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Fredholm Number
Fredholm is a Swedish surname. Notable people with the surname include: *Erik Ivar Fredholm (1866–1927), Swedish mathematician **Fredholm alternative, in mathematics **Fredholm determinant, in mathematics **Fredholm integral equation, in mathematics **Fredholm kernel, in mathematics **Fredholm module, In noncommutative geometry ** Fredholm number, in number theory, apparently not in fact studied by Fredholm **Fredholm operator, in mathematics **Fredholm's theorem, in mathematics ** Analytic Fredholm theorem, in mathematics **Fredholm theory In mathematics, Fredholm theory is a theory of integral equations. In the narrowest sense, Fredholm theory concerns itself with the solution of the Fredholm integral equation. In a broader sense, the abstract structure of Fredholm's theory is given ..., in mathematics ** Fredholm (crater), a small lunar impact crater ** 21659 Fredholm (1999 PR3), main-belt asteroid discovered in 1999 by P. G. Comba * (1830–1891), Swedish industrialist * Gert ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Erik Ivar Fredholm
Erik Ivar Fredholm (7 April 1866 – 17 August 1927) was a Swedish mathematician whose work on integral equations and operator theory foreshadowed the theory of Hilbert spaces. Biography Fredholm was born in Stockholm in 1866. He obtained his PhD at Uppsala University in 1898, under the supervision of Gösta Mittag-Leffler. He was ''docent'' at Stockholm University from 1898 to 1906 and professor from 1906 until his death. In 1914 he was elected a member of the Royal Swedish Academy of Sciences. Beside his academic career he was recruited to the Swedish Social Insurance Agency when it was founded in 1902. He later served as an actuary at the insurance company Skandia (1904-1927), where his Fredholm equation was used to calculate buyback-prices. From 1911, he was married to Agnes Maria Liljeblad, the secretary of Mittag-Leffler. Mathematical work In , Fredholm introduced and analysed a class of integral equations now called Fredholm equations. His analysis included the c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fredholm Alternative
In mathematics, the Fredholm alternative, named after Ivar Fredholm, is one of Fredholm's theorems and is a result in Fredholm theory. It may be expressed in several ways, as a theorem of linear algebra, a theorem of integral equations, or as a theorem on Fredholm operators. Part of the result states that a non-zero complex number in the spectrum of a compact operator is an eigenvalue. Linear algebra If ''V'' is an ''n''-dimensional vector space and T:V\to V is a linear transformation, then exactly one of the following holds: #For each vector ''v'' in ''V'' there is a vector ''u'' in ''V'' so that T(u) = v. In other words: ''T'' is surjective (and so also bijective, since ''V'' is finite-dimensional). #\dim(\ker(T)) > 0. A more elementary formulation, in terms of matrices, is as follows. Given an ''m''×''n'' matrix ''A'' and a ''m''×1 column vector b, exactly one of the following must hold: #''Either:'' ''A'' x = b has a solution x #''Or:'' ''A''T y = 0 has a solution ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fredholm Determinant
In mathematics, the Fredholm determinant is a complex-valued function which generalizes the determinant of a finite dimensional linear operator. It is defined for bounded operators on a Hilbert space which differ from the identity operator by a trace-class operator. The function is named after the mathematician Erik Ivar Fredholm. Fredholm determinants have had many applications in mathematical physics, the most celebrated example being Gábor Szegő's limit formula, proved in response to a question raised by Lars Onsager and C. N. Yang on the spontaneous magnetization of the Ising model. Definition Let ''H'' be a Hilbert space and ''G'' the set of bounded invertible operators on ''H'' of the form ''I'' + ''T'', where ''T'' is a trace-class operator. ''G'' is a group because (I+T)^ - I = - T(I+T)^, so (''I''+''T'')−1−''I'' is trace class if ''T'' is. It has a natural metric given by , where is the trace-class norm. If ''H'' is a Hilbert space with inner product (\cd ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fredholm Integral Equation
In mathematics, the Fredholm integral equation is an integral equation whose solution gives rise to Fredholm theory, the study of Fredholm kernels and Fredholm operators. The integral equation was studied by Ivar Fredholm. A useful method to solve such equations, the Adomian decomposition method, is due to George Adomian. Equation of the first kind A Fredholm equation is an integral equation in which the term containing the kernel function (defined below) has constants as integration limits. A closely related form is the Volterra integral equation which has variable integral limits. An inhomogeneous Fredholm equation of the first kind is written as and the problem is, given the continuous kernel function K and the function g, to find the function f. An important case of these types of equation is the case when the kernel is a function only of the difference of its arguments, namely K(t,s)=K(ts), and the limits of integration are ±∞, then the right hand side of the e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fredholm Kernel
In mathematics, a Fredholm kernel is a certain type of a kernel on a Banach space, associated with nuclear operators on the Banach space. They are an abstraction of the idea of the Fredholm integral equation and the Fredholm operator, and are one of the objects of study in Fredholm theory. Fredholm kernels are named in honour of Erik Ivar Fredholm. Much of the abstract theory of Fredholm kernels was developed by Alexander Grothendieck and published in 1955. Definition Let ''B'' be an arbitrary Banach space, and let ''B''* be its dual, that is, the space of bounded linear functionals on ''B''. The tensor product B^*\otimes B has a completion under the norm :\Vert X \Vert_\pi = \inf \sum_ \Vert e^*_i\Vert \Vert e_i \Vert where the infimum is taken over all finite representations :X=\sum_ e^*_i \otimes e_i \in B^*\otimes B The completion, under this norm, is often denoted as :B^* \widehat_\pi B and is called the projective topological tensor product. The elements of thi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fredholm Module
In noncommutative geometry, a Fredholm module is a mathematical structure used to quantize the differential calculus. Such a module is, up to trivial changes, the same as the abstract elliptic operator introduced by . Definition If ''A'' is an involutive algebra over the complex numbers C, then a Fredholm module over ''A'' consists of an involutive representation of ''A'' on a Hilbert space ''H'', together with a self-adjoint operator ''F'', of square 1 and such that the commutator In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory. Group theory The commutator of two elements, a ... : 'F'', ''a'' is a compact operator, for all ''a'' in ''A''. References The paper by Atiyah is reprinted in volume 3 of his collected works, * * *{{citation, last= Atiyah, first= Michael, authorlink=Michael Atiyah, title= Collected works. Vol. 3. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fredholm Number
Fredholm is a Swedish surname. Notable people with the surname include: *Erik Ivar Fredholm (1866–1927), Swedish mathematician **Fredholm alternative, in mathematics **Fredholm determinant, in mathematics **Fredholm integral equation, in mathematics **Fredholm kernel, in mathematics **Fredholm module, In noncommutative geometry ** Fredholm number, in number theory, apparently not in fact studied by Fredholm **Fredholm operator, in mathematics **Fredholm's theorem, in mathematics ** Analytic Fredholm theorem, in mathematics **Fredholm theory In mathematics, Fredholm theory is a theory of integral equations. In the narrowest sense, Fredholm theory concerns itself with the solution of the Fredholm integral equation. In a broader sense, the abstract structure of Fredholm's theory is given ..., in mathematics ** Fredholm (crater), a small lunar impact crater ** 21659 Fredholm (1999 PR3), main-belt asteroid discovered in 1999 by P. G. Comba * (1830–1891), Swedish industrialist * Gert ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fredholm Operator
In mathematics, Fredholm operators are certain Operator (mathematics), 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 (algebra), kernel \ker T and finite-dimensional (algebraic) cokernel \mathrm\,T = Y/\mathrm\,T, and with closed range of a function, range \mathrm\,T. The last condition is actually redundant. The ''Linear transform#Index, 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fredholm's Theorem
In mathematics, Fredholm's theorems are a set of celebrated results of Ivar Fredholm in the Fredholm theory of integral equations. There are several closely related theorems, which may be stated in terms of integral equations, in terms of linear algebra, or in terms of the Fredholm operator on Banach spaces. The Fredholm alternative is one of the Fredholm theorems. Linear algebra Fredholm's theorem in linear algebra is as follows: if ''M'' is a matrix, then the orthogonal complement of the row space of ''M'' is the null space of ''M'': :(\operatorname M)^\bot = \ker M. Similarly, the orthogonal complement of the column space of ''M'' is the null space of the adjoint: :(\operatorname M)^\bot = \ker M^*. Integral equations Fredholm's theorem for integral equations is expressed as follows. Let K(x,y) be an integral kernel, and consider the homogeneous equations :\int_a^b K(x,y) \phi(y) \,dy = \lambda \phi(x) and its complex adjoint :\int_a^b \psi(x) \overline \, dx = \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Analytic Fredholm Theorem
In mathematics, the analytic Fredholm theorem is a result concerning the existence of bounded inverses for a family of bounded linear operators on a Hilbert space. It is the basis of two classical and important theorems, the Fredholm alternative and the Hilbert–Schmidt theorem. The result is named after the Swedish mathematician Erik Ivar Fredholm. Statement of the theorem Let be a domain (an open and connected set). Let be a real or complex Hilbert space and let Lin(''H'') denote the space of bounded linear operators from ''H'' into itself; let I denote the identity operator. Let be a mapping such that * ''B'' is analytic on ''G'' in the sense that the limit \lim_ \frac exists for all ; and * the operator ''B''(''λ'') is a compact operator for each . Then either * does not exist for any ; or * exists for every , where ''S'' is a discrete subset of ''G'' (i.e., ''S'' has no limit point In mathematics, a limit point, accumulation point, or cluster point of a set ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fredholm Theory
In mathematics, Fredholm theory is a theory of integral equations. In the narrowest sense, Fredholm theory concerns itself with the solution of the Fredholm integral equation. In a broader sense, the abstract structure of Fredholm's theory is given in terms of the spectral theory of Fredholm operators and Fredholm kernels on Hilbert space. The theory is named in honour of Erik Ivar Fredholm. Overview The following sections provide a casual sketch of the place of Fredholm theory in the broader context of operator theory and functional analysis. The outline presented here is broad, whereas the difficulty of formalizing this sketch is, of course, in the details. Fredholm equation of the first kind Much of Fredholm theory concerns itself with the following integral equation for ''f'' when ''g'' and ''K'' are given: :g(x)=\int_a^b K(x,y) f(y)\,dy. This equation arises naturally in many problems in physics and mathematics, as the inverse of a differential equation. That is, one is aske ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fredholm (crater)
Fredholm is a small lunar impact crater that is located in the rugged ground to the west of the Mare Crisium. It was named after Swedish mathematician Erik I. Fredholm. It was previously designated Macrobius D. It lies midway between the prominent craters Macrobius to the north and Proclus almost due south. This is a circular, symmetrical formation with a bowl-shaped interior. The inner walls gradually slope down towards the small, central floor, which is less than one quarter the total diameter of the crater. Neatly attached to the northern rim is the smaller Macrobius E. References * * * * * * * * * * * External links LTO-43C3 Proclus— L&PI topographic map In modern mapping, a topographic map or topographic sheet is a type of map characterized by large-scale detail and quantitative representation of relief features, usually using contour lines (connecting points of equal elevation), but histori ... {{Craters on the Moon: C-F Impact craters on t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
