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Abstract Wiener Space
The concept of an abstract Wiener space is a mathematical construction developed by Leonard Gross to understand the structure of Gaussian measures on infinite-dimensional spaces. The construction emphasizes the fundamental role played by the Cameron–Martin space. The classical Wiener space is the prototypical example. The structure theorem for Gaussian measures states that all Gaussian measures can be represented by the abstract Wiener space construction. Motivation Let H be a real Hilbert space, assumed to be infinite dimensional and separable. In the physics literature, one frequently encounters integrals of the form :\frac\int_H f(v) e^ Dv, where Z is supposed to be a normalization constant and where Dv is supposed to be the non-existent Lebesgue measure on H. Such integrals arise, notably, in the context of the Euclidean path-integral formulation of quantum field theory. At a mathematical level, such an integral cannot be interpreted as integration against a measure ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Leonard Gross
Leonard Gross (born February 24, 1931) is an American mathematician and Professor Emeritus of Mathematics at Cornell University. Gross has made fundamental contributions to mathematics and the mathematically rigorous study of quantum field theory. Education and career Leonard Gross graduated from James Madison High School in December 1948. He was awarded an Emil Schweinberg scholarship that enabled him to attend college. He studied at City College of New York for one term and then studied electrical engineering at Cooper Union for two years. He then transferred to the University of Chicago, where he obtained a master's degree in physics and mathematics (1954) and a Ph.D. in mathematics (1958). Gross taught at Yale University and was awarded a National Science Foundation Fellowship in 1959. He joined the faculty of the mathematics department of Cornell University in 1960. Gross was a member of the Institute for Advanced Study in 1959 and in 1983 and has held other visiting pos ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gaussian Integral
The Gaussian integral, also known as the Euler–Poisson integral, is the integral of the Gaussian function f(x) = e^ over the entire real line. Named after the German mathematician Carl Friedrich Gauss, the integral is \int_^\infty e^\,dx = \sqrt. Abraham de Moivre originally discovered this type of integral in 1733, while Gauss published the precise integral in 1809. The integral has a wide range of applications. For example, with a slight change of variables it is used to compute the normalizing constant of the normal distribution. The same integral with finite limits is closely related to both the error function and the cumulative distribution function of the normal distribution. In physics this type of integral appears frequently, for example, in quantum mechanics, to find the probability density of the ground state of the harmonic oscillator. This integral is also used in the path integral formulation, to find the propagator of the harmonic oscillator, and in statistical ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wiener Process
In mathematics, the Wiener process is a real-valued continuous-time stochastic process named in honor of American mathematician Norbert Wiener for his investigations on the mathematical properties of the one-dimensional Brownian motion. It is often also called Brownian motion due to its historical connection with the physical process of the same name originally observed by Scottish botanist Robert Brown. It is one of the best known Lévy processes ( càdlàg stochastic processes with stationary independent increments) and occurs frequently in pure and applied mathematics, economics, quantitative finance, evolutionary biology, and physics. The Wiener process plays an important role in both pure and applied mathematics. In pure mathematics, the Wiener process gave rise to the study of continuous time martingales. It is a key process in terms of which more complicated stochastic processes can be described. As such, it plays a vital role in stochastic calculus, diffusion ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inner Product
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often denoted with angle brackets such as in \langle a, b \rangle. Inner products allow formal definitions of intuitive geometric notions, such as lengths, angles, and orthogonality (zero inner product) of vectors. Inner product spaces generalize Euclidean vector spaces, in which the inner product is the dot product or ''scalar product'' of Cartesian coordinates. Inner product spaces of infinite dimension are widely used in functional analysis. Inner product spaces over the field of complex numbers are sometimes referred to as unitary spaces. The first usage of the concept of a vector space with an inner product is due to Giuseppe Peano, in 1898. An inner product naturally induces an associated norm, (denoted , x, and , y, in the pictu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Path (topology)
In mathematics, a path in a topological space X is a continuous function from the closed unit interval , 1/math> into X. Paths play an important role in the fields of topology and mathematical analysis. For example, a topological space for which there exists a path connecting any two points is said to be path-connected. Any space may be broken up into path-connected components. The set of path-connected components of a space X is often denoted \pi_0(X). One can also define paths and loops in pointed spaces, which are important in homotopy theory. If X is a topological space with basepoint x_0, then a path in X is one whose initial point is x_0. Likewise, a loop in X is one that is based at x_0. Definition A ''curve'' in a topological space X is a continuous function f : J \to X from a non-empty and non-degenerate interval J \subseteq \R. A in X is a curve f : , b\to X whose domain , b/math> is a compact non-degenerate interval (meaning a is homeomorphic to , 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kolmogorov's Zero–one Law
In probability theory, Kolmogorov's zero–one law, named in honor of Andrey Nikolaevich Kolmogorov, specifies that a certain type of event, namely a ''tail event of independent σ-algebras'', will either almost surely happen or almost surely not happen; that is, the probability of such an event occurring is zero or one. Tail events are defined in terms of countably infinite families of σ-algebras. For illustrative purposes, we present here the special case in which each sigma algebra is generated by a random variable X_k for k\in\mathbb N. Let \mathcal be the sigma-algebra generated jointly by all of the X_k. Then, a tail event F \in \mathcal is an event which is probabilistically independent of each finite subset of these random variables. (Note: F belonging to \mathcal implies that membership in F is uniquely determined by the values of the X_k, but the latter condition is strictly weaker and does not suffice to prove the zero-one law.) For example, the event that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Measure Zero
In mathematical analysis, a null set N \subset \mathbb is a measurable set that has measure zero. This can be characterized as a set that can be covered by a countable union of intervals of arbitrarily small total length. The notion of null set should not be confused with the empty set as defined in set theory. Although the empty set has Lebesgue measure zero, there are also non-empty sets which are null. For example, any non-empty countable set of real numbers has Lebesgue measure zero and therefore is null. More generally, on a given measure space M = (X, \Sigma, \mu) a null set is a set S\in\Sigma such that \mu(S) = 0. Example Every finite or countably infinite subset of the real numbers is a null set. For example, the set of natural numbers and the set of rational numbers are both countably infinite and therefore are null sets when considered as subsets of the real numbers. The Cantor set is an example of an uncountable null set. Definition Suppose A is a subset ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cartesian Product
In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\times B = \. A table can be created by taking the Cartesian product of a set of rows and a set of columns. If the Cartesian product is taken, the cells of the table contain ordered pairs of the form . One can similarly define the Cartesian product of ''n'' sets, also known as an ''n''-fold Cartesian product, which can be represented by an ''n''-dimensional array, where each element is an ''n''- tuple. An ordered pair is a 2-tuple or couple. More generally still, one can define the Cartesian product of an indexed family of sets. The Cartesian product is named after René Descartes, whose formulation of analytic geometry gave rise to the concept, which is further generalized in terms of direct product. Examples A deck of cards A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Translation (geometry)
In Euclidean geometry, a translation is a geometric transformation that moves every point of a figure, shape or space by the same distance in a given direction. A translation can also be interpreted as the addition of a constant vector to every point, or as shifting the origin of the coordinate system. In a Euclidean space, any translation is an isometry. As a function If \mathbf is a fixed vector, known as the ''translation vector'', and \mathbf is the initial position of some object, then the translation function T_ will work as T_(\mathbf)=\mathbf+\mathbf. If T is a translation, then the image of a subset A under the function T is the translate of A by T . The translate of A by T_ is often written A+\mathbf . Horizontal and vertical translations In geometry, a vertical translation (also known as vertical shift) is a translation of a geometric object in a direction parallel to the vertical axis of the Cartesian coordinate system. Often, vertical translation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Functional
In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers). If is a vector space over a field , the set of all linear functionals from to is itself a vector space over with addition and scalar multiplication defined pointwise. This space is called the dual space of , or sometimes the algebraic dual space, when a topological dual space is also considered. It is often denoted , p. 19, §3.1 or, when the field is understood, V^*; other notations are also used, such as V', V^ or V^. When vectors are represented by column vectors (as is common when a basis is fixed), then linear functionals are represented as row vectors, and their values on specific vectors are given by matrix products (with the row vector on the left). Examples * The constant zero function, mapping every vector to zero, is trivially a linear functional. * Index ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Open Set
In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are sufficiently near to (that is, all points whose distance to is less than some value depending on ). More generally, one defines open sets as the members of a given collection of subsets of a given set, a collection that has the property of containing every union of its members, every finite intersection of its members, the empty set, and the whole set itself. A set in which such a collection is given is called a topological space, and the collection is called a topology. These conditions are very loose, and allow enormous flexibility in the choice of open sets. For example, ''every'' subset can be open (the discrete topology), or no set can be open except the space itself and the empty set (the indiscrete topology). In practice, howe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Borel Algebra
In mathematics, a Borel set is any set in a topological space that can be formed from open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement. Borel sets are named after Émile Borel. For a topological space ''X'', the collection of all Borel sets on ''X'' forms a σ-algebra, known as the Borel algebra or Borel σ-algebra. The Borel algebra on ''X'' is the smallest σ-algebra containing all open sets (or, equivalently, all closed sets). Borel sets are important in measure theory, since any measure defined on the open sets of a space, or on the closed sets of a space, must also be defined on all Borel sets of that space. Any measure defined on the Borel sets is called a Borel measure. Borel sets and the associated Borel hierarchy also play a fundamental role in descriptive set theory. In some contexts, Borel sets are defined to be generated by the compact sets of the topological s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
