Fixed Point Theory and Variational Principles in Metric Spaces
The book is designed for undergraduates, graduates, and researchers of mathematics studying fixed point theory or nonlinear analysis. It deals with the fixed point theory for not only single-valued maps but also set-valued maps. The text is divided into three parts: fixed point theory for single-valued mappings, continuity and fixed point aspects of set-valued analysis, and variational principles and their equilibrium problems. It comprises a comprehensive study of these topics and includes all important results derived from them. The applications of fixed point principles and variational principles, and their generalizations to differential equations and optimization are covered in the text. An elementary treatment of the theory of equilibrium problems and equilibrium version of Ekeland's variational principle is also provided. New topics such as equilibrium problems, variational principles, Caristi's fixed point theorem, and Takahashi's minimization theorem with their applications are also included.
Linear Models and the Relevant Distributions and Matrix Algebra
Linear Models and the Relevant Distributions and Matrix Algebra: A Unified Approach, Volume 2 covers several important topics that were not included in the first volume. The second volume complements the first, providing detailed solutions to the exercises in both volumes.
Optimization and Applications
This book constitutes the refereed proceedings of the 14th International Conference on Optimization and Applications, OPTIMA 2023, held in Petrovac, Montenegro, during September 18-22, 2023.The 27 full papers included in this book were carefully reviewed and selected from 68 submissions. They were organized in topical sections as follows: ​mathematical programming; global optimization; discrete and combinatorial optimization; game theory and mathematical economics; optimization in economics and finance; and applications.
Math Mammoth Grade 4 Answer Keys, Canadian Version
This book includes answers for the Math Mammoth Grade 4-A and 4-B student worktexts (Canadian version), for all the chapter tests, for the end-of-year test, and for the cumulative review lessons in Math Mammoth Grade 4 curriculum (Canadian version).
Practical Applications of Stochastic Modelling
This book constitutes the referred proceedings of the 11th International Workshop on Practical Applications of Stochastic Modelling, PASM 2022, was held in Alicante, Spain, in September 2022.The 7 full papers presented in this volume were carefully reviewed and selected from 9 submissions. The papers demonstrate a diverse set of applications and approaches of stochastic modelling.
Recent Progress in Mathematics
This book consists of five chapters presenting problems of current research in mathematics, with its history and development, current state, and possible future direction. Four of the chapters are expository in nature while one is based more directly on research. All deal with important areas of mathematics, however, such as algebraic geometry, topology, partial differential equations, Riemannian geometry, and harmonic analysis. This book is addressed to researchers who are interested in those subject areas. Young-Hoon Kiem discusses classical enumerative geometry before string theory and improvements after string theory as well as some recent advances in quantum singularity theory, Donaldson-Thomas theory for Calabi-Yau 4-folds, and Vafa-Witten invariants. Dongho Chae discusses the finite-time singularity problem for three-dimensional incompressible Euler equations. He presents Kato's classicallocal well-posedness results, Beale-Kato-Majda's blow-up criterion, and recent studies on the singularity problem for the 2D Boussinesq equations. Simon Brendle discusses recent developments that have led to a complete classification of all the singularity models in a three-dimensional Riemannian manifold. He gives an alternative proof of the classification of noncollapsed steady gradient Ricci solitons in dimension 3. Hyeonbae Kang reviews some of the developments in the Neumann-Poincare operator (NPO). His topics include visibility and invisibility via polarization tensors, the decay rate of eigenvalues and surface localization of plasmon, singular geometry and the essential spectrum, analysis of stress, and the structure of the elastic NPO.Danny Calegari provides an explicit description of the shift locus as a complex of spaces over a contractible building. He describes the pieces in terms of dynamically extended laminations and of certain explicit "discriminant-like" affine algebraic varieties.
Singular Integral Operators, Quantitative Flatness, and Boundary Problems
This monograph provides a state-of-the-art, self-contained account on the effectiveness of the method of boundary layer potentials in the study of elliptic boundary value problems with boundary data in a multitude of function spaces. Many significant new results are explored in detail, with complete proofs, emphasizing and elaborating on the link between the geometric measure-theoretic features of an underlying surface and the functional analytic properties of singular integral operators defined on it. Graduate students, researchers, and professionals interested in a modern account of the topic of singular integral operators and boundary value problems - as well as those more generally interested in harmonic analysis, PDEs, and geometric analysis - will find this text to be a valuable addition to the mathematical literature.
Tropical and Logarithmic Methods in Enumerative Geometry
This book is based on the lectures given at the Oberwolfach Seminar held in Fall 2021. Logarithmic Gromov-Witten theory lies at the heart of modern approaches to mirror symmetry, but also opens up a number of new directions in enumerative geometry of a more classical flavour. Tropical geometry forms the calculus through which calculations in this subject are carried out. These notes cover the foundational aspects of this tropical calculus, geometric aspects of the degeneration formula for Gromov-Witten invariants, and the practical nuances of working with and enumerating tropical curves. Readers will get an assisted entry route to the subject, focusing on examples and explicit calculations.
Random Walk and Diffusion Models
This book offers an accessible introduction to random walk and diffusion models at a level consistent with the typical background of students in the life sciences. In recent decades these models have become widely used in areas far beyond their traditional origins in physics, for example, in studies of animal behavior, ecology, sociology, sports science, population genetics, public health applications, and human decision making. Developing the main formal concepts, the book provides detailed and intuitive step-by-step explanations, and moves smoothly from simple to more complex models. Finally, in the last chapter, some successful and original applications of random walk and diffusion models in the life and behavioral sciences are illustrated in detail. The treatment of basic techniques and models is consolidated and extended throughout by a set of carefully chosen exercises.
Problem Posing and Solving for Mathematically Gifted and Interested Students
Mathematics and mathematics education research have an ongoing interest in improving our understanding of mathematical problem posing and solving. This book focuses on problem posing in a context of mathematical giftedness. The contributions particularly address where such problems come from, what properties they should have, and which differences between school mathematics and more complex kinds of mathematics exist. These perspectives are examined internationally, allowing for cross-national insights.
Geometrical Kaleidoscope (Second Edition)
The goal of the book is to provide insight into many enjoyable and fascinating aspects of geometry, and to reveal interesting geometrical properties. The emphasis is on the practical applications of theory in the problem-solving process. The chapters cover a myriad of topics among which are the classic theorems and formulas such as Archimedes' Law of the Lever, the Pythagorean Theorem, Heron's formula, Brahmagupta's formula, Appollonius's Theorem, Euler's line properties, the Nine-Point Circle, Fagnano's Problem, the Steiner-Lehmus Theorem, Napoleon's Theorem, Ceva's Theorem, Menelaus's Theorem, Pompeiu's Theorem, and Morley's Miracle. The book focuses on geometric thinking -- what it means, how to develop it, and how to recognize it. 'Geometrical Kaleidoscope' consists of a kaleidoscope of topics that seem to not be related at first glance. However, that perception disappears as you go from chapter to chapter and explore the multitude of surprising relationships, unexpected connections, and links. Readers solving a chain of problems will learn from them general techniques, rather than isolated instances of the application of a technique. In spite of the many problems' challenging character, their solutions require no more than a basic knowledge covered in a high school geometry curriculum. There are plenty of problems for readers to work out for themselves (solutions are provided at the end of the book).In the 2nd edition of the book there are many new ideas and additional explanations that help the reader better understand the solutions of problems and connect the chapters to one another. A new chapter 'Alternative proofs of the Pythagorean Theorem' is added. It covers seven different proofs of the famous theorem and discusses its generalizations and applications. There is also Appendix and Index added, which were missing in the first edition of the book.
Stein Estimation
This book provides a self-contained introduction of Stein/shrinkage estimation for the mean vector of a multivariate normal distribution. The book begins with a brief discussion of basic notions and results from decision theory such as admissibility, minimaxity, and (generalized) Bayes estimation. It also presents Stein's unbiased risk estimator and the James-Stein estimator in the first chapter. In the following chapters, the authors consider estimation of the mean vector of a multivariate normal distribution in the known and unknown scale case when the covariance matrix is a multiple of the identity matrix and the loss is scaled squared error. The focus is on admissibility, inadmissibility, and minimaxity of (generalized) Bayes estimators, where particular attention is paid to the class of (generalized) Bayes estimators with respect to an extended Strawderman-type prior. For almost all results of this book, the authors present a self-contained proof. The book is helpful for researchers and graduate students in various fields requiring data analysis skills as well as in mathematical statistics.
Multivariate Statistical Analysis in the Real and Complex Domains
This book explores topics in multivariate statistical analysis, relevant in the real and complex domains. It utilizes simplified and unified notations to render the complex subject matter both accessible and enjoyable, drawing from clear exposition and numerous illustrative examples. The book features an in-depth treatment of theory with a fair balance of applied coverage, and a classroom lecture style so that the learning process feels organic. It also contains original results, with the goal of driving research conversations forward.This will be particularly useful for researchers working in machine learning, biomedical signal processing, and other fields that increasingly rely on complex random variables to model complex-valued data. It can also be used in advanced courses on multivariate analysis. Numerous exercises are included throughout.
Reachability Problems
This book constitutes the refereed proceedings of the 17th International Conference on Reachability Problems, RP 2023, held in Nice, France, during October 11-13, 2023.The 13 full papers included in this book were carefully reviewed and selected from 19 submissions. They present recent research on reachability problems to promote the exploration of new approaches for the modeling and analysis of computational processes by combining mathematical, algorithmic, and computational techniques.
Geometrical Kaleidoscope (Second Edition)
The goal of the book is to provide insight into many enjoyable and fascinating aspects of geometry, and to reveal interesting geometrical properties. The emphasis is on the practical applications of theory in the problem-solving process. The chapters cover a myriad of topics among which are the classic theorems and formulas such as Archimedes' Law of the Lever, the Pythagorean Theorem, Heron's formula, Brahmagupta's formula, Appollonius's Theorem, Euler's line properties, the Nine-Point Circle, Fagnano's Problem, the Steiner-Lehmus Theorem, Napoleon's Theorem, Ceva's Theorem, Menelaus's Theorem, Pompeiu's Theorem, and Morley's Miracle. The book focuses on geometric thinking -- what it means, how to develop it, and how to recognize it. 'Geometrical Kaleidoscope' consists of a kaleidoscope of topics that seem to not be related at first glance. However, that perception disappears as you go from chapter to chapter and explore the multitude of surprising relationships, unexpected connections, and links. Readers solving a chain of problems will learn from them general techniques, rather than isolated instances of the application of a technique. In spite of the many problems' challenging character, their solutions require no more than a basic knowledge covered in a high school geometry curriculum. There are plenty of problems for readers to work out for themselves (solutions are provided at the end of the book).In the 2nd edition of the book there are many new ideas and additional explanations that help the reader better understand the solutions of problems and connect the chapters to one another. A new chapter 'Alternative proofs of the Pythagorean Theorem' is added. It covers seven different proofs of the famous theorem and discusses its generalizations and applications. There is also Appendix and Index added, which were missing in the first edition of the book.
Optimal Fractionation in Radiotherapy
This monograph provides a mathematically rigorous overview of optimal fractionation in cancer radiotherapy, which seeks to address the fundamental tradeoff of maximizing tumor-kill while protecting nearby healthy tissue from toxic effects. Most mathematical research on exact solutions to this problem is scattered across journals in applied mathematics, operations research, radiobiology, medicine, and medical physics. These works rarely include rigorous proofs or detailed derivations of their methodologies. Moreover, computer programs utilized for conducting numerical experiments seldom accompany these publications, thereby jeopardizing reproducibility. This monograph aims to provide a comprehensive pedagogical reference that brings researchers up to speed on optimal fractionation, utilizing and illustrating analytical techniques from linear algebra, calculus, linear programming, quadratic and nonlinear programming, robust optimization, and inverse optimization. Its purpose is to help readers understand the mathematics behind the optimal fractionation problem, empowering them to contribute original work to this field.
Numerical Methods for Science and Engineering
Numerical Methods is the go-to textbook for B.Sc and B.Tech students in search of a comprehensive guide to numerical analysis. This self-contained classroom text offers an in-depth exploration of key topics such as errors, difference operators, and interpolation with both equal and unequal intervals. With detailed explanations of methods for solving linear algebraic and transcendental equations, numerical integration, differentiation, and ordinary differential equations.Additional topics covered in this text include central difference interpolation formulas, inverse interpolation, and the Guass-Jacobi and Gauss Seidel methods. Whether you are a student or a professional in the field of numerical analysis, Numerical Methods provides the solid foundation you need to succeed.This book is an essential resource for students seeking to master the principles and techniques of numerical analysis.
Engineering Mathematics and Computing
This book contains select papers presented at the 3rd International Conference on Engineering Mathematics and Computing (ICEMC 2020), held at the Haldia Institute of Technology, Purba Midnapur, West Bengal, India, from 5-7 February 2020. The book discusses new developments and advances in the areas of neural networks, connectionist systems, genetic algorithms, evolutionary computation, artificial intelligence, cellular automata, self-organizing systems, soft computing, fuzzy systems, hybrid intelligent systems, etc. The book, containing 19 chapters, is useful to the researchers, scholars, and practising engineers as well as graduate students of engineering and applied sciences.
The Shape of Knowledge
The Shape of Knowledge is the outcome of a meaningful experience that occurred in 2012. In it are developed the foundations of a new science of philosophy, which promises to provide a solution to the disparity preventing our discourse from progress. Through the language of the Western canon, The Shape of Knowledge exposes the ubiquitous structure that conditions our capacity to reason the truth for our world. Then, through an investigation of the phenomenon of self-reference, in both the processes and products of thought, this structure is shown to necessitate its own existence. Underscoring it all is a principle of complementarity, which arises as the modality of the rationalisation of paradox. Experience is shown to be a relative process of making sense of the nonsensical nature of reality, and the emergence of paraphilosophy is our means of reconciling the present war of opposites--having now served its purpose--with the nondual nature of self-consciousness. Paraphilosophy is not an idea to be believed--it is the idea of the idea, which is our creative spirit. So this work is at root an inquiry into oneself.
Functional Estimation for Density, Regression Models and Processes
Nonparametric kernel estimators apply to the statistical analysis of independent or dependent sequences of random variables and for samples of continuous or discrete processes. The optimization of these procedures is based on the choice of a bandwidth that minimizes an estimation error and the weak convergence of the estimators is proved. This book introduces new mathematical results on statistical methods for the density and regression functions presented in the mathematical literature and for functions defining more complex models such as the models for the intensity of point processes, for the drift and variance of auto-regressive diffusions and the single-index regression models.This second edition presents minimax properties with Lp risks, for a real p larger than one, and optimal convergence results for new kernel estimators of function defining processes: models for multidimensional variables, periodic intensities, estimators of the distribution functions of censored and truncated variables, estimation in frailty models, estimators for time dependent diffusions, for spatial diffusions and for diffusions with stochastic volatility.
Essays in Constructive Mathematics
This collection of essays aims to promote constructive mathematics, not by defining it or formalizing it, but by practicing it. All definitions and proofs are based on finite algorithms, which pave illuminating paths to nontrivial results, primarily in algebra, number theory, and the theory of algebraic curves. The second edition adds a new set of essays that reflect and expand upon the first. The topics covered derive from classic works of nineteenth-century mathematics, among them Galois's theory of algebraic equations, Gauss's theory of binary quadratic forms, and Abel's theorems about integrals of rational differentials on algebraic curves. Other topics include Newton's diagram, the fundamental theorem of algebra, factorization of polynomials over constructive fields, and the spectral theorem for symmetric matrices, all treated using constructive methods in the spirit of Kronecker. In this second edition, the essays of the first edition are augmented with newessays that give deeper and more complete accounts of Galois's theory, points on an algebraic curve, and Abel's theorem. Readers will experience the full power of Galois's approach to solvability by radicals, learn how to construct points on an algebraic curve using Newton's diagram, and appreciate the amazing ideas introduced by Abel in his 1826 Paris memoir on transcendental functions. Mathematical maturity is required of the reader, and some prior knowledge of Galois theory is helpful. But experience with constructive mathematics is not necessary; readers should simply be willing to set aside abstract notions of infinity and explore deep mathematics via explicit constructions.
Core Mathematics for High Schools
Core Mathematics for High School gives a fresh and easy approach for solving difficult high school level mathematics problems. By going through basic concepts in algebra, numbers, trigonometry, kinematics and more, this book allows the student to gradually build on their understanding and confidence in tackling tricky mathematics problems, regardless of the topic. Each chapter has several practice questions that will ensure the student has a clear understanding of key concepts and a variety of approaches to solving problems. Finally, at the back of the book there are step-by-step solutions to all practice questions so that the student can verify their answers.
Math Mammoth Grade 4 Tests and Cumulative Reviews, Canadian Version
Math Mammoth Grade 4 Tests and Cumulative Revisions (Canadian version) contains the end-of-chapter tests, end-of-year test, and cumulative review lessons for Math Mammoth grade 4 curriculum. Note: this book does not contain answers. The answer key book for grade 4 may be purchased separately.
Math Mammoth Grade 4-B Worktext, Canadian Version
Math Mammoth Grade 4-B worktext (Canadian version) is the student worktext for the latter half of the 4th grade math, and part of Math Mammoth complete curriculum for fourth grade. The book starts out with the topic of division. The focus is on learning long division and using division in word problems.In the geometry chapter, students first revise area and perimeter, and then concentrate on the topic of angles. Students measure and draw angles, solve simple angle problems and classify triangles according to their angles. They also study parallel and perpendicular lines.Fractions and decimals are presented last in the school year. These two chapters practise only some of the basic operations with fractions and decimals. The focus is still on conceptual understanding and on building a good foundation towards 5th grade math, where fractions and decimals will be in focus. FeaturesMath Mammoth focuses on conceptual understanding. It explains the "WHY", so your children can understand the math, not just learn "HOW" to do it.The curriculum is mastery-oriented. This means it concentrates fairly long on a topic, delving into its various aspects. This promotes conceptual understanding, as opposed to spiral curricula that often tend to jump from topic to topic too much.There is a strong emphasis on mental math and number sense.It requires very little teacher preparation, which is a big help to most parents.: ) The curriculum has no separate teacher's manual nor is it scripted. The introduction to each chapter has some notes for the teacher concerning the material in the chapter. All the instruction is written directly to the student.
Math Mammoth Grade 4-A Worktext, Canadian Version
Math Mammoth Grade 4-A worktext (Canadian version) is the student worktext for the first half of the 4th grade math, and part of Math Mammoth complete curriculum for fourth grade. This book starts out with a review of addition and subtraction, patterns, and graphs. We illustrate word problems with bar diagrams and study finding missing addends, which teaches algebraic thinking. Children also learn addition and subtraction terminology, the order of operations, and statistical graphs.Next come large numbers-up to millions, and the place value concept. At first the student reviews thousands and some mental math with them. Next are presented numbers up to one million, calculations with them, the concept of place value and comparing. In the end of the chapter we find out more about millions and an introduction to multiples of 10, 100, and 1000. The third chapter is all about multiplication. After briefly reviewing the concept and the times tables, the focus is on learning multi-digit multiplication (multiplication algorithm). The children also learn why it works when they multiply in parts. We also study the order of operations again, touch on proportional reasoning, and do more money and change related word problems.The last chapter in the 4-A worktext is about time, temperature, length, weight, and volume. Students will learn to solve more complex problems using various measuring units and to convert between measuring units. FeaturesMath Mammoth focuses on conceptual understanding. It explains the "WHY", so your children can understand the math, not just learn "HOW" to do it.The curriculum is mastery-oriented. This means it concentrates fairly long on a topic, delving into its various aspects. This promotes conceptual understanding, as opposed to spiral curricula that often tend to jump from topic to topic too much.There is a strong emphasis on mental math and number sense.It requires very little teacher preparation, which is a big help to most parents.: )The curriculum has no separate teacher's manual nor is it scripted. The introduction to each chapter has some notes for the teacher concerning the material in the chapter. All the instruction is written directly to the student.
Math Mammoth Grade 5 Tests and Cumulative Reviews, Canadian Version
Math Mammoth Grade 5 Tests and Cumulative Reviews (Canadian Version ) contains the chapter tests, end-of-year test, and cumulative review worksheets for Math Mammoth grade 5 curriculum. It does not contain answers; the answer key book may be purchased separately.
Potential Functions of Random Walks in ℤ With Infinite Variance
This book studies the potential functions of one-dimensional recurrent random walks on the lattice of integers with step distribution of infinite variance. The central focus is on obtaining reasonably nice estimates of the potential function. These estimates are then applied to various situations, yielding precise asymptotic results on, among other things, hitting probabilities of finite sets, overshoot distributions, Green functions on long finite intervals and the half-line, and absorption probabilities of two-sided exit problems.The potential function of a random walk is a central object in fluctuation theory. If the variance of the step distribution is finite, the potential function has a simple asymptotic form, which enables the theory of recurrent random walks to be described in a unified way with rather explicit formulae. On the other hand, if the variance is infinite, the potential function behaves in a wide range of ways depending on the step distribution, which the asymptotic behaviour of many functionals of the random walk closely reflects.In the case when the step distribution is attracted to a strictly stable law, aspects of the random walk have been intensively studied and remarkable results have been established by many authors. However, these results generally do not involve the potential function, and important questions still need to be answered. In the case where the random walk is relatively stable, or if one tail of the step distribution is negligible in comparison to the other on average, there has been much less work. Some of these unsettled problems have scarcely been addressed in the last half-century. As revealed in this treatise, the potential function often turns out to play a significant role in their resolution. Aimed at advanced graduate students specialising in probability theory, this book will also be of interest to researchers and engineers working with random walks and stochastic systems.
Research in Mathematics of Materials Science
This volume highlights contributions of women mathematicians in the study of complex materials and includes both original research papers and reviews. The featured topics and methods draw on the fields of Calculus of Variations, Partial Differential Equations, Functional Analysis, Differential Geometry and Topology, as well as Numerical Analysis and Mathematical Modelling. Areas of applications include foams, fluid-solid interactions, liquid crystals, shape-memory alloys, magnetic suspensions, failure in solids, plasticity, viscoelasticity, homogenization, crystallization, grain growth, and phase-field models.
Math Mammoth Grade 5-A Worktext, Canadian Version
Math Mammoth Grade 5-A Worktext (Canadian version) is the student book for the first half of grade 5, and part of the Math Mammoth Grade 5 complete curriculum. This is the Canadian version, which means the materials use mostly metric units, it is formatted to letter size, and large numbers are formatted with a space (e.g. 214 000). However, keep in mind that the "Grade 5" refers to the U.S. grade 5, and corresponds roughly to Canadian school system's year 6. The main areas of study in Math Mammoth Grade 5-A worktext are the four operations with whole numbers, place value with large numbers and the judicious use of calculator, problem solving & simple equations, decimal arithmetic (operations with decimals), and graphing. The year starts out with a study of whole numbers and their operations. Students get to review multi-digit multiplication and learn long division with two-digit divisors. We also review divisibility and factors from fourth grade, and study prime factorization.In the second chapter, the focus is on large numbers and using a calculator. This is the first time a calculator is introduced in Math Mammoth complete curriculum-thus far, all calculations have been done mentally or with paper and pencil. I want students to learn to be critical in their use of the calculator and use it with good judgment. Every exercise where calculator use is to be allowed is marked with a little calculator symbol.The third chapter is about equations and problem solving. We study simple equations with the help of a balance and bar models. The main idea is to get students used to the idea of an equation and what it means to solve an equation. Students also do a fair amount of problem solving using the visual bar model. The fourth chapter is about decimals and some of the operations with decimals (the rest will be studied in part 5-B). In this chapter, we focus on place value with decimals, addition and subtraction of decimals and multiplication and division of decimals by whole numbers. In chapter 5 we study graphing in the coordinate grid, line graphs, and the concept of average. Please note this is the student worktext and does not contain the answers. The answer key book is available separately
Symmetric Cycles
This original research monograph concerns various aspects of how (based on the decompositions of vertices of hypercube graphs with respect to their symmetric cycles) the vertex sets of related discrete hypercubes, as well as the power sets of the corresponding ground sets, emerge from rank 2 oriented matroids, from underlying rank 2 systems of linear inequalities, and thus literally from arrangements of straight lines crossing a common point on a piece of paper. It reveals some beautiful and earlier-hidden fragments in the true foundations of discrete mathematics. The central observation made and discussed in the book from various viewpoints consists in that 2t subsets of a finite t-element set Et, which form in a natural way a cyclic structure (well, just t subsets that are the vertices of a path in the cycle suffice), allow us to construct any of 2t subsets of the set Et by means of a more than elementary voting procedure expressed in basic linear algebraic terms. The monograph will be of interest to researchers, students, and readers in the fields of discrete mathematics, theoretical computer science, Boolean function theory, enumerative combinatorics and combinatorics on words, combinatorial optimization, coding theory, and discrete and computational geometry.
Pre-Calculus All-In-One for Dummies
The easy way to understand and retain all the concepts taught in pre-calculus classes Pre-Calculus All-in-One For Dummies is a great resource if you want to do you best in Pre-Calculus. Packed with lessons, examples, and practice problems in the book, plus extra chapter quizzes online, it gives you absolutely everything you need to succeed in pre-calc. Unlike your textbook, this book presents the essential topics clearly and concisely, so you can really understand the stuff you learn in class, score high on your tests (including the AP Pre-Calculus exam!), and get ready to confidently move ahead to upper-level math courses. And if you need a refresher before launching into calculus, look no further--this book has your back. Review what you learned in algebra and geometry, then dig into pre-calculus Master logarithms, exponentials, conic sections, linear equations, and beyond Get easy-to-understand explanations that match the methods your teacher uses Learn clever shortcuts, test-taking tips, and other hacks to make your life easierPre-Calculus All-in-One For Dummies is the must-have resource for students who need to review for exams or just want a little (or a lot of!) extra help understanding what's happening in class.
The Splendors and Miseries of Martingales
Over the past eighty years, martingales have become central in the mathematics of randomness. They appear in the general theory of stochastic processes, in the algorithmic theory of randomness, and in some branches of mathematical statistics. Yet little has been written about the history of this evolution. This book explores some of the territory that the history of the concept of martingales has transformed. The historian of martingales faces an immense task. We can find traces of martingale thinking at the very beginning of probability theory, because this theory was related to gambling, and the evolution of a gambler's holdings as a result of following a particular strategy can always be understood as a martingale. More recently, in the second half of the twentieth century, martingales became important in the theory of stochastic processes at the very same time that stochastic processes were becoming increasingly important in probability, statistics and more generally invarious applied situations. Moreover, a history of martingales, like a history of any other branch of mathematics, must go far beyond an account of mathematical ideas and techniques. It must explore the context in which the evolution of ideas took place: the broader intellectual milieux of the actors, the networks that already existed or were created by the research, even the social and political conditions that favored or hampered the circulation and adoption of certain ideas. This books presents a stroll through this history, in part a guided tour, in part a random walk. First, historical studies on the period from 1920 to 1950 are presented, when martingales emerged as a distinct mathematical concept. Then insights on the period from 1950 into the 1980s are offered, when the concept showed its value in stochastic processes, mathematical statistics, algorithmic randomness and various applications.
Models for Multi-State Survival Data
Multi-state models provide a statistical framework for studying longitudinal data on subjects when focus is on the occurrence of events that the subjects may experience over time. They find application particularly in biostatistics, medicine, and public health.
Lectures on Linear Algebra and its Applications
The present book is based on the extensive lecture notes of the author and contains a concise course on Linear Algebra. The sections begin with an intuitive presentation, aimed at the beginners, and then often include rather non-trivial topics and exercises. This makes the book suitable for introductory as well as advanced courses on Linear Algebra.The first part of the book deals with the general idea of systems of linear equations, matrices and eigenvectors. Linear systems of differential equations are developed carefully and in great detail. The last chapter gives an overview of applications to other areas of Mathematics, like calculus and differential geometry. A large number of exercises with selected solutions make this a valuable textbook for students of the topic as well as lecturers, preparing a course on Linear Algebra.
Introduction to Combinatorial Optimization
Introductory courses in combinatorial optimization are popular at the upper undergraduate/graduate levels in computer science, industrial engineering, and business management/OR, owed to its wide applications in these fields. There are several published textbooks that treat this course and the authors have used many of them in their own teaching experiences. This present text fills a gap and is organized with a stress on methodology and relevant content, providing a step-by-step approach for the student to become proficient in solving combinatorial optimization problems. Applications and problems are considered via recent technology developments including wireless communication, cloud computing, social networks, and machine learning, to name several, and the reader is led to the frontiers of combinatorial optimization. Each chapter presents common problems, such as minimum spanning tree, shortest path, maximum matching, network flow, set-cover, as well as key algorithms, such as greedy algorithm, dynamic programming, augmenting path, and divide-and-conquer. Historical notes, ample exercises in every chapter, strategically placed graphics, and an extensive bibliography are amongst the gems of this textbook.
Numerical Methods for Black-Box Software in Computational Continuum Mechanics
The organization of the material is presented as follows: This introductory chapter I represents a theoretical analysis of the computational algorithms for a numerical solution of the basic equations in continuum mechanics. In this chapter, the general requirements for computational grids, discretization, and iterative methods for black-box software are examined. Finally, a concept of a two-grid algorithm for (de-)coupled solving multidimensional non-linear (initial-)boundary value problems in continuum mechanics (multiphysics simulation) in complex domains is presented. Chapter II contains descriptions of the sequential Robust Multigrid Technique which is developed as a general-purpose solver in black-box codes. This chapter presents the main components of the Robust Multigrid Technique (RMT) used in the two-grid algorithm (Chapter I) to compute the auxiliary (structured) grid correction. This includes the generation of multigrid structures, computation of index mapping, and integral evaluation. Finite volume discretization on the multigrid structures will be explained by studying a 1D linear model problem. In addition, the algorithmic complexity of RMT and black-box optimization of the problem-dependent components of RMT are analysed. Chapter III provides a description of parallel RMT. This chapter introduces parallel RMT-based algorithms for solving the boundary value problems and initial-boundary value problems in unified manner. Section 1 presents a comparative analysis of the parallel RMT and the sequential V-cycle. Sections 2 and 3 present a geometric and an algebraic parallelism of RMT, i.e. parallelization of the smoothing iterations on the coarse and the levels. A parallel multigrid cycle will be considered in Section 4. A parallel RMT for the time-dependent problems is given in Section 5. Finally, the basic properties of parallel RMT will be summarized in Section 6. Theoretical aspects of the used algorithms for solving multidimensional problems are discussed in Chapters IV. This chapter contains the theoretical aspects of the algorithms used for the numerical solving of the resulting system of linear algebraic equations obtained from discrete multidimensional (initial-)boundary value problems.
Combined Measure and Shift Invariance Theory of Time Scales and Applications
This monograph is devoted to developing a theory of combined measure and shift invariance of time scales with the related applications to shift functions and dynamic equations. The study of shift closeness of time scales is significant to investigate the shift functions such as the periodic functions, the almost periodic functions, the almost automorphic functions, and their generalizations with many relevant applications in dynamic equations on arbitrary time scales.First proposed by S. Hilger, the time scale theory-a unified view of continuous and discrete analysis-has been widely used to study various classes of dynamic equations and models in real-world applications. Measure theory based on time scales, in its turn, is of great power in analyzing functions on time scales or hybrid domains. As a new and exciting type of mathematics-and more comprehensive and versatile than the traditional theories of differential and difference equations-, the time scale theory can precisely depict the continuous-discrete hybrid processes and is an optimal way forward for accurate mathematical modeling in applied sciences such as physics, chemical technology, population dynamics, biotechnology, and economics and social sciences.Graduate students and researchers specializing in general dynamic equations on time scales can benefit from this work, fostering interest and further research in the field. It can also serve as reference material for undergraduates interested in dynamic equations on time scales. Prerequisites include familiarity with functional analysis, measure theory, and ordinary differential equations.
Wave Scattering by Small Bodies
The book is a research monograph. An asymptotically exact solution of the many-body scattering problem is given under the assumption a ≪ d ≪ λ, where a is the characteristic size of a small particle, d is the smallest distance between particles and λ is the wavelength in the medium in which the particles are embedded. Scattering of scalar and electromagnetic waves is considered. Heat transfer theory in the medium in which many small bodies are embedded is developed. Quantum-mechanical theory of scattering by many potentials with small support is constructed.On the basis of these theoretical results, important applications are presented. First, a method for creating materials with a desired refraction coefficient is given. Secondly, a method for creating wave-focusing materials is developed. Technological problems to be solved for practical usage of these applied results are discussed.This book contains the contents of the author's earlier monograph, published in 2013. New appendices, based on the author's review papers published after 2013, are added.
Tesla Cybertruck
In addition to producing electric vehicles, tesla has also made significant investments in energy storage and solar power. Tesla's powerwall and powerpack products allow households and businesses to store energy and use it when needed, reducing their dependence on the grid. The solar roof, a revolutionary new product that integrates solar panels into the roof of a home, has also been well received by customers.Tesla has also built a network of superchargers, fast charging stations that allow tesla owners to travel long distances with ease. In this book you will learn: Elon musk fans who are interested in the details of everything elon muskAnyone who wants to learn more about elon musk and his companiesJournalists and podcasters who want to get their facts straightElon detractors who wonder why so many think he is the real iron manInvestors who want to get in on the potential for stock prices going to the moonTesla started off by producing high-performance electric sports cars and quickly expanded into the mass market with the launch of the model s, a luxury electric sedan. The model s quickly became a hit, winning awards and setting new standards for electric vehicles in terms of performance, range, and charging. With the launch of the model x, a luxury electric suv, tesla continued to expand its product lineup and reach new customers.
Reshaping Convex Polyhedra
^ the="" study="" of="" convex="" polyhedra="" in="" ordinary="" space="" is="" a="" central="" piece="" classical="" and="" modern="" geometry="" that="" has="" had="" significant="" impact="" on="" many="" areas="" mathematics="" also="" computer="" science.="" present="" book="" project="" by="" joseph="" o'rourke="" costin="" v簾lcu="" brings="" together="" two="" important="" strands="" subject="" --="" combinatorics="" polyhedra, ="" intrinsic="" underlying="" surface.="" this="" leads="" to="" remarkable="" interplay="" concepts="" come="" life="" wide="" range="" very="" attractive="" topics="" concerning="" polyhedra.="" gets="" message="" across="" thetheory="" although="" with="" roots, ="" still="" much="" alive="" today="" continues="" be="" inspiration="" basis="" lot="" current="" research="" activity.="" work="" presented="" manuscript="" interesting="" applications="" discrete="" computational="" geometry, ="" as="" well="" other="" mathematics.="" treated="" detail="" include="" unfolding="" onto="" surfaces, ="" continuous="" flattening="" convexity="" theory="" minimal="" length="" enclosing="" polygons.="" along="" way, ="" open="" problems="" suitable="" for="" graduate="" students="" are="" raised, ="" both="" aThe focus of this monograph is converting--reshaping--one 3D convex polyhedron to another via an operation the authors call "tailoring." A convex polyhedron is a gem-like shape composed of flat facets, the focus of study since Plato and Euclid. The tailoring operation snips off a corner (a "vertex") of a polyhedron and sutures closed the hole. This is akin to Johannes Kepler's "vertex truncation," but differs in that the hole left by a truncated vertex is filled with new surface, whereas tailoring zips the hole closed. A powerful "gluing" theorem of A.D. Alexandrov from 1950 guarantees that, after closing the hole, the result is a new convex polyhedron. Given two convex polyhedra P, and Q inside P, repeated tailoringallows P to be reshaped to Q. Rescaling any Q to fit inside P, the result is universal: any P can be reshaped to any Q. This is one of the main theorems in Part I, with unexpected theoretical consequences.Part II carries out a systematic study of "vertex-merging," a technique that can be viewed as a type of inverse operation to tailoring. Here the start is P which is gradually enlarged as much as possible, by inserting new surface along slits. In a sense, repeated vertex-merging reshapes P to be closer to planarity. One endpoint of such a process leads to P being cut up and "pasted" inside a cylinder. Then rolling the cylinder on a plane achieves an unfolding of P. The underlying subtext is a question posed by Geoffrey Shephard in 1975 and already implied by drawings by Albrecht D羹rer in the 15th century: whether every convex polyhedron can be unfolded to a planar "net." Toward this end, the authors initiate an exploration of convexity on convex polyhedra, a topic rarely studiedin the literature but with considerable promise for future development.This monograph uncovers new research directions and reveals connections among several, apparently distant, topics in geometry: Alexandrov's Gluing Theorem, shortest paths and cut loci, Cauchy's Arm Lemma, domes, quasigeodesics, convexity, and algorithms throughout. The interplay between these topics and the way the main ideas develop throughout the book could make the "journey" worthwhile for students and researchers in geometry, even if not directly interested in specific topics. Parts of the material will be of interest and accessible even to undergraduates. Although the proof difficulty varies from simple to quite intricate, with some proofs spanning several chapters, many examples and 125 figures help ease the exposition and illustrate the concepts.^>
The Fate of AI Society
Hackers who exploit binary computers become expert cybercriminals. A vicious cycle of undetected attacks by criminal gangs, spies, and foreign enemies fuels skilled staff shortages and escalating costs. Ken Hamer-Hodges, explains why outdated computers cannot stop malware and how democracy is undermined by corrupt dictators. Digital convergence subverts yesterday's binary computer, allowing advanced malware, pervasive cameras, misinformation, AI, and deep-fakes to destroy our culture and civilization. His inspiring examples explain the perfection of computer science that all can grasp. How malware thrives and why operating systems lead to Orwellian dictatorship. To prevent catastrophe computer hardware must catch up with software progress, preventing malware and stopping AI breakout. He explains how to transition to a well engineered, crime free, global cybersociety. How machine code achieves Alonzo Church's vision of networked function abstractions that avoid disaster by accelerating scientific progress. Plotting the path for radical improvement is vital for civilization to flourish as democratically controlled, AI-empowered, global cyber societies. Ken shows how science drives high performance with high reliability for independent applications needed in a world run by superhuman software. Join the author as he explores the fix to computer science. He shows how nations can thrive in a world run by dubious software, governed by superhuman AI, working as functional democracies kept safe from criminals, spies, and dictators.
Lesson Study with Mathematics and Science Preservice Teachers
This insightful volume offers an overview of the fundamentals of lesson student practice in US teacher education as well as examples from math and science teacher educators using lesson study in their local contexts.The number of teacher educators using lesson study with preservice teachers is small but growing. This book is aimed at teacher educators who may want to try lesson study in university contexts without the challenge of translating the practice from the K-12 context on their own. In this volume, lesson study is broadly overviewed, attention is given to its constituent steps, and examples of lesson study in preservice contexts are shared. Given the broad array of teacher education program designs, numerous contingencies guide teacher educators in their implementation of lesson study, given their contextual affordances and limitations. The lesson study descriptions and cases in this book will support teacher educators and scholars across subject specialities and geographic lines, as they seek instructional frameworks to advance their pedagogical goals.
