Download or read book Investigating Polygons and Polyhedra with Googolplex written by Allen W. Banbury and published by . This book was released on 1988 with total page 60 pages. Available in PDF, EPUB and Kindle. Book excerpt: Explains how to use this toy in the classroom to teach basic geometry concepts to students in grades K-6.
Download or read book The Science Teacher written by and published by . This book was released on 1996 with total page 986 pages. Available in PDF, EPUB and Kindle. Book excerpt: Some issues are accompanied by a CD-ROM on a selected topic.
Download or read book Canadian Books in Print written by and published by . This book was released on 1996 with total page 760 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Mathematics for Computer Science written by Eric Lehman and published by . This book was released on 2017-03-08 with total page 988 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book covers elementary discrete mathematics for computer science and engineering. It emphasizes mathematical definitions and proofs as well as applicable methods. Topics include formal logic notation, proof methods; induction, well-ordering; sets, relations; elementary graph theory; integer congruences; asymptotic notation and growth of functions; permutations and combinations, counting principles; discrete probability. Further selected topics may also be covered, such as recursive definition and structural induction; state machines and invariants; recurrences; generating functions.
Download or read book The Arithmetic Teacher written by and published by . This book was released on 1989 with total page 568 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book International Bulletin of Bibliography on Education written by and published by . This book was released on 1989 with total page 580 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Mind Tools written by Rudy Rucker and published by Courier Corporation. This book was released on 2013-11-21 with total page 337 pages. Available in PDF, EPUB and Kindle. Book excerpt: Originally published: Boston: Houghton Mifflin, 1987.
Download or read book Convex Optimization Euclidean Distance Geometry written by Jon Dattorro and published by Meboo Publishing USA. This book was released on 2005 with total page 776 pages. Available in PDF, EPUB and Kindle. Book excerpt: The study of Euclidean distance matrices (EDMs) fundamentally asks what can be known geometrically given onlydistance information between points in Euclidean space. Each point may represent simply locationor, abstractly, any entity expressible as a vector in finite-dimensional Euclidean space.The answer to the question posed is that very much can be known about the points;the mathematics of this combined study of geometry and optimization is rich and deep.Throughout we cite beacons of historical accomplishment.The application of EDMs has already proven invaluable in discerning biological molecular conformation.The emerging practice of localization in wireless sensor networks, the global positioning system (GPS), and distance-based pattern recognitionwill certainly simplify and benefit from this theory.We study the pervasive convex Euclidean bodies and their various representations.In particular, we make convex polyhedra, cones, and dual cones more visceral through illustration, andwe study the geometric relation of polyhedral cones to nonorthogonal bases biorthogonal expansion.We explain conversion between halfspace- and vertex-descriptions of convex cones,we provide formulae for determining dual cones,and we show how classic alternative systems of linear inequalities or linear matrix inequalities and optimality conditions can be explained by generalized inequalities in terms of convex cones and their duals.The conic analogue to linear independence, called conic independence, is introducedas a new tool in the study of classical cone theory; the logical next step in the progression:linear, affine, conic.Any convex optimization problem has geometric interpretation.This is a powerful attraction: the ability to visualize geometry of an optimization problem.We provide tools to make visualization easier.The concept of faces, extreme points, and extreme directions of convex Euclidean bodiesis explained here, crucial to understanding convex optimization.The convex cone of positive semidefinite matrices, in particular, is studied in depth.We mathematically interpret, for example,its inverse image under affine transformation, and we explainhow higher-rank subsets of its boundary united with its interior are convex.The Chapter on "Geometry of convex functions",observes analogies between convex sets and functions:The set of all vector-valued convex functions is a closed convex cone.Included among the examples in this chapter, we show how the real affinefunction relates to convex functions as the hyperplane relates to convex sets.Here, also, pertinent results formultidimensional convex functions are presented that are largely ignored in the literature;tricks and tips for determining their convexityand discerning their geometry, particularly with regard to matrix calculus which remains largely unsystematizedwhen compared with the traditional practice of ordinary calculus.Consequently, we collect some results of matrix differentiation in the appendices.The Euclidean distance matrix (EDM) is studied,its properties and relationship to both positive semidefinite and Gram matrices.We relate the EDM to the four classical axioms of the Euclidean metric;thereby, observing the existence of an infinity of axioms of the Euclidean metric beyondthe triangle inequality. We proceed byderiving the fifth Euclidean axiom and then explain why furthering this endeavoris inefficient because the ensuing criteria (while describing polyhedra)grow linearly in complexity and number.Some geometrical problems solvable via EDMs,EDM problems posed as convex optimization, and methods of solution arepresented;\eg, we generate a recognizable isotonic map of the United States usingonly comparative distance information (no distance information, only distance inequalities).We offer a new proof of the classic Schoenberg criterion, that determines whether a candidate matrix is an EDM. Our proofrelies on fundamental geometry; assuming, any EDM must correspond to a list of points contained in some polyhedron(possibly at its vertices) and vice versa.It is not widely known that the Schoenberg criterion implies nonnegativity of the EDM entries; proved here.We characterize the eigenvalues of an EDM matrix and then devisea polyhedral cone required for determining membership of a candidate matrix(in Cayley-Menger form) to the convex cone of Euclidean distance matrices (EDM cone); \ie,a candidate is an EDM if and only if its eigenspectrum belongs to a spectral cone for EDM^N.We will see spectral cones are not unique.In the chapter "EDM cone", we explain the geometric relationship betweenthe EDM cone, two positive semidefinite cones, and the elliptope.We illustrate geometric requirements, in particular, for projection of a candidate matrixon a positive semidefinite cone that establish its membership to the EDM cone. The faces of the EDM cone are described,but still open is the question whether all its faces are exposed as they are for the positive semidefinite cone.The classic Schoenberg criterion, relating EDM and positive semidefinite cones, isrevealed to be a discretized membership relation (a generalized inequality, a new Farkas''''''''-like lemma)between the EDM cone and its ordinary dual. A matrix criterion for membership to the dual EDM cone is derived thatis simpler than the Schoenberg criterion.We derive a new concise expression for the EDM cone and its dual involvingtwo subspaces and a positive semidefinite cone."Semidefinite programming" is reviewedwith particular attention to optimality conditionsof prototypical primal and dual conic programs,their interplay, and the perturbation method of rank reduction of optimal solutions(extant but not well-known).We show how to solve a ubiquitous platonic combinatorial optimization problem from linear algebra(the optimal Boolean solution x to Ax=b)via semidefinite program relaxation.A three-dimensional polyhedral analogue for the positive semidefinite cone of 3X3 symmetricmatrices is introduced; a tool for visualizing in 6 dimensions.In "EDM proximity"we explore methods of solution to a few fundamental and prevalentEuclidean distance matrix proximity problems; the problem of finding that Euclidean distance matrix closestto a given matrix in the Euclidean sense.We pay particular attention to the problem when compounded with rank minimization.We offer a new geometrical proof of a famous result discovered by Eckart \& Young in 1936 regarding Euclideanprojection of a point on a subset of the positive semidefinite cone comprising all positive semidefinite matriceshaving rank not exceeding a prescribed limit rho.We explain how this problem is transformed to a convex optimization for any rank rho.
Download or read book Teaching Advanced Learners in the General Education Classroom written by Joan Franklin Smutny and published by Corwin Press. This book was released on 2011-08-22 with total page 201 pages. Available in PDF, EPUB and Kindle. Book excerpt: Low-stress tips for challenging high-ability learners Many teachers ask: “What do I do for students who finish their work before everyone else?” If you would like to do more for gifted students and need simple strategies that you can use tomorrow, this book is for you. Inside are helpful methods for challenging students who need more than the regular curriculum can provide. The authors provide practical tools, including: Tips for using existing resources and potential A progression from simpler to more complex adjustments for advanced learners Specific lessons for language arts, math, science, social studies, and the arts
Download or read book The Concise Oxford Dictionary of Mathematics written by Christopher Clapham and published by OUP Oxford. This book was released on 2014-05-22 with total page 545 pages. Available in PDF, EPUB and Kindle. Book excerpt: Authoritative and reliable, this A-Z provides jargon-free definitions for even the most technical mathematical terms. With over 3,000 entries ranging from Achilles paradox to zero matrix, it covers all commonly encountered terms and concepts from pure and applied mathematics and statistics, for example, linear algebra, optimisation, nonlinear equations, and differential equations. In addition, there are entries on major mathematicians and on topics of more general interest, such as fractals, game theory, and chaos. Using graphs, diagrams, and charts to render definitions as comprehensible as possible, entries are clear and accessible. Almost 200 new entries have been added to this edition, including terms such as arrow paradox, nested set, and symbolic logic. Useful appendices follow the A-Z dictionary and include lists of Nobel Prize winners and Fields' medallists, Greek letters, formulae, and tables of inequalities, moments of inertia, Roman numerals, a geometry summary, additional trigonometric values of special angles, and many more. This edition contains recommended web links, which are accessible and kept up to date via the Dictionary of Mathematics companion website. Fully revised and updated in line with curriculum and degree requirements, this dictionary is indispensable for students and teachers of mathematics, and for anyone encountering mathematics in the workplace.
Download or read book Who s who in American Education written by and published by . This book was released on 1992 with total page 1006 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Infinity and the Mind written by Rudy Rucker and published by Princeton University Press. This book was released on 2019-07-23 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: A dynamic exploration of infinity In Infinity and the Mind, Rudy Rucker leads an excursion to that stretch of the universe he calls the “Mindscape,” where he explores infinity in all its forms: potential and actual, mathematical and physical, theological and mundane. Using cartoons, puzzles, and quotations to enliven his text, Rucker acquaints us with staggeringly advanced levels of infinity, delves into the depths beneath daily awareness, and explains Kurt Gödel’s belief in the possibility of robot consciousness. In the realm of infinity, mathematics, science, and logic merge with the fantastic. By closely examining the paradoxes that arise, we gain profound insights into the human mind, its powers, and its limitations. This Princeton Science Library edition includes a new preface by the author.
Download or read book Moments in Mathematics written by Henry J. Landau and published by American Mathematical Soc.. This book was released on 1987 with total page 170 pages. Available in PDF, EPUB and Kindle. Book excerpt: Function theory, spectral decomposition of operators, probability, approximation, electrical and mechanical inverse problems, prediction of stochastic processes, the design of algorithms for signal-processing VLSI chips--these are among a host of important theoretical and applied topics illuminated by the classical moment problem. To survey some of these ramifications and the research which derives from them, the AMS sponsored the Short Course Moments in Mathematics at the Joint Mathematics Meetings, held in San Antonio, Texas, in January 1987. This volume contains the six lectures presented during that course. The papers are likely to find a wide audience, for they are expository, but nevertheless lead the reader to topics of current research. In his paper, Henry J. Landau sketches the main ideas of past work related to the moment problem by such mathematicians as Caratheodory, Herglotz, Schur, Riesz, and Krein and describes the way the moment problem has interconnected so many diverse areas of research. J. H. B. Kemperman examines the moment problem from a geometric viewpoint which involves a certain natural duality method and leads to interesting applications in linear programming, measure theory, and dilations. Donald Sarason first provides a brief review of the theory of unbounded self-adjoint operators then goes on to sketch the operator-theoretic treatment of the Hamburger problem and to discuss Hankel operators, the Adamjan-Arov-Krein approach, and the theory of unitary dilations. Exploring the interplay of trigonometric moment problems and signal processing, Thomas Kailath describes the role of Szego polynomials in linear predictive coding methods, parallel implementation, one-dimensional inverse scattering problems, and the Toeplitz moment matrices. Christian Berg contrasts the multi-dimensional moment problem with the one-dimensional theory and shows how the theory of the moment problem may be viewed as part of harmonic analysis on semigroups. Starting from a historical survey of the use of moments in probability and statistics, Persi Diaconis illustrates the continuing vitality of these methods in a variety of recent novel problems drawn from such areas as Wiener-Ito integrals, random graphs and matrices, Gibbs ensembles, cumulants and self-similar processes, projections of high-dimensional data, and empirical estimation.
Download or read book CRC Standard Mathematical Tables and Formulae 32nd Edition written by Daniel Zwillinger and published by CRC Press. This book was released on 2011-06-22 with total page 792 pages. Available in PDF, EPUB and Kindle. Book excerpt: With over 6,000 entries, CRC Standard Mathematical Tables and Formulae, 32nd Edition continues to provide essential formulas, tables, figures, and descriptions, including many diagrams, group tables, and integrals not available online. This new edition incorporates important topics that are unfamiliar to some readers, such as visual proofs and sequences, and illustrates how mathematical information is interpreted. Material is presented in a multisectional format, with each section containing a valuable collection of fundamental tabular and expository reference material. New to the 32nd Edition A new chapter on Mathematical Formulae from the Sciences that contains the most important formulae from a variety of fields, including acoustics, astrophysics, epidemiology, finance, statistical mechanics, and thermodynamics New material on contingency tables, estimators, process capability, runs test, and sample sizes New material on cellular automata, knot theory, music, quaternions, and rational trigonometry Updated and more streamlined tables Retaining the successful format of previous editions, this comprehensive handbook remains an invaluable reference for professionals and students in mathematical and scientific fields.
Download or read book The Handy Math Answer Book written by Patricia Barnes-Svarney and published by Visible Ink Press. This book was released on 2012-05-01 with total page 528 pages. Available in PDF, EPUB and Kindle. Book excerpt: From modern-day challenges such as balancing a checkbook, following the stock market, buying a home, and figuring out credit card finance charges to appreciating historical developments by Pythagoras, Archimedes, Newton, and other mathematicians, this engaging resource addresses more than 1,000 questions related to mathematics. Organized into chapters that cluster similar topics in an easily accessible format, this reference provides clear and concise explanations about the fundamentals of algebra, calculus, geometry, trigonometry, and other branches of mathematics. It contains the latest mathematical discoveries, including newly uncovered historical documents and updates on how science continues to use math to make cutting-edge innovations in DNA sequencing, superstring theory, robotics, and computers. With fun math facts and illuminating figures, The Handy Math Answer Book explores the uses of math in everyday life and helps the mathematically challenged better understand and enjoy the magic of numbers.
Download or read book Four Colors Suffice written by Robin J. Wilson and published by Princeton University Press. This book was released on 2002 with total page 284 pages. Available in PDF, EPUB and Kindle. Book excerpt: On October 23, 1852, Professor Augustus De Morgan wrote a letter to a colleague, unaware that he was launching one of the most famous mathematical conundrums in history--one that would confound thousands of puzzlers for more than a century. This is the amazing story of how the "map problem" was solved. The problem posed in the letter came from a former student: What is the least possible number of colors needed to fill in any map (real or invented) so that neighboring counties are always colored differently? This deceptively simple question was of minimal interest to cartographers, who saw little need to limit how many colors they used. But the problem set off a frenzy among professional mathematicians and amateur problem solvers, among them Lewis Carroll, an astronomer, a botanist, an obsessive golfer, the Bishop of London, a man who set his watch only once a year, a California traffic cop, and a bridegroom who spent his honeymoon coloring maps. In their pursuit of the solution, mathematicians painted maps on doughnuts and horseshoes and played with patterned soccer balls and the great rhombicuboctahedron. It would be more than one hundred years (and countless colored maps) later before the result was finally established. Even then, difficult questions remained, and the intricate solution--which involved no fewer than 1,200 hours of computer time--was greeted with as much dismay as enthusiasm. Providing a clear and elegant explanation of the problem and the proof, Robin Wilson tells how a seemingly innocuous question baffled great minds and stimulated exciting mathematics with far-flung applications. This is the entertaining story of those who failed to prove, and those who ultimately did prove, that four colors do indeed suffice to color any map.
