Mathematics

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.

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

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.

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.

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.

Die Numerische Mathematik ist einer der Grundpfeiler des Mathematik-, Ingenieur-, Physik- und Informatikstudiums. Dieses zweibändige Lehrbuch ist für Einführungsvorlesungen konzipiert und legt eine solide Basis für weiterführende Lerneinheiten. Der Text ist aus Vorlesungsmanuskripten hervorgegangen, die der Verfasser seit etwa 30 Jahren für seine Grundvorlesungen auf dem Gebiet der Numerischen Mathematik und des Wissenschaftlichen Rechnens an der Friedrich-Schiller-Universität Jena verwendet.

Das Buch deckt den gesamten Bereich der Numerischen Mathematik von den klassischen Techniken wie Gaußscher Algorithmus und Newtonsches Verfahren bis hin zu modernen Algorithmen wie kubische Spline-Interpolation, Kleinste-Quadrate-Approximation mittels Householder- und Givens-Transformationen sowie Deflationstechniken ab. Die Verfahren werden mathematisch exakt beschrieben, in MATLAB-Codes implementiert und anhand von Beispielen demonstriert. Die MATLAB-Codes sind auf der Webseite des Verlages zum Download bereitgestellt, so dass der Leser seine eigenen Experimente mit den numerischen Verfahren durchführen kann.

Durch seinen didaktischen Aufbau und die zahlreichen anschaulichen Beispiele und Übungsaufgaben eignet sich dieses Buch hervorragend als vorlesungsbegleitende Lektüre und als Grundlage für ein erfolgreiches Selbststudium. Gleichzeitig kann es von Mathematikern, Naturwissenschaftlern und Ingenieuren als profundes Nachschlagewerk herangezogen werden.

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./span>

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.

Algebra in the Middle Grades addresses topics that are formalized in the first half of an Algebra 1 course, focusing on linear equations, their graphs and their applications to problem solving. The book is intended to be used by students in the middle school in preparation for the formal study of Algebra 1. In this, it is a truly Pre-Algebra program.

Algebra in the Middle Grades is an approach that brings Algebra to the students. It is written to have the essential elements of Algebra be understood on an inductive level so that when students later address the topic formally, there will be an informal background to create deep understanding.

The approach is extremely visual using both algebra tile-type diagrams along with graphs for equations and grids for ratio. In this way, the text gives students 3 aspects of a topic, the Numerical, the Geometric and the Algebraic. The text places the Numerical and Geometric first having understanding in these contexts leading to understanding in an Algebraic, symbolic context.

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.

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.

Die Numerische Mathematik ist einer der Grundpfeiler des Mathematik-, Ingenieur-, Physik- und Informatikstudiums. Dieses zweibändige Lehrbuch ist für Einführungsvorlesungen konzipiert und legt eine solide Basis für weiterführende Lerneinheiten. Der Text ist aus Vorlesungsmanuskripten hervorgegangen, die der Verfasser seit etwa 30 Jahren für seine Grundvorlesungen auf dem Gebiet der Numerischen Mathematik und des Wissenschaftlichen Rechnens an der Friedrich-Schiller-Universität Jena verwendet.

Das Buch deckt den gesamten Bereich der Numerischen Mathematik von den klassischen Techniken wie Gaußscher Algorithmus und Newtonsches Verfahren bis hin zu modernen Algorithmen wie kubische Spline-Interpolation, Kleinste-Quadrate-Approximation mittels Householder- und Givens-Transformationen sowie Deflationstechniken ab. Die Verfahren werden mathematisch exakt beschrieben, in MATLAB-Codes implementiert und anhand von Beispielen demonstriert. Die MATLAB-Codes sind auf der Webseite des Verlages zum Download bereitgestellt, so dass der Leser seine eigenen Experimente mit den numerischen Verfahren durchführen kann.

Durch seinen didaktischen Aufbau und die zahlreichen anschaulichen Beispiele und Übungsaufgaben eignet sich dieses Buch hervorragend als vorlesungsbegleitende Lektüre und als Grundlage für ein erfolgreiches Selbststudium. Gleichzeitig kann es von Mathematikern, Naturwissenschaftlern und Ingenieuren als profundes Nachschlagewerk herangezogen werden.

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.

Sponsored by the American Educational Research Association's Special Interest Group for Educational Statisticians

This volume is the second edition of Hancock and Mueller’s highly-successful 2006 volume, with all of the original chapters updated as well as four new chapters. The second edition, like the first, is intended to serve as a didactically-oriented resource for graduate students and research professionals, covering a broad range of advanced topics often not discussed in introductory courses on structural equation modeling (SEM). Such topics are important in furthering the understanding of foundations and assumptions underlying SEM as well as in exploring SEM, as a potential tool to address new types of research questions that might not have arisen during a first course. Chapters focus on the clear explanation and application of topics, rather than on analytical derivations, and contain materials from popular SEM software.

Also available in a color version

AMTE, in the Standards for Preparing Teachers of Mathematics (SPTM), puts forward a national vision of initial preparation for all Pre-K–12 teachers who teach mathematics. SPTM contains critical messages for all who teach mathematics, including elementary school teachers teaching all disciplines, middle and high school mathematics teachers who may teach mathematics exclusively, special education teachers, teachers of emergent multilingual students, and other teaching professionals and administrators who have responsibility for students’ mathematical learning. SPTM has broad implications for teacher preparation programs, in which stakeholders include faculty and administrators in both education and mathematics at the university level; teachers, principals, and district leaders in the schools with which preparation programs partner; and the communities in which preparation programs and their school partners are situated.

SPTM is intended as a national guide that articulates a vision for mathematics teacher preparation and supports the continuous improvement of teacher preparation programs. Such continuous improvement includes changes to preparation program courses and structures, partnerships involving schools and universities and their leaders, the ongoing accreditation of such programs regionally and nationally, and the shaping of state and national mathematics teacher preparation policy. SPTM is also designed to inform assessment practices for mathematics teacher preparation programs, to influence policies related to preparation of teachers of mathematics, and to promote national dialogue around preparing teachers of mathematics. The vision articulated in SPTM is aspirational in that it describes a set of high expectations for developing a well-prepared beginning teacher of mathematics who can support meaningful student learning. The vision is research-based and establishes a set of goals for the continued development and refinement of a mathematics teacher preparation program and a research agenda for the study of the effects of such a program. SPTM contains detailed depictions of what a well-prepared beginning teacher knows and is able to do related to content, pedagogy, and disposition, and what a strong preparation program entails with respect to learning experiences, assessments, and partnerships. Stakeholders in mathematics teacher preparation will find messages related to their roles.

Standards for Preparing Teachers of Mathematics includes standards and indicators for teacher candidates and for the design of teacher preparation programs. SPTM outlines assessment practices related to overall quality, program effectiveness, and candidate performance. SPTM describes specific focal practices by grade band and provides guidance to stakeholders regarding processes for productive change.

Also available in a black + white version

AMTE, in the Standards for Preparing Teachers of Mathematics (SPTM), puts forward a national vision of initial preparation for all Pre-K–12 teachers who teach mathematics. SPTM contains critical messages for all who teach mathematics, including elementary school teachers teaching all disciplines, middle and high school mathematics teachers who may teach mathematics exclusively, special education teachers, teachers of emergent multilingual students, and other teaching professionals and administrators who have responsibility for students’ mathematical learning. SPTM has broad implications for teacher preparation programs, in which stakeholders include faculty and administrators in both education and mathematics at the university level; teachers, principals, and district leaders in the schools with which preparation programs partner; and the communities in which preparation programs and their school partners are situated.

SPTM is intended as a national guide that articulates a vision for mathematics teacher preparation and supports the continuous improvement of teacher preparation programs. Such continuous improvement includes changes to preparation program courses and structures, partnerships involving schools and universities and their leaders, the ongoing accreditation of such programs regionally and nationally, and the shaping of state and national mathematics teacher preparation policy. SPTM is also designed to inform assessment practices for mathematics teacher preparation programs, to influence policies related to preparation of teachers of mathematics, and to promote national dialogue around preparing teachers of mathematics. The vision articulated in SPTM is aspirational in that it describes a set of high expectations for developing a well-prepared beginning teacher of mathematics who can support meaningful student learning. The vision is research-based and establishes a set of goals for the continued development and refinement of a mathematics teacher preparation program and a research agenda for the study of the effects of such a program. SPTM contains detailed depictions of what a well-prepared beginning teacher knows and is able to do related to content, pedagogy, and disposition, and what a strong preparation program entails with respect to learning experiences, assessments, and partnerships. Stakeholders in mathematics teacher preparation will find messages related to their roles.

Standards for Preparing Teachers of Mathematics includes standards and indicators for teacher candidates and for the design of teacher preparation programs. SPTM outlines assessment practices related to overall quality, program effectiveness, and candidate performance. SPTM describes specific focal practices by grade band and provides guidance to stakeholders regarding processes for productive change.

The teaching and learning of mathematics in British Columbia has a long and storied history. An integral part of the past 50 years (1962-2012) of this history has been Vector: Journal of the British Columbia Association of Mathematics Teachers. This volume, which presents ten memorable articles from each of the past five decades, that is, 50 articles from the past 50 years of the journal, provides an opportunity to share this rich history with a wide range of individuals interested in the teaching and learning of mathematics and mathematics education. Each decade begins with an introduction, providing a historical context, and concludes with a commentary from a prominent member of the British Columbia mathematics education community. As a result, this monograph provides a historical account as well as a contemporary view of many of the trends and issues in the teaching and learning of mathematics. This volume is meant to serve as a resource for a variety of individuals including: teachers of mathematics, mathematics teacher educators, mathematics education researchers, historians, and undergraduate and graduate students. Most importantly, this volume is a celebratory retrospective on the work of the British Columbia Association of Mathematics Teachers.

Reflecting the World: A Guide to Incorporating Equity in Mathematics Teacher Education is a guide for mathematics teacher educators interested in incorporating equity concerns into their teaching. The book draws on the authors’ research and experience integrating issues of equity, diversity, and social justice into their work as mathematics teacher educators of preservice and inservice preK-9 teachers. Reflecting the World includes both a framework for integrating issues of equity into mathematics teacher education courses and professional development and example lessons. The lessons are organized by content area and include guidance for using them effectively.

Elementary and middle grades pre-service teachers are often uncomfortable with mathematics, uncertain about their ability to teach it, and unsure of how it connects to the real world. For many preservice teachers a focus on the real world—and in particular on issues of equity, diversity, and social justice—is more engaging than their past experiences with mathematics and can help lessen their mathematical anxieties.

Reflecting the World will assist teacher educators in designing and teaching mathematics content and methods courses in ways that support future teachers to see the relevance of mathematics to our world and in becoming critical, questioning citizens in an increasingly mathematical world. The book provides a set of tools for helping future teachers connect mathematics to the lives, interests, and political realities of an increasingly diverse student body, and in doing so it provides a meaningful answer to the question, “when will I ever use this?”

The diversity of research in mathematics education has been addressed as both, a problem and a strength. When manifested through adherence to different intellectual roots and theoretical orientations, diversions constitute ‘refractions’ of mathematics education. The collection and analysis of empirical data in a study are by necessity refracted through the specific analytical lens employed, as well as the aim of the study itself. Refractions can also refer to looking at old phenomena through new lenses.

The chapters in this book are refracted through philosophical, political, mathematical and personal lenses by distinguished authors in the field, addressing issues about the elusive experience of doing mathematics, purification of texts, refractions, mathematics and ethnomathematics, political messages in textbook tasks, mathematics education policy debate, the political in mathematics education research, philosophy and mathematics, meanings and representations, identity of mathematical modeling, and dilemmas in the teaching of calculus.

An ancient Sanskrit adage states that Knowledge is something that grows when shared, but shrinks when hoarded. Academics engaged in the generation of new Knowledge are blessed with both the time and the freedom to engage in pursuits that allow for intellectual pleasure. As a phenomenon of the Zeitgeist many have succumbed to the increased corporatization of academic work, engaging in activities for monetary and self advancement purposes. Are there any real intellectuals left in academia, a là Adorno, Bourdieu, Chomsky, Foucault, among others? This Festschrift is dedicated to academics that don't bother with self promotion or aggrandizement of themselves or their ideas in simplistic terms.

Mathematics teacher education has a critical role to play in preparing teachers to put at center stage goals to support equity in mathematics education and to diversify student interest and participation in mathematics. These goals must also resonate with broader public interest goals to improve educational and social conditions both in the U.S. and abroad. The Mathematics Teacher Education in the Public Interest book aims to support mathematics teacher educators to prepare teachers with new knowledge and skills to support all students to learn mathematics and to become informed, engaged, and critical citizens within their community, nation, and world. While internationally there is considerable interest among mathematics educators in issues of equity and social justice, the literature on mathematics teacher education for equity and social justice thus far has been very limited.The book provides theoretical discussions on the need for equity and social justice emphases in mathematics teacher education, as well as practical examples from mathematics teacher educators, documenting their own professional efforts to center practices on equity and social justice. Section emphases include critical perspectives on mathematics teacher education, the use of equity and social justice-themed activities in mathematics teacher preparation courses, and issues of identity and community and cultural contexts in mathematics teacher education. In addition syntheses of major ideas of the book are offered by experienced researchers.

The Association of Mathematics Teacher Educators (AMTE) in its 2015 position paper on Equity in Mathematics Teacher Education provides a list of actions for mathematics teacher educators (MTE’s) to help them develop and implement equitable practices. The position paper states it is critical that mathematics teacher educators: “Model equity-based pedagogy that emphasizes rich and rigorous mathematics; elicit and build on children’s and young adults’ mathematical thinking; connect to P-12 students’ cultural/linguistic knowledge and backgrounds as well as individual interests; facilitate mathematical discourse; minimize status issues by expanding broader participation and engagement where varied mathematical strengths are valued; and promote positive mathematical identity and agency (p. 2)”.

Cases for Mathematics Teacher Educators: Facilitating Conversations about Inequities in Mathematics Classrooms provides an excellent resource to start conversations describing the enactment of these actions. The book is organized into three main sections: (1) Conversations About Inequities in Mathematics Methods Courses, (2) Conversations About Inequities in Mathematics Content Courses, and (3) Conversations about Inequities in Graduate and Professional Development Contexts. Across these sections there are 19 cases and 57 corresponding commentaries focused on dilemmas that arise when mathematics teacher educators foreground equity in their work. This book of cases provides a needed resource for MTEs to engage prospective teachers, practicing teachers, and future teacher educators in discussions about inequities, privilege, and oppression in society, in schools, and in the mathematics classroom. It is the product of the thinking and experiences of 87 authors who are committed to the improvement of mathematics teacher education.