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Accretive Operator
In mathematics, a dissipative operator is a linear operator ''A'' defined on a linear subspace ''D''(''A'') of Banach space ''X'', taking values in ''X'' such that for all ''λ'' > 0 and all ''x'' ∈ ''D''(''A'') :\, (\lambda I-A)x\, \geq\lambda\, x\, . A couple of equivalent definitions are given below. A dissipative operator is called maximally dissipative if it is dissipative and for all ''λ'' > 0 the operator ''λI'' − ''A'' is surjective, meaning that the range when applied to the domain ''D'' is the whole of the space ''X''. An operator that obeys a similar condition but with a plus sign instead of a minus sign (that is, the negation of a dissipative operator) is called an accretive operator. The main importance of dissipative operators is their appearance in the Lumer–Phillips theorem which characterizes maximally dissipative operators as the generators of contraction semigroups. Properties A dissipative operator has the following properties: * From the inequality g ... [...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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert Space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturally and frequently in mathematics and physics, typically as function spaces. Formally, a Hilbert space is a vector space equipped with an inner product that defines a distance function for which the space is a complete metric space. The earliest Hilbert spaces were studied from this point of view in the first decade of the 20th century by David Hilbert, Erhard Schmidt, and Frigyes Riesz. They are indispensable tools in the theories of partial differential equations, quantum mechanics, Fourier analysis (which includes applications to signal processing and heat transfer), and ergodic theory (which forms the mathematical underpinning of thermodynamics). John von Neumann coined the term ''Hilbert space'' for the abstract concept that under ... [...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, however, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Almost Everywhere
In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion to the concept of measure zero, and is analogous to the notion of ''almost surely'' in probability theory. More specifically, a property holds almost everywhere if it holds for all elements in a set except a subset of measure zero, or equivalently, if the set of elements for which the property holds is conull. In cases where the measure is not complete, it is sufficient that the set be contained within a set of measure zero. When discussing sets of real numbers, the Lebesgue measure is usually assumed unless otherwise stated. The term ''almost everywhere'' is abbreviated ''a.e.''; in older literature ''p.p.'' is used, to stand for the equivalent French language phrase ''presque partout''. A set with full measure is one whose complement i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integration By Parts
In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivative. It is frequently used to transform the antiderivative of a product of functions into an antiderivative for which a solution can be more easily found. The rule can be thought of as an integral version of the product rule of differentiation. The integration by parts formula states: \begin \int_a^b u(x) v'(x) \, dx & = \Big (x) v(x)\Biga^b - \int_a^b u'(x) v(x) \, dx\\ & = u(b) v(b) - u(a) v(a) - \int_a^b u'(x) v(x) \, dx. \end Or, letting u = u(x) and du = u'(x) \,dx while v = v(x) and dv = v'(x) \, dx, the formula can be written more compactly: \int u \, dv \ =\ uv - \int v \, du. Mathematician Brook Taylor discovered integration by parts, first publishing the idea in 1715. More general formulations of integration by parts ex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sobolev Space
In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of ''Lp''-norms of the function together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense to make the space complete, i.e. a Banach space. Intuitively, a Sobolev space is a space of functions possessing sufficiently many derivatives for some application domain, such as partial differential equations, and equipped with a norm that measures both the size and regularity of a function. Sobolev spaces are named after the Russian mathematician Sergei Sobolev. Their importance comes from the fact that weak solutions of some important partial differential equations exist in appropriate Sobolev spaces, even when there are no strong solutions in spaces of continuous functions with the derivatives understood in the classical sense. Motivation In this section and throughout the article \Omega is an open subset of \R^n. There are many c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weak Derivative
In mathematics, a weak derivative is a generalization of the concept of the derivative of a function (''strong derivative'') for functions not assumed differentiable, but only integrable, i.e., to lie in the L''p'' space L^1( ,b. The method of integration by parts holds that for differentiable functions u and \varphi we have :\begin \int_a^b u(x) \varphi'(x) \, dx & = \Big (x) \varphi(x)\Biga^b - \int_a^b u'(x) \varphi(x) \, dx. \\ pt \end A function ''u''' being the weak derivative of ''u'' is essentially defined by the requirement that this equation must hold for all infinitely differentiable functions ''φ'' vanishing at the boundary points (\varphi(a)=\varphi(b)=0). Definition Let u be a function in the Lebesgue space L^1( ,b. We say that v in L^1( ,b is a weak derivative of u if :\int_a^b u(t)\varphi'(t) \, dt=-\int_a^b v(t)\varphi(t) \, dt for ''all'' infinitely differentiable functions \varphi with \varphi(a)=\varphi(b)=0. Generalizing to n dimensions, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square-integrable Function
In mathematics, a square-integrable function, also called a quadratically integrable function or L^2 function or square-summable function, is a real- or complex-valued measurable function for which the integral of the square of the absolute value is finite. Thus, square-integrability on the real line (-\infty,+\infty) is defined as follows. One may also speak of quadratic integrability over bounded intervals such as ,b/math> for a \leq b. An equivalent definition is to say that the square of the function itself (rather than of its absolute value) is Lebesgue integrable. For this to be true, the integrals of the positive and negative portions of the real part must both be finite, as well as those for the imaginary part. The vector space of square integrable functions (with respect to Lebesgue measure) forms the ''Lp'' space with p=2. Among the ''Lp'' spaces, the class of square integrable functions is unique in being compatible with an inner product, which allows notions lik ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quadratic Form
In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example, :4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong to a fixed field , such as the real or complex numbers, and one speaks of a quadratic form over . If K=\mathbb R, and the quadratic form takes zero only when all variables are simultaneously zero, then it is a definite quadratic form, otherwise it is an isotropic quadratic form. Quadratic forms occupy a central place in various branches of mathematics, including number theory, linear algebra, group theory (orthogonal group), differential geometry (Riemannian metric, second fundamental form), differential topology ( intersection forms of four-manifolds), and Lie theory (the Killing form). Quadratic forms are not to be confused with a quadratic equation, which has only one variable and includes terms of degree two or less. A quadratic form is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adjoint Matrix
In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an m \times n complex matrix \boldsymbol is an n \times m matrix obtained by transposing \boldsymbol and applying complex conjugate on each entry (the complex conjugate of a+ib being a-ib, for real numbers a and b). It is often denoted as \boldsymbol^\mathrm or \boldsymbol^* or \boldsymbol'. H. W. Turnbull, A. C. Aitken, "An Introduction to the Theory of Canonical Matrices," 1932. For real matrices, the conjugate transpose is just the transpose, \boldsymbol^\mathrm = \boldsymbol^\mathsf. Definition The conjugate transpose of an m \times n matrix \boldsymbol is formally defined by where the subscript ij denotes the (i,j)-th entry, for 1 \le i \le n and 1 \le j \le m, and the overbar denotes a scalar complex conjugate. This definition can also be written as :\boldsymbol^\mathrm = \left(\overline\right)^\mathsf = \overline where \boldsymbol^\mathsf denotes the transpose and \overline denotes t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Identity Operator
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Dot Product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors), and returns a single number. In Euclidean geometry, the dot product of the Cartesian coordinates of two vectors is widely used. It is often called the inner product (or rarely projection product) of Euclidean space, even though it is not the only inner product that can be defined on Euclidean space (see Inner product space for more). Algebraically, the dot product is the sum of the products of the corresponding entries of the two sequences of numbers. Geometrically, it is the product of the Euclidean magnitudes of the two vectors and the cosine of the angle between them. These definitions are equivalent when using Cartesian coordinates. In mo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
