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Carré Du Champ Operator
The (French for ''square of a field'' operator) is a bilinear, symmetric operator from analysis and probability theory. The measures how far an infinitesimal generator is from being a derivation. The operator was introduced in 1969 by and independently discovered in 1976 by Jean-Pierre Roth in his doctoral thesis. The name ''"carré du champ"'' comes from electrostatics. Carré du champ operator for a Markov semigroup Let (X,\mathcal,\mu) be a σ-finite measure space, \_ a Markov semigroup of non-negative operators on L^2(X,\mu), A the infinitesimal generator of \_ and \mathcal the algebra of functions in \mathcal(A), i.e. a vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ... such that for all f,g\in \mathcal also fg\in \mathcal. Carré du champ operator The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bilinear Map
In mathematics, a bilinear map is a function combining elements of two vector spaces to yield an element of a third vector space, and is linear in each of its arguments. Matrix multiplication is an example. A bilinear map can also be defined for modules. For that, see the article pairing. Definition Vector spaces Let V, W and X be three vector spaces over the same base field F. A bilinear map is a function B : V \times W \to X such that for all w \in W, the map B_w v \mapsto B(v, w) is a linear map from V to X, and for all v \in V, the map B_v w \mapsto B(v, w) is a linear map from W to X. In other words, when we hold the first entry of the bilinear map fixed while letting the second entry vary, the result is a linear operator, and similarly for when we hold the second entry fixed. Such a map B satisfies the following properties. * For any \lambda \in F, B(\lambda v,w) = B(v, \lambda w) = \lambda B(v, w). * The map B is additive in both components: if v_1, v_2 \in V an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetric Map
In mathematics, symmetrization is a process that converts any function in n variables to a symmetric function in n variables. Similarly, antisymmetrization converts any function in n variables into an antisymmetric function. Two variables Let S be a set and A be an additive abelian group. A map \alpha : S \times S \to A is called a if \alpha(s,t) = \alpha(t,s) \quad \text s, t \in S. It is called an if instead \alpha(s,t) = - \alpha(t,s) \quad \text s, t \in S. The of a map \alpha : S \times S \to A is the map (x,y) \mapsto \alpha(x,y) + \alpha(y,x). Similarly, the or of a map \alpha : S \times S \to A is the map (x,y) \mapsto \alpha(x,y) - \alpha(y,x). The sum of the symmetrization and the antisymmetrization of a map \alpha is 2 \alpha. Thus, away from 2, meaning if 2 is invertible, such as for the real numbers, one can divide by 2 and express every function as a sum of a symmetric function and an anti-symmetric function. The symmetrization of a symmetric map is its dou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (mathematics), series, and analytic functions. These theories are usually studied in the context of Real number, real and Complex number, complex numbers and Function (mathematics), functions. Analysis evolved from calculus, which involves the elementary concepts and techniques of analysis. Analysis may be distinguished from geometry; however, it can be applied to any Space (mathematics), space of mathematical objects that has a definition of nearness (a topological space) or specific distances between objects (a metric space). History Ancient Mathematical analysis formally developed in the 17th century during the Scientific Revolution, but many of its ideas can be traced back to earlier mathematicians. Early results in analysis were ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Theory
Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms of probability, axioms. Typically these axioms formalise probability in terms of a probability space, which assigns a measure (mathematics), measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the sample space. Any specified subset of the sample space is called an event (probability theory), event. Central subjects in probability theory include discrete and continuous random variables, probability distributions, and stochastic processes (which provide mathematical abstractions of determinism, non-deterministic or uncertain processes or measured Quantity, quantities that may either be single occurrences or evolve over time in a random fashion). Although it is no ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
C0-semigroup
In mathematical analysis, a ''C''0-semigroup, also known as a strongly continuous one-parameter semigroup, is a generalization of the exponential function. Just as exponential functions provide solutions of scalar linear constant coefficient ordinary differential equations, strongly continuous semigroups provide solutions of linear constant coefficient ordinary differential equations in Banach spaces. Such differential equations in Banach spaces arise from e.g. delay differential equations and partial differential equations. Formally, a strongly continuous semigroup is a representation of the semigroup (R+, +) on some Banach space ''X'' that is continuous in the strong operator topology. Formal definition A strongly continuous semigroup on a Banach space X is a map T : \mathbb_+ \to L(X) (where L(X) is the space of bounded operators on X) such that # T(0) = I , (the identity operator on X) # \forall t,s \ge 0 : \ T(t + s) = T(t) T(s) # \forall x_0 \in X: ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Derivation (differential Algebra)
In mathematics, a derivation is a function on an algebra that generalizes certain features of the derivative operator. Specifically, given an algebra ''A'' over a ring or a field ''K'', a ''K''-derivation is a ''K''-linear map that satisfies Leibniz's law: : D(ab) = a D(b) + D(a) b. More generally, if ''M'' is an ''A''-bimodule, a ''K''-linear map that satisfies the Leibniz law is also called a derivation. The collection of all ''K''-derivations of ''A'' to itself is denoted by Der''K''(''A''). The collection of ''K''-derivations of ''A'' into an ''A''-module ''M'' is denoted by . Derivations occur in many different contexts in diverse areas of mathematics. The partial derivative with respect to a variable is an R-derivation on the algebra of real-valued differentiable functions on R''n''. The Lie derivative with respect to a vector field is an R-derivation on the algebra of differentiable functions on a differentiable manifold; more generally it is a derivation on the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jean-Pierre Roth (mathematician)
Jean-Pierre Roth (born on 28 April 1946) is a Swiss banker who served as chairman of the Swiss National Bank from 1 January 2001 until 31 December 2009. He joined the Swiss National Bank in 1979, working in Zürich and Bern. He became vice-chairman of the governing board in 1996. In 2001, he became chairman of the governing board. Between 2001 and 2009 he was Governor of the Washington-based International Monetary Fund (IMF) for Switzerland and chairman of the board of directors of the Bank for International Settlements (BIS) in Basel. Roth has been credited with the measures taken to restore public confidence in the Swiss banking system. He has been a member of the Board of Directors of Swatch Group since 2010, of Swiss Re from 2010 to 2016 and of Nestlé from 2010 to 2019. He was Chairman of Banque cantonale de Genève from 2010 to 2017. He has been Vice Chairman of Arab Bank Switzerland since 2017 and a member of the Board of Directors of MKS (Switzerland) since 2014. He wa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Electrostatics
Electrostatics is a branch of physics that studies slow-moving or stationary electric charges. Since classical antiquity, classical times, it has been known that some materials, such as amber, attract lightweight particles after triboelectric effect, rubbing. The Greek language, Greek word (), meaning 'amber', was thus the Root (linguistics), root of the word ''electricity''. Electrostatic phenomena arise from the forces that electric charges exert on each other. Such forces are described by Coulomb's law. There are many examples of electrostatic phenomena, from those as simple as the attraction of plastic wrap to one's hand after it is removed from a package, to the apparently spontaneous explosion of grain silos, the damage of electronic components during manufacturing, and photocopier and laser printing, laser printer operation. The electrostatic model accurately predicts electrical phenomena in "classical" cases where the velocities are low and the system is macroscopic so ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
σ-finite Measure
In mathematics, given a positive or a signed measure \mu on a measurable space (X, \mathcal F), a \sigma-finite subset is a measurable subset which is the union of a countable number of measurable subsets of finite measure. The measure \mu is called a \sigma-finite measure if the set X is \sigma-finite. A finite measure, for instance a probability measure, is always \sigma-finite. A different but related notion that should not be confused with \sigma-finiteness is s-finiteness. Definition Let (X, \mathcal) be a measurable space and \mu a measure on it. The measure \mu is called a σ-finite measure, if it satisfies one of the four following equivalent criteria: # the set X can be covered with at most countably many measurable sets with finite measure. This means that there are sets A_1, A_2, \ldots \in \mathcal A with \mu\left(A_n\right) < \infty for all that satisfy . # the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Measure Space
A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes. It contains an underlying set, the subsets of this set that are feasible for measuring (the -algebra) and the method that is used for measuring (the measure). One important example of a measure space is a probability space. A measurable space consists of the first two components without a specific measure. Definition A measure space is a triple (X, \mathcal A, \mu), where * X is a set * \mathcal A is a -algebra on the set X * \mu is a measure on (X, \mathcal) In other words, a measure space consists of a measurable space (X, \mathcal) together with a measure on it. Example Set X = \. The \sigma-algebra on finite sets such as the one above is usually the power set, which is the set of all subsets (of a given set) and is denoted by \wp(\cdot). Sticking with this convention, we set \mathcal = \wp(X) In this simple case, the power set can be writ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Markov Semigroup
In probability theory and ergodic theory, a Markov operator is an operator on a certain function space that conserves the mass (the so-called Markov property). If the underlying measurable space is topologically sufficiently rich enough, then the Markov operator admits a kernel representation. Markov operators can be linear or non-linear. Closely related to Markov operators is the Markov semigroup. The definition of Markov operators is not entirely consistent in the literature. Markov operators are named after the Russian mathematician Andrey Markov. Definitions Markov operator Let (E,\mathcal) be a measurable space and V a set of real, measurable functions f:(E,\mathcal)\to (\mathbb,\mathcal(\mathbb)). A linear operator P on V is a Markov operator if the following is true # P maps bounded, measurable function on bounded, measurable functions. # Let \mathbf be the constant function x\mapsto 1, then P(\mathbf)=\mathbf holds. (''conservation of mass'' / ''Markov property'') # If ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
