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Israel Gelfand
Israel Moiseevich Gelfand, also written Israïl Moyseyovich Gel'fand, or Izrail M. Gelfand ( yi, ישראל געלפֿאַנד, russian: Изра́иль Моисе́евич Гельфа́нд, uk, Ізраїль Мойсейович Гельфанд; – 5 October 2009) was a prominent Soviet-American mathematician. He made significant contributions to many branches of mathematics, including group theory, representation theory and functional analysis. The recipient of many awards, including the Order of Lenin and the first Wolf Prize, he was a Foreign Fellow of the Royal Society and professor at Moscow State University and, after immigrating to the United States shortly before his 76th birthday, at Rutgers University. Gelfand is also a 1994 MacArthur Fellow. His legacy continues through his students, who include Endre Szemerédi, Alexandre Kirillov, Edward Frenkel, Joseph Bernstein, David Kazhdan, as well as his own son, Sergei Gelfand. Early years A native of Kherson Go ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Okny
Okny ( uk, Окни, russian: Окны) is an urban-type settlement in the west of Odesa Oblast, Ukraine. It served as the administrative center of Okny Raion. Population: Okny is located on the banks of the Yahorlik River, a left tributary of the Dniester. History Okny was founded in the end of the 18th century by Moldovan settlers. The area was settled after 1792, when the lands between the Southern Bug and the Dniester were transferred to Russia according to the Iasi Peace Treaty. The area was included in Tiraspol Uyezd, which belonged to Yekaterinoslav Viceroyalty until 1795, Voznesensk Viceroyalty until 1796, Novorossiya Governorate until 1803, and Kherson Governorate until 1920. In 1834, the area was transferred to newly established Ananyiv Uyezd. In 1919, Okny was renamed Krasni Okny. On 16 April 1920, Odessa Governorate split off, and Ananyivsky Uyezd was moved to Odessa Governorate, where it was abolished in 1921. In 1923, uyezds in Ukrainian Soviet Socialist R ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alexandre Kirillov
Alexandre Aleksandrovich Kirillov (russian: Алекса́ндр Алекса́ндрович Кири́ллов, born 1936) is a Soviet and Russian mathematician, known for his works in the fields of representation theory, topological groups and Lie groups. In particular he introduced the orbit method into representation theory. He is an emeritus professor at the University of Pennsylvania. Career Kirillov studied at Moscow State University where he was a student of Israel Gelfand. His Ph.D. (kandidat) dissertation ''Unitary representations of nilpotent Lie groups'' was published in 1962. He was awarded the degree of Doctor of Science. At the time he was the youngest Doctor of Science in the Soviet Union. He worked at the Moscow State University until 1994 when he became the Francis J. Carey Professor of Mathematics at the University of Pennsylvania. During his school years, Kirillov was a winner of many mathematics competitions, and he is still an active organizer of Russian ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Liouville–Bratu–Gelfand Equation
: ''For Liouville's equation in differential geometry, see Liouville's equation.'' In mathematics, Liouville–Bratu–Gelfand equation or Liouville's equation is a non-linear Poisson equation, named after the mathematicians Joseph Liouville, G. Bratu and Israel Gelfand. The equation reads :\nabla^2 \psi + \lambda e^\psi = 0 The equation appears in thermal runaway as Frank-Kamenetskii theory, astrophysics for example, Emden–Chandrasekhar equation. This equation also describes space charge of electricity around a glowing wire and describes planetary nebula. Liouville's solution In two dimension with Cartesian Coordinates (x,y), Joseph Liouville proposed a solution in 1853 as :\lambda e^\psi (u^2 + v^2 + 1) ^2 = 2 \left left(\frac\right)^2 + \left(\frac\right)^2\right/math> where f(z)=u + i v is an arbitrary analytic function with z=x+iy. In 1915, G.W. Walker found a solution by assuming a form for f(z). If r^2=x^2+y^2, then Walker's solution is :8 e^ = \lambda \left left(\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gelfand–Naimark Theorem
In mathematics, the Gelfand–Naimark theorem states that an arbitrary C*-algebra ''A'' is isometrically *-isomorphic to a C*-subalgebra of bounded operators on a Hilbert space. This result was proven by Israel Gelfand and Mark Naimark in 1943 and was a significant point in the development of the theory of C*-algebras since it established the possibility of considering a C*-algebra as an abstract algebraic entity without reference to particular realizations as an operator algebra. Details The Gelfand–Naimark representation π is the direct sum of representations π''f'' of ''A'' where ''f'' ranges over the set of pure states of A and π''f'' is the irreducible representation associated to ''f'' by the GNS construction. Thus the Gelfand–Naimark representation acts on the Hilbert direct sum of the Hilbert spaces ''H''''f'' by : \pi(x) bigoplus_ H_f= \bigoplus_ \pi_f(x)H_f. π(''x'') is a bounded linear operator since it is the direct sum of a family of operators, each one h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gelfand Representation
In mathematics, the Gelfand representation in functional analysis (named after I. M. Gelfand) is either of two things: * a way of representing commutative Banach algebras as algebras of continuous functions; * the fact that for commutative C*-algebras, this representation is an isometric isomorphism. In the former case, one may regard the Gelfand representation as a far-reaching generalization of the Fourier transform of an integrable function. In the latter case, the Gelfand–Naimark representation theorem is one avenue in the development of spectral theory for normal operators, and generalizes the notion of diagonalizing a normal matrix. Historical remarks One of Gelfand's original applications (and one which historically motivated much of the study of Banach algebras) was to give a much shorter and more conceptual proof of a celebrated lemma of Norbert Wiener (see the citation below), characterizing the elements of the group algebras ''L''1(R) and \ell^1() whose translates ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gelfand-Pettis Integral
In mathematics, the Pettis integral or Gelfand–Pettis integral, named after Israel M. Gelfand and Billy James Pettis, extends the definition of the Lebesgue integral to vector-valued functions on a measure space, by exploiting duality. The integral was introduced by Gelfand for the case when the measure space is an interval with Lebesgue measure. The integral is also called the weak integral in contrast to the Bochner integral, which is the strong integral. Definition Let f : X \to V where (X,\Sigma,\mu) is a measure space and V is a topological vector space (TVS) with a continuous dual space V' that separates points (that is, if x \in Vis nonzero then there is some l \in V' such that l(x) \neq 0), for example, V is a normed space or (more generally) is a Hausdorff locally convex TVS. Evaluation of a functional may be written as a duality pairing: \langle \varphi, x \rangle = \varphi The map f : X \to V is called if for all \varphi \in V', the scalar-valued map \v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gelfand–Levitan–Marchenko Integral Equation
In mathematical physics, more specifically the one-dimensional inverse scattering problem, the Marchenko equation (or Gelfand-Levitan-Marchenko equation or GLM equation), named after Israel Gelfand, Boris Levitan and Vladimir Marchenko, is derived by computing the Fourier transform of the scattering relation: : K(r,r^\prime) + g(r,r^\prime) + \int_r^ K(r,r^) g(r^,r^\prime) \mathrmr^ = 0 Where g(r,r^\prime)\,is a symmetric kernel, such that g(r,r^\prime)=g(r^\prime,r),\,which is computed from the scattering data. Solving the Marchenko equation, one obtains the kernel of the transformation operator K(r,r^\prime) from which the potential can be read off. This equation is derived from the Gelfand–Levitan integral equation, using the Povzner–Levitan representation. See also * Lax pair References * Integral equations Scattering theory {{scattering-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Representation Theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and their algebraic operations (for example, matrix addition, matrix multiplication). The theory of matrices and linear operators is well-understood, so representations of more abstract objects in terms of familiar linear algebra objects helps glean properties and sometimes simplify calculations on more abstract theories. The algebraic objects amenable to such a description include groups, associative algebras and Lie algebras. The most prominent of these (and historically the first) is the representation theory of groups, in which elements of a group are represented by invertible matrices in such a way that the group operation i ... [...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 i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integral Geometry
In mathematics, integral geometry is the theory of measures on a geometrical space invariant under the symmetry group of that space. In more recent times, the meaning has been broadened to include a view of invariant (or equivariant) transformations from the space of functions on one geometrical space to the space of functions on another geometrical space. Such transformations often take the form of integral transforms such as the Radon transform and its generalizations. Classical context Integral geometry as such first emerged as an attempt to refine certain statements of geometric probability theory. The early work of Luis Santaló and Wilhelm Blaschke was in this connection. It follows from the classic theorem of Crofton expressing the length of a plane curve as an expectation of the number of intersections with a random line. Here the word 'random' must be interpreted as subject to correct symmetry considerations. There is a sample space of lines, one on which the affin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field (mathematics), fields, and vector spaces, can all be seen as groups endowed with additional operation (mathematics), operations and axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right. Various physical systems, such as crystals and the hydrogen atom, and Standard Model, three of the four known fundamental forces in the universe, may be modelled by symmetry groups. Thus group theory and the closely related representation theory have many important applications in physics, chemistry, and materials science. Group theory is also ce ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vitalii Ditkin
Vitalii Arsenievich Ditkin (2 May 1910, Bogorodsk (now Noginsk), Russia – 17 October 1987, Moscow) was a Soviet mathematician who introduced Ditkin sets. Biography Studied at the Moscow State University in 1932–1935; in 1938 got PhD degree (advisor – Abraham Plessner). From 1943 to 1948 he was with the Steklov Institute of Mathematics; from 1948 to 1955, with the Lebedev Institute of Precision Mechanics and Computer Engineering. In 1949, got the Doctor of Sciences degree. In 1955, he became a deputy director of newly formed Computing Centre of the Academy of Sciences of the USSR. He remained with the Computing Centre till his death. In 1978 was awarded the USSR State Prize The USSR State Prize (russian: links=no, Государственная премия СССР, Gosudarstvennaya premiya SSSR) was the Soviet Union's state honor. It was established on 9 September 1966. After the dissolution of the Soviet Union, t ... in sciences. References * * * {{DEFAULTSORT:D ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
