Pages: 610
ISBN: 978-981-98-2057-3 (hardcover)
This monograph offers a compelling exploration of structures induced by smooth vector flows on the boundaries of compact manifolds, with a particular focus on flows that admit Lyapunov functions — strictly monotonic along each trajectory. These traversing flows reveal rich and intricate interactions between dynamics and geometry, especially at the manifold boundaries.
At the heart of this work lies a central theme: how trajectories interact with boundaries, and how these interactions shape and are shaped by the manifold's topology. This perspective opens a natural gateway to a wide array of mathematical disciplines, including Singularity Theory, Combinatorics, Differential and Contact Geometry, Differential Topology, and Dynamical Systems.
Written with a view toward both depth and interconnection, the monograph highlights the fundamental interplay between boundary value problems for ordinary differential equations and the topological structures they inhabit. Scholars and advanced students across geometry, topology, and dynamical systems will find in these pages a rich landscape of ideas and a fresh perspective on a classical subject.
Holography of Traversing Flows:
Flows in 2D-land, their Holography and Combinatorics
Trivia about Vector Fields on Manifolds with Boundary
Causal Holography of Traversing Flows
Causal Holography in Application to the Inverse Scattering Problems and Billiards
Homology of Traversing Flows:
Doodles and Blobs on a Ruled Page: Convex Quasi-envelops of Traversing Flows on Surfaces
Topology of Spaces of Polynomials with Constrained Real Zero Divisors
Spaces of Polynomials as Grassmanians for Immersions and Embeddings
Spaces of Polynomials as Grassmanians for Traversing Flows
Researches and graduate students interested in the fields of Algebraic
and Geometric Topology, Dynamical Systems and Ergodic Theory, and Combinatorics.
Pages: 400
ISBN: 978-981-98-1766-5 (hardcover)
This book is devoted to description of different geometric structures, including the classical structures of differential geometry like de Rham complex, Dolbeaux complex and Dirac complex and also some structures that have not attracted yet an attention of mathematicians in the language of supersymmetric quantum mechanics. The book is addressed both to physicists and mathematicians.
The first part is addressed mainly to physicists and describes basic properties of smooth manifolds of different type: real, complex, Kaehler, hyperkaehler and HKT. The second part is addressed to mathematicians and describes basic properties of classical and quantum mechanical systems, including supersymmetric systems with Grassmann dynamic variables. The third part is called Synthesis: we show how the physical methods allow one to describe in a simple way and understand many nontrivial geometric facts. For example, the famous Atiyah-Singer theorem admits a rather natural and simple supersymmetric interpretation.
This book is an updated and expanded version of the book Differential geometry through supersymmetric glasses published in 2020 by World Scientific. New material on hyperkaehler geometry and its supersymmetric description and on the gauge fields in CPn manifolds is added.
Part I:
Real Manifolds
Complex Manifolds
Hyperkaehler and HKT Manifolds
Part II:
Dynamical Systems with and without Grassmann Variables
Supersymmetry
Path Integral and the Witten Index
Superspace and Superfields
Part III:
Supersymmetric Description of the de Rham Complex
Supersymmetric Description of the Dolbeault Complex
Sigma Models with Extended Supersymmetries: N=4
Sigma Models with Extended Supersymmetries: N=8
Taming the Zoo of Models
HK and HKT through Harmonic Glasses
Gauge Fields on Manifolds
Atiyah-Singer Theorem
Researchers and students in differential geometry and theoretical and mathematical physics.
Pages: 340
ISBN: 978-981-98-1878-5 (hardcover)
This book explores advanced topics in Geometric Function Theory and Complex Analysis, with a focus on analytic functions, differential operators, and their geometric properties. It draws from the rich tradition of the Romanian Mathematical School and highlights foundational results, contemporary research, and novel methods involving fractional calculus, q-calculus, and fuzzy analysis.
Readers will find comprehensive coverage of:
Classical and modern differential subordinations and superordinations
Strong and fuzzy differential operators
Key differential and integral operators, including q-analogues of well-known transformations
Applications of operators in defining new subclasses of analytic functions and analyzing their geometric features: starlikeness, convexity, distortion bounds, and more
The book builds on pioneering work by S S Miller, P T Mocanu, H M Srivastava, and others, offering new theoretical insights and practical tools for researchers in complex analysis and applied mathematics.
Designed for graduate students, mathematics researchers, and theoretical physicists, this volume serves as both a reference and a research springboard for further study in geometric function theory, fractional and quantum calculus, and operator theory.
Preliminaries on Differential Subordinations and Superordinations Theories
Defining New Operators
Recent Developments in the Classical Differential Subordinations and Superordinations Theory
Recent Developments in the Strong Differential Subordinations and Superordinations Theory
Recent Developments in the Fuzzy Differential Subordinations and Superordinations Theory
Researchers, postgraduate and undergraduate students, practitioners and seminars of Complex Analysis, Fuzzy Mathematics, Analysis and Differential Equations. Also to be in all Science and Engineering libraries.
Pages: 260
ISBN: 978-981-98-1945-4 (hardcover)
In this book, we study the shape of different geometric objects (curves, surfaces, and knots) through small deformation: infinitesimal bending. The shape of geometric objects is determined by their geometric curvature as well as curvature-based functionals, which are referred to as energies. Different kinds of curvatures, their properties, and their influence on shape and energy are discussed. For the investigation of shape and energy, we use our own software visualization tools.
We aim to offer a mathematical way of considering shape. That is, all the geometrical information that is invariant to translations, rotations, and size changes. A more flexible definition of shape takes into consideration the fact that we often deal with deformable shapes in reality. By allowing also isometric (or near-isometric) deformations, such as infinitesimal bending, the intrinsic geometry of the object will stay the same, while sub-parts might be located at very different positions in space.
A special part of the book presents the construction of examples of unknots, as well as knots that can resist simplification by energy gradient and force evolution methods in programming models.
On Shape and Energies
Curves and Knots
Surfaces
Surfaces of Revolution
Shape and Energy of Knots-Topological Topics
Advanced undergraduates, graduate students, researchers, and practitioners in mathematics, biology, physics, mechanics, and architecture, especially those interested in the applications of differential geometry and topology.
Pages: 165
ISBN: 978-981-98-2341-3 (hardcover)
ISBN: 978-981-98-2236-2 (softcover)
This book is the result of a collaboration between three passionate puzzlers, inviting readers to challenge themselves with a diverse collection of puzzles and brainteasers from across the mathematical landscape. While the solutions are accessible to younger puzzlers, solving them independently can still pose an enjoyable challenge for adults. Beyond puzzles and solutions, the authors aim to teach problem-solving methods, interspersed with fascinating mathematical anecdotes that provide refreshing insights and breaks for dedicated solvers.
Featuring a blend of never-before-seen puzzles and classic favorites with new twists, this unique collection offers challenges and curiosities to satisfy puzzlers, enrich mathematicians, and welcome novices eager to explore the beauty of mathematical thinking. Covering topics from logic and probability to geometry and number theory, this book has something for everyone who loves a good puzzle.
Whether you seek puzzles for the pure joy of solving or wish to sharpen your logic skills while uncovering surprising mathematical insights, this book is a must-have addition to your collection.
This Title has Twenty ̲ Letters
Knights, Nuts & Knaves
Unusual Numbers
Dicey Dice
Ants and their Ways
On Prophets and Sages
Can You Recognize the Pattern?
ImProbabilities
Unusual Geometry
Acknowledgments
Bibliography
Index
The book is mainly aimed at readers who enjoy mathematical puzzles and brainteasers, including general puzzle enthusiasts, STEM students, and educators. The solutions are accessible to high-school students, but finding them is challenging even for seasoned puzzlers. The book can also be appealing to professionals and students in fields such as mathematics, computer science, physics, and engineering, and is relevant to both academic and industry audiences who enjoy creative problem-solving.
Pages: 770
ISBN: 978-981-98-2169-3 (hardcover)
ISBN: 978-981-98-2251-5 (softcover)
This second edition to A First Course in Partial Differential Equations provides a clear, rigorous, and student-friendly introduction to the core theory and solution techniques for partial differential equations (PDEs), making it an ideal text for upper-level undergraduates in mathematics, physics, engineering, and the applied sciences.
This volume builds on the strengths of the first edition by integrating new topics that bridge classical theory with modern applications. In addition to comprehensive treatments of standard second-order linear PDEs — the heat equation, wave equation, and Laplace's equation — this edition includes substantial new content:
A new chapter on the Fourier Transform, providing students with powerful tools to analyze PDEs in the frequency domain, along with practical examples relevant to physics and engineering.
A new chapter on Green's Functions, illustrating their construction and use in solving nonhomogeneous boundary value problems, thereby deepening understanding of linear operators and solution representations.
Expanded content on nonhomogeneous equations and boundary conditions, with methods such as Duhamel's principle fully developed.
Enhanced coverage of numerical methods, especially finite difference approximations, to offer a practical introduction to computational approaches in solving PDEs.
More than 400 new exercises, now organized by section, promoting targeted practice and easier integration into coursework.
Many chapters conclude with open-ended explorations and project suggestions, making the text ideal for undergraduate theses, research projects, or independent study.
Core topics also include first-order linear and nonlinear PDEs arising in the physical and life sciences, Fourier series, Sturm–Liouville problems, and special functions of mathematical physics. Appendices review essential background in complex analysis and linear algebra, ensuring accessibility for students from a broad range of STEM disciplines.
With its flexible structure, this textbook supports both one- and two-semester courses, and provides a solid foundation for students preparing for graduate-level PDE courses. It is equally valuable as a reference text for researchers and practitioners seeking practical methods for solving PDEs in scientific and engineering contexts.