High Quality Content by WIKIPEDIA articles! Bezout's theorem is a statement in algebraic geometry concerning the number of common points, or intersection points, of two plane algebraic curves. The theorem claims that the number of common points of two such curves X and Y is equal to the product of their degrees. This statement must be qualified in several important ways, by considering points at infinity, allowing complex coordinates (or more generally, coordinates from the algebraic closure of the ground field), assigning an appropriate multiplicity to each intersection point, and excluding a degenerate case when X and Y have a common component. A simpler special case is that if X and Y are both real or complex irreducible curves, X has...

ISBN: 978-6-1311-9149-7
Издательство: Книга по требованию
Дата выхода: июль 2011

Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematics and its applications, a coordinate (or co-ordinate) system is a system for assigning an n-tuple of numbers, scalars or variables to each point in an n-dimensional space. This concept is part of the theory of manifolds. "Scalars" in many cases means real numbers, but, depending on context, can mean complex numbers or elements of some other commutative ring. For complicated spaces, it is often not possible to provide one consistent practical coordinate system for the entire space. In this case, a collection of coordinate systems, called graphs, are put together to form an atlas covering the whole...

ISBN: 978-6-1312-2032-6
Издательство: Книга по требованию
Дата выхода: июль 2011

Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In algebraic geometry, an algebraic curve is an algebraic variety of dimension one. The theory of these curves in general was quite fully developed in the nineteenth century, after many particular examples had been considered, starting with circles and other conic sections.An algebraic curve defined over a field F may be considered as the locus of points in Fn determined by at least n – 1 independent polynomial functions in n variables with coefficients in F, gi(x1, …, xn), where the curve is defined by setting each gi = 0. Данное издание не является оригинальным....

ISBN: 978-6-1312-2289-4
Издательство: Книга по требованию
Дата выхода: июль 2011

Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. An n-agonal bifrustum (or truncated bipyramid) is a polyhedron composed of three parallel planes of n-agons, with the middle plane largest and usually the top and bottom congruent. It can be constructed as two congruent frustum combined across a plane of symmetry, and also as a bipyramid with the two polar vertices truncated. They are duals to the family of elongated bipyramids. Данное издание не является оригинальным. Книга печатается по технологии принт-он-деманд после получения заказа.

ISBN: 978-6-1312-9818-9
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In the mathematical theory of functions of one or more complex variables, and also in complex algebraic geometry, a biholomorphism or biholomorphic function is a bijective holomorphic function whose inverse is also holomorphic. Formally, a biholomorphic function is a function ? defined on an open subset U of the n-dimensional complex space Cn with values in Cn which is holomorphic and one-to-one, such that its image is an open set V in Cn and the inverse phi^{-1}:Vto U is also holomorphic. More generally, U and V can be complex manifolds. One can prove that it is enough for ? to be holomorphic and one-to-one in order for it to be biholomorphic onto its image. Данное издание не...

ISBN: 978-6-1311-9190-9
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! Bipolar coordinates are a two-dimensional orthogonal coordinate system. There are two commonly defined types of bipolar coordinates. The other system is two-center bipolar coordinates. There is also third coordinate system that based on two poles (biangular coordinates). First is based on the Apollonian circles. The curves of constant ? and of ? are circles that intersect at right angles. The coordinates have two foci F1 and F2, which are generally taken to be fixed at (?a, 0) and (a, 0), respectively, on the x-axis of a Cartesian coordinate system. Данное издание не является оригинальным. Книга печатается по технологии...

ISBN: 978-6-1312-2036-4
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Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! Bipolar cylindrical coordinates are a three-dimensional orthogonal coordinate system that results from projecting the two-dimensional bipolar coordinate system in the perpendicular z-direction. The two lines of foci F1 and F2 of the projected Apollonian circles are generally taken to be defined by x = ? a and x = + a, respectively, (and by y = 0) in the Cartesian coordinate system. The term "bipolar" is often used to describe other curves having two singular points (foci), such as ellipses, hyperbolas, and Cassini ovals. However, the term bipolar coordinates is never used to describe coordinates associated with those curves, e.g., elliptic coordinates. Данное издание не...

ISBN: 978-6-1312-2042-5
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! An n-gonal bipyramid or dipyramid is a polyhedron formed by joining an n-gonal pyramid and its mirror image base-to-base. The referenced n-gon in the name of the bipyramids is not an external face but an internal one, existing on the primary symmetry plane which connects the two pyramid halves. The face-transitive bipyramids are the dual polyhedra of the uniform prisms and will generally have isosceles triangle faces. A bipyramid can be projected on a sphere or globe as n equally spaced lines of longitude going from pole to pole, and bisected by a line around the equator. Данное издание не является оригинальным. Книга печатается по...

ISBN: 978-6-1312-8561-5
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Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In mathematics, birational geometry is a part of the subject of algebraic geometry, that deals with the geometry of an algebraic variety that is dependent only on its function field. In the case of dimension two, the birational geometry of algebraic surfaces was largely worked out by the Italian school of algebraic geometry in the years 1890–1910. From about 1970 advances have been made in higher dimensions, giving a good theory of birational geometry for dimension three. Birational geometry is largely a geometry of transformations, but it doesn't fit exactly with the Erlangen programme. One reason is that its nature is to deal with transformations that are only defined on an open, dense...

ISBN: 978-6-1311-5737-0
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! The bisection method in mathematics, is a root- finding algorithm which repeatedly bisects an interval then selects a subinterval in which a root must lie for further processing. It is a very simple and robust method, but it is also relatively slow. Данное издание не является оригинальным. Книга печатается по технологии принт-он-деманд после получения заказа.

ISBN: 978-6-1341-0066-3
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Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! Bispherical coordinates are a three-dimensional orthogonal coordinate system that results from rotating the two-dimensional bipolar coordinate system about the axis that connects the two foci. Thus, the two foci F1 and F2 in bipolar coordinates remain points (on the z-axis, the axis of rotation) in the bispherical coordinate system. The classic applications of bispherical coordinates are in solving partial differential equations, e.g., Laplace's equation, for which bispherical coordinates allow a separation of variables. However, the Helmholtz equation is not separable in bispherical coordinates. A typical example would be the electric field surrounding two conducting spheres of different...

ISBN: 978-6-1312-2047-0
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In mathematics, a bitangent to a curve C is a line L that touches C in two distinct points P and Q and that has the same direction to C at these points. That is, L is an tangent line at P and at Q. In general, an algebraic curve will have infinitely many secant lines, but only finitely many bitangents. Bezout's theorem implies that a plane curve with a bitangent must have degree at least 4. The case of the 28 bitangents of a quartic was a celebrated piece of geometry of the nineteenth century, a relationship being shown to the 27 lines on the cubic surface. Данное издание не является оригинальным. Книга печатается по технологии...

ISBN: 978-6-1312-3407-1
Издательство: Книга по требованию
Дата выхода: июль 2011

ISBN: 9781849684347
Издательство: Packt Publishing

Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. Blau space consists of the multidimensional coordinate system, created by considering the set of socio-demographic variables as dimensions. All socio-demographic characteristics are potential elements of Blau space, including continuous characteristics such as age, years of education, income, occupational prestige, geographic location, and so forth. In addition, categorical measures of socio-demographic characteristics such as race, sex, religion, birthplace, and others are Blau dimensions. "Blau space" is a theoretical construct, named after Peter Blau, first presented in McPherson's "Ecology of Affiliation" in...

ISBN: 978-6-1312-2067-8
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f : U -> R (where U is an open subset of Rn) which satisfies Laplace's equation, i.e. In Riemannian geometry, a Riemannian manifold or Riemannian space (M,g) is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, in a manner which varies smoothly from point to point. The metric g is a positive definite symmetric tensor: a metric tensor. This allows one to define various notions such as angles, lengths of curves, areas (or volumes), curvature, gradients of functions and divergence of...

ISBN: 978-6-1312-3437-8
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High Quality Content by WIKIPEDIA articles! In mathematics — specifically, differential geometry — the Bochner identity is an identity concerning harmonic maps between Riemannian manifolds. The identity is named after the American mathematician Salomon Bochner.Let M and N be Riemannian manifolds and let u : M ? N be a harmonic map. Let d denote the exterior derivative, ? the gradient, ? the Laplace-Beltrami operator, RiemN the Riemann curvature tensor on N and RicM the Ricci curvature tensor on M. Данное издание не является оригинальным. Книга печатается по технологии принт-он-деманд после получения заказа.

ISBN: 978-6-1312-3426-2
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Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In mathematics, the Bogomolov–Miyaoka–Yau inequality is the inequality between Chern numbers of compact complex surfaces of general type. Its major interest is the way it restricts the possible topological types of the underlying real 4-manifold. It was proved independently by S.-T. Yau (1977, 1978) and Yoichi Miyaoka (1977), after Van de Ven (1966) and Fedor Bogomolov (1978) proved weaker versions with the constant 3 replaced by 8 and 4. Borel and Hirzebruch showed that the inequality is best possible by finding infinitely many cases where equality holds. The inequality is false in positive characteristic: (Lang 1983) and Easton (2008) gave examples of surfaces in characteristic p, such as...

ISBN: 978-6-1312-3458-3
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In geometry, the Bolyai–Gerwien theorem states that any two simple polygons of equal area are equidecomposable; i.e. one can cut the first into finitely many polygonal pieces and rearrange the pieces to obtain the second polygon. "Rearrangement" means that one may apply a translation and a rotation to every polygonal piece. Unlike the generalized solution to Tarski's circle-squaring problem, the axiom of choice is not required for the proof, and the decomposition and reassembly can actually be carried out "physically"; the pieces can, in theory, be cut with scissors from paper and reassembled by hand. Данное издание не является оригинальным. Книга...

ISBN: 978-6-1312-5508-3
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Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In mathematics, the Bolza surface is a compact Riemann surface of genus 2 with the highest possible order of the conformal automorphism group in this genus, namely 48.The Bolza surface is a (2,3,8) triangle surface. More specifically, the Fuchsian group defining the Bolza surface is a subgroup of the group generated by reflections in the sides of a hyperbolic triangle with angles More specifically, it is a subgroup of the index-two subgroup of the group of reflections, which consists of products of an even number of reflections, which has an abstract presentation in terms of generators s2,s3,s8 and relations s22 = s33 = s88 = 1 as well as s2s3 = s8. The Fuchsian group defining the Bolza surface...

ISBN: 978-6-1312-3477-4
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Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! A Boolean network consists of a set of Boolean variables whose state is determined by other variables in the network. They are a particular case of discrete dynamical networks, where time and states are discrete, i.e. they have a bijection onto an integer series. Boolean and elementary cellular automata are particular cases of Boolean networks, where the state of a variable is determined by its spatial neighbors. The first Boolean networks were proposed by Stuart A. Kauffman in 1969, as random models of genetic regulatory networks (Kauffman 1969, 1993). Данное издание не является оригинальным. Книга печатается по технологии...

ISBN: 978-6-1311-7408-7
Издательство: Книга по требованию
Дата выхода: июль 2011

ISBN: 9781849514880
Издательство: Packt Publishing

ISBN: 9781782163268
Издательство: Packt Publishing

High Quality Content by WIKIPEDIA articles! In mathematics, the Borel fixed-point theorem is a fixed-point theorem in algebraic geometry. The result was proved by the Swiss mathematician Armand Borel in 1956. Geometry "Earth-Measuring" is a part of mathematics concerned with questions of size, shape, relative position of figures, and the properties of space. Geometry is one of the oldest sciences. Initially a body of practical knowledge concerning lengths, areas, and volumes, in the 3rd century BC geometry was put into an axiomatic form by Euclid, whose treatment—Euclidean geometry—set a standard for many centuries to follow. The field of astronomy, especially mapping the positions of the stars and planets on the celestial sphere,...

ISBN: 978-6-1311-9178-7
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In mathematics, the Borell–Brascamp–Lieb inequality is an integral inequality due to many different mathematicians but named after Christer Borell, Herm Jan Brascamp and Elliott Lieb. The result was proved for p > 0 by Henstock and Macbeath in 1953. The case p = 0 is known as the Prekopa–Leindler inequality and was re-discovered by Brascamp and Lieb in 1976, when they proved the general version below; working independently, Borell had done the same in 1975. The nomenclature of "Borell–Brascamp–Lieb inequality" is due to Cordero-Erausquin, McCann and Schmuckenschlager, who in 2001 generalized the result to Riemannian manifolds such as the sphere and hyperbolic space. Данное...

ISBN: 978-6-1312-6873-1
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! The Borsuk problem in geometry, for historical reasons incorrectly called a Borsuk conjecture, is a question in discrete geometry.In 1932 Karol Borsuk has shown that an ordinary 3-dimensional ball in Euclidean space can be easily dissected into 4 solids, each of which has a smaller diameter than the ball, and generally d-dimensional ball can be covered with d + 1 compact sets of diameters smaller than the ball. At the same time he proved that d subsets are not enough in general. The proof is based on the Borsuk–Ulam theorem. Данное издание не является оригинальным. Книга печатается по технологии принт-он-деманд...

ISBN: 978-6-1312-5509-0
Издательство: Книга по требованию
Дата выхода: июль 2011