High Quality Content by WIKIPEDIA articles! Sr?niv?sa Aiyang?r R?m?nujan FRS, better known as Srinivasa Iyengar Ramanujan (22 December 1887 – 26 April 1920) was an Indian mathematician and self taught genius who, with almost no formal training in pure mathematics, made substantial contributions to mathematical analysis, number theory, infinite series and continued fractions. Born and raised in Erode, Tamil Nadu, India, Ramanujan first encountered formal mathematics at age 10. He demonstrated a natural ability, and was given books on advanced trigonometry written by S L Loney. He had mastered them by age 12, and even discovered theorems of his own. He demonstrated unusual mathematical skills at school, winning accolades and awards. By...

ISBN: 978-6-1311-6228-2
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High Quality Content by WIKIPEDIA articles! In computer science and computational complexity theory, st-connectivity or STCON is a decision problem asking, for vertices s and t in a directed graph, if t is reachable from s. Formally, the decision problem is given by PATH = {?D, s, t? | D is a directed graph with a path from vertex s to t}. The problem can be shown to be in NL, as a non-deterministic Turing machine can guess the next node of the path, while the only information which has to be stored is which node is the node currently under consideration. The algorithm terminates if either the target node t is reached, or the path so far exceeds n, the number of nodes in the graph. The complement of st-connectivity, known as...

ISBN: 978-6-1312-3623-5
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ISBN: 978-3-8383-8469-6
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High Quality Content by WIKIPEDIA articles! In model theory, a branch of mathematical logic, a complete first-order theory T is called stable in ? (an infinite cardinal number), if the Stone space of every model of T of size ? ? has itself size ? ?. T is called a stable theory if there is no upper bound for the cardinals ? such that T is stable in ?. The stability spectrum of T is the class of all cardinals ? such that T is stable in ?. For countable theories there are only four possible stability spectra. The corresponding dividing lines are those for total transcendentality, superstability and stability. This result is due to Saharon Shelah, who also defined stability and superstability. Данное издание не...

ISBN: 978-6-1312-3644-0
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High Quality Content by WIKIPEDIA articles! The stability radius of a continuous function f (in a functional space F) with respect to an open stability domain D is the distance between f and the set of unstable functions (with respect to D). We say that a function is stable with respect to D if its spectrum is in D. Here, the notion of spectrum is defined on a case by case basis, as explained below.The notion of stability radius is generally applied to special functions as polynomials (the spectrum is then the roots) and matrices (the spectrum is the eigenvalues). The case where C is a proper subset of F permits us to consider structured perturbations (e.g. for a matrix, we could only need perturbations on the last row). It is an...

ISBN: 978-6-1312-3639-6
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High Quality Content by WIKIPEDIA articles! In probability theory, the stability of a random variable is the property that a linear combination of two independent copies of the variable has the same distribution, up to location and scale parameters. The distributions of random variables having this property are said to be "stable distributions". Results available in probability theory show that all possible distributions having this property are members of a four-parameter family of distributions. The article on the stable distribution describes this family together with some of the properties of these distributions. Данное издание не является оригинальным. Книга печатается по...

ISBN: 978-6-1312-3630-3
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High Quality Content by WIKIPEDIA articles! In mathematics, and in particular the study of dynamical systems, the idea of stable and unstable sets or stable and unstable manifolds give a formal mathematical definition to the general notions embodied in the idea of an attractor or repellor. In the case of hyperbolic dynamics, the corresponding notion is that of the hyperbolic set.The following provides a definition for the case of a system that is either an iterated function or has discrete-time dynamics. Similar notions apply for systems whose time evolution is given by a flow. Данное издание не является оригинальным. Книга печатается по технологии...

ISBN: 978-6-1312-3923-6
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High Quality Content by WIKIPEDIA articles! In mathematics, specifically in symplectic topology and algebraic geometry, one can construct the moduli space of stable maps, satisfying specified conditions, from Riemann surfaces into a given symplectic manifold. This moduli space is the essence of the Gromov–Witten invariants, which find application in enumerative geometry and type IIA string theory. Because the construction is lengthy and difficult, it is carried out here rather than in the Gromov–Witten invariants article itself. Данное издание не является оригинальным. Книга печатается по технологии принт-он-деманд после получения заказа.

ISBN: 978-6-1312-3941-0
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High Quality Content by WIKIPEDIA articles! In mathematics, especially in the fields of game theory and combinatorics, the stable roommate problem (SRP) is the problem of finding a stable matching — a matching in which there is no pair of elements, each from a different matched set, where each member of the pair prefers the other to their match. This is different from the stable marriage problem in that the stable roommates problem does not require that a set is broken up into male and female subsets. Any person can prefer anyone in the same set.In a given instance of the Stable Roommates problem (SRP), each of 2n participants ranks the others in strict order of preference. A matching is a set of n disjoint (unordered) pairs of...

ISBN: 978-6-1312-4059-1
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High Quality Content by WIKIPEDIA articles! In representation theory, the stable module category is a category in which projectives are "factored out." Let R be a ring. For two modules M and N, define underline{mathrm{Hom}}(M,N) to be the set of R-linear maps from M to N modulo the relation that f~g if f-g factors through a projective module. The stable module category is defined by setting the objects to be the R-modules, and the morphisms are the equivalence classes underline{mathrm{Hom}}(M,N). Given a module M, let P be a projective module with a surjection p colon P to M. Then set ?(M) to be the kernel of p. Suppose we are given a morphism f colon M to N and a surjection q colon Q to N where Q is projective. Then one can lift f to a...

ISBN: 978-6-1312-3966-3
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High Quality Content by WIKIPEDIA articles! In model theory, a stable group is a group that is stable in the sense of stability theory. An important class of examples is provided by groups of finite Morley rank (see below).The Cherlin–Zilber conjecture, due to Gregory Cherlin (1979) and Boris Zilber (1977), suggests that infinite (?-stable) simple groups of finite Morley rank are simple algebraic groups over algebraically closed fields. The conjecture would have followed from Zilber trichotomy conjecture. Cherlin posed the question for all ?-stable simple groups, but remarked that even the case of groups of finite Morley rank seemed hard. Данное издание не является оригинальным. Книга...

ISBN: 978-6-1312-3903-8
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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 surgery theory, a branch of mathematics, the stable normal bundle of a differentiable manifold is an invariant which encodes the stable normal (dually, tangential) data. There are analogs for generalizations of manifold, notably PL-manifolds and topological manifolds. There is also an analogue in homotopy theory for Poincare spaces, the Spivak spherical fibration, named after Michael Spivak (reference below).Given an embedding of a manifold in Euclidean space (provided by the theorem of Whitney), it has a normal bundle. The embedding is not unique, but for high dimension of the Euclidean space it is unique up...

ISBN: 978-6-1312-3989-2
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High Quality Content by WIKIPEDIA articles! In mathematics, especially in the study of dynamical systems and differential equations, the stable manifold theorem is an important result about the structure of the set of orbits approaching a given hyperbolic fixed point. The dynamical system concept is a mathematical formalization for any fixed "rule" which describes the time dependence of a point's position in its ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each spring in a lake. Данное издание не является оригинальным. Книга печатается по технологии...

ISBN: 978-6-1312-3930-4
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High Quality Content by WIKIPEDIA articles! Negation as failure (NAF, for short) is a non-monotonic inference rule in logic programming, used to derive mathrm{not}~p (i.e. that ~p is assumed not to hold) from failure to derive ~p. Note that mathrm{not}~p is different from the stronger statement neg p of the logical negation of ~p. Negation as failure has been an important feature of logic programming since the earliest days of both Planner and Prolog. In Prolog, it is usually implemented using Prolog's extralogical constructs. Данное издание не является оригинальным. Книга печатается по технологии принт-он-деманд после получения заказа.

ISBN: 978-6-1303-1546-7
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High Quality Content by WIKIPEDIA articles! In mathematics, the stable marriage problem (SMP) is the problem of finding a stable matching — a matching in which no element of the first matched set prefers an element of the second matched set that also prefers the first element. Given n men and n women, where each person has ranked all members of the opposite sex with a unique number between 1 and n in order of preference, marry the men and women off such that there are no two people of opposite sex who would both rather have each other than their current partners. If there are no such people, all the marriages are "stable". Данное издание не является оригинальным. Книга печатается по...

ISBN: 978-6-1312-3951-9
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High Quality Content by WIKIPEDIA articles! The first condition defines Hurwitz (or continuous-time) stability and the second one Schur (or discrete-time) stability. Stable polynomials arise in various mathematical fields, for example in control theory and differential equations. Indeed, a linear, time-invariant system (see LTI system theory) is said to be BIBO stable if and only if bounded inputs produce bounded outputs; this is equivalent to requiring that the denominator of its transfer function (which can be proven to be rational) is stable. The denominator is required to be Hurwitz stable if the system is in continuous-time and Schur stable if it is in discrete-time. Stable polynomials are sometimes called Hurwitz polynomials and...

ISBN: 978-6-1312-4020-1
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High Quality Content by WIKIPEDIA articles! In algebraic geometry, a stable curve is an algebraic curve that is asymptotically stable in the sense of geometric invariant theory. This is equivalent to the condition that it is a complete connected curve whose only singularities are ordinary double points and whose automorphism group is finite. The condition that the automorphism group is finite can be replaced by the condition that it is not of arithmetic genus one and every non-singular rational component meets the other components in at least 3 points (Deligne Данное издание не является оригинальным. Книга печатается по технологии принт-он-деманд после...

ISBN: 978-6-1312-3672-3
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High Quality Content by WIKIPEDIA articles! In model theory, a complete theory is called stable if it does not have too many types. One goal of classification theory is to divide all complete theories into those whose models can be classified and those whose models are too complicated to classify, and to classify all models in the cases where this can be done. Roughly speaking, if a theory is not stable then its models are too complicated and numerous to classify, while if a theory is stable there might be some hope of classifying its models, especially if the theory is superstable or totally transcendental. Данное издание не является оригинальным. Книга печатается по...

ISBN: 978-6-1312-4065-2
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High Quality Content by WIKIPEDIA articles! In mathematics a stack is a concept used to formalise some of the main constructions of descent theory. Descent theory is concerned with generalisations of situations where geometrical objects (such as vector bundles on topological spaces) can be "glued together" when they are isomorphic (in a compatible way) when restricted to intersections of the sets in an open covering of a space. In more general set-up the restrictions are replaced with general pull-backs, and fibred categories form the right framework to discuss the possibility of such "glueing". The intuitive meaning of a stack is that it is a fibred category such that "all possible glueings work". The specification of glueings requires...

ISBN: 978-6-1311-3186-8
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High Quality Content by WIKIPEDIA articles! The stalk of a sheaf is a mathematical construction capturing the behaviour of a sheaf around a given point.Sheaves are defined on open sets, but the underlying topological space X consists of points. It is reasonable to attempt to isolate the behavior of a sheaf at a single fixed point x of X. Conceptually speaking, we do this by looking at small neighborhoods of the point. If we look at a sufficiently small neighborhood of x, the behavior of the sheaf mathcal{F} on that small neighborhood should be the same as the behavior of mathcal{F} at that point. Of course, no single neighborhood will be small enough, so we will have to take a limit of some sort. Данное издание не...

ISBN: 978-6-1312-4087-4
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High Quality Content by WIKIPEDIA articles! In probability theory and statistics, the kth standardized moment of a probability distribution is frac{mu_k}{sigma^k}! where ?k is the kth moment about the mean and ? is the standard deviation. It is the normalization of the kth moment with respect to standard deviation. The power of k is because moments scale as xk, meaning that ?k = ?k?k: they are homogeneous polynomials of degree k, thus the standardized moment is scale invariant. This can also be understood as being because moments have dimension, but in the ratio defining standardized moments, the dimensions cancel, so they are dimensionless numbers. Данное издание не является оригинальным. Книга...

ISBN: 978-6-1303-6118-1
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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. A standard normal table also called the "Unit Normal Table" is a mathematical table for the values of , the cumulative distribution function of the normal distribution. They are used to find the probability that a statistic is observed below, above, or between values on the standard normal distribution, and by extension, any normal distribution. Normal distributions are symmetrical, bell-shaped distributions that are useful in describing real-world data. The standard normal distribution, represented by the letter Z, is the normal distribution having a mean of 0 and a standard deviation of 1. Since probability...

ISBN: 978-6-1312-5197-9
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High Quality Content by WIKIPEDIA articles! In mathematics, the standard basis (also called natural basis or canonical basis) for a Euclidean space consists of one unit vector pointing in the direction of each axis of the Cartesian coordinate system. For example, the standard basis for the Euclidean plane are the vectors mathbf{e}_x = (1,0),quad mathbf{e}_y = (0,1), and the standard basis for three-dimensional space are the vectors mathbf{e}_x = (1,0,0),quad mathbf{e}_y = (0,1,0),quad mathbf{e}_z=(0,0,1). Here the vector ex points in the x direction, the vector ey points in the y direction, and the vector ez points in the z direction. There are several common notations for these vectors, including {ex, ey, ez}, {e1, e2, e3}, {i, j, k},...

ISBN: 978-6-1312-4164-2
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High Quality Content by WIKIPEDIA articles! The standard error of a method of measurement or estimation is the standard deviation of the sampling distribution associated with the estimation method. The term may also be used to refer to an estimate of that standard deviation, derived from a particular sample used to compute the estimate.For example, the sample mean is the usual estimator of a population mean. However, different samples drawn from that same population would in general have different values of the sample mean. The standard error of the mean (i.e., of using the sample mean as a method of estimating the population mean) is the standard deviation of those sample means over all possible samples (of a given size) drawn from the...

ISBN: 978-6-1304-9867-2
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High Quality Content by WIKIPEDIA articles! In coding theory, a standard array (or Slepian array) is a qn ? k by qk array that lists all elements of a particular mathbb{F}_q^n vector space. Standard arrays are used to decode linear codes; i.e. to find the corresponding codeword for any received vector. Note that the above is only one possibility for the standard array; had 00011 been chosen as the first coset leader of weight two, another standard array representing the code would have been constructed. Note that the first row contains the 0 vector and the codewords of C3 (0 itself being a codeword). Also, the leftmost column contains the vectors of minimum weight enumerating vectors of weight 1 first and then using vectors of weight 2....

ISBN: 978-6-1312-4152-9
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Дата выхода: июль 2011