Modern Mathematical Statistics with Applications
This 3rd edition of Modern Mathematical Statistics with Applications tries to strike a balance between mathematical foundations and statistical practice. The book provides a clear and current exposition of statistical concepts and methodology, including many examples and exercises based on real data gleaned from publicly available sources. Here is a small but representative selection of scenarios for our examples and exercises based on information in recent articles: Use of the "Big Mac index" by the publication The Economist as a humorous way to compare product costs across nationsVisualizing how the concentration of lead levels in cartridges varies for each of five brands of e-cigarettesDescribing the distribution of grip size among surgeons and how it impacts their ability to use a particular brand of surgical staplerEstimating the true average odometer reading of used Porsche Boxsters listed for sale on www.cars.comComparing head acceleration after impact when wearing a football helmet with acceleration without a helmetInvestigating the relationship between body mass index and foot load while running The main focus of the book is on presenting and illustrating methods of inferential statistics used by investigators in a wide variety of disciplines, from actuarial science all the way to zoology. It begins with a chapter on descriptive statistics that immediately exposes the reader to the analysis of real data. The next six chapters develop the probability material that facilitates the transition from simply describing data to drawing formal conclusions based on inferential methodology. Point estimation, the use of statistical intervals, and hypothesis testing are the topics of the first three inferential chapters. The remainder of the book explores the use of these methods in a variety of more complex settings. This edition includes manynew examples and exercises as well as an introduction to the simulation of events and probability distributions. There are more than 1300 exercises in the book, ranging from very straightforward to reasonably challenging. Many sections have been rewritten with the goal of streamlining and providing a more accessible exposition. Output from the most common statistical software packages is included wherever appropriate (a feature absent from virtually all other mathematical statistics textbooks). The authors hope that their enthusiasm for the theory and applicability of statistics to real world problems will encourage students to pursue more training in the discipline.
SAS for Elementary StatisticsGetting Started
SAS for Elementary Statistics: Getting Started provides an introduction to SAS programming for those who have experience with introductory statistical methods. It is also an excellent programming supplement for an introductory statistics course. It is appropriate for the beginning programmer with no prior SAS experience and the researcher who would like to refresh SAS programming skills. These lessons are those the author has found successful in the classroom. Strengths of this book include the following: Examples are easy to follow and understand. Chapters have user-friendly text and objectives. Each chapter has clear objectives with SAS syntax and output results given. Objectives are stated as tasks with detailed step-by-step instructions. Programming notes based on the author's experience occur throughout the book. The author assists the reader in making sense of the error messages in the SAS log. Brief reviews of statistical methods are included in chapters accompanying the corresponding SAS procedures. Easy transition from user terminology to SAS terminology is provided. The ability to select or suppress results using Output Delivery System (ODS) is made simple. Reading and writing to external files are among the most used SAS skills, and these concepts are clearly presented. The IMPORT and EXPORT procedures and ODS are used to accomplish these tasks. Statistical Graphics procedures and SAS/GRAPH can be quite challenging to learn, but these are presented in a very achievable format. Basic graph construction is first introduced then readers learn how to add color, pattern, and other enhancements to graphics images.
Gauge Field Theory Without Groups
Gauge Field theory in Natural Geometric Language addresses the need to clarify basic mathematical concepts at the crossroad between gravitation and quantum physics. Selected mathematical and theoretical topics are exposed within a brief, integrated approach that exploits standard and non-standard notions, as well as recent advances, in a natural geometric language in which the role of structure groups can be regarded as secondary even in the treatment of the gauge fields themselves. In proposing an original bridge between physics and mathematics, this text will appeal not only to mathematicians who wish to understand some of the basic ideas involved in quantum particle physics, but also to physicists who are not satisfied with the usual mathematical presentations of their field.
Advances on Links Between Mathematics and Industry
This book results from the talks presented at the First Conference on Transfer between Mathematics & Industry (CTMI 2019). Its goal is to promote and disseminate the mathematical tools for Statistics & Big Data, MSO (Modeling, Simulation and Optimization) and their industrial applications. In this volume, the reader will find innovative advances in the automotive, energy, railway, logistics, and materials sectors. In addition, Advances CTMI 2019 promotes the opening of new research lines aiming to provide suitable solutions for the industrial and societal challenges. Fostering effective interaction between Academia and Industry is our main purpose with this book. CTMI conferences are one of the main forums where significant advances in industrial mathematics are presented, bringing together outstanding leaders from business, science and academia to promote the use of mathematics for an innovative industry.
Nonstandard Finite Difference Schemes: Methodology and Applications
"This book contains a clear presentation of nonstandard finite difference schemes for the numerical integration of differential equations. A set of rules for constructing nonstandard finite difference schemes is also presented. An important feature of the book is the illustration of the various discrete modeling principles, by their application to a large number of both ordinary and partial differential equations."Mathematical ReviewsThis second edition of Nonstandard Finite Difference Models of Differential Equations provides an update on the progress made in both the theory and application of the NSFD methodology during the past two and a half decades. In addition to discussing details related to the determination of the denominator functions and the nonlocal discrete representations of functions of dependent variables, we include many examples illustrating just how this should be done.Of real value to the reader is the inclusion of a chapter listing many exact difference schemes, and a chapter giving NSFD schemes from the research literature. The book emphasizes the critical roles played by the "principle of dynamic consistency" and the use of sub-equations for the construction of valid NSFD discretizations of differential equations.
Probability Theory
This popular textbook, now in a revised and expanded third edition, presents a comprehensive course in modern probability theory.Probability plays an increasingly important role not only in mathematics, but also in physics, biology, finance and computer science, helping to understand phenomena such as magnetism, genetic diversity and market volatility, and also to construct efficient algorithms. Starting with the very basics, this textbook covers a wide variety of topics in probability, including many not usually found in introductory books, such as: limit theorems for sums of random variables martingales percolation Markov chains and electrical networks construction of stochastic processes Poisson point process and infinite divisibility large deviation principles and statistical physics Brownian motion stochastic integrals and stochastic differential equations. The presentation is self-contained and mathematically rigorous, with the material on probability theory interspersed with chapters on measure theory to better illustrate the power of abstract concepts.This third edition has been carefully extended and includes new features, such as concise summaries at the end of each section and additional questions to encourage self-reflection, as well as updates to the figures and computer simulations. With a wealth of examples and more than 290 exercises, as well as biographical details of key mathematicians, it will be of use to students and researchers in mathematics, statistics, physics, computer science, economics and biology.
Resilience and Stability of Ecological and Social Systems
This monograph, co-authored by three longtime collaborators, aims to promote the interdisciplinary field of mathematical biology by providing accessible new approaches to study natural systems. As there is currently scarce literature on the applications of mathematical modelling for biology research, this book presents a new way of studying interactions at the level of populations, societies, ecosystems, and biomes through open-sourced modeling platforms. It offers an interdisciplinary approach to analyzing natural phenomena-for example, by showing how master equations developed to describe electrical circuits can also describe biological systems mathematically. Ultimately it promotes a method of study based on modelling and mathematical principles, facilitating collaboration between mathematicians, biologists, engineers, and other researchers to enrich knowledge of the world's ecosystems.
Random Forests with R
This book offers an application-oriented guide to random forests: a statistical learning method extensively used in many fields of application, thanks to its excellent predictive performance, but also to its flexibility, which places few restrictions on the nature of the data used. Indeed, random forests can be adapted to both supervised classification problems and regression problems. In addition, they allow us to consider qualitative and quantitative explanatory variables together, without pre-processing. Moreover, they can be used to process standard data for which the number of observations is higher than the number of variables, while also performing very well in the high dimensional case, where the number of variables is quite large in comparison to the number of observations. Consequently, they are now among the preferred methods in the toolbox of statisticians and data scientists. The book is primarily intended for students in academic fields such as statistical education, but also for practitioners in statistics and machine learning. A scientific undergraduate degree is quite sufficient to take full advantage of the concepts, methods, and tools discussed. In terms of computer science skills, little background knowledge is required, though an introduction to the R language is recommended. Random forests are part of the family of tree-based methods; accordingly, after an introductory chapter, Chapter 2 presents CART trees. The next three chapters are devoted to random forests. They focus on their presentation (Chapter 3), on the variable importance tool (Chapter 4), and on the variable selection problem (Chapter 5), respectively. After discussing the concepts and methods, we illustrate their implementation on a running example. Then, various complements are provided before examining additional examples. Throughout the book, each result is given together with the code (in R) that can be used to reproduce it. Thus, the book offers readersessential information and concepts, together with examples and the software tools needed to analyse data using random forests.
Introduction to Applied Optimization
Intended for advanced undergraduate/graduate students as well as scientists and engineers, this textbook presents a multi-disciplinary view of optimization, providing a thorough examination of algorithms, methods, techniques, and tools from diverse areas of optimization. Linear programming, nonlinear programming, discrete optimization, global optimization, optimization under uncertainty, multi-objective optimization, optimal control, and stochastic optimal control are introduced in each self-contained chapter, with exercises, examples, and case studies, the true gems of this text. This third edition includes additional content in each chapter designed to clarify or enhance the exposition, and update methodologies and solutions. A new real-world case study related to sustainability is added in Chapters 2--7. GAMS, AIMMS, and MATLAB(R) files of case studies for Chapters 2, 3, 4, 5, and 7 are freely accessible electronically as extra source materials. A solutions manual is available to instructors who adopt the textbook for their course.From the reviews: This work is definitely a welcome addition to the existing optimization literature, given its emphasis on modeling and solution practice, as well as its 'user-friendly' style of exposition. -- J獺nos D. Pint矇r, European Journal of Operations Research, Vol. 177, 2007 Urmila Diwekar's book on applied optimization is one of the few books on the subject that combines impressive breadth of coverage with delightful readability. In her exposition of concepts and algorithms in the major areas of optimization, she always goes to the heart of the matter and illustrates her explanations with simple diagrams and numerical examples. Graduate and undergraduate students, who constitute part of the target audience, should find this a very useful book. -- Jamshed A. Modi, Interfaces, Vol.36 (1), 2006Optimization is a rich field with a strong history; this book nicely introduces both, moving from very introductory material to challenging techniques toward the end ... Examples range from quite simplistic through realistic difficult scheduling problems. Some examples resurface in different chapter with twists to demonstrate how different techniques are required for differing data and constraints. -- CHOICE, September 2004
Spatial Relationships Between Two Georeferenced Variables
This book offers essential, systematic information on the assessment of the spatial association between two processes from a statistical standpoint. Divided into eight chapters, the book begins with preliminary concepts, mainly concerning spatial statistics. The following seven chapters focus on the methodologies needed to assess the correlation between two or more processes; from theory introduced 35 years ago, to techniques that have only recently been published. Furthermore, each chapter contains a section on R computations to explore how the methodology works with real data. References and a list of exercises are included at the end of each chapter. The assessment of the correlation between two spatial processes has been tackled from several different perspectives in a variety of applications fields. In particular, the problem of testing for the existence of spatial association between two georeferenced variables is relevant for posterior modeling and inference. One evident application in this context is the quantification of the spatial correlation between two images (processes defined on a rectangular grid in a two-dimensional space). From a statistical perspective, this problem can be handled via hypothesis testing, or by using extensions of the correlation coefficient. In an image-processing framework, these extensions can also be used to define similarity indices between images.
Introduction to Statistics in Metrology
This book provides an overview of the application of statistical methods to problems in metrology, with emphasis on modelling measurement processes and quantifying their associated uncertainties. It covers everything from fundamentals to more advanced special topics, each illustrated with case studies from the authors' work in the Nuclear Security Enterprise (NSE). The material provides readers with a solid understanding of how to apply the techniques to metrology studies in a wide variety of contexts. The volume offers particular attention to uncertainty in decision making, design of experiments (DOEx) and curve fitting, along with special topics such as statistical process control (SPC), assessment of binary measurement systems, and new results on sample size selection in metrology studies. The methodologies presented are supported with R script when appropriate, and the code has been made available for readers to use in their own applications. Designed to promote collaboration between statistics and metrology, this book will be of use to practitioners of metrology as well as students and researchers in statistics and engineering disciplines.
Crowd Dynamics, Volume 2
This contributed volume explores innovative research in the modeling, simulation, and control of crowd dynamics. Chapter authors approach the topic from the perspectives of mathematics, physics, engineering, and psychology, providing a comprehensive overview of the work carried out in this challenging interdisciplinary research field. After providing a critical analysis of the current state of the field and an overview of the current research perspectives, chapters focus on three main research areas: pedestrian interactions, crowd control, and multiscale modeling. Specific topics covered in this volume include: crowd dynamics through conservation lawsrecent developments in controlled crowd dynamicsmixed traffic modelinginsights and applications from crowd psychology Crowd Dynamics, Volume 2 is ideal for mathematicians, engineers, physicists, and other researchers working in therapidly growing field of modeling and simulation of human crowds.
Mathematical Modelling in Real Life ProblemsCase Studies from Ecmi-Modelling Weeks
This book is intended to be a useful contribution for the modern teaching of applied mathematics, educating Industrial Mathematicians that will meet the growing demand for such experts. It covers many applications where mathematics play a fundamental role, from biology, telecommunications, medicine, physics, finance and industry. It is presented in such a way that can be useful in Modelation, Simulation and Optimization courses, targeting master and PhD students. Its content is based on many editions from the successful series of Modelling Weeks organized by the European Consortium of Mathematics in Industry (ECMI). Each chapter addresses a particular problem, and is written in a didactic way, providing the description of the problem, the particular way of approaching it and the proposed solution, along with the results obtained.
Space-Time Geometries and Movement in the Brain and in the Arts
Introduction.- Perception and Memory.- Action and Emotion.- Music.- Drawing and Painting.- Performing arts.- Digital Arts.
Nonparametric Statistical Inference
Since its first publication in 1971, Nonparametric Statistical Inference has been widely regarded as the source for learning about nonparametrics. The sixth edition carries on this tradition and incorporates computer solutions based on R.
Michele Sce's Works in Hypercomplex Analysis
This book presents English translations of Michele Sce's most important works, originally written in Italian during the period 1955-1973, on hypercomplex analysis and algebras of hypercomplex numbers. Despite their importance, these works are not very well known in the mathematics community because of the language they were published in. Possibly the most remarkable instance is the so-called Fueter-Sce mapping theorem, which is a cornerstone of modern hypercomplex analysis, and is not yet understood in its full generality.This volume is dedicated to revealing and describing the framework Sce worked in, at an exciting time when the various generalizations of complex analysis in one variable were still in their infancy. In addition to faithfully translating Sce's papers, the authors discuss their significance and explain their connections to contemporary research in hypercomplex analysis. They also discuss many concrete examples that can serve as a basis for further research. The vast majority of the results presented here will be new to readers, allowing them to finally access the original sources with the benefit of comments from fellow mathematicians active in the field of hypercomplex analysis. As such, the book offers not only an important chapter in the history of hypercomplex analysis, but also a roadmap for further exciting research in the field.
Research in Mathematics and Public Policy
This volume features a variety of research projects at the intersection of mathematics and public policy. The topics included here fall in the areas of cybersecurity and climate change, two broad and impactful issues that benefit greatly from mathematical techniques. Each chapter in the book is a mathematical look into a specific research question related to one of these issues, an approach that offers the reader insight into the application of mathematics to important public policy questions. The articles in this volume are papers inspired by a Workshop for Women in Mathematics and Public Policy, held January 22-25, 2019 at the Institute for Pure and Applied Mathematics and the Luskin Center at the University of California, Los Angeles. The workshop was created to promote and develop women at all levels of their careers as researchers in mathematics and public policy. The idea was modeled after other successful Research Collaboration Conferences for Women, where junior and senior women come together at week-long conferences held at mathematics institutes to work on pre-defined research projects. The workshop focused on how mathematics can be used in public policy research and was designed to foster collaborative networks for women to help address the gender gap in mathematics and science.
Equidistribution of Dynamical Systems
We know very little about the time-evolution of many-particle dynamical systems, the subject of our book. Even the 3-body problem has no explicit solution (we cannot solve the corresponding system of differential equations, and computer simulation indicates hopelessly chaotic behaviour). For example, what can we say about the typical time evolution of a large system starting from a stage far from equilibrium? What happens in a realistic time scale? The reader's first reaction is probably: What about the famous Second Law (of thermodynamics)?Unfortunately, there are plenty of notorious mathematical problems surrounding the Second Law. (1) How to rigorously define entropy? How to convert the well known intuitions (like "disorder" and "energy spreading") into precise mathematical definitions? (2) How to express the Second Law in forms of a rigorous mathematical theorem? (3) The Second Law is a "soft" qualitative statement about entropy increase, but does not say anything about the necessary time to reach equilibrium.The object of this book is to answer questions (1)-(2)-(3). We rigorously prove a Time-Quantitative Second Law that works on a realistic time scale. As a by product, we clarify the Loschmidt-paradox and the related reversibility/irreversibility paradox.
Spectral Theory and Mathematical Physics
Commutator methods for N-body Schr繹dinger operators.- Resolvent estimates and resonance free regions for Schr繹dinger operators with matrix-valued potentials.- One-dimensional discrete Anderson model in a decaying random potential: From a.c. spectrum to dynamical localization.- On non-selfadjoint operators with finite discrete spectrum.- Pseudo-differential perturbations of the Landau Hamiltonian.- Semiclassical surface wave tomography of isotropic media.- Persistence of point spectrum for perturbations of one-dimensional operators with discrete spectra.- Resonances for a system of Schr繹dinger operators above an energy-level crossing.- Nonexistence result for wave operators in massive relativistic system.- Quantized calculus for perturbed massive Dirac operator on noncommutative Euclidian space.- On the explicit semiclassical limiting eigenvalue (resonance) distribution for the Zeeman (Stark) hydrogen atom Hamiltonian.- Negative spectrum of the Robin Laplacian.- On some integral operators appearing in scattering theory, and their resolutions.- The strong Scott conjecture: The density of heavy atoms close to the nucleous.
Learning Microeconometrics with R
This book provides an introduction to the field of microeconometrics through the use of R. The focus is on applying current learning from the field to real world problems. It uses R to both teach the concepts of the field and show the reader how the techniques can be used.
Progress in Mathematical Fluid Dynamics
- A Heuristic Approach to Convex Integration for the Euler Equations. - Fluid-Structure Interaction with Incompressible Fluids. - Regularity and Inviscid Limits in Hydrodynamic Models. - Small Scale Creation in Active Scalars.
Mathematik F羹r Das Erste Semester
Zum Anfang des Studiums sind Studierende der Ingenieurwissenschaften haupts瓣chlich mit Grundlagen besch瓣ftigt, zu denen wesentlich die Mathematik geh繹rt. Hier sind insbesondere die Analysis (in einer Variablen) und Lineare Algebra zu nennen, die zu oft eine gro?e H羹rde darstellen. Mit unserem Buch wollen wir den Weg ebnen, indem wir Sie ausf羹hrlich - und ohne Umwege - mit dem genannten Stoff vertraut machen. In einem verbindlichen, aber dennoch entspannten Stil bringen wir Ihnen die wichtigen Methoden und Begriffe bei. Besonderheiten: Zahlreiche Bilder und Beispiele. Viele begleitende Aufgaben mit vollst瓣ndigen L繹sungen. Klausuraufgaben mit kompletten L繹sungen. Motivation und Verst瓣ndnisfragen f羹r jedes Kapitel. "Erste-Hilfe-Kurs" f羹r Pr羹fungen.F羹r die 2. Auflage wurden viele Stellen didaktisch verbessert und korrigiert. Durch zus瓣tzliche Erkl瓣rungen, Grafiken und das Eingehen auf Leserkommentare ist das Buch nun noch verst瓣ndlicher und hervorragend als freundlicher Begleiter f羹r Ihr erstes mathematisches Semester geeignet.
Shape Optimization Problems
This book provides theories on non-parametric shape optimization problems, systematically keeping in mind readers with an engineering background. Non-parametric shape optimization problems are defined as problems of finding the shapes of domains in which boundary value problems of partial differential equations are defined. In these problems, optimum shapes are obtained from an arbitrary form without any geometrical parameters previously assigned. In particular, problems in which the optimum shape is sought by making a hole in domain are called topology optimization problems. Moreover, a problem in which the optimum shape is obtained based on domain variation is referred to as a shape optimization problem of domain variation type, or a shape optimization problem in a limited sense. Software has been developed to solve these problems, and it is being used to seek practical optimum shapes. However, there are no books explaining such theories beginning with their foundations.The structure of the book is shown in the Preface. The theorems are built up using mathematical results. Therefore, a mathematical style is introduced, consisting of definitions and theorems to summarize the key points. This method of expression is advanced as provable facts are clearly shown. If something to be investigated is contained in the framework of mathematics, setting up a theory using theorems prepared by great mathematicians is thought to be an extremely effective approach. However, mathematics attempts to heighten the level of abstraction in order to understand many things in a unified fashion. This characteristic may baffle readers with an engineering background. Hence in this book, an attempt has been made to provide explanations in engineering terms, with examples from mechanics, after accurately denoting the provable facts using definitions and theorems.
Network Science
This book provides an overview of network science from the perspective of diverse academic fields, offering insights into the various research areas within network science. The authoritative contributions on statistical network analysis, mathematical network science, genetic networks, Bayesian networks, network visualisation, and systemic risk in networks explore the main questions in the respective fields: What has been achieved to date? What are the research challenges and obstacles? What are the possible interconnections with other fields? And how can cross-fertilization between these fields be promoted? Network science comprises numerous scientific disciplines, including computer science, economics, mathematics, statistics, social sciences, bioinformatics, and medicine, among many others. These diverse research areas require and use different data-analytic and numerical methods as well as different theoretical approaches. Nevertheless, they all examine and describe interdependencies, associations, and relationships of entities in different kinds of networks. The book is intended for researchers as well as interested readers working in network science who want to learn more about the field - beyond their own research or work niche. Presenting network science from different perspectives without going into too much technical detail, it allows readers to gain an overview without having to be a specialist in any or all of these disciplines.
High Dimensional Probability VIII
This volume collects selected papers from the 8th High Dimensional Probability meeting held at Casa Matem獺tica Oaxaca (CMO), Mexico. High Dimensional Probability (HDP) is an area of mathematics that includes the study of probability distributions and limit theorems in infinite-dimensional spaces such as Hilbert spaces and Banach spaces. The most remarkable feature of this area is that it has resulted in the creation of powerful new tools and perspectives, whose range of application has led to interactions with other subfields of mathematics, statistics, and computer science. These include random matrices, nonparametric statistics, empirical processes, statistical learning theory, concentration of measure phenomena, strong and weak approximations, functional estimation, combinatorial optimization, random graphs, information theory and convex geometry. The contributions in this volume show that HDP theory continues to thrive and develop new tools, methods, techniques and perspectives to analyze random phenomena.
Approximation Techniques for Engineers
This book's specific goal is to provide a working knowledge of the various approximation techniques for engineering practice. Therefore, many sections are illuminated by either a computational example or an algorithm to enhance understanding and provide a template for the reader's own application of the technique. The more advanced techniques ar
Gibbs Semigroups
This book focuses on the theory of the Gibbs semigroups, which originated in the 1970s and was motivated by the study of strongly continuous operator semigroups with values in the trace-class ideal. The book offers an up-to-date, exhaustive overview of the advances achieved in this theory after half a century of development. It begins with a tutorial introduction to the necessary background material, before presenting the Gibbs semigroups and then providing detailed and systematic information on the Trotter-Kato product formulae in the trace-norm topology. In addition to reviewing the state-of-art concerning the Trotter-Kato product formulae, the book extends the scope of exposition from the trace-class ideal to other ideals. Here, special attention is paid to results on semigroups in symmetrically normed ideals and in the Dixmier ideal. By examining the progress made in Gibbs semigroup theory and in extensions of the Trotter-Kato product formulae to symmetrically normed and Dixmier ideals, the book shares timely and valuable insights for readers interested in pursuing these subjects further. As such, it will appeal to researchers, undergraduate and graduate students in mathematics and mathematical physics.
Generators of Markov Chains
Elementary treatments of Markov chains, especially those devoted to discrete-time and finite state-space theory, leave the impression that everything is smooth and easy to understand. This exposition of the works of Kolmogorov, Feller, Chung, Kato, and other mathematical luminaries, which focuses on time-continuous chains but is not so far from being elementary itself, reminds us again that the impression is false: an infinite, but denumerable, state-space is where the fun begins. If you have not heard of Blackwell's example (in which all states are instantaneous), do not understand what the minimal process is, or do not know what happens after explosion, dive right in. But beware lest you are enchanted: 'There are more spells than your commonplace magicians ever dreamed of.'
Mathematical Labyrinths. Pathfinding
Mathematical Labyrinths. Pathfinding provides an overview of various non-standard problems and the approaches to their solutions. The essential idea is a framework laid upon the reader on how to solve nonconventional problems - particularly in the realm of mathematics and logic. It goes over the key steps in approaching a difficult problem, contemplating a plan for its solution, and discusses set of mental models to solve math problems.The book is not a routine set of problems. It is rather an entertaining and educational journey into the fascinating world of mathematical reasoning and logic. It is about finding the best path to a solution depending on the information given, asking and answering the right questions, analyzing and comparing alternative approaches to problem solving, searching for generalizations and inventing new problems. It also considers as an important pedagogical tool playing mathematical and logical games, deciphering mathematical sophisms, and interpreting mathematical paradoxes.It is suitable for mathematically talented and curious students in the age range 10-20. There are many 'Eureka'- type, out of the ordinary, fun problems that require bright idea and insight. These intriguing and thought-provoking brainteasers and logic puzzles should be enjoyable by the audience of almost any age group, from 6-year-old children to 80-year-old and older adults.
Singular Spectrum Analysis for Time Series
This book gives an overview of singular spectrum analysis (SSA). SSA is a technique of time series analysis and forecasting combining elements of classical time series analysis, multivariate statistics, multivariate geometry, dynamical systems and signal processing. SSA is multi-purpose and naturally combines both model-free and parametric techniques, which makes it a very special and attractive methodology for solving a wide range of problems arising in diverse areas. Rapidly increasing number of novel applications of SSA is a consequence of the new fundamental research on SSA and the recent progress in computing and software engineering which made it possible to use SSA for very complicated tasks that were unthinkable twenty years ago. In this book, the methodology of SSA is concisely but at the same time comprehensively explained by two prominent statisticians with huge experience in SSA. The book offers a valuable resource for a very wide readership, including professional statisticians, specialists in signal and image processing, as well as specialists in numerous applied disciplines interested in using statistical methods for time series analysis, forecasting, signal and image processing. The second edition of the book contains many updates and some new material including a thorough discussion on the place of SSA among other methods and new sections on multivariate and multidimensional extensions of SSA.
Mathematical Labyrinths. Pathfinding
Mathematical Labyrinths. Pathfinding provides an overview of various non-standard problems and the approaches to their solutions. The essential idea is a framework laid upon the reader on how to solve nonconventional problems - particularly in the realm of mathematics and logic. It goes over the key steps in approaching a difficult problem, contemplating a plan for its solution, and discusses set of mental models to solve math problems.The book is not a routine set of problems. It is rather an entertaining and educational journey into the fascinating world of mathematical reasoning and logic. It is about finding the best path to a solution depending on the information given, asking and answering the right questions, analyzing and comparing alternative approaches to problem solving, searching for generalizations and inventing new problems. It also considers as an important pedagogical tool playing mathematical and logical games, deciphering mathematical sophisms, and interpreting mathematical paradoxes.It is suitable for mathematically talented and curious students in the age range 10-20. There are many 'Eureka'- type, out of the ordinary, fun problems that require bright idea and insight. These intriguing and thought-provoking brainteasers and logic puzzles should be enjoyable by the audience of almost any age group, from 6-year-old children to 80-year-old and older adults.
Frontiers in Analysis and Probability
The volume presents extensive research devoted to a broad spectrum of mathematical analysis and probability theory. Subjects discussed in this Work are those treated in the so-called Strasbourg-Z羹rich Meetings. These meetings occur twice yearly in each of the cities, Strasbourg and Z羹rich, venues of vibrant mathematical communication and worldwide gatherings. The topical scope of the book includes the study of monochromatic random waves defined for general Riemannian manifolds, notions of entropy related to a compact manifold of negative curvature, interacting electrons in a random background, lp-cohomology (in degree one) of a graph and its connections with other topics, limit operators for circular ensembles, polyharmonic functions for finite graphs and Markov chains, the ETH-Approach to Quantum Mechanics, 2-dimensional quantum Yang-Mills theory, Gibbs measures of nonlinear Schr繹dinger equations, interfaces in spectral asymptotics and nodal sets. Contributionsin this Work are composed by experts from the international community, who have presented the state-of-the-art research in the corresponding problems treated. This volume is expected to be a valuable resource to both graduate students and research mathematicians working in analysis, probability as well as their interconnections and applications.
The Generalized Fourier Series Method
The book presents and explains a general, efficient, and elegant method of approximate solution for boundary value problems for an elliptic system of partial differential equations arising in elasticity theory. The methodology for constructing generalized Fourier series based on the structure of the problem is shown in detail, and all the attending mathematical properties are derived with full rigor. A numerical scheme directly related to the series method is developed and employed to compute approximate solutions, illustrated by a variety of examples.
Landscapes of Time-Frequency Analysis
Radon transform: dual pairs and irreducible representations.- Data approximation with time-frequency invariant systems.- The Shearlet transform and Lizorkin spaces.- Time-frequency localization operators: state of the art.- Time-frequency analysis: what we know and what we don't.- Some notes about distribution frame multipliers.- Generalized Anti-Wick quantum states.- Signal analysis and quantum formalism. Quantizations with no Planck constant.- Quantization methods in ocular fundus imaging: analysis of retinal microvasculature.- A time-frequency analysis perspective on Feynman path integrals.
Shallow Water Hydraulics
This book presents the theory and computation of open channel flows, using detailed analytical, numerical and experimental results. The fundamental equations of open channel flows are derived by means of a rigorous vertical integration of the RANS equations for turbulent flow. In turn, the hydrostatic pressure hypothesis, which forms the core of many shallow water hydraulic models, is scrutinized by analyzing its underlying assumptions. The book's main focus is on one-dimensional models, including detailed treatments of unsteady and steady flows. The use of modern shock capturing finite difference and finite volume methods is described in detail, and the quality of solutions is carefully assessed on the basis of analytical and experimental results.The book's unique features include: - Rigorous derivation of the hydrostatic-based shallow water hydraulic models- Detailed treatment of steady open channel flows, including the computation of transcritical flow profiles- General analysis of gate maneuvers as the solution of a Riemann problem- Presents modern shock capturing finite volume methods for the computation of unsteady free surface flows- Introduces readers to movable bed and sediment transport in shallow water models- Includes numerical solutions of shallow water hydraulic models for non-hydrostatic steady and unsteady free surface flowsThis book is suitable for both undergraduate and graduate level students, given that the theory and numerical methods are progressively introduced starting with the basics. As supporting material, a collection of source codes written in Visual Basic and inserted as macros in Microsoft Excel(R) is available. The theory is implemented step-by-step in the codes, and the resulting programs are used throughout the book to produce the respective solutions.
Elon Lima - Selected Papers
This book contains all research papers published by the distinguished Brazilian mathematician Elon Lima. It includes the papers from his PhD thesis on homotopy theory, which are hard to find elsewhere. Elon Lima wrote more than 40 books in the field of topology and dynamical systems. He was a profound mathematician with a genuine vocation to teach and write mathematics.
Nonlinear Optimization
Introduction.- Unconstrained Optimization.- Applications of Nonlinear Programming.- Optimality Conditions and Duality Theory.- General Solution Methods for Constrained Optimization.- Methods for Linearly Constrained Problems.- Methods for Nonlinearly Constrained Problems.- Geometric Programming.
Fundamentals of Data Analytics
This book introduces the basic methodologies for successful data analytics. Matrix optimization and approximation are explained in detail and extensively applied to dimensionality reduction by principal component analysis and multidimensional scaling. Diffusion maps and spectral clustering are derived as powerful tools. The methodological overlap between data science and machine learning is emphasized by demonstrating how data science is used for classification as well as supervised and unsupervised learning.
World Women in Mathematics 2018
The first World Meeting for Women in Mathematics - (WM)簡 - was a satellite event of the International Congress of Mathematicians (ICM) 2018 in Rio de Janeiro. With a focus on Latin America, the first (WM)簡 brought together mathematicians from all over the world to celebrate women mathematicians, and also to reflect on gender issues in mathematics, challenges, initiatives, and perspectives for the future. Its activities were complemented by a panel discussion organized by the Committee for Women in Mathematics (CWM) of the International Mathematical Union (IMU) inside the ICM 2018 entitled "The gender gap in mathematical and natural sciences from a historical perspective".This historical proceedings book, organized by CWM in coordination with the Association for Women in Mathematics, records the first (WM)簡 and the CWM panel discussion at ICM 2018. The first part of the volume includes a report of activities with pictures of the first (WM)簡 and a tribute to Maryam Mirzakhani, the first woman to be awarded the Fields medal. It also comprises survey research papers from invited lecturers, which provide panoramic views of different fields in pure and applied mathematics. The second part of the book contains articles from the panelists of the CWM panel discussion, which consider the historical context of the gender gap in mathematics. It includes an analysis of women lecturers in the ICM since its inception. This book is dedicated to the memory of Maryam Mirzakhani.
Cut the Knot
He who untied the Gordian knot would rule all of Asia So goes the legend of the tricky knot of Gordius, king of Phrygia.Many had tried; many had failed, but Alexander the Great simplycut the knot with his sword. He went on to conquer most of Asia, eventually reaching as far east as Northern India.Cut the Knot is a book of probability riddles curated to challenge the mind andexpand mathematical and logical thinking skills. First housed on cut-the-knot.org, these puzzles and their solutions represent the efforts of great minds around theworld. Follow along as Alexander Bogomolny presents these selected riddles bytopical progression. Try them for yourself before reading their solutions. Just like itwas for Alexander the Great, the non-trivial, unexpected solution might be exactlythe one you need.
Atomicity Through Fractal Measure Theory
This book presents an exhaustive study of atomicity from a mathematics perspective in the framework of multi-valued non-additive measure theory. Applications to quantum physics and, more generally, to the fractal theory of the motion, are highlighted. The study details the atomicity problem through key concepts, such as the atom/pseudoatom, atomic/nonatomic measures, and different types of non-additive set-valued multifunctions. Additionally, applications of these concepts are brought to light in the study of the dynamics of complex systems.The first chapter prepares the basics for the next chapters. In the last chapter, applications of atomicity in quantum physics are developed and new concepts, such as the fractal atom are introduced. The mathematical perspective is presented first and the discussion moves on to connect measure theory and quantum physics through quantum measure theory. New avenues of research, such as fractal/multifractal measure theory with potentialapplications in life sciences, are opened.
Complex Analysis with Applications to Number Theory
The book discusses major topics in complex analysis with applications to number theory. This book is intended as a text for graduate students of mathematics and undergraduate students of engineering, as well as to researchers in complex analysis and number theory. This theory is a prerequisite for the study of many areas of mathematics, including the theory of several finitely and infinitely many complex variables, hyperbolic geometry, two and three manifolds and number theory. In additional to solved examples and problems, the book covers most of the topics of current interest, such as Cauchy theorems, Picard's theorems, Riemann-Zeta function, Dirichlet theorem, gamma function and harmonic functions.
Multi-Variable Calculus
The book deals with functions of many variables: differentiation and integration, extrema with a number of digressions to related subjects such as curves, surfaces and Morse theory. The background needed for understanding the examples and how to compute in Mathematica(R) will also be discussed.
Mathematical Economics
Chapter 1. Logic and Proof.- Chapter 2. Sets and Relations.- Chapter 3. Basic Topology.- Chapter 4. Linear Algebra.- Chapter 5. Vector Calculus.- Chapter 6. Convex Analysis.- Chapter 7. Optimization.- Chapter 8. Probability.- Chapter 9. Dynamic Modeling.
Basics of Probability and Stochastic Processes
Combinatorial Analysis.- Basic Concepts in Probability.- Conditional Probability, Bayes's Formula, Independent Events.- Introduction to Random Variables.- Discrete Random Variables.- Continuous Random Variables.- Other Selected Topics in Basic Probability.- A Brief Introduction to Stochastic Processes.- A Brief Introduction to Point Process, Counting Process, Renewal Process, Regenerative Process, Poisson Process.- Poisson Process.- Renewal Process.- An Introduction to Markov Chains.- Special Discrete-Time Markov Chains.- Continuous-Time Markov Chains.- An Introduction to Queueing Models.- Introduction to Brownian Motion.- Basics of Martingales.- Basics of Reliability Theory.
Analytical and Stochastic Modelling Techniques and Applications
This book constitutes the refereed proceedings of the 25th International Conference on Analytical and Stochastic Modelling Techniques and Applications, ASMTA 2019, held in Moscow, Russia, in October 2019. Methods of analytical and stochastic modelling are widely used in engineering to assess and design various complex systems, like computer and communication networks, and manufacturing systems. The 13 full papers presented in this book were carefully reviewed and selected from 22 submissions. The papers detail a diverse range of analysis techniques, including Markov processes, queueing theoretical results, reliability of stochastic systems, stochastic network calculus, and wide variety of applications.
Dispersive Shallow Water Waves
This monograph presents cutting-edge research on dispersive wave modelling, and the numerical methods used to simulate the propagation and generation of long surface water waves. Including both an overview of existing dispersive models, as well as recent breakthroughs, the authors maintain an ideal balance between theory and applications. From modelling tsunami waves to smaller scale coastal processes, this book will be an indispensable resource for those looking to be brought up-to-date in this active area of scientific research.Beginning with an introduction to various dispersive long wave models on the flat space, the authors establish a foundation on which readers can confidently approach more advanced mathematical models and numerical techniques. The first two chapters of the book cover modelling and numerical simulation over globally flat spaces, including adaptive moving grid methods along with the operator splitting approach, which was historically proposed at the Institute of Computational Technologies at Novosibirsk. Later chapters build on this to explore high-end mathematical modelling of the fluid flow over deformed and rotating spheres using the operator splitting approach. The appendices that follow further elaborate by providing valuable insight into long wave models based on the potential flow assumption, and modified intermediate weakly nonlinear weakly dispersive equations.Dispersive Shallow Water Waves will be a valuable resource for researchers studying theoretical or applied oceanography, nonlinear waves as well as those more broadly interested in free surface flow dynamics.
Mathematische Lehr-Lernprozesse Im Kontext Digitaler Medien
Der vorliegenden Band zeichnet sich durch eine gro?e Bandbreite an Zug瓣ngen zum Themenkomplex "Digitales im Mathematikunterricht" aus; diese reicht von Theorieartikeln zur Fundierung des Einsatzes digitaler Werkzeuge und Medien in der Mathematikdidaktik, 羹ber Anwendungsperspektiven f羹r die Mathematiklehrerinnen- und Mathematiklehrerausbildung, bis hin zu aussagekr瓣ftigen Praxisberichten aus der Schule. Das nun entstandene Werk ist Ausdruck einer lebendigen multiperspektivischen Auseinandersetzung mit dem Gegenstand der digitalen Bildung in der Mathematikdidaktik. Diese wird im vorliegenden Buch getragen von einer positiven Grundeinstellung zu den M繹glichkeiten, die digitale Werkzeuge und Medien f羹r den Mathematikunterricht entfalten k繹nnen, werden aber in kritischer Abw瓣gung wissenschaftlich betrachtet um auszuloten wann, wo und wie ein Einsatz einen fachinhaltlichen und fachdidaktischen Mehrwert entfalten kann.
