|
PaleyâWiener Integral
In mathematics, the PaleyâWiener integral is a simple stochastic integral. When applied to classical Wiener space, it is less general than the ItĹ integral, but the two agree when they are both defined. The integral is named after its discoverers, Raymond Paley and Norbert Wiener. Definition Let i : H \to E be an abstract Wiener space with abstract Wiener measure \gamma on E. Let j : E^* \to H be the adjoint of i. (We have abused notation slightly: strictly speaking, j : E^* \to H^*, but since H is a Hilbert space, it is isometrically isomorphic to its dual space H^*, by the Riesz representation theorem.) It can be shown that j is an injective function and has dense image in H. Furthermore, it can be shown that every linear functional f \in E^* is also square-integrable: in fact, :\, f \, _ = \, j(f) \, _ This defines a natural linear map from j(E^*) to L^2(E, \gamma; \mathbb), under which j(f) \in j(E^*) \subseteq H goes to the equivalence class /math> of f in L ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of t ... [...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] |
Proceedings Of The American Mathematical Society
''Proceedings of the American Mathematical Society'' is a monthly peer-reviewed scientific journal of mathematics published by the American Mathematical Society. As a requirement, all articles must be at most 15 printed pages. According to the ''Journal Citation Reports'', the journal has a 2018 impact factor of 0.813. Scope ''Proceedings of the American Mathematical Society'' publishes articles from all areas of pure and applied mathematics, including topology, geometry, analysis, algebra, number theory, combinatorics, logic, probability and statistics. Abstracting and indexing This journal is indexed in the following databases: 2011. American Mathematical Society. * |
Stratonovich Integral
In stochastic processes, the Stratonovich integral (developed simultaneously by Ruslan Stratonovich and Donald Fisk) is a stochastic integral, the most common alternative to the ItĂ´ integral. Although the ItĂ´ integral is the usual choice in applied mathematics, the Stratonovich integral is frequently used in physics. In some circumstances, integrals in the Stratonovich definition are easier to manipulate. Unlike the ItĂ´ calculus, Stratonovich integrals are defined such that the chain rule of ordinary calculus holds. Perhaps the most common situation in which these are encountered is as the solution to Stratonovich stochastic differential equations (SDEs). These are equivalent to ItĂ´ SDEs and it is possible to convert between the two whenever one definition is more convenient. Definition The Stratonovich integral can be defined in a manner similar to the Riemann integral, that is as a limit of Riemann sums. Suppose that W : , T\times \Omega \to \mathbb is a Wiener pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Skorokhod Integral
In mathematics, the Skorokhod integral (also named Hitsuda-Skorokhod integral), often denoted \delta, is an operator of great importance in the theory of stochastic processes. It is named after the Ukrainian mathematician Anatoliy Skorokhod and japanese mathematician Masuyuki Hitsuda. Part of its importance is that it unifies several concepts: * \delta is an extension of the ItĂ´ integral to non-adapted processes; * \delta is the adjoint of the Malliavin derivative, which is fundamental to the stochastic calculus of variations (Malliavin calculus); * \delta is an infinite-dimensional generalization of the divergence operator from classical vector calculus. The integral was introduced by Hitsuda in 1972 and by Skorokhod in 1975. Definition Preliminaries: the Malliavin derivative Consider a fixed probability space (\Omega, \Sigma, \mathbf) and a Hilbert space H; \mathbf denotes expectation with respect to \mathbf \mathbf := \int_ X(\omega) \, \mathrm \mathbf(\omega). ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
CameronâMartin Theorem
In mathematics, the CameronâMartin theorem or CameronâMartin formula (named after Robert Horton Cameron and W. T. Martin) is a theorem of measure theory that describes how abstract Wiener measure changes under translation by certain elements of the CameronâMartin Hilbert space. Motivation The standard Gaussian measure \gamma^n on n-dimensional Euclidean space \mathbf^n is not translation- invariant. (In fact, there is a unique translation invariant Radon measure up to scale by Haar's theorem: the n-dimensional Lebesgue measure, denoted here dx.) Instead, a measurable subset A has Gaussian measure :\gamma_n(A) = \frac\int_A \exp\left(-\tfrac12\langle x, x\rangle_\right)\,dx. Here \langle x,x\rangle_ refers to the standard Euclidean dot product in \mathbf^n. The Gaussian measure of the translation of A by a vector h \in \mathbf^n is :\begin \gamma_n(A-h) &= \frac\int_A \exp\left(-\tfrac12\langle x-h, x-h\rangle_\right)\,dx\\ pt&=\frac\int_A \exp\left(\frac\right)\exp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abuse Of Notation
In mathematics, abuse of notation occurs when an author uses a mathematical notation in a way that is not entirely formally correct, but which might help simplify the exposition or suggest the correct intuition (while possibly minimizing errors and confusion at the same time). However, since the concept of formal/syntactical correctness depends on both time and context, certain notations in mathematics that are flagged as abuse in one context could be formally correct in one or more other contexts. Time-dependent abuses of notation may occur when novel notations are introduced to a theory some time before the theory is first formalized; these may be formally corrected by solidifying and/or otherwise improving the theory. ''Abuse of notation'' should be contrasted with ''misuse'' of notation, which does not have the presentational benefits of the former and should be avoided (such as the misuse of constants of integration). A related concept is abuse of language or abuse of termi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Function (mathematics)
In mathematics, a function from a set to a set assigns to each element of exactly one element of .; the words map, mapping, transformation, correspondence, and operator are often used synonymously. The set is called the domain of the function and the set is called the codomain of the function.Codomain ''Encyclopedia of Mathematics'Codomain. ''Encyclopedia of Mathematics''/ref> The earliest known approach to the notion of function can be traced back to works of Persian mathematicians Al-Biruni and Sharaf al-Din al-Tusi. Functions were originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet is a ''function'' of time. Historically, the concept was elaborated with the infinitesimal calculus at the end of the 17th century, and, until the 19th century, the functions that were considered were differentiable (that is, they had a high degree of regularity). The concept of a function was formalized at the end of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Banach Space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well-defined limit that is within the space. Banach spaces are named after the Polish mathematician Stefan Banach, who introduced this concept and studied it systematically in 1920â1922 along with Hans Hahn and Eduard Helly. Maurice RenĂŠ FrĂŠchet was the first to use the term "Banach space" and Banach in turn then coined the term " FrĂŠchet space." Banach spaces originally grew out of the study of function spaces by Hilbert, FrĂŠchet, and Riesz earlier in the century. Banach spaces play a central role in functional analysis. In other areas of analysis, the spaces under study are often Banach spaces. Definition A Banach space is a complete n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuous Function
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value, known as '' discontinuities''. More precisely, a function is continuous if arbitrarily small changes in its value can be assured by restricting to sufficiently small changes of its argument. A discontinuous function is a function that is . Up until the 19th century, mathematicians largely relied on intuitive notions of continuity, and considered only continuous functions. The epsilonâdelta definition of a limit was introduced to formalize the definition of continuity. Continuity is one of the core concepts of calculus and mathematical analysis, where arguments and values of functions are real and complex numbers. The concept has been generalized to functions between metric spaces and between topological spaces. The latter are t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isometry
In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: áź´ĎÎżĎ ''isos'' meaning "equal", and ÎźÎĎĎον ''metron'' meaning "measure". Introduction Given a metric space (loosely, a set and a scheme for assigning distances between elements of the set), an isometry is a transformation which maps elements to the same or another metric space such that the distance between the image elements in the new metric space is equal to the distance between the elements in the original metric space. In a two-dimensional or three-dimensional Euclidean space, two geometric figures are congruent if they are related by an isometry; the isometry that relates them is either a rigid motion (translation or rotation), or a composition of a rigid motion and a reflection. Isometries are often used in constructions where one space ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equivalence Class
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a and b belong to the same equivalence class if, and only if, they are equivalent. Formally, given a set S and an equivalence relation \,\sim\, on S, the of an element a in S, denoted by is the set \ of elements which are equivalent to a. It may be proven, from the defining properties of equivalence relations, that the equivalence classes form a partition of S. This partitionâthe set of equivalence classesâis sometimes called the quotient set or the quotient space of S by \,\sim\,, and is denoted by S / \sim. When the set S has some structure (such as a group operation or a topology) and the equivalence relation \,\sim\, is compatible with this structure, the quotient set often inherits a similar structure from its parent set. Exa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
