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Mark Krein
Mark Grigorievich Krein (, ; 3 April 1907 – 17 October 1989) was a Soviet mathematician, one of the major figures of the Soviet school of functional analysis. He is known for works in operator theory (in close connection with concrete problems coming from mathematical physics), the problem of moments, classical analysis and representation theory. He was born in Kyiv, leaving home at age 17 to go to Odesa. He had a difficult academic career, not completing his first degree and constantly being troubled by anti-Semitic discrimination. His supervisor was Nikolai Chebotaryov. He was awarded the Wolf Prize in Mathematics in 1982 (jointly with Hassler Whitney), but was not allowed to attend the ceremony. David Milman, Mark Naimark, Israel Gohberg, Vadym Adamyan, Mikhail Livsic and other known mathematicians were his students. He died in Odesa. On 14 January 2008, the memorial plaque of Mark Krein was unveiled on the main administration building of I.I. Mechnikov Odesa Na ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kyiv
Kyiv, also Kiev, is the capital and most populous List of cities in Ukraine, city of Ukraine. Located in the north-central part of the country, it straddles both sides of the Dnieper, Dnieper River. As of 1 January 2022, its population was 2,952,301, making Kyiv the List of European cities by population within city limits, seventh-most populous city in Europe. Kyiv is an important industrial, scientific, educational, and cultural center. It is home to many High tech, high-tech industries, higher education institutions, and historical landmarks. The city has an extensive system of Transport in Kyiv, public transport and infrastructure, including the Kyiv Metro. The city's name is said to derive from the name of Kyi, one of its four legendary founders. During History of Kyiv, its history, Kyiv, one of the oldest cities in Eastern Europe, passed through several stages of prominence and obscurity. The city probably existed as a commercial center as early as the 5th century. A Slav ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Krein Space
In mathematics, in the field of functional analysis, an indefinite inner product space :(K, \langle \cdot,\,\cdot \rangle, J) is an infinite-dimensional complex vector space K equipped with both an indefinite inner product :\langle \cdot,\,\cdot \rangle \, and a positive semi-definite inner product :(x,\,y) \ \stackrel\ \langle x,\,Jy \rangle, where the metric operator J is an endomorphism of K obeying :J^3 = J. \, The indefinite inner product space itself is not necessarily a Hilbert space; but the existence of a positive semi-definite inner product on K implies that one can form a quotient space on which there is a positive definite inner product. Given a strong enough topology on this quotient space, it has the structure of a Hilbert space, and many objects of interest in typical applications fall into this quotient space. An indefinite inner product space is called a Krein space (or J''-space'') if (x,\,y) is positive definite and K possesses a majorant topology. K ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Moment Problem
In mathematics, a moment problem arises as the result of trying to invert the mapping that takes a measure \mu to the sequence of moments :m_n = \int_^\infty x^n \,d\mu(x)\,. More generally, one may consider :m_n = \int_^\infty M_n(x) \,d\mu(x)\,. for an arbitrary sequence of functions M_n. Introduction In the classical setting, \mu is a measure on the real line, and M is the sequence \. In this form the question appears in probability theory, asking whether there is a probability measure having specified mean, variance and so on, and whether it is unique. There are three named classical moment problems: the Hamburger moment problem in which the support of \mu is allowed to be the whole real line; the Stieltjes moment problem, for ,\infty); and the Hausdorff moment problem for a bounded interval, which without loss of generality may be taken as ,1/math>. The moment problem also extends to complex analysis as the trigonometric moment problem in which the Hankel matric ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Physics
Mathematical physics is the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories". An alternative definition would also include those mathematics that are inspired by physics, known as physical mathematics. Scope There are several distinct branches of mathematical physics, and these roughly correspond to particular historical parts of our world. Classical mechanics Applying the techniques of mathematical physics to classical mechanics typically involves the rigorous, abstract, and advanced reformulation of Newtonian mechanics in terms of Lagrangian mechanics and Hamiltonian mechanics (including both approaches in the presence of constraints). Both formulations are embodied in analytical mechanics and lead ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Operator Theory
In mathematics, operator theory is the study of linear operators on function spaces, beginning with differential operators and integral operators. The operators may be presented abstractly by their characteristics, such as bounded linear operators or closed operators, and consideration may be given to nonlinear operators. The study, which depends heavily on the topology of function spaces, is a branch of functional analysis. If a collection of operators forms an algebra over a field, then it is an operator algebra. The description of operator algebras is part of operator theory. Single operator theory Single operator theory deals with the properties and classification of operators, considered one at a time. For example, the classification of normal operators in terms of their spectra falls into this category. Spectrum of operators The spectral theorem is any of a number of results about linear operators or about matrices. In broad terms the spectral theorem provides cond ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Functional Analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, Inner product space#Definition, inner product, Norm (mathematics)#Definition, norm, or Topological space#Definitions, topology) and the linear transformation, linear functions defined on these spaces and suitably respecting these structures. The historical roots of functional analysis lie in the study of function space, spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining, for example, continuous function, continuous or unitary operator, unitary operators between function spaces. This point of view turned out to be particularly useful for the study of differential equations, differential and integral equations. The usage of the word ''functional (mathematics), functional'' as a noun goes back to the calculus of v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wolf Prize In Mathematics
The Wolf Prize in Mathematics is awarded almost annually by the Wolf Foundation in Israel. It is one of the six Wolf Prizes established by the Foundation and awarded since 1978; the others are in Agriculture, Chemistry, Medicine, Physics and Arts. The Wolf Prize includes a monetary award of $100,000. According to a reputation survey conducted in 2013 and 2014, the Wolf Prize in Mathematics is the third most prestigious international academic award in mathematics, after the Abel Prize and the Fields Medal. Laureates Laureates per country Below is a chart of all laureates per country (updated to 2024 laureates). Some laureates are counted more than once if they have multiple citizenships. Notes See also * List of mathematics awards References External links * * * Israel-Wolf-Prizes 2015Jerusalempost Wolf Prizes 2017Jerusalempost Wolf Prizes 2018Wolf Prize 2019 {{DEFAULTSORT:Wolf Prize In Mathematics Mathematics Mathematics is a field of study th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tannaka–Krein Duality
In mathematics, Tannaka–Krein duality theory concerns the interaction of a compact topological group and its category of linear representations. It is a natural extension of Pontryagin duality, between compact and discrete commutative topological groups, to groups that are compact but noncommutative. The theory is named after Tadao Tannaka and Mark Grigorievich Krein. In contrast to the case of commutative groups considered by Lev Pontryagin, the notion dual to a noncommutative compact group is not a group, but a category of representations Π(''G'') with some additional structure, formed by the finite-dimensional representations of ''G''. Duality theorems of Tannaka and Krein describe the converse passage from the category Π(''G'') back to the group ''G'', allowing one to recover the group from its category of representations. Moreover, they in effect completely characterize all categories that can arise from a group in this fashion. Alexander Grothendieck later showed th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Markov–Krein Theorem
In probability theory, the Markov–Krein theorem gives the best upper and lower bounds on the expected values of certain functions of a random variable where only the first moments of the random variable are known. The result is named after Andrey Markov and Mark Krein Mark Grigorievich Krein (, ; 3 April 1907 – 17 October 1989) was a Soviet mathematician, one of the major figures of the Soviet school of functional analysis. He is known for works in operator theory (in close connection with concrete problems .... The theorem can be used to bound average response times in the M/G/k queueing system. References Theorems in probability theory {{probability-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Favard Constant
In mathematics, the Favard constant (also called the Akhiezer–Krein–Favard constant) of order r is defined as K_r = \frac \sum\limits_^ \left \frac \right. The particular values of Favard constant are K_0 = 1, K_1 = \frac, K_2 = \frac. This constant is named after the French mathematician Jean Favard, and after the Soviet mathematicians Naum Akhiezer and Mark Krein. Uses This constant is used in solutions of several extremal problems, for example * Favard's constant is the sharp constant in Jackson's inequality for trigonometric polynomials * the sharp constants in the Landau–Kolmogorov inequality are expressed via Favard's constants * Norms of periodic perfect splines. *The second Favard constant, K_2 = \frac is the same as the value of the internal 4-dimensional equivalent of the angles in a tesseract In geometry, a tesseract or 4-cube is a four-dimensional hypercube, analogous to a two-dimensional square and a three-dimensional cube. Just as th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Krein–Smulian Theorem
In mathematics, particularly in functional analysis, the Krein-Smulian theorem can refer to two theorems relating the closed convex hull and compactness in the weak topology. They are named after Mark Krein Mark Grigorievich Krein (, ; 3 April 1907 – 17 October 1989) was a Soviet mathematician, one of the major figures of the Soviet school of functional analysis. He is known for works in operator theory (in close connection with concrete problems ... and Vitold Shmulyan, who published them in 1940. Statement Both of the following theorems are referred to as the Krein-Smulian Theorem. See also * * References Bibliography * * * * * Further reading * https://www.math.ias.edu/~lurie/261ynotes/lecture12.pdf Banach spaces Topological vector spaces Theorems in functional analysis {{hyperbolic-geometry-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Krein–Rutman Theorem
In functional analysis, the Krein–Rutman theorem is a generalisation of the Perron–Frobenius theorem to infinite-dimensional Banach spaces. It was proved by Krein and Rutman in 1948. Statement Let X be a Banach space, and let K\subset X be a convex cone such that K\cap -K = \, and K-K is dense in X, i.e. the closure of the set \=X. K is also known as a total cone. Let T:X\to X be a non-zero compact operator, and assume that it is ''positive'', meaning that T(K)\subset K, and that its spectral radius r(T) is strictly positive. Then r(T) is an eigenvalue of T with positive eigenvector In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by ..., meaning that there exists u\in K\setminus such that T(u)=r(T)u. De Pagter's theorem If the positive operator T is assumed to be ideal '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
