981-12-8327-7 (hardcover)
This book is a comprehensive exploration of the interplay between Stochastic Analysis, Geometry, and Partial Differential Equations (PDEs). It aims to investigate the influence of geometry on diffusions induced by underlying structures, such as Riemannian or sub-Riemannian geometries, and examine the implications for solving problems in PDEs, mathematical finance, and related fields. The book aims to unify the relationships between PDEs, nonholonomic geometry, and stochastic processes, focusing on a specific condition shared by these areas known as the bracket-generating condition or Hormander's condition. The main objectives of the book are:
To unify the relationship between PDEs, nonholonomic geometry, and stochastic processes by examining the common condition imposed on vector fields in both fields.
To explore diffusions induced by underlying geometry, whether Riemannian or sub-Riemannian, and study how curvature affects the diffusion of Brownian movement along curves.
To compute heat kernels and fundamental solutions for various operators, using stochastic methods, and analyze their properties.
To investigate the dynamics of elliptic and sub-elliptic diffusions on different geometric structures and their applications.
To explore the connections between sub-elliptic differential systems and sub-Riemannian geometry.
To analyze the dynamics of LC-circuits using variational approaches and establish their relationship with stochastic analysis and geometric analysis.
The intended audience for this book includes researchers and practitioners in mathematics, physics, and engineering, who are interested in stochastic techniques applied to geometry and PDEs, as well as their applications in mathematical finance and electrical circuits.
Topics of Stochastics Calculus
Stochastic Geometry in Euclidean Space
Hypoelliptic Operators
Heat Kernels with Applications
Fundamental Solutions
Elliptic Diffusions
Sub-Elliptic Diffusions
Systems of Sub-Elliptic Differential Equations
Applications to LC-Circuits
Advanced undergraduate and graduate students, researchers and practitioners in the fields of mathematics,
physics, and engineering who are interested in the application of stochastic techniques in geometry and PDEs
and how it relates to mathematical finance and electrical circuits.
Pages: 250
ISBN: 978-981-128-394-9 (hardcover)
Differential Geometry is one of the major branches of current Mathematics, and it is an unavoidable language in modern Physics. The main characters in Differential Geometry are smooth manifolds: a class of geometric objects that locally behave like the standard Euclidean space.
The book provides a first introduction to smooth manifolds, aimed at undergraduate students in Mathematics and Physics. The only prerequisites are the Linear Algebra and Calculus typically covered in the first two years. The presentation is as simple as possible, but it does not sacrifice the rigor.
The lecture notes are divided into 10 chapters, with gradually increasing difficulty. The first chapters cover basic material, while the last ones present more sophisticated topics. The definitions, propositions, and proofs are complemented by examples and exercises. The exercises, which include part of the proofs, are designed to help the reader learn the language of Differential Geometry and develop their problem-solving skills in the area. The exercises are also aimed at promoting an active learning process. Finally, the book contains pictures which are useful aids for the visualization of abstract geometric situations. The lecture notes can be used by instructors as teaching material in a one-semester course on smooth manifolds.
Charts, Atlases and Smooth Manifolds
Smooth Maps and Submanifolds
Tangent Vectors
Full Rank Smooth Maps
Vector Fields
Flows and Symmetries
Covectors and Differential 1-Forms
Differential Forms and Cartan Calculus
Vector Bundles
Integration on Manifolds
Undergraduate students in Mathematics and Physics, as textbook for a one-semester intro course on differential geometry;
master, PhD students and researchers in physics and mathematics needing a quick introduction to smooth manifolds and calculus thereon.
Pages: 308
ISBN: 978-981-12-8481-6 (hardcover)
ISBN: 978-981-12-8501-1 (softcover)
Graph theory is an area in discrete mathematics which studies configurations (called graphs) involving a set of vertices interconnected by edges. This book is intended as a general introduction to graph theory.
The book builds on the verity that graph theory even at high school level is a subject that lends itself well to the development of mathematical reasoning and proof.
This is an updated edition of two books already published with World Scientific, i.e., Introduction to Graph Theory: H3 Mathematics & Introduction to Graph Theory: Solutions Manual. The new edition includes solutions and hints to selected problems. This combination allows the book to be used as a textbook for undergraduate students. Professors can select unanswered problems for tutorials while students have solutions for reference.
Fundamental Concepts and Basic Results
Graph Isomorphism, Subgraphs, the Complement of a Graph
Bipartite Graphs and Trees
Vertex-Colourings of Graphs
Matchings in Bipartite Graphs
Eulerian Multigraphs and Hamiltonian Graphs
Digraphs and Tournaments
Solutions to Selected Problems
Junior college students, teachers, and undergraduates studying mathematics and computer science.
Pages: 180
ISBN: 978-981-12-8333-8 (hardcover)
This book is based on a series of lectures at the Mathematics Department of the University of Jena, developed in the period from 1995 up to 2015. It is completed by additional material and extensions of some basic results from the literature to more general metric spaces.
This book provides a clear introduction to classical fields of fractal geometry, which provide some background for modern topics of research and applications. Some basic knowledge on general measure theory and on topological notions in metric spaces is presumed.
Measure Theoretic Foundations
Hausdorff and Packing Measures
Upper and Lower Densities of Measures and Comparison with Hausdorff and Packing Measures
Hausdorff Dimension and Potential Theory
Other Fractal Dimensions
Dimensions of Borel Measures
Attractors of Iterated Function Systems
An Example From the Theory of Dynamical Systems
Graphs of Functions and Stochastic Processes
The book is suited for Master and PhD students, but also for mathematicians from other fields interested in fractals.
While basic knowledge on general measure theory and on topological notions in metric spaces is presumed,
for courses the material can also be restricted to the Euclidean setting. Some of the exercises are included.
Monographs in Number Theory: Volume 12
Pages: 650
ISBN: 978-981-12-7736-8 (hardcover)
This volume reflects the contributions stemming from the conference Analytic and Combinatorial Number Theory: The Legacy of Ramanujan which took place at the University of Illinois at Urbana-Champaign on June 6?9, 2019. The conference included 26 plenary talks, 71 contributed talks, and 170 participants. As was the case for the conference, this book is in honor of Bruce C Berndt and in celebration of his mathematics and his 80th birthday.
Along with a number of papers previously appearing in Special Issues of the International Journal of Number Theory, the book collects together a few more papers, a biography of Bruce by Atul Dixit and Ae Ja Yee, a preface by George Andrews, a gallery of photos from the conference, a number of speeches from the conference banquet, the conference poster, a list of Bruce's publications at the time this volume was created, and a list of the talks from the conference.
A Triple Integral Analog of a Multiple Zeta Value (T Amdeberhan, V H Moll, A Straub and C Vignat)
The Ramanujan?Dyson Identities and George Beck's Congruence Conjectures (G E Andrews)
On Entry II.16.12: A Continued Fraction of Ramanujan (G Bhatnagar and M E H Ismail)
General Fine Transformations I (D Bowman and S Wesley)
Combinatorics of Two Second-Order Mock Theta Functions (H Burson)
Chasing After Cancellations: Revisiting a Classic Identity That Implies the Rogers?Ramanujan Identities (H-C Chan)
A Mock Theta Function Identity Related to the Partition Rank Modulo 3 and 9 (S H Chan, N Hong, Jerry and J Lovejoy)
The Congruence Equation ? + b ≡ c (mod p) (T H Chan)
Weighted Partition Rank and Crank Moments I Andrews?Beck Type Congruences (S Chern)
Hankel Determinants of Linear Combinations of Moments of Orthogonal Polynomials (J Cigler and C Krattenthaler)
Polynomial Analogues of Restricted Multicolor b-ary Partition Functions (K Dilcher and L Ericksen)
On Hurwitz Zeta Function and Lommel Functions (A Dixit and R Kumar)
An Inequality Between Finite Analogues of Rank and Crank Moments (P Eyyunni, B Maji and G Sood)
Asymptotic Expansions, Partial Theta Functions, and Radial Limit Differences of Mock Modular and Modular Forms (A Folsom)
On Conjectures of Koike and Somos for Modular identities for the Rogers?Ramanujan Functions (C Gugg)
Proof of a Rational Ramanujan-type Series for 1/?? The Fastest One in Level 3 (J Guillera)
Poles of Triple Product L-Functions Involving Monomial Representations (H Hahn)
Polynomials and Reciprocals of Eisenstein Series (B Heim and M Neuhauser)
Arithmetic Properties of Septic Partition Functions (T Huber, M Huerta and N Mayes)
A Twisted Generalization of the Classical Dedekind Sum (B Isaacson)
Stability of Asai Local Factors for GL(2) (Y Jo and M Krishnamurthy)
Local Densities of Diagonal Integral Ternary Quadratic Forms at Odd Primes (E Jones)
Divisibility Properties of the Fourier Coefficients of (Mock) Modular Functions and Ramanujani (S-Y Kang)
Restricted k-color partitions, II (W J Keith)
On Weighted Overpartitions Related to Some q-series in Ramanujan's Lost Notebook (B Kim, E Kim and J Lovejoy)
Twin Nicomachean q-identities and Conjectures for the Associated Discriminants, Polynomials, and Inequalities (S-H Kim and K B Stolarsky)
A Combinatorial Proof of a Recurrence Relation for the Sum of Divisors Function (S Kim)
(k, ?)-Regular Partitions (L W Kolitsch)
A Combinatorial Construction for Two Formulas in Slater's List (K Kur?ungaz)
Some Observations on Lambert Series, Vanishing Coefficients and Dissections of Infinite Products and Series (J Mc Laughlin)
On the Sum of Parts with Multiplicity at Least 2 in All the Partitions of n (M Merca and A J Yee)
Mock Modular Eisenstein Series with Nebentypus (M H Mertens, K Ono and L Rolen)
The Largest Size of an (s, s + 1)-Core Partition with Parts of the Same Parity (H Nam and M Yu)
Holonomic Relations for Modular Functions and Forms: First Guess, then Prove (P Paule and C-S Radu)
Equal Sums of Two Cubes of Quadratic Forms (B Reznick)
Log-concavity Results for a Biparametric and an Elliptic Extension of the q-binomial Coefficients (M J Schlosser, K Senapati and A K Uncu)
Analysis and Combinatorics of Partition Zeta Functions (R Schneider and A V Sills)
Automatic Discovery of Irrationality Proofs and Irrationality Measures (D Zeilberger and W Zudilin)
Graduate students and academics interested in the research of Number Theory, especially on Ramanujan's work.