High Quality Content by WIKIPEDIA articles! High Quality Content by WIKIPEDIA articles! Shing-Tung Yau ( born April 4, 1949) is a Chinese American mathematician working in differential geometry. He was born in Shantou, Guangdong Province, China into a family of scholars from Jiaoling, Guangdong Province. Yau has impacts on both physics and mathematics. Calabi-Yau manifolds are among the ‘standard toolkit' for string theorists today. He has been active at the interface between geometry and theoretical physics. His proof of the positive energy theorem in general relativity demonstrated—sixty years after its discovery—that Einstein's theory is consistent and stable. His proof of the Calabi conjecture allowed physicists—using...

ISBN: 978-6-1329-9232-1
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Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In mathematics, a Shintani zeta function or Shintani L-function is a generalization of the Riemann zeta function. They were studied studied by Takuro Shintani (1976). They include Hurwitz zeta functions, Barnes zeta functions, and Witten zeta functions as special cases. The Shintani zeta function of (s1, ..., sk) is given by sum_{n_1,dots,n_mge 0}frac{1}{L_1^{s_1} cdots L_k^{s_k}} where each Lj is an inhomogeneous linear function of (n1, ... ,nm). The special case when k = 1 is the Barnes zeta function. Данное издание не является оригинальным. Книга печатается по технологии принт-он-деманд после получения...

ISBN: 978-6-1311-7351-6
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Дата выхода: июль 2011

The initial data of a general Riemann problem is constant along radial directions from an origin and it is piecewise constant as a function of angle. In this thesis, the special case with initial data of piecewise constant solutions joined by four forward rarefaction waves is considered. A transonic shock is discovered through refined numerical simulations and generalized characteristic analysis, and the mathematical mechanism of shock formation for four rarefactions cases is clarified. To our best knowledge, shock formation in a rarefaction wave-only 2D Riemann problem, had not been previously observed. In this thesis, front tracking method is implemented as a numerical method based on the Riemann solution and Glimm's method; the...

ISBN: 978-3-6392-5886-8
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Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! The shoelace formula, or shoelace algorithm, is a mathematical algorithm to determine the area of a polygon whose vertices are described by ordered pairs in the plane. The user cross-multiplies corresponding coordinates to find the area encompassing the polygon, and subtracts it from the surrounding polygon to find the area of the polygon within. It is called the shoelace formula because of the constant cross-multiplying for the coordinates making up the polygon, like tying shoelaces. It is also sometimes called the shoelace method. In mathematics, an ordered pair is a collection of objects having two coordinates (or entries or projections), such that one can always uniquely determine the...

ISBN: 978-6-1311-7712-5
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High Quality Content by WIKIPEDIA articles! Shor's algorithm is important because it can, using a quantum computer, be used to break the widely used public-key cryptography scheme known as RSA. RSA is based on the assumption that factoring large numbers is computationally infeasible. So far as is known, this assumption is valid for classical (non-quantum) computers; no classical algorithm is known that can factor in polynomial time. However, Shor's algorithm shows that factoring is efficient on a quantum computer, so an appropriately large quantum computer can break RSA. It was also a powerful motivator for the design and construction of quantum computers and for the study of new quantum computer algorithms. Данное издание...

ISBN: 978-6-1305-7878-7
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High Quality Content by WIKIPEDIA articles! In arithmetic, short division is a procedure which breaks down a division problem into a series of easier steps. As in all division problems, one number, called the dividend, is divided by another, called the divisor, producing a result called the quotient. Short division requires no advanced technology nor mental gymnastics, merely paper and pencil (or any similar means for writing). It is very powerful, enabling computations involving arbitrarily large numbers to be performed by following a series of simple steps. Short division is an abbreviated form of long division. As short division relies on mental arithmetic, it is only suitable if the divisor is small - typically less than 10....

ISBN: 978-6-1305-7885-5
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Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In mathematics, especially homological algebra and other applications of abelian category theory, the short five lemma is a special case of the five lemma. It states that for the following commutative diagram (in any abelian category, or in the category of groups), if the rows are short exact sequences, and if g and h are isomorphisms, then f is an isomorphism as well.The essence of the lemma can be summarized as follows: if you have a homomorphism f from an object B to an object B?, and this homomorphism induces an isomorphism from a subobject A of B to a subobject A? of B? and also an isomorphism from the factor object B/A to B?/A?'', then f itself is an isomorphism. Note however that the...

ISBN: 978-6-1312-3983-0
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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. Short-term business statistics (STS) are economic statistics published by Eurostat, the European Commission's statistical office. They cover the single countries of the European Union and provide aggregated results for the euro area and the EU. Short-term business statistics provide information on the economic development of four major domains: industry, construction, retail trade and other services. These domains are defined in relation to a classification of economic activities, the NACE Rev.2 (Statistical Classification of Economic Activities in the European Community, second revision). Данное...

ISBN: 978-6-1348-2005-9
Издательство: Книга по требованию
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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In the context of interest rate derivatives, a short-rate model is a mathematical model that describes the future evolution of interest rates by describing the future evolution of the short rate.Throughout this section Wt represents a standard Brownian motion under a risk-neutral probability measure and dWt its differential. Other than Rendleman–Bartter and Ho–Lee, which do not capture the mean reversion of interest rates, these models can be thought of as special cases of Ornstein–Uhlenbeck processes. Данное издание не является оригинальным. Книга...

ISBN: 978-6-1348-0878-1
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High Quality Content by WIKIPEDIA articles! The short-time Fourier transform (STFT), or alternatively short-term Fourier transform, is a Fourier-related transform used to determine the sinusoidal frequency and phase content of local sections of a signal as it changes over time. Simply, in the continuous-time case, the function to be transformed is multiplied by a window function which is nonzero for only a short period of time. The Fourier transform (a one-dimensional function) of the resulting signal is taken as the window is slid along the time axis, resulting in a two-dimensional representation of the signal. Данное издание не является оригинальным. Книга печатается по...

ISBN: 978-6-1303-1917-5
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High Quality Content by WIKIPEDIA articles! The short ton is a unit of weight equal to 2,000 pounds (907.18474 kg). In the United States it is often called simply ton without distinguishing it from the metric ton (or tonne, 1,000 kilograms) or the long ton (2,240 pounds (1,016.0469088 kg)); rather, the other two are specifically noted. There are, however, some U.S. applications for which unspecified tons normally means long tons (for example, Navy ships) or metric tons (world grain production figures). Both the long and short ton are defined as 20 hundredweights, but a hundredweight is 112 pounds (50.80234544 kg) (which is equal to 8 stone, 1 stone/6.35029318 kilograms) in the Imperial system (long or gross hundredweight) and 100 pounds...

ISBN: 978-6-1303-3033-0
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High Quality Content by WIKIPEDIA articles! In mathematics, a shrewd cardinal is a certain kind of large cardinal number introduced by (Rathjen 1995)., extending the definition of indescribable cardinals. A cardinal number is called -shrewd if for every proposition , and set A V with (V + , , A) there exists an , ' < with (V + ', , A V ) . It is called shrewd if it is -shrewd for every (including > ). This definition extends the concept of indescribability to transfinite levels. A -shrewd cardinal is also -shrewd for any ordinal < . Shrewdness was developed by Michael Rathjen as part of his ordinal analysis of 12-comprehension. It is essentially the nonrecursive analog to the stability property for admissible ordinals. More...

ISBN: 978-6-1311-7785-9
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Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In category theory, a branch of mathematics, certain unusual functors are denoted f! and f!, with the exclamation mark used to indicate that they are exceptional in some way. They are thus accordingly sometimes called shriek maps, with "shriek" being slang for an exclamation mark, though other terms are used, depending on context. In algebraic geometry, these arise in image functors for sheaves, particularly Verdier duality, where f! is a "less usual" functor. In algebraic topology, these arise particularly in fiber bundles, where they yield maps that have the opposite of the usual variance. They are thus called wrong way maps, Gysin maps, as they originated in the Gysin sequence, or transfer...

ISBN: 978-6-1311-7797-2
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High Quality Content by WIKIPEDIA articles! In the mathematical field of graph theory, the Shrikhande graph is a named graph discovered by S. S. Shrikhande in 1959. It is a strongly regular graph with 16 vertices and 48 edges, with each vertex having a degree of 6. In the Shrikhande graph, any two vertices I and J have two distinct neighbors in common (excluding the two vertices I and J themselves), which holds true whether or not I is adjacent to J. In other words, its parameters for being strongly regular are: {16,6,2,2}, with ? = ? = 2, this equality implying that the graph is associated with a symmetric BIBD. It shares these parameters with a different graph, the 4x4 rook's graph. The Shrikhande graph is locally hexagonal; that is,...

ISBN: 978-6-1311-7804-7
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In mathematics, more specifically algebraic topology, a covering map is a continuous surjective function p a from a topological space, C, to a topological space, X, such that each point in X has a neighbourhood evenly covered by p. This means that for each point x in X, there is associated an ordered pair, (K, U), where U is a neighborhood of x and where K is a collection of disjoint open sets in C, each of which gets mapped homeomorphically, via p, to U (as shown in the image). In particular, this means that every covering map is necessarily a local homeomorphism. Under this definition, C is called the covering space of X. Данное издание не является...

ISBN: 978-6-1303-3234-1
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High Quality Content by WIKIPEDIA articles! The Shulba Sutras or ?ulbas?tras (Sanskrit ?ulba: "string, cord, rope") are sutra texts belonging to the ?rauta ritual and containing geometry related to fire-altar construction. The Shulba Sutras are part of the larger corpus of texts called the Shrauta Sutras, considered to be appendices to the Vedas. They are the only sources of knowledge of Indian mathematics from the Vedic period. Unique fire-altar shapes were associated with unique gifts from the Gods. For instance, "he who desires heaven is to construct a fire-altar in the form of a falcon"; "a fire-altar in the form of a tortoise is to be constructed by one desiring to win the world of Brahman" and "those who wish to destroy existing...

ISBN: 978-6-1311-7829-0
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High Quality Content by WIKIPEDIA articles! The Shusaku number represents the "distance" between a Go player and Honinbo Shusaku, measured in Go opponents. A person's Shusaku number can be determined, as follows: * Shusaku himself has the Shusaku number 0. * Any other Go player has the number 1 + the smallest Shusaku number of the player's opponents (provided at least one opponent has a Shusaku number). For example, if a player has played Shusaku himself, that player would have a Shusaku number of 1. * For all other people, the Shusaku number is undefined. To extend the analogy to Erd?s numbers even further, one can define Shusaku numbers of the second kind to only consider players who have defeated a lower-numbered opponent. That is,...

ISBN: 978-6-1311-7837-5
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Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! The International System of Units (SI) specifies a set of unit prefixes, known as SI prefixes, also known as metric prefixes. An SI prefix is a name that precedes a basic unit of measure to indicate a decimal multiple or fraction of the unit. Each prefix has a unique symbol that is prepended to the unit symbol. The SI prefixes are standardized by the International Bureau of Weights and Measures in resolution dating from 1960 to 1991. SI prefixes are used to reduce the number of zeros shown in numerical quantities before or after the decimal point. For example, an electrical current of 0.000000001ampere, or one-billionth (short scale) of an ampere, is written by using the SI-prefix nano as 1...

ISBN: 978-6-1305-7897-8
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High Quality Content by WIKIPEDIA articles! Sicherman dice are the only pair of 6-sided dice which are not normal dice, bear only positive integers, and have the same probability distribution for the sum as normal dice. The faces on the dice are numbered 1, 2, 2, 3, 3, 4 and 1, 3, 4, 5, 6, 8. Crazy dice refers to a standard mathematical problem or puzzle in elementary combinatorics, involving a re-labeling the faces of a pair of six-sided dice to reproduce the same frequency of sums as the standard labeling. It is a standard exercise in elementary combinatorics to calculate the number of ways of rolling any given value with a pair of fair six-sided dice (by taking the sum of the two rolls). These dice were discovered by Colonel George...

ISBN: 978-6-1311-7858-0
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Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In mathematics, LHS is informal shorthand for the left-hand side of an equation. Similarly, RHS is the right-hand side. Each is solely a name for a term as part of an expression; and they are in practice interchangeable, since equality is symmetric. This abbreviation is seldom if ever used in print; it is very informal. More generally, these terms may apply to an inequation or inequality. In the inequality case, there is no symmetry. The right-hand side is everything on the right side of a test operator in an expression. Conversely, the left-hand side is everything on the left side. The equation on the right side/right part of the "=" sign is the right side of the equation and the left of the...

ISBN: 978-6-1311-7893-1
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Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In number theory, a Sidon sequence (or Sidon set), named after the Hungarian mathematician Simon Sidon, is a sequence A = {a0, a1, a2, ...} of natural numbers in which all pairwise sums ai + aj (i ? j) are different. Sidon introduced the concept in his investigations of Fourier series. The main problem, posed by Paul Erd?s and Pal Turan, is how many elements can A have up to some number x. Despite a large body of research this question remains, as yet, unanswered. Erd?s and Turan proved that the number of elements of A up to x is at most sqrt{x}+O(sqrt[4]{x}) and using a construction of J. Singer they get a sqrt{x}(1-o(1)) lower bound. All finite Sidon sets are Golomb rulers, and vice-versa. To...

ISBN: 978-6-1311-7923-5
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Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! Given a holomorphic endomorphism f:Sto S on a Riemann surface S we consider the dynamical system generated by the iterates of f denoted by f^n=fcircstackrel{left.nright)}{cdots}circ f. We then call the orbit mathcal{O}^+(z_0) of z0 as the set of forward iterates of z0. We are interested in the asymptotic behavior of the orbits in S (which will usually be mathbb{C}, the complex plane or mathbb{hat C}=mathbb{C}cup{infty}, the Riemann sphere), and we call S the phase plane or dynamical plane. One possible asymptotic behavior for a point z0 is to be a fixed point, or in general a periodic point. In this last case fp(z0) = z0 where p is the period and p = 1 means z0 is a fixed point. We can then...

ISBN: 978-6-1311-7939-6
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Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In mathematics, more specifically in the field of analytic number theory, a Siegel zero, named after Carl Ludwig Siegel, is a type of potential counterexample to the generalized Riemann hypothesis, on the zeroes of Dirichlet L-function. The importance of the possible Siegel zeroes is seen in all known results on the zero-free regions of L-functions: they show a kind of 'indentation' near s = 1, while otherwise generally resembling that for the Riemann zeta function — that is, they are to the left of the line Re(s) = 1, and asymptotic to it. Because of the analytic class number formula, data on Siegel zeroes have a direct impact on the class number problem, of giving lower bounds for class...

ISBN: 978-6-1311-7959-4
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Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In mathematics, Siegel modular forms are a major type of automorphic form. These stand in relation to the conventional elliptic modular forms as abelian varieties do in relation to elliptic curves; the complex manifolds constructed as in the theory are basic models for what a moduli space for abelian varieties (with some extra level structure) should be, as quotients of the Siegel upper half-space rather than the upper half-plane by discrete groups. The modular forms of the theory are holomorphic functions on the set of symmetric n x n matrices with positive definite imaginary part; the forms must satisfy an automorphy condition. Siegel modular forms can be thought of as multivariable modular...

ISBN: 978-6-1311-7946-4
Издательство: Книга по требованию
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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. The Sierpi?ski arrowhead curve is a fractal curve similar in appearance and identical in limit to the Sierpi?ski triangle.The Sierpinski triangle (also with the original orthography Sierpi?ski), also called the Sierpinski gasket or the Sierpinski Sieve, is a fractal named after the Polish mathematician Wac?aw Sierpi?ski who described it in 1915.Originally constructed as a curve, this is one of the basic examples of self-similar sets, i.e. it is a mathematically generated pattern that can be reproducible at any magnification or reduction.Comparing the Sierpinski triangle or the Sierpinski carpet to equivalent...

ISBN: 978-6-1312-4083-6
Издательство: Книга по требованию
Дата выхода: июль 2011