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An Interactive Introduction to Knot Theory
by Allison K. Henrich Inga JohnsonThis well-written and engaging volume, intended for undergraduates, introduces knot theory, an area of growing interest in contemporary mathematics. The hands-on approach features many exercises to be completed by readers. Prerequisites are only a basic familiarity with linear algebra and a willingness to explore the subject in a hands-on manner.The opening chapter offers activities that explore the world of knots and links — including games with knots — and invites the reader to generate their own questions in knot theory. Subsequent chapters guide the reader to discover the formal definition of a knot, families of knots and links, and various knot notations. Additional topics include combinatorial knot invariants, knot polynomials, unknotting operations, and virtual knots.
An Introduction To Braille Mathematics: Based On The Nemeth Braille Code For Mathematics And Science Notation, 1972
by Library Of Congress Helen Roberts Bernard Marvin Krebs Barbara Taffet Abraham Nemeth National Library Service for the Blind and Physically Handicapped Staff<P><P><i>Advisory: Bookshare has learned that this book offers only partial accessibility. We have kept it in the collection because it is useful for some of our members. Benetech is actively working on projects to improve accessibility issues such as thesse.</i>
An Introduction To Chaotic Dynamical Systems
by Robert L. DevaneyThere is an explosion of interest in dynamical systems in the mathematical community as well as in many areas of science. The results have been truly exciting: systems which once seemed completely intractable from an analytic point of view can now be understood in a geometric or qualitative sense rather easily. Scientists and engineers realize the power and the beauty of the geometric and qualitative techniques. These techniques apply to a number of important nonlinear problems ranging from physics and chemistry to ecology and economics. Computer graphics have allowed us to view the dynamical behavior geometrically. The appearance of incredibly beautiful and intricate objects such as the Mandelbrot set, the Julia set, and other fractals have really piqued interest in the field. This is text is aimed primarily at advanced undergraduate and beginning graduate students. Throughout, the author emphasizes the mathematical aspects of the theory of discrete dynamical systems, not the many and diverse applications of this theory. The field of dynamical systems and especially the study of chaotic systems has been hailed as one of the important breakthroughs in science in the past century and its importance continues to expand. There is no question that the field is becoming more and more important in a variety of scientific disciplines. New to this edition: •Greatly expanded coverage complex dynamics now in Chapter 2•The third chapter is now devoted to higher dimensional dynamical systems.•Chapters 2 and 3 are independent of one another.•New exercises have been added throughout.
An Introduction To Experimental Design And Statistics For Biology
by David HeathThis illustrated textbook for biologists provides a refreshingly clear and authoritative introduction to the key ideas of sampling, experimental design, and statistical analysis. The author presents statistical concepts through common sense, non-mathematical explanations and diagrams. These are followed by the relevant formulae and illustrated by w
An Introduction To Linear Algebra For Science And Engineering (Second Edition)
by Daniel Norman Dan WolczukNorman/Wolczuk's An Introduction to Linear Algebra for Science and Engineering has been widely respected for its unique approach, which helps students understand and apply theory and concepts by combining theory with computations and slowly bringing students to the difficult abstract concepts. This approach includes an early treatment of vector spaces and complex topics in a simpler, geometric context. An Introduction to Linear Algebra for Science and Engineering promotes advanced thinking and understanding by encouraging students to make connections between previously learned and new concepts and demonstrates the importance of each topic through applications. NEW! MyMathLab is now available for this text. The course features assignable homework exercises plus the complete eBook, in addition to tutorial and assessment tools that make it easy to manage your course online.
An Introduction To Probability And Inductive Logic
by Ian HackingThis is an introductory textbook on probability and induction written by one of the world's foremost philosophers of science. The book has been designed to offer maximal accessibility to the widest range of students (not only those majoring in philosophy) and assumes no formal training in elementary symbolic logic.
An Introduction To Stata Programming
by Christopher BaumChristopher F. Baum's An Introduction to Stata Programming, Second Edition, is a great reference for anyone that wants to learn Stata programming. For those learning, Baum assumes familiarity with Stata and gradually introduces more advanced programming tools. For the more advanced Stata programmer, the book introduces Stata's Mata programming language and optimization routines. <P><P>This new edition of the book reflects some of the most important statistical tools added since Stata 10, when the book was introduced. Of note are factor variables and operators, the computation of marginal effects, marginal means, and predictive margins using margins, the use of gmm to implement generalized method of moments estimation, and the use of suest for seemingly unrelated estimation. As in the previous edition of the book, Baum steps the reader through the three levels of Stata programming. He starts with do-files. Do-files are powerful batch files that support loops and conditional statements and are ideal to automate your workflow as well as to guarantee reproducibility of your work. <P><P>While giving examples of do-file programming, Baum introduces useful programming tips and advice. He then delves into ado-files, which are used to extend Stata by creating new commands that share the syntax and behavior of official commands. Baum gives an example of how to write a simple additional command for Stata, complete with documentation and certification. After writing the simple command, users can then learn how to write their own custom estimation commands by using both Stata's built-in numerical maximum-likelihood estimation routine, ml, its built-in nonlinear least-squares routines, nl and nlsur, and its built-in generalized method of moments estimation routine. <P><P>Finally, he introduces Mata, Stata's matrix programming language. Mata programs are integrated into ado-files to build a custom estimation routine that is optimized for speed and numerical stability. While discussing Mata, Baum presents useful topics for advanced programming such as structures and pointers and likelihood-function evaluators using Mata. Baum introduces concepts by providing the background and importance for the topic, presents common uses and examples, and then concludes with larger, more applied examples he refers to as "cookbook recipes". <P><P>Many of the examples in the book are of particular interest because they arose from frequently asked questions from Stata users. If you want to understand basic Stata programming or want to write your own routines and commands using advanced Stata tools, Baum's book is a great reference.
An Introduction To Statistical Learning: With Applications In R (Springer Texts In Statistics Ser.)
by Trevor Hastie Gareth James Robert Tibshirani Daniela WittenAn Introduction to Statistical Learning provides an accessible overview of the field of statistical learning, an essential toolset for making sense of the vast and complex data sets that have emerged in fields ranging from biology to finance to marketing to astrophysics in the past twenty years. This book presents some of the most important modeling and prediction techniques, along with relevant applications. Topics include linear regression, classification, resampling methods, shrinkage approaches, tree-based methods, support vector machines, clustering, deep learning, survival analysis, multiple testing, and more. Color graphics and real-world examples are used to illustrate the methods presented. Since the goal of this textbook is to facilitate the use of these statistical learning techniques by practitioners in science, industry, and other fields, each chapter contains a tutorial on implementing the analyses and methods presented in R, an extremely popular open source statistical software platform. Two of the authors co-wrote The Elements of Statistical Learning (Hastie, Tibshirani and Friedman, 2nd edition 2009), a popular reference book for statistics and machine learning researchers. An Introduction to Statistical Learning covers many of the same topics, but at a level accessible to a much broader audience. This book is targeted at statisticians and non-statisticians alike who wish to use cutting-edge statistical learning techniques to analyze their data. The text assumes only a previous course in linear regression and no knowledge of matrix algebra. This Second Edition features new chapters on deep learning, survival analysis, and multiple testing, as well as expanded treatments of naïve Bayes, generalized linear models, Bayesian additive regression trees, and matrix completion. R code has been updated throughout to ensure compatibility.
An Introduction To The Atomic And Radiation Physics Of Plasmas
by G. J. TallentsPlasmas comprise more than 99% of the observable universe. They are important in many technologies and are key potential sources for fusion power. Atomic and radiation physics is critical for the diagnosis, observation and simulation of astrophysical and laboratory plasmas, and plasma physicists working in a range of areas from astrophysics, magnetic fusion, and inertial fusion utilise atomic and radiation physics to interpret measurements. This text develops the physics of emission, absorption and interaction of light in astrophysics and in laboratory plasmas from first principles using the physics of various fields of study including quantum mechanics, electricity and magnetism, and statistical physics. Linking undergraduate level atomic and radiation physics with the advanced material required for postgraduate study and research, this text adopts a highly pedagogical approach and includes numerous exercises within each chapter for students to reinforce their understanding of the key concepts.
An Introduction To Viscosity Solutions for Fully Nonlinear PDE with Applications to Calculus of Variations in L (SpringerBriefs in Mathematics)
by Nikos KatzourakisThe purpose of this book is to give a quick and elementary, yet rigorous, presentation of the rudiments of the so-called theory of Viscosity Solutions which applies to fully nonlinear 1st and 2nd order Partial Differential Equations (PDE). For such equations, particularly for 2nd order ones, solutions generally are non-smooth and standard approaches in order to define a "weak solution" do not apply: classical, strong almost everywhere, weak, measure-valued and distributional solutions either do not exist or may not even be defined. The main reason for the latter failure is that, the standard idea of using "integration-by-parts" in order to pass derivatives to smooth test functions by duality, is not available for non-divergence structure PDE.
An Introduction to Acceptance Sampling and SPC with R
by John LawsonAn Introduction to Acceptance Sampling and SPC with R is an introduction to statistical methods used in monitoring, controlling and improving quality. Topics covered include acceptance sampling; Shewhart control charts for Phase I studies; graphical and statistical tools for discovering and eliminating the cause of out-of-control-conditions; Cusum and EWMA control charts for Phase II process monitoring; and the design and analysis of experiments for process troubleshooting and discovering ways to improve process output. Origins of statistical quality control and the technical topics presented in the remainder of the book are those recommended in the ANSI/ASQ/ISO guidelines and standards for industry. The final chapter ties everything together by discussing modern management philosophies that encourage the use of the technical methods presented earlier. In the modern world sampling plans and the statistical calculations used in statistical quality control are done with the help of computers. As an open source high-level programming language with flexible graphical output options, R runs on Windows, Mac and Linux operating systems, and has add-on packages that equal or exceed the capability of commercial software for statistical methods used in quality control. In this book, we will focus on several R packages. In addition to demonstrating how to use R for acceptance sampling and control charts, this book will concentrate on how the use of these specific tools can lead to quality improvements both within a company and within their supplier companies. This would be a suitable book for a one-semester undergraduate course emphasizing statistical quality control for engineering majors (such as manufacturing engineering or industrial engineering), or a supplemental text for a graduate engineering course that included quality control topics.
An Introduction to Algebraic Statistics with Tensors (UNITEXT #118)
by Cristiano Bocci Luca ChiantiniThis book provides an introduction to various aspects of Algebraic Statistics with the principal aim of supporting Master’s and PhD students who wish to explore the algebraic point of view regarding recent developments in Statistics. The focus is on the background needed to explore the connections among discrete random variables. The main objects that encode these relations are multilinear matrices, i.e., tensors. The book aims to settle the basis of the correspondence between properties of tensors and their translation in Algebraic Geometry. It is divided into three parts, on Algebraic Statistics, Multilinear Algebra, and Algebraic Geometry. The primary purpose is to describe a bridge between the three theories, so that results and problems in one theory find a natural translation to the others. This task requires, from the statistical point of view, a rather unusual, but algebraically natural, presentation of random variables and their main classical features. The third part of the book can be considered as a short, almost self-contained, introduction to the basic concepts of algebraic varieties, which are part of the fundamental background for all who work in Algebraic Statistics.
An Introduction to Algebraic Structures (Dover Books on Mathematics)
by Joseph LandinAs the author notes in the preface, "The purpose of this book is to acquaint a broad spectrum of students with what is today known as 'abstract algebra.'" Written for a one-semester course, this self-contained text includes numerous examples designed to base the definitions and theorems on experience, to illustrate the theory with concrete examples in familiar contexts, and to give the student extensive computational practice.The first three chapters progress in a relatively leisurely fashion and include abundant detail to make them as comprehensible as possible. Chapter One provides a short course in sets and numbers for students lacking those prerequisites, rendering the book largely self-contained. While Chapters Four and Five are more challenging, they are well within the reach of the serious student.The exercises have been carefully chosen for maximum usefulness. Some are formal and manipulative, illustrating the theory and helping to develop computational skills. Others constitute an integral part of the theory, by asking the student to supply proofs or parts of proofs omitted from the text. Still others stretch mathematical imaginations by calling for both conjectures and proofs.Taken together, text and exercises comprise an excellent introduction to the power and elegance of abstract algebra. Now available in this inexpensive edition, the book is accessible to a wide range of students, who will find it an exceptionally valuable resource.
An Introduction to Algebraic Topology (Dover Books on Mathematics)
by Andrew H. WallaceThis self-contained treatment of algebraic topology assumes only some knowledge of real numbers and real analysis. The first three chapters focus on the basics of point-set topology, offering background to students approaching the subject with no previous knowledge. Readers already familiar with point-set topology can proceed directly to Chapter 4, which examines the fundamental group as well as homology groups and continuous mapping, barycentric subdivision and excision, the homology sequence, and simplicial complexes.Exercises form an integral part of the text; they include theorems that are as valuable as some of those whose proofs are given in full. Author Andrew H. Wallace, Professor Emeritus at the University of Pennsylvania, concludes the text with a guide to further reading.
An Introduction to Analysis (Textbooks in Mathematics)
by James R. KirkwoodThe third edition of this widely popular textbook is authored by a master teacher. This book provides a mathematically rigorous introduction to analysis of realvalued functions of one variable. This intuitive, student-friendly text is written in a manner that will help to ease the transition from primarily computational to primarily theoretical mathematics. The material is presented clearly and as intuitive as possible while maintaining mathematical integrity. The author supplies the ideas of the proof and leaves the write-up as an exercise. The text also states why a step in a proof is the reasonable thing to do and which techniques are recurrent. Examples, while no substitute for a proof, are a valuable tool in helping to develop intuition and are an important feature of this text. Examples can also provide a vivid reminder that what one hopes might be true is not always true. Features of the Third Edition: Begins with a discussion of the axioms of the real number system. The limit is introduced via sequences. Examples motivate what is to come, highlight the need for hypothesis in a theorem, and make abstract ideas more concrete. A new section on the Cantor set and the Cantor function. Additional material on connectedness. Exercises range in difficulty from the routine "getting your feet wet" types of problems to the moderately challenging problems. Topology of the real number system is developed to obtain the familiar properties of continuous functions. Some exercises are devoted to the construction of counterexamples. The author presents the material to make the subject understandable and perhaps exciting to those who are beginning their study of abstract mathematics. Table of Contents Preface Introduction The Real Number System Sequences of Real Numbers Topology of the Real Numbers Continuous Functions Differentiation Integration Series of Real Numbers Sequences and Series of Functions Fourier Series Bibliography Hints and Answers to Selected Exercises Index Biography James R. Kirkwood holds a Ph.D. from University of Virginia. He has authored fifteen, published mathematics textbooks on various topics including calculus, real analysis, mathematical biology and mathematical physics. His original research was in mathematical physics, and he co-authored the seminal paper in a topic now called Kirkwood-Thomas Theory in mathematical physics. During the summer, he teaches real analysis to entering graduate students at the University of Virginia. He has been awarded several National Science Foundation grants. His texts, Elementary Linear Algebra, Linear Algebra, and Markov Processes, are also published by CRC Press.
An Introduction to Analysis of Financial Data With R
by Ruey S. TsayA complete set of statistical tools for beginning financial analysts from a leading authority Written by one of the leading experts on the topic, An Introduction to Analysis of Financial Data with R explores basic concepts of visualization of financial data. Through a fundamental balance between theory and applications, the book supplies readers with an accessible approach to financial econometric models and their applications to real-world empirical research. The author supplies a hands-on introduction to the analysis of financial data using the freely available R software package and case studies to illustrate actual implementations of the discussed methods. The book begins with the basics of financial data, discussing their summary statistics and related visualization methods. Subsequent chapters explore basic time series analysis and simple econometric models for business, finance, and economics as well as related topics including: Linear time series analysis, with coverage of exponential smoothing for forecasting and methods for model comparison Different approaches to calculating asset volatility and various volatility models High-frequency financial data and simple models for price changes, trading intensity, and realized volatility Quantitative methods for risk management, including value at risk and conditional value at risk Econometric and statistical methods for risk assessment based on extreme value theory and quantile regression Throughout the book, the visual nature of the topic is showcased through graphical representations in R, and two detailed case studies demonstrate the relevance of statistics in finance. A related website features additional data sets and R scripts so readers can create their own simulations and test their comprehension of the presented techniques. An Introduction to Analysis of Financial Data with R is an excellent book for introductory courses on time series and business statistics at the upper-undergraduate and graduate level. The book is also an excellent resource for researchers and practitioners in the fields of business, finance, and economics who would like to enhance their understanding of financial data and today's financial markets.
An Introduction to Analytic Functions: With Theoretical Implications
by John Sheridan Mac NerneyWhen first published in 1959, this book was the basis of a two-semester course in complex analysis for upper undergraduate and graduate students. J. S. Mac Nerney was a proponent of the Socratic, or “do-it-yourself” method of learning mathematics, in which students are encouraged to engage in mathematical problem solving, including theorems at every level which are often regarded as “too difficult” for students to prove for themselves. Accordingly, Mac Nerney provides no proofs. What he does instead is to compose and arrange the investigation in his own unique style, so that a contextual proof is always available to the persistent student who enjoys a challenge. The central idea is to empower students by allowing them to discover and rely on their own mathematical abilities. This text may be used in a variety of settings, including: the usual classroom or seminar, but with the teacher acting mainly as a moderator while the students present their discoveries, a small-group setting in which the students present their discoveries to each other, and independent study. The Editors, William E. Kaufman (who was Mac Nerney’s last PhD student) and Ryan C. Schwiebert, have composed the original typed Work into LaTeX ; they have updated the notation, terminology, and some of the prose for modern usage, but the organization of content has been strictly preserved. About this Book, some new exercises, and an index have also been added.
An Introduction to Analytical Fuzzy Plane Geometry (Studies in Fuzziness and Soft Computing #381)
by Debdas Ghosh Debjani ChakrabortyThis book offers a rigorous mathematical analysis of fuzzy geometrical ideas. It demonstrates the use of fuzzy points for interpreting an imprecise location and for representing an imprecise line by a fuzzy line. Further, it shows that a fuzzy circle can be used to represent a circle when its description is not known precisely, and that fuzzy conic sections can be used to describe imprecise conic sections. Moreover, it discusses fundamental notions on fuzzy geometry, including the concepts of fuzzy line segment and fuzzy distance, as well as key fuzzy operations, and includes several diagrams and numerical illustrations to make the topic more understandable. The book fills an important gap in the literature, providing the first comprehensive reference guide on the fuzzy mathematics of imprecise image subsets and imprecise geometrical objects. Mainly intended for researchers active in fuzzy optimization, it also includes chapters relevant for those working on fuzzy image processing and pattern recognition. Furthermore, it is a valuable resource for beginners interested in basic operations on fuzzy numbers, and can be used in university courses on fuzzy geometry, dealing with imprecise locations, imprecise lines, imprecise circles, and imprecise conic sections.
An Introduction to Applied Multivariate Analysis
by George A. Marcoulides Tenko RaykovThis comprehensive text introduces readers to the most commonly used multivariate techniques at an introductory, non-technical level. By focusing on the fundamentals, readers are better prepared for more advanced applied pursuits, particularly on topics that are most critical to the behavioral, social, and educational sciences. Analogies betwe
An Introduction to Applied Statistics: With Step-By-Step SPSS Instructions
by Edward T. Vieira, Jr.An Introduction to Applied Statistics offers a comprehensive and accessible foundation in applied statistics, empowering students with the essential concepts and practical skills necessary for data-driven decision-making in today's world. Thoroughly covering key topics – including data management, probability fundamentals, data screening, descriptive statistics, and a broad spectrum of inferential analysis techniques – this indispensable guide demystifies statistical concepts and equips students to confidently apply statistical analysis in real-world contexts.With a systematic, beginner-friendly approach, the author assumes no prior knowledge, making complex statistical foundations accessible to students from a variety of disciplines. Concise, digestible chapters build statistical competencies within a practical, evidence-based framework, minimizing technical jargon to facilitate comprehension. Now in its latest edition, the book is fully updated with SPSS v29.0 instructions and screenshots, ensuring compatibility with the most recent software developments. It also includes expanded content on addressing nonrandom sampling issues, such as case weighting, and delves into advanced topics like factor analysis, logistic regression, cluster analysis, and discriminant analysis, catering to the evolving needs of students and professionals alike.An invaluable resource for introductory quantitative research methods courses in psychology, social sciences, business, and marketing, this text combines practical examples, online resources, and an approachable format to support both learning and application.Key Features: Concise chapters integrating real-world applications: Seamlessly blends statistical skills with practical scenarios, illustrating the flexible use of statistics in evidence-based decision-making. Accessible presentation: Offers practical explanations of statistical procedures with minimal technical jargon, enhancing understanding and retention. Foundational preparation: Early chapters are designed to equip students for advanced statistical procedures, building a strong foundational knowledge. Step-by-step SPSS instructions: Provides detailed SPSS v29.0 guidance with screenshots to reinforce comprehension and hands-on skills. Real-world exercises with answers: Includes practical exercises complete with solutions to facilitate active learning and application. Comprehensive instructor resources: Offers extensive teaching support with chapter PowerPoints and test banks to enhance the educational experience.
An Introduction to Artificial Intelligence Based on Reproducing Kernel Hilbert Spaces (Compact Textbooks in Mathematics)
by Sergei PereverzyevThis textbook provides an in-depth exploration of statistical learning with reproducing kernels, an active area of research that can shed light on trends associated with deep neural networks. The author demonstrates how the concept of reproducing kernel Hilbert Spaces (RKHS), accompanied with tools from regularization theory, can be effectively used in the design and justification of kernel learning algorithms, which can address problems in several areas of artificial intelligence. Also provided is a detailed description of two biomedical applications of the considered algorithms, demonstrating how close the theory is to being practically implemented. Among the book’s several unique features is its analysis of a large class of algorithms of the Learning Theory that essentially comprise every linear regularization scheme, including Tikhonov regularization as a specific case. It also provides a methodology for analyzing not only different supervised learning problems, such as regression or ranking, but also different learning scenarios, such as unsupervised domain adaptation or reinforcement learning. By analyzing these topics using the same theoretical framework, rather than approaching them separately, their presentation is streamlined and made more approachable.An Introduction to Artificial Intelligence Based on Reproducing Kernel Hilbert Spaces is an ideal resource for graduate and postgraduate courses in computational mathematics and data science.
An Introduction to Automorphic Representations: With a view toward trace formulae (Graduate Texts in Mathematics #300)
by Jayce R. Getz Heekyoung HahnThe goal of this textbook is to introduce and study automorphic representations, objects at the very core of the Langlands Program. It is designed for use as a primary text for either a semester or a year-long course, for the independent study of advanced topics, or as a reference for researchers. The reader is taken from the beginnings of the subject to the forefront of contemporary research. The journey provides an accessible gateway to one of the most fundamental areas of modern mathematics, with deep connections to arithmetic geometry, representation theory, harmonic analysis, and mathematical physics.The first part of the text is dedicated to developing the notion of automorphic representations. Next, it states a rough version of the Langlands functoriality conjecture, motivated by the description of unramified admissible representations of reductive groups over nonarchimedean local fields. The next chapters develop the theory necessary to make the Langlands functoriality conjecture precise. Thus supercuspidal representations are defined locally, cuspidal representations and Eisenstein series are defined globally, and Rankin-Selberg L-functions are defined to give a link between the global and local settings. This preparation complete, the global Langlands functoriality conjectures are stated and known cases are discussed.This is followed by a treatment of distinguished representations in global and local settings. The link between distinguished representations and geometry is explained in a chapter on the cohomology of locally symmetric spaces (in particular, Shimura varieties). The trace formula, an immensely powerful tool in the Langlands Program, is discussed in the final chapters of the book. Simple versions of the general relative trace formulae are treated for the first time in a textbook, and a wealth of related material on algebraic group actions is included. Outlines for several possible courses are provided in the Preface.
An Introduction to Bartlett Correction and Bias Reduction (SpringerBriefs in Statistics)
by Gauss M. Cordeiro Francisco Cribari-NetoThis book presents a concise introduction to Bartlett and Bartlett-type corrections of statistical tests and bias correction of point estimators. The underlying idea behind both groups of corrections is to obtain higher accuracy in small samples. While the main focus is on corrections that can be analytically derived, the authors also present alternative strategies for improving estimators and tests based on bootstrap, a data resampling technique and discuss concrete applications to several important statistical models.
An Introduction to Bayesian Inference, Methods and Computation
by Nick HeardThese lecture notes provide a rapid, accessible introduction to Bayesian statistical methods. The course covers the fundamental philosophy and principles of Bayesian inference, including the reasoning behind the prior/likelihood model construction synonymous with Bayesian methods, through to advanced topics such as nonparametrics, Gaussian processes and latent factor models. These advanced modelling techniques can easily be applied using computer code samples written in Python and Stan which are integrated into the main text. Importantly, the reader will learn methods for assessing model fit, and to choose between rival modelling approaches.
An Introduction to Benford's Law
by Theodore P. Hill Arno BergerThis book provides the first comprehensive treatment of Benford's law, the surprising logarithmic distribution of significant digits discovered in the late nineteenth century. Establishing the mathematical and statistical principles that underpin this intriguing phenomenon, the text combines up-to-date theoretical results with overviews of the law's colorful history, rapidly growing body of empirical evidence, and wide range of applications.An Introduction to Benford's Law begins with basic facts about significant digits, Benford functions, sequences, and random variables, including tools from the theory of uniform distribution. After introducing the scale-, base-, and sum-invariance characterizations of the law, the book develops the significant-digit properties of both deterministic and stochastic processes, such as iterations of functions, powers of matrices, differential equations, and products, powers, and mixtures of random variables. Two concluding chapters survey the finitely additive theory and the flourishing applications of Benford's law.Carefully selected diagrams, tables, and close to 150 examples illuminate the main concepts throughout. The text includes many open problems, in addition to dozens of new basic theorems and all the main references. A distinguishing feature is the emphasis on the surprising ubiquity and robustness of the significant-digit law. This text can serve as both a primary reference and a basis for seminars and courses.