This textbook is designed for students. Rather than the typical definition-theorem-proof-repeat style, this text includes much more commentary, motivation and explanation. The proofs are not terse, and aim for understanding over economy. Furthermore, dozens of proofs are preceded by scratch work or a proof sketch to give students a big-picture view and an explanation of how they would come up with it on their own. Examples often drive the narrative and challenge the intuition of the reader. The text also aims to make the ideas visible, and contains over 200 illustrations. The writing is relaxed and includes interesting historical notes, periodic attempts at humor, and occasional diversions into other interesting areas of mathematics. The text covers the real numbers, cardinality, sequences, series, the topology of the reals, continuity, differentiation, integration, and sequences and series of functions. Each chapter ends with exercises, and nearly all include some open questions. The first appendix contains a construction the reals, and the second is a collection of additional peculiar and pathological examples from analysis. The author believes most textbooks are extremely overpriced and endeavors to help change this.Hints and solutions to select exercises can be found at LongFormMath.com.

This is the 2 + epsilon edition of this book. The second edition was published in July 2019. In January 2024, an epsilon of changes were made and the manuscript was updated, without officially creating a new edition.

From the bestselling author of Quantum Computing for Everyone, a concise, accessible, and elegant approach to mathematics that not only illustrates concepts but also conveys the surprising nature of the digital information age.

Most of us know something about the grand theories of physics that transformed our views of the universe at the start of the twentieth century: quantum mechanics and general relativity. But we are much less familiar with the brilliant theories that make up the backbone of the digital revolution. In Beautiful Math, Chris Bernhardt explores the mathematics at the very heart of the information age. He asks questions such as: What is information? What advantages does digital information have over analog? How do we convert analog signals into digital ones? What is an algorithm? What is a universal computer? And how can a machine learn?
The four major themes of Beautiful Math are information, communication, computation, and learning. Bernhardt typically starts with a simple mathematical model of an important concept, then reveals a deep underlying structure connecting concepts from what, at first, appear to be unrelated areas. His goal is to present the concepts using the least amount of mathematics, but nothing is oversimplified. Along the way, Bernhardt also discusses alphabets, the telegraph, and the analog revolution; information theory; redundancy and compression; errors and noise; encryption; how analog information is converted into digital information; algorithms; and, finally, neural networks. Historical anecdotes are included to give a sense of the technology at that time, its impact, and the problems that needed to be solved.
Taking its readers by the hand, regardless of their math background, Beautiful Math is a fascinating journey through the mathematical ideas that undergird our everyday digital interactions.

This lively introductory text exposes the student to the rewards of a rigorous study of functions of a real variable. In each chapter, informal discussions of questions that give analysis its inherent fascination are followed by precise, but not overly formal, developments of the techniques needed to make sense of them. By focusing on the unifying themes of approximation and the resolution of paradoxes that arise in the transition from the finite to the infinite, the text turns what could be a daunting cascade of definitions and theorems into a coherent and engaging progression of ideas. Acutely aware of the need for rigor, the student is much better prepared to understand what constitutes a proper mathematical proof and how to write one.

Fifteen years of classroom experience with the first edition of Understanding Analysis have solidified and refined the central narrative of the second edition. Roughly 150 new exercises join a selection of the best exercises fromthe first edition, and three more project-style sections have been added. Investigations of Euler's computation of ζ(2), the Weierstrass Approximation - Theorem, and the gamma function are now among the book's cohort of seminal results serving as motivation and payoff for the beginning student to master the methods of analysis.

The human brain is made up of 85 billion neurons, which are connected by over 100 trillion synapses.

For more than a century, a diverse array of researchers searched for a language that could be used to capture the essence of what these neurons do and how they communicate - and how those communications create thoughts, perceptions and actions. The language they were looking for was mathematics, and we would not be able to understand the brain as we do today without it.

In Models of the Mind, author and computational neuroscientist Grace Lindsay explains how mathematical models have allowed scientists to understand and describe many of the brain's processes, including decision-making, sensory processing, quantifying memory, and more. She introduces readers to the most important concepts in modern neuroscience, and highlights the tensions that arise when the abstract world of mathematical modelling collides with the messy details of biology.

Each chapter of Models of the Mind focuses on mathematical tools that have been applied in a particular area of neuroscience, progressing from the simplest building block of the brain - the individual neuron - through to circuits of interacting neurons, whole brain areas and even the behaviours that brains command.

Lindsay examines the history of the field, starting with experiments done on frog legs in the late eighteenth century and building to the large models of artificial neural networks that form the basis of modern artificial intelligence. Throughout, she reveals the value of using the elegant language of mathematics to describe the machinery of neuroscience.

French doctoral student, Louis Bachelier, successfully defended his thesis Th orie de la Sp culation at the Sorbonne.

This book provides a new translation of Bachelier's seminal work.

Bachelier's thesis is a remarkable document on two counts. In mathematical terms Bachelier's achievement was to introduce many of the concepts of what is now known as stochastic analysis. His purpose, however, was to give a theory for the valuation of financial options. He came up with a formula that is both correct on its own terms and surprisingly close to the Nobel Prize-winning solution to the option pricing problem by Fischer Black, Myron Scholes, and Robert Merton in 1973.

This book offers great insight into the historical developments of probability and mathematical finance.

Classic graduate-level text discusses the Fourier series in Hilbert space, examines further properties of trigonometrical Fourier series, and concludes with a detailed look at the applications of previously outlined theorems. 1956 edition.

This comprehensive introduction to global spectral methods for fractional differential equations from leaders of this emerging field is designed to be accessible to graduate students and researchers across math, science, and engineering. The book begins by covering the foundational fractional calculus concepts needed to understand and model anomalous transport phenomena. The authors proceed to introduce a series of new spectral theories and new families of orthogonal and log orthogonal functions, then present corresponding spectral and spectral element methods for fractional differential equations. The book also covers the fractional Laplacian in unbounded and bounded domains and major developments in time-integration of fractional models. It ends by sampling the wide variety of real-world applications of fractional modeling, including concentration transport in surface/subsurface dynamics, complex rheology and material damage, and fluid turbulence and geostrophic transport.

This classroom-tested volume offers a definitive look at modern analysis, with views of applications to statistics, numerical analysis, Fourier series, differential equations, mathematical analysis, and functional analysis. Upper-level undergraduate students with a background in calculus will benefit from its teachings, along with beginning graduate students seeking a firm grounding in modern analysis.
A self-contained text, it presents the necessary background on the limit concept, and the first seven chapters could constitute a one-semester introduction to limits. Subsequent chapters discuss differential calculus of the real line, the Riemann-Stieltjes integral, sequences and series of functions, transcendental functions, inner product spaces and Fourier series, normed linear spaces and the Riesz representation theorem, and the Lebesgue integral. Supplementary materials include an appendix on vector spaces and more than 750 exercises of varying degrees of difficulty. Hints and solutions to selected exercises, indicated by an asterisk, appear at the back of the book.

Many applied mathematical disciplines, such as dynamical systems and optimization theory as well as classical mathematical disciplines like differential geometry and the theory of Lie groups, have a common foundation in general topology and multivariate calculus in normed vector spaces. In this book, students from both pure and applied subjects are offered an opportunity to work seriously with fundamental notions from mathematical analysis that are important not only from a mathematical point of view but also occur frequently in the theoretical parts of, for example, the engineering sciences. The book provides complete proofs of the basic results from topology and differentiability of mappings in normed vector spaces. It is a useful resource for students and researchers in mathematics and the many sciences that depend on fundamental techniques from mathematical analysis.In this second edition, the notions of compactness and sequentially compactness are developed with independent proofs for the main results. Thereby the material on compactness is apt for direct applications also in functional analysis, where the notion of sequentially compactness prevails. This edition also covers a new section on partial derivatives, and new material has been incorporated to make a more complete account of higher order derivatives in Banach spaces, including full proofs for symmetry of higher order derivatives and Taylor's formula. The exercise material has been reorganized from a collection of problem sets at the end of the book to a section at the end of each chapter with further results. Readers will find numerous new exercises at different levels of difficulty for practice.

A reader looking for interesting problems tackled often by highly original methods, for precise results fully proved, and for procedures fully motivated, will be delighted. -- Mathematical Reviews.
Asymptotics is not new. Its importance in many areas of pure and applied mathematics has been recognized since the days of Laplace. Asymptotic estimates of series, integrals, and other expressions are commonly needed in physics, engineering, and other fields. Unfortunately, for many years there was a dearth of literature dealing with this difficult but important topic. Then, in 1958, Professor N. G. de Bruijn published this pioneering study. Widely considered the first text on the subject -- and the first comprehensive coverage of this broad field -- the book embodied an original and highly effective approach to teaching asymptotics. Rather than trying to formulate a general theory (which, in the author's words, leads to stating more and more about less and less) de Bruijn teaches asymptotic methods through a rigorous process of explaining worked examples in detail.
Most of the important asymptotic methods are covered here with unusual effectiveness and clarity: Every step in the mathematical process is explained, its purpose and necessity made clear, with the result that the reader not only has no difficulty in following the rigorous proofs, but even turns to them with eager expectation. (Nuclear Physics).
Part of the attraction of this book is its pleasant, straightforward style of exposition, leavened with a touch of humor and occasionally even using the dramatic form of dialogue. The book begins with a general introduction (fundamental to the whole book) on O and o notation and asymptotic series in general. Subsequent chapters cover estimation of implicit functions and the roots of equations; various methods of estimating sums; extensive treatment of the saddle-point method with full details and intricate worked examples; a brief introduction to Tauberian theorems; a detailed chapter on iteration; and a short chapter on asymptotic behavior of solutions of differential equations. Most chapters progress from simple examples to difficult problems; and in some cases, two or more different treatments of the same problem are given to enable the reader to compare different methods. Several proofs of the Stirling theorem are included, for example, and the problem of the iterated sine is treated twice in Chapter 8. Exercises are given at the end of each chapter.
Since its first publication, Asymptotic Methods in Analysis has received widespread acclaim for its rigorous and original approach to teaching a difficult subject. This Dover edition, with corrections by the author, offers students, mathematicians, engineers, and physicists not only an inexpensive, comprehensive guide to asymptotic methods but also an unusually lucid and useful account of a significant mathematical discipline.

Fourier's discovery that it is possible to build complex signals out of basic sine waves revolutionized the way we look at data. It was years ahead of its time. In today's digital age, understanding the data we collect is vital. Fourier's idea is now in use more than ever before.

In my last book: How the Fourier Series Works we studied Fourier's original theory, published in 1822. However, the Fourier Series can only model repeating signals. It took the further work of the German mathematician, Peter Gustav Lejeune Dirichlet, to expand the capabilities of the Fourier Series so that it could model non-repeating signals.
In this book, we'll follow the journey of the development of the Fourier Transform from Fourier's original idea, through Dirichlet's changes, and into the digital age of sampled signals, all the way to the Fast Fourier Transform that is used so widely today.

In addition to producing electric vehicles, tesla has also made significant investments in energy storage and solar power. Tesla's powerwall and powerpack products allow households and businesses to store energy and use it when needed, reducing their dependence on the grid. The solar roof, a revolutionary new product that integrates solar panels into the roof of a home, has also been well received by customers.

Tesla has also built a network of superchargers, fast charging stations that allow tesla owners to travel long distances with ease.

In this book you will learn:

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Tesla started off by producing high-performance electric sports cars and quickly expanded into the mass market with the launch of the model s, a luxury electric sedan. The model s quickly became a hit, winning awards and setting new standards for electric vehicles in terms of performance, range, and charging. With the launch of the model x, a luxury electric suv, tesla continued to expand its product lineup and reach new customers.

The book Sequences and Series in Calculus is designed as the first college/university calculus course for students who take and do well on the AP AB exam in high school and who are interested in a more proof-oriented treatment of calculus. The text begins with an ε-ℕ treatment of sequence convergence, then builds on this to discuss convergence of series--first series of real numbers, then series of functions. The difference between uniform and pointwise convergence is discussed in some detail. This is followed by a discussion of calculus on power series and Taylor series. Finally improper integrals, integration by parts and partial fractions integration all are introduced.

This book is designed both to teach calculus, and to give the readers and students a taste of analysis to help them determine if they wish to study this material even more deeply. It might be used by colleges and universities who teach special versions of calculus courses for their most mathematically advanced entering first-year students, as might its older sibling text Multivariable and Vector Calculus which appeared in 2020 and is intended for students who take and do well on the AP BC exam.

Understanding the natural numbers, which we use to count things, comes naturally. Meanwhile, the real numbers, which include a wide range of numbers from whole numbers to fractions to exotic ones like π, are, frankly, really difficult to describe rigorously. Instead of waiting to take a theorem-proof graduate course to appreciate the real numbers, readers new to university-level mathematics can explore the core ideas behind the construction of the real numbers in this friendly introduction. Beginning with the intuitive notion of counting, the book progresses step-by-step to the real numbers. Each sort of number is defined in terms of a simpler kind by developing an equivalence relation on a previous idea. We find the finite sets' equivalence classes are the natural numbers. Integers are equivalence classes of pairs of natural numbers. Modular numbers are equivalence classes of integers. And so forth. Exercises and their solutions are included.

Geospatial Analysis (6th edition) provides a compehensive guide to concepts, methods and tools, with many examples being provided using a variety of software tools such as ArcGIS, Idrisi, Grass, Surfer and many others to clarify the concepts discussed. Topics covered in detail include:

• Geospatial analysis concepts
• Analytical methodologies and model building
• Core components of geospatial analysis, including distance and directional analysis, geometrical processing, map algebra, and grid models
• Exploratory Spatial and Spatio-temporal Data Analysis (ESDA, ESTDA) and spatial statistics, including spatial autocorrelation and spatial regression
• Surface analysis, including surface form and flow analysis, gridding and interpolation methods, and visibility analysis
• Network and locational analysis, including shortest path calculation, travelling salesman problems, facility location and arc routing
• Geocomputational methods, including agent-based modelling, artifical neural networks and evolutionary computing
• Big Data - lessons for researchers

Fourier's discovery that almost any signal could be built out of basic sine waves, was years ahead of its time. The Fourier Series, and its later developments: the Fourier Transform and the Fast Fourier Transform, lie at the heart of so many of our electronic devices today. From the mobile phone sitting in your pocket, to early warning systems that keep aircraft engines running safely, Fourier is there, telling you which sine waves are present in your signal.

The standard approach to teaching complicated mathematical concepts in schools and universities relies more on mathematical analysis and the student's capability to manipulate equations, than on his or her intuition. A good diagram and a clear analogy are often worth a thousand words.

This book will guide you systematically, visually, and intuitively through each of the core concepts making up the Fourier Series; granting you the knowledge you need to confidently use one of the cornerstones of signal analysis.

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Gilbert Strang's Calculus textbook is ideal both as a course companion and for self study. The author has a direct style. His book presents detailed and intensive explanations. Many diagrams and key examples are used to aid understanding, as well as the application of calculus to physics and engineering and economics. The text is well organized, and it covers single variable and multivariable calculus in depth. An instructor's manual and student guide are available online at http: //ocw.mit.edu/ans7870/resources/Strang/strangtext.htm.