A groundbreaking treatise by one of the great mathematicians of our age, who outlines a style of thinking by which great ideas are conceived.What inspires and spurs on a great idea? Can we train ourselves to think in a way that will enable world-changing understandings and insights to emerge?Richard Hamming said we can. He first inspired a generation of engineers, scientists, and researchers in 1986 with “You and Your Research,” an electrifying sermon on why some scientists do great work, why most don’t, why he did, and why you can—and should—too. The Art of Doing Science and Engineering is the full expression of what “You and Your Research” outlined. It's a book about thinking; more specifically, a style of thinking by which great ideas are conceived.The book is filled with stories of great people performing mighty deeds—but they are not meant simply to be admired. Instead, they are to be aspired to, learned from, and surpassed. Hamming consistently returns to Shannon’s information theory, Einstein’s theory of relativity, Grace Hopper’s work on high-level programming, Kaiser’s work on digital filters, and his own work on error-correcting codes. He also recounts a number of his spectacular failures as clear examples of what to avoid.Originally published in 1996 and adapted from a course that Hamming taught at the US Naval Postgraduate School, this edition includes an all-new foreword by designer, engineer, and founder of Dynamicland Bret Victor, plus more than 70 redrawn graphs and charts.The Art of Doing Science and Engineering is a reminder that a capacity for learning and creativity are accessible to everyone. Hamming was as much a teacher as a scientist, and having spent a lifetime forming and confirming a theory of great people and great ideas, he prepares the next generation for even greater distinction. Read more

The classic book on statistical graphics, charts, tables. Theory and practice in the design of data graphics, 250 illustrations of the best (and a few of the worst) statistical graphics, with detailed analysis of how to display data for precise, effective, quick analysis. Design of the high-resolution displays, small multiples. Editing and improving graphics. The data-ink ratio. Time-series, relational graphics, data maps, multivariate designs. Detection of graphical deception: design variation vs. data variation. Sources of deception. Aesthetics and data graphical displays. This is the second edition of The Visual Display of Quantitative Information. This edition provides excellent color reproductions of the many graphics of William Playfair, adds color to other images, and includes all the changes and corrections accumulated during 17 printings of the first edition. Read more

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. Read more

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 from the 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. Read more

This first volume, a three-part introduction to the subject, is intended for students with a beginning knowledge of mathematical analysis who are motivated to discover the ideas that shape Fourier analysis. It begins with the simple conviction that Fourier arrived at in the early nineteenth century when studying problems in the physical sciences--that an arbitrary function can be written as an infinite sum of the most basic trigonometric functions. The first part implements this idea in terms of notions of convergence and summability of Fourier series, while highlighting applications such as the isoperimetric inequality and equidistribution. The second part deals with the Fourier transform and its applications to classical partial differential equations and the Radon transform; a clear introduction to the subject serves to avoid technical difficulties. The book closes with Fourier theory for finite abelian groups, which is applied to prime numbers in arithmetic progression. In organizing their exposition, the authors have carefully balanced an emphasis on key conceptual insights against the need to provide the technical underpinnings of rigorous analysis. Students of mathematics, physics, engineering and other sciences will find the theory and applications covered in this volume to be of real interest. The Princeton Lectures in Analysis represents a sustained effort to introduce the core areas of mathematical analysis while also illustrating the organic unity between them. Numerous examples and applications throughout its four planned volumes, of which Fourier Analysis is the first, highlight the far-reaching consequences of certain ideas in analysis to other fields of mathematics and a variety of sciences. Stein and Shakarchi move from an introduction addressing Fourier series and integrals to in-depth considerations of complex analysis; measure and integration theory, and Hilbert spaces; and, finally, further topics such as functional analysis, distributions and elements of probability theory. Read more

Taken literally, the title "All of Statistics" is an exaggeration. But in spirit, the title is apt, as the book does cover a much broader range of topics than a typical introductory book on mathematical statistics. This book is for people who want to learn probability and statistics quickly. It is suitable for graduate or advanced undergraduate students in computer science, mathematics, statistics, and related disciplines. The book includes modern topics like non-parametric curve estimation, bootstrapping, and classification, topics that are usually relegated to follow-up courses. The reader is presumed to know calculus and a little linear algebra. No previous knowledge of probability and statistics is required. Statistics, data mining, and machine learning are all concerned with collecting and analysing data. Read more

Data is collected constantly: how far we travel, who we interact with online and where we spend our money. Every bit of data has a story to tell but isolated, these morsels of information lie dormant and useless, like unattached Lego blocks. Written by the author of Amazon Best Seller Machine Learning for Absolute Beginners, this book guides you through the fundamentals of inferential and descriptive statistics with a mix of practical demonstrations, visual examples, historical origins, and plain English explanations. As a resource for beginners, this book won't teach you how to beat the market or predict the next U.S. election but ensures a concise and simple-to-understand supplement to a standard textbook.This includes an introduction to important techniques used to infer predictions from data, such as hypothesis testing, linear regression analysis, confidence intervals, probability theory, and data distribution. Descriptive statistics techniques such as central tendency measures and standard deviation are also covered in this book.Full Overview of Book Themes• The Historical Development of Statistics• Data Sampling • Central Tendency Measures • Measures Of Spread • Measures Of Position• Designing Hypothesis Tests• Probability and Bayes Theory• Regression Analysis• Clustering AnalysisAs the launchpad to quantitative research, business optimization or a promising career in data science, it's never been a better time to brush up on statistics or learn these concepts for the very first time. Read more

Master essential logarithm and exponential skills through helpful explanations, instructive examples, and plenty of practice exercises with answers. Authored by experienced teacher, Chris McMullen, Ph.D., this self-study math workbook covers:logarithms of various bases and natural logarithmsthe change of base formulalogarithm rules like the sum and difference formulasexponential functionshyperbolic functions and their inversesgraphs of logarithms, exponentials, and hyperbolic functionsa concise review of exponents in the first chapterEuler’s numberapplications such as population growth, continuously compounded interest, and radioactive nuclear decaysan introduction to complex numbers in the last chapteran optional chapter covering the calculus of logarithms, exponentials, and hyperbolic functionsThe author, Chris McMullen, Ph.D., has over twenty years of experience teaching math skills to physics students. He prepared this workbook of the Improve Your Math Fluency series to share his strategies for working with logarithms and exponentials. Read more

Numerical methods are ubiquitous in scientific research, often working quietly behind the scenes in algorithmic black boxes. Practitioners who use such black boxes don't always know what's happening inside them, sometimes leading to inaccurate or inefficient solutions and occasionally flat-out wrong ones. This book breaks open the algorithms to explain how they work and why they can fail. It helps develop both the intuitive understanding of the underlying mathematical theory and practical skills for research.This book teaches not only how to be a critical user of scientific computing algorithms but also a knowledgeable creator of them. Ideal as an introductory text for senior undergraduate and first-year graduate students, as a self-study for anyone with a working knowledge of multivariate calculus and linear algebra, and as a modern reference on numerical methods for researchers.This revised edition, extensively rewritten and expanded, provides a comprehensive guide to using numerical methods in linear algebra, analysis, and differential equations. Examples and exercises are worked out in detail. This book includes extensive commentary and code for three essential scientific computing languages: Julia, Python, and Matlab/OctaveKyle Novak is an applied mathematician, data scientist, and decision analyst with twenty-five years of experience on topics ranging from autonomous systems and cryptanalysis to complex networks and federal policy. He is the author of Special Functions of Mathematical Physics: A Tourist's Guidebook and is featured in the American Mathematical Society's 101 Careers in Mathematics. Read more

With this second volume, we enter the intriguing world of complex analysis. From the first theorems on, the elegance and sweep of the results is evident. The starting point is the simple idea of extending a function initially given for real values of the argument to one that is defined when the argument is complex. From there, one proceeds to the main properties of holomorphic functions, whose proofs are generally short and quite illuminating: the Cauchy theorems, residues, analytic continuation, the argument principle. With this background, the reader is ready to learn a wealth of additional material connecting the subject with other areas of mathematics: the Fourier transform treated by contour integration, the zeta function and the prime number theorem, and an introduction to elliptic functions culminating in their application to combinatorics and number theory. Thoroughly developing a subject with many ramifications, while striking a careful balance between conceptual insights and the technical underpinnings of rigorous analysis, Complex Analysis will be welcomed by students of mathematics, physics, engineering and other sciences. The Princeton Lectures in Analysis represents a sustained effort to introduce the core areas of mathematical analysis while also illustrating the organic unity between them. Numerous examples and applications throughout its four planned volumes, of which Complex Analysis is the second, highlight the far-reaching consequences of certain ideas in analysis to other fields of mathematics and a variety of sciences. Stein and Shakarchi move from an introduction addressing Fourier series and integrals to in-depth considerations of complex analysis; measure and integration theory, and Hilbert spaces; and, finally, further topics such as functional analysis, distributions and elements of probability theory. Read more

Can networks unlock secrets of AI or make sense of a social media mess? A behind-the-scenes look at how networks reveal reality.According to mathematician Anthony Bonato, the hidden world of networks permeates our lives in astounding ways. From Bitcoin transactions to neural connections, his book explains how networks shape everything from political landscapes to climate patterns and how deceptively simple dots and lines can unveil the wonders of technology, society, and even nature. From a fresh and startling look at the true impact of clever keywords in politicians' social media posts to a fun breakdown of survival strategies in reality TV shows, Bonato shows us how network theory operates everywhere. Each chapter focuses on a unique aspect of networks to reveal how they provide a captivating lens for bringing diverse phenomena into clearer focus.The book offers an accessible snapshot of networks for anyone curious about what makes the modern world tick. Bonato's insights will give readers a deeper appreciation and understanding of networks and their relevance to our everyday lives. Read more

This text provides the fundamental concepts and techniques of real analysis for students in all of these areas. It helps one develop the ability to think deductively, analyze mathematical situations, and extend ideas to a new context. Like the first three editions, this edition maintains the same spirit and user-friendly approach with additional examples and expansion on Logical Operations and Set Theory. There is also content revision in the following areas: Introducing point-set topology before discussing continuity, including a more thorough discussion of limsup and limimf, covering series directly following sequences, adding coverage of Lebesgue Integral and the construction of the reals, and drawing student attention to possible applications wherever possible. Read more

Real Analysis is the third volume in the Princeton Lectures in Analysis, a series of four textbooks that aim to present, in an integrated manner, the core areas of analysis. Here the focus is on the development of measure and integration theory, differentiation and integration, Hilbert spaces, and Hausdorff measure and fractals. This book reflects the objective of the series as a whole: to make plain the organic unity that exists between the various parts of the subject, and to illustrate the wide applicability of ideas of analysis to other fields of mathematics and science. After setting forth the basic facts of measure theory, Lebesgue integration, and differentiation on Euclidian spaces, the authors move to the elements of Hilbert space, via the L2 theory. They next present basic illustrations of these concepts from Fourier analysis, partial differential equations, and complex analysis. The final part of the book introduces the reader to the fascinating subject of fractional-dimensional sets, including Hausdorff measure, self-replicating sets, space-filling curves, and Besicovitch sets. Each chapter has a series of exercises, from the relatively easy to the more complex, that are tied directly to the text. A substantial number of hints encourage the reader to take on even the more challenging exercises. As with the other volumes in the series, Real Analysis is accessible to students interested in such diverse disciplines as mathematics, physics, engineering, and finance, at both the undergraduate and graduate levels. Also available, the first two volumes in the Princeton Lectures in Analysis: Read more

FROM THE AUTHOR:The new Foreword by Sir Roger Penrose is only available through the "Read Sample" attached to the Kindle edition; for some reason, it is not included in the free sample for the paperback.All legitimate copies of VCA produced by Oxford University Press are crisply printed on high-quality paper.  If you obtain a shoddily printed copy, it's a fake: please return it and purchase a genuine OUP copy.****************************************************************************************************The fundamental advance in the new 25th Anniversary Edition is that the original 501 diagrams now include brand-new captions that fully explain the geometrical reasoning, making it possible to read the work in an entirely new way―as a highbrow comic book!The 25th Anniversary Edition also features a new Foreword by Sir Roger Penrose, as well as a new Preface by the author.Complex Analysis is the powerful fusion of the complex numbers (involving the 'imaginary' square root of -1) with ordinary calculus, resulting in a tool that has been of central importance to science for more than 200 years.This book brings this majestic and powerful subject to life by consistently using geometry (not calculation) as the means of explanation. The 501 diagrams of the original edition embodied geometrical arguments that (for the first time) replaced the long and often opaque computations of the standard approach, in force for the previous 200 years, providing direct, intuitive, visual access to the underlying mathematical reality. REVIEWS FROM THE BACK COVER:"I was delighted when I came across Visual Complex Analysis. As soon as I thumbed through it, I realized that this was the book I was looking for ten years ago."Ed Catmull, former president of Pixar and Disney Animation Studios "Visual Complex Analysis is a delight, and a book after my own heart. By his innovative and exclusive use of the geometrical perspective, Tristan Needham uncovers many surprising and largely unappreciated aspects of the beauty of complex analysis."Sir Roger Penrose "If your budget limits you to only buying one mathematics book in a year then make sure that this is the one that you buy."Mathematical Gazette ". . . it is comparable with Feynman's Lectures on Physics. At every point it asks "why" and finds a beautiful visual answer."Newsletter of the European Mathematical Society ". . . an engaging, broad, thorough, and often deep, development of undergraduate complex analysis and related areas from a geometric point of view. The style is lucid, informal, reader-friendly, and rich with helpful images (e.g., the complex derivative as an "amplitwist"). A truly unusual and notably creative look at a classical subject. "Paul Zorn, American Mathematical Monthly "Newton would have approved . . . a fascinating and refreshing look at a familiar subject . . . essential reading for anybody with any interest at all in this absorbing area of mathematics. "Times Higher Education Supplement "It is . . . a special pleasure to see Tristan Needham's Visual Complex Analysis with its elegantly illustrated visual approach."Ian Stewart, New Scientist "The new ideas and exercises bring together a body of information potentially invaluable to researchers in fields from topology to number theory . . . this is only the beginning of a long list of famous facts for which Needham offers attractive visual proofs . . . Cauchy's theorem is a satisfying example: you can see the contribution to the integral from each infinitesimal square vanish before your eyes."Frank Farris, American Mathematical Monthly "The arguments constructed are highly innovative; even veterans of the field will find new ideas here. . . .This is a special book. Tristan Needham has not only completely rethought a classical field of mathematics but has presented it in a clear and compelling way.  Visual Complex Analysis is worthy of the accolades it has received."MAA Reviews Read more

Based on an honors course taught by the author at UC Berkeley, this introduction to undergraduate real analysis gives a different emphasis by stressing the importance of pictures and hard problems. Topics include: a natural construction of the real numbers, four-dimensional visualization, basic point-set topology, function spaces, multivariable calculus via differential forms (leading to a simple proof of the Brouwer Fixed Point Theorem), and a pictorial treatment of Lebesgue theory. Over 150 detailed illustrations elucidate abstract concepts and salient points in proofs. The exposition is informal and relaxed, with many helpful asides, examples, some jokes, and occasional comments from mathematicians, such as Littlewood, Dieudonné, and Osserman. This book thus succeeds in being more comprehensive, more comprehensible, and more enjoyable, than standard introductions to analysis.New to the second edition of Real Mathematical Analysis is a presentation of Lebesgue integration done almost entirely using the undergraph approach of Burkill. Payoffs include: concise picture proofs of the Monotone and Dominated Convergence Theorems, a one-line/one-picture proof of Fubini's theorem from Cavalieri’s Principle, and, in many cases, the ability to see an integral result from measure theory. The presentation includes Vitali’s Covering Lemma, density points ― which are rarely treated in books at this level ― and the almost everywhere differentiability of monotone functions. Several new exercises now join a collection of over 500 exercises that pose interesting challenges and introduce special topics to the student keen on mastering this beautiful subject. Read more

GETTING A DOCTOR’S APPOINTMENT SHOULDN’T BE HARDER THAN BOOKING A VACATION The US healthcare system excels in research, innovation, and clinical care, but is failing to keep up with the operational challenges of the digital age. Today’s healthcare organizations face immense financial challenges, and their most valuable resources―people, rooms, and equipment―are being used inefficiently. The result? Long wait times for patients, overstressed staff, underused assets, and poor ROI for organizations. Why do health systems struggle with optimization? The fundamental problem is one of matching an unpredictable demand for services with a constrained supply. The math being used to solve this problem is a holdover from the paper-and-pencil era. In Better Healthcare Through Math, authors Mohan Giridharadas and Sanjeev Agrawal show you that there is a better way. Healthcare systems can harness the power of sophisticated, analytics-driven mathematics to optimize the matching of supply and demand. By upgrading to software systems built on better math, they can enable staff to make data-based decisions to flatten peaks of demand and create smoother patient flow. Read more

This text is geared toward assisting engineering and physical science students in cultivating comprehensive skills in linear static and dynamic finite element methodology. Based on courses taught at Stanford University and the California Institute of Technology, it ranges from fundamental concepts to practical computer implementations. Additional sections touch upon the frontiers of research, making the book of potential interest to more experienced analysts and researchers working in the finite element field. In addition to its examination of numerous standard aspects of the finite element method, the volume includes many unique components, including a comprehensive presentation and analysis of algorithms of time-dependent phenomena, plus beam, plate, and shell theories derived directly from three-dimensional elasticity theory. It also contains a systematic treatment of "weak," or variational, formulations for diverse classes of initial/boundary-value problems. Directed toward students without in-depth mathematical training, the text incorporates introductory material on the mathematical theory of finite elements and many important mathematical results, making it an ideal primer for more advanced works on this subject. Read more

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. Read more

This introduction to combinatorial analysis defines the subject as "the number of ways there are of doing some well-defined operation." Chapter 1 surveys that part of the theory of permutations and combinations that finds a place in books on elementary algebra, which leads to the extended treatment of generation functions in Chapter 2, where an important result is the introduction of a set of multivariable polynomials.Chapter 3 contains an extended treatment of the principle of inclusion and exclusion which is indispensable to the enumeration of permutations with restricted position given in Chapters 7 and 8. Chapter 4 examines the enumeration of permutations in cyclic representation and Chapter 5 surveys the theory of distributions. Chapter 6 considers partitions, compositions, and the enumeration of trees and linear graphs.Each chapter includes a lengthy problem section, intended to develop the text and to aid the reader. These problems assume a certain amount of mathematical maturity. Equations, theorems, sections, examples, and problems are numbered consecutively in each chapter and are referred to by these numbers in other chapters. Read more

This book introduces the reader to solving partial differential equations (PDEs) numerically using element-based Galerkin methods. Although it draws on a solid theoretical foundation (e.g. the theory of interpolation, numerical integration, and function spaces), the book’s main focus is on how to build the method, what the resulting matrices look like, and how to write algorithms for coding Galerkin methods. In addition, the spotlight is on tensor-product bases, which means that only line elements (in one dimension), quadrilateral elements (in two dimensions), and cubes (in three dimensions) are considered. The types of Galerkin methods covered are: continuous Galerkin methods (i.e., finite/spectral elements), discontinuous Galerkin methods, and hybridized discontinuous Galerkin methods using both nodal and modal basis functions. In addition, examples are included (which can also serve as student projects) for solving hyperbolic and elliptic partial differential equations, includingboth scalar PDEs and systems of equations. Read more