Format: Paperback / softback, 176 pages, height x width: 235x155 mm, 7 Illustrations, color; 1 Illustrations, black and white; X, 176 p. 8 illus., 7 illus. in color., 1 Paperback / softback
Series: Lecture Notes in Mathematics 2377
Pub. Date: 17-Sep-2025
ISBN-13: 9783031968082
The aim of this book is to present a self-contained account of discrete weak KAM theory. Putting aside its intrinsic elegance, this theory also provides a toy model for classical weak KAM theory, where many technical difficulties disappear, but where the central ideas and results persist. It therefore serves as a good introduction to (continuous) weak KAM theory. The first three chapters give a general exposition of the general abstract theory, concluding with a discussion of the relations between the results proved in the discrete setting and the analogous theorems of classical weak KAM theory. Several examples are studied and some key differences between the discrete and classical theory are highlighted. The final chapter is devoted to the historical problem of conservative twist maps of the annulus.
Chapter 1. Introduction.
Chapter 2. The discrete weak KAM setting.-
Chapter 3. Characterizations of the Aubry sets.
Chapter 4. Mather measures, discounted semigroups.
Chapter 5. A family of examples.
Chapter 6. Twist maps.
Format: Paperback / softback, 240 pages, height x width: 235x155 mm, 8 Illustrations, color; 26 Illustrations, black and white; X, 240 p. 34 illus., 8 illus. in color., 1 Paperback / softback
Series: Lecture Notes in Mathematics 2378
Pub. Date: 09-Sep-2025
ISBN-13: 9783031974410
This book introduces a new fast high-order method for approximating volume potentials and other integral operators with singular kernel. These operators arise naturally in many fields, including physics, chemistry, biology, and financial mathematics. A major impediment to solving real world problems is the so-called curse of dimensionality, where the cubature of these operators requires a computational complexity that grows exponentially in the physical dimension. The development of separated representations has overcome this curse, enabling the treatment of higher-dimensional numerical problems. The method of approximate approximations discussed here provides high-order semi-analytic cubature formulas for many important integral operators of mathematical physics. By using products of Gaussians and special polynomials as basis functions, the action of the integral operators can be written as one-dimensional integrals with a separable integrand. The approximation of a separated representation of the density combined with a suitable quadrature of the one-dimensional integrals leads to a separated approximation of the integral operator. This method is also effective in high-dimensional cases. The book is intended for graduate students and researchers interested in applied approximation theory and numerical methods for solving problems of mathematical physics.
Chapter 1. Introduction.
Chapter 2. Quasi-interpolation.
Chapter 3. Approximation of integral operators.
Chapter 4. Some other cubature problems.
Chapter 5. Approximate solution of non-stationary problems.-
Chapter 6. Integral operators over hyper-rectangular domains.
Format: Hardback, 336 pages, height x width: 254x178 mm, 38 Illustrations, color; 59 Illustrations, black and white; X, 336 p. 97 illus., 38 illus. in color., 1 Hardback
Series: Contemporary Mathematicians
Pub. Date: 01-Oct-2025
ISBN-13: 9783031914546
Walter Gautschi has written extensively on topics ranging from special functions, quadrature and orthogonal polynomials to difference and differential equations, software implementations, and the history of mathematics. He is world renowned for his pioneering work in numerical analysis and constructive orthogonal polynomials, including a definitive textbook in the former, and a monograph in the latter area.
This four-volume set, Walter Gautschi: Selected Works with Commentaries, is a compilation of Gautschis most influential papers and includes commentaries by leading experts. The work begins with a detailed biographical section, and Part III includes a section commemorating Walters prematurely deceased twin brother, Werner. This title will appeal to graduate students and researchers in numerical analysis, as well as to historians of science.
Publications.- Special Functions.- Approximation.- Orthogonal
Polynomials on the Real Line.- Gauss type Quadrature.- Ordinary Differential
Equations.- History.- Miscellanea.- Papers on Special Functions.- Paper on
Approximation.- Papers on Orthogonal Polynomials on the Real Line.- Paper on
Gauss type Quadrature.- Papers on Ordinary Differential Equations.- Papers on
History.- Paper on Miscellanea.- Public Lecture on Expatriate Swiss
Mathematician.
ISBN: 978-1-80061-659-2 (hardcover)
ISBN: 978-1-80061-678-3 (softcover)
Geometric Mechanics: Part III is a textbook presented in a lecture notes format, providing precise definitions and practical examples across a series of 31 lectures that have been developed from the author's extensive experience of teaching and research. Geometric mechanics is an incredibly rich field of study: beyond its mathematical depth and beauty, it provides a robust framework for exploring the geometric structures underpinning many dynamical systems crucial to physics.
The first part introduces undergraduate mathematics and physics students to the applications of geometric mechanics in finite dimensional dynamical systems of ordinary differential equations. The second part covers the essential theory of manifolds and Lie groups to prepare senior undergraduates and graduate students for the modern applications of geometric mechanics. These applications are introduced in the third part, which delves into the geometric mechanics of partial differential equations that govern the dynamics of ideal continuum mechanics, including fluids and plasmas, at the cutting edge of current research.
This textbook is designed to facilitate both course learning and individual study. With focused notes, numerous examples, and nearly 200 exercises, it serves as a valuable resource for postgraduate students, course instructors, and researchers.
Introduction
Counterpoints between Mathematics and Physics
Particle Mechanics of Newton, Lagrange and Hamilton
Matrix Lie Groups and Lie Algebras
The Rigid Body in R3
Broken Symmetry: Heavy Top Equations
Lagrangian and Hamiltonian Methods for Geometric Ray Optics in Translation-Invariant, Axisymmetric Material
Rigid Body Equations on SO(n)
Exercises: Inside the Geometric Mechanics Cube
Geometric Structure of Classical Mechanics
Introduction to Vector Fields
Derivatives of Differentiable Maps ? The Tangent Lift
Lifted Actions and the Jacobi?Lie Bracket on Vector Fields
Lie Group Action on Its Tangent Bundle
Hamilton's Principle on Manifolds
Euler?Lagrange Equations on Manifolds
Momentum Maps
Hamiltonian Vector Fields and Differential Forms
More About Vector Fields and Differential Forms
Euler?Poincare Reduction Theorem
EPDiff: An Euler?Poincare Equation on the Diffeomorphisms
EPDiff Solution Behaviour in 1D
Diffeons: Singular Momentum Solutions of the EPDiff Equation for Geodesic Motion in Higher Dimensions
The Geometry of the Momentum Map
Euler?Poincare Framework of Fluid Dynamics
Euler?Poincare Theory of Geophysical Fluid Dynamics
Five More Continuum Applications
Dispersive Shallow Water (DSW) Equations in 1D and 2D
Rotating Shallow Magnetised Water (RSW-MHD)
Incompressible 2D MHD Alfven Wave Turbulence
Lust Hall Magnetohydrodynamics
Appendix A Geometric Mechanics ? Definitions and Topics
This book is suitable for adoption for courses in mathematics, physics and mathematical physics, as well as for related fields, including robotics, control theory, engineering, and computing. It is also an excellent resource for self study for advanced undergraduates and graduate students in these areas.
Pages: 348
ISBN: 978-981-98-0197-8 (hardcover)
ISBN: 978-981-98-0544-0 (softcover)
Unlock the mysteries of Calculus with a fresh approach rooted in simplicity and historical insight. This book reintroduces a nearly forgotten idea from Rene Descartes (1596?1650), showing how the fundamental concepts of Calculus can be understood using just basic algebra. Starting with rational functions ? the core of early Calculus ? this method allows the reader to grasp the rules for derivatives without the intimidating concepts of limits or real numbers, making the subject more accessible than ever.
But the journey doesn't stop there. While attempting to apply this algebraic approach to exponential functions, the reader will encounter the limitations of simple methods, revealing the necessity for more advanced mathematical tools. This natural progression leads to the discovery of continuity, the approximation process, and ultimately, the introduction of real numbers and limits. These deeper concepts pave the way for understanding differentiable functions, seamlessly bridging the gap between elementary algebra and the profound ideas that underpin Calculus.
Whether you're a student, educator, or math enthusiast, this book offers a unique pathway to mastering Calculus. By connecting historical context with modern mathematical practice, it provides a richer, more motivating learning experience. For those looking to dive even deeper, the author's 2015 book, What is Calculus? From Simple Algebra to Deep Analysis, is the perfect next step.
Preface
Why Do We Need a New Approach?
The New Approach
About the Author
The Main Characters:
The Rational Numbers
Functions and Their Graphs
Linear Functions and Slope
Simple Algebra and Tangents:
Quadratic Equations and Functions
Double Roots and Tangents
Motion with Variable Speed
Tangents to Graphs of Polynomials
Simple Differentiation Rules for Polynomials
The Differential Calculus of Rational Functions:
Rational Expressions and Functions
Tangents and Simple Differentiation Rules
Product and Quotient Rule
Continuity and Approximation of Derivatives:
Local Boundedness and Continuity
Rates of Change
Approximation of Algebraic Derivatives
A Look Beyond Algebraic Functions
Exercises
The Heart of Real Analysis:
Completeness of the Real Numbers
Limits and Continuity
Exponential Functions for Real Numbers
Derivatives of Exponential Functions
Differentiable Functions
Some Basic Properties of Differentiable Functions
Applications of Derivatives: A Brief Introduction:
Acceleration and Motion with Constant Acceleration
The Inverse Problem and Antiderivatives
Exponential Models
"Explosive Growth" Models
Periodic Motions
Epilogue
Index
High school math teachers and high school students interested in the introduction to calculus; mathematics educators in volved in math curriculum development; first year college students, college calculus instructors; college math students; mathematics historians and general science readers.
Pages: 230
ISBN: 978-981-12-8062-7 (hardcover)
ISBN: 978-981-12-8157-0 (softcover)
The compendium provides an introduction to the theory of deep learning, from basic principles of neural network modeling and optimization to more advanced topics of neural networks as Gaussian processes, neural tangent and information theory.
This unique reference text complements a largely missing theoretical introduction to neural networks without being overwhelmingly technical in a level accessible to upper-level undergraduate engineering students.
Advanced chapters were designed to offer an additional intuition into the field by explaining deep learning from statistical and information theory perspectives. The book further provides additional intuition to the field by relating it to other statistical and information modeling approaches.
Neural Network Basics:
Introduction
Neural Networks in Use
Optimization Methods
Representation Learning:
Autoencoders and PCA
Probabilistic PCA and VAE
Variational Inference and Information
Convolutional, Recurrent and Transformer Neural Networks:
Convolutional Neural Networks
CNN Applications in Vision and Audio
RNN
RNN Applications in Speech and Audio
Attention and Transformers
Generative Models:
Generative Adversarial Networks (GAN)
Wasserstein GAN
Normalizing Flows and Diffusion Models
Deeper Understanding of Deep Learning:
Information Theory of Learning
NN as Gaussian Processes
Neural Tangent Kernel
Further Topics:
Transfer Learning
Domain Adversarial (Adaptive) AI
Explainable AI
Deep Reinforcement Learning
Bibliography
Researchers, professionals, academics, and undergraduate and graduate students in artificial intelligence and data bases/info science.
Pages: 288
ISBN: 978-981-98-0209-8 (hardcover)
Recent breakthroughs in volatility modelling have brought fractional stochastic calculus to a groundbreaking position. Readers of Fractional S(P)DEs will find a unique and comprehensive overview encompassing the theory and the numerics of both ordinary and partial differential equations (SDEs and SPDEs, respectively), driven by fractional Brownian motion.
Within this book, both differential equations are considered with fractional noise, while also considering fractional derivatives in the case of SPDEs. Three primary aspects are pursued: Theory and numerics for rough SPDEs; Optimal control of both SDEs and SPDEs driven by fractional Brownian motions (and their applications); And numerics for time-fractional SPDEs driven by both Gaussian and non-Gaussian noises.
This series of complementary articles, compiled by two internationally renowned scientists, is united by a common application-oriented view of fractional Brownian motion and its stochastic calculus. As such, this book will be particularly useful for mathematicians working in the fields of stochastics applied in Finance and Natural Sciences, as well as those preparing courses on advanced stochastic processes.
Preface
About the Editors
About the Contributors
State-of-the-Art Numerical Schemes for Solving Rough Differential Equations (Martin Redmann and Justus Werner)
Mild Solutions to Semilinear Rough Partial Differential Equations (Stefan Tappe)
Fractional Noise-Perturbed Nonlinear Schrodinger Equations: Stochastic Minimization Problems (Wilfried Grecksch and Hannelore Lisei)
Calibration of Non-Semimartingale Models: An Adjoint Approach (Christian Bender and Matthias Thiel)
Strong Convergence Analysis of a Fractional Exponential Integrator and Finite Element Method for Time-Fractional SPDEs Driven by Gaussian and Non-Gaussian Noises (Aurelien Junior Noupelah and Antoine Tambue)
Index
Graduate students, researchers and practitioners in the fields of stochastics,
numerics, physics, and finance. Secondarily, scientists with a strong background
in mathematics, and undergraduate students in the field of applied mathematics.
Pages: 184
Monographs in Number Theory: Volume 14
ISBN: 978-981-98-1316-2 (hardcover)
The focus of this monograph is on the Jacobsthal sums of the title. These are quadratic character sums with polynomial arguments of a certain simple form. In addition to studying Jacobsthal sums on their own, the monograph explores their role in several topics of number-theoretical interest. A prominent theme is their use in counting solutions to equations over prime fields. Another aim is to construct representations of primes as sums of squares using Jacobsthal sums. Finally, Jacobsthal sums are applied to evaluate other quadratic character sums with polynomial arguments.
This text is self-contained, with minimal technical prerequisites. We have strived for an engaging exposition, incorporating numerous examples and applications, carefully selected exercises, and historical notes and perspectives. Complete solutions to all exercises are provided.
This monograph should be of interest to researchers studying character sums or counting solutions to equations over finite fields, as well as to graduate students and advanced undergraduates interested in these topics.
Preface
About the Author
Preliminaries
Quadratic Character Sums
Jacobsthal Sums
Solution Counting with Jacobsthal Sums
Further Quadratic Character Sums
Solutions to Exercises
Bibliography
Index
Graduate and advanced undergraduate students, researchers in Number Theory.
Pages: 423
ISBN: 978-981-98-1306-3 (hardcover)
ISBN: 978-981-98-1394-0 (softcover)
This book offers a captivating journey through the world of elliptic partial differential equations (PDEs) and their surprising appearances across science, engineering, and even everyday life. Blending mathematical rigor with an accessible and engaging style, it introduces the Laplace operator and elliptic PDEs through diverse real-world phenomena, from physics and biology to artificial intelligence and image processing.
Readers will explore fundamental topics such as diffusion, wave propagation, and fluid dynamics, alongside unexpected applications ? including pattern formation in nature, traffic jams, quantum mechanics, and even the mathematics behind fighting pandemics and cancer. With a focus on both intuition and formalism, the book provides a wealth of examples, historical insights, and thought-provoking problems that make the subject come alive.
Awarded the prestigious Book Prize of the Unione Matematica Italiana, this work has been praised as "a remarkable journey through delightful applications of elliptic PDEs, all while maintaining the highest standards of mathematical rigor." Whether you are a student, researcher, or simply curious about how mathematics shapes the world around us, this book will challenge and inspire you.
The Heat Equation
Population Dynamics, Chemotaxis and Random Walks
Pattern Formation, Or How the Leopard Gets Its Spots
Space Invaders
Equations from Hydrodynamics
Irrotational Fluids
Propagation of Sound Waves
Hydrodynamics Doesn't Always Work Right
Lift of an Airfoil
Surfing the Waves
Plasma Physics
Galaxy Dynamics
How to Count What We Cannot See
Good Vibrations
Elastic Membranes
Elasticity Theory and Torsion of Bars
Bending Beams and Plates
Gravitation and Electrostatics
Classical Electromagnetism
Quantum-Mechanical Systems
Bouncing Balls and Whispers
Strange Attractions
Who Wants to be a Millionaire?
Diffusion of Transition Probabilities
Phase Coexistence Models
Growth of Interfaces
What to Do When You're Stuck in a Traffic Jam
Reaching Equilibrium. or Maybe Not?
Definitions Come Later On
Nerd Sniping
The Streetsweeper Problem
Image Processing
Artificial Intelligence and Machine Learning
Cutting Networks
The Wear of the Rolling Stones
Bushrangers and Outlaws
Fighting Cancer Using Differential Equations
When You are a Mathematician
If You Are Shy, Do as the Romulans Do
Fighting a Pandemic Using Differential Equations
Bibliography
Index
Undergraduate Students, Postgraduate Students, Academics, Researchers.