Mathematics

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 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.

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 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.

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.

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 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.

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 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.

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.

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.

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.

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.

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.

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

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.

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.

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 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.

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.

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.

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.

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

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 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.

This volume presents selected contributions from the seminar “Mathematical Models and Integration Methods,” highlighting recent advances in differential equations, mathematical modeling, and modern analytical techniques. The topics cover a wide range of problems, including vortex structures in ferromagnets, wave propagation in inhomogeneous media, reciprocal transformations of heat equations, nonlinear resonance phenomena, and composite beam analysis. The collection also includes studies on the exponential map of vector fields, sedeonic models in superconductivity, and transformations in relativistic systems. Most contributions are from leading Russian researchers, complemented by papers from international participants. The book is intended for specialists in applied mathematics, mathematical physics, and engineering disciplines interested in contemporary modeling approaches and integration methods.

Logic and Deduction differs from other titles and offers fresh viewpoints and suggestions for more effective and better utilization of logic in the sciences and mathematics, such as theoretical physics, number theory, sociology, and economics. Much of the title touches on scientific and mathematical topics, e.g., mathematical proof, Godel's incompleteness theorems, set theory, infinity, the work of well-known logicians such as Frege and Russell, calculus, relativity, quantum theory, chaos, unified field theory, speed of light, gravity, etc., including the soft sciences such as economics and sociology. The coverage is comprehensive, as logic applies not only to the hard sciences but also to the soft sciences. The book could be useful to scholars, researchers, students of the hard and soft sciences, intellectuals, professionals such as physicists working on unified field theory & mathematicians working on number theory, logicians, philosophers, lawyers arguing criminal law cases, writers, economists, sociologists, government bureaucrats & policy-makers, corporate executives, activists aiming for a better society, and anyone keen on increasing brainpower or mind-expansion. It could inspire and spur further thought on logic.

This book aims to highlight the latest developments in fixed point theory and functional analysis by presenting insights from renowned scientists, physicists, and mathematicians worldwide.

The book offers a comprehensive overview of the latest advancements in the field, featuring original contributions and surveys. Readers will find a wealth of useful tools and techniques to deepen their understanding of recent advances in mathematical and functional analysis, as well as their applications in physics and engineering. Each chapter highlights new research avenues, making this book an ideal resource for graduate students, faculty, and researchers seeking to expand their knowledge of fixed point theory and functional analysis, as well as their practical applications. The computational aspects in Banach and Hilbert spaces are well explored. Only a basic knowledge of analysis, topology, basic computation, and functional analysis is required to fully appreciate the material.

In today’s data-driven era, the convergence of mathematics, computing, artificial intelligence, and blockchain is emerging as a significant area at the intersection of applied mathematics and computer science, particularly in decision-making. This book explores the applications of advanced mathematical models and computational algorithms to AI-driven strategies and blockchain technologies.
It covers advanced linear algebra techniques, probability theory, optimization methods, game theory, cryptography, and statistical learning, providing deep mathematical insights into AI, blockchain, and data-driven decision-making. The book delves into matrix computations and eigenvalue problems relevant to deep learning, Bayesian inference for predictive modeling, and reinforcement learning for dynamic decision-making.
Additionally, optimization methods such as convex programming and Lagrangian multipliers enhance resource allocation, while cryptographic protocols ensure the security of blockchain systems. By integrating these mathematical frameworks, this book provides researchers, professionals, and students with practical tools for addressing complex business challenges ranging from fraud detection to automated contract execution.

This book explores a wide range of methods for advanced solutions in the field of oncology. It explores the transformative role of AI in advancing cancer care, focusing on its applications in drug discovery, surgical progress, and digital twin technology. The book provides an in-depth analysis of AI's impact on various cancers, including breast, colorectal, brain, and others, and highlights its potential to enhance early detection, accurate diagnosis, and personalized treatment. The book will investigate the development and optimization of AI algorithms to achieve high accuracy in detecting malignant cells at different stages. The technologies encompass machine learning techniques to identify patterns in medical imaging, natural language processing to evaluate patient histories, and deep learning models to forecast treatment outcomes. Specific methodologies such as supervised and unsupervised learning models, convolutional neural networks for image analysis, and reinforcement learning for adaptive treatment strategies are provided. The utilization of case studies and real-world examples in cancer care will demonstrate every approach, enabling readers to gain a practical understanding of how these technologies might be implemented.

The premise of this book is that by understanding the initial conditions at the origins of reality, we can comprehend how all of our observable physical systems (including chemical and biological systems) have emerged. The book focuses on familiar principles of quantum physics and provides very simple derivations of the Heisenberg Uncertainty Principle, Pauli's Exclusion Principle, Schrodinger's wave equation, Noether's theorem, and the Continuity Equation. Background is provided in group theory and symmetries. Using these concepts, the book explores the nature of vacuum, of fundamental constants and particles, and of physical laws. Building on these basic principles, the book explains the emergence of atoms, the natural elements, chemical reactions, and life. In the process, numerous paradoxes are encountered and solved.

The aim of the Expositions is to present new and important developments in pure and applied mathematics. Well established in the community over more than two decades, the series offers a large library of mathematical works, including several important classics.

The volumes supply thorough and detailed expositions of the methods and ideas essential to the topics in question. In addition, they convey their relationships to other parts of mathematics. The series is addressed to advanced readers interested in a thorough study of the subject.

Editorial Board

Lev Birbrair, Universidade Federal do Ceará, Fortaleza, Brasil
Markus J. Pflaum, University of Colorado, Boulder, USA
Dierk Schleicher, Aix-Marseille Université, France
Katrin Wendland, Trinity College Dublin, Dublin, Ireland

Data science, quantitative analysis, computer optimisation, and machine learning are all explored in this book. Insightful mathematical analysis of ML models is presented in this work. Various fields, including healthcare, banking, and e-commerce, are covered in this book through case studies. To choose features, hyperparameters, and models, the latter shows how optimisation can be implemented. Practical insights on the assessment and improvement of machine learning models in real-world situations, such as e-commerce, healthcare, banking, and agriculture, are included. Professionals in the fields of research, engineering, mathematics, and computer science are the target audience for this book. Through the integration of theoretical principles with practical applications, the usage of computational optimisation techniques in real-time settings is to be enhanced. 

The publication of this book is motivated by the need to present the latest research & advancements in the fields of Fluid/Solid Mechanics, Nonlinear Dynamics, and Differential Equations in Applied Mathematics. This book gathers the work of leading experts, offering cutting-edge findings addressing existing challenges in the field. It covers a broad spectrum of topics, including advanced computational methods/mathematical modeling/different Mathematical methods & their applications in many scientific disciplines like predicting the nature and behavior of physical systems in engineering. These problems often require solving differential equations governing fluid flow, heat transfer, and structural deformation, often simultaneously, among other topics. Each chapter delves into specific problems, showcasing interdisciplinary approaches & demonstrating the practical impact of mathematical research on real-world issues. This book is a great resource for scholars, professionals, and researchers as it offers a comprehensive overview of cutting-edge methodologies & innovative solutions. It aims to stimulate additional investigation, promote interdisciplinary collaboration & make substantial contributions to advancing knowledge.

Optimization should be the science of making the best possible decisions. Making decisions is a virtually universal human activity encountered by professionals (in any field) or people in their everyday lives. You would think, then, that the study of making good decisions is a subject that should be taught broadly to students throughout engineering, the physical and social sciences, business, and policy. Yet today, “optimization” is widely taught as a mathematically sophisticated subject, often limited to graduate students in specialized fields. In operations research (or industrial engineering), “optimization” is equivalent to deterministic math programming, starting with linear programs (and the simplex algorithm), and then transitioning through integer linear programs and nonlinear programs. If you are in departments like electrical or mechanical engineering, optimization means teaching optimal control. And if you are in computer science, optimization today could be interpreted in the context of machine learning (such as fitting models to data) or as reinforcement learning.

This book claims that the traditional style of teaching optimization is misguided and out of date. First, while the simplex algorithm is a powerful strategy for solving linear programs, the details of the simplex algorithm are completely inappropriate in an introductory course in optimization. Second, while linear programs are appropriate for solving many problems, they are only applicable to a tiny fraction of all decisions. Third, linear programs (along with integer and nonlinear programs) are static models for problems with (typically) vector-valued decisions. By contrast, most decisions are sequential since they are made periodically over time as new information is arriving. In addition, the vast majority of these decisions are scalar (possibly continuous or discrete).

This book is designed for instructors (or potential instructors) looking to introduce the science of making good decisions to the broadest possible audience. It should also be of interest to anyone who has already had a traditional course in optimization of any type. The presentation is organized around a series of topics that suggest a fundamentally different approach to teaching “optimization” spanning both sequential decision problems (which offer the simplest problem settings) before transitioning to more complex vector-valued decisions. It also makes the case that most problems which are modeled as linear (or integer, or nonlinear programs) are actually methods for making decisions in a sequential setting. For this reason, these topics are introduced with much less emphasis on algorithms than is traditionally used, both in static and sequential settings.

The mathematical modeling of many problems from economics, computer science, engineering, biological neural networks and others leads to the consideration of nonlinear difference equations. Many authors have studied such problems by using various methods, such as: the fixed point theory, the fixed point index theory, variational methods, the critical point theory, different transformations, extensions of Perron's second theorem, diversified criteria for the stability of solutions, and so on. 

This monograph studies the existence of positive solutions for some classes of second-order nonlinear finite difference equations, and systems of second-order nonlinear finite difference equations, subject to various multi-point boundary conditions. In the case of systems, these boundary conditions may be uncoupled or coupled. It also investigates a class of nonlinear ????th order Atıcı-Eloe fractional difference equations supplemented with varied boundary conditions, and some systems of generalized second-order difference equations in Hilbert spaces with multi-point boundary conditions. The book draws together our results that have been obtained in the last years. Chapter 1 deals with the existence of positive solutions for two second-order finite difference equations which contain a linear term and a sign-changing nonlinearity, with or without parameters, subject to multi-point boundary conditions. Chapter 2 is focused on the existence and multiplicity of positive solutions for two systems of nonlinear second-order difference equations with uncoupled multi-point boundary conditions. The nonlinearities from the systems are nonnegative functions and satisfy some assumptions containing concave functions, or they are sign-changing functions. Chapter 3 studies the existence and nonexistence of positive solutions for two systems of nonlinear second-order difference equations supplemented with coupled multi-point boundary conditions, with positive parameters in the systems or in the boundary conditions. The nonlinearities of the systems are nonnegative functions and satisfy various assumptions. Chapter 4 is concerned with the existence and multiplicity of positive solutions for two systems of nonlinear second-order difference equations subject to coupled multi-point boundary conditions, without parameters. The nonlinearities of the systems are nonnegative functions and satisfy various assumptions. Chapter 5 is devoted to the existence of positive solutions for a system of nonlinear second-order difference equations with parameters and sign-changing nonlinearities, supplemented with multi-point coupled boundary conditions. Chapter 6 deals with the existence of nontrivial solutions, nonnegative solutions and positive solutions for a class of nonlinear ????th order Atıcı-Eloe fractional difference equations with left focal boundary conditions or Dirichlet boundary conditions. In each chapter, various examples are presented which support the main results.

Finally, the new Chapter 7 investigates the existence and uniqueness of solutions for some nonlinear systems of generalized second-order difference equations in Hilbert spaces, subject to multi-point boundary conditions containing monotone operators. Some applications to initial-boundary value problems for nonlinear first-order differential systems with monotone operators are also addressed. The methods used in the proof of our theorems include results from the fixed point theory, the fixed point index theory, the theory of monotone operators and nonlinear evolution equations of monotone type in Hilbert spaces.

This monograph can serve as a good resource for the mathematical and scientific researchers, and for the graduate students in mathematics and science interested in the existence of solutions and positive solutions for finite difference equations and systems.

Der zweite Band beschäftigt sich mit erprobten numerischen Verfahren zur Lösung analytischer Problemstellungen. Im Mittelpunkt stehen dabei numerische Techniken für die ein- und zweidimensionale Interpolation, die Kleinste-Quadrate Approximation sowie für die Berechnung der Ableitung und des bestimmten Integrals reeller Funktionen. Des Weiteren werden die wichtigsten Verfahren zur numerischen Behandlung von Anfangs- und Randwertproblemen gewöhnlicher Differentialgleichungen vorgestellt. Viele dieser numerischen Verfahren sind in Form von m-Files für die Software-Umgebung MATLAB implementiert, sodass die Leser selbstständig mit den numerischen Techniken experimentieren und damit deren Vor- und Nachteile unmittelbar am Rechner erleben können.

Die wichtigsten Verfahren sind als MATLAB-Programme auf der Webseite des Verlages zum Download bereit

Complex Engineering Systems - Modeling and Optimization offers an in-depth exploration of the foundational principles, advanced methodologies, and interdisciplinary applications essential for understanding and managing intricate engineering systems. Spanning mathematical modeling, numerical simulation, optimization strategies, and AI-assisted techniques, the book presents a rich blend of theory and real-world problem-solving tools. This comprehensive volume is tailored for researchers, professionals, and graduate students engaged in engineering, applied mathematics, and computational sciences.

Covering diverse themes-from system-of-systems behavior and multiphysics modeling to nanofluid dynamics, fractional heat transfer, queuing theory, and machine learning integration - the chapters collectively emphasize the interplay between complexity, adaptability, and innovation. The contributors shed light on modern challenges like cryogenic flow analysis, pressure sensing in microfluidics, MHD flow behavior, and AI-driven predictive modeling.

This book bridges theoretical concepts in fuzzy clustering with practical applications. It elucidates the mathematical foundations of fuzzy clustering methods, serving as a much-needed resource in the field. By explicating complex concepts, it will aid researchers, practitioners, and students in the analysis, design, and optimization of algorithms. This valuable resource will contributes to the utilization and enhancement of fuzzy clustering techniques across various domains. The book delves into the mathematical underpinnings, methods, and applications of fuzzy clustering. This book will enable academics, practitioners, and students to grasp the mathematical significance of fuzzy clustering in contemporary data analysis and to explore creative applications. It is an essential volume for data analysis professionals and students alike. Using advanced mathematical methods, it instructs readers on understanding, designing, and applying these algorithms. Its unique blend of theory and practice bridges the mathematical foundation with practical application, empowering readers to effectively analyze large datasets and uncover hidden patterns.

This book uncovers the innovative features of complex algebra i.e., the octonion -sparse concept in image processing and how it can help in improving the computational efficiency of image processing.
It deals with the most common octonion convolution neural network, quaternion wavelet, and sparse representation-based techniques in multispectral image processing.
The book focusses on how evolution in algebraic concepts i.e. octonion and sparse helps in improving accuracy and efficiency of various color image restoration, reconstruction and recognition.
It provides examples of research and real-world applications related to biometric security, medical image processing, face recognition, and how crucial characteristics are extracted. 
The main focus of this book is to bring all the related technologies, novel findings and managerial applications of Colored Medical Imaging on a single platform to provide great readability, easy understanding, and smooth adaptability of various basic and advanced concepts to the Undergraduate and Postgraduate students, Researchers, Managers, Designers, Academicians, and Practitioners doing work in this domain.

Dieser Titel aus dem De Gruyter-Verlagsarchiv ist digitalisiert worden, um ihn der wissenschaftlichen Forschung zugänglich zu machen. Da der Titel erstmals im Nationalsozialismus publiziert wurde, ist er in besonderem Maße in seinem historischen Kontext zu betrachten. Mehr erfahren Sie hier.

Dieses Lehrbuch liefert eine kompakte und dennoch sehr ansprechend geschriebene Einführung in die Wahrscheinlichkeitstheorie. Neben den Grundlagen werden auch stochastische Prozesse, Martingale und Anwendungsbeispiele behandelt. Die notwendigen Grundlagen über Maßtheorie und metrische Räume werden bereit gestellt. Der Text kann als eine hochklassige und motivierende Grundlage für einen ambitionierten dreisemestrigen einführenden Kurs über Wahrscheinlichkeitstheorie dienen.

This volume consists of seven papers related in various matters to the research work of Kostia Beidar †, a distinguished ring theorist and professor of National Ching Kung University (NCKU). Written by leading experts in these areas, the papers also emphasize important applications to other fields of mathematics. Most papers are based on talks that were presented at the memorial conference which was held in March 2005 at NCKU.

This monograph provides a thorough treatment of parameter-dependent extremal problems with local minimum values that remain unchanged under changes of the parameter.

The authors consider the theory as well the practical treatment of those problems, both in finite-dimensional as well as in infinite-dimensional spaces. Various applications are considered, e.g., variational calculus, control theory and bifurcations theory.

• Thorough treatment of parameter-dependent extremal problems with local minimum values.
• Includes many applications, e.g., variational calculus, control theory and bifurcations theory.
• Intended for specialists in the field of nonlinear analysis and its applications as well as for students specializing in these subjects.
  

Die zweite, überarbeitete und erweiterte Auflage dieses Lehrbuchs liefert eine fundierte mathematische Einführung in die Thematik. Besonderer Wert wird auf möglichst einfache Beweise gelegt, die zugleich eine geometrische Anschauung erlauben. Zahlreiche Übungsaufgaben und Beispiele ergänzen den Inhalt.

• Überarbeitete Neuauflage eines bekannten Lehrbuchs
• Fundierte mathematische Einführung in die Thematik
• Mit einfachen und anschaulichen Beweisen

This textbook provides a broad and solid introduction to mathematical statistics, including the classical subjects hypothesis testing, normal regression analysis, and normal analysis of variance. In addition, non-parametric statistics and vectorial statistics are considered, as well as applications of stochastic analysis in modern statistics, e.g., Kolmogorov-Smirnov testing, smoothing techniques, robustness and density estimation.

• For students with some elementary mathematical background.
• With many exercises.
• Updated, complete Solutions Manual available on request
• Prerequisites from measure theory and linear algebra are presented.
  

This book contains four survey papers related to different topics in computational mechanics, in particular (1) novel discretization and solver techniques in mechanics and (2) inverse, control, and optimization problems in mechanics. These topics were considered in lectures, seminars, tutorials, and workshops at the Special Semester on Computational Mechanics held at the Johann Radon Institute for Computational and Applied Mathematics (RICAM), Linz, Austria, in December 2005.

Dieses Lehrbuch liefert eine elementare, auch für Studierende der Ingenieurwissenschaften geeignete Einführung in die grundlegenden numerischen Verfahren zur Lösung von Anfangs- und Randwertaufgaben für gewöhnliche Differentialgleichungen. Das Studium der Inhalte wird durch ergänzende Modellbeispiele sowie zahlreiche theoretische und numerische Übungsaufgaben mit Lösungshinweisen erleichtert. Ein kurzer Abschnitt liefert die notwendigen Grundlagen zur Theorie der gewöhnlichen Differentialgleichungen.

Dieses Lehrbuch zur gleichnamigen Grundvorlesung behandelt ausgewählte Themen der mehrdimensionalen Analysis. Es führt außerdem in die Theorie der gewöhnlichen Differentialgleichungen ein, und die für die Anwendungen in der Bildverarbeitung und Akustik wichtige Fouriertheorie wird ebenfalls vorgestellt.

• Themen: Fixpunktsätze, gewöhnliche Differentialgleichungen, Theorie der Fourierreihen, Mannigfaltigkeiten und Differentialformen.
• Die ausführliche Darstellung der Themen erleichtert das Verständnis.
• Mit zahlreichen grafischen Illustrationen und Übungsaufgaben.

This book is a translation of the third edition of the well accepted German textbook 'Stochastik', which presents the fundamental ideas and results of both probability theory and statistics, and comprises the material of a one-year course. The stochastic concepts, models and methods are motivated by examples and problems and then developed and analysed systematically.

Heinz Bauer (1928-2002) was one of the prominent figures in Convex Analysis and Potential Theory in the second half of the 20th century. The Bauer minimum principle and Bauer's work on Silov's boundary and the Dirichlet problem are milestones in convex analysis. Axiomatic potential theory owes him what is known by now as Bauer harmonic spaces.

These Selecta collect more than twenty of Bauer's research papers including his seminal papers in Convex Analysis and Potential Theory. Above his research contributions Bauer is best known for his art of writing survey articles. Five of his surveys on different topics are reprinted in this volume. Among them is the well-known article Approximation and Abstract Boundary, for which he was awarded with the Chauvenet Price by the American Mathematical Association in 1980.

The Proceedings contain twenty selected, refereed contributions arising from the International Conference on Public-Key Cryptography and Computational Number Theory held in Warsaw, Poland, on September 11-15, 2000.

The conference, attended by eightyfive mathematicians from eleven countries, was organized by the Stefan Banach International Mathematical Center.

This volume contains articles from leading experts in the world on cryptography and computational number theory, providing an account of the state of research in a wide variety of topics related to the conference theme. It is dedicated to the memory of the Polish mathematicians Marian Rejewski (1905-1980), Jerzy Róøycki (1909-1942) and Henryk Zygalski (1907-1978), who deciphered the military version of the famous Enigma in December 1932 – January 1933. A noteworthy feature of the volume is a foreword written by Andrew Odlyzko on the progress in cryptography from Enigma time until now.

In the design of a neural network, either for biological modeling, cognitive simulation, numerical computation or engineering applications, it is important to investigate the network's computational performance which is usually described by the long-term behaviors, called dynamics, of the model equations. The purpose of this book is to give an introduction to the mathematical modeling and analysis of networks of neurons from the viewpoint of dynamical systems.

In this book the theory of binary systems is considered as a part of group theory and, in particular, within the framework of Lie groups. The novelty is the consequent treatment of topological and differentiable loops as topological and differentiable sections in Lie groups. The interplay of methods and tools from group theory, differential geometry and topology, symmetric spaces, topological geometry, and the theory of foliations is what gives a special flavour to the results presented in this book. It is the first monograph devoted to the study of global loops. So far books on differentiable loops deal with local loops only. This theory can only be used partially for the theory of global loops since non-associative local structures have, in general, no global forms.

The text is addressed to researchers in non-associative algebra and foundations of geometry. It should prove enlightening to a broad range of readers, including mathematicians working in group theory, the theory of Lie groups, in differential and topological geometry, and in algebraic groups.

The authors have produced a text that is suitable not only for a graduate course, but also for selfstudy in the subjectby interested graduate students. Moreover, the material presented can be used for lectures and seminars in algebra, topological algebra and geometry.

This book gives a straightforward introduction to the field as it is nowadays required in many branches of analysis and especially in probability theory. The first three chapters (Measure Theory, Integration Theory, Product Measures) basically follow the clear and approved exposition given in the author's earlier book on "Probability Theory and Measure Theory". Special emphasis is laid on a complete discussion of the transformation of measures and integration with respect to the product measure, convergence theorems, parameter depending integrals, as well as the Radon-Nikodym theorem.

The final chapter, essentially new and written in a clear and concise style, deals with the theory of Radon measures on Polish or locally compact spaces. With the main results being Luzin's theorem, the Riesz representation theorem, the Portmanteau theorem, and a characterization of locally compact spaces which are Polish, this chapter is a true invitation to study topological measure theory.

The text addresses graduate students, who wish to learn the fundamentals in measure and integration theory as needed in modern analysis and probability theory. It will also be an important source for anyone teaching such a course.

This book is an outgrowth of a Research Symposium on the Modular Representation Theory of Finite Groups, held at the University of Virginia in May 1998. The main themes of this symposium were representations of groups of Lie type in nondefining (or cross) characteristic, and recent developments in block theory.

Series of lectures were given by M. Geck, A. Kleshchev and R. Rouquier, and their brief was to present material at the leading edge of research but accessible to graduate students working in the field. The first three articles are substantial expansions of their lectures, and each provides a complete account of a significant area of the subject together with an extensive bibliography. The remaining articles are based on some of the other lectures given at the symposium; some again are full surveys of the topic covered while others are short, but complete, research articles.

The opportunity has been taken to produce a book of enduring value so that this is not a conference proceedings in the conventional sense. Material has been updated so that this book, through its own content and in its extensive bibliographies, will serve as an invaluable resource for all those working in the area, whether established researchers or graduate students who wish to gain a general knowledge of the subject starting from a single source.

The book discusses the stability, observability, and controllability of nonlinear systems of PDEs (such as Wave, Heat, Euler-Bernoulli beam, Petrovsky, Kirchhoff, equations, and more). Methods based on the theory of classical weak functions analysis and movements in Sobolev spaces are used to analyze nonlinear systems of evolutionary partial differential equations. With the unifying theme of evolutionary dynamic equations, both linear and nonlinear, in more complex environments with different approaches, the book presents a multidisciplinary blend of topics, spanning the fields of PDEs applied to various models coming from theoretical physics, biology, engineering, and natural sciences.
This comprehensive book is prepared for a diverse audience interested in applied mathematics. With its broad applicability, this book aims to foster interdisciplinary collaboration and facilitate a deeper understanding of complex phenomenon concepts, practically in electromagnetic waves, the acoustic model for seismic waves, waves in blood vessels, wind drag on space, the linear shallow water equations, sound waves in liquids and gases, non-elastic effects in the string.