New Advancements in Pure and Applied Mathematics via Fractals and Fractional Calculus
This book focuses on exploring new developments in both pure and applied mathematics as a result of fractional behaviour. It covers the range of ongoing activities in the context of fractional calculus by offering alternate viewpoints, workable solutions, new derivatives, and methods to solve real-world problems.It is impossible to deny that fractional behaviour exists in nature. Any phenomenon that has a pulse, rhythm, or pattern appears to be a fractal. The 17 papers that were published and are part of this volume provide credence to that claim. A variety of topics illustrate the use of fractional calculus in a range of disciplines and offer sufficient coverage to pique every reader's attention.
Note Taking Activities in E-Learning Environments
The main focus of this book is presenting practical procedures for improving learning effectiveness using note taking activities during e-learning courses. Although presentation of e-learning activities recently has been spreading to various education sectors, some practical problems have been discussed such as evaluation of learning performance and encouragement of students. The authors introduce note taking activity as a conventional learning tool in order to promote individual learning activity and learning efficacy. The effectiveness of note taking has been measured in practical teaching in a Japanese university using techniques of learning analytics, and the results are shown here. The relationships between note taking activity and students' characteristics, the possibility of predicting the final learning performance using metrics of students' note taking, and the effectiveness for individual emotional learning factors are evaluated. Some differences between blended learning and fully online learning courses are also discussed. The authors provide novel analytical procedures and ideas to manage e-learning courses. In particular, the assessment of note taking activity may help to track individual learning progress and to encourage learning motivation.
Recent Advances in Mechanics and Fluid-Structure Interaction with Applications
This volume examines current research in mechanics and its applications to various disciplines, with a particular focus on fluid-structure interaction (FSI). The topics have been chosen in commemoration of Dr. Bong Jae Chung and with respect to his wide range of research interests. This volume stands apart because of this diversity of interests, featuring an interdisciplinary and in-depth analysis of FSI that is difficult to find conveniently collected elsewhere in the literature. Contributors include mathematicians, physicists, mechanical and biomechanical engineers, and psychologists. This volume is structured into four thematic areas in order to increase its accessibility: theory, computations, experiments, and applications. Recent Advances in Mechanics and Fluid-Structure Interaction with Applications will appeal to established researchers as well as postdocs and graduate students interested in this active area of research.
Statistical Foundations, Reasoning and Inference
This textbook provides a comprehensive introduction to statistical principles, concepts and methods that are essential in modern statistics and data science. The topics covered include likelihood-based inference, Bayesian statistics, regression, statistical tests and the quantification of uncertainty. Moreover, the book addresses statistical ideas that are useful in modern data analytics, including bootstrapping, modeling of multivariate distributions, missing data analysis, causality as well as principles of experimental design. The textbook includes sufficient material for a two-semester course and is intended for master's students in data science, statistics and computer science with a rudimentary grasp of probability theory. It will also be useful for data science practitioners who want to strengthen their statistics skills.
Modeling, Dynamics, Optimization and Bioeconomics IV
This book, following the three published volumes of the book, provides the main purpose to collect research papers and review papers to provide an overview of the main issues, results, and open questions in the cutting-edge research on the fields of modeling, optimization, and dynamics and their applications to biology, economy, energy, industry, physics, psychology and finance. Assuming the scientific relevance of the presenting innovative applications as well as merging issues in these areas, the purpose of this book is to collect papers of the world experts in mathematics, economics, and other applied sciences that is seminal to the future research developments. The majority of the papers presented in this book is authored by the participants in The Joint Meeting 6th International Conference on Dynamics, Games, and Science - DGSVI - JOLATE and in the 21st ICABR Conference. The scientific scope of the conferences is focused on the fields ofmodeling, optimization, and dynamics and their applications to biology, economy, energy, industry, physics, psychology, and finance. Assuming the scientific relevance of the presenting innovative applications as well as merging issues in these areas, the purpose of the conference is to bring together some of the world experts in mathematics, economics, and other applied sciences that reinforce ongoing projects and establish future works and collaborations.
Belief Functions: Theory and Applications
This book constitutes the refereed proceedings of the 7th International Conference on Belief Functions, BELIEF 2022, held in Paris, France, in October 2022.The theory of belief functions is now well established as a general framework for reasoning with uncertainty, and has well-understood connections to other frameworks such as probability, possibility, and imprecise probability theories. It has been applied in diverse areas such as machine learning, information fusion, and pattern recognition. The 29 full papers presented in this book were carefully selected and reviewed from 31 submissions. The papers cover a wide range on theoretical aspects on mathematical foundations, statistical inference as well as on applications in various areas including classification, clustering, data fusion, image processing, and much more.
Topological Dynamics of Enveloping Semigroups
This book introduces the theory of enveloping semigroups--an important tool in the field of topological dynamics--introduced by Robert Ellis. The book deals with the basic theory of topological dynamics and touches on the advanced concepts of the dynamics of induced systems and their enveloping semigroups. All the chapters in the book are well organized and systematically dealing with introductory topics through advanced research topics. The basic concepts give the motivation to begin with, then the theory, and finally the new research-oriented topics. The results are presented with detailed proof, plenty of examples and several open questions are put forward to motivate for future research. Some of the results, related to the enveloping semigroup, are new to the existing literature. The enveloping semigroups of the induced systems is considered for the first time in the literature, and some new results are obtained. The book has a research-oriented flavour in the field of topological dynamics.
Applied Mathematics and Fractional Calculus
In the last three decades, fractional calculus has broken into the field of mathematical analysis, both at the theoretical level and at the level of its applications. In essence, the fractional calculus theory is a mathematical analysis tool applied to the study of integrals and derivatives of arbitrary order, which unifies and generalizes the classical notions of differentiation and integration. These fractional and derivative integrals, which until not many years ago had been used in purely mathematical contexts, have been revealed as instruments with great potential to model problems in various scientific fields, such as: fluid mechanics, viscoelasticity, physics, biology, chemistry, dynamical systems, signal processing or entropy theory. Since the differential and integral operators of fractional order are nonlinear operators, fractional calculus theory provides a tool for modeling physical processes, which in many cases is more useful than classical formulations. This is why the application of fractional calculus theory has become a focus of international academic research. This Special Issue "Applied Mathematics and Fractional Calculus" has published excellent research studies in the field of applied mathematics and fractional calculus, authored by many well-known mathematicians and scientists from diverse countries worldwide such as China, USA, Canada, Germany, Mexico, Spain, Poland, Portugal, Iran, Tunisia, South Africa, Albania, Thailand, Iraq, Egypt, Italy, India, Russia, Pakistan, Taiwan, Korea, Turkey, and Saudi Arabia.
Discrete Choice Analysis with R
This book is designed as a gentle introduction to the fascinating field of choice modeling and its practical implementation using the R language. Discrete choice analysis is a family of methods useful to study individual decision-making. With strong theoretical foundations in consumer behavior, discrete choice models are used in the analysis of health policy, transportation systems, marketing, economics, public policy, political science, urban planning, and criminology, to mention just a few fields of application. The book does not assume prior knowledge of discrete choice analysis or R, but instead strives to introduce both in an intuitive way, starting from simple concepts and progressing to more sophisticated ideas. Loaded with a wealth of examples and code, the book covers the fundamentals of data and analysis in a progressive way. Readers begin with simple data operations and the underlying theory of choice analysis and conclude by working with sophisticated models including latent class logit models, mixed logit models, and ordinal logit models with taste heterogeneity. Data visualization is emphasized to explore both the input data as well as the results of models. This book should be of interest to graduate students, faculty, and researchers conducting empirical work using individual level choice data who are approaching the field of discrete choice analysis for the first time. In addition, it should interest more advanced modelers wishing to learn about the potential of R for discrete choice analysis. By embedding the treatment of choice modeling within the R ecosystem, readers benefit from learning about the larger R family of packages for data exploration, analysis, and visualization.
Nonparametric Statistics with Applications to Science and Engineering with R
NONPARAMETRIC STATISTICS WITH APPLICATIONS TO SCIENCE AND ENGINEERING WITH R Introduction to the methods and techniques of traditional and modern nonparametric statistics, incorporating R code Nonparametric Statistics with Applications to Science and Engineering with R presents modern nonparametric statistics from a practical point of view, with the newly revised edition including custom R functions implementing nonparametric methods to explain how to compute them and make them more comprehensible. Relevant built-in functions and packages on CRAN are also provided with a sample code. R codes in the new edition not only enable readers to perform nonparametric analysis easily, but also to visualize and explore data using R's powerful graphic systems, such as ggplot2 package and R base graphic system. The new edition includes useful tables at the end of each chapter that help the reader find data sets, files, functions, and packages that are used and relevant to the respective chapter. New examples and exercises that enable readers to gain a deeper insight into nonparametric statistics and increase their comprehension are also included. Some of the sample topics discussed in Nonparametric Statistics with Applications to Science and Engineering with R include Basics of probability, statistics, Bayesian statistics, order statistics, Kolmogorov-Smirnov test statistics, rank tests, and designed experiments Categorical data, estimating distribution functions, density estimation, least squares regression, curve fitting techniques, wavelets, and bootstrap sampling EM algorithms, statistical learning, nonparametric Bayes, WinBUGS, properties of ranks, and Spearman coefficient of rank correlation Chi-square and goodness-of-fit, contingency tables, Fisher exact test, MC Nemar test, Cochran's test, Mantel-Haenszel test, and Empirical Likelihood Nonparametric Statistics with Applications to Science and Engineering with R is a highly valuable resource for graduate students in engineering and the physical and mathematical sciences, as well as researchers who need a more comprehensive, but succinct understanding of modern nonparametric statistical methods.
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.
Optimization in Banach Spaces
The book is devoted to the study of constrained minimization problems on closed and convex sets in Banach spaces with a Frechet differentiable objective function. Such problems are well studied in a finite-dimensional space and in an infinite-dimensional Hilbert space. When the space is Hilbert there are many algorithms for solving optimization problems including the gradient projection algorithm which is one of the most important tools in the optimization theory, nonlinear analysis and their applications. An optimization problem is described by an objective function and a set of feasible points. For the gradient projection algorithm each iteration consists of two steps. The first step is a calculation of a gradient of the objective function while in the second one we calculate a projection on the feasible set. In each of these two steps there is a computational error. In our recent research we show that the gradient projection algorithm generates a good approximate solution, if all the computational errors are bounded from above by a small positive constant. It should be mentioned that the properties of a Hilbert space play an important role. When we consider an optimization problem in a general Banach space the situation becomes more difficult and less understood. On the other hand such problems arise in the approximation theory. The book is of interest for mathematicians working in optimization. It also can be useful in preparation courses for graduate students. The main feature of the book which appeals specifically to this audience is the study of algorithms for convex and nonconvex minimization problems in a general Banach space. The book is of interest for experts in applications of optimization to the approximation theory.In this book the goal is to obtain a good approximate solution of the constrained optimization problem in a general Banach space under the presence of computational errors. It is shown that the algorithm generates a good approximate solution, if the sequence of computational errors is bounded from above by a small constant. The book consists of four chapters. In the first we discuss several algorithms which are studied in the book and prove a convergence result for an unconstrained problem which is a prototype of our results for the constrained problem. In Chapter 2 we analyze convex optimization problems. Nonconvex optimization problems are studied in Chapter 3. In Chapter 4 we study continuous algorithms for minimization problems under the presence of computational errors. The algorithm generates a good approximate solution, if the sequence of computational errors is bounded from above by a small constant. The book consists of four chapters. In the first we discuss several algorithms which are studied in the book and prove a convergence result for an unconstrained problemwhich is a prototype of our results for the constrained problem. In Chapter 2 we analyze convex optimization problems. Nonconvex optimization problems are studied in Chapter 3. In Chapter 4 we study continuous algorithms for minimization problems under the presence of computational errors.
Step Into the World of Mathematics
Foreword: Mathematics, our invisible friend.- 1. My adventures in the world of mathematics.- 2. Examples of mathematical models.- 3. The large models of Earth and space.- 4. Mathematics in the service of doctors.- 5. Is humanity mathematics as well?.- Mathematics belongs to everyone.
Harmonic Analysis on the Real Line
- Introduction. - Classes of Functions. - Fourier Series. - Fourier Transform. - Hilbert Transform. - Hardy Spaces and their Subspaces. - Hardy Inequalities. - Certain Applications.
Permutation Statistical Methods with R
This book takes a unique approach to explaining permutation statistics by integrating permutation statistical methods with a wide range of classical statistical methods and associated R programs. It opens by comparing and contrasting two models of statistical inference: the classical population model espoused by J. Neyman and E.S. Pearson and the permutation model first introduced by R.A. Fisher and E.J.G. Pitman. Numerous comparisons of permutation and classical statistical methods are presented, supplemented with a variety of R scripts for ease of computation. The text follows the general outline of an introductory textbook in statistics with chapters on central tendency and variability, one-sample tests, two-sample tests, matched-pairs tests, completely-randomized analysis of variance, randomized-blocks analysis of variance, simple linear regression and correlation, and the analysis of goodness of fit and contingency. Unlike classical statistical methods, permutation statistical methods do not rely on theoretical distributions, avoid the usual assumptions of normality and homogeneity, depend only on the observed data, and do not require random sampling. The methods are relatively new in that it took modern computing power to make them available to those working in mainstream research. Designed for an audience with a limited statistical background, the book can easily serve as a textbook for undergraduate or graduate courses in statistics, psychology, economics, political science or biology. No statistical training beyond a first course in statistics is required, but some knowledge of, or some interest in, the R programming language is assumed.
Geometric Integrators for Differential Equations with Highly Oscillatory Solutions
1 Oscillation-preserving integrators for highly oscillatory systems of second-order ODEs2 Continuous-stage ERKN integrators for second-order ODEs with highly oscillatory solutions3 Stability and convergence analysis of ERKN integrators for second-order ODEs with highly oscillatory solutions4 Functionally-fitted energy-preserving integrators for Poisson systems 5 Exponential collocation methods for conservative or dissipative systems 6 Volume-preserving exponential integrators 7 Global error bounds of one-stage explicit ERKN integrators for semilinear wave equations 8 Linearly-fitted conservative (dissipative) schemes for nonlinear wave equations9 Energy-preserving schemes for high-dimensional nonlinear KG equations 10 High-order symmetric Birkhoff-Hermite time integrators for semilinear KG equations 11 Symplectic approximations for efficiently solving semilinear KG equations12 Continuous-stage leap-frog schemes for semilinear Hamiltonian wave equations13 Semi-analytical ERKN integrators for solving high-dimensional nonlinear wave equations 14 Long-time momentum and actions behaviour of energy-preserving methods for wave equations
Introduction to Reaction-Diffusion Equations
This book introduces some basic mathematical tools in reaction-diffusion models, with applications to spatial ecology and evolutionary biology. It is divided into four parts.The first part is an introduction to the maximum principle, the theory of principal eigenvalues for elliptic and periodic-parabolic equations and systems, and the theory of principal Floquet bundles.The second part concerns the applications in spatial ecology. We discuss the dynamics of a single species and two competing species, as well as some recent progress on N competing species in bounded domains. Some related results on stream populations and phytoplankton populations are also included. We also discuss the spreading properties of a single species in an unbounded spatial domain, as modeled by the Fisher-KPP equation.The third part concerns the applications in evolutionary biology. We describe the basic notions of adaptive dynamics, such as evolutionarily stable strategies and evolutionarybranching points, in the context of a competition model of stream populations. We also discuss a class of selection-mutation models describing a population structured along a continuous phenotypical trait.The fourth part consists of several appendices, which present a self-contained treatment of some basic abstract theories in functional analysis and dynamical systems. Topics include the Krein-Rutman theorem for linear and nonlinear operators, as well as some elements of monotone dynamical systems and abstract competition systems.Most of the book is self-contained and it is aimed at graduate students and researchers who are interested in the theory and applications of reaction-diffusion equations.
Timed Multiplication Drill Facts Grades 4-6
40+ reproducible drill worksheets. Strengthen students' speed and accuracy on their multiplication facts. Each drill page concentrates on a specific area. In multiplication fact recall, building students' skills as they progress from the easiest level to the most difficult.Each level has a daily practice page, a home practice page, an extra practice page, and a review test page.The extra practice drill sheet is to be used with students still having difficulty recalling facts quickly and accurately. It is a different approach to the timed drill method. This requires the student to complete the fact with its missing number. This different approach helps the students remember the facts more quickly.The multiplication fact drills have been designated to help strengthen students' speed and accuracy through practice during a specified time or each student could be timed individually. This practice will help to strengthen the process of memorization which is a skill needed to recall facts quickly.Some of the drills are shorter and are to be completed on a specified day. Each drill page concentrates on a specific area in multiplication fact recall. The drills proceed from the easiest level to the most difficult level. Each level has a daily practice page, a home practice page, an extra practice page, and a review test page.The daily practice page is divided into five days. Each day of the week the student is to complete a drill. The date, score, and time it took are to be recorded in each section. This page could be glued into the students' workbooks or kept in individual file folders.The home practice page is to be sent home to practice fact recall with parent supervision. Once completed it is to be returned to school and signed by a parent. A letter of explanation should be sent home with the first practice page explaining how it is to be completed.The extra practice drill sheet is to be used with students who are still having difficultyrecalling facts quickly and accurately. It is a different approach to the timed drill method. The student must complete the fact with its missing number. There is no extra practice page for multiplying with zero and one.The review page or test page is to be used to test the speed and accuracy within a given length of time. Begin with five minutes graduating down to one minute. Tell students when to begin and when to stop. Have the student circle the last completed question with a red crayon or a red pencil crayon. The students are to exchange their papers and to mark each incorrect answer with a red dot as you read the answers aloud. Have the students count the number of correct answers. No credit is to be given to incomplete answers. The student is to record the number of correct answers, the time, and the date on each sheet where indicated. On each review test have all incomplete answers finished for extra practice after the completed answers have been marked.There are two timed review tests for each section that may be used after each section has been practiced successfully. These test will evaluate students' speed and accuracy in each section.The final drill pages test all the facts in the multiplication tables. These pages are to be used in the same manner as the other drills.The results of the various drills may be recorded on the Score Record Sheets provided in this book.
Timed Division Drill Facts Grades 4-6
40+ reproducible drill worksheets. Strengthen students' speed and accuracy on their division facts.Each drill page concentrates on a specific area. In division fact recall, building students' skills as they progress from the easiest level to the most difficult. Each level has a daily practice page, a home practice page, an extra practice page, and a review test page. The extra practice drill sheet is to be used with students still having difficulty recalling facts quickly and accurately. It is a different approach to the timed drill method. This requires the student to complete the fact with its missing number. This different approach helps the students remember the facts more quickly.Supports Common Core Standards for Mathematics Gr. 4-5.The division fact drills have been designed to help strengthen students' speed and accuracy through practice during a specified time or each student could be timed individually.Some of the drills are shorter and are to be completed on a specified day. Each drill page concentrates on a specific area in division fact recall. The drills proceed from the easiest level to the most difficult level. Each level has a daily practise page, a home practise page, an extra practise page, and a review test page.The daily practice page is divided into five days. Each day of the week the student is tocomplete a drill, the date, score, and time it took to do the drill are to be recorded in each section. This page could be glued into the students' work books or kept in individual file folders.The home practice page is to be sent home to practice fact recall with parent supervision. Once completed, it is to be returned to school signed by a parent. A letter of explanation should be sent home with the first home practice page explaining how it is to be completed.The extra practice drill sheet is to be used with students who are still having difficultyrecalling facts quickly and accurately. It is a different approach to the time drill method. The student must complete the fact with its missing number.The review page or test page is to be used to test speed and accuracy within a given length of time. Begin with five minutes graduating down to one minute.Tell students when to begin and when to stop. Have the students circle the last completed question with a red crayon or red pencil crayon. The students are to exchange their papers and to mark each incorrect answer with a red dot as you read the answers aloud. Have the students count the number of correct answers. No credit is to be given for incomplete answers. The student is to record the number of correct answers, time, and date on each sheet where indicated. On each review test have all incomplete answers finished for extra practice after the completed answers have been marked.There are two timed review tests for each section that may be used after each section has been practiced successfully. These tests will evaluate students' speed and accuracy in each section.The final drill pages test all the division facts. These pages are to be used in the same manner as the other drills.The results of the various drills may be recorded on the Score Record Sheets provided in this book.
Control Problems for Conservation Laws with Traffic Applications
Introduction.- Boundary Control.- Decentralized Control.- Distributed Control.- Lagrangian Control.- Hamilton-Jacobi Equations.- Appendix A: Balance Laws with Boundary.- Conservation Laws on Networks.
Notes on Real Analysis and Measure Theory
Preface.- 1. Real-Valued Semicontinuous Functions.- 2. The Oscillations of Real-Valued Functions.- 3. Monotone and Continuous Restrictions of Real-Valued Functions.- 4. Bijective Continuous Images of Absolute Null Sets.- 5. Projective Absolutely Nonmeasurable Functions.- 6. Borel Isomorphisms of Analytic Sets.- 7. Iterated Integrals of Real-Valued Functions of Two Real Variables.- 8. The Steinhaus Property, Ergocidity, and Density Points.- 9. Measurability Properties of H-Selectors and Partial H-Selectors.- 10. A Decomposition of an Uncountable Solvable Group into Three Negligible Sets.- 11. Negligible Sets Versus Absolutely Nonmeasurable Sets.- 12. Measurability Properties of Mazurkiewicz Sets.- 13. Extensions of Invariant Measures on R.- A. A Characterization of Uncountable Sets in Terms of their Self-Mappings.- B. Some Applications of Peano Type Functions.- C. Almost Rigid Mathematical Structures.- D. Some Unsolved Problems in Measure Theory.- Bibliography.- Index.
Locating Eigenvalues in Graphs
This book focuses on linear time eigenvalue location algorithms for graphs. This subject relates to spectral graph theory, a field that combines tools and concepts of linear algebra and combinatorics, with applications ranging from image processing and data analysis to molecular descriptors and random walks. It has attracted a lot of attention and has since emerged as an area on its own.Studies in spectral graph theory seek to determine properties of a graph through matrices associated with it. It turns out that eigenvalues and eigenvectors have surprisingly many connections with the structure of a graph. This book approaches this subject under the perspective of eigenvalue location algorithms. These are algorithms that, given a symmetric graph matrix M and a real interval I, return the number of eigenvalues of M that lie in I. Since the algorithms described here are typically very fast, they allow one to quickly approximate the value of any eigenvalue, which is a basic step in most applications of spectral graph theory. Moreover, these algorithms are convenient theoretical tools for proving bounds on eigenvalues and their multiplicities, which was quite useful to solve longstanding open problems in the area. This book brings these algorithms together, revealing how similar they are in spirit, and presents some of their main applications.This work can be of special interest to graduate students and researchers in spectral graph theory, and to any mathematician who wishes to know more about eigenvalues associated with graphs. It can also serve as a compact textbook for short courses on the topic.
Classical Hopf Algebras and Their Applications
This book is dedicated to the structure and combinatorics of classical Hopf algebras. Its main focus is on commutative and cocommutative Hopf algebras, such as algebras of representative functions on groups and enveloping algebras of Lie algebras, as explored in the works of Borel, Cartier, Hopf and others in the 1940s and 50s.The modern and systematic treatment uses the approach of natural operations, illuminating the structure of Hopf algebras by means of their endomorphisms and their combinatorics. Emphasizing notions such as pseudo-coproducts, characteristic endomorphisms, descent algebras and Lie idempotents, the text also covers the important case of enveloping algebras of pre-Lie algebras. A wide range of applications are surveyed, highlighting the main ideas and fundamental results.Suitable as a textbook for masters or doctoral level programs, this book will be of interest to algebraists and anyone working in one of the fields of application of Hopf algebras.
Mirzakhani's Curve Counting and Geodesic Currents
1. Introduction.- 2. Read Me.- 3. Geodesic Currents.- 4. Train Tracks.- 5. Radallas.- 6. Subconvergence of Measures.- 7. Approximating the Thurston Measure.- 8. The Main Theorem.- 9. Counting Curves.- 10. Counting Square Tiled Surfaces.- 11. Statistics of Simple Curves.- 12. Sm繹rg疇sbord.- A. Radon Measures.- B. Computing Thurston Volumes.- References.- Index.
From Great Discoveries in Number Theory to Applications
Foreword.- 1. Divisibility and Congruence.- 2. Prime and Composite Numbers.- 3. Properties of Prime Numbers.- 4. Special Types of Primes.- 5. On a Connection of Number Theory with Graph Theory.- 6. Pseudoprimes.- 7. Fibonacci and Lucas Numbers.- 8. Further Special Types of Integers.- 9. Magic and Latin Squares.- 10. The Mathematics Behind Prague's Horologe.- 11. Applications of Primes.- 12. Further Applications of Number Theory.- Tables.- References.
Landscape of 21st Century Mathematics
1 Number Theory.- 2 Combinatorics.- 3 Analysis.- 4 Algebra.- 5 Geometry and Topology.- 6 Probability and Statistics.- 7 Algorithms and Complexity.- 8 Logic and Set Theory.- References.- Index.
Crime Mapping and Spatial Data Analysis Using R
Crime mapping and analysis sit at the intersection of geocomputation, data visualisation and cartography, spatial statistics, environmental criminology, and crime analysis. This book brings together relevant knowledge from these fields into a practical, hands-on guide, providing a useful introduction and reference material for topics in crime mapping, the geography of crime, environmental criminology, and crime analysis. It can be used by students, practitioners, and academics alike, whether to develop a university course, to support further training and development, or to hone skills in self-teaching R and crime mapping and spatial data analysis. It is not an advanced statistics textbook, but rather an applied guide and later useful reference books, intended to be read and for readers to practice the learnings from each chapter in sequence. In the first part of this volume we introduce key concepts for geographic analysis and representation and provide the reader with the foundations needed to visualise spatial crime data. We then introduce a series of tools to study spatial homogeneity and dependence. A key focus in this section is how to visualise and detect local clusters of crime and repeat victimisation. The final chapters introduce the use of basic spatial models, which account for the distribution of crime across space. In terms of spatial data analysis the focus of the book is on spatial point pattern analysis and lattice or area data analysis. 
Making up Numbers
Making up Numbers: A History of Invention in Mathematics offers a detailed but accessible account of a wide range of mathematical ideas. Starting with elementary concepts, it leads the reader towards aspects of current mathematical research.The book explains how conceptual hurdles in the development of numbers and number systems were overcome in the course of history, from Babylon to Classical Greece, from the Middle Ages to the Renaissance, and so to the nineteenth and twentieth centuries. The narrative moves from the Pythagorean insistence on positive multiples to the gradual acceptance of negative numbers, irrationals and complex numbers as essential tools in quantitative analysis.Within this chronological framework, chapters are organised thematically, covering a variety of topics and contexts: writing and solving equations, geometric construction, coordinates and complex numbers, perceptions of 'infinity' and its permissible uses in mathematics, number systems, and evolving views of the role of axioms.Through this approach, the author demonstrates that changes in our understanding of numbers have often relied on the breaking of long-held conventions to make way for new inventions at once providing greater clarity and widening mathematical horizons. Viewed from this historical perspective, mathematical abstraction emerges as neither mysterious nor immutable, but as a contingent, developing human activity. Making up Numbers will be of great interest to undergraduate and A-level students of mathematics, as well as secondary school teachers of the subject. In virtue of its detailed treatment of mathematical ideas, it will be of value to anyone seeking to learn more about the development of the subject.
Making up Numbers
Making up Numbers: A History of Invention in Mathematics offers a detailed but accessible account of a wide range of mathematical ideas. Starting with elementary concepts, it leads the reader towards aspects of current mathematical research.The book explains how conceptual hurdles in the development of numbers and number systems were overcome in the course of history, from Babylon to Classical Greece, from the Middle Ages to the Renaissance, and so to the nineteenth and twentieth centuries. The narrative moves from the Pythagorean insistence on positive multiples to the gradual acceptance of negative numbers, irrationals and complex numbers as essential tools in quantitative analysis.Within this chronological framework, chapters are organised thematically, covering a variety of topics and contexts: writing and solving equations, geometric construction, coordinates and complex numbers, perceptions of 'infinity' and its permissible uses in mathematics, number systems, and evolving views of the role of axioms.Through this approach, the author demonstrates that changes in our understanding of numbers have often relied on the breaking of long-held conventions to make way for new inventions at once providing greater clarity and widening mathematical horizons. Viewed from this historical perspective, mathematical abstraction emerges as neither mysterious nor immutable, but as a contingent, developing human activity. Making up Numbers will be of great interest to undergraduate and A-level students of mathematics, as well as secondary school teachers of the subject. In virtue of its detailed treatment of mathematical ideas, it will be of value to anyone seeking to learn more about the development of the subject.
How to Cheat with Statistics - And Get Away with It: From Data Snooping Over Kitchen Sink Regression to Creative Reporting
The book explains how to identify and catch statistical cheaters. The author came across many weaknesses and flaws in statistics through 30 years of teaching. These weaknesses allow a malevolent researcher to manipulate the inputs, the calculations, and the reporting of results to derive a desired outcome.This book should be valuable to everyone who wants to gain a deeper understanding of the weaknesses in statistics and learn how to evaluate statistical research to catch a statistical cheater!The math is explained in simple terms and should be easy to follow. In addition, the book comes with 18 Excel spreadsheets and 7 Python codes. There are also questions and problems at the end of each chapter, which should facilitate the usage in a classroom. Answers to the questions and problems are available to instructors upon request.
Epistemic Processes
This book discusses a link between statistical theory and quantum theory based on the concept of epistemic processes. The latter are processes, such as statistical investigations or quantum mechanical measurements, that can be used to obtain knowledge about something. Various topics in quantum theory are addressed, including the construction of a Hilbert space from reasonable assumptions and an interpretation of quantum states. Separate derivations of the Born formula and the one-dimensional Schr繹dinger equation are given. In concrete terms, a Hilbert space can be constructed under some technical assumptions associated with situations where there are two conceptual variables that can be seen as maximally accessible. Then to every accessible conceptual variable there corresponds an operator on this Hilbert space, and if the variables take a finite number of values, the eigenspaces/eigenvectors of these operators correspond to specific questions in nature together with sharp answers to these questions. This paves a new way to the foundations of quantum theory. The resulting interpretation of quantum mechanics is related to Herv矇 Zwirn's recent Convivial Solipsism, but it also has some relations to Quantum Bayesianism and to Rovelli's relational quantum mechanics. Niels Bohr's concept of complementarity plays an important role. Philosophical implications of this approach to quantum theory are discussed, including consequences for macroscopic settings.The book will benefit a broad readership, including physicists and statisticians interested in the foundations of their disciplines, philosophers of science and graduate students, and anyone with a reasonably good background in mathematics and an open mind.
Solomon Golomb's Course on Undergraduate Combinatorics
0. Basic Tools.- 1. Combinations.- 2. Recurrence Relations and Generating Functions.- 3. Permutations.- 4. Special Numbers.- 5. Counting Under Symmetries.- 6. Combinatorial Structures.- A. Additional Exercises.- B. Additional Examples.- C. Solutions to Odd-numbered Exercises.- Bibliography.- Index.
How to Cheat with Statistics - And Get Away with It: From Data Snooping Over Kitchen Sink Regression to Creative Reporting
The book explains how to identify and catch statistical cheaters. The author came across many weaknesses and flaws in statistics through 30 years of teaching. These weaknesses allow a malevolent researcher to manipulate the inputs, the calculations, and the reporting of results to derive a desired outcome.This book should be valuable to everyone who wants to gain a deeper understanding of the weaknesses in statistics and learn how to evaluate statistical research to catch a statistical cheater!The math is explained in simple terms and should be easy to follow. In addition, the book comes with 18 Excel spreadsheets and 7 Python codes. There are also questions and problems at the end of each chapter, which should facilitate the usage in a classroom. Answers to the questions and problems are available to instructors upon request.
Analysis of Data from Randomized Controlled Trials
This book provides a practical guide to the analysis of data from randomized controlled trials (RCT). It gives an answer to the question of how to estimate the intervention effect in an appropriate way. This problem is examined for different RCT designs, such as RCTs with one follow-up measurement, RCTs with more than one follow-up measurement, cluster RCTs, cross-over trials, stepped wedge trials, and N-of-1 trials. The statistical methods are explained in a non-mathematical way and are illustrated by extensive examples. All datasets used in the book are available for download, so readers can reanalyse the examples to gain a better understanding of the methods used. Although most examples are taken from epidemiological and clinical studies, this book is also highly recommended for researchers working in other fields.
Modeling of Transportation Aviation Processes
This book presents the fundamental methods of modeling, analysis and synthesis of transport processes which allow: - to optimize the work of transport organizations according to different criteria of their functioning; - to evaluate the quality of decisions made in conditions of high entropy and uncertainty of technological processes when applying new technologies and systems; -to reduce the negative impact of the human factor on the safety of ongoing transport processes by increasing the motivational component of the training processes for the operator of active transport systems. The structural diagrams presented in this book make it possible to visualize the processes of training, self-training of operators and the mechanisms of the formation of volitional motivational efforts under various external influences of the environment and the teacher-instructor. The obtained functional dependencies serve as an assessment for determining and forming the dynamics of motivation and making a decision about the readiness for safe work of the operator of active transport systems and the decision-maker in the new conditions of the technological process.
Elementary Methods of Graph Ramsey Theory
This book is intended to provide graduate students and researchers in graph theory with an overview of the elementary methods of graph Ramsey theory. It is especially targeted towards graduate students in extremal graph theory, graph Ramsey theory, and related fields, as the included contents allow the text to be used in seminars. It is structured in thirteen chapters which are application-focused and largely independent, enabling readers to target specific topics and information to focus their study. The first chapter includes a true beginner's overview of elementary examples in graph Ramsey theory mainly using combinatorial methods. The following chapters progress through topics including the probabilistic methods, algebraic construction, regularity method, but that's not all. Many related interesting topics are also included in this book, such as the disproof for a conjecture of Borsuk on geometry, intersecting hypergraphs, Tur獺n numbers and communication channels, etc.
illpha Math Book (Level 2)
The 2nd book of illpha series for learning Children writing numbers and counting With fun and interesting way For children from 4 to 5 years old.
Mathematical Methods in Dynamical Systems
Mathematical Methods in Dynamical Systems offers problem solving techniques and includes different analytical, semi-analytical, numerical and machine intelligence methods for finding exact and/or approximate solutions of governing equations arising in dynamical systems.
Representation Theory and Algebraic Geometry
The chapters in this volume explore the influence of the Russian school on the development of algebraic geometry and representation theory, particularly the pioneering work of two of its illustrious members, Alexander Beilinson and Victor Ginzburg, in celebration of their 60th birthdays. Based on the work of speakers and invited participants at the conference "Interactions Between Representation Theory and Algebraic Geometry", held at the University of Chicago, August 21-25, 2017, this volume illustrates the impact of their research and how it has shaped the development of various branches of mathematics through the use of D-modules, the affine Grassmannian, symplectic algebraic geometry, and other topics. All authors have been deeply influenced by their ideas and present here cutting-edge developments on modern topics. Chapters are organized around three distinct themes: Groups, algebras, categories, and representation theoryD-modules and perverse sheavesAnalogous varieties defined by quivers Representation Theory and Algebraic Geometry will be an ideal resource for researchers who work in the area, particularly those interested in exploring the impact of the Russian school.
Probability and Random Variables: Theory and Applications
Preface.- Chapter 1 - Preliminaries.- Chapter 2 - Fundamentals of Probability.- Chapter 3 - Random Variables.- Chapter 4 - Random Vectors.- Chapter 5 - Normal Random Vectors.- Chapter 6 - Convergence of Random Variables.- Answers to Selected Exercises.- Index.
Novice Mathematics Teachers' Professional Learning
This book investigates three novice high school mathematics teachers' professional learning processes in the early stages of their careers at schools in Shanghai, China. Teacher professional learning is examined as a complex and dynamic system that connects both cognitive and situated perspectives on learning theory. Inspiring mathematics teachers to adopt student-focused pedagogies is challenging, particularly in China where tensions in teacher-centred, content-focused and examination-oriented practices are predominant. The three novice teachers who participated in this study brought different beliefs and knowledge derived from their different individual experiences to bear on their teaching practices. However, they were strongly influenced by the environments in which they taught and mainly adopted a professional learning approach to teacher-centred practices, despite reporting that they favoured student-centred teaching practices. The study also observed professional learning towards student-centred pedagogical aspects in a single teacher case with mentorship support, indicating that student-centred pedagogies may be promoted within the constraints of the existing dominant teaching practice.
Research in Computational Topology 2
This second volume of Research in Computational Topology is a celebration and promotion of research by women in applied and computational topology, containing the proceedings of the second workshop for Women in Computational Topology (WinCompTop) as well as papers solicited from the broader WinCompTop community. The multidisciplinary and international WinCompTop workshop provided an exciting and unique opportunity for women in diverse locations and research specializations to interact extensively and collectively contribute to new and active research directions in the field. The prestigious senior researchers that signed on to head projects at the workshop are global leaders in the discipline, and two of them were authors on some of the first papers in the field. Some of the featured topics include topological data analysis of power law structure in neural data; a nerve theorem for directional graph covers; topological or homotopical invariantsfor directed graphs encoding connections among a network of neurons; and the issue of approximation of objects by digital grids, including precise relations between the persistent homology of dual cubical complexes.
Multivariate Data Analysis on Matrix Manifolds
This graduate-level textbook aims to give a unified presentation and solution of several commonly used techniques for multivariate data analysis (MDA). Unlike similar texts, it treats the MDA problems as optimization problems on matrix manifolds defined by the MDA model parameters, allowing them to be solved using (free) optimization software Manopt. The book includes numerous in-text examples as well as Manopt codes and software guides, which can be applied directly or used as templates for solving similar and new problems. The first two chapters provide an overview and essential background for studying MDA, giving basic information and notations. Next, it considers several sets of matrices routinely used in MDA as parameter spaces, along with their basic topological properties. A brief introduction to matrix (Riemannian) manifolds and optimization methods on them with Manopt complete the MDA prerequisite. The remaining chapters study individual MDA techniques in depth. The number of exercises complement the main text with additional information and occasionally involve open and/or challenging research questions. Suitable fields include computational statistics, data analysis, data mining and data science, as well as theoretical computer science, machine learning and optimization. It is assumed that the readers have some familiarity with MDA and some experience with matrix analysis, computing, and optimization.
Stochastic Analysis, Filtering, and Stochastic Optimization
This volume is a collection of research works to honor the late Professor Mark H.A. Davis, whose pioneering work in the areas of Stochastic Processes, Filtering, and Stochastic Optimization spans more than five decades. Invited authors include his dissertation advisor, past collaborators, colleagues, mentees, and graduate students of Professor Davis, as well as scholars who have worked in the above areas. Their contributions may expand upon topics in piecewise deterministic processes, pathwise stochastic calculus, martingale methods in stochastic optimization, filtering, mean-field games, time-inconsistency, as well as impulse, singular, risk-sensitive and robust stochastic control.
Optimal Control Problems Arising in Mathematical Economics
This book is devoted to the study of two large classes of discrete-time optimal control problems arising in mathematical economics. Nonautonomous optimal control problems of the first class are determined by a sequence of objective functions and sequence of constraint maps. They correspond to a general model of economic growth. We are interested in turnpike properties of approximate solutions and in the stability of the turnpike phenomenon under small perturbations of objective functions and constraint maps. The second class of autonomous optimal control problems corresponds to another general class of models of economic dynamics which includes the Robinson-Solow-Srinivasan model as a particular case. In Chap. 1 we discuss turnpike properties for a large class of discrete-time optimal control problems studied in the literature and for the Robinson-Solow-Srinivasan model. In Chap. 2 we introduce the first class of optimal control problems and study its turnpike property. This class of problems is also discussed in Chaps. 3-6. In Chap. 3 we study the stability of the turnpike phenomenon under small perturbations of the objective functions. Analogous results for problems with discounting are considered in Chap. 4. In Chap. 5 we study the stability of the turnpike phenomenon under small perturbations of the objective functions and the constraint maps. Analogous results for problems with discounting are established in Chap. 6. The results of Chaps. 5 and 6 are new. The second class of problems is studied in Chaps. 7-9. In Chap. 7 we study the turnpike properties. The stability of the turnpike phenomenon under small perturbations of the objective functions is established in Chap. 8. In Chap. 9 we establish the stability of the turnpike phenomenon under small perturbations of the objective functions and the constraint maps. The results of Chaps. 8 and 9 are new. In Chap. 10 we study optimal control problems related to a model of knowledge-based endogenous economic growth and show the existence of trajectories of unbounded economic growth and provide estimates for the growth rate.
New Sinc Methods of Numerical Analysis
This contributed volume honors the 80th birthday of Frank Stenger who established new Sinc methods in numerical analysis.The contributions, written independently from each other, show the new developments in numerical analysis in connection with Sinc methods and approximations of solutions for differential equations, boundary value problems, integral equations, integrals, linear transforms, eigenvalue problems, polynomial approximations, computations on polyhedra, and many applications. The approximation methods are exponentially converging compared with standard methods and save resources in computation. They are applicable in many fields of science including mathematics, physics, and engineering.The ideas discussed serve as a starting point in many different directions in numerical analysis research and applications which will lead to new and unprecedented results. This book will appeal to a wide readership, from students to specialized experts.
