These singularities are places where space-time begins or ends, and the presently known laws of physics break down. They will occur inside black holes, and in the past are what might be construed as the beginning of the universe. To show how these predictions arise, the authors discuss the General Theory of Relativity in the large. Starting with a precise formulation of the theory and an account of the necessary background of differential geometry, the significance of space-time curvature is discussed and the global properties of a number of exact solutions of Einstein's field equations are examined. The theory of the causal structure of a general space-time is developed, and is used to study black holes and to prove a number of theorems establishing the inevitability of singualarities under certain conditions. A discussion of the Cauchy problem for General Relativity is also included in this 1973 book.

    Delving into an immense range of topics, from philosophy and literature to social criticism to mathematics and science, with essays that date from 1930s to the 1990s, Martin Gardner has astounded readers with his insight and erudition. The Night Is Large is the crowning achievement of his extraordinary career.

    Never before in history have we been able to acquire, record, communicate, and use information in so many different forms. Never before have we had access to such vast quantities of data of every kind. This revolution goes far beyond the limitless content that fills our lives, because information also underlies our understanding of ourselves, the natural world, and the universe. It is the key that unites fields as different as linguistics, cryptography, neuroscience, genetics, economics, and quantum mechanics. And the fact that information bears no necessary connection to meaning makes it a profound puzzle that people with a passion for philosophy have pondered for centuries.Table of ContentsLECTURE 1The Transformability of Information 4LECTURE 2Computation and Logic Gates 17LECTURE 3Measuring Information 26LECTURE 4Entropy and the Average Surprise 34LECTURE 5Data Compression and Prefix-Free Codes 44LECTURE 6Encoding Images and Sounds 57LECTURE 7Noise and Channel Capacity 69LECTURE 8Error-Correcting Codes 82LECTURE 9Signals and Bandwidth 94LECTURE 10Cryptography and Key Entropy 110LECTURE 11Cryptanalysis and Unraveling the Enigma 119LECTURE 12Unbreakable Codes and Public Keys 130LECTURE 13What Genetic Information Can Do 140LECTURE 14Life’s Origins and DNA Computing 152LECTURE 15Neural Codes in the Brain 169LECTURE 16Entropy and Microstate Information 185LECTURE 17Erasure Cost and Reversible Computing 198LECTURE 18Horse Races and Stock Markets 213LECTURE 19Turing Machines and Algorithmic Information 226LECTURE 20Uncomputable Functions and Incompleteness 239LECTURE 21Qubits and Quantum Information 253LECTURE 22Quantum Cryptography via Entanglement 266LECTURE 23It from Bit: Physics from Information 281LECTURE 24The Meaning of Information 293

    Twenty-two years later, he did. April 1945 – Berlin. The world had been at war for more than five-and-a-half years – approximately seventy million people were dead across the globe. The epicentre of the twelve-year-old Third Reich was now surrounded, enveloped by bitter Soviet forces hardened by Nazi barbarity in the east over the last four years. As the buildings were blasted into rubble, pounded by Russian guns and bombs, before their troops and tanks, Hitler was hunkered down in his last headquarters – the dark and damp bunker under the Reich Chancellery. As the Third Reich began to crumble as fast as the city’s buildings, what was the state of mind of the tyrant? Only his closest and fanatical allies saw the collapse, none more so than Hitler’s servants, Otto Gunsche and Heinz Linge – two individuals which witnessed the final act of their regime. An act tinged over the last ten days in late April with selfish betrayal, increasingly forlorn hope, pleas, desperation and eventually suicide. As the Soviets closed in with impending vigour, in the concrete tomb below ground and under the thunderous booms of the petrifying battle for Berlin, the mind of the dictator disintegrated into drugs, delusion and a determination to die. Not by the enemy bullet but one of his own. This is the story of the people who held a unique place in world history – the ones who were there when the nightmare of Nazism and the horrors which accompanied it was finally banished as a dark chapter in the story of the human race.

    Featuring Gerber's signature easy-to-understand, easy-to-implement style, The E-Myth Attorney features:A complete start-up guide you can use to get your practice off the ground quickly, as well as comprehensive action steps for maximizing the performance of an existing practice Industry specific advice from two recognized legal experts that have developed a highly successful legal practice using Gerber's principles Gerber's universal appeal as a recognized expert on small businesses who has coached, taught, and trained over 60,000 small businesses The E-Myth Attorney is the last guide you'll ever need to make the difference in building or developing your successful legal practice.

    David Quammen, together with series editor Burkhard Bilger, has assembled a remarkable group of writers whose selections appeared in periodicals from NATIONAL GEOGRAPHIC, SCIENCE, and THE NEW YORKER to PUERTO DEL SOL and DOUBLETAKE. Among the acclaimed writers represented in this volume are Richard Preston on “The Demon in the Freezer,” John McPhee bidding “Farewell to the Nineteeth Century,” Oliver Sacks remembering the “Brilliant Light” of his boyhood, and Wendell Berry going “Back to the Land.” Also including such literary lights as Anne Fadiman, David Guterson, Edward Hoagland, Natalie Angier, and Peter Matthiessen, this new collection presents selections bound together by their timelessness.

    Casti contemplates an imaginary evening of intellectual inquiry—a sort of “My Dinner with” not Andre, but five of the most brilliant thinkers of the twentieth century.Imagine, if you will, one stormy summer evening in 1949, as novelist and scientist C. P. Snow, Britain’s distinguished wartime science advisor and author of The Two Cultures, invites four singular guests to a sumptuous seven-course dinner at his alma mater, Christ’s College, Cambridge, to discuss one of the emerging scientific issues of the day: Can we build a machine that could duplicate human cognitive processes? The distinguished guest list for Snow’s dinner consists of physicist Erwin Schrodinger, inventor of wave mechanics; Ludwig Wittgenstein, the famous twentieth-century philosopher of language, who posited two completely contradictory theories of human thought in his lifetime; population geneticist/science popularizer J.B.S. Haldane; and Alan Turing, the mathematician/codebreaker who formulated the computing scheme that foreshadowed the logical structure of all modern computers. Capturing not only their unique personalities but also their particular stands on this fascinating issue, Casti dramatically shows what each of these great men might have argued about artificial intelligence, had they actually gathered for dinner that midsummer evening.With Snow acting as referee, a lively intellectual debate unfolds. Philosopher Wittgenstein argues that in order to become conscious, a machine would have to have life experiences similar to those of human beings—such as pain, joy, grief, or pleasure. Biologist Haldane offers the idea that mind is a separate entity from matter, so that regardless of how sophisticated the machine, only flesh can bond with that mysterious force called intelligence. Both physicist Schrodinger and, of course, computer pioneer Turing maintain that it is not the substance, but rather the organization of that substance, that makes a mind conscious.With great verve and skill, Casti recreates a unique and thrilling moment of time in the grand history of scientific ideas. Even readers who have already formed an opinion on artificial intelligence will be forced to reopen their minds on the subject upon reading this absorbing narrative. After almost four decades, the solutions to the epic scientific and philosophical problems posed over this meal in C. P. Snow’s old rooms at Christ’s College remains tantalizingly just out of reach, making this adventure into scientific speculation as valid today as it was in 1949.

    Aczel turns his sights on probability theory -- the branch of mathematics that measures the likelihood of a random event. He explains probability in clear, layman's terms, and shows its practical applications. What is commonly called luck has mathematical roots and in Chance, you'll learn to increase your odds of success in everything from true love to the stock market. For thousands of years, the twin forces of chance and mischance have beguiled humanity like none other. Why does fortune smile on some people, and smirk on others? What is luck, and why does it so often visit the undeserving? How can we predict the random events happening around us? Even better, how can we manipulate them? In this delightful and lucid voyage through the realm of the random, Dr. Aczel once again makes higher mathematics intelligible to us.

    Julia Diggins masterfully recreates the atmosphere of ancient times, when men, using three simple tools, the string, the straightedge, and the shadow, discovered the basic principles and constructions of elementary geometry. Her book reveals how these discoveries related to the early civilizations of Mesopotamia, Egypt, and Greece.The fabric of the story is woven out of archeological and historical records and legends about the major men of mathematics. By reconstructing the events as they might have happened, Diggins enables the attentive reader to easily follow the pattern of reasoning that leads to an ingenious proof of the Pythagorean theorem, an appreciation of the significance of the Golden Mean in art and architecture, and the construction of the five regular solids.Out of print for 34 years, Julia Diggins' classic book is back and is a must-read for middle school students or for parents helping their children through their first geometry course. You will be fascinated with the graphic illustrations and written depiction of how the knowledge and wisdom of so many cultures helped shape our civilization today. This book is popular with teachers and parents who use Jamie York's Making Math Meaningful curriculum books.

    'Big data', 'data science', and 'machine learning' have become familiar terms in the news, as statistical methods are brought to bear upon the enormous data sets of modern science and commerce. How did we get here? And where are we going? This book takes us on an exhilarating journey through the revolution in data analysis following the introduction of electronic computation in the 1950s. Beginning with classical inferential theories - Bayesian, frequentist, Fisherian - individual chapters take up a series of influential topics: survival analysis, logistic regression, empirical Bayes, the jackknife and bootstrap, random forests, neural networks, Markov chain Monte Carlo, inference after model selection, and dozens more. The distinctly modern approach integrates methodology and algorithms with statistical inference. The book ends with speculation on the future direction of statistics and data science.

    But they wanted the one thing that no amount of money could buy: life. They wanted to create the Perfect Crime--to kidnap and murder a 14-year-old boy for the thrill of getting away with murder.The crime was so horrifying that even legendary filmmaker Alfred Hitchcock took notice, and directed his version of the story: Rope. But the real story of the Rope is much more brutal and suspenseful than even Hitchcock could do justice to. Read the real history in this thrilling true crime book.

    But now, with the aid of high-speed computers, scientists have been able to penetrate a reality that is changing the way we perceive the universe. Their findings -- the basis for chaos theory -- represent one of the most exciting scientific pursuits of our time.No better introduction to this find could be found than John Briggs and F. David Peat's Turbulent Mirror. Together, they explore the many faces of chaos and reveal how its law direct most of the processes of everyday life and how it appears that everything in the universe is interconnected -- discovering an "emerging science of wholeness."Turbulent Mirror introduces us to the scientists involved in study this endlessly strange field; to the theories that are turning our perception of the world on its head; and to the discoveries in mathematics, biology, and physics that are heralding a revolution more profound than the one responsible for producing the atomic bomb. With practical applications ranging from the control of traffic flow and the development of artifical intelligence to the treatment of heart attacks and schizophrenia, chaos promises to be an increasingly rewarding area of inquiry -- of interest to everyone.

    For the calculus-based General Physics course primarily taken by engineers and scientists.

    This groundbreaking and powerful theory now forms the basis of computer science. In Turing's Vision, Chris Bernhardt explains the theory, Turing's most important contribution, for the general reader. Bernhardt argues that the strength of Turing's theory is its simplicity, and that, explained in a straightforward manner, it is eminently understandable by the nonspecialist. As Marvin Minsky writes, -The sheer simplicity of the theory's foundation and extraordinary short path from this foundation to its logical and surprising conclusions give the theory a mathematical beauty that alone guarantees it a permanent place in computer theory.- Bernhardt begins with the foundation and systematically builds to the surprising conclusions. He also views Turing's theory in the context of mathematical history, other views of computation (including those of Alonzo Church), Turing's later work, and the birth of the modern computer.In the paper, -On Computable Numbers, with an Application to the Entscheidungsproblem, - Turing thinks carefully about how humans perform computation, breaking it down into a sequence of steps, and then constructs theoretical machines capable of performing each step. Turing wanted to show that there were problems that were beyond any computer's ability to solve; in particular, he wanted to find a decision problem that he could prove was undecidable. To explain Turing's ideas, Bernhardt examines three well-known decision problems to explore the concept of undecidability; investigates theoretical computing machines, including Turing machines; explains universal machines; and proves that certain problems are undecidable, including Turing's problem concerning computable numbers.

    He decried the internment of Japanese-Americans in World War Two, agitated against nuclear weapons, promoted vitamin C as a cure for the common cold and researched the idea of DNA.