High Quality Content by WIKIPEDIA articles! Ribbon diagrams, also known as Richardson Diagrams, are 3D schematic representations of protein structure and are one of the most common methods of protein depiction used today. Ribbon diagrams are generated by interpolating a smooth curve through the polypeptide backbone. ?-helices are shown as coiled ribbons or thick tubes, ?-strands as arrows, and lines or thin tubes for random coils. The direction of the polypeptide chain may be indicated by a colour ramp along the length of the ribbon. Ribbon diagrams are simple, yet powerful, in expressing the visual basics of a molecular structure (twist, fold and unfold). This method has successfully portrayed the overall organization of the protein...

ISBN: 978-6-1310-6512-5
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In mathematics, Ribet's theorem (earlier called the epsilon conjecture or ?-conjecture) is a statement in number theory concerning properties of Galois representations associated with modular forms. It was proposed by Jean-Pierre Serre and proved by Ken Ribet. The proof of epsilon conjecture was a significant step towards the proof of Fermat's Last Theorem. As shown by Serre and Ribet, the Taniyama–Shimura conjecture (whose status was unresolved at the time) and the epsilon conjecture together imply that Fermat's Last Theorem is true. Ribet mentions in a NOVA program entitled "The Proof" that Barry Mazur casually assisted him in the completion of his proof by suggesting that he add a...

ISBN: 978-6-1312-6171-8
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In mathematics, an ordinary differential equation (or ODE) is a relation that contains functions of only one independent variable, and one or more of its derivatives with respect to that variable. Ordinary differential equations are distinguished from partial differential equations, which involve partial derivatives of several variables. Ordinary differential equations arise in many different contexts including geometry, mechanics, astronomy and population modelling. Many famous mathematicians have studied differential equations and contributed to the field, including Newton, Leibniz, the Bernoulli family, Riccati, Clairaut, d'Alembert and Euler. Much study has been devoted to the solution of...

ISBN: 978-6-1311-5461-4
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In differential geometry, the Ricci flow is an intrinsic geometric flow (a process which deforms the metric of a Riemannian manifold) in this case in a manner formally analogous to the diffusion of heat, thereby smoothing out irregularities in the metric. It plays an important role in Grigori Perelman's solution of the Poincare conjecture; in this context is also called the Ricci–Hamilton flow. The Ricci flow has been intensively studied since 1981.For instance, acertain class of solutions to the Ricci flow demonstrates that neckpinch singularities will form on an evolving n-dimensional metric Riemannian manifold having a certain topological property (positive Euler characteristic), as the...

ISBN: 978-6-1329-5845-7
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In computability theory, Rice's theorem states that, for any non-trivial property of partial functions, there is no general and effective method to decide whether an algorithm computes a partial function with that property. Here, a property of partial functions is called trivial if it holds for all partial computable functions or for none, and an effective decision method is called general if it decides correctly for every algorithm. The theorem is named after Henry Gordon Rice, and is also known as the Rice-Myhill-Shapiro theorem after Rice, John Myhill, and Norman Shapiro. Данное издание не является оригинальным. Книга печатается по...

ISBN: 978-6-1312-6241-8
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In computability theory, the Rice–Shapiro theorem is a generalization of Rice's theorem, and is named after Henry Gordon Rice and Norman Shapiro.The statement can be expressed informally as follows: given that a computable function (viewed as a black-box) is an infinite object, and we cannot hope to develop a general algorithm studying a property of function on infinite input-output pairs; there is in general no truly better way than to apply the function on some inputs and hope to get their outputs. Данное издание не является оригинальным. Книга печатается по технологии принт-он-деманд после получения...

ISBN: 978-6-1312-6228-9
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In logic, Richard's paradox is a semantical antinomy in set theory and natural language first described by the French mathematician Jules Richard in 1905. Today, the paradox is ordinarily used in order to motivate the importance of carefully distinguishing between mathematics and metamathematics. The paradox was also a motivation in the development of predicative mathematics.The original statement of the paradox, due to Richard (1905), has a relation to Cantor's diagonal argument on the uncountability of the set of real numbers. The paradox begins with the observation that certain expressions in English unambiguously define real numbers, while other expressions in English do not. For example,...

ISBN: 978-6-1312-6313-2
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In numerical analysis, Richardson extrapolation is a sequence acceleration method, used to improve the rate of convergence of a sequence. It is named after Lewis Fry Richardson, who introduced the technique in the early 20th century. In the words of Birkhoff and Rota, "... its usefulness for practical computations can hardly be overestimated." Practical applications of Richardson extrapolation include Romberg integration, which applies Richardson extrapolation to the trapezium rule, and the Bulirsch–Stoer algorithm for solving ordinary differential equations. Данное издание не является оригинальным. Книга печатается по технологии...

ISBN: 978-6-1312-6344-6
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In logic, Richard's paradox is a semantical antinomy in set theory and natural language first described by the French mathematician Jules Richard in 1905. Today, the paradox is ordinarily used in order to motivate the importance of carefully distinguishing between mathematics and metamathematics. The paradox was also a motivation in the development of predicative mathematics.The original statement of the paradox, due to Richard (1905), has a relation to Cantor's diagonal argument on the uncountability of the set of real numbers. Данное издание не является оригинальным. Книга печатается по технологии принт-он-деманд...

ISBN: 978-6-1329-5914-0
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! The Richardson–Lucy algorithm, also known as Richardson–Lucy deconvolution, is an iterative procedure for recovering a latent image that has been blurred by a known point spread function.In mathematics, deconvolution is an algorithm-based process used to reverse the effects of convolution on recorded data. The concept of deconvolution is widely used in the techniques of signal processing and image processing. Because these techniques are in turn widely used in many scientific and engineering disciplines, deconvolution finds many applications.In this case ? is noise that has entered our recorded signal. If we assume that a noisy signal or image is noiseless when we try to make a statistical...

ISBN: 978-6-1312-6353-8
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! The Richardson number is named after Lewis Fry Richardson (1881 – 1953). It is the dimensionless number that expresses the ratio of potential to kinetic energy Ri = {ghover u^2} If the Richardson number is much less than unity, buoyancy is unimportant in the flow. If it is much greater than unity, buoyancy is dominant (in the sense that there is insufficient kinetic energy to homogenize the fluids).In aviation, the Richardson number is used as a rough measure of expected air turbulence. A lower score indicates a higher degree of turbulence. Values in the range 10 to 0.1 are typical, with values below unity indicating significant turbulence. Данное издание не является...

ISBN: 978-6-1305-1280-4
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In numerical analysis, Richardson extrapolation is a sequence acceleration method, used to improve the rate of convergence of a sequence. It is named after Lewis Fry Richardson, who introduced the technique in the early 20th century.In the words of Birkhoff and Rota, "... its usefulness for practical computations can hardly be overestimated."Practical applications of Richardson extrapolation include Romberg integration, which applies Richardson extrapolation to the trapezium rule, and the Bulirsch–Stoer algorithm for solving ordinary differential equations. Данное издание не является оригинальным. Книга печатается по технологии...

ISBN: 978-6-1329-5949-2
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! The ridges (or the ridge set) of a smooth function of two variables is a set of curves whose points are, in one or more ways to be made precise below, local maxima of the function in at least one dimension. For a function of N variables, its ridges are a set of curves whose points are local maxima in N ? 1 dimensions. In this respect, the notion of ridge points extends the concept of a local maximum. Correspondingly, the notion of valleys for a function can be defined by replacing the condition of a local maximum with the condition of a local minimum. The union of ridge sets and valley sets, together with a related set of points called the connector set form a connected set of curves that...

ISBN: 978-6-1329-5947-8
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! The ridges (or the ridge set) of a smooth function of two variables is a set of curves whose points are, loosely speaking, local maxima in at least one dimension. For a function of N variables, its ridges are a set of curves whose points are local maxima in N ? 1 dimensions. (A more precise definition is given below). In this respect, the notion of ridge points can be seen as an extension of the concept of a local maximum. Correspondingly, the notion of valleys for a function can be defined by replacing the condition of a local maximum with the condition of a local minimum. The union of ridge sets and valley sets, together with a related set of points called the connector set form a connected...

ISBN: 978-6-1312-6368-2
Издательство: Книга по требованию
Дата выхода: июль 2011

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
Издательство: Книга по требованию
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
Издательство: Книга по требованию
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
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. 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
Издательство: Книга по требованию
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
Издательство: Книга по требованию
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
Издательство: Книга по требованию
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
Издательство: Книга по требованию
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
Издательство: Книга по требованию
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
Издательство: Книга по требованию
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
Издательство: Книга по требованию
Дата выхода: июль 2011