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The Golden Ratio: The Story of PHI, the World's Most Astonishing Number
by Mario LivioThroughout history, thinkers from mathematicians to theologians have pondered the mysterious relationship between numbers and the nature of reality. In this fascinating book, Mario Livio tells the tale of a number at the heart of that mystery: phi, or 1.6180339887...This curious mathematical relationship, widely known as "The Golden Ratio," was discovered by Euclid more than two thousand years ago because of its crucial role in the construction of the pentagram, to which magical properties had been attributed. Since then it has shown a propensity to appear in the most astonishing variety of places, from mollusk shells, sunflower florets, and rose petals to the shape of the galaxy. Psychological studies have investigated whether the Golden Ratio is the most aesthetically pleasing proportion extant, and it has been asserted that the creators of the Pyramids and the Parthenon employed it. It is believed to feature in works of art from Leonardo da Vinci's Mona Lisa to Salvador Dali's The Sacrament of the Last Supper, and poets and composers have used it in their works. It has even been found to be connected to the behavior of the stock market!The Golden Ratio is a captivating journey through art and architecture, botany and biology, physics and mathematics. It tells the human story of numerous phi-fixated individuals, including the followers of Pythagoras who believed that this proportion revealed the hand of God; astronomer Johannes Kepler, who saw phi as the greatest treasure of geometry; such Renaissance thinkers as mathematician Leonardo Fibonacci of Pisa; and such masters of the modern world as Goethe, Cezanne, Bartok, and physicist Roger Penrose. Wherever his quest for the meaning of phi takes him, Mario Livio reveals the world as a place where order, beauty, and eternal mystery will always coexist.From the Hardcover edition.
The Golden Ratio: The Divine Beauty of Mathematics
by Gary B. MeisnerThis enlightening and gorgeously illustrated book explores the beauty and mystery of the divine proportion in art, architecture, nature, and beyond.From the pyramids of Giza, to quasicrystals, to the proportions of the human face, the golden ratio has an infinite capacity to generate shapes with exquisite properties. Author Gary Meisner has spent decades researching the subject, investigating and collaborating with people across the globe in dozens of professions and walks of life. In The Golden Ratio, he shares his enlightening journey. Exploring the long history of this fascinating number, as well as new insights into its power and potential applications, The Golden Ratio invites you to take a new look at this timeless topic.
The Golden Ratio: Geometric and Number Theoretical Considerations
by Hans WalserThis book illustrates key mathematical aspects of the Golden Ratio: It particularly delves into geometric and number theoretical connections and examples, and makes further considerations and generalizations accessible. The book is primarily aimed at students, pupils, mathematics teachers, and interested laypeople. It is modular in structure, so the individual chapters can be read independently of one another. The reading is intended to encourage one's own geometric activities. Tips and procedural hints from the craft-creative realm are also provided. Supplementary animations can be accessed with the SN More Media App: simply download the SN More Media App for free, scan a picture or link with the play button, and immediately play the animation on your smartphone or tablet. The translation was done with the help of artificial intelligence. A subsequent human revision was done primarily in terms of content.
The Golden Rule of Ethics: A Dynamic Game-Theoretic Framework Based on Berge Equilibrium (Communications in Cybernetics, Systems Science and Engineering #9)
by Vladislav I. Zhukovskiy Mindia E. SalukvadzeThis book synthesizes the game-theoretic modeling of decision-making processes and an ancient moral requirement called the Golden Rule of ethics (GR). This rule states "Behave to others as you would like them to behave to you." The GR is one of the oldest, most widespread, and specific moral requirements that appear in Christianity, Islam, Judaism, Buddhism, and Confucianism. This book constructs and justifies mathematical models of dynamic socio-economic processes and phenomena that reveal the mechanism of the GR and are based on the concept of Berge equilibrium. The GR can be naturally used for resolving or balancing conflicts, and its "altruistic character" obviously excludes wars, blood-letting, and armed clashes. The previous book by the authors, The Berge Equilibrium: A Game-Theoretic Framework for the Golden Rule of Ethics, covers the static case of the GR. In this book, the dynamic case of the GR is investigated using the altruistic concept of Berge equilibrium and three factors as follows: 1) a modification of N.N. Krasovskii’s mathematical formalization of differential positional games (DPGs), in view of the counterexamples given by A.I. Subbotin and A.F. Kononenko; 2) the method of guiding control, proposed by N.N. Krasovskii; and 3) the Germier convolution of the payoff functions of different players. Additionally, this book features exercises, problems, and solution tips collected together in Appendix 1, as well as new approaches to conflict resolution as presented in Appendices 2 to 4. This book will be of use to undergraduate and graduate students and experts in the field of decision-making in complex control and management systems, as well as anyone interested in game theory and applications.
The Golden Ticket: P, NP, and the Search for the Impossible
by Lance FortnowThe computer science problem whose solution could transform life as we know itThe P-NP problem is the most important open problem in computer science, if not all of mathematics. Simply stated, it asks whether every problem whose solution can be quickly checked by computer can also be quickly solved by computer. The Golden Ticket provides a nontechnical introduction to P-NP, its rich history, and its algorithmic implications for everything we do with computers and beyond. Lance Fortnow traces the history and development of P-NP, giving examples from a variety of disciplines, including economics, physics, and biology. He explores problems that capture the full difficulty of the P-NP dilemma, from discovering the shortest route through all the rides at Disney World to finding large groups of friends on Facebook. The Golden Ticket explores what we truly can and cannot achieve computationally, describing the benefits and unexpected challenges of this compelling problem.
Golf by the Numbers: How Stats, Math, and Physics Affect Your Game
by Roland MintonHow do the world’s greatest golfers improve their game? Practice, sure, but Roland Minton says mathematics and statistics are also key to their success. Golf by the Numbers analyzes the mathematical strategies behind the sport, giving fans a behind-the-scenes look at how numbers drive the game.Computers, GPS trackers, swing simulators, and high-speed cameras have introduced new and exciting ways of seeing and understanding the complicated and endlessly fascinating game of golf. Players like Phil Mickelson are so good because they review the results of every swing they take. Minton’s comprehensive analysis of statistics taken from the PGA Tour’s ShotLink system walks readers through the mountains of data that pros use to inform and refine their play. The result is an insider’s perspective of how the world’s greatest golfers apply mathematics to the sport.Minton discusses randomness in golf (especially how much luck is involved in putting) as well as aggressive and cautious strategies both on and off the greens, and he explains, by the numbers, just how Tiger Woods was so dominant from 2004 to 2009.Here is a book that tells some truly engaging stories of modern golf, featuring famous players and memorable tournaments, all through the lens of elementary probability theory. Minton’s informal style and clear and direct explanations make even the most detailed discussions accessible to all curious-minded golfers. His mathematical morsels are not only enjoyable to read—they may even help you improve your game.
Good Math: A Geek's Guide to the Beauty of Numbers, Logic, and Computation
by Mark C. Chu-CarrollMathematics is beautiful--and it can be fun and exciting as well as practical. Good Math is your guide to some of the most intriguing topics from two thousand years of mathematics: from Egyptian fractions to Turing machines; from the real meaning of numbers to proof trees, group symmetry, and mechanical computation. If you've ever wondered what lay beyond the proofs you struggled to complete in high school geometry, or what limits the capabilities of computer on your desk, this is the book for you.Why do Roman numerals persist? How do we know that some infinities are larger than others? And how can we know for certain a program will ever finish? In this fast-paced tour of modern and not-so-modern math, computer scientist Mark Chu-Carroll explores some of the greatest breakthroughs and disappointments of more than two thousand years of mathematical thought. There is joy and beauty in mathematics, and in more than two dozen essays drawn from his popular "Good Math" blog, you'll find concepts, proofs, and examples that are often surprising, counterintuitive, or just plain weird.Mark begins his journey with the basics of numbers, with an entertaining trip through the integers and the natural, rational, irrational, and transcendental numbers. The voyage continues with a look at some of the oddest numbers in mathematics, including zero, the golden ratio, imaginary numbers, Roman numerals, and Egyptian and continuing fractions. After a deep dive into modern logic, including an introduction to linear logic and the logic-savvy Prolog language, the trip concludes with a tour of modern set theory and the advances and paradoxes of modern mechanical computing.If your high school or college math courses left you grasping for the inner meaning behind the numbers, Mark's book will both entertain and enlighten you.
Good Vibrations: The Physics of Music
by Barry ParkerWhy does a harpsichord sound different from a piano? For that matter, why does middle C on a piano differ from middle C on a tuning fork, a trombone, or a flute? Good Vibrations explains in clear, friendly language the out-of-sight physics responsible not only for these differences but also for the whole range of noises we call music.The physical properties and history of sound are fascinating to study. Barry Parker's tour of the physics of music details the science of how instruments, the acoustics of rooms, electronics, and humans create and alter the varied sounds we hear. Using physics as a base, Parker discusses the history of music, how sounds are made and perceived, and the various effects of acting on sounds. In the process, he demonstrates what acoustics can teach us about quantum theory and explains the relationship between harmonics and the theory of waves.Peppered throughout with anecdotes and examples illustrating key concepts, this invitingly written book provides a firm grounding in the actual and theoretical physics of music.
Goodness-of-Fit-Techniques
by RalphB. D'AgostinoConveniently grouping methods by techniques, such as chi-squared and empirical distributionfunction , and also collecting methods of testing for specific famous distributions, this usefulreference is the fust comprehensive.review of the extensive literature on the subject. It surveysthe leading methods of testing fit . .. provides tables to make the tests available . .. assessesthe comparative merits of different test procedures . .. and supplies numerical examples to aidin understanding these techniques.Goodness-of-Fit Techniques shows how to apply the techniques . .. emphasizes testing for thethree major distributions, normal, exponential, and uniform . .. discusses the handling of censoreddata .. . and contains over 650 bibliographic citations that cover the field.Illustrated with tables and drawings, this volume is an ideal reference for mathematical andapplied statisticians, and biostatisticians; professionals in applied science fields, including psychologists,biometricians , physicians, and quality control and reliability engineers; advancedundergraduate- and graduate-level courses on goodness-of-fit techniques; and professional seminarsand symposia on applied statistics, quality control, and reliability.
Google's PageRank and Beyond: The Science of Search Engine Rankings
by Amy N. Langville Carl D. MeyerWhy doesn't your home page appear on the first page of search results, even when you query your own name? How do other web pages always appear at the top? What creates these powerful rankings? And how? The first book ever about the science of web page rankings, Google's PageRank and Beyond supplies the answers to these and other questions and more. The book serves two very different audiences: the curious science reader and the technical computational reader. The chapters build in mathematical sophistication, so that the first five are accessible to the general academic reader. While other chapters are much more mathematical in nature, each one contains something for both audiences. For example, the authors include entertaining asides such as how search engines make money and how the Great Firewall of China influences research. The book includes an extensive background chapter designed to help readers learn more about the mathematics of search engines, and it contains several MATLAB codes and links to sample web data sets. The philosophy throughout is to encourage readers to experiment with the ideas and algorithms in the text. Any business seriously interested in improving its rankings in the major search engines can benefit from the clear examples, sample code, and list of resources provided. Many illustrative examples and entertaining asides MATLAB code Accessible and informal style Complete and self-contained section for mathematics review
GOP GPS: How to Find the Millennials and Urban Voters the Republican Party Needs to Survive
by Evan Siegfried Robert A. GeorgeEditors’ Pick for the Conservative Book ClubA roadmap to the future for Republicans With each passing election cycle, Republicans face stiffer and stiffer odds for winning elections at all levels. They have long relied on a combination of Baby Boomers and rural voters to propel them to victory in races for office. However, both of these populations will decline in the future. At the same time, the ranks of millennials and urban voters have exploded. Millennials (now the largest generation in the United States) and urban residents (the growing majority of the population) are the two groups that are the key to the future?both for the United States and for any major political party. Unfortunately, the Republican Party has largely put off bringing these vital groups into the fold. In fact, it faces a real identity crisis with them. Mention the words ?GOP” or ?Republican” and they immediately react negatively. They see the party and its members as being out of touch and focused on the past. What possible reason should they have to vote for them, or even listen to their ideas?GOP GPS offers a roadmap for Republicans to win over these key groups, showing how to do so using conservative values and principles. It addresses a wide array of issues, including social justice, education, marriage equality, debt, and the family. It will challenge all and show that the Republican Party is not the caricature the Left or the media make it out to be.
Gorenstein Homological Algebra
by Alina IacobGorenstein homological algebra is an important area of mathematics, with applications in commutative and noncommutative algebra, model category theory, representation theory, and algebraic geometry. While in classical homological algebra the existence of the projective, injective, and flat resolutions over arbitrary rings are well known, things are a little different when it comes to Gorenstein homological algebra. The main open problems in this area deal with the existence of the Gorenstein injective, Gorenstein projective, and Gorenstein flat resolutions. Gorenstein Homological Algebra is especially suitable for graduate students interested in homological algebra and its applications.
Gottfried Wilhelm Leibniz: The Polymath Who Brought Us Calculus
by M. B. W. TentGottfried Wilhelm Leibniz: The Polymath Who Brought Us Calculus focuses on the life and accomplishments of one of the seventeenth century's most influential mathematicians and philosophers. The book, which draws on Leibniz's written works and translations, and reconstructs dialogues Leibniz may have had based on the historical record of his life ex
Gottlieb and Whitehead Center Groups of Spheres, Projective and Moore Spaces
by Marek Golasiński Juno MukaiThis is a monograph that details the use of Siegel's method and the classical results of homotopy groups of spheres and Lie groups to determine some Gottlieb groups of projective spaces or to give the lower bounds of their orders. Making use of the properties of Whitehead products, the authors also determine some Whitehead center groups of projective spaces that are relevant and new within this monograph.
Governance and Opportunity in Metropolitan America
by National Research CouncilAmerica's cities have symbolized the nation's prosperity, dynamism, and innovation. Even with the trend toward suburbanization, many central cities attract substantial new investment and employment. Within this profile of health, however, many urban areas are beset by problems of economic disparity, physical deterioration, and social distress.This volume addresses the condition of the city from the perspective of the larger metropolitan region. It offers important, thought-provoking perspectives on the structure of metropolitan-level decisionmaking, the disadvantages faced by cities and city residents, and expanding economic opportunity to all residents in a metropolitan area. The book provides data, real-world examples, and analyses in key areas: Distribution of metropolitan populations and what this means for city dwellers, suburbanites, whites, and minorities.How quality of life depends on the spatial structure of a community and how problems are based on inequalities in spatialopportunity--with a focus on the relationship between taxes and services.The role of the central city today, the rationale for revitalizing central cities, and city-suburban interdependence. The book includes papers that provide in-depth examinations of zoning policy in relation to patterns of suburban development; regionalism in transportation and air quality; the geography of economic and social opportunity; social stratification in metropolitan areas; and fiscal and service disparities within metropolitan areas.
Government Consolidation and Economic Development in Allegheny County and the City of Pittsburgh
by Sally Sleeper Rae W. ArchibaldThis report concludes that, although evidence is mixed and effects difficult to measure, consolidating the City of Pittsburgh and Allegheny County could enhance economic development by unifying leadership, improving policy direction and coordination, and sharpening economic-development initiatives. Increased collaboration with the private sector also is important, and combining only two governments still leaves the region highly fragmented.
GPU Parallel Program Development Using CUDA (Chapman & Hall/CRC Computational Science)
by Tolga SoyataGPU Parallel Program Development using CUDA teaches GPU programming by showing the differences among different families of GPUs. This approach prepares the reader for the next generation and future generations of GPUs. The book emphasizes concepts that will remain relevant for a long time, rather than concepts that are platform-specific. At the same time, the book also provides platform-dependent explanations that are as valuable as generalized GPU concepts. The book consists of three separate parts; it starts by explaining parallelism using CPU multi-threading in Part I. A few simple programs are used to demonstrate the concept of dividing a large task into multiple parallel sub-tasks and mapping them to CPU threads. Multiple ways of parallelizing the same task are analyzed and their pros/cons are studied in terms of both core and memory operation. Part II of the book introduces GPU massive parallelism. The same programs are parallelized on multiple Nvidia GPU platforms and the same performance analysis is repeated. Because the core and memory structures of CPUs and GPUs are different, the results differ in interesting ways. The end goal is to make programmers aware of all the good ideas, as well as the bad ideas, so readers can apply the good ideas and avoid the bad ideas in their own programs. Part III of the book provides pointer for readers who want to expand their horizons. It provides a brief introduction to popular CUDA libraries (such as cuBLAS, cuFFT, NPP, and Thrust),the OpenCL programming language, an overview of GPU programming using other programming languages and API libraries (such as Python, OpenCV, OpenGL, and Apple’s Swift and Metal,) and the deep learning library cuDNN.
GPU Pro 360 Guide to Geometry Manipulation
by Wolfgang EngelWolfgang Engel’s GPU Pro 360 Guide to Geometry Manipulation gathers all the cutting-edge information from his previous seven GPU Pro volumes into a convenient single source anthology that covers geometry manipulation in computer graphics. This volume is complete with 19 articles by leading programmers that focus on the ability of graphics processing units to process and generate geometry in exciting ways. GPU Pro 360 Guide to Geometry Manipulation is comprised of ready-to-use ideas and efficient procedures that can help solve many computer graphics programming challenges that may arise. <P><P> Key Features: <li> Presents tips and tricks on real-time rendering of special effects and visualization data on common consumer software platforms such as PCs, video consoles, mobile devices <li> Covers specific challenges involved in creating games on various platforms <li> Explores the latest developments in the rapidly evolving field of real-time rendering <li> Takes a practical approach that helps graphics programmers solve their daily challenges
Grade 1 Math Speed Drills: Beginning Arithmetic
by Rod and Staff PublishersThis small tablet of 72 speed drills provides timed practice of addition and subtractions facts. These drills begin at lesson 28 and continue through lesson 170. Speed drills are to be given every other day. Directions for using this tablet are in the teacher's manual.
Grade 5: Getting Ready for High Stakes Assessments
by Houghton Mifflin Harcourt Publishing CompanyGo Math! Getting Ready for High Stakes Assessments Student Edition Grade 5
Grade 7 Common Core Assessment Readiness
by Holt McdougalMathematics: Assessment Readiness Workbook Grade 7
Graded Finite Element Methods for Elliptic Problems in Nonsmooth Domains (Surveys and Tutorials in the Applied Mathematical Sciences #10)
by Hengguang LiThis book develops a class of graded finite element methods to solve singular elliptic boundary value problems in two- and three-dimensional domains. It provides an approachable and self-contained presentation of the topic, including both the mathematical theory and numerical tools necessary to address the major challenges imposed by the singular solution. Moreover, by focusing upon second-order equations with constant coefficients, it manages to derive explicit results that are accessible to the broader computation community. Although written with mathematics graduate students and researchers in mind, this book is also relevant to applied and computational mathematicians, scientists, and engineers in numerical methods who may encounter singular problems.
Graded Rings and Graded Grothendieck Groups
by Roozbeh HazratThis study of graded rings includes the first systematic account of the graded Grothendieck group, a powerful and crucial invariant in algebra which has recently been adopted to classify the Leavitt path algebras. The book begins with a concise introduction to the theory of graded rings and then focuses in more detail on Grothendieck groups, Morita theory, Picard groups and K-theory. The author extends known results in the ungraded case to the graded setting and gathers together important results which are currently scattered throughout the literature. The book is suitable for advanced undergraduate and graduate students, as well as researchers in ring theory.
The Gradient Discretisation Method (Mathématiques et Applications #82)
by Jérôme Droniou Robert Eymard Thierry Gallouët Cindy Guichard Raphaèle HerbinThis monograph presents the Gradient Discretisation Method (GDM), which is a unified convergence analysis framework for numerical methods for elliptic and parabolic partial differential equations. The results obtained by the GDM cover both stationary and transient models; error estimates are provided for linear (and some non-linear) equations, and convergence is established for a wide range of fully non-linear models (e.g. Leray–Lions equations and degenerate parabolic equations such as the Stefan or Richards models). The GDM applies to a diverse range of methods, both classical (conforming, non-conforming, mixed finite elements, discontinuous Galerkin) and modern (mimetic finite differences, hybrid and mixed finite volume, MPFA-O finite volume), some of which can be built on very general meshes.
Gradually-varied Flow Profiles in Open Channels
by Chyan-Deng JanGradually-varied flow (GVF) is a steady non-uniform flow in an open channel with gradual changes in its water surface elevation. The evaluation of GVF profiles under a specific flow discharge is very important in hydraulic engineering. This book proposes a novel approach to analytically solve the GVF profiles by using the direct integration and Gaussian hypergeometric function. Both normal-depth- and critical-depth-based dimensionless GVF profiles are presented. The novel approach has laid the foundation to compute at one sweep the GVF profiles in a series of sustaining and adverse channels, which may have horizontal slopes sandwiched in between them.