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Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, Volume III
This title is part of UC Press's Voices Revived program, which commemorates University of California Press's mission to seek out and cultivate the brightest minds and give them voice, reach, and impact. Drawing on a backlist dating to 1893, Voices Revived makes high-quality, peer-reviewed scholarship accessible once again using print-on-demand technology. This title was originally published in 1972.
Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, Volume II, Part II
This title is part of UC Press's Voices Revived program, which commemorates University of California Press's mission to seek out and cultivate the brightest minds and give them voice, reach, and impact. Drawing on a backlist dating to 1893, Voices Revived makes high-quality, peer-reviewed scholarship accessible once again using print-on-demand technology. This title was originally published in 1967.
Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, Volume II, Part II
This title is part of UC Press's Voices Revived program, which commemorates University of California Press's mission to seek out and cultivate the brightest minds and give them voice, reach, and impact. Drawing on a backlist dating to 1893, Voices Revived makes high-quality, peer-reviewed scholarship accessible once again using print-on-demand technology. This title was originally published in 1967.
Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, Volume III
This title is part of UC Press's Voices Revived program, which commemorates University of California Press's mission to seek out and cultivate the brightest minds and give them voice, reach, and impact. Drawing on a backlist dating to 1893, Voices Revived makes high-quality, peer-reviewed scholarship accessible once again using print-on-demand technology. This title was originally published in 1972.
Game Theory and Applications
This textbook provides an overview of the fundamentals of game theory and its applications in various fields. It introduces game theory as an established toolkit for the mathematical analysis and evaluation of strategic decisions. Through applied exercises, it introduces the basic concepts of game theory and offers students from various disciplines the opportunity to practice the concepts through in-depth training. The textbook addresses advanced students of economics, business administration, and related disciplines, university graduates with basic mathematical training as well as interested readers from all fields. For this, it provides student-friendly explanations, a variety of exercises and problems, and useful references to further reading. The book is divided into a beginner-friendly theory section, in which the most important aspects are presented in a compact and clear manner, and an application-oriented problem section, in which the readers can directly check what they have learned and find many application examples. The latter can also be used as a source of inspiration for instructors.
Finite Difference Methods on Irregular Networks
No detailed description available for "Finite Difference Methods on Irregular Networks".
Theory of Functions on Complex Manifolds
No detailed description available for "Theory of Functions on Complex Manifolds".
Parametric Optimization and Related Topics
No detailed description available for "Parametric Optimization and Related Topics".
A Model Theoretic Oriented Approach to Partial Algebras
No detailed description available for "A Model Theoretic Oriented Approach to Partial Algebras".
The Arithmetic of Life and Death
Whether you realize it or not, numbers are everywhere--and integral to almost every facet of your life . . . from your next raise in pay to the inevitable rise of inflation, your weekly family budget to your end of the national debt. And as George Shaffner amazingly reveals, there are discerning answers (and a great measure of comfort) in numbers. In The Arithmetic of Life, he applies the basic principles of mathematics--addition, subtraction, multiplication, and division--to some of the most profound and just plain puzzling questions of our time. Illuminated with anecdotes, humor, and insight, each chapter explains a unique part of life that can be understood only through the magic of numbers. Whether it's an unconventional theory on why more things go wrong than right, a simple calculation of how much it will cost you to smoke for a lifetime, why crime (accumulatively) doesn't pay, or a glimpse into the probability of life after death, this enlightening and lucidly reasoned book will forever change the way you think about numbers--and the world around you.
Introductory Banach Space Operators
This book provides a concise introduction to the analysis of Banach space operators within the framework of Hilbert space theory. The guiding notion in this approach is that of operator properties. At the same time the notion of function of an operator is emphasized. The formal aspects of these concepts are explained in all chapters. Each chapter consists of varying sections and exercises at the end of each chapter.
Geometric Gems
Our physical world is embedded in a geometric environment. Plane geometry has many amazing wonders beyond those that are briefly touched on in school curriculums. The triangle, one of the basic instruments in geometry, has a plethora of unexpected curiosities. Geometric Gems presents one of the largest collections of triangle curiosities currently available, which the authors discuss in an easily understood fashion, requiring nothing more of readers other than the very basics of school geometry to appreciate these curiosities and their justifications or proofs.The book is intended to be widely appreciated by a general audience, and their love for geometry should be greatly enhanced through exploring these many unexpected relationships in geometry. Geometric Gems is also suitable for mathematics teachers, to enhance the education of their students with these highly motivating triangle properties.
A Compendium of Musical Mathematics
The purpose of this book is to provide a concise introduction to the mathematical theory of music, opening each chapter to the most recent research. Despite the complexity of some sections, the book can be read by a large audience. Many examples illustrate the concepts introduced. The book is divided into 9 chapters.In the first chapter, we tackle the question of the classification of chords and scales. Chapter 2 is a mathematical presentation of David Lewin's Generalized Interval Systems. Chapter 3 offers a new theory of diatonicity in equal-tempered universes. Chapter 4 presents the Neo-Riemannian theories based on the work of David Lewin, Richard Cohn and Henry Klumpenhouwer. Chapter 5 is devoted to the application of word combinatorics to music. Chapter 6 studies the rhythmic canons and the tessellation of the line. Chapter 7 is devoted to serial knots. Chapter 8 presents combinatorial designs and their applications to music. The last chapter, chapter 9, is dedicated to the study of tuning systems.
Discrete-Valued Time Series
The analysis and modeling of time series has been an active research area for more than 100 years, with the main focus on time series having a continuous range consisting of real numbers or real vectors. It took until the 1980s for the first papers on discrete-valued time series to appear. In the 2000s, a rapid increase in research activity was noted, but only in the last few years was a certain maturity and consolidation of the area of discrete-valued time series observed. This reprint is a collection of articles on a wide range of topics on discrete-valued time series (especially count time series), covering stochastic models and methods for their analysis, univariate and multivariate time series, applications of time series methods to risk analysis, statistical process control, and many more. The proposed approaches and concepts are thoroughly discussed and illustrated with several real-world data examples.
Introduction to Iso- Euclidean Geometry
The main purpose in this book is to represent some recent researches of Santilli iso-mathematics in the area of the plane geometry. This book is devoted to the iso-plane geometry. It summarizes the most recent contributions in this area. The book is intended for senior undergraduate students and beginning graduate students of engineering and science courses. The book contains five chapters. The chapters in the book are pedagogically organized. Each chapter concludes with a section with practical problems. In Chapter 1 we introduce iso-real numbers with one and several iso-units. They are defined the basic operations with them and they are deducted some of their basic properties. In the chapter they are defined iso-matrices, iso-determinants and iso-trigonometric functions. Chapter 2 deals with straight iso-lines. It is defined iso-angle between two iso-vectors. They are introduced iso-lines and they are deducted the main equations of iso-lines. They are given criteria for iso-perpendicularity and iso-parallel of iso-lines. In Chapter 3 we introduce iso-motions: iso-reflections, iso-rotations, iso-translations and iso-glide iso-reflections. Chapter 4 is devoted on iso-circles. They are given the iso- parametric iso-representations of the iso-circles. In Chapter 5 they are introduced iso-parabolas, iso-ellipses and iso-hyperbolas. The aim of this book is to present a clear and well-organized treatment of the concept behind the development of iso-mathematics as well as solution techniques. The text material of this book is presented in a readable and mathematically solid format.
Geometric Gems
Our physical world is embedded in a geometric environment. Plane geometry has many amazing wonders beyond those that are briefly touched on in school curriculums. The triangle, one of the basic instruments in geometry, has a plethora of unexpected curiosities. Geometric Gems presents one of the largest collections of triangle curiosities currently available, which the authors discuss in an easily understood fashion, requiring nothing more of readers other than the very basics of school geometry to appreciate these curiosities and their justifications or proofs.The book is intended to be widely appreciated by a general audience, and their love for geometry should be greatly enhanced through exploring these many unexpected relationships in geometry. Geometric Gems is also suitable for mathematics teachers, to enhance the education of their students with these highly motivating triangle properties.
Objective Bayesian Inference
Bayesian analysis is today understood to be an extremely powerful method of statistical analysis, as well an approach to statistics that is particularly transparent and intuitive. It is thus being extensively and increasingly utilized in virtually every area of science and society that involves analysis of data.A widespread misconception is that Bayesian analysis is a more subjective theory of statistical inference than what is now called classical statistics. This is true neither historically nor in practice. Indeed, objective Bayesian analysis dominated the statistical landscape from roughly 1780 to 1930, long before 'classical' statistics or subjective Bayesian analysis were developed. It has been a subject of intense interest to a multitude of statisticians, mathematicians, philosophers, and scientists. The book, while primarily focusing on the latest and most prominent objective Bayesian methodology, does present much of this fascinating history.The book is written for four different audiences. First, it provides an introduction to objective Bayesian inference for non-statisticians; no previous exposure to Bayesian analysis is needed. Second, the book provides an overview of the development and current state of objective Bayesian analysis and its relationship to other statistical approaches, for those with interest in the philosophy of learning from data. Third, the book presents a careful development of the particular objective Bayesian approach that we recommend, the reference prior approach. Finally, the book presents as much practical objective Bayesian methodology as possible for statisticians and scientists primarily interested in practical applications.
Essays on Coding Theory
Critical coding techniques have developed over the past few decades for data storage, retrieval and transmission systems, significantly mitigating costs for governments and corporations that maintain server systems containing large amounts of data. This book surveys the basic ideas of these coding techniques, which tend not to be covered in the graduate curricula, including pointers to further reading. Written in an informal style, it avoids detailed coverage of proofs, making it an ideal refresher or brief introduction for students and researchers in academia and industry who may not have the time to commit to understanding them deeply. Topics covered include fountain codes designed for large file downloads; LDPC and polar codes for error correction; network, rank metric, and subspace codes for the transmission of data through networks; post-quantum computing; and quantum error correction. Readers are assumed to have taken basic courses on algebraic coding and information theory.
Comparative Genomics
This book constitutes the proceedings of the 21st International Conference on Comparative Genomics, RECOMB-CG 2024, which was held in Boston, MA, USA, during April 27-28, 2024. The 13 full papers presented in this book were carefully reviewed and selected from 21 submissions. The papers are divided into the following topical sections: phylogenetic networks; homology and phylogenetic reconstruction; tools for evolution reconstruction; genome rearrangements; and genome evolution.
Complex Heterogeneous Systems
The author's research on energy storage systems generally was confronted with five characteristics, i.e., complex, interacting, transporting, reacting, and heterogeneous systems. Hence, we refer to these kind of systems as Complex Heterogeneous Systems (CHeSs). The work considers interacting systems that exchange energy, mass, information, etc. in various ways. The elementary building blocks of CHeSs are based on fundamental thermodynamic, chemical, material, physical, and mathematical principles such as variational and graph-theoretic concepts. It investigates ways of defining complexity, computing percolation thresholds, making smart decisions also by learning from data/past experiences (e.g., providing a systematic approach towards battery management systems), and identifying battery life (e.g., by blow-up analysis of highly nonlinear concentrated solutions). Ultimately, the elaborated tools shall allow the reader to obtain a general understanding for simulating (also on quantum computers), controlling, and developing CHeSs as well as to pave the way for a general theory on CHeSs generalizing the view on complexity, measurement, estimation, and control.
Volatility Estimation
These notes have been written with the precisely purpose of summarizing the more often encountered and implemented volatility estimation techniques, to describe the realized volatility surface and its term structure, for example in developing Option Pricing libraries. The common and accepted assumptions behind the random fashion, that each quoted and traded asset follows, there are stochastic differential equations (SDEs) characterized by two main terms: one is the drift and the other one is the diffusion term or volatility. If the drift term is set uniquely by the definition of the martingale measure, imposing the drift's value under such risk neutral measure, to be equal to the free interest rate; on the other side, the diffusion term or volatility is not estimated or defined uniquely. Indeed, the latter is estimated involving several different approaches, that over the time have been developed, trying to catch a better fit with the observed options' quotation.
So You Think You Have Brains?
Really Challenging Mathematical Puzzles Newspapers and similar media often feature crossword puzzles, word-searches, general knowledge quizzes, problems on word games like Scrabble and other puzzles, typically on 'brain' games like chess and bridge. As one who has always been fascinated by mathematics, the author offers testing examples from the world of numbers which will require a fair amount of brain-work if you are to find correct solutions. The world of numbers displays much variety and some strange traits. That world forms the first of our group of problems and then the book focuses attention on the three "p's" palindromes, primes, and powers. Later, geometry problems and examples where numbers can be substituted for letters conclude the book. We hope readers enjoy being challenged!
Polytopes and Graphs
This book introduces convex polytopes and their graphs, alongside the results and methodologies required to study them. It guides the reader from the basics to current research, presenting many open problems to facilitate the transition. The book includes results not previously found in other books, such as: the edge connectivity and linkedness of graphs of polytopes; the characterisation of their cycle space; the Minkowski decomposition of polytopes from the perspective of geometric graphs; Lei Xue's recent lower bound theorem on the number of faces of polytopes with a small number of vertices; and Gil Kalai's rigidity proof of the lower bound theorem for simplicial polytopes. This accessible introduction covers prerequisites from linear algebra, graph theory, and polytope theory. Each chapter concludes with exercises of varying difficulty, designed to help the reader engage with new concepts. These features make the book ideal for students and researchers new to the field.
A Primer on Smooth Manifolds
Differential Geometry is one of the major branches of current Mathematics, and it is an unavoidable language in modern Physics. The main characters in Differential Geometry are smooth manifolds: a class of geometric objects that locally behave like the standard Euclidean space.The book provides a first introduction to smooth manifolds, aimed at undergraduate students in Mathematics and Physics. The only prerequisites are the Linear Algebra and Calculus typically covered in the first two years. The presentation is as simple as possible, but it does not sacrifice the rigor.The lecture notes are divided into 10 chapters, with gradually increasing difficulty. The first chapters cover basic material, while the last ones present more sophisticated topics. The definitions, propositions, and proofs are complemented by examples and exercises. The exercises, which include part of the proofs, are designed to help the reader learn the language of Differential Geometry and develop their problem-solving skills in the area. The exercises are also aimed at promoting an active learning process. Finally, the book contains pictures which are useful aids for the visualization of abstract geometric situations. The lecture notes can be used by instructors as teaching material in a one-semester course on smooth manifolds.
Understanding the Language of Mathematics
Alexander Firestone always wanted to be a teacher but felt that in order to know what was important to teach, he should be out in the real world to see what he was able to do with his present education. Upon graduation from the University, he secured a position as a Research Physicist working on new types of rocket propulsion for deep space exploration. In the first week, he realized that his present education ill-equipped him as a problem-solver working on new ideas. This was the beginning of What, How, and Why. After successfully working on the projects he was assigned, he realized he was ready for teaching. Over the last 50 years he has used his teaching and classroom experiences as a laboratory, developing What, How, and Why learning.I still get telephone calls this very evening (student from Westmount College in Christchurch) former students wanting to know how I'm doing and sharing their classroom experiences with me as a teacher. That was nearly 40 years ago. I'm a very passionate teacher who has taught for over 40 years and still teaches casually full-time. I am probably the oldest Mathematics teacher in Australia who is a passionate Mathematics teacher and is still able to teach full time. My teaching positions include classroom teacher, HOD mathematics, Principal, University lecturer in China. I have over 8 years of part-time experience doing post-graduate university study on What, How, and Why. Three in China and five at Griffith University in Queensland. I have presented papers and given Talks at International Education Conferences in Australia, and New Zealand.
Functional Analysis Revisited
'Functional Analysis Revisited' is not a first course in functional analysis - although it covers the basic notions of functional analysis, it assumes the reader is somewhat acquainted with them. It is by no means a second course either: there are too many deep subjects that are not within scope here. Instead, having the basics under his belt, the author takes the time to carefully think through their fundamental consequences. In particular, the focus is on the notion of completeness and its implications, yet without venturing too far from areas where the description 'elementary' is still valid. The author also looks at some applications, perhaps just outside the core of functional analysis, that are not completely trivial. The aim is to show how functional analysis influences and is influenced by other branches of contemporary mathematics. This is what we mean by 'Functional Analysis Revisited.'
Functional Analysis Revisited
'Functional Analysis Revisited' is not a first course in functional analysis - although it covers the basic notions of functional analysis, it assumes the reader is somewhat acquainted with them. It is by no means a second course either: there are too many deep subjects that are not within scope here. Instead, having the basics under his belt, the author takes the time to carefully think through their fundamental consequences. In particular, the focus is on the notion of completeness and its implications, yet without venturing too far from areas where the description 'elementary' is still valid. The author also looks at some applications, perhaps just outside the core of functional analysis, that are not completely trivial. The aim is to show how functional analysis influences and is influenced by other branches of contemporary mathematics. This is what we mean by 'Functional Analysis Revisited.'
The Art of Finding Hidden Risks
This text gives a comprehensive, largely self-contained treatment of multivariate heavy tail analysis. Emphasizing regular variation of measures means theory can be presented systematically and without regard to dimension. Tools are developed that allow a flexible definition of "extreme" in higher dimensions and permit different heavy tails to coexist on the same state space leading to "hidden regular variation" and "steroidal regular variation". This emphasizes when estimating risks, it is important to choose the appropriate heavy tail. Theoretical foundations lead naturally to statistical techniques; examples are drawn from risk estimation, finance, climatology and network analysis. Treatments target a broad audience in insurance, finance, data analysis, network science and probability modeling. The prerequisites are modest knowledge of analysis and familiarity with the definition of a measure; regular variation of functions is reviewed but is not a focal point.
Upper Bounds for Grothendieck Constants, Quantum Correlation Matrices and CCP Functions
This book concentrates on the famous Grothendieck inequality and the continued search for the still unknown best possible value of the real and complex Grothendieck constant (an open problem since 1953). It describes in detail the state of the art in research on this fundamental inequality, including Krivine's recent contributions, and sheds light on related questions in mathematics, physics and computer science, particularly with respect to the foundations of quantum theory and quantum information theory. Unifying the real and complex cases as much as possible, the monograph introduces the reader to a rich collection of results in functional analysis and probability. In particular, it includes a detailed, self-contained analysis of the multivariate distribution of complex Gaussian random vectors. The notion of Completely Correlation Preserving (CCP) functions plays a particularly important role in the exposition.The prerequisites are a basic knowledge of standard functional analysis, complex analysis, probability, optimisation and some number theory and combinatorics. However, readers missing some background will be able to consult the generous bibliography, which contains numerous references to useful textbooks. The book will be of interest to PhD students and researchers in functional analysis, complex analysis, probability, optimisation, number theory and combinatorics, in physics (particularly in relation to the foundations of quantum mechanics) and in computer science (quantum information and complexity theory).
Fundamentals of Abstract Algebra
This is a primary textbook for a one year first course in Abstract Algebra. The book helps in explorations interesting applications, such as Galois theory. Replete with exercises and examples, the book is geared towards careful pedagogy and accessibility, and requires only minimal prerequisites.
Extreme Values in Random Sequences
The main subject is the probabilistic extreme value theory. The purpose is to present recent results related to limiting distributions of maxima in incomplete samples from stationary sequences, and results related to extremal properties of different combinatorial configurations. The necessary contents related to regularly varying functions and basic results of extreme value theory are included in the first two chapters with examples, exercises and supplements. The motivation for consideration maxima in incomplete samples arises from the fact that real data are often incomplete. A sequence of observed random variables from a stationary sequence is also stationary only in very special cases. Hence, the results provided in the third chapter are also related to non-stationary sequences. The proof of theorems related to joint limiting distribution of maxima in complete and incomplete samples requires a non-trivial combination of combinatorics and point process theory. Chapter four provides results on the asymptotic behavior of the extremal characteristics of random permutations, the coupon collector's problem, the polynomial scheme, random trees and random forests, random partitions of finite sets, and the geometric properties of samples of random vectors. The topics presented here provide insight into the natural connections between probability theory and algebra, combinatorics, graph theory and combinatorial geometry. The contents of the book may be useful for graduate students and researchers who are interested in probabilistic extreme value theory and its applications.
Harmonic Numbers and Open problems in Series
Problems involving harmonic numbers are so strange in solution way and those approaches or results are attracting the attention of students interested in mathematics as well as mathematicians and engineers, as there are many areas of application. In scientific and technological calculations that occur in various fields of mathematics and engineering, computational problems associated with various mathematical constants are often encountered, some of which are important tools for solving scientific and technological problems. There have been many open problems that have not been addressed before, due to the high level of scientific theory and the high performance of computer-based computing tools. Nevertheless, the world of mathematics still has a lot of problems to be discovered, so new and diverse methods are needed to solve these problems. This book contains open problems and challenging problems presented in several mathematical reference books and add their new generalizations written by us.
Krasner Hyperring Theory
The theory of algebraic hyperstructures, in particular the theory of Krasner hyperrings, has seen a spectacular development in the last 20 years, which is why a book dedicated to the study of these is so vital. Krasner hyperrings are a generalization of hyperfields, introduced by Krasner in order to study complete valued fields. A Krasner hyperring (R, +, .) is an algebraic structure, where (R, +) is a canonical hypergroup, (R, .) is a semigroup having zero as a bilaterally absorbing element and the multiplication is distributive with respect to the hyperoperation +.Krasner Hyperring Theory presents an elaborate study on hyperstructures, particularly Krasner hyperrings, across 10 chapters with extensive examples. It contains the results of the authors, but also of other researchers in the field, focusing especially on recent research. This book is especially addressed to doctoral students or researchers in the field, as well as to all those interested in this interesting part of algebra, with applications in other fields.
Elements of Stochastic Modelling
This is a thoroughly revised and expanded third edition of a successful university textbook that provides a broad introduction to key areas of stochastic modelling. The previous edition was developed from lecture notes for two one-semester courses for third-year science and actuarial students at the University of Melbourne.This book reviews the basics of probability theory and presents topics on Markov chains, Markov decision processes, jump Markov processes, elements of queueing theory, basic renewal theory, elements of time series and simulation. It also features elements of stochastic calculus and introductory mathematical finance. This makes the book suitable for a larger variety of university courses presenting the fundamentals of modern stochastic modelling.To make the text covering a lot of material more appealing and accessible to the reader, instead of rigorous proofs we often give only sketches of the arguments, with indications as to why a particular result holds and also how it is related to other results, and illustrate them by examples. It is in this aspect that the present, third edition differs from the second one: the included background material and argument sketches have been extended, made more graphical and informative. The whole text was reviewed and streamlined wherever possible to make the book more attractive and useful for readers. Where appropriate, the book includes references to more specialised texts on respective topics that contain both complete proofs and more advanced material.
Elements of Stochastic Modelling (Third Edition)
This is a thoroughly revised and expanded third edition of a successful university textbook that provides a broad introduction to key areas of stochastic modelling. The previous edition was developed from lecture notes for two one-semester courses for third-year science and actuarial students at the University of Melbourne.This book reviews the basics of probability theory and presents topics on Markov chains, Markov decision processes, jump Markov processes, elements of queueing theory, basic renewal theory, elements of time series and simulation. It also features elements of stochastic calculus and introductory mathematical finance. This makes the book suitable for a larger variety of university courses presenting the fundamentals of modern stochastic modelling.To make the text covering a lot of material more appealing and accessible to the reader, instead of rigorous proofs we often give only sketches of the arguments, with indications as to why a particular result holds and also how it is related to other results, and illustrate them by examples. It is in this aspect that the present, third edition differs from the second one: the included background material and argument sketches have been extended, made more graphical and informative. The whole text was reviewed and streamlined wherever possible to make the book more attractive and useful for readers. Where appropriate, the book includes references to more specialised texts on respective topics that contain both complete proofs and more advanced material.
The Daring Invention of Logarithm Tables
In the early 17th century, both Jost B羹rgi and John Napier dared to invent a logarithm table whose construction required tens of thousands of computing steps. These tables reduced computing effort for multiplication and division by an order of magnitude. Indeed, their invention launched a computing revolution that continues to this day. The book, which is the color edition of the original black and white edition published in 2020, tells the story of B羹rgi's and Napier's work, and how Henry Briggs built on Napier's idea, creating a table of logarithms that was easier to use. John Napier and Henry Briggs described their methods in detail; distribution of their results was widespread. In contrast, Jost B羹rgi did not leave detailed records of his work. Just a few copies of his table and terse handwritten instructions for its use have survived. To fill this gap, the book reconstructs B羹rgi's thinking leading up to his table. The reader looks over his shoulder, so to speak, and learns how B羹rgi came upon the idea, how he decided on the specific format of the table, and how his instructions should be interpreted. And so the reader experiences the magic of the invention of logarithms. The final chapters examine the question "Who invented logarithms?". For centuries, few people were aware of B羹rgi's work; John Napier was considered to be the sole inventor. This changed at the middle of the 19th century when Jost B羹rgi's work became more widely known. Since then there has been extensive debate whether B羹rgi should be considered an independent co-inventor. Careful parsing of the history of logarithm going back to Archimedes of antiquity then reveals that, without doubt, John Napier and Jost B羹rgi are independent co-inventors of logarithms.
Notes on Tug-Of-War Games and the P-Laplace Equation
This book addresses the interplay between stochastic processes and partial differential equations. More specifically, it focuses on the connection between the nonlinear p-Laplace equation and the stochastic game called tug-of-war with noise. The connection in this context was discovered approximately 15 years ago and has since provided new insights and approaches. These lecture notes provide a brief but detailed and accessible introduction to the subject and to the more research-oriented literature. The book also presents the parabolic case side by side with the elliptic case, highlighting the fact that elliptic and parabolic equations are close in spirit in certain aspects. Moreover, it covers some parts of the regularity theory for these problems. Graduate students and advanced undergraduate students with a basic understanding of probability and partial differential equations will find this book useful.
Primes and Particles
Many philosophers, physicists, and mathematicians have wondered about the remarkable relationship between mathematics with its abstract, pure, independent structures on one side, and the wilderness of natural phenomena on the other. Famously, Wigner found the "effectiveness" of mathematics in defining and supporting physical theories to be unreasonable, for how incredibly well it worked. Why, in fact, should these mathematical structures be so well-fitting, and even heuristic in the scientific exploration and discovery of nature? This book argues that the effectiveness of mathematics in physics is reasonable. The author builds on useful analogies of prime numbers and elementary particles, elementary structure kinship and the structure of systems of particles, spectra and symmetries, and for example, mathematical limits and physical situations. The two-dimensional Ising model of a permanent magnet and the proofs of the stability of everyday matter exemplify such effectiveness, and the power of rigorous mathematical physics. Newton is our original model, with Galileo earlier suggesting that mathematics is the language of Nature.
Supportive Foundation for High School Mathematics
This book is written to help students acquire a solid foundation in high school mathematics and enable them to develop problem-solving skills. Considerable effort has been devoted to ensure that the text is understandable and covers areas where learners have common difficulty. Numerous carefully selected examples are provided with detailed solutions to illustrate the mathematical concepts and enhance students understanding of the subject matter. The book has five chapters: (1) Relations and graphs of equations; (2) Functions with emphasis on linear, quadratic, and polynomial functions; (3) Exponential and logarithmic functions; (4) Topics of geometry that are basic to high school; (5) Trigonometric functions, trigonometric equations, and applications involving triangles.Undoubtedly, students who work through this manuscript will find that the book presents a comprehensive foundation to successfully understand high school mathematics. The other book of the author, namely, "Supportive Foundation for Basic Algebra" provides a comprehensive approach to the fundamental concepts and techniques of algebra and offers a useful supplementary background for enhanced mathematical competence.
Lagrangian Floer Theory and Its Deformations
A-infinity structure was introduced by Stasheff in the 1960s in his homotopy characterization of based loop space, which was the culmination of earlier works of Sugawara's homotopy characterization of H-spaces and loop spaces. At the beginning of the 1990s, a similar structure was introduced by Fukaya in his categorification of Floer homology in symplectic topology. This structure plays a fundamental role in the celebrated homological mirror symmetry proposal by Kontsevich and in more recent developments of symplectic topology.A detailed construction of A-infinity algebra structure attached to a closed Lagrangian submanifold is given in Fukaya, Oh, Ohta, and Ono's two-volume monograph Lagrangian Intersection Floer Theory (AMS-IP series 46 I & II), using the theory of Kuranishi structures-a theory that has been regarded as being not easily accessible to researchers in general. The present lecture note is provided by one of the main contributors to the Lagrangian Floer theory and is intended to provide a quick, reader-friendly explanation of the geometric part of the construction. Discussion of the Kuranishi structures is minimized, with more focus on the calculations and applications emphasizing the relevant homological algebra in the filtered context.The book starts with a quick explanation of Stasheff polytopes and their two realizations-one by the rooted metric ribbon trees and the other by the genus-zero moduli space of open Riemann surfaces-and an explanation of the A-infinity structure on the motivating example of the based loop space. It then provides a description of the moduli space of genus-zero bordered stable maps and continues with the construction of the (curved) A-infinity structure and its canonical models. Included in the explanation are the (Landau-Ginzburg) potential functions associated with compact Lagrangian submanifolds constructed by Fukaya, Oh, Ohta, and Ono. The book explains calculations of potential functions for toric fibers in detail and reviews several explicit calculations in the literature of potential functions with bulk as well as their applications to problems in symplectic topology via the critical point theory thereof. In the Appendix, the book also provides rapid summaries of various background materials such as the stable map topology, Kuranishi structures, and orbifold Lagrangian Floer theory.
Data and Process Visualisation for Graphic Communication
This book guides the reader through the process of graphic communication with a particular focus on representing data and processes. It considers a variety of common graphic communication scenarios among those that arise most frequently in practical applications. The book is organized in two parts: representing data (Part I) and representing processes (Part II). The first part deals with the graphical representation of data. It starts with an introductory chapter on the types of variables, then guides the reader through the most common data visualization scenarios - i.e.: representing magnitudes, proportions, one variable as a function of the other, groups, relations, bivariate, trivariate and geospatial data. The second part covers various tools for the visual representation of processes; these include timelines, flow-charts, Gantt charts and PERT diagrams. In addition, the book also features four appendices which cover cross-chapter topics: mathematics and statistics review, Matplotlib primer, color representation and usage, and representation of geospatial data. Aimed at junior and senior undergraduate students in various technical, scientific, and economic fields, this book is also a valuable aid for researchers and practitioners in data science, marketing, entertainment, media and other fields.
Diophantine M-Tuples and Elliptic Curves
This book provides an overview of the main results and problems concerning Diophantine m-tuples, i.e., sets of integers or rationals with the property that the product of any two of them is one less than a square, and their connections with elliptic curves. It presents the contributions of famous mathematicians of the past, like Diophantus, Fermat and Euler, as well as some recent results of the author and his collaborators. The book presents fragments of the history of Diophantine m-tuples, emphasising the connections between Diophantine m-tuples and elliptic curves. It is shown how elliptic curves are used in solving some longstanding problems on Diophantine m-tuples, such as the existence of infinite families of rational Diophantine sextuples. On the other hand, rational Diophantine m-tuples are used to construct elliptic curves with interesting Mordell-Weil groups, including curves of record rank with agiven torsion group. The book contains concrete algorithms and advice on how to use the software package PARI/GP for solving computational problems. This book is primarily intended for researchers and graduate students in Diophantine equations and elliptic curves. However, it can be of interest to other mathematicians interested in number theory and arithmetic geometry. The prerequisites are on the level of a standard first course in elementary number theory. Background in elliptic curves, Diophantine equations and Diophantine approximations is provided.
Calculus for Beginners
Foundations of Calculus: A Beginner's Comprehensive GuideCalculus for Beginners stands out as a pioneering educational tool, designed to demystify the complexities of calculus for novices. This book is crafted to cater to the needs of beginners, providing a clear and thorough foundation in calculus concepts, principles, and applications. Its unique structure and comprehensive coverage make it an indispensable resource for anyone looking to grasp the fundamentals of calculus.Empowering Learners with a Multifaceted ApproachCentral to this guide is its multifaceted approach to teaching calculus, which addresses the subject's inherent complexity and variations through a meticulously designed curriculum. Each chapter unfolds logically, guiding learners from basic principles to more intricate concepts. This pedagogical strategy is augmented by a wealth of resources aimed at reinforcing understanding and facilitating practical application.Highlights of the Book: Interactive Learning Experience: For each topic covered, the book includes a QR code and a dedicated link to an accompanying online course. This feature provides learners with instant access to detailed lessons, examples, exercises, video lessons, and worksheets, creating a rich, interactive learning environment.Comprehensive Curriculum: The book covers all fundamental aspects of calculus, including limits, derivatives, integrals, and differential equations, ensuring a solid foundation in each area. The content is presented in a clear, understandable language, making complex concepts accessible to beginners.Practical Application and Examples: Real-world applications and examples are woven throughout the text, illustrating how calculus principles are applied in various fields. This approach not only enhances comprehension but also demonstrates the relevance of calculus in solving practical problems.Enhanced Learning Tools: Each section is supplemented with exercises and worksheets, allowing learners to practice and apply what they have learned. Video lessons offer visual explanations of complex topics, catering to different learning styles.Accessible Solutions: A complete set of solutions for all exercises and questions is provided, enabling learners to check their work and understand the rationale behind each answer. This feedback mechanism is crucial for self-assessment and improvement. Calculus for Beginners is more than just a textbook; it is a comprehensive learning system that integrates traditional teaching methods with innovative digital resources. By offering a blend of written content, interactive online components, and practical exercises, this book ensures that learners not only understand calculus concepts but also know how to apply them in real-world contexts. Whether you are a student, a professional seeking to refresh your knowledge, or a curious mind venturing into the world of calculus for the first time, this guide offers the tools and guidance necessary to master calculus with confidence and ease. Ideal for self-study and classroom usage!
The Logic of Partitions
This book is an introduction to the logic of partitions on a set as well as the (quantum) logic of partitions (direct-sum decompositions or DSDs) on a vector space. Partitions of a set are categorically dual to subsets of a set. Thus the logic of partitions is, in that sense, the dual to the Boolean logic of subsets (usually presented as the special case of propositional logic).Since partitions can be seen as the inverse image partitions of random variables or numerical attributes, partition logic is the logic of random variables or numerical attributes (abstracted from the actual values). On the lattice of partitions of an arbitrary unstructured set, there is a rich algebraic structure of dual operations of implication and co-implication - resembling a non-distributive version of Heyting and co-Heyting algebras.
Four Famous Numbers
WE DELVE DEEP INTO THE SCIENCE AND INTER-RELATIONSHIPS OF FOUR FAMOUS MATHEMATICAL CONSTANTS: EULER'S NUMBER, e, THE LUDOLPHINE CONSTANT, ( pi ), PYTHAGORAS' CONSTANT AND THE RATIO OF PHIDIAS ( phi ). ALONG THE WAY WE ASSESS THE USEFULNESS OF DIVERSE METHODS OF COMPUTING THESE NUMBERS, TECHNIQUES DATING FROM THE BRONZE AGE TO THE TWENTY-FIRST CENTURY. FROM THE EARLIEST DAYS OF VISIBLE LIFE TO OUR OWN TIMES, TINY ANIMALS HAVE PLOWED AND BURROWED THE DEEP SEA FLOOR IN SYSTEMATIC, GEOMETRICAL PATHS. THEY OFTEN SEEMED TO HAVE CONFORMED TO THE MATHEMATICS OF OUR FAMOUS NUMBERS! CAN THIS REALLY BE TRUE? WE CONSIDER THE EVIDENCE. THIS SECOND EDITION, CORRECTED AND EXTENDED, IS PROFUSELY ILLUSTRATED AND HAS ORIGINAL RESEARCH, ALGEBRAIC DERIVATIONS, FULL ACADEMIC REFERENCES, AND A COMPREHENSIVE INDEX. "FOUR FAMOUS NUMBERS" WILL APPEAL TO ENTHUSIASTS, UNDERGRADUATES AND TEACHERS IN HIGHER EDUCATION.
