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This digital document is an article from Science and Its Times, brought to you by Gale®, a part of Cengage Learning, a world leader in e-research and educational publishing for libraries, schools and businesses.The length of the article is 1013 words.The article is delivered in HTML format and is available in your Amazon.com Digital Locker immediately after purchase.You can view it with any web browser.The histories of science, technology, and mathematics merge with the study of humanities and social science in this interdisciplinary reference work. Essays on people, theories, discoveries, and concepts are combined with overviews, bibliographies of primary documents, and chronological elements to offer students a fascinating way to understand the impact of science on the course of human history and how science affects everyday life. Entries represent people and developments throughout the world, from about 2000 B.C. through the end of the twentieth century. ... Read more

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Historically, nonclassical physics developed in three stages. First came a collection of ad hoc assumptions and then a cookbook of equations known as "quantum mechanics". The equations and their philosophical underpinnings were then collected into a model based on the mathematics of Hilbert space. From the Hilbert space model came the abstaction of "quantum logics". This book explores all three stages, but not in historical order. Instead, in an effort to illustrate how physics and abstract mathematics influence each other we hop back and forth between a purely mathematical development of Hilbert space, and a physically motivated definition of a logic, partially linking the two throughout, and then bringing them together at the deepest level in the last two chapters. This book should be accessible to undergraduate and beginning graduate students in both mathematics and physics. The only strict prerequisites are calculus and linear algebra, but the level of mathematical sophistication assumes at least one or two intermediate courses, for example in mathematical analysis or advanced calculus. No background in physics is assumed. ... Read more

Customer Reviews (1)

Effective introduction
This book gives a nice introduction to the mathematical formalism behind quantum physics and the logic of measurement.The first chapter gives an introduction to measure theory with emphasis on probabilities of measurement outcomes. The author is careful to point out that the calculation of the Lebesgue integral presents more difficulties than in the Riemann integral case, since the fundamental theorem of calculus does not apply to Lebesgue integrals.

This is followed by an elementary introduction to Hilbert space in Chapter 2. This is standard material and most of the proofs of the main results are omitted and left to the reader as projects.

Chapter 3 is more controversial, and attempts to formulate a logic of experimentation for "non-classical" systems. This is done by use of what the author calls a "manual", which is viewed as an abstraction of the experimenters knowledge about a physical system. A manual is a collection of experiments, and an "event" is a subset of an experiment. Orthogonality of events is defined, along with the notion of a collection of events being "compatible", meaning that there is an experiment that contains all of these events. A manual is called "classical" if every pair of events is compatible. The author then exhibits systems that are not classical via the double-slit and Stern-Gerlach experiments. A logic of events is then developed in the next section, where quantum logic is defined explicitly. The author defines a pure state that is not dispersion-free as a state of ontological uncertainty as opposed to "epistemic" uncertainty. Quantum systems have states that are ontologically uncertain according to the author. The author chooses not to engage in the debate about the actual existence of these states and, accordingly, no real-world experiments are given to illustrate the relevance of the concepts and definitions.

The next chapter covers the geometry of infinite-dimensional Hilbert spaces. The structure of the collection of these subspaces is defined in terms of the quantum logic defined earlier. This is followed by a discussion of maps on Hilbert spaces, as preparation for defining observables in quantum systems. The important Riesz representation theorem is stated but the proof left to the reader. Projection operators are defined also with the eventual goal of relating them to the compatibility of two propositions.

Gleason's theorem is discussed in Chapter 6, along with a discussion of the geometry of state space. The proof of Gleason's theorem is omitted, the author emphasizing its difficulty. The proof in the literature is non-constructive and thus the theorem is suspect according to some schools of thought.

The spectral theorem, so important in quantum physics, is discussed in the next chapter. Once again the proofs are left to the reader for most of the results. The spectral theorem allows the author to define another notion of compatibility in terms of the commutativity of two Hermitian operators.

The books ends with a overview of the EPR dilemna and is naturally more controversial than the rest of the book. This topic has provoked much philosophical debate, and the author gives the reader a small taste of this in this chapter.

The book does serve its purpose well, and regardless of one's philosophical position on quantum physics, the mathematical formulations of quantum physics and measurement theory are nicely expounded in this book. ... Read more

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Chapters: P Versus Np Problem, Poincaré Conjecture, Collatz Conjecture, Hilbert's Problems, Catalan's Conjecture, Sierpinski Number, Generalized Riemann Hypothesis, Langlands Program, Weil Conjectures, Geometrization Conjecture, Aanderaa-karp-rosenberg Conjecture, Kepler Conjecture, Hadwiger Conjecture, Hodge Conjecture, Unsolved Problems in Mathematics, Bieberbach Conjecture, Cycle Double Cover, Farrell-jones Conjecture, Abc Conjecture, Lindelöf Hypothesis, Birch and Swinnerton-Dyer Conjecture, Erdős-straus Conjecture, Highly Composite Number, Unique Games Conjecture, Vizing's Conjecture, Sato-tate Conjecture, Schanuel's Conjecture, Carathéodory Conjecture, Baum-connes Conjecture, Hilbert-pólya Conjecture, List of Conjectures, Erdős-burr Conjecture, No-Three-In-Line Problem, Union-Closed Sets Conjecture, Generalized Poincaré Conjecture, Dyson Conjecture, Reconstruction Conjecture, Albertson Conjecture, Scheinerman's Conjecture, Erdős-faber-lovász Conjecture, Hedetniemi's Conjecture, Homological Conjectures in Commutative Algebra, Hilbert's Twelfth Problem, Standard Conjectures on Algebraic Cycles, Calogero Conjecture, Road Coloring Problem, Segal Conjecture, Oppenheim Conjecture, Singmaster's Conjecture, Artin's Conjecture on Primitive Roots, Pierce-birkhoff Conjecture, N!-Conjecture, Weinstein Conjecture, Graceful Labeling, Stark Conjectures, Brumer-stark Conjecture, Unsolved Problems in Computer Science, New Digraph Reconstruction Conjecture, Heawood Conjecture, Littlewood Conjecture, Serre's Multiplicity Conjectures, Jacobian Conjecture, Borel Conjecture, Novikov Conjecture, Conway's Thrackle Conjecture, Filling Area Conjecture, Montgomery's Pair Correlation Conjecture, Erdős Conjecture, Edgeworth Conjecture, Beal's Conjecture, Hirsch Conjecture, Leopoldt's Conjecture, Grothendieck-katz P-Curvature Conjecture, Tameness Conjecture, Erdős-graham Conjecture, Space Form, Weil Conjecture on Tamagawa Numbers, Erdős-gyárfás C...More: http://booksllc.net/?id=19344125 ... Read more

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Chapters: Hilbert's Third Problem, Conic Section, Gömböc, Solid Modeling, Soddy's Hexlet, 3d Projection, Dihedral Angle, Dandelin Spheres, Skew Lines, Cylinder, Constructive Solid Geometry, Reuleaux Tetrahedron, Three-Dimensional Space, Conical Surface, Surface of Constant Width, Steinmetz Solid, Unit Cube, Apex. Source: Wikipedia. Pages: 94. Not illustrated. Free updates online. Purchase includes a free trial membership in the publisher's book club where you can select from more than a million books without charge. Excerpt: In mathematics, a conic section (or just conic) is a curve obtained by intersecting a cone (more precisely, a right circular conical surface) with a plane. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2. It can be defined as the locus of points whose distances are in a fixed ratio to some point, called the focus, and some line, called the directrix. The conic sections were named and studied as long ago as 200 BC, when Apollonius of Perga undertook a systematic study of their properties. It is believed that the first definition of a conic section is due to Menaechmus. This work does not survive, however, and is only known through secondary accounts. The definition used at that time differs from the one commonly used today in that it requires the plane cutting the cone to be perpendicular to the line that generates the cone as a surface of revolution. Thus the shape of the conic is determined by the angle formed at the vertex of the cone; If the angle is acute then the conic is an ellipse, if the angle is right then the conic is a parabola, and if the angle is obtuse then the conic is a hyperbola. Note that the circle cannot be defined this way and was not considered a conic at this time. Euclid is said to have written four books on conics but these were lost as well. Archimedes is known to have studied conics, having determined the area bounded by a parabola and an e...More: http://booksllc.net/?id=19008673 ... Read more

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High Quality Content by WIKIPEDIA articles! Hilbert'stwelfth problem, of the 23 Hilbert's problems, is theextension of Kronecker-Weber theorem on abelianextensions of the rational numbers, to any base numberfield. The classical theory of complex multiplicationdoes this for any imaginary quadratic field. The moregeneral cases, now often known as the KroneckerJugendtraum (although not so accurately), are stillopen as of 2005. Leopold Kronecker is supposed to havedescribed the complex multiplication issue as hisliebster Jugendtraum or "dearest dream of his youth". ... Read more

Customer Reviews (1)

A Book about Polyhedra and Hilbert's Third Problem Intended for Undergraduate and Below
A. R. Rajwade's "Convex Polyhedra with Regularity Conditions and Hilbert's Third Problem" has thirteen chapters. The last chapter, Hilbert's third problem, caught my attention the most, because my primary reason to purchase the book was to know more about the Hilbert's third problem. In short, the third problem asks whether or not A can be divided into finitely many polyhedral pieces which can be rejoined to give B (or whether A and B are equidecomposable), if A is the regular tetrahedron and B is the cube with volume A = volume B. The answer to the problem is NO even though the answer to the two-dimensional analogue is positive. That is two polygons A and B of equal area are equidecomposable.

Rajwde, the author, presents a simple proof of Hilbert's third problem on the last chapter. The proof is primarily from Boltianski.The peculiarity is to avoid the use of the axiom of choice, which implies the Banach-Tarski Paradox. The presented proof includes (1) a function f named Dehn invariant, (2) the statement--if P and Q are equidecomposable, then f(P)=f(Q), and (3) the result: f(regular tetrahedron) not equal to f(cube), which implies that a regular tetrahedron and the cube with the same volume are non-equidecomposable.

One of the aims of the book is to introduce the importance of the four regularities of polyhedra. They are (I) all faces are regular polygons, (II) all faces are congruent polygons, (III) all solid angles are regular, and (IV) all solid angles are congruent. By imposing various combinations of these regularities, we could obtain the Platonic solids (I & III, I & IV), the semi-regular solids (I & IV), the deltahedra (I & II), and the regular faced polyhedra (I). Another aim is to present the enumeration of the convex polyhedra. There are only five regular Platonic solids, thirteen semi-regular solids, eight deltahedra, and ninety-two regular faced polyhedra. The book also includes a "recreation" chapter on plane tessellations. One might consider reading the chapter prior to tile one's own floor or even a three-dimensional object.

This is the most economical and up-to-date book available in the market with "Hilbert's third problem" shown in the title. ... Read more

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Purchase includes free access to book updates online and a free trial membership in the publisher's book club where you can select from more than a million books without charge. Chapters: Hilbert's Fifth Problem, Differentiable Manifold, Smooth Function, Exotic Sphere, Differential Structure, Rokhlin's Theorem, Clutching Construction, Exotic R4. Excerpt:In topology , a branch of mathematics, the clutching construction is a way of constructing fiber bundles, particularly vector bundles on spheres. Definition Consider the sphere S n as the union of the upper and lower hemispheres and along their intersection, the equator, an S n 1. Given trivialized fiber bundles with fiber F and structure group G over the two disks, then given a map (called the clutching map ), glue the two trivial bundles together via f . Formally, it is the coequalizer of the inclusions via and : glue the two bundles together on the boundary, with a twist. Thus we have a map : clutching information on the equator yields a fiber bundle on the total space. In the case of vector bundles, this yields , and indeed this map is an isomorphism (under connect sum of spheres on the right). Generalization The above can be generalized by replacing the disks and sphere with any closed triad ( X ; A , B ), that is, a space X , together with two closed subsets A and B whose union is X . Then a clutching map on gives a vector bundle on X . Classifying map construction Let be a fibre bundle with fibre F . Let be a collection of pairs ( U i , f i ) such that is a local trivialization of p over . Moreover, we demand that the union of all the sets U i is N (ie: the collection is an atlas of trivializations ). Consider the space modulo the equivalence relation is equivalent to if and only if and . By design, the local trivializations q i give a fibrewise equivalence between this quotient space and the fibre bundle p . Consider the space modulo the equivalence relation is equivalent to if and only if and consider t... ... Read more

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Chapters: Bitangents of a Quartic, Hilbert's Sixteenth Problem, Nash Functions, Mnev's Universality Theorem, Tarski-seidenberg Theorem, Ragsdale Conjecture, Semialgebraic Set, Harnack's Curve Theorem, Semialgebraic Space. Source: Wikipedia. Pages: 36. Not illustrated. Free updates online. Purchase includes a free trial membership in the publisher's book club where you can select from more than a million books without charge. Excerpt: In real algebraic geometry, a general quartic plane curve has 28 bitangent lines, lines that are tangent to the curve in two places. These lines exist in the complex projective plane, but it is possible to define curves for which all 28 of these lines have real numbers as their coordinates and therefore belong to the Euclidean plane. An explicit quartic with twenty-eight real bitangents was first given by Plücker (1839) As Plücker showed, the number of real bitangents of any quartic must be 28, 16, or a number less than 9. Another quartic with 28 real bitangents can be formed by the locus of centers of ellipses with fixed axis lengths, tangent to two non-parallel lines. Shioda (1995) gave a different construction of a quartic with twenty-eight bitangents, formed by projecting a cubic surface; twenty-seven of the bitangents to Shioda's curve are real while the twenty-eighth is the line at infinity in the projective plane. The Trott curve, another curve with 28 real bitangents, is the set of points (x,y) satisfying the degree four polynomial equation These points form a nonsingular quadric curve that has genus three and that has twenty-eight real bitangents. Like the examples of Plücker and of Blum and Guinand, the Trott curve has four separated ovals, the maximum number for a curve of degree four, and hence is an M-curve. The four ovals can be grouped into six different pairs of ovals; for each pair of ovals there are four bitangents touching both ovals in the pair, two that separate the two ova...More: http://booksllc.net/?id=5204204 ... Read more

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Chapters: Lie Group, Lie Algebra, Pauli Matrices, Haar Measure, General Linear Group, Special Linear Group, Hilbert's Fifth Problem, Lattice, Möbius Transformation, Lorentz Group, an Exceptionally Simple Theory of Everything, Symmetric Space, Projective Linear Group, Root System, List of Simple Lie Groups, Orthogonal Group, Matrix Exponential, Heisenberg Group, Sl₂(r), Special Unitary Group, Ping-Pong Lemma, So(4), Complex Reflection Group, Freudenthal Magic Square, Exponential Map, Rotation Group, Table of Lie Groups, Euclidean Group, Projective Unitary Group, Bianchi Classification, Maurer-cartan Form, Baker-campbell-hausdorff Formula, Covering Group, Principal Homogeneous Space, Circle Group, E6, Group Algebra, Spin Group, Representation of a Lie Group, Classical Group, Symplectic Group, Ade Classification, Nilmanifold, Automorphic Form, Wess-zumino-witten Model, Cartan Decomposition, Weyl Group, Compact Group, Hermitian Symmetric Space, E7, Charts on So(3), Affine Group, Real Form, Killing Form, Darboux Derivative, Theta Representation, Indefinite Orthogonal Group, Pseudogroup, Adjoint Endomorphism, Klein Geometry, Maximal Compact Subgroup, Kleinian Group, Oppenheim Conjecture, Representation Ring, Coxeter Number, Glossary of Semisimple Groups, Maximal Torus, Triality, Reductive Group, Poincaré Group, One-Parameter Group, Gell-Mann Matrices, Bruhat Decomposition, List of Lie Group Topics, Ratner's Theorems, Quaternion-Kähler Symmetric Space, Vector Flow, Infinitesimal Transformation, G2, F4, Fundamental Representation, So(8), Lie Product Formula, Improper Rotation, Iwasawa Decomposition, Hitchin System, Lie Group Decomposition, Identity Component, Weakly Symmetric Space, Lie Group Homomorphism, Poisson-lie Group, Lie Theory, Lie Subgroup, Cartan Subgroup, E7½, Cartan's Theorem, Schottky Group, Continuous Symmetry, P-Compact Group, Analytic Subgroup, Lie's Third Theorem, Dunkl Operator, Iwasawa Manifold, So(5), Langlands ...More: http://booksllc.net/?id=314493 ... Read more

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The book gives the first complete presentation of two closely connected subjects that are now used in various branches of complex analysis. In the first part the theory of analytic functions of completely regular growth in an angle is developed. Being a natural extension of the classical theory of entire functions of completely regular growth, this theory possesses principally new features due to the influence of boundary values on the sides of the angle. The second part contains the theory of the Riemann boundary value problem with power type infinite index. Using the results obtained in the first part of the book, the author gives complete and efficient solutions to the problem in various natural classes of analytic functions. The solution to the well-known Paley problem on the growth of entire functions of finite order is given as one possible application. ... Read more

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The Riemann problem is the most fundamental problem in the entire field of non-linear hyperbolic conservation laws. Since first posed and solved in 1860, great progress has been achieved in the one-dimensional case. However, the two-dimensional case is substantially different. Although research interest in it has lasted more than a century, it has yielded almost no analytical demonstration. It remains a great challenge for mathematicians.This volume presents work on the two-dimensional Riemann problem carried out over the last 20 years by a Chinese group. The authors explore four models: scalar conservation laws, compressible Euler equations, zero-pressure gas dynamics, and pressure-gradient equations. They use the method of generalized characteristic analysis plus numerical experiments to demonstrate the elementary field interaction patterns of shocks, rarefaction waves, and slip lines. They also discover a most interesting feature for zero-pressure gas dynamics: a new kind of elementary wave appearing in the interaction of slip lines-a weighted Dirac delta shock of the density function.The Two-Dimensional Riemann Problem in Gas Dynamics establishes the rigorous mathematical theory of delta-shocks and Mach reflection-like patterns for zero-pressure gas dynamics, clarifies the boundaries of interaction of elementary waves, demonstrates the interesting spatial interaction of slip lines, and proposes a series of open problems. With applications ranging from engineering to astrophysics, and as the first book to examine the two-dimensional Riemann problem, this volume will prove fascinating to mathematicians and hold great interest for physicists and engineers. ... Read more

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The Riemann-Hilbert problem deals with linear systems of ordinary differential euqations in the complex domain. Namely, the question is whether there is a Fuchsian system with prescribed singularities and monodromy. Hilbert was, in fact, convinced that such a system does always exist. However, as it turned out only recently this is one of the very rare cases of a wrong prediction by him. In 1989, the second author of this book, A. Bolibruch, discovered a counterexample, thus giving a negative answer to Hilbert's famous 21st problem in its original form. This fact has immediately changed the point of view. Now, one has to ask for conditions on singularities and monodromy implying existence and respectively nonexistence of corresponding Fuchsian systems. This book treats all known results on the problem, both positive and negative. Besides this, it also contains other related results on scalar linear ordinary differential equations in the complex domain. Many examples are given. In the book, it is only assumed that the reader is acquainted with the basics from linear algebra, ordinary differential equations and functions of one complex variable.The more complicated material needed for the treatment of the Riemann-Hilbert problem is presented in the first chapters. They contain: the local theory, including its new version due to A.H.M. Levelt; Birkhoff-Grothendieck's theorem on vector bundles over the Riemann sphere or, equivalently, Birkhoff's factorization result for certain matrix functions; Plemelj's theorem providing a positive answer to a question similar to Hilbert's 21st problem concerning so-called regular systems instead of Fuchsian ones. The exposition of this "preliminary" material turns the book also into a useful introduction to several important chapters of the contemporary theory of ordinary differential equations in the complex domain. ... Read more