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Fredholm Operator
In mathematics, Fredholm operators are certain operators that arise in the Fredholm theory of integral equations. They are named in honour of Erik Ivar Fredholm. By definition, a Fredholm operator is a bounded linear operator ''T'' : ''X'' → ''Y'' between two Banach spaces with finite-dimensional kernel \ker T and finite-dimensional (algebraic) cokernel \mathrm\,T = Y/\mathrm\,T, and with closed range \mathrm\,T. The last condition is actually redundant. The '' index'' of a Fredholm operator is the integer : \mathrm\,T := \dim \ker T - \mathrm\,\mathrm\,T or in other words, : \mathrm\,T := \dim \ker T - \mathrm\,\mathrm\,T. Properties Intuitively, Fredholm operators are those operators that are invertible "if finite-dimensional effects are ignored." The formally correct statement follows. A bounded operator ''T'' : ''X'' → ''Y'' between Banach spaces ''X'' and ''Y'' is Fredholm if and only if it is invertible modulo compact ... [...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] |
Hermitian Adjoint
In mathematics, specifically in operator theory, each linear operator A on a Euclidean vector space defines a Hermitian adjoint (or adjoint) operator A^* on that space according to the rule :\langle Ax,y \rangle = \langle x,A^*y \rangle, where \langle \cdot,\cdot \rangle is the inner product on the vector space. The adjoint may also be called the Hermitian conjugate or simply the Hermitian after Charles Hermite. It is often denoted by in fields like physics, especially when used in conjunction with bra–ket notation in quantum mechanics. In finite dimensions where operators are represented by matrices, the Hermitian adjoint is given by the conjugate transpose (also known as the Hermitian transpose). The above definition of an adjoint operator extends verbatim to bounded linear operators on Hilbert spaces H. The definition has been further extended to include unbounded '' densely defined'' operators whose domain is topologically dense in—but not necessarily equal to— ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homotopy Class
In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deformation being called a homotopy (, ; , ) between the two functions. A notable use of homotopy is the definition of homotopy groups and cohomotopy groups, important invariants in algebraic topology. In practice, there are technical difficulties in using homotopies with certain spaces. Algebraic topologists work with compactly generated spaces, CW complexes, or spectra. Formal definition Formally, a homotopy between two continuous functions ''f'' and ''g'' from a topological space ''X'' to a topological space ''Y'' is defined to be a continuous function H: X \times ,1\to Y from the product of the space ''X'' with the unit interval , 1to ''Y'' such that H(x,0) = f(x) and H(x,1) = g(x) for all x \in X. If we think of the second p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological K-theory
In mathematics, topological -theory is a branch of algebraic topology. It was founded to study vector bundles on topological spaces, by means of ideas now recognised as (general) K-theory that were introduced by Alexander Grothendieck. The early work on topological -theory is due to Michael Atiyah and Friedrich Hirzebruch. Definitions Let be a compact Hausdorff space and k= \R or \Complex. Then K_k(X) is defined to be the Grothendieck group of the commutative monoid of isomorphism classes of finite-dimensional -vector bundles over under Whitney sum. Tensor product of bundles gives -theory a commutative ring structure. Without subscripts, K(X) usually denotes complex -theory whereas real -theory is sometimes written as KO(X). The remaining discussion is focused on complex -theory. As a first example, note that the -theory of a point is the integers. This is because vector bundles over a point are trivial and thus classified by their rank and the Grothendieck group of the natur ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Atiyah-Jänich Theorem
In mathematics, Kuiper's theorem (after Nicolaas Kuiper) is a result on the topology of operators on an infinite-dimensional, complex Hilbert space ''H''. It states that the topological space, space GL(''H'') of invertible bounded operator, bounded linear operator, endomorphisms of ''H'' is such that all maps from any CW complex, finite complex ''Y'' to GL(''H'') are homotopic to a constant, for the norm topology on operators. A significant corollary, also referred to as ''Kuiper's theorem'', is that this group is weakly contractible, ''ie.'' all its homotopy groups are trivial. This result has important uses in topological K-theory. General topology of the general linear group For finite dimensional ''H'', this group would be a complex general linear group and not at all contractible. In fact it is homotopy equivalent to its maximal compact subgroup, the unitary group ''U'' of ''H''. The proof that the complex general linear group and unitary group have the same homotopy typ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parametrix
In mathematics, and specifically the field of partial differential equations (PDEs), a parametrix is an approximation to a fundamental solution of a PDE, and is essentially an approximate inverse to a differential operator. A parametrix for a differential operator is often easier to construct than a fundamental solution, and for many purposes is almost as good. It is sometimes possible to construct a fundamental solution from a parametrix by iteratively improving it. Overview and informal definition It is useful to review what a fundamental solution for a differential operator with constant coefficients is: it is a distribution on ℝ''n'' such that :P(D) = \delta(x)~, in the weak sense, where is the Dirac delta distribution. In a similar way, a parametrix for a variable coefficient differential operator is a distribution such that :P(x,D) = \delta(x) + \omega(x) ~, where is some function with compact support. The parametrix is a useful concept in the study of e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Differential Equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to how is thought of as an unknown number to be solved for in an algebraic equation like . However, it is usually impossible to write down explicit formulas for solutions of partial differential equations. There is, correspondingly, a vast amount of modern mathematical and scientific research on methods to Numerical methods for partial differential equations, numerically approximate solutions of certain partial differential equations using computers. Partial differential equations also occupy a large sector of pure mathematics, pure mathematical research, in which the usual questions are, broadly speaking, on the identification of general qualitative features of solutions of various partial differential equations, such a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elliptic Operator
In the theory of partial differential equations, elliptic operators are differential operators that generalize the Laplace operator. They are defined by the condition that the coefficients of the highest-order derivatives be positive, which implies the key property that the principal symbol is invertible, or equivalently that there are no real characteristic directions. Elliptic operators are typical of potential theory, and they appear frequently in electrostatics and continuum mechanics. Elliptic regularity implies that their solutions tend to be smooth functions (if the coefficients in the operator are smooth). Steady-state solutions to hyperbolic and parabolic equations generally solve elliptic equations. Definitions Let L be linear differential operator of order ''m'' on a domain \Omega in R''n'' given by Lu = \sum_ a_\alpha(x)\partial^\alpha u where \alpha = (\alpha_1, \dots, \alpha_n) denotes a multi-index, and \partial^\alpha u = \partial^_1 \cdots \partial_n^ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Winding Number
In mathematics, the winding number or winding index of a closed curve in the plane around a given point is an integer representing the total number of times that curve travels counterclockwise around the point, i.e., the curve's number of turns. The winding number depends on the orientation of the curve, and it is negative if the curve travels around the point clockwise. Winding numbers are fundamental objects of study in algebraic topology, and they play an important role in vector calculus, complex analysis, geometric topology, differential geometry, and physics (such as in string theory). Intuitive description Suppose we are given a closed, oriented curve in the ''xy'' plane. We can imagine the curve as the path of motion of some object, with the orientation indicating the direction in which the object moves. Then the winding number of the curve is equal to the total number of counterclockwise turns that the object makes around the origin. When counting the total nu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Toeplitz Operator
In operator theory, a Toeplitz operator is the compression of a multiplication operator on the circle to the Hardy space. Details Let ''S''1 be the circle, with the standard Lebesgue measure, and ''L''2(''S''1) be the Hilbert space of square-integrable functions. A bounded measurable function ''g'' on ''S''1 defines a multiplication operator ''Mg'' on ''L''2(''S''1). Let ''P'' be the projection from ''L''2(''S''1) onto the Hardy space ''H''2. The ''Toeplitz operator with symbol g'' is defined by :T_g = P M_g \vert_, where " , " means restriction. A bounded operator on ''H''2 is Toeplitz if and only if its matrix representation, in the basis , has constant diagonals. Theorems * Theorem: If g is continuous, then T_g - \lambda is Fredholm if and only if \lambda is not in the set g(S^1). If it is Fredholm, its index is minus the winding number of the curve traced out by g with respect to the origin. For a proof, see . He attributes the theorem to Mark Krein, Harold Widom, and A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hardy Space
In complex analysis, the Hardy spaces (or Hardy classes) ''Hp'' are certain spaces of holomorphic functions on the unit disk or upper half plane. They were introduced by Frigyes Riesz , who named them after G. H. Hardy, because of the paper . In real analysis Hardy spaces are certain spaces of distributions on the real line, which are (in the sense of distributions) boundary values of the holomorphic functions of the complex Hardy spaces, and are related to the ''Lp'' spaces of functional analysis. For 1 ≤ ''p'' < ∞ these real Hardy spaces ''Hp'' are certain s of ''Lp'', while for ''p'' < 1 the ''Lp'' spaces have some undesirable properties, and the Hardy spaces are much better behaved. There are also higher-dimensional generalizations, consisting of certain holomorphic functions on |
