In mathematics, the Gateaux differential or Gateaux derivative is a generalisation of the concept of directional derivative in differential calculus. Named after Rene Gateaux, a French mathematician who died young in World War I, it is defined for functions between locally convex topological vector spaces such as Banach spaces. Like the Frechet derivative on a Banach space, the Gateaux differential is often used to formalize the functional derivative commonly used in the calculus of variations and physics. Unlike other forms of derivatives, the Gateaux differential of a function may be nonlinear. However, often the definition of the Gateaux differential also requires that it be a continuous linear transformation. Some authors, such as...

ISBN: 978-6-1302-9977-4
Издательство: Книга по требованию
Дата выхода: июль 2011

The gauge of a firearm is a unit of measurement used to express the diameter of the barrel. Gauge is determined from the weight of a solid sphere of lead that will fit the bore of the firearm, and is expressed as the multiplicative inverse of the sphere's weight as a fraction of a pound (e.g. a 1/12th pound ball fits a 12-gauge bore). The term is related to the measurement of cannon, which were also measured by the weight of their iron round shot; an 8 pounder would fire an 8 pound (3.6 kg) spherical cast iron ball and had a bore diameter of about 91 mm (3.6 inches). Commonly used today in reference to shotguns, though historically it was also used in large double rifles, which were made in sizes up to 4 bore during their heyday in the...

ISBN: 978-6-1306-5891-5
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In mathematics, Gaussian measure is a Borel measure on finite-dimensional Euclidean space Rn, closely related to the normal distribution in statistics. There is also a generalization to infinite- dimensional spaces. Gaussian measures are named after the German mathematician Carl Friedrich Gauss. Let n N and let B0(Rn) denote the completion of the Borel -algebra on Rn. Let n : B0(Rn) -> [0, + ] denote the usual n-dimensional Lebesgue measure Данное издание не является оригинальным. Книга печатается по технологии принт-он-деманд после получения заказа.

ISBN: 978-6-1337-1178-5
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! Gaussian adaptation is an evolutionary algorithm designed for the maximization of manufacturing yield due to statistical deviation of component values of signal processing systems. In short, GA is a stochastic adaptive process where a number of samples of an n- dimensional vector x[xT = (x1, x2, ..., xn)] are taken from a multivariate Gaussian distribution, N(m, M), having mean m and moment matrix M. The samples are tested for fail or pass. The first- and second-order moments of the Gaussian restricted to the pass samples are m* and M*. The outcome of x as a pass sample is determined by a function s(x), 0 < s(x) < q ? 1, such that s(x) is the probability that x will be selected as a pass...

ISBN: 978-6-1325-4293-9
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. The expected utility model developed by John von Neumann and Oskar Morgenstern dominated decision theory from its formulation in 1944 until the late 1970s, not only as a prescriptive, but also as a descriptive model, despite powerful criticism from Maurice Allais and Daniel Ellsberg who showed that, in certain choice problems, decisions were usually inconsistent with the axioms of expected utility theory. These problems are usually referred to as the Allais paradox and Ellsberg paradox. Данное издание не является оригинальным. Книга печатается по...

ISBN: 978-6-1329-0142-2
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In mathematics, the Fibonacci numbers form a sequence defined recursively by: F(0) = 0 F(1) = 1 F(n) = F(n-1) + F(n-2), for integer n > 1. That is, after two starting values, each number is the sum of the two preceding numbers. The Fibonacci sequence has been studied extensively and generalized in many ways, for example, by starting with other numbers than 0 and 1, by adding more than two numbers to generate the next number, or by adding objects other than numbers. Данное издание не является оригинальным. Книга печатается по технологии принт-он-деманд после получения заказа.

ISBN: 978-6-1308-7780-4
Издательство: Книга по требованию
Дата выхода: июль 2011

Problems that are considered in this book model various mechanical phenomena, especially from rod theory and viscoelasticity. Differential equations that arise have coefficients which are not smooth, can be even distributions (Dirac delta and Heaviside functions), and some equations are coupled with fractional differential equations. This is why one has to go beyond classical spaces and look for solutions within generalized function spaces such as spaces of distributions, Sobolev spaces and Colombeau's generalized functions. Different kinds of techniques are presented and afterwards are employed to show existence and uniqueness of solutions. Данное издание не является оригинальным. Книга...

ISBN: 978-3-6392-0612-8
Издательство: Книга по требованию
Дата выхода: июль 2011

Various (putative) generalisations of the Contraction Mapping Theorem in metric space theory in the literature are considered. In particular, generalisations due to Kannan and Suzuki are studied, as is a result of Merryfield, Rothschild and Stein with interesting links to Ramsey theory. The main subsequent business is a question of Stein about when there is a finite family of maps of the metric space, at least one of which contracts any particular pair of points: does some composition of members of the family have to have a unique fixed point? The answer is shown to be 'yes' in the special case of two commuting continuous maps of the metric space, but to be 'no' in general, by the counterexample of Austin based on combinatorics on words....

ISBN: 978-3-8433-8479-7
Издательство: Книга по требованию
Дата выхода: июль 2011

The essential Mathematics has the same importance and the same youthfulness through all times. Finding ways to apply and develop it especially in Math Education may continuously open new doors for an excellent prospective in educational process. The content of this book wants to be an example in this matter and keeps the track of study in Foundations of Mathematics and Education. Данное издание не является оригинальным. Книга печатается по технологии принт-он-деманд после получения заказа.

ISBN: 978-3-8364-9488-5
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In mathematical logic, generalization (also universal generalization, GEN) is an inference rule of predicate calculus. It states that if vdash P (x) has been derived, then vdash forall x , P (x) can be derived. The full generalization rule allows for hypotheses to the left of the turnstile, but with restrictions. Assume ? is a set of formulas, ? a formula, and Gamma vdash phi(y) has been derived. The generalization rule states that Gamma vdash forall x phi(x) can be derived if x is not mentioned in ? and x does not occur in ?. Данное издание не является оригинальным. Книга печатается по технологии принт-он-деманд...

ISBN: 978-6-1308-7772-9
Издательство: Книга по требованию
Дата выхода: июль 2011

In mathematics, given an infinite sequence of numbers { an }, a series is informally the result of adding all those terms together: a1 + a2 + a3 + · · ·. These can be written more compactly using the summation symbol ?.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, infinite series are also widely used in other quantitative disciplines such as physics and computer science. Данное издание не...

ISBN: 978-6-1311-0876-1
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 the mathematical field of set theory, a generic filter is a kind of object used in the theory of forcing, a technique used for many purposes, but especially to establish the independence of certain propositions from certain formal theories, such as ZFC. Данное издание не является оригинальным. Книга печатается по технологии принт-он-деманд после получения заказа.

ISBN: 978-6-1337-5575-8
Издательство: Книга по требованию
Дата выхода: июль 2011

This book is about "generalized quadratic variations of Gaussian processes". Generalized quadratic variations means that we consider mathematical objects wich are generalization of the classical quadratic variations used in semi-martingale theory. The Gaussian processes studied in this work are of fractional type, that is they resemble the fractional Brownian motion (FBM). In FBM theory, mathematicians consider second order quadratic variations which yield good estimation of the Hurst index of FBM. The aim of this work is to generalize this results to other fractional processes, without stationarity or harmonizable representation assumptions. Данное издание не является оригинальным. Книга...

ISBN: 978-3-6391-8800-4
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! The generalization error of a machine learning model is a function that measures how far the student machine is from the teacher machine in average over the entire set of possible data that can be generated by the teacher after each iteration of the learning process. It has this name because this function indicates the capacity of a machine that learns with the specified algorithm to infer a rule (or generalize) that is used by the teacher machine to generate data based only on a few examples. The performance of a machine learning algorithm is measured by plots of the generalization error values through the learning process and are called learning curves. The generalization error of a...

ISBN: 978-6-1308-7767-5
Издательство: Книга по требованию
Дата выхода: июль 2011

Two topics are in the focus of this book: the generic construction of versatile data structures for numerical linear algebra, as well as the application of algebraic multigrid methods for stabilized flow problems. The first part demonstrates how generic programming and template techniques of C++ can be used to automatically generate highly efficient matrix classes and algorithms. Benchmarks show that this abstract programming approach offers competitive performance in comparison with traditional libraries. In the second and third part, the application of algebraic multigrid to scalar convection-diffusion and linearized Navier-Stokes equations is investigated. A specially adapted method for Oseen problems is introduced, utilizing the...

ISBN: 978-3-8364-3301-3
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In mathematics, a generalized inverse or pseudoinverse of a matrix A is a matrix that has some properties of the inverse matrix of A but not necessarily all of them. The term "the pseudoinverse" commonly means the Moore–Penrose pseudoinverse. The purpose of constructing a generalized inverse is to obtain a matrix that can serve as the inverse in some sense for a wider class of matrices than invertible ones. Typically, the generalized inverse exists for an arbitrary matrix, and when a matrix has an inverse, then its inverse and the generalized inverse are the same. Some generalized inverses can be defined in any mathematical structure that involves associative multiplication, that is, in a...

ISBN: 978-6-1337-6198-8
Издательство: Книга по требованию
Дата выхода: июль 2011

This is a dissertation presented to the faculty of department of mathematics, Brandeis University, in partial fulfillment of the requirements for the degree Doctor of Philosophy. In this thesis, we apply the stack sorting operator to r-permutations and construct the functional equation for the generating function of two-stack-sortable permutations counted by descents by using a factorization similar to Zeilberger's. We solve the functional equation and give explicit formulas for the number of two-stack-sortable permutations. Данное издание не является оригинальным. Книга печатается по технологии принт-он-деманд после получения заказа.

ISBN: 978-3-8383-0621-6
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! A generalization of a concept is an extension of the concept to less-specific criteria. It is a foundational element of logic and human reasoning. Generalization posits the existence of a domain or set of elements, as well as one or more common characteristics shared by those elements. As such, it is the essential basis of all valid deductive inference. The process of verification is necessary to determine whether a generalization holds true for any given situation. Данное издание не является оригинальным. Книга печатается по технологии принт-он-деманд после получения заказа.

ISBN: 978-6-1311-8203-7
Издательство: Книга по требованию
Дата выхода: июль 2011

Within dynamical systems theory, homogeneity (of systems) has previously been defined with respect to a class of functions named dilations. This book treats homogeneity in a more general way by extending the classical definition through a broader class of functions, named generalized dilations. The new way of looking at homogeneity yields the following implications which constitute the most of the material studied in the book. Generalized dilations make possible to devise optimization-based numerical algorithms to obtain offline-computed, globally stabilizing feedback laws for a general class of discrete-time systems. Also, by nonstandard sample-and-hold techniques (which take homogeneity into account) such algorithms tailored for...

ISBN: 978-3-6390-8668-3
Издательство: Книга по требованию
Дата выхода: июль 2011

Der menschliche Organismus besteht aus einer Vielzahl verschiedener Zellarten. Diese unterschiedlichen Zellen zeichnen sich dadurch aus, dass sie bestimmte fur sie spezifische Gene exprimieren, was ihnen ihren speziellen Charakter verleiht. Alle Zellen eines Organismus stammen von Stammzellen ab, wobei zwischen unterschiedlichen Stammzellarten zu unterscheiden ist. Neuronale Zellen stammen von den sogenannten neuralen Stammzellen ab, welche den gewebsspezifischen Stammzellen zuzuordnen sind. Die Zelllinie HNSC.100 ist eine humane Stammzelllinie, die aus dem Hippokampus eines fetalen menschlichen Aborts prapariert wurde. Da bestimmte neuronale Gene in diesen Zellen exprimiert werden, dient die Zelllinie als optimales Modellsystem, um das...

ISBN: 978-3-6390-2612-2
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! A geoboard is a mathematical manipulative often used to explore basic concepts in plane geometry such as perimeter, area or the characteristics of triangles and other polygons. Consisting of a physical board with a certain number of nails half driven in, in a symmetrical square five-by-five array, students are encouraged to place rubber bands around the pegs to model various geometric concepts or to solve other mathematical puzzles. Two-dimensional representations of the geoboard may be applied to ordinary paper using rubber stamps or special "geoboard paper" with diagrams of geoboards may be used to help capture a student's explanations of the concept they have discovered or illustrated on the...

ISBN: 978-6-1311-3761-7
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In general relativity, the geodesic deviation equation is an equation involving the Riemann curvature tensor, which measures the change in separation of neighbouring geodesics or, equivalently, the tidal force experienced by a rigid body moving along a geodesic. In the language of mechanics it measures the rate of relative acceleration of two particles moving forward on neighbouring geodesics. In differential geometry, the geodesic deviation equation is more commonly known as the Jacobi equation. Let T a be the tangent vector to a given geodesic ?, and X a a vector field along ? connecting it to an infinitesimally near geodesic (the deviation vector). Данное издание не...

ISBN: 978-6-1311-4720-3
Издательство: Книга по требованию
Дата выхода: июль 2011

En cualquier analisis estadistico en ciencias ambientales, el analisis y determinacion de la estructura espacial subyacente es absolutamente necesaria. La geoestadistica espacial es una parte de procesos estocasticos que se encarga de este tipo de analisis y que caracteriza las dependencias espaciales del proceso continuo muestreado en una cierta region para modelar dichas dependencias y predecir con un cierto grado de incertidumbre el proceso de interes en zonas no muestreadas. Se presenta la metodologia necesaria subyacente a la teoria de la geoestadistica espacial. Al mismo tiempo se hace un enfasis particular en el desarrollo del espectro y de la teoria espectral dentro de este contexto. El analisis de la independencia espacial y el...

ISBN: 978-3-6392-1398-0
Издательство: Книга по требованию
Дата выхода: июль 2011

Geometric analysis is a mathematical discipline at the interface of differential geometry and differential equations. It includes both the use of geometrical methods in the study of partial differential equations (when it is also known as "geometric PDE"), and the application of the theory of partial differential equations to geometry. It incorporates problems involving curves and surfaces, or domains with curved boundaries, but also the study of Riemannian manifolds in arbitrary dimension. The calculus of variations is sometimes regarded as part of geometric analysis, because differential equations arising from variational principles have a strong geometric content. Geometric analysis also includes global analysis, which concerns the...

ISBN: 978-6-1311-0884-6
Издательство: Книга по требованию
Дата выхода: июль 2011

In mechanics, the Geometric phase, or the Pancharatnam-Berry phase (named after S. Pancharatnam and Sir Michael Berry), also known as the Pancharatnam phase or Berry phase, is a phase acquired over the course of a cycle, when the system is subjected to cyclic adiabatic processes, resulting from the geometrical properties of the parameter space of the Hamiltonian. The phenomenon was first discovered in 1956, and rediscovered in 1984. It can be seen in the Aharonov-Bohm effect and in the conical intersection of potential energy surfaces. In the case of the Aharonov-Bohm effect, the adiabatic parameter is the magnetic field inside the solenoid, and cyclic means that the difference involved in measuring the effect by interference corresponds...

ISBN: 978-6-1306-4674-5
Издательство: Книга по требованию
Дата выхода: июль 2011