Mathematics

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.

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 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 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.

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.

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.

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

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.

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.

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.

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.

The book includes recent articles on various topics studied recently in Banach algebras and abstract harmonic analysis. On the Banach algebra side, the reader will find idempotents and socle of a Banach algebra; closed subspaces of a Banach space, including the classical sequence spaces, which are realized as the kernel of a bounded operator; the connection between the stable rank one and Dedekind-finite property of the algebra of operators on a Banach space; spectral synthesis properties in convolution Sobolev algebras on the real line.

The harmonic analysis side includes a generalization of the famous Beurling theorem; groups with few finite-dimensional unitary representations; relations between ideals of the Figa-Talamanca Herz algebra of a locally compact group and ideals of Figa-Talamanca Herz algebra of its closed subgroup; the tame functionals on Banach algebras and in harmonic analysis; and cancellation, factorization, and isometries in algebras on a locally compact group.

The book also includes four surveys written by leaders in the area of full sheaf cohomology theory for noncommutative C - algebras; one-parameter semigroups of bounded operators on a Banach space which are weakly continuous in the sense of Arveson.

The book summarizes recent results on problems with uni-variate polynomials. The first of them reads: given the signs of the coefficients of a real polynomial (i. e. its sign pattern), for which pairs of prescribed numbers of positive and negative roots (compatible with Descartes’ rule of signs) can one find such a polynomial? For each degree greater or equal to 4, there are non-realizable cases. The problem is resolved for degree less or equal to 8. In another realization problem (resolved for degree less or equal to 5), one fixes the pairs (compatible with Rolle’s theorem) of numbers of positive and negative roots of the polynomial and its non-constant derivatives. A third problem concerns polynomials with all roots real. One considers the sign pattern and the order in which the moduli of its positive and negative roots are arranged on the positive half-line. There are examples of pairs (sign pattern, order of moduli) compatible with Descartes’ rule of signs that are not realizable. And there are various questions about the discriminant of the general family of uni-variate polynomials. The non-trivial answers to these simply formulated problems will give students and scholars a better understanding of uni-variate polynomials.

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

Kirchhoff graphs are a new type of graph, one whose edges are vectors. They depict the dependencies in sets of vectors, and in this regard, they are akin to matroids. Indeed every binary matroid is Kirchhoff graphic. If the vectors in the set represent the stoichiometry of a reaction network (chemical, electrochemical or otherwise), a Kirchhoff graph is a circuit diagram for that reaction network. This book discusses the basic definitions for Kirchhoff graphs, their connections to matroids and finite fields, their connections to reaction networks and use as circuit diagrams, Kirchhoff graph uniformity, prime, composite and fundamental Kirchhoff graphs, and the structure of families of Kirchhoff graphs. They been studied by the author and his collaborators since 2005, and this book presents what we have learned in the past twenty years.

Methods for solving problems of mathematical physics can be divided into the following four classes.

Analytical methods (the method of separation of variables, the method of characteristics, the method of Green's functions, etc.) methods have a relatively low degree of universality, i.e. focused on solving rather narrow classes of problems.

Approximate analytical methods (projection, variational methods, small parameter method, operational methods, various iterative methods) are more versatile than analytical ones.

Numerical methods (finite difference method, direct method, control volume method, finite element method, etc.) are very universal methods.

Probabilistic methods (Monte Carlo methods) are highly versatile. Can be used to calculate discontinuous solutions. However, they require large amounts of calculations and, as a rule, they lose with the computational complexity of the above methods when solving such problems to which these methods are applicable.

Comparing methods for solving problems of mathematical physics, it is impossible to give unconditional primacy to any of them. Any of them may be the best for solving problems of a certain class.

The proposed method of moving nodes for boundary value problems of differential equations combines a combination of numerical and analytical methods. In this case, we can obtain, on the one hand, an approximately analytical solution of the problem, which is not related to the methods listed above. On the other hand, this method allows one to obtain compact discrete approximations of the original problem. Note that obtaining an approximately analytical solution of differential equations is based on numerical methods. The nature of numerical methods also allows obtaining an approximate analytical expression for solving differential equations

While the Cobordism Hypothesis provides a translation between topological and categorical structures, the subject of fusion categories arising from representations of finite groups has shown the need for a robust theory of duality in monoidal 2-categories. This book intertwines 3-dimensional category theory with enriched and symmetric monoidal 2-categories to present a comprehensive foundation for duals in dimension two. This framework is presented using wire diagrams, a typographical tool that combines the geometric appeal of ordinary string diagrams with the clarity of standard commutative diagrams in category theory. Chapters on enriched 2-categories, closed 2-categories, and compact closed 2-categories build the theory from the ground up and in full detail. Shorter appendices on technical topics from higher category theory such as icons, strictification theorems, and computads provide readers with concise explanations of key results, as well as references for researchers interested in more depth. The book concludes with a chapter of examples from algebra and topology, including a detailed construction for the example of cobordisms.

This book is intended for a technical audience with advanced learning and in-depth analysis of the latest real-world developments in the field of quantum computing. The book is focused on modeling emerging drone-based applications. The methods of developing faster drive with the new trending technology of quantum genetic heuristics, artificial intelligence, and machine-based applications are discussed. Researchers will find novel ways to secure the data and Quantum drone networks. The book examines the most promising exploratory quantum-computed drone that optimizes, secures, and dynamically analyses various sectors, including healthcare, the educational industry, finance, transportation, and manufacturing.

The book focuses on the theory of fixed points, which is a foundation for many branches of pure and applied mathematics. Fixed point theorems have been studied in various function spaces. The book contains modern results on these theorems, investigated in generalized spaces such as S-metric spaces, convex metric spaces, and bipolar metric spaces, with applications in medical imaging. The nonlinear analysis presented in the book is valuable for modeling and solving real-world problems. It includes work on specific nonlinear operators and nonlinear fractional integral equations in Banach spaces. Relevant studies are also included on statistical convergence, inventory model modeling, computational techniques for Sentiment Analysis on Twitter Data, and Blood Management applications. The book is intended for young researchers interested in nonlinear analysis, fixed-point theory, and computational techniques.

Wolmer Vasconcelos was one of the giants in the development of Commutative Algebra in the latter half of the twentieth century and the first decades of the twenty-first century. This work collects in one place essays illustrating the important developments of his work particularly in commutative algebra that permits the reader to see the development of his important ideas and how they influence the development of mathematics today.

The introduction of cross diffusivity opens many questions in the theory of reactiondiffusion systems. This book will be the first to investigate such problems presenting new findings for researchers interested in studying parabolic and elliptic systems where classical methods are not applicable. In addition, The Gagliardo-Nirenberg inequality involving BMO norms is improved and new techniques are covered that will be of interest. This book also provides many open problems suitable for interested Ph.D students.

The recent advent of rapid technological changes, scientific developments, and educational expansions have created complex heterogeneities, environmental uncertainties, and socio-economic-ecological inequalities globally. The innovative beneficial resources upgrade and update the existing varieties of structural features such as hereditary, random environmental, spatial and atmospheric perturbations in human population dynamics processes and predator-prey systems. The highly interconnected system under operating random environmental conditions is represented by nonlinear nonstationary large-scale multi-level hierarchical network-centric dynamic processes of Ito-Doob and finite Markovian types with network-centric structural perturbations. For instance, complex spatial, behavioral, and epidemiological structures in human populations vary from citizen to visitor; practicing and adhering to different disease preventive measures at sites in meta-populations; and different ages, stages and resistance levels to infections, respectively. An advantage of the presented results in simple algebraic system parameters form is easy verification and application to planning, prevention, policies, stabilization, monitoring and diseases management.