Turing Mathematician Alan Turing invented an imaginary computer known as the Turing Machine; in an age before computers, he explored the concept of what it meant to be "computable," creating the field of computability theory in the process, a foundation of present-day computer programming.The book expands Turing's original 36-page paper with additional background chapters and extensive annotations; the author elaborates on and clarifies many of Turing's statements, making the original difficult-to-read document accessible to present day programmers, computer science majors, math geeks, and others.Interwoven into the narrative are the highlights of Turing's own life: his years at Cambridge and Princeton, his secret work in cryptanalysis during World War II, his involvement in seminal computer projects, his speculations about artificial intelligence, his arrest and prosecution for the crime of "gross indecency," and his early death by apparent suicide at the age of 41.

    Includes many examples and figures. GENERAL TOPOLOGY. Set Theory and Logic. Topological Spaces and Continuous Functions. Connectedness and Compactness. Countability and Separation Axioms. The Tychonoff Theorem. Metrization Theorems and paracompactness. Complete Metric Spaces and Function Spaces. Baire Spaces and Dimension Theory. ALGEBRAIC TOPOLOGY. The Fundamental Group. Separation Theorems. The Seifert-van Kampen Theorem. Classification of Surfaces. Classification of Covering Spaces. Applications to Group Theory. For anyone needing a basic, thorough, introduction to general and algebraic topology and its applications.

    But more than this, there was gravity. After that, all hell broke loose… In A Universe from Nothing, Krauss revealed how our entire universe could arise from nothing. Now, he reveals what that something—reality—is. And, reality is not what we think or sense—it’s weird, wild, and counterintuitive; it’s hidden beneath everyday experience; and its inner workings seem even stranger than the idea that something can come from nothing. In a landmark, unprecedented work of scientific history, Krauss leads us to the furthest reaches of space and time, to scales so small they are invisible to microscopes, to the birth and rebirth of light, and into the natural forces that govern our existence. His unique blend of rigorous research and engaging storytelling invites us into the lives and minds of the remarkable, creative scientists who have helped to unravel the unexpected fabric of reality—with reason rather than superstition and dogma. Krauss has himself been an active participant in this effort, and he knows many of them well. The Greatest Story challenges us to re-envision ourselves and our place within the universe, as it appears that “God” does play dice with the universe. In the incisive style of his scintillating essays for The New Yorker, Krauss celebrates the greatest intellectual adventure ever undertaken—to understand why we are here in a universe where fact is stranger than fiction.

    From the months each of us spent suspended in the womb anticipating birth to the moments when we wait for sleep to transport us to other realities, we are always aware of gravity. In On Gravity, physicist A. Zee combines profound depth with incisive accessibility to take us on an original and compelling tour of Einstein's general theory of relativity.Inspired by Einstein's audacious suggestion that spacetime could ripple, Zee begins with the stunning discovery of gravity waves. He goes on to explain how gravity can be understood in comparison to other classical field theories, presents the idea of curved spacetime and the action principle, and explores cutting-edge topics, including black holes and Hawking radiation. Zee travels as far as the theory reaches, leaving us with tantalizing hints of the utterly unknown, from the intransigence of quantum gravity to the mysteries of dark matter and energy.Concise and precise, and infused with Zee's signature warmth and freshness of style, On Gravity opens a unique pathway to comprehending relativity and gaining deep insight into gravity, spacetime, and the workings of the universe.

    This wonderfully accessible book illuminates the professional and personal paths that led him to this remarkable conclusion.All interactions between particles in the universe, Lloyd explains, convey not only energy but also information—in other words, particles not only collide, they compute. And what is the entire universe computing, ultimately? “Its own dynamical evolution,” he says. “As the computation proceeds, reality unfolds.”To elucidate his theory, Lloyd examines the history of the cosmos, posing questions that in other hands might seem unfathomably complex: How much information is there in the universe? What information existed at the moment of the Big Bang and what happened to it? How do quantum mechanics and chaos theory interact to create our world? Could we attempt to re-create it on a giant quantum computer? Programming the Universe presents an original and compelling vision of reality, revealing our world in an entirely new light.

    Most know Richard Feynman for the hilarious anecdotes and exploits in his best-selling books Surely You're Joking, Mr. Feynman! and What DoYou Care What Other People Think? But not always obvious in those stories was his brilliance as a pure scientist—one of the century's greatest physicists. With this book and CD, we hear the voice of the great Feynman in all his ingenuity, insight, and acumen for argument. This breathtaking lecture—"The Motion of the Planets Around the Sun"—uses nothing more advanced than high-school geometry to explain why the planets orbit the sun elliptically rather than in perfect circles, and conclusively demonstrates the astonishing fact that has mystified and intrigued thinkers since Newton: Nature obeys mathematics. David and Judith Goodstein give us a beautifully written short memoir of life with Feynman, provide meticulous commentary on the lecture itself, and relate the exciting story of their effort to chase down one of Feynman's most original and scintillating lectures.

    But when Feynman said he didn’t understand quantum mechanics, he didn’t mean that he couldn’t do it – he meant that’s all he could do. He didn’t understand what the maths was saying: what quantum mechanics tells us about reality.Over the past decade or so, the enigma of quantum mechanics has come into sharper focus. We now realise that quantum mechanics is less about particles and waves, uncertainty and fuzziness, than a theory about information: about what can be known and how.This is more disturbing than our bad habit of describing the quantum world as ‘things behaving weirdly’ suggests. It calls into question the meanings and limits of space and time, cause and effect, and knowledge itself.The quantum world isn’t a different world: it is our world, and if anything deserves to be called ‘weird’, it’s us. This exhilarating book is about what quantum maths really means – and what it doesn’t mean.

    This book gives a lucid, exciting, and accessible account of the surprising and counterintuitive ideas that shape our understanding of the sub-atomic world. It does not disguise the problems of interpretation that still remain unsettled 75 years after the initial discoveries. The main text makes no use of equations, but there is a Mathematical Appendix for those desiring stronger fare. Uncertainty, probabilistic physics, complementarity, the problematic character of measurement, and decoherence are among the many topics discussed. This volume offers the reader access to one of the greatest discoveries in the history of physics and one of the outstanding intellectual achievements of the twentieth century.About the Series: Combining authority with wit, accessibility, and style, Very Short Introductions offer an introduction to some of life's most interesting topics. Written by experts for the newcomer, they demonstrate the finest contemporary thinking about the central problems and issues in hundreds of key topics, from philosophy to Freud, quantum theory to Islam.

    These topics lie at the heart of many exciting areas of contemporary science and engineering - communication, signal processing, data mining, machine learning, pattern recognition, computational neuroscience, bioinformatics, and cryptography. This textbook introduces theory in tandem with applications. Information theory is taught alongside practical communication systems, such as arithmetic coding for data compression and sparse-graph codes for error-correction. A toolbox of inference techniques, including message-passing algorithms, Monte Carlo methods, and variational approximations, are developed alongside applications of these tools to clustering, convolutional codes, independent component analysis, and neural networks. The final part of the book describes the state of the art in error-correcting codes, including low-density parity-check codes, turbo codes, and digital fountain codes -- the twenty-first century standards for satellite communications, disk drives, and data broadcast. Richly illustrated, filled with worked examples and over 400 exercises, some with detailed solutions, David MacKay's groundbreaking book is ideal for self-learning and for undergraduate or graduate courses. Interludes on crosswords, evolution, and sex provide entertainment along the way. In sum, this is a textbook on information, communication, and coding for a new generation of students, and an unparalleled entry point into these subjects for professionals in areas as diverse as computational biology, financial engineering, and machine learning.

    Our Big Bang, our distant future, parallel worlds, the sub-atomic and intergalactic - none of them are what they seem. But there is a way to understand this immense strangeness - mathematics. Seeking an answer to the fundamental puzzle of why our universe seems so mathematical, Tegmark proposes a radical idea: that our physical world not only is described by mathematics, but that it is mathematics. This may offer answers to our deepest questions: How large is reality? What is everything made of? Why is our universe the way it is?Table of ContentsPreface 1 What Is Reality? Not What It Seems • What’s the Ultimate Question? • The Journey Begins Part One: Zooming Out 2 Our Place in Space Cosmic Questions • How Big Is Space? • The Size of Earth • Distance to the Moon • Distance to the Sun and the Planets • Distance to the Stars • Distance to the Galaxies • What Is Space? 3 Our Place in TimeWhere Did Our Solar System Come From? • Where Did theGalaxies Come From? • Where Did the Mysterious MicrowavesCome From? • Where Did the Atoms Come From? 4 Our Universe by NumbersWanted: Precision Cosmology • Precision Microwave-Background Fluctuations • Precision Galaxy Clustering • The Ultimate Map of Our Universe • Where Did Our Big Bang Come From? 5 Our Cosmic Origins What’s Wrong with Our Big Bang? • How Inflation Works • The Gift That Keeps on Giving • Eternal Inflation 6 Welcome to the Multiverse The Level I Multiverse • The Level II Multiverse • Multiverse Halftime Roundup Part Two: Zooming In 7 Cosmic Legos Atomic Legos • Nuclear Legos • Particle-Physics Legos • Mathematical Legos • Photon Legos • Above the Law? • Quanta and Rainbows • Making Waves • Quantum Weirdness • The Collapse of Consensus • The Weirdness Can’t Be Confined • Quantum Confusion 8 The Level III Multiverse The Level III Multiverse • The Illusion of Randomness • Quantum Censorship • The Joys of Getting Scooped • Why Your Brain Isn’t a Quantum Computer • Subject, Object and Environment • Quantum Suicide • Quantum Immortality? • Multiverses Unified • Shifting Views: Many Worlds or Many Words? Part Three: Stepping Back 9 Internal Reality, External Reality and Consensus Reality External Reality and Internal Reality • The Truth, the Whole Truth and Nothing but the Truth • Consensus Reality • Physics: Linking External to Consensus Reality 10 Physical Reality and Mathematical Reality Math, Math Everywhere! • The Mathematical Universe Hypothesis • What Is a Mathematical Structure? 11 Is Time an Illusion? How Can Physical Reality Be Mathematical? • What Are You? • Where Are You? (And What Do You Perceive?) • When Are You? 12 The Level IV Multiverse Why I Believe in the Level IV Multiverse • Exploring the Level IV Multiverse: What’s Out There? • Implications of the Level IV Multiverse • Are We Living in a Simulation? • Relation Between the MUH, the Level IV Multiverse and Other Hypotheses •Testing the Level IV Multiverse 13 Life, Our Universe and Everything How Big Is Our Physical Reality? • The Future of Physics • The Future of Our Universe—How Will It End? • The Future of Life •The Future of You—Are You Insignificant? Acknowledgments Suggestions for Further Reading Index

    We're all looking for new ways of thinking that can bring about real solutions to modern problems, from the pursuit of inner serenity to solving world conflicts. In Do You QuantumThink? author Dianne Collins shares her ingenious discovery that reveals a critical missing link to make sense of our changing times. Her discovery provides us with the understanding and methodology to rise above problems of today by laying the foundation for an entirely new way to think.Part science, part philosophy, part spirituality, Do You QuantumThink? draws on a wide spectrum of sources, from cutting edge innovations in the sciences to the insights of the world's greatest spiritual leaders. This book will make you laugh, free you from limiting ideas, and introduce you to the most advanced principles and practical methods for living. Do You QuantumThink? will rock your world in the best of ways as you experience one revelation after another.

    This is done in a way that is not only easy to understand, but is actually fun! Professors and engineers, with high school and college students following closely, comprise the largest percentage of our readers. It is a must-have for anyone interested in music, mathematics, physics, engineering, or complex science. Dr. Yoichiro Nambu, 2008 Nobel Prize Winner in Physics, served as a senior adviser to the English version of Who is Fourier? A Mathematical Adventure.

    The more we discover, the more puzzling the universe appears to be. How and why are the laws of nature what they are? A philosopher and a physicist, world-renowned for their radical ideas in their fields, argue for a revolution. To keep cosmology scientific, we must replace the old view in which the universe is governed by immutable laws by a new one in which laws evolve. Then we can hope to explain them. The revolution that Roberto Mangabeira Unger and Lee Smolin propose relies on three central ideas. There is only one universe at a time. Time is real: everything in the structure and regularities of nature changes sooner or later. Mathematics, which has trouble with time, is not the oracle of nature and the prophet of science; it is simply a tool with great power and immense limitations. The argument is readily accessible to non-scientists as well as to the physicists and cosmologists whom it challenges.

    Life remains the only way to make life. Are we still missing a vital ingredient in its creation?      Like Richard Dawkins' The Selfish Gene, which provided a new perspective on how evolution works, Life on the Edge alters our understanding of our world's fundamental dynamics. Bringing together first-hand experience at the cutting edge of science with unparalleled gifts of explanation, Jim Al-Khalili and Johnjoe Macfadden reveal that missing ingredient to be quantum mechanics; the phenomena that lie at the heart of this most mysterious of sciences. Drawing on recent ground-breaking experiments around the world, each chapter in Life on the Edge engages by illustrating one of life's puzzles: How do migrating birds know where to go? How do we really smell the scent of a rose? How do our genes copy themselves with such precision? Life on the Edge accessibly reveals how quantum mechanics can answer these probing questions of the universe. Guiding the reader through the rapidly unfolding discoveries of the last few years, Al-Khalili and McFadden communicate the excitement of the explosive new field of quantum biology and its potentially revolutionary applications, while offering insights into the biggest puzzle of all: what is life? As they brilliantly demonstrate in these groundbreaking pages, life exists on the quantum edge.

    The universe is all about extremes, and in this engaging and thought-provoking book, astronomer Bryan Gaensler gives a whirlwind tour of the galaxies, with an emphasis on its fastest, hottest, heaviest, brightest, oldest, densest, and even loudest elements. From supernova explosions a billion times brighter than the sun to an asteroid the size of a beach ball, Extreme Cosmos offers a fascinating, fresh, and informed perspective of the remarkable richness of the universe, and the incredible physics that modern astronomy has revealed.