Mathematics

This book covers multivariable and vector calculus. It can be used as a textbook for a one-semester course or self-study. It includes worked-through exercises, with answers provided for many of the basic computational ones and hints for the more complex ones.. This second edition features new exercises, new sections on twist and binormal vectors for curves in space, linear approximations, and the Laplace and Poisson equations.

This book is an introduction to machine learning, with a strong focus on the mathematics behind the standard algorithms and techniques in the field, aimed at senior undergraduates and early graduate students of Mathematics.

There is a focus on well-known supervised machine learning algorithms, detailing the existing theory to provide some theoretical guarantees, featuring intuitive proofs and exposition of the material in a concise and precise manner. A broad set of topics is covered, giving an overview of the field. A summary of the topics covered is: statistical learning theory, approximation theory, linear models, kernel methods, Gaussian processes, deep neural networks, ensemble methods and unsupervised learning techniques, such as clustering and dimensionality reduction.

This book is suited for students who are interested in entering the field, by preparing them to master the standard tools in Machine Learning. The reader will be equipped to understand the main theoretical questions of the current research and to engage with the field.

An accessible explanation of Kurt Gödel's groundbreaking work in mathematical logic

In 1931 Kurt Gödel published his fundamental paper, "On Formally Undecidable Propositions of Principia Mathematica and Related Systems." This revolutionary paper challenged certain basic assumptions underlying much research in mathematics and logic. Gödel received public recognition of his work in 1951 when he was awarded the first Albert Einstein Award for achievement in the natural sciences—perhaps the highest award of its kind in the United States. The award committee described his work in mathematical logic as "one of the greatest contributions to the sciences in recent times."

However, few mathematicians of the time were equipped to understand the young scholar's complex proof. Ernest Nagel and James Newman provide a readable and accessible explanation to both scholars and non-specialists of the main ideas and broad implications of Gödel's discovery. It offers every educated person with a taste for logic and philosophy the chance to understand a previously difficult and inaccessible subject.

New York University Press is proud to publish this special edition of one of its bestselling books. With a new introduction by Douglas R. Hofstadter, this book will appeal students, scholars, and professionals in the fields of mathematics, computer science, logic and philosophy, and science.

Progress in low-dimensional topology has been very quick in the last three decades, leading to the solutions of many difficult problems. Among the earlier highlights of this period was Casson's λ-invariant that was instrumental in proving the vanishing of the Rohlin invariant of homotopy 3-spheres. The proof of the three-dimensional Poincaré conjecture has rendered this application moot but hardly made Casson's contribution less relevant: in fact, a lot of modern day topology, including a multitude of Floer homology theories, can be traced back to his λ-invariant.

The principal goal of this book, now in its second revised edition, remains providing an introduction to the low-dimensional topology and Casson's theory; it also reaches out, when appropriate, to more recent research topics. The book covers some classical material, such as Heegaard splittings, Dehn surgery, and invariants of knots and links. It then proceeds through the Kirby calculus and Rohlin's theorem to Casson's invariant and its applications, and concludes with a brief overview of recent developments.

The book will be accessible to graduate students in mathematics and theoretical physics familiar with some elementary algebraic and differential topology, including the fundamental group, basic homology theory, transversality, and Poincaré duality on manifolds.

This book introduces the theory and application of multigrid methods for the fast numerical solution of linear and weakly nonlinear elliptic PDE. No previous exposure to numerical discretization methods is assumed. All that is required of the reader is curiosity and some basic knowledge of matrix theory and the theory of finite-dimensional vector spaces.

We use an axiomatic, mostly-matrix-based approach in the book, both as a way of presenting the theory in a natural and simple setting, and as a means for translating the theory into practical codes. We deviate a little from the matrix-based-approach in the presentation of the framework for nonlinear problems in the latter part of the book. That nonlinear analysis, based on subspace decompositions, represents an area of current research. In fact, the book takes the reader all the way from the basics and simple implementation issues to the front lines of multigrid research.

Coding the multigrid method is notoriously difficult. The current book, which contains several sample codes in the finite element and cell-centered finite difference frameworks, will train the interested reader in the construction of sophisticated, efficient multigrid codes using the simple but powerful MATLAB© programming environment.

This book provides a self-contained account of continuous-parameter time series, starting with second-order models. Integration with respect to orthogonal increment processes, spectral theory and linear prediction are treated in detail. Lévy-driven models are incorporated, extending coverage to allow for infinite variance, a variety of marginal distributions and sample paths having jumps. The necessary theory of Lévy processes and integration of deterministic functions with respect to these processes is developed at length. Special emphasis is given to the analysis of continuous-time ARMA processes.

This graduate-level mathematics textbook provides an in-depth and readable exposition of selected topics in complex analysis. The material spans both the standard theory at a level suitable for a first-graduate class on the subject and several advanced topics delving deeper into the subject and applying the theory in different directions. The focus is on beautiful applications of complex analysis to geometry and number theory. The text is accompanied by beautiful figures illustrating many of the concepts and proofs.

Among the topics covered are asymptotic analysis; conformal mapping and the Riemann mapping theory; the Euler gamma function, the Riemann zeta function, and a proof of the prime number theorem; elliptic functions, and modular forms. The final chapter gives the first detailed account in textbook format of the recent solution to the sphere packing problem in dimension 8, published by Maryna Viazovska in 2016 — a groundbreaking proof for which Viazovska was awarded the Fields Medal in 2022.

The book is suitable for self-study by graduate students or advanced undergraduates with an interest in complex analysis and its applications, or for use as a textbook for graduate mathematics classes, with enough material for 2-3 semester-long classes. Researchers in complex analysis, analytic number theory, modular forms, and the theory of sphere packing, will also find much to enjoy in the text, including new material not found in standard textbooks.

This is the fi rst textbook leading coherently from classical character theory to the theory of lattices over orders and integral representations of fi nite groups. Character theory is developed in a highly pedagogical way including many examples and exercises covering at once the fi rst defi nitions up to Clifford theory, Brauer’s induction theorem and the splitting fi eld theorem, as well as self-dual simple modules allowing bilinear forms. This latter part is done step by step using the approach given by Sin and Willems. Dirichlet characters and Dirichlet’s result on primes in arithmetic progressions are given as an application. Examples of integral representations of fi nite groups are already detailed at a quite early stage where appropriate, so that the more systematic treatment of lattices over orders is natural. After that, the necessary number theory and homological algebra is developed as far as needed. Maximal as well as hereditary orders are introduced and the Auslander- Buchsbaum theorem is proved. The Jordan-Zassenhaus theorem on the fi niteness of lattices in a given vector space is fully proved. Then the development and properties of class groups of orders is a fi rst focus. As a further highlight Swan’s example of a stably free but not free ideal over the integral group ring of the generalised quaternion group of order 32 is developed in great detail.

• A student friendly introduction to ordinary representation theory
• Many examples and exercises, including solutions for some of them, make the book well suited for self-study
• Leads coherently from ordinary character theory to the integral representation theory of lattices over orders
• Several topics appear for the fi rst time in a textbook, such as Sin-Willems’ approach to self-dual simple modules and Swan‘s example of a stably free non free ideal.

Affine algebraic geometry has progressed remarkably in the last half a century, and its central topics are affine spaces and affine space fibrations. This authoritative book is aimed at graduate students and researchers alike, and studies the geometry and topology of morphisms of algebraic varieties whose general fibers are isomorphic to the affine space while describing structures of algebraic varieties with such affine space fibrations.

The philosophy of the book, which makes it quite distinct from many existing texts on the subject, is based on treating the concepts of measure and integration starting with the most general abstract setting and then introducing and studying the Lebesgue measure and integration on the real line as an important particular case.　

The book consists of nine chapters and appendix, with the material flowing from the basic set classes, through　measures, outer measures and the general procedure of measure extension, through measurable functions　and　various types of convergence of sequences of such based on the idea of measure, to the fundamentals of the abstract Lebesgue integration, the basic limit theorems, and the comparison of the Lebesgue and Riemann integrals. Also, studied are Lp spaces, the basics of normed vector spaces, and signed measures. The novel approach based on the Lebesgue measure and integration theory is applied to develop a better understanding of differentiation and extend the classical total change formula linking differentiation with integration to a substantially wider class of functions.

Being designed as a text to be used in a classroom, the book constantly calls for the student's actively mastering the knowledge of the subject matter. There are problems at the end of each chapter, starting with Chapter 2 and totaling at 125. Many important statements are given as problems and frequently referred to in the main body. There are also 358 Exercises throughout the text, including Chapter 1 and the Appendix, which require of the student to prove or verify a statement or an example, fill in certain details in a proof, or provide an intermediate step or a counterexample. They are also an inherent part of the material. More difficult problems are marked with an asterisk, many problems and exercises are supplied with ``existential'' hints.　

The book is generous on Examples and contains numerous Remarks accompanying definitions, examples, and statements to discuss certain subtleties, raise questions on whether the converse assertions are true, whenever appropriate, or whether the conditions are essential.　 　　

With　plenty of examples, problems, and exercises, this well-designed text is ideal for a one-semester Master's level graduate course on real analysis with emphasis on the measure and integration theory for students majoring in mathematics, physics, computer science, and engineering.　

• A concise but profound and detailed presentation of the basics of real analysis with emphasis on the measure and integration theory.
• Designed　for a one-semester graduate course, with plethora of examples, problems, and exercises.
• Is of interest to students and instructors in mathematics, physics, computer science, and engineering.
• Prepares the students for more advanced courses in functional analysis and operator theory.　　

Contents
Preliminaries
Basic Set Classes
Measures
Extension of Measures
Measurable Functions
Abstract Lebesgue Integral
Lp Spaces
Differentiation and Integration
Signed Measures
The Axiom of Choice and Equivalents

The artist Markus Roskar and Professor of Mathematics, Georg Glaeser, (who both teach at the University for Applied Arts in Vienna) have written a mathematics book of the other kind. In this book, mathematics sheds its image as an inaccessible, cumbersome science, and acquires an open and emotional dimension.

The book is structured on the double-page principle: each double page includes an almost always humorous professional drawing that relates to the text on the other page. This text is written in an easy-going, often humorous, but always mathematically correct manner. It mostly deals with an everyday subject that has a mathematical background or an unorthodox mathematical question. Readers can count on a wealth of surprising solutions to often complex problems.

The book is focused on physical interpretation and visualization of the obtained invariant solutions for nonlinear mathematical modeling of atmospheric and ocean waves. This volume represents a unique blend of analytical and numerical methods complemented by the author's developments in ocean and atmospheric sciences and it is meant for researchers and graduate students interested in applied mathematics and mathematical modeling.

In the two-volume set ‘A Selection of Highlights’ we present basics of mathematics in an exciting and pedagogically sound way. This volume examines fundamental results in Algebra and Number Theory along with their proofs and their history. In the second edition, we include additional material on perfect and triangular numbers. We also added new sections on elementary Group Theory, p-adic numbers, and Galois Theory.

• A true collection of mathematical gems in Algebra and Number Theory, including the integers, the reals, and the complex numbers, along with beautiful results from Galois Theory and associated geometric applications.
• Valuable for lecturers, teachers and students of mathematics as well as for all who are mathematically interested.

This course aims at providing a mathematical perspective to some key elements of the so-called deep neural networks (DNNs).  Much of the interest on deep learning has focused on the implementation of DNN-based algorithms.   Our hope is that this compact textbook will offer a complementary point of view that emphasizes the underlying mathematical ideas. We believe that a more foundational perspective will help to answer important questions that have only received empirical answers so far.

Our goal is to introduce basic concepts from deep learning in a rigorous mathematical fashion, e.g. introduce mathematical definitions of deep neural networks (DNNs), loss functions, the backpropagation algorithm, etc. 

We   attempt to identify for each concept the simplest setting that minimizes technicalities but still contains the key mathematics.

The book focuses on deep learning techniques and introduces them almost immediately. Other techniques such as regression and SVM are briefly introduced and used as a steppingstone for explaining basic ideas of deep learning.

Throughout these notes, the rigorous definitions and statements are supplemented by heuristic explanations and figures. The book is organized so that each chapter introduces a key concept.  When teaching this course, some chapters could be presented as a part of a single lecture whereas the others have more material and would take several lectures.

This volume is an introduction to inner model theory, an area of set theory which is concerned with fine structural inner models reflecting large cardinal properties of the set theoretic universe.

The monograph contains a detailed presentation of general fine structure theory as well as a modern approach to the construction of small core models, namely those models containing at most one strong cardinal, together with some of their applications. The final part of the book is devoted to a new approach encompassing large inner models which admit many Woodin cardinals.

The exposition is self-contained and does not assume any special prerequisities, which should make the text comprehensible not only to specialists but also to advanced students in Mathematical Logic and Set Theory.

The book consists of eight chapters, each focusing on different aspects of multiple integrals and related topics in mathematical analysis.
In Chapter 1, multiple integrals are defined and developed. The Jordan measure in n-dimensional unit balls is introduced, along with the definition and criteria for multiple integrals, as well as their properties. 
Chapter 2 delves into advanced techniques for computing multiple integrals. It introduces the Taylor formula, discusses linear maps on measurable sets, and explores the metric properties of differentiable maps. 
In Chapter 3, we focus on improper multiple integrals and their properties. The chapter deduces criteria for the integrability of functions of several variables and develops concepts such as improper integrals of nonnegative functions, comparison criteria, and absolute convergence. 
Chapter 4 investigates the Stieltjes integral and its properties. Topics covered include the differentiation of monotone functions of finite variation and the Helly principle of choice, as well as continuous functions of finite variation. 
Chapter 5 addresses curvilinear integrals, defining line integrals of both the first and second kinds. It also discusses the independence of line integrals from the path of integration. 
In Chapter 6, surface integrals of the first and second kinds are introduced. The chapter presents the Gauss-Ostrogradsky theorem and Stokes’ formulas, along with advanced practical problems to practice these concepts.

Error Freed CFD Mathematics analytically derives and validates nonlinear continuum calculus alterations to Navier-Stokes partial differential equation systems that completely annihilate the legacy CFD theory/practice intrinsic error mechanisms 

• spatial-temporal discretization generated instability
• discrete algebra theorization limitations
• physics-based isotropic Reynolds stress tensor modeling
• weak linear algebra admitted non-convergence

that persist to compromise physics of fluids prediction fidelity.  Weak formulation continuous Galerkin finite element (FE) basis theorization identifies cubically nonlinear continuum calculus tensor product functionals that totally eliminate the need for code phake physics stabilization. also stabilized shock capture.   Resultant is classic tri-diagonal stencil equivalent generation of strictly monotone discrete approximations that are 4^th order accurate in physical space, wave number space and implicit time on any mesh.  Summarily, matrix differential calculus identifies all nonlinear contributions to the quadratic convergent Newton iteration algorithm to eliminate generation of non-converged solutions.

• covers incompressible/compressible laminar, turbulent, transitional thermal-fluid dynamics processes in multiply connected domains with shocks, contact surfaces
• rigorous theory derived asymptotic convergence, local and global error estimates, error quantification, stopping criterion for regular solution adapted nonuniform mesh refinement “on-the-fly” code execution at the optimal mesh solution
•  mathematical complexity of TEA theory  unstagnation advancements are keyed to ready alteration of current practice finite volume commercial/government and FE CFD codes

This book focuses on functional phase space methods in quantum optics and provides a comprehensive introduction to the Wigner functional formalism. The book shows how this formalism can be used for analyzing and designing practical photonic quantum information systems. It also explains in detail the application of the Wigner functional formalism to parametric down-conversion, an important process in quantum optics.

A computer algebra system such as Mathematica® is able to do much more than just numerics: This revised text shows how to tackle real mathematical problems from basic analysis. The reader learns how Mathematica® represents domains, qualifiers and limits to implement actual proofs – a requirement to unlock the huge potential of Mathematica® for a variety of applications.

The book contains six chapters.  In Chapter 1, the set Rn is defined.  n-dimensional ball, sphere, and rectangular neighborhood of a point are defined. Sequences in Rn are defined by their properties.  Open, closed and compact sets in Rn are investigated. The Heine-Borel theorem is formulated and proved. In Chapter 2, functions of several variables are introduced. Limits of functions of several variables are defined with their properties are deducted. Continuous functions of several variables are investigated. Uniform continuity is introduced and explored. In Chapter 3, partial derivatives of higher order and differentials for functions of several variables and their properties are introduced. Criteria for the differentiability of functions of several variables are deducted. Gradient of a function explored. Directional derivatives are defined and investigated. In Chapter 4, higher order partial derivatives of functions of several variables are investigated. The minimum and maximum of functions of several variables are introduced and investigated. Implicit functions are defined and explored. The method of Lagrange multipliers is introduced. The book contains proofs, numerous examples, and exercises with solutions in the two last chapters.

This book provides a comprehensive exploration of Mean Field Games (MFG) theory, a mathematical framework for modeling the collective behavior of rational agents in complex systems. MFG theory can govern a range of societal phenomena, including finance, sociology, machine learning, and economics. The focus is on the system of two coupled nonlinear parabolic partial differential equations (PDEs) that define the Mean Field Games System. The book covers key theoretical topics such as solution stability and uniqueness, with a particular emphasis on Carleman estimates, which are used to estimate solution errors based on noise in the input data. It also introduces the theory of Ill-Posed and Inverse Problems within MFG theory. Both theoretical and numerical aspects of forward and inverse problems are explored through Carleman estimates, offering a rigorous foundation for researchers and practitioners in applied mathematics and related fields.

This book offers a rigorous approach to Carleman estimates, a key element of Mean Field Games theory, making it an essential resource for researchers, graduate students, and professionals looking to apply this powerful framework to complex, real-world systems.

Designed for a broad spectrum of mathematics majors, not only those pursuing graduate school, this book also provides a thorough explanation of undergraduate Real Analysis. Through a developmentally appropriate narrative that integrates informal discussion, motivation, and basic proof writing approaches with mathematical rigor and clarity, the aim is to assist all students in learning more about the real number system and calculus theory.

This book provides an introduction to probabilistic methods in finance, based on stochastic models in discrete time. It is aimed primarily at graduate students in mathematics but may also benefit mathematicians in academia and the financial industry.　

In this fifth edition, the entire text has been thoroughly revised to enhance clarity and completeness. This includes new sections on

The aim of the textbook is two-fold: first to serve as an introductory graduate course in Algebraic Topology and then to provide an application-oriented presentation of some fundamental concepts in Algebraic Topology to the fixed point theory.

A simple approach based on point-set Topology is used throughout to introduce many standard constructions of fundamental and homological groups of surfaces and topological spaces. The approach does not rely on Homological Algebra. The constructions of some spaces using the quotient spaces such as the join, the suspension, and the adjunction spaces are developed in the setting of Topology only.

The computations of the fundamental and homological groups of many surfaces and topological spaces occupy large parts of the book (sphere, torus, projective space, Mobius band, Klein bottle, manifolds, adjunctions spaces). Borsuk's theory of retracts which is intimately related to the problem of the extendability of continuous functions is developed in details. This theory together with the homotopy theory, the lifting and covering maps may serve as additional course material for students involved in General Topology.

The book comprises 280 detailed worked examples, 320 exercises (with hints or references), 80 illustrative figures, and more than 80 commutative diagrams to make it more oriented towards applications (maps between spheres, Borsuk-Ulam Theory, Fixed Point Theorems, …) As applications, the book offers some existence results on the solvability of some nonlinear differential equations subject to initial or boundary conditions.

The book is suitable for students primarily enrolled in Algebraic Topology, General Topology, Homological Algebra, Differential Topology, Differential Geometry, and Topological Geometry. It is also useful for advanced undergraduate students who aspire to grasp easily some new concepts in Algebraic Topology and Applications. The textbook is practical both as a teaching and research document for Bachelor, Master students, and first-year PhD students since it is accessible to any reader with a modest understanding of topological spaces.

The book aspires to fill a gap in the existing literature by providing a research and teaching document which investigates both the theory and the applications of Algebraic Topology in an accessible way without missing the main results of the topics covered.

The idea of the book is to present a text that is useful for both students of quantitative sciences and practitioners who work with univariate or multivariate probabilistic models. Since the text should also be suitable for self-study, excessive formalism is avoided though mathematical rigor is retained. A deeper insight into the topics is provided by detailed examples and illustrations. The book covers the standard content of a course in probability and statistics. However, the second edition includes two new chapters about distribution theory and exploratory data analysis. The first-mentioned chapter certainly goes beyond the standard material. It is presented to reflect the growing practical importance of developing new distributions. The second new chapter studies intensively one- and bidimensional concepts like assymetry, kurtosis, correlation and determination coefficients. In particular, examples are intended to enable the reader to take a critical look at the appropriateness of the geometrically motivated concepts.

This book is intended as an introduction to Probability Theory and Mathematical Statistics for students in mathematics, the physical sciences, engineering, and related fields. It is based on the author’s 25 years of experience teaching probability and is squarely aimed at helping students overcome common difficulties in learning the subject. The focus of the book is an explanation of the theory, mainly by the use of many examples. Whenever possible, proofs of stated results are provided. All sections conclude with a short list of problems. The book also includes several optional sections on more advanced topics. This textbook would be ideal for use in a first course in Probability Theory.

Contents:
Probabilities
Conditional Probabilities and Independence
Random Variables and Their Distribution
Operations on Random Variables
Expected Value, Variance, and Covariance
Normally Distributed Random Vectors
Limit Theorems

Introduction to Stochastic Processes
Mathematical Statistics
Appendix
Bibliography
Index

This book is a concise introduction to the stochastic calculus of variations for processes with jumps. The author provides many results on this topic in a self-contained way for e.g., stochastic differential equations (SDEs) with jumps. The book also contains some applications of the stochastic calculus for processes with jumps to the control theory, mathematical finance and so. This third and entirely revised edition of the work is updated to reflect the latest developments in the theory and some applications with graphics.

This book offers an introduction to concepts of probability theory, probability distributions relevant in the applied sciences, as well as basics of sampling distributions, estimation and hypothesis testing. As a companion for classes for engineers and scientists, the book also covers applied topics such as model building and experiment design.

Contents
Random phenomena
Probability
Random variables
Expected values
Commonly used discrete distributions
Commonly used density functions
Joint distributions
Some multivariate distributions
Collection of random variables
Sampling distributions
Estimation
Interval estimation
Tests of statistical hypotheses
Model building and regression
Design of experiments and analysis of variance
Questions and answers

This textbook is devoted to the general asymptotic theory of statistical experiments. Local asymptotics for statistical models in the sense of local asymptotic (mixed) normality or local asymptotic quadraticity make up the core of the book. Numerous examples deal with classical independent and identically distributed models and with stochastic processes.

The book can be read in different ways, according to possibly different mathematical preferences of the reader. One reader may focus on the statistical theory, and thus on the chapters about Gaussian shift models, mixed normal and quadratic models, and on local asymptotics where the limit model is a Gaussian shift or a mixed normal or a quadratic experiment (LAN, LAMN, LAQ). Another reader may prefer an introduction to stochastic process models where given statistical results apply, and thus concentrate on subsections or chapters on likelihood ratio processes and some diffusion type models where LAN, LAMN or LAQ occurs. Finally, readers might put together both aspects.

The book is suitable for graduate students starting to work in statistics of stochastic processes, as well as for researchers interested in a precise introduction to this area.

This book on finite element-based computational methods for solving incompressible viscous fluid flow problems shows readers how to apply operator splitting techniques to decouple complicated computational fluid dynamics problems into a sequence of relatively simpler sub-problems at each time step, such as hemispherical cavity flow, cavity flow of an Oldroyd-B viscoelastic flow, and particle interaction in an Oldroyd-B type viscoelastic fluid. Efficient and robust numerical methods for solving those resulting simpler sub-problems are introduced and discussed. Interesting computational results are presented to show the capability of methodologies addressed in the book.

Mathematical models cannot be solved using the traditional analytical methods for dynamic equations on time scales. These models must be dealt with using computational methods. This textbook introduces numerical methods for initial value problems for dynamic equations on time scales. Hands-on examples utilizing MATLAB and practical problems illustrate a wide variety of solution techniques.

The latest advancements in time scale calculus are the focus of this book. New types of time-scale integral transforms are discussed in the book, along with how they can be used to solve dynamic equations. Novel numerical techniques for partial dynamic equations on time scales are described. New time scale inequalities for exponentially convex functions are introduced as well.

This book is a concise yet complete calculus textbook covering all essential topics in multi-variable calculus, including geometry in three-dimensional space, partial derivatives, maximum/minimum, multiple integrals and vector calculus as well as a chapter for ODE. All the chapters are constructed in a logical way to outline the essence of each topic and to address potential difficulties arising from learning.

This book is devoted to recent developments of linear and nonlinear integral inequalities on time scales. The book is intended for the use in the field of dynamic calculus on time scales, dynamic equation and integral equations on time scales. It is also suitable for graduate courses in the above fields. The book is designed for those who have mathematical background on time scales calculus.

The book is about strong axioms of infi nity in set theory (also known as large cardinal axioms), and the ongoing search for natural models of these axioms. Assuming the Ultrapower Axiom, a combinatorial principle conjectured to hold in all such natural models, we solve various classical problems in set theory (for example, the Generalized Continuum Hypothesis) and uncover a theory of large cardinals that is much clearer than the one that can be developed using only the standard axioms.

Subsurface flow problems are inherently multiscale in space due to the large variability of material properties and in time due to the coupling of many different physical processes, such as advection, diffusion, reaction and phase exchange. Subsurface flow models still need considerable development. For example, nonequilibrium effects, entrapped air, anomalous dispersion and hysteresis effects can still not be adequately described. Moreover, parameters of the models are diffcult to access and often uncertain. Computational issues in subsurface flows include the treatment of strong heterogeneities and anisotropies in the models, the effcient solution of transport-reaction problems with many species, treatment of multiphase-multicomponent flows and the coupling of subsurface flow models to surface flow models given by shallow water or Stokes equations. With respect to energy and the environment, in particular the modelling and simulation of radioactive waste management and sequestration of CO₂ underground have gained high interest in the community in recent years. Both applications provide unique challenges ranging from modelling of clay materials to treating very large scale models with high-performance computing.

This book brings together key numerical mathematicians whose interest is in the analysis and computation of multiscale subsurface flow and practitioners from engineering and industry whose interest is in the applications of these core problems.

This book's subject lies in the nexus of partial differential equations, operator theory, and complex analysis. The spectral analysis of the complex Laplacian and the compactness of the d-bar-Neumann operator are primary topics.The revised 2nd edition explores updates to Schrödinger operators with magnetic fields and connections to the Segal Bargmann space (Fock space), to quantum mechanics, and the uncertainty principle.

This book contains surveys and research articles on the state-of-the-art in finitely presented groups for researchers and graduate students. Overviews of current trends in exponential groups and of the classification of finite triangle groups and finite generalized tetrahedron groups are complemented by new results on a conjecture of Rosenberger and an approximation theorem. A special emphasis is on algorithmic techniques and their complexity, both for finitely generated groups and for finite Z-algebras, including explicit computer calculations highlighting important classical methods. A further chapter surveys connections to mathematical logic, in particular to universal theories of various classes of groups, and contains new results on countable elementary free groups. Applications to cryptography include overviews of techniques based on representations of p-groups and of non-commutative group actions. Further applications of finitely generated groups to topology and artificial intelligence complete the volume. All in all, leading experts provide up-to-date overviews and current trends in combinatorial group theory and its connections to cryptography and other areas.

This book is ideal as an introduction to algebraic topology and applied algebraic topology featuring a streamlined approach including coverage of basic categorical notions, simplicial, cellular, and singular homology, persistent homology, cohomology groups, cup products, Poincare Duality, homotopy theory, and spectral sequences. The focus is on examples and computations, and there are many end of chapter exercises and extensive student projects.

This book offers an introduction to differential geometry for the non-specialist. It includes most of the required material from multivariable calculus, linear algebra, and basic analysis. An intuitive approach and a minimum of prerequisites make it a valuable companion for students of mathematics and physics.

The main focus is on manifolds in Euclidean space and the metric properties they inherit from it. Among the topics discussed are curvature and how it affects the shape of space, and the generalization of the fundamental theorem of calculus known as Stokes' theorem.

Continuous model theory is an extension of classical first order logic which is best suited for classes of structures which are endowed with a metric. Applications have grown considerably in the past decade. This book is dedicated to showing how the techniques of continuous model theory are used to study C*-algebras and von Neumann algebras. This book geared to researchers in both logic and functional analysis provides the first self-contained collection of articles surveying the many applications of continuous logic to operator algebras that have been obtained in the last 15 years.

In this monograph we study the problem of construction of asymptotic solutions of equations for functions whose number of arguments tends to infinity as the small parameter tends to zero. Such equations arise in statistical physics and in quantum theory of a large number of fi elds. We consider the problem of renormalization of quantum field theory in the Hamiltonian formalism, which encounters additional difficulties related to the Stückelberg divergences and the Haag theorem. Asymptotic methods for solving pseudodifferential equations with small parameter multiplying the derivatives, as well as the asymptotic methods developed in the present monograph for solving problems in statistical physics and quantum field theory, can be considered from a unified viewpoint if one introduces the notion of abstract canonical operator. The book can be of interest for researchers – specialists in asymptotic methods, statistical physics, and quantum fi eld theory as well as for graduate and undergraduate students of these specialities.

Derivations on von Neumann algebras are well understood and are always inner, meaning that they act as commutators with a fixed element from the algebra itself. The purpose of this book is to provide a complete description of derivations on algebras of operators affiliated with a von Neumann algebra. The book is designed to serve as an introductory graduate level to various measurable operators affiliated with a von Neumann algebras and their properties. These classes of operators form their respective algebras and the problem of describing derivations on these algebras was raised by Ayupov, and later by Kadison and Liu. A principal aim of the book is to fully resolve the Ayupov-Kadison-Liu problem by proving a necessary and sufficient condition of the existence of non-inner derivation of algebras of measurable operators. It turns out that only for a finite type I von Neumann algebra M may there exist a non-inner derivation on the algebra of operators affiliated with M. In particular, it is established that the classical derivation d/dt of functions of real variables can be extended up to a derivation on the algebra of all measurable functions. This resolves a long-standing problem in classical analysis.

With applications in quantum field theory, elementary particle physics and general relativity, this two-volume work studies invariance of differential operators under Lie algebras, quantum groups, superalgebras including infinite-dimensional cases, Schrödinger algebras, applications to holography. This first volume covers the general aspects of Lie algebras and group theory supplemented by many concrete examples for a great variety of noncompact semisimple Lie algebras and groups.

Contents:
Introduction
Lie Algebras and Groups
Real Semisimple Lie Algebras
Invariant Differential Operators
Case of the Anti-de Sitter Group
Conformal Case in 4D
Kazhdan–Lusztig Polynomials, Subsingular Vectors, and Conditionally Invariant Equations
Invariant Differential Operators for Noncompact Lie Algebras Parabolically Related to Conformal Lie Algebras
Multilinear Invariant Differential Operators from New Generalized Verma Modules
Bibliography
Author Index
Subject Index

With applications in quantum field theory, general relativity and elementary particle physics, this three-volume work studies the invariance of differential operators under Lie algebras, quantum groups and superalgebras. This second volume covers quantum groups in their two main manifestations: quantum algebras and matrix quantum groups. The exposition covers both the general aspects of these and a great variety of concrete explicitly presented examples. The invariant q-difference operators are introduced mainly using representations of quantum algebras on their dual matrix quantum groups as carrier spaces. This is the first book that covers the title matter applied to quantum groups.

Contents
Quantum Groups and Quantum Algebras
Highest-Weight Modules over Quantum Algebras
Positive-Energy Representations of Noncompact Quantum Algebras
Duality for Quantum Groups
Invariant q-Difference Operators
Invariant q-Difference Operators Related to GLq(n)
q-Maxwell Equations Hierarchies

This challenging book argues that a new way of speaking of mathematics and describing it emerged at the end of the sixteenth century. Leading mathematicians like Hariot, Stevin, Galileo, and Cavalieri began referring to their field in terms drawn from the exploration accounts of Columbus and Magellan. As enterprising explorers in search of treasures of knowledge, these mathematicians described themselves as sailing the treacherous seas of mathematics, facing shipwreck on the shoals of paradox, and seeking shelter and refuge on the shores of geometrical demonstrations. Mathematics, formerly praised for its logic, clarity, and inescapable truths, was for them a hazardous voyage in inhospitable geometrical lands.

Significantly, many of the same practitioners who promoted the vision of mathematics as heroic exploration also played central roles in developing the most important mathematical innovation of the period—the infinitesimal methods. This was no coincidence: the heroic tales of exploration and discovery helped shape a new form of mathematical practice, complete with new questions, new acceptable answers, and new standards of evidence. It was this new vision of mathematics as a grand adventure that allowed for the development of the new techniques that led to the Newtonian calculus.

In demonstrating this, the book moves from real voyages to imaginary ones, from the coasts of the Canadian Arctic to the tropical forests of Guyana, and from the inner structure of matter to the intricacies of the mathematical continuum. Throughout, a common rhetoric and imagery of exploration and discovery run like a thread through these diverse elements and bind them together.

The second edition of this well-received textbook is devoted to Combinatorics and Graph Theory, which are cornerstones of Discrete Mathematics. Every section begins with simple model problems. Following their detailed analysis, the reader is led through the derivation of definitions, concepts, and methods for solving typical problems. Theorems then are formulated, proved, and illustrated by more problems of increasing difficulty.

Quantum Yang-Mills theory is now the foundation of most of elementary particle theory, and its predictions have been tested at many experimental laboratories, but its mathematical foundation is still unclear. The ''mass gap'' property has been discovered by physicists from experiment, but it still has not been understood from a theoretical point of view. Proposed book describes author's approach to solution of this problem on base of Mathematics with Observers (removing from arithmetic infinity idea, taking into account Observers dependent ascending chain of embedded sets of finite decimal fractions with arithmetic operations locally coinciding with standard operations, and getting new calculus, diff geometry, etc), including interpretations of vector fields and differential forms, generalization of Yang-Mills equations, proof of mass gap existing, consideration the theory of matrix Lie groups and algebras, and this point of view gives the possibilities to make new approach and establish the existence of the Yang-Mills theory and a mass gap, Grand unified theories and Standard model of particle physics.

The book concerns with solving about 650 ordinary and partial differential equations. Each equation has at least one solution and each solution has at least one coloured graph. The coloured graphs reveal different features of the solutions. Some graphs are dynamical as for Clairaut differential equations. Thus, one can study the general and the singular solutions. All the equations are solved by Mathematica. The first chapter contains mathematical notions and results that are used later through the book. Thus, the book is self-contained that is an advantage for the reader. The ordinary differential equations are treated in Chapters 2 to 4, while the partial differential equations are discussed in Chapters 5 to 10. The book is useful for undergraduate and graduate students, for researchers in engineering, physics, chemistry, and others. Chapter 9 treats parabolic partial differential equations while Chapter 10 treats third and higher order nonlinear partial differential equations, both with modern methods. Chapter 10 discusses the Korteweg-de Vries, Dodd-Bullough-Mikhailov, Tzitzeica-Dodd-Bullough, Benjamin, Kadomtsev-Petviashvili, Sawada-Kotera, and Kaup-Kupershmidt equations.

This book presents a projector analysis of dynamic systems on time scales. The dynamic systems are classified as first, second, third and fourth kinds. For each classes of dynamic systems the basic matrix chains are constructed. The proposed theory is applied for decoupling of dynamic equations on time scales. Properly involved derivatives, constraints and consistent initial values for the considered equations are defined. A linearization for nonlinear dynamic systems is introduced and the total derivative for regular linearized equations with tractability index one is investigated.

This book presents a topological approach to combinatorial configurations, in particular graphs, by introducing a new pair of homology and cohomology via polyhedra. On this basis, a number of problems are solved using a new approach, such as the embeddability of a graph on a surface (orientable and nonorientable) with given genus, the Gauss crossing conjecture, the graphicness and cographicness of a matroid, and so forth. Notably, the specific case of embeddability on a surface of genus zero leads to a number of corollaries, including the theorems of Lefschetz (on double coverings), of MacLane (on cycle bases), and of Whitney (on duality) for planarity. Relevant problems include the Jordan axiom in polyhedral forms, efficient methods for extremality and for recognizing a variety of embeddings (including rectilinear layouts in VLSI), and pan-polynomials, including those of Jones, Kauffman (on knots), and Tutte (on graphs), among others.

Contents

Homology on Polyhedra

Polyhedra on the Sphere

Automorphisms of a Polyhedron

Gauss Crossing Sequences

Cohomology on Graphs

Embeddability on Surfaces

Embeddings on Sphere

Orthogonality on Surfaces

Net Embeddings

Extremality on Surfaces

Matroidal Graphicness

Knot Polynomials

The idea behind this book is to provide the mathematical foundations for assessing modern developments in the Information Age. It deepens and complements the basic concepts, but it also considers instructive and more advanced topics. The treatise starts with a general chapter on algebraic structures; this part provides all the necessary knowledge for the rest of the book. The next chapter gives a concise overview of cryptography. Chapter 3 on number theoretic algorithms is important for developping cryptosystems, Chapter 4 presents the deterministic primality test of Agrawal, Kayal, and Saxena. The account to elliptic curves again focuses on cryptographic applications and algorithms. With combinatorics on words and automata theory, the reader is introduced to two areas of theoretical computer science where semigroups play a fundamental role.The last chapter is devoted to combinatorial group theory and its connections to automata.

Contents:
Algebraic structures
Cryptography
Number theoretic algorithms
Polynomial time primality test
Elliptic curves
Combinatorics on words
Automata
Discrete infinite groups

Nonlinear functional analysis is a central subject of mathematics with applications in many areas of geometry, analysis, fl uid and elastic mechanics, physics, chemistry, biology, control theory, optimization, game theory, economics etc. This work is devoted, in a self-contained way, to several subjects of this topic such as theory of accretive operators in Banach spaces, theory of abstract Cauchy problem, metric and topological fixed point theory. Special emphasis is given to the study how these theories can be used to obtain existence and uniqueness of solutions for several types of evolution and stationary equations. In particular, equations arising in dynamical population and neutron transport equations are discussed.

This book is intended to be used as a rather informal, and surely not complete, textbook on the subjects indicated in the title. It collects my Lecture Notes held during three academic years at the University of Siena for a one semester course on "Basic Mathematical Physics", and is organized as a short presentation of few important points on the arguments indicated in the title.

It aims at completing the students' basic knowledge on Ordinary Differential Equations (ODE) - dealing in particular with those of higher order - and at providing an elementary presentation of the Partial Differential Equations (PDE) of Mathematical Physics, by means of the classical methods of separation of variables and Fourier series. For a reasonable and consistent discussion of the latter argument, some elementary results on Hilbert spaces and series expansion in othonormal vectors are treated with some detail in Chapter 2.

Prerequisites for a satisfactory reading of the present Notes are not only a course of Calculus for functions of one or several variables, but also a course in Mathematical Analysis where - among others - some basic knowledge of the topology of normed spaces is supposed to be included. For the reader's convenience some notions in this context are explicitly recalled here and there, and in particular as an Appendix in Section 1.4. An excellent reference for this general background material is W. Rudin's classic Principles of Mathematical Analysis. On the other hand, a complete discussion of the results on ODE and PDE that are here just sketched are to be found in other books, specifically and more deeply devoted to these subjects, some of which are listed in the Bibliography.

In conclusion and in brief, my hope is that the present Notes can serve as a second quick reading on the theme of ODE, and as a first introductory reading on Fourier series, Hilbert spaces, and PDE

Analysis and Probability on graphs is an introduction to random graphs, Markov chains on digraphs, entropy of Markov Chains, and discrete Lyapunov exponents and Hausdorff dimension, requiring only minimal background in probability, mathematical analysis, and graphs. This textbook includes constructive discussions about the motivation of basic concepts, and many worked-out problems in each chapter, making it ideal for classroom use or self-study.

This book gives an advanced overview of several topics in infinite group theory. It can also be considered as a rigorous introduction to combinatorial and geometric group theory. The philosophy of the book is to describe the interaction between these two important parts of infinite group theory. In this line of thought, several theorems are proved multiple times with different methods either purely combinatorial or purely geometric while others are shown by a combination of arguments from both perspectives.

The first part of the book deals with Nielsen methods and introduces the reader to results and examples that are helpful to understand the following parts. The second part focuses on covering spaces and fundamental groups, including covering space proofs of group theoretic results. The third part deals with the theory of hyperbolic groups. The subjects are illustrated and described by prominent examples and an outlook on solved and unsolved problems.

New edition now includes the topics on universal free groups, quasiconvex subgroups and hyperbolic groups, and also Stallings foldings and subgroups of free groups. New results on groups of F-types are added.

Algebraic combinatorics is the study of combinatorial objects as an extension of the study of finite permutation groups, or, in other words, group theory without groups. In the spirit of Delsarte's theory, this book studies combinatorial objects such as graphs, codes, designs, etc. in the general framework of association schemes, providing a comprehensive overview of the theory as well as pointing out to extensions.

In this textbook, a concise approach to complex analysis of one and several variables is presented. After an introduction of Cauchy‘s integral theorem general versions of Runge‘s approximation theorem and Mittag-Leffler‘s theorem are discussed. The fi rst part ends with an analytic characterization of simply connected domains. The second part is concerned with functional analytic methods: Fréchet and Hilbert spaces of holomorphic functions, the Bergman kernel, and unbounded operators on Hilbert spaces to tackle the theory of several variables, in particular the inhomogeneous Cauchy-Riemann equations and the d-bar Neumann operator.

Contents
Complex numbers and functions
Cauchy’s Theorem and Cauchy’s formula
Analytic continuation
Construction and approximation of holomorphic functions
Harmonic functions
Several complex variables
Bergman spaces
The canonical solution operator to
Nuclear Fréchet spaces of holomorphic functions
The -complex
The twisted -complex and Schrödinger operators

This textbook offers a high-level introduction to multi-variable differential calculus. Differential forms are introduced incrementally in the narrative, eventually leading to a unified treatment of Green’s, Stokes’ and Gauss’ theorems. Furthermore, the presentation offers a natural route to differential geometry.

Contents:
Calculus of Vector Functions
Tangent Spaces and 1-forms
Line Integrals
Differential Calculus of Mappings
Applications of Differential Calculus
Double and Triple Integrals
Wedge Products and Exterior Derivatives
Integration of Forms
Stokes’ Theorem and Applications

Provides the first systematic study of geometry and topology of locally symmetric rank one manifolds and dynamics of discrete action of their fundamental groups. In addition to geometry and topology, this study involves several other areas of Mathematics – from algebra of varieties of groups representations and geometric group theory, to geometric analysis including classical questions from function theory.

The most difficult computational problems nowadays are those of higher dimensions. This research monograph offers an introduction to tensor numerical methods designed for the solution of the multidimensional problems in scientific computing. These methods are based on the rank-structured approximation of multivariate functions and operators by using the appropriate tensor formats. The old and new rank-structured tensor formats are investigated. We discuss in detail the novel quantized tensor approximation method (QTT) which provides function-operator calculus in higher dimensions in logarithmic complexity rendering super-fast convolution, FFT and wavelet transforms.

This book suggests the constructive recipes and computational schemes for a number of real life problems described by the multidimensional partial differential equations. We present the theory and algorithms for the sinc-based separable approximation of the analytic radial basis functions including Green’s and Helmholtz kernels. The efficient tensor-based techniques for computational problems in electronic structure calculations and for the grid-based evaluation of long-range interaction potentials in multi-particle systems are considered. We also discuss the QTT numerical approach in many-particle dynamics, tensor techniques for stochastic/parametric PDEs as well as for the solution and homogenization of the elliptic equations with highly-oscillating coefficients.

Contents
Theory on separable approximation of multivariate functions
Multilinear algebra and nonlinear tensor approximation
Superfast computations via quantized tensor approximation
Tensor approach to multidimensional integrodifferential equations

This book is a comprehensive guide to Linear Algebra and covers all the fundamental topics such as vector spaces, linear independence, basis, linear transformations, matrices, determinants, inner products, eigenvectors, bilinear forms, and canonical forms. It also introduces concepts such as fields, rings, group homomorphism, and binary operations early on, which gives students a solid foundation to understand the rest of the material.

Unlike other books on Linear Algebra that are either too theory-oriented with fewer solved examples or too problem-oriented with less good quality theory, this book strikes a balance between the two. It provides easy-to-follow theorem proofs and a considerable number of worked examples with various levels of difficulty. The fundamentals of the subject are explained in a methodical and straightforward way.

This book is aimed at undergraduate and graduate students of Mathematics and Engineering Mathematics who are studying Linear Algebra. It is also a useful resource for students preparing for exams in higher education competitions such as NET, GATE, lectureships, etc. The book includes some of the most recent and challenging questions from these exams.

This book introduces quadratic ideal numbers as objects of study with applications to binary quadratic forms and other topics. The text requires only minimal background in number theory, much of which is reviewed as needed. Computational methods are emphasized throughout, making this subject appropriate for individual study or research at the undergraduate level or above.

• Uses a unique programmed approach that takes readers through the mathematics in a step-by-step fashion with a wealth of worked examples and exercises.
• Contains many Quizzes, Learning Outcomes, and Can You? checklists.
• Ideal as a classroom textbook or a self-learning manual.

The subject of applied complex variables is so fundamental that most of the other topics in advanced engineering mathematics (AEM) depend on it. The present book contains complete coverage of the subject, summarizing the more elementary aspects that you find in most AEM textbooks and delving into the more specialized topics that are less commonplace. The book represents a one-stop reference for complex variables in engineering analysis. The applications of conformal mapping in this book are significantly more extensive than in other AEM textbooks. The treatments of complex integral transforms enable a much larger class of functions that can be transformed, resulting in an expanded use of complex-transform techniques in engineering analysis. The inclusion of the asymptotics of complex integrals enables the analysis of models with irregular singular points. The book, which has more than 300 illustrations, is generous with realistic example problems.

This updated edition of a classic title studies identical relations in Lie algebras and also in other classes of algebras, a theory with over 40 years of development in which new methods and connections with other areas of mathematics have arisen. New topics covered include graded identities, identities of algebras with actions and coactions of various Hopf algebras, and the representation theory of the symmetric and general linear group.

This text offers a unique balance of theory and a variety of standard and new applications along with solved technology-aided problems.

The book includes the fundamental mathematical theory, as well as a wide range of applications, numerical methods, projects, and technology-assisted problems and solutions in Maple, Mathematica, and MATLAB. Some of the applications are new, some are unique, and some are discussed in an essay. There is a variety of exercises which include True/False questions, questions that require proofs, and questions that require computations.

The goal is to provide the student with is a solid foundation of the mathematical theory and an appreciation of some of the important real-life applications. Emphasis is given on geometry, matrix transformations, orthogonality, and least-squares.

Designed for maximum flexibility, it is written for a one-semester/two semester course at the sophomore or junior level for students of mathematics or science.

The objective of this book is to look at certain commutative graded algebras that appear frequently in algebraic geometry. By studying classical constructions from geometry from the point of view of modern commutative algebra, this carefully-written book is a valuable source of information, offering a careful algebraic systematization and treatment of the problems at hand, and contributing to the study of the original geometric questions. In greater detail, the material covers aspects of rational maps (graph, degree, birationality, specialization, combinatorics), Cremona transformations, polar maps, Gauss maps, the geometry of Fitting ideals, tangent varieties, joins and secants, Aluffi algebras. The book includes sections of exercises to help put in practice the theoretic material instead of the mere complementary additions to the theory.

This is a high level introduction to abstract algebra which is aimed at readers whose interests lie in mathematics and the information and physical sciences. In addition to introducing the main concepts of modern algebra – groups, rings, modules and fields – the book contains numerous applications, which are intended to illustrate the concepts and to show the utility and relevance of algebra today. In particular applications to Polya coloring theory, latin squares, Steiner systems, error correcting codes and economics are described. There is ample material here for a two semester course in abstract algebra. Proofs of almost all results are given. The reader led through the proofs in gentle stages. There are more than 500 problems, of varying degrees of diffi culty. The book should be suitable for advanced undergraduate students in their fi nal year of study and for fi rst or second year graduate students at a university in Europe or North America. In this third edition three new chapters have been added: an introduction to the representation theory of fi nite groups, free groups and presentations of groups, an introduction to category theory.

Elwyn Berlekamp, John Conway, and Richard Guy wrote ‘Winning Ways for your Mathematical Plays’ and turned a recreational mathematics topic into a full mathematical fi eld. They combined set theory, combinatorics, codes, algorithms, and a smattering of other fi elds, leavened with a liberal dose of humor and wit. Their legacy is a lively fi eld of study that still produces many surprises. Despite being experts in other areas of mathematics, in the 50 years since its publication, they also mentored, talked, and played games, giving their time, expertise, and guidance to several generations of mathematicians.

This volume is dedicated to Elwyn Berlekamp, John Conway, and Richard Guy. It includes 20 contributions from colleagues that refl ect on their work in combinatorial game theory.

Students completing a year of Calculus can find novel and practical challenges in mathematics using numerical analysis. This text follows the mathematician's process of examining a problem and deciding which algebraic or calculus techniques lead to reliable solutions. Each algorithm includes all developmental steps, illustrating the beauty of reducing complex systems to streamlined formulae. Accessible proofs using techniques beyond those in geometry are fully explained. All algorithms contain detailed examples of success and situational pitfalls. An exhaustive set of practice problems was meticulously developed by hand, not AI-generated, ensuring deviations at each stage of a particular algorithm. Many exercises highlight advantages and disadvantages of algorithms through comparison.

Most modern textbooks on partial differential equations either contain proofs of fundamental results only in special cases, or proof outlines, or do not contain proofs at all. This textbook, despite its relatively small volume, provides all the main results with complete proofs.

The foundation for the material in the textbook is the functional approach associated with the concept of a weak solution and Sobolev spaces. In this textbook, such an approach is demonstrated on second-order model equations: the Poisson equation, the heat equation, and the wave equation. However, it allows one to generalize the main results presented in the textbook to the case of more general equations of the corresponding type with variable coefficients.

The theory of partial differential equations constructed on the basis of Sobolev spaces was earlier presented in a number of remarkable books. Although many of these sources have been used in preparing this textbook, its main difference however is the presentation of the semigroup theory with the application to evolutionary equations (thanks to this, mixed problems and the Cauchy problem are studied using a unified method), as well as the use of modern techniques, which made it possible to simplify the majority of proofs.

The book will contain refereed articles by the invited speakers on the 2024 AMP conference to be held during August 5-17, 2024 online. AMP stands for Analysis and Mathematical Physics. It brings leading experts in their fields, delivering 1-hour lectures in their field of expertise. There are 28 invited speakers, and there will also be many participants attending the online conference without presenting talks. The conference will concentrate on the following topics:

Inverse problems for nonlocal operators

Orthogonal polynomials, Jacobi and CMV matrices

Quantum graphs

Applications of spectral and scattering theory to quantum mechanics and plasma physics

Differential operators on closed sets

Differential operators on spatial networks

Direct and inverse spectral and scattering theory for differential and difference equations and for systems of such equations

The aim of the conference is to bring together leading experts and young researchers from all over the world who work in the area of mathematical physics. It promotes exchange of ideas and develops future scientific collaborations and strengthens existing collaborations. The online conference will be broadcast live on a zoom webinar broadcast on Facebook and Youtube channels of IIMAS of UNAM.

The list of the speakers and the details for the conference can be found at the conference webpage:

https://www.iimas.unam.mx/amp2024/

Mathematical modeling and computational techniques are crucial for solving complex real-world problems across diverse domains. This book presents cutting-edge research in applied mathematics, covering topics such as fractional differential equations, fuzzy numerical techniques, nonlinear programming, disease modeling, social network analysis, deep learning, and optimization algorithms. It provides rigorous mathematical formulations, algorithmic implementations, and real-world applications in agriculture, medicine, environmental sciences, and engineering. Readers will gain insights into innovative problem-solving approaches, from stability analysis and bifurcations to AI-driven disease detection and forensic analysis. With a balance of theoretical foundations and practical applications, this volume is an essential resource for researchers, academics, and professionals seeking to advance their understanding of modern mathematical techniques and computational advancements.

This textbook is a real contribution to mathematical analysis, a discipline that requires a great deal of study and attention and finds interesting applications in many disciplines of sciences and engineering. This book is designed for undergraduate students in mathematics and engineering. From the first chapters, the book introduces real and complex numbers, the cornerstones of analysis. It explains in detail the properties of real numbers, such as the square root, absolute value. The analysis of sequences and series occupies a central place in this textbook. The reader discovers, step by step, the convergence through the study of real sequences, drawing on the concepts of Cauchy sequences and monotone sequences. The book also covers numerical series, presenting a variety of convergence criteria. The field of functions and asymptotic study of functions is treated with particular attention.   The book contains proofs, numerous examples, and exercises with hints and solutions.  Our proposal differs from all similar books in that it is sequential, simplified concepts, and includes numerous solved examples and exercises that serve the reader, enabling them to understand the content well. 

The De Gruyter Series in Mathematics and Life Sciences is dedicated to publishing monographs that explore the connection between mathematical, physical, computational, stochastic, and statistical modeling in all areas of life sciences and natural phenomena research. Apart from a focus on complex systems, the series also addresses new types of mathematical problems that emerge with biological, environmental, and real-world mathematical modeling research.

The main objectives is to make such expositions available to and accessible by an interdisciplinary, growing readership hailing from all disciplines involved. The volumes shall convey the context of the given topic and enable these readers to understand, apply and develop further mathematical methods to given problems in life science. For this reason, works with up to four authors are preferred over edited volumes.

Therefore, contributions which are on the borderline of mathematics and life sciences and which stimulate further research at the crossroads of these areas are particularly welcome. In addition, use of electronic media to demonstrate, visualize and model the methods presented are very welcome, especially when interwoven with the written text.

Editor-in-Chief Vitaly Volpert, CNRS, University of Lyon, FR, Editorial Board Andrey Chtcheprov, USA, Anna Marciniak-Czochra, Heidelberg University, GR, Benoît Perthame, Sorbonne-Université, FR, Olga Vasieva, UK, Roderick Melnik, Interdisciplinary Research Institute, Canada

The book presents much creative, original and stimulating ideas on the various approaches which could be taken to tackle five important open problems in mathematics, viz., the twin primes, Goldbach and Polignac’s conjectures, and, the Navier-Stokes equations and turbulence, and the Riemann hypothesis which are both Millennium Prize problems. Technical areas such as calculus, differential equations, group theory/symmetry and complex analysis are covered, as well as logic and mathematical reasoning in general, and they are presented as clearly as possible for easier assimilation. For those who are interested in mathematical reasoning and logic, and researchers working on the open problems in mathematics, the book will offer much food for thought and inspire further advancement of the ideas presented in the book.

This book aims to engage the Young Science—Talented and Ambitious, for a lasting collaboration in holistic mathematical modeling of “how the body works”under permanent impact from variant life-sphere surroundings. It is then a road-map for challenging research in all aspects of human biology and health.  
Following up on its predecessor, BIOKYBERNETIKA (2024), this book “grounds” an axiomatic functional physiology in biology, and elaborates on the roles of thermo¬dynamic entropy in evolution, of allostatic load at the verge of health and disease, and of environmental and social factors in senescence and frailty. 

For cellular systems, enzyme structures, virus infection progression, and tumor growth in vitro are modeled, while dynamics in physiology address vascular transports with liver detoxification interaction, and cortisol rhythmicity. Epidemiological virus spread and self-organization in urban eco-systems touch on population dynamics.  

AI applications focus on medical data bases and on video supervision of sperm injection; direct therapeutical impact have systems endocrinology in precision medicine and optimization in adaptive cancer therapy. 

Interested are systems-oriented researchers in all sciences related to human health. 

This book offers a unified framework for the analysis of function spaces associated with non-additive measures. Motivated by questions arising in nonlinear potential theory and super-critical PDEs, it develops the calculus of Choquet integration and studies the structural properties of Lorentz-type spaces defined via general capacities. Particular attention is given to Bessel capacities, including their role in describing fine properties of Sobolev functions and the normability of Choquet integral spaces. Building on this foundation, the second part introduces and characterizes Sobolev multiplier spaces defined through capacities, establishing their preduals and embedding properties. The boundedness of maximal operators is analyzed using tools from nonlinear potential theory, yielding vector-valued estimates in this setting. The monograph is intended for researchers in analysis and PDEs interested in the interplay between capacity theory, function spaces, and operator estimates.

The series De Gruyter Studies in Mathematics was founded in 1982 by the late Professor Heinz Bauer and Professor Peter Gabriel.

The series is devoted to the publication of monographs and high-level textbooks in mathematics, mathematical methods and their applications. Apart from covering important areas of current interest, a major aim is to make topics of an interdisciplinary nature accessible to the non-specialist.

The works in this series are addressed to advanced students and researchers in mathematics and theoretical physics. In addition, it can serve as a guide for lectures and seminars on a graduate level.

The series De Gruyter Studies in Mathematics is indexed in MathSciNet (Mathematical Reviews) and Scopus.

Editor-in-Chief

Guozhen Lu, University of Connecticut, USA

Editorial Board

Carstensen Carsten, Humboldt-Universitat zu Berlin, Germany

Gavril Farkas, Humboldt-Universitat zu Berlin, Germany

Nicola Fusco, Università di Napoli "Federico II", Italy

Fritz Gesztesy, Baylor University, USA

Zenghu Li, Beijing Normal University, China

Karl-Hermann Neeb, Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany

René L. Schilling, Technische Universität Dresden, Germany

Volkmar Welker, Philipps-Universität Marburg, Germany

Please submit book proposals to Guozhen Lu

The monograph represents peer-reviewed original studies that make significant contributions to various branches of Computational Mathematics and Engineering. Each chapter highlights current trends in its respective research field and offers valuable insights, innovative methodologies, and recent advances in theories and computational applications. Key topics include artificial neural networks, data sciences, fuzzy set theory, numerical optimization techniques, soft computing, computational fluid dynamics and mathematical modelling. Additionally, several AI and machine learning algorithms are included, showcasing noteworthy computational results. This monograph features mathematical content validated by simulation results, enhancing the significance of the book. It serves as a valuable resource for both early-career and experienced researchers tackling complex real-world problems, such as addressing COVID-19 dynamics and solving transportation issues in engineering designs that traditional methods fail to resolve.

This book explores advanced mathematical computational algorithms in artificial intelligence and robotics, with a focus on areas such as machine learning, optimization, and digital simulations. Key topics include the mathematical foundations of AI, robotics control systems, computational geometry for image processing, and their applications in real-world problem-solving, such as autonomous systems and smart manufacturing.

This book explores cryptographic techniques by integrating Elliptic and Jacobian Elliptic Curve Cryptography with Ancient Indian Mathematics, optimizing security and computational efficiency. As digital security becomes vital, the rich mathematical heritage of ancient India offers profound insights into cryptography, bridging historical wisdom with cutting-edge security solutions.
Key frameworks- algebra, analysis, and optimization- enhance point operations like addition and doubling in cryptographic protocols. The book explores trilinear pairing and cryptographic schemes over specialized rings, leveraging projective geometry and algebraic structures to enhance encryption and decryption methods. Covering foundational concepts, algorithms, and applications, the book highlights how ancient Indian insights can improve Elliptic Curve Cryptography, Jacobian Elliptic Curve Cryptography, and trilinear pairing.
Applications span secure vehicular communication using fog computing, blockchain-based authentication, and cloud security. It is targeted at cryptographers, mathematicians, and computer scientists and provides algorithmic innovations, security proofs, structured exercises, and real-world case studies.

This book explores the blend of metaheuristic optimization algorithms and fuzzy decision-making paradigms to reinforce supply chain sustainability and viability. The combination of mathematical and metaheuristic optimization methods based on natural processes and fuzzy decision-making allows models to be built that can effectively deal with the multifaceted issues of contemporary supply chains. This book provides various techniques of decision-making, such as entropy measures, distance measures, coefficient correlation, aggregating operators, TOPSIS, EDAS, etc. This books overcomes the drawbacks of existing   fuzzy decision-making methods This book also provides case studies in which metaheuristic decision-making algorithms have been applied to enhance the sustainability and viability of supply chains. One such notable example is green supplier selection in supply chain management. The changing supply chain management landscape offers many areas for future research, like investigating the intersection of mathematical and metaheuristic fuzzy decision-making algorithms with new technologies such as the Internet of Things (IoT) and artificial intelligence, which can create more responsive and adaptive supply chains.

Observability in Mathematics was developed based on the denial of the concept of infinity. The book introduces  Observers into arithmetic, and arithmetic becomes dependent on Observers. And after that, the basic mathematical parts also become dependent on Observers. One of such parts is arithmetic itself and algebra. Arithmetic and Algebra play important roles not only in pure Mathematics but in contemporary Physics, for example, in Relativity theory and Quantum Mechanics. They will be called New Arithmetic and Algebra, both observers at the logical level and in arithmetic and algebra. The book reconsiders the foundations of classic arithmetic and algebra from this mathematical perspective. The relationships between numbers, polynomials, quaternions, groups, and algebras are discovered and exhibit new properties.  It is shown that almost all classic arithmetic and algebra theorems are satisfied in Mathematics with Observers' arithmetic and algebra, where probabilities are less than 1.

In Chapter 1, time scales are defined and some examples are given. Forward jump operators, backward jump operators, forward graininess functions and backward graininess functions are defined and some of their properties are given. The induction principle and dual induction principle are introduced. Chapter 2 deals with the delta differential calculus for one variable functions on time scales. The basic definition of delta differentiation is due to Hilger. We have included several examples on delta differentiation and the delta Leibniz formula for the nth delta derivative of a product of two functions. Nabla derivatives are introduced. We present delta mean value results. They are given sufficient conditions for delta convexity and delta concavity of one variable functions. It is stated a sufficient condition for completely delta differentiability of one variable function. Several versions of
delta chain rules and delta L’Hˆopitals rules, which do not appear in the usual form, are included. In Chapter 3 are introduced the main concepts for regulated, delta rd-continuous and delta pre-differentiable functions. They are defined indefinite delta integral and the Riemann delta integral and they are deducted some of their properties. The basic delta monomials are defined and investigated. In the chapter are represented different variants of Taylor’s formula. Improper integrals of the first and the second kind are defined and some of their properties are deduced. A survey on nabla integrals is done. Chapter 4 is devoted to the Hilger complex plane and the basic operations circle plus and circle minus. They are defined the basic delta elementary functions: delta exponential function, delta trigonometric functions and delta hyperbolic functions. Some of their properties are deduced.

In chapter 1 N dimensional spaces, contravariant vectors, covariant vectors and invariants are introduced. Some of their properties are deduced. Transformations of coordinates are investigated. Chapter 2 provides an informative introduction concerning the origin and nature of the tensor concept and the scope of the tensor calculus. The tensor algebra has been developed in an N dimensional space. Contravariant, covariant and mixed tensors of arbitrary order are defined. Some of their properties are deduced. The quotient law of the tensors is formulated and proved. Outer product and contractions are introduced. In Chapter 3, an N dimensional Riemannian space has been chosen for the development of tensor calculus. Metric and associated tensors are defines and some of their properties are explored. Affine and curvalinear coordinates are introduced. In Chapter 4, the Christoffel symbols of the first and second kinds are defined. Some of their basic properties are established. Covariant derivatives are introduced. The divergence, Laplace operator and curl are defined and explored. Intrinsic differentiation is studied. Chapter 5 is devoted to Riemann-Christoffel tensor, Ricci tensor, covariant curvature tensor, Riemann curvature and Einstein tensor and we deduct some of their properties. In Chapter 6, we represent some applications of tensor calculus in relativistic dynamics and relativistic kinematics. Lorentz transformations are derived on arbitrary time scales.

Velocity and acceleration vectors are defined and developed. Lagrange equations are deducted. Conservation laws for the energy momentum vector and angular momentum tensor are obtained.

The aim of this book is to present a clear and well-organized treatment of the concept behind the development of mathematics and solution techniques. The text material of this book is presented in highly readable, mathematically solid format. Many practical problems are illustrated displaying a wide variety of solution techniques.