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How Groups Grow--
by Avinoam MannGrowth of groups is an innovative new branch of group theory. This is the first book to introduce the subject from scratch. It begins with basic definitions and culminates in the seminal results of Gromov and Grigorchuk and more. The proof of Gromov's theorem on groups of polynomial growth is given in full, with the theory of asymptotic cones developed on the way. Grigorchuk's first and general groups are described, as well as the proof that they have intermediate growth, with explicit bounds, and their relationship to automorphisms of regular trees and finite automata. Also discussed are generating functions, groups of polynomial growth of low degrees, infinitely generated groups of local polynomial growth, the relation of intermediate growth to amenability and residual finiteness, and conjugacy class growth. This book is valuable reading for researchers, from graduate students onward, working in contemporary group theory.
How Humans Learn to Think Mathematically
by David TallHow Humans Learn to Think Mathematically describes the development of mathematical thinking from the young child to the sophisticated adult. Professor David Tall reveals the reasons why mathematical concepts that make sense in one context may become problematic in another. For example, a child's experience of whole number arithmetic successively affects subsequent understanding of fractions, negative numbers, algebra, and the introduction of definitions and proof. Tall's explanations for these developments are accessible to a general audience while encouraging specialists to relate their areas of expertise to the full range of mathematical thinking. The book offers a comprehensive framework for understanding mathematical growth, from practical beginnings through theoretical developments, to the continuing evolution of mathematical thinking at the highest level.
How James Watt Invented the Copier
by René SchilsFeatures 25 different scientists and the ideas which may not have made them famous, but made history...Typically, we remember our greatest scientists from one single invention, one new formula or one incredible breakthrough. This narrow perspective does not give justice to the versatility of many scientists who also earned a reputation in other areas of science. James Watt, for instance, is known for inventing the steam engine, yet most people do not know that he also invented the copier. Alexander Graham Bell of course invented the telephone, but only few know that he invented artificial breathing equipment, a prototype of the 'iron lung'. Edmond Halley, whose name is associated with the comet that visits Earth every 75 years, produced the first mortality tables, used for life insurances. This entertaining book is aimed at anyone who enjoys reading about inventions and discoveries by the most creative minds. Detailed illustrations of the forgotten designs and ideas enrich the work throughout.
How Labor Powers the Global Economy: A Labor Theory of Capitalism (New Economic Windows)
by Emmanuel D. Farjoun Moshé Machover David ZachariahThis book presents a probabilistic approach to studying the fundamental role of labor in capitalist economies and develops a non-deterministic theoretical framework for the foundations of political economy. By applying the framework to real-world data, the authors offer new insights into the dynamics of growth, wages, and accumulation in capitalist development around the globe. The book demonstrates that a probabilistic political economy based on labor inputs enables us to describe central organizing principles in modern capitalism. Starting from a few basic assumptions, it shows that the working time of employees is the main regulating variable for determining strict numerical limits on the rate of economic growth, the range of wages, and the pace of accumulation under the present global economic system. This book will appeal to anyone interested in how the capitalist mode of production works and its inherent limitations; in particular, it will be useful to scholars and students of Marxian economics.“Emmanuel Farjoun and Moshé Machover, follow up their pathbreaking work on the application of statistical physics methods to political economy in this book with David Zachariah, in which they develop methods for making educated and structured estimates of stylized facts applicable to capitalist economies. There’s a lot for economists and anyone interested in the political economy of capitalism to learn from their reasoning on these issues, including their novel and challenging suggestion of bounds on the rates of increase of use-value productivity of labor, and on the range of variation of the wage share.”Duncan K. Foley, Leo Model Professor of Economics, New School for Social Research
How Likely is it? Understanding Probability
by Glenda Lappan James T. Fey William M. Fitzgerald Susan N. Friel Elizabeth Difanis PhillipsNIMAC-sourced textbook
How Many? (Talking Math Ser.)
by Christopher DanielsonTalking math with your child is fun and easy with this better approach to counting!Written by a math educator, this innovative book encourages critical thinking and sparks memorable mathematical conversations. You and your child decide what to count on each page. You have many choices, and the longer you look, the more possibilities you'll notice. There are no wrong answers in this book. As long as you're talking about what you see, think, and wonder, you're talking math!
How Many Dolphins In a Pod?: Counting By 10's (Nature Numbers)
by Ruth MusgraveIn Nature Numbers, math is beautiful, recognizable, and all around us! Highly engaging pictures of animals and nature scenes, along with cool chalk illustrations, are used to introduce basic math concepts and encourage kids to see a world of numbers all around them.K-2 math concepts include power of 10. This book encourages kids to learn about multiples of 10 as schools of fish swim by and coral polyps build reefs, all with amazing nature pictures and chalk illustrations!
How Many Fish? (I Can Read! #My First Shared Reading)
by Caron Lee CohenA story of six fish and six feet interact underneath the water. When one fish gets lost under a bucket, the feet unknowingly saves the fish.
How Many Licks?: Or, How to Estimate Damn Near Anything
by Aaron SantosHow many licks to the center of a Tootsie Pop? How many people are having sex at this moment? How long would it take a monkey on a typewriter to produce the plays of Shakespeare? For all those questions that keep you up at night, here's the way to answer them. And the beauty of it is that it's all approximate!Using Enrico Fermi's theory of approximation, Santos brings the world of numbers into perspective. For puzzle junkies and trivia fanatics, these 70 word puzzles will show the reader how to take a bit of information, add what they already know, and extrapolate an answer.Santos has done the impossible: make math and the multiple possibilities of numbers fun and informative. Can you really cry a river? Is it possible to dig your way out of jail with just a teaspoon and before your life sentence is up?Taking an academic subject and using it as the prism to view everyday off-the-wall questions as math problems to be solved is a natural step for the lovers of sudoku, cryptograms, word puzzles, and other thought-provoking games.
How Many Penguins?: Counting Animals 0-100 (Nature Numbers)
by Jill EsbaumIn Nature Numbers, math is beautiful, recognizable, and all around us! Highly engaging pictures of animals and nature scenes, along with cool chalk illustrations, are used to introduce basic math concepts and encourage kids to see a world of numbers all around them.K-2 math concepts include counting 1-100. This book encourages kids to count groups of 10 animals with amazing nature pictures and chalk illustrations!
How Many Pennies Make A Dollar?
by Rebecca Wingard-NelsonHow many pennies are in a dollar? How many nickels make a dime? In this great book of money combinations, young readers can learn how all of the different denominations of money relate to each other. Free worksheets along with full-color photos and easy-to-read text make this a great way to reinforce math concepts.
How Many Seeds in a Pumpkin?
by Margaret Mcnamara"How many seeds in a pumpkin?" Mr. Tiffin asks his class as they gather around the big, medium, and small pumpkins on his desk. Robert, the biggest kid, guesses that the largest one has a million seeds; Elinor, sounding like she knows what she's talking about, guesses the medium one has 500 seeds; and Anna, who likes even numbers better than odd ones, guesses that the little one has 22. Charlie, the smallest boy in the class, doesn't have a guess. Counting pumpkin seeds is messy business, but once the slimy job is done, to everyone's surprise, the smallest pumpkin has the most seeds! As Charlie happily exclaims, "Small things have a lot going on inside of them."
How Many Zeroes?: Counting Solutions of Systems of Polynomials via Toric Geometry at Infinity (CMS/CAIMS Books in Mathematics #2)
by Pinaki MondalThis graduate textbook presents an approach through toric geometry to the problem of estimating the isolated solutions (counted with appropriate multiplicity) of n polynomial equations in n variables over an algebraically closed field. The text collects and synthesizes a number of works on Bernstein’s theorem of counting solutions of generic systems, ultimately presenting the theorem, commentary, and extensions in a comprehensive and coherent manner. It begins with Bernstein’s original theorem expressing solutions of generic systems in terms of the mixed volume of their Newton polytopes, including complete proofs of its recent extension to affine space and some applications to open problems. The text also applies the developed techniques to derive and generalize Kushnirenko's results on Milnor numbers of hypersurface singularities, which has served as a precursor to the development of toric geometry. Ultimately, the book aims to present material in an elementary format, developing all necessary algebraic geometry to provide a truly accessible overview suitable to second-year graduate students.
How Math Explains the World: A Guide to the Power of Numbers, from Car Repair to Modern Physics
by D. SteinIn How Math Explains the World, mathematician Stein reveals how seemingly arcane mathematical investigations and discoveries have led to bigger, more world-shaking insights into the nature of our world. In the four main sections of the book, Stein tells the stories of the mathematical thinkers who discerned some of the most fundamental aspects of our universe. From their successes and failures, delusions, and even duels, the trajectories of their innovations—and their impact on society—are traced in this fascinating narrative. Quantum mechanics, space-time, chaos theory and the workings of complex systems, and the impossibility of a "perfect" democracy are all here. Stein's book is both mind-bending and practical, as he explains the best way for a salesman to plan a trip, examines why any thought you could have is imbedded in the number p , and—perhaps most importantly—answers one of the modern world's toughest questions: why the garage can never get your car repaired on time.Friendly, entertaining, and fun, How Math Explains the World is the first book by one of California's most popular math teachers, a veteran of both "math for poets" and Princeton's Institute for Advanced Studies. And it's perfect for any reader wanting to know how math makes both science and the world tick.
How Mathematicians Think: Using Ambiguity, Contradiction, and Paradox to Create Mathematics
by William ByersTo many outsiders, mathematicians appear to think like computers, grimly grinding away with a strict formal logic and moving methodically--even algorithmically--from one black-and-white deduction to another. Yet mathematicians often describe their most important breakthroughs as creative, intuitive responses to ambiguity, contradiction, and paradox. A unique examination of this less-familiar aspect of mathematics, How Mathematicians Think reveals that mathematics is a profoundly creative activity and not just a body of formalized rules and results. Nonlogical qualities, William Byers shows, play an essential role in mathematics. Ambiguities, contradictions, and paradoxes can arise when ideas developed in different contexts come into contact. Uncertainties and conflicts do not impede but rather spur the development of mathematics. Creativity often means bringing apparently incompatible perspectives together as complementary aspects of a new, more subtle theory. The secret of mathematics is not to be found only in its logical structure. The creative dimensions of mathematical work have great implications for our notions of mathematical and scientific truth, and How Mathematicians Think provides a novel approach to many fundamental questions. Is mathematics objectively true? Is it discovered or invented? And is there such a thing as a "final" scientific theory? Ultimately, How Mathematicians Think shows that the nature of mathematical thinking can teach us a great deal about the human condition itself.
How Much Inequality Is Fair?: Mathematical Principles of a Moral, Optimal, and Stable Capitalist Society
by Venkat VenkatasubramanianMany in the United States feel that the nation’s current level of economic inequality is unfair and that capitalism is not working for 90% of the population. Yet some inequality is inevitable. The question is: What level of inequality is fair? Mainstream economics has offered little guidance on fairness and the ideal distribution of income. Political philosophy, meanwhile, has much to say about fairness yet relies on qualitative theories that cannot be verified by empirical data. To address inequality, we need to know what the goal is—and for this, we need a quantitative, testable theory of fairness for free-market capitalism.How Much Inequality Is Fair? synthesizes concepts from economics, political philosophy, game theory, information theory, statistical mechanics, and systems engineering into a mathematical framework for a fair free-market society. The key to this framework is the insight that maximizing fairness means maximizing entropy, which makes it possible to determine the fairest possible level of pay inequality. The framework therefore provides a moral justification for capitalism in mathematical terms. Venkat Venkatasubramanian also compares his theory’s predictions to actual inequality data from various countries—showing, for instance, that Scandinavia has near-ideal fairness, while the United States is markedly unfair—and discusses the theory’s implications for tax policy, social programs, and executive compensation.
How Much Is A Million?
by David M. SchwartzAn attempt to help children conceptualize the immensity of numbers is aided immeasurably by the artist's jovial, detailed, whimsical illustrations.
How Not to Be Wrong: The Power of Mathematical Thinking
by Jordan Ellenberg<P>The Freakonomics of math—a math-world superstar unveils the hidden beauty and logic of the world and puts its power in our handsThe math we learn in school can seem like a dull set of rules, laid down by the ancients and not to be questioned. In How Not to Be Wrong, Jordan Ellenberg shows us how terribly limiting this view is: Math isn’t confined to abstract incidents that never occur in real life, but rather touches everything we do—the whole world is shot through with it. <P>Math allows us to see the hidden structures underneath the messy and chaotic surface of our world. It’s a science of not being wrong, hammered out by centuries of hard work and argument. Armed with the tools of mathematics, we can see through to the true meaning of information we take for granted: How early should you get to the airport? What does “public opinion” really represent? Why do tall parents have shorter children? Who really won Florida in 2000? And how likely are you, really, to develop cancer? <P>How Not to Be Wrong presents the surprising revelations behind all of these questions and many more, using the mathematician’s method of analyzing life and exposing the hard-won insights of the academic community to the layman—minus the jargon. Ellenberg chases mathematical threads through a vast range of time and space, from the everyday to the cosmic, encountering, among other things, baseball, Reaganomics, daring lottery schemes, Voltaire, the replicability crisis in psychology, Italian Renaissance painting, artificial languages, the development of non-Euclidean geometry, the coming obesity apocalypse, Antonin Scalia’s views on crime and punishment, the psychology of slime molds, what Facebook can and can’t figure out about you, and the existence of God. <P>Ellenberg pulls from history as well as from the latest theoretical developments to provide those not trained in math with the knowledge they need. Math, as Ellenberg says, is “an atomic-powered prosthesis that you attach to your common sense, vastly multiplying its reach and strength.” With the tools of mathematics in hand, you can understand the world in a deeper, more meaningful way. How Not to Be Wrong will show you how. <P><b>A New York Times Bestseller</b>
How Numbers Work: Discover the strange and beautiful world of mathematics (New Scientist Instant Expert Ser.)
by New ScientistThink of a number between one and ten. No, hang on, let's make this interesting. Between zero and infinity. Even if you stick to the whole numbers, there are a lot to choose from - an infinite number in fact. Throw in decimal fractions and infinity suddenly gets an awful lot bigger (is that even possible?) And then there are the negative numbers, the imaginary numbers, the irrational numbers like pi which never end. It literally never ends. <P><P>The world of numbers is indeed strange and beautiful. Among its inhabitants are some really notable characters - pi, e, the "imaginary" number i and the famous golden ratio to name just a few. Prime numbers occupy a special status. Zero is very odd indeed: is it a number, or isn't it? <P><P>How Numbers Work takes a tour of this mind-blowing but beautiful realm of numbers and the mathematical rules that connect them. Not only that, but take a crash course on the biggest unsolved problems that keep mathematicians up at night, find out about the strange and unexpected ways mathematics influences our everyday lives, and discover the incredible connection between numbers and reality itself. <P><P>ABOUT THE SERIES <P><P>New Scientist Instant Expert books are definitive and accessible entry points to the most important subjects in science; subjects that challenge, attract debate, invite controversy and engage the most enquiring minds. Designed for curious readers who want to know how things work and why, the Instant Expert series explores the topics that really matter and their impact on individuals, society, and the planet, translating the scientific complexities around us into language that's open to everyone, and putting new ideas and discoveries into perspective and context.
How Numbers Work: Discover the strange and beautiful world of mathematics (Instant Expert)
by New ScientistThink of a number between one and tenNo, hang on, let's make this interesting. Between zero and infinity. Even if you stick to the whole numbers, there are a lot to choose from - an infinite number in fact. Throw in decimal fractions and infinity suddenly gets an awful lot bigger (is that even possible?) And then there are the negative numbers, the imaginary numbers, the irrational numbers like p which never end. It literally never ends.The world of numbers is indeed strange and beautiful. Among its inhabitants are some really notable characters - pi, e, the square root of minus two and the famous golden ratio to name just a few. Prime numbers occupy a special status. Zero is very odd indeed. And even some apparently common-or-garden integers such as 37 have special properties. Adventures In Mathematics takes a tour of this mind-blowing but beautiful world of numbers and the mathematical rules that connect them. Find out mathematicians' favourite numbers, and the ones they are afraid of (spoiler: it isn't 13). Discover the incredible connection between numbers and the rules of nature. And learn some amazing mathematical tricks that will keep you amused for hours.
How Numbers Work: Discover the strange and beautiful world of mathematics (New Scientist Instant Expert)
by New ScientistHow Numbers Work is a tour of the mind-blowing but beautiful realm of numbers and the mathematical rules that connect them.Think of a number between one and ten. No, hang on, let's make this interesting. Between zero and infinity. Even if you stick to the whole numbers, there are a lot to choose from - an infinite number in fact. Throw in decimal fractions and infinity suddenly gets an awful lot bigger (is that even possible?) And then there are the negative numbers, the imaginary numbers, the irrational numbers like pi which never end. It literally never ends.The world of numbers is indeed strange and beautiful. Among its inhabitants are some really notable characters - pi, e, the "imaginary" number i and the famous golden ratio to name just a few. Prime numbers occupy a special status. Zero is very odd indeed: is it a number, or isn't it?How Numbers Work takes a tour of this mind-blowing but beautiful realm of numbers and the mathematical rules that connect them. Not only that, but take a crash course on the biggest unsolved problems that keep mathematicians up at night, find out about the strange and unexpected ways mathematics influences our everyday lives, and discover the incredible connection between numbers and reality itself. ABOUT THE SERIESNew Scientist Instant Expert books are definitive and accessible entry points to the most important subjects in science; subjects that challenge, attract debate, invite controversy and engage the most enquiring minds. Designed for curious readers who want to know how things work and why, the Instant Expert series explores the topics that really matter and their impact on individuals, society, and the planet, translating the scientific complexities around us into language that's open to everyone, and putting new ideas and discoveries into perspective and context.(P)2018 Hodder & Stoughton Limited
How Pi Can Save Your Life: Using Math to Survive Plane Crashes, Zombie Attacks, Alien Encounters, and Other Improbable Real-World Situations
by Chris WaringDiscover how you can guard a priceless painting in the Louvre, pilot a space shuttle back to Earth, and survive other ridiculous-yet-possible scenarios using the all-encompassing power of mathematical equations.Whether you paid much attention in math class or not, the inescapable truth is that real life is full of equations. You use differentiation when driving from point A to B and apply basic geometry when you&’re crossing the road between traffic, even if you don&’t realize it. But what if you were plummeting to your death inside a plane with no engine and you needed to know what size parachute to make from your cabinmate&’s sari in order to jump and survive? How Pi Can Save Your Life tackles some frankly ridiculous scenarios with essential, bullet-proof equations that you'll probably never apply in real life. But before you get into the life-saving mathematics, this book will start you off with the essential math that will be the key to taking on the wild scenarios later on in this book. Using only the power of equations, you will be able to: Communicate with an alien civilization Save your town from a zombie apocalypse Contain a major oil spill Excavate a fossil that could be a major scientific discovery Perform Hollywood stunts in a blockbuster action film And more! How Pi Can Save Your Life will take your basic understanding of mathematical equations to the next level, teaching you that the real key to surviving in an unpredictable world is understanding the natural forces and relationships that rule everything we do.
How Qualitative Data Analysis Happens: Moving Beyond "Themes Emerged"
by Áine Humble Elise RadinaWinner of the 2020 Anselm Strauss Award for Qualitative Family Research, National Council on Family Relations. How is qualitative data actually collected, analyzed, and accomplished? Real stories of How Qualitative Data Analysis Occurs: Moving Beyond "Themes Emerged" offers an in-depth look into how qualitative social science researchers studying family issues and dynamics approach their data analyses. It moves beyond the usual vague statement of "themes emerged from the data" to show readers how researchers actively and consciously arrive at their themes and conclusions, revealing the complexity and time involved in making sense of thousands of pages of interview data, multiple data sources, and diverse types of data. How Qualitative Data Analysis Occurs focuses on a diversity of topics in family research across the life course. The various authors provide detailed narratives into how they analyzed their data from previous publications, and what methodologies they used, ranging from arts-based research, autoethnography, community-based participatory research, ethnography, grounded theory, to narrative analysis. Supplemental figures, images, and screenshots which are referred to in the chapters, are included in an accompanying eResource, as well as links to the previously published work on which the chapters are based. This book is an invaluable resource for experienced and novice qualitative researchers throughout the social sciences.
How Round Is Your Circle?: Where Engineering and Mathematics Meet
by John Bryant Chris SangwinHow do you draw a straight line? How do you determine if a circle is really round? These may sound like simple or even trivial mathematical problems, but to an engineer the answers can mean the difference between success and failure. How Round Is Your Circle? invites readers to explore many of the same fundamental questions that working engineers deal with every day--it's challenging, hands-on, and fun. John Bryant and Chris Sangwin illustrate how physical models are created from abstract mathematical ones. Using elementary geometry and trigonometry, they guide readers through paper-and-pencil reconstructions of mathematical problems and show them how to construct actual physical models themselves--directions included. It's an effective and entertaining way to explain how applied mathematics and engineering work together to solve problems, everything from keeping a piston aligned in its cylinder to ensuring that automotive driveshafts rotate smoothly. Intriguingly, checking the roundness of a manufactured object is trickier than one might think. When does the width of a saw blade affect an engineer's calculations--or, for that matter, the width of a physical line? When does a measurement need to be exact and when will an approximation suffice? Bryant and Sangwin tackle questions like these and enliven their discussions with many fascinating highlights from engineering history. Generously illustrated, How Round Is Your Circle? reveals some of the hidden complexities in everyday things.
How Slow Is a Sloth?: Measure the Rainforest (Nature Numbers)
by Jill EsbaumIn Nature Numbers, math is beautiful, recognizable, and all around us! Highly engaging pictures of animals and nature scenes, along with cool chalk illustrations, are used to introduce basic math concepts and encourage kids to see a world of numbers all around them.K-2 math concepts include measuring. This book explores measuring elements of the rainforest with amazing nature pictures and chalk illustrations!