Riemann Hypothesis. Bernhard Riemann, Riemann Zeta Function, Conjecture, Hilbert–Palya Conjecture, Generalized Riemann Hypothesis, Lee–Yang Theorem, Local Zeta-Function, Explicit Formula, Logarithmic Integral Function, Leonhard Euler, Euler Product, On the Number of Primes Less Than a Given Magnitude, Article Sources and Contributors, Image Sources, Licenses and Contributors Данное издание не является оригинальным. Книга печатается по технологии принт-он-деманд после получения заказа.

ISBN: 978-6-1300-4861-7
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Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In the mathematical field of differential geometry, the Riemann curvature tensor, or Riemann–Christoffel tensor after Bernhard Riemann and Elwin Bruno Christoffel, is the most standard way to express curvature of Riemannian manifolds. It associates a tensor to each point of a Riemannian manifold (i.e., it is a tensor field), that measures the extent to which the metric tensor is not locally isometric to a Euclidean space. The curvature tensor can also be defined for any pseudo-Riemannian manifold, or indeed any manifold equipped with an affine connection. It is a central mathematical tool in the theory of general relativity, the modern theory of gravity, and the curvature of spacetime is in...

ISBN: 978-6-1312-6382-8
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High Quality Content by WIKIPEDIA articles! In mathematics, a Riemann sum is a method for approximating the total area underneath a curve on a graph, otherwise known as an integral. It may also be used to define the integration operation. The method was named after German mathematician Bernhard Riemann. Any Riemann sum on a given partition (that is, for any choice of yi between xi-1 and xi) is contained between the lower and the upper Riemann sums. A function is defined to be Riemann integrable if the lower and upper Riemann sums get ever closer as the partition gets finer and finer. This fact can also be used for numerical integration. Данное издание не является оригинальным. Книга...

ISBN: 978-6-1329-6235-5
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High Quality Content by WIKIPEDIA articles! In mathematics, a Riemann–Roch theorem for smooth manifolds is a version of results such as the Hirzebruch–Riemann–Roch theorem or Grothendieck–Riemann–Roch theorem (GRR) without a hypothesis making the smooth manifolds involved carry a complex structure. Results of this kind were obtained by Michael Atiyah and Friedrich Hirzebruch in 1959, reducing the requirements to something like a spin structure. Данное издание не является оригинальным. Книга печатается по технологии принт-он-деманд после получения заказа.

ISBN: 978-6-1312-6607-2
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High Quality Content by WIKIPEDIA articles! Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i.e. with an inner product on the tangent space at each point which varies smoothly from point to point. This gives in particular local notions of angle, length of curves, surface area, and volume. From those some other global quantities can be derived by integrating local contributions.Riemannian geometry originated with the vision of Bernhard Riemann expressed in his inaugurational lecture Uber die Hypothesen, welche der Geometrie zu Grunde liegen (German: On the hypotheses on which geometry is based). It is a very broad and abstract generalization of the...

ISBN: 978-6-1305-1292-7
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Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval. While the Riemann integral is unsuitable for many theoretical purposes, it is one of the easiest integrals to define. Some of these technical deficiencies can be remedied by the Riemann–Stieltjes integral, and most of them disappear in the Lebesgue integral. Loosely speaking, the Riemann integral is the limit of the Riemann sums of a function as the partitions get finer. If the limit exists then the function is said to be integrable (or more specifically Riemann-integrable). The Riemann sum can be made as close...

ISBN: 978-6-1303-3135-1
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Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In mathematics, particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional complex manifold. Riemann surfaces can be thought of as "deformed versions" of the complex plane: locally near every point they look like patches of the complex plane, but the global topology can be quite different. For example, they can look like a sphere or a torus or a couple of sheets glued together. Данное издание не является оригинальным. Книга печатается по технологии принт-он-деманд после получения заказа.

ISBN: 978-6-1329-6265-2
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Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In the mathematical field of differential geometry, the Riemann curvature tensor, or Riemann–Christoffel tensor after Bernhard Riemann and Elwin Bruno Christoffel, is the most standard way to express curvature of Riemannian manifolds. It associates a tensor to each point of a Riemannian manifold (i.e., it is a tensor field), that measures the extent to which the metric tensor is not locally isometric to a Euclidean space. The curvature tensor can also be defined for any pseudo-Riemannian manifold, or indeed any manifold equipped with an affine connection. It is a central mathematical tool in the theory of general relativity, the modern theory of gravity, and the curvature of spacetime is in...

ISBN: 978-6-1329-6005-4
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In mathematics, the Riemann Xi function is a variant of the Riemann zeta function, and is defined so as to have a particularly simple functional equation. The function is named in honour of Bernhard Riemann. The Riemann zeta function is a function of complex argument s that analytically continues the sum of the infinite series sum_{n=1}^{infty}frac{1}{n^s}, quad Re(s)>1. It plays a pivotal role in analytic number theory and has applications in physics, probability theory, and applied statistics. First results about this function were obtained by Leonhard Euler in the eighteenth century. It is named after Bernhard Riemann, who in the memoir "On the Number of Primes Less Than a Given...

ISBN: 978-6-1312-6532-7
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High Quality Content by WIKIPEDIA articles! A Riemann problem, named after Bernhard Riemann, consists of a conservation law together with piecewise constant data having a single discontinuity. The Riemann problem is very useful for the understanding of hyperbolic partial differential equations like the Euler equations because all properties, such as shocks and rarefaction waves, appear as characteristics in the solution. It also gives an exact solution to some complex nonlinear equations, such as the Euler equations. In numerical analysis Riemann problems appear in a natural way in finite volume methods for the solution of equation of conservation laws due to the discreteness of the grid. For that it is widely used in computational fluid...

ISBN: 978-6-1312-6425-2
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Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! n mathematics, the Riemann–Lebesgue lemma, named after Bernhard Riemann and Henri Lebesgue, is of importance in harmonic analysis and asymptotic analysis. The lemma says that the Fourier transform or Laplace transform of an L1 function vanishes at infinity. Let :R ? C be a measurable function. If is L1 integrable, that is to say if the Lebesgue integral of | | is finite, then int^infty_{-infty} f(x) e^{izx},dx rightarrow 0text{ as } zrightarrow pminfty. This says that the Fourier transform of tends to 0 as z tends to infinity. In fact, the same holds for the Laplace transform of if is supported on (0, ?), i.e. the above holds as |z| ? ?, Im(z) ? 0 if (x) = 0 for x < 0. If instead, is a...

ISBN: 978-6-1312-6567-9
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Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In their paper they define, for oriented smooth closed manifolds X and Y and a continuous mapping f: Y -> X that f is a c1-map if there is c1 in the integral cohomology group H2(Y, Z) such that for the Stiefel-Whitney classes w2 we have c1 = w2(Y) – f*(w2(X)) modulo 2 in H2(Y, Z/2Z). Writing ch(X) for the image in H*(X, Q) they showed that for f a c1-map there is f!: ch(Y) -> ch(X) which is a homomorphism of abelian groups, and satisfying f!(y)A^(X) = f*(y.exp(c1)/2)A^(Y)), where A^ is the A-hat genus and f* the Gysin homomorphism. This mimics the GRR theorem; but f! has only an implicit definition. This they specialised and refined in the case X = a point, where the condition becomes the...

ISBN: 978-6-1312-6617-1
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Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In mathematics, the Riemann–Hurwitz formula, named after Bernhard Riemann and Adolf Hurwitz, describes the relationship of the Euler characteristics of two surfaces when one is a ramified covering of the other. It therefore connects ramification with algebraic topology, in this case. It is a prototype result for many others, and is often applied in the theory of Riemann surfaces (which is its origin) and algebraic curves. Now assume that S and S? are Riemann surfaces, and that the map ? is complex analytic. The map ? is said to be ramified at a point P in S? if there exist analytic coordinates near P and ?(P) such that ? takes the form ?(z) = zn, and n > 1. An equivalent way of thinking...

ISBN: 978-6-1312-6564-8
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Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In mathematics, the Riemann–Stieltjes integral is a generalization of the Riemann integral, named after Bernhard Riemann and Thomas Joannes Stieltjes. The Riemann–Stieltjes integral of a real-valued function of a real variable with respect to a real function g is denoted by int_a^b f(x) , dg(x) and defined to be the limit, as the mesh of the partition P={ a = x_0 < x_1 < cdots < x_n = b} of the interval [a, b] approaches zero, of the approximating sum S(P,f,g) = sum_{i=0}^{n-1} f(c_i)(g(x_{i+1})-g(x_i)) where ci is in the i-th subinterval [xi, xi+1]. The two functions and g are respectively called the integrand and the integrator. The "limit" is here understood to be a number A...

ISBN: 978-6-1312-6637-9
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In mathematics, the Riemann–Hurwitz formula, named after Bernhard Riemann and Adolf Hurwitz, describes the relationship of the Euler characteristics of two surfaces when one is a ramified covering of the other. It therefore connects ramification with algebraic topology, in this case. It is a prototype result for many others, and is often applied in the theory of Riemann surfaces (which is its origin) and algebraic curves. Данное издание не является оригинальным. Книга печатается по технологии принт-он-деманд после получения заказа.

ISBN: 978-6-1329-6287-4
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Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! 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 vector fields. In other words, a Riemannian manifold is a differentiable manifold in which the tangent space at each point is a finite-dimensional Euclidean space. The terms are named after German mathematician Bernhard Riemann....

ISBN: 978-6-1305-1309-2
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Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In complex analysis, the Riemann mapping theorem states that if U is a non-empty simply connected open subset of the complex number plane Bbb C which is not all of Bbb C, then there exists a biholomorphic (bijective and holomorphic) mapping f, from U, onto the open unit disk D={zin {Bbb C} :|z|<1}. Intuitively, the condition that U be simply connected means that U does not contain any “holes”. The fact that f is biholomorphic implies that it is a conformal map and therefore angle-preserving. Intuitively, such a map preserves the shape of any sufficiently small figure, while possibly rotating and scaling (but not reflecting) it. Данное издание не является...

ISBN: 978-6-1329-6042-9
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Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! The Riemann–Roch theorem is an important tool in mathematics, specifically in complex analysis and algebraic geometry, for the computation of the dimension of the space of meromorphic functions with prescribed zeroes and allowed poles. It relates the complex analysis of a connected compact Riemann surface with the surface's purely topological genus g, in a way that can be carried over into purely algebraic settings. Initially proved as Riemann's inequality, the theorem reached its definitive form for Riemann surfaces after work of Riemann's short-lived student Gustav Roch in the 1850s. It was later generalized to algebraic curves, to higher-dimensional varieties and beyond. Данное...

ISBN: 978-6-1312-6600-3
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Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In mathematics, the Riemann series theorem (also called the Riemann rearrangement theorem), named after 19th-century German mathematician Bernhard Riemann, says that if an infinite series is conditionally convergent, then its terms can be arranged in a permutation so that the series converges to any given value, or even diverges.The terms of the series are often produced according to a certain rule, such as by a formula, or by an algorithm. As there are an infinite number of terms, this notion is often called an infinite series. Unlike finite summations, infinite series need tools from mathematical analysis to be fully understood and manipulated. In addition to their ubiquity in mathematics,...

ISBN: 978-6-1312-6434-4
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Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In mathematics, the Riemann series theorem (also called the Riemann rearrangement theorem), named after 19th-century German mathematician Bernhard Riemann, says that if an infinite series is conditionally convergent, then its terms can be arranged in a permutation so that the series converges to any given value, or even diverges. More generally, given a converging series of vectors in a finite dimensional real vector space E, the set of sums of converging rearranged series is an affine subspace of E. Данное издание не является оригинальным. Книга печатается по технологии принт-он-деманд после получения...

ISBN: 978-6-1329-6129-7
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High Quality Content by WIKIPEDIA articles! In the mathematical theory of harmonic analysis, the Riesz transforms are a family of generalizations of the Hilbert transform to Euclidean spaces of dimension d > 1. They are a type of singular integral operator, meaning that they are given by a convolution of one function with another function having a singularity at the origin. The Riesz transforms arises in the study of differentiability properties of harmonic potentials in potential theory and harmonic analysis. In particular, they arise in the proof of the Calderon-Zygmund inequality (Gilbarg Данное издание не является оригинальным. Книга печатается по технологии...

ISBN: 978-6-1329-6500-4
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High Quality Content by WIKIPEDIA articles! In mathematics, the Riesz–Thorin theorem, often referred to as the Riesz–Thorin interpolation theorem or the Riesz–Thorin convexity theorem is a result about interpolation of operators. It is named after Marcel Riesz and his student G. Olof Thorin.This theorem bounds the norms of linear maps acting between Lp spaces. Its usefulness stems from the fact that some of these spaces have rather simpler structure than others. Usually that refers to L2 which is a Hilbert space, or to L1 and L?. Therefore one may prove theorems about the more complicated cases by proving them in two simple cases and then using the Riesz–Thorin theorem to pass from the simple cases to the complicated cases. The...

ISBN: 978-6-1329-6566-0
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Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! There are several well-known theorems in functional analysis known as the Riesz representation theorem. They are named in honour of Frigyes Riesz.This theorem establishes an important connection between a Hilbert space and its (continuous) dual space: if the underlying field is the real numbers, the two are isometrically isomorphic; if the field is the complex numbers, the two are isometrically anti-isomorphic. The (anti-) isomorphism is a particular natural one as will be described next. Данное издание не является оригинальным. Книга печатается по технологии принт-он-деманд после получения заказа.

ISBN: 978-6-1329-6470-0
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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, the Riesz mean is a certain mean of the terms in a series. They were introduced by Marcel Riesz in 1911 as an improvement over the Cesaro mean. The Riesz mean should not be confused with the Bochner–Riesz mean or the Strong–Riesz mean. In statistics, mean has two related meanings: * the arithmetic mean (and is distinguished from the geometric mean or harmonic mean). * the expected value of a random variable, which is also called the population mean. There are other statistical measures that use samples that some people confuse with averages - including 'median' and 'mode.' Other simple...

ISBN: 978-6-1312-6726-0
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Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In mathematics, the Riesz function is an entire function defined by Marcel Riesz in connection with the Riemann hypothesis, by means of the power series {rm Riesz}(x) = -sum_{k=1}^infty frac{(-x)^k}{(k-1)! zeta(2k)}. If we set F(x) = frac12 {rm Riesz}(4 pi^2 x) we may define it in terms of the coefficients of the Laurent series development of the hyperbolic (or equivalently, the ordinary) cotangent around zero. The Maclaurin series coefficients of F increase in absolute value until they reach their maximum at the 40th term of -1.753 x 1017. By the 109th term they have dropped below one in absolute value. Taking the first 1000 terms suffices to give a very accurate value for F(z) for | z | <...

ISBN: 978-6-1312-6684-3
Издательство: Книга по требованию
Дата выхода: июль 2011