Shows how any proof can be understood as a sequence of techniques. Covers the full range of techniques used in proofs, such as the contrapositive, induction, and proof by contradiction. Explains how to identify which techniques are used and how they are applied in the specific problem. Illustrates how to read written proofs with many step-by-step examples. Includes new, expanded appendices related to discrete mathematics, linear algebra, modern algebra and real analysis.

    Yet geometry is so much more than shapes and numbers; indeed, it governs much of our lives—from architecture and microchips to car design, animated movies, the molecules of food, even our own body chemistry. And as Siobhan Roberts elegantly conveys in The King of Infinite Space, there can be no better guide to the majesty of geometry than Donald Coxeter, perhaps the greatest geometer of the twentieth century.Many of the greatest names in intellectual history—Pythagoras, Plato, Archimedes, Euclid— were geometers, and their creativity and achievements illuminate those of Coxeter, revealing geometry to be a living, ever-evolving endeavor, an intellectual adventure that has always been a building block of civilization. Coxeter's special contributions—his famed Coxeter groups and Coxeter diagrams—have been called by other mathematicians "tools as essential as numbers themselves," but his greatest achievement was to almost single-handedly preserve the tradition of classical geometry when it was under attack in a mathematical era that valued all things austere and rational.Coxeter also inspired many outside the field of mathematics. Artist M. C. Escher credited Coxeter with triggering his legendary Circle Limit patterns, while futurist/inventor Buckminster Fuller acknowledged that his famed geodesic dome owed much to Coxeter's vision. The King of Infinite Space is an elegant portal into the fascinating, arcane world of geometry.

    "More concretely," the authors explain, "it is the controlled manipulation of mathematical formulas, using a collection of techniques for solving problems."

    A. Zee, a distinguished physicist and skillful expositor, tells the exciting story of how today's theoretical physicists are following Einstein in their search for the beauty and simplicity of Nature. Animated by a sense of reverence and whimsy, the book describes the majestic sweep and accomplishments of twentieth-century physics. In the end, we stand in awe before the grand vision of modern physics--one of the greatest chapters in the intellectual history of humankind.

    Its easy-to-read treatment offers an intuitive approach, featuring informal discussions followed by thematically arranged exercises. Intended for undergraduate courses in abstract algebra, it is suitable for junior- and senior-level math majors and future math teachers. This second edition features additional exercises to improve student familiarity with applications. An introductory chapter traces concepts of abstract algebra from their historical roots. Succeeding chapters avoid the conventional format of definition-theorem-proof-corollary-example; instead, they take the form of a discussion with students, focusing on explanations and offering motivation. Each chapter rests upon a central theme, usually a specific application or use. The author provides elementary background as needed and discusses standard topics in their usual order. He introduces many advanced and peripheral subjects in the plentiful exercises, which are accompanied by ample instruction and commentary and offer a wide range of experiences to students at different levels of ability.

    It requires, in short, an understanding of probability. In this Very Short Introduction, John Haigh introduces the ideas of probability--and the different philosophical approaches to probability--and gives a brief account of the history of development of probability theory, from Galileo and Pascal to Bayes, Laplace, Poisson, and Markov. He describes the basic probability distributions and discusses a wide range of applications in science, economics, and a variety of other contexts such as games and betting. He concludes with an intriguing discussion of coincidences and some curious paradoxes.

    Bringing together the best and most interesting science stories appearing in Quanta Magazine over the past five years, Alice and Bob Meet the Wall of Fire reports on some of the greatest scientific minds as they test the limits of human knowledge. Quanta, under editor-in-chief Thomas Lin, is the only popular publication that offers in-depth coverage of today's challenging, speculative, cutting-edge science. It communicates science by taking it seriously, wrestling with difficult concepts and clearly explaining them in a way that speaks to our innate curiosity about our world and ourselves.In the title story, Alice and Bob--beloved characters of various thought experiments in physics--grapple with gravitational forces, possible spaghettification, and a massive wall of fire as Alice jumps into a black hole. Another story considers whether the universe is impossible, in light of experimental results at the Large Hadron Collider. We learn about quantum reality and the mystery of quantum entanglement; explore the source of time's arrow; and witness a eureka moment when a quantum physicist exclaims: "Finally, we can understand why a cup of coffee equilibrates in a room." We reflect on humans' enormous skulls and the Brain Boom; consider the evolutionary benefits of loneliness; peel back the layers of the newest artificial-intelligence algorithms; follow the "battle for the heart and soul of physics"; and mourn the disappearance of the "diphoton bump," revealed to be a statistical fluctuation rather than a revolutionary new particle. These stories from Quanta give us a front-row seat to scientific discovery.ContributorsPhilip Ball, K. C. Cole, Robbert Dijkgraaf, Dan Falk, Courtney Humphries, Ferris Jabr, Katia Moskvitch, George Musser, Michael Nielsen, Jennifer Ouellette, John Pavlus, Emily Singer, Andreas von Bubnoff, Frank Wilczek, Natalie Wolchover, Carl Zimmer

    Switching from thin tailed to fat tailed distributions requires more than "changing the color of the dress." Traditional asymptotics deal mainly with either n=1 or n=∞, and the real world is in between, under the "laws of the medium numbers"-which vary widely across specific distributions. Both the law of large numbers and the generalized central limit mechanisms operate in highly idiosyncratic ways outside the standard Gaussian or Levy-Stable basins of convergence. A few examples: - The sample mean is rarely in line with the population mean, with effect on "na�ve empiricism," but can be sometimes be estimated via parametric methods. - The "empirical distribution" is rarely empirical. - Parameter uncertainty has compounding effects on statistical metrics. - Dimension reduction (principal components) fails. - Inequality estimators (Gini or quantile contributions) are not additive and produce wrong results. - Many "biases" found in psychology become entirely rational under more sophisticated probability distributions. - Most of the failures of financial economics, econometrics, and behavioral economics can be attributed to using the wrong distributions. This book, the first volume of the Technical Incerto, weaves a narrative around published journal articles.

    And proves once and for all that maths isn't just for old men with white hair and beards who associate with elves.Maths has never been merrier.

    It features a visual presentation, designed to encourage learning; revised exercises to ensure clarity, balance and relevance; and clear commentary on the difficult subject of critical multivariable calculus topics.

    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. Recently published, this new 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.

    The book includes full discussions of many problems of current interest which are not treated in any extant book, and all these matters are considered with perception and understanding."—S. Chandrasekhar "A tour de force: lucid, straightforward, mathematically rigorous, exacting in the analysis of the theory in its physical aspect."—L. P. Hughston, Times Higher Education Supplement"Truly excellent. . . . A sophisticated text of manageable size that will probably be read by every student of relativity, astrophysics, and field theory for years to come."—James W. York, Physics Today

    Bridging the gap from geometry to the latest work in observational cosmology, the book illustrates the connection between geometry and the behavior of the physical universe and explains how radiation remaining from the big bang may reveal the actual shape of the universe.

    Boundless works with subject matter experts to select the best open educational resources available on the web, review the content for quality, and create introductory, college-level textbooks designed to meet the study needs of university students.<br><br>This textbook covers:<br><br><b>Human Anatomy and Physiology Introduction</b> -- Anatomy and Physiology Overview, Life, Homeostasis, Anatomical Terms, Clinical Cases<br><br><b>General Chemistry</b> -- Matter and Energy, Element Properties: Atomic structure, Chemical Bonds, Chemical Reactions, Inorganic Compounds, Organic Compounds<br><br><b>Cellular Structure and Function</b> -- the study of cells, Cell membranes and the fluid mosaic model, Transport across membranes, How reception works in cell signaling, Nucleus and Ribosomes, Organelles, The Cytoskeleton, External cellular components, Cell division: process and importance, The cell cycle, Transcription and translation, RNA processing, Translation to a polypeptide, Transcription, Apoptosis signals an orderly cell death<br><br><b>Tissues</b> -- Epithelial Tissue, Cell Junctions, Clinical Cases, Tissue Repair, Tissue Development, Cancer, Connective Tissue, Membranes, Nervous Tissue<br><br><b>The Integumentary System</b> -- The Skin, Accessory Structures of the Skin, Functions of the Integumentary System, Wound Healing, Integumentary System Development, Skin Disorders, Imbalances, Diseases, and Clinical Cases<br><br><b>Skeletal Tissue</b> -- Cartilage, Bone Classification, Bone Formation, Bone and Calcium, Bone Development, Bone Diseases, Disorders, Imbalances, and Clinical Cases<br><br><b>The Skeletal System</b> -- Overview of the Musculoskeletal system, Divisions of the Skeletal System, The Axial Skeleton, Skull, Hyoid Bone, Vertebral Column, Thorax, Clinical Cases of the Axial Skeleton, The Appendicular Skeleton, The Pectoral (Shoulder) Girdle, Upper Limb, The Pelvic (Hip) Girdle, Lower Limb, Skeletal System Development, Clinical Cases of the Appendicular Skeleton<br><br><b>Joints</b> -- Classification of Joints, Synovial Joints, Joint Development, Clinical Cases<br><br><b>Muscle Tissue</b> -- Overview of Muscle Tissue, Skeletal Muscle, Control of Muscle Tension, Muscle Metabolism, Exercise and Skeletal Muscle Tissue, Smooth Muscle, Clinical Cases: Muscle Disorders, Development of Muscle<br><br><b>The Muscular System</b> -- Overview of the Muscular System, Head And Neck Muscles, Trunk Muscles, Muscles of the Upper Limb, Muscles of the Lower Limb, Clinical Cases and Muscular System Disorders<br><br><b>Nervous Tissue</b> -- Overview of the Nervous System, Neuroglia, Neurons, Collections of Nervous Tissue, Neurophysiology<br><br><b>Central Nervous System (CNS)</b> -- The Brain, Consciousness, Sleep, Language, and Memory, Protection of the Brain, Parts of The Brain Stem, The Cerebellum, The Diencephalon, Cerebral Cortex (or Cerebral Hemispheres), Functional Systems of the Cerebral Cortex, Development of the CNS, Brain Disorders and Clinical Cases, The Spinal Cord, Spinal Cord Anatomy, Disorders and Clinical Cases of the Spinal Cord<br><br><b>Peripheral Nervous System (PNS)</b> -- Sensation, Sensory Receptors, Somatosensory System, Nerves, Cranial Nerves, Spinal Nerves, Distribution of Spinal Nerves, Motor Activity, Motor Pathways, Reflexes, Pain, Development of the Nervous System, Disorders of Spinal Nerves and Clinical Cases<br><br><b>Autonomic Nervous System (ANS)</b> -- Autonomic Nervous System, ANS Anatomy, Physiology of ANS,

    Requiring only basic algebra and trigonometry, the new edition is notable for its readability, learning features and numerous full-color photos and illustrations. A systems approach is used to cover state-of-the-art communications technologies, to best reflect current industry practice. This edition contains greatly expanded and updated material on the Internet, cell phones, and wireless technologies. Practical skills like testing and troubleshooting are integrated throughout. A brand-new Laboratory & Activities Manual provides both hands-on experiments and a variety of other activities, reflecting the variety of skills now needed by technicians. A new Online Learning Center web site is available, with a wealth of learning resources for students. An Instructor Productivity Center CD-ROM features solutions to all problems, PowerPoint lessons, and ExamView test banks for each chapter.