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Mark Grigorievich Krein
Mark Grigorievich Krein ( uk, Марко́ Григо́рович Крейн, russian: Марк Григо́рьевич Крейн; 3 April 1907 – 17 October 1989) was a Soviet mathematician, one of the major figures of the Soviet Union, Soviet school of functional analysis. He is known for works in operator theory (in close connection with concrete problems coming from mathematical physics), moment problem, the problem of moments, classical analysis and representation theory. He was born in Kyiv, leaving home at age 17 to go to Odessa. 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 Samuilovich Livsic, Mikhail Livsic and other known mathematicians were his studen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kyiv
Kyiv, also spelled Kiev, is the capital and most populous city of Ukraine. It is in north-central Ukraine along the Dnieper, Dnieper River. As of 1 January 2021, its population was 2,962,180, 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 in Eastern Europe. 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 Slavs, Slavic settlement on the great trade ... [...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 that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
1989 Deaths
File:1989 Events Collage.png, From left, clockwise: The Cypress Street Viaduct, Cypress structure collapses as a result of the 1989 Loma Prieta earthquake, killing motorists below; The proposal document for the World Wide Web is submitted; The Exxon Valdez oil tanker runs aground in Prince William Sound, Alaska, causing a large Exxon Valdez oil spill, oil spill; The Fall of the Berlin Wall begins the downfall of Communism in Eastern Europe, and heralds German reunification; The United States United States invasion of Panama, invades Panama to depose Manuel Noriega; The Singing Revolution led to the independence of the Baltic states of Estonia, Latvia, and Lithuania from the Soviet Union; The stands of Hillsborough Stadium in Sheffield, Yorkshire, where the Hillsborough disaster occurred; 1989 Tiananmen Square protests and massacre, Students demonstrate in Tiananmen Square, Beijing; many are killed by forces of the Chinese Communist Party., 300x300px, thumb rect 0 0 200 200 1989 Loma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
1907 Births
Nineteen or 19 may refer to: * 19 (number), the natural number following 18 and preceding 20 * one of the years 19 BC, AD 19, 1919, 2019 Films * ''19'' (film), a 2001 Japanese film * ''Nineteen'' (film), a 1987 science fiction film Music * 19 (band), a Japanese pop music duo Albums * ''19'' (Adele album), 2008 * ''19'', a 2003 album by Alsou * ''19'', a 2006 album by Evan Yo * ''19'', a 2018 album by MHD * ''19'', one half of the double album ''63/19'' by Kool A.D. * ''Number Nineteen'', a 1971 album by American jazz pianist Mal Waldron * ''XIX'' (EP), a 2019 EP by 1the9 Songs * "19" (song), a 1985 song by British musician Paul Hardcastle. * "Nineteen", a song by Bad4Good from the 1992 album '' Refugee'' * "Nineteen", a song by Karma to Burn from the 2001 album ''Almost Heathen''. * "Nineteen" (song), a 2007 song by American singer Billy Ray Cyrus. * "Nineteen", a song by Tegan and Sara from the 2007 album '' The Con''. * "XIX" (song), a 2014 song by Slipk ... [...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 which is ''positive'', meaning that T(K)\subset K, and assume 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 of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ..., 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 ''irreduci ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hassler Whitney
Hassler Whitney (March 23, 1907 – May 10, 1989) was an American mathematician. He was one of the founders of singularity theory, and did foundational work in manifolds, embeddings, immersions, characteristic classes, and geometric integration theory. Biography Life Hassler Whitney was born on March 23, 1907, in New York City, where his father Edward Baldwin Whitney was the First District New York Supreme Court judge. His mother, A. Josepha Newcomb Whitney, was an artist and active in politics. He was the paternal nephew of Connecticut Governor and Chief Justice Simeon Eben Baldwin, his paternal grandfather was William Dwight Whitney, professor of Ancient Languages at Yale University, linguist and Sanskrit scholar. Whitney was the great-grandson of Connecticut Governor and US Senator Roger Sherman Baldwin, and the great-great-grandson of American founding father Roger Sherman. His maternal grandparents were astronomer and mathematician Simon Newcomb (1835-1909), a Steeves desce ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Anti-Semitic
Antisemitism (also spelled anti-semitism or anti-Semitism) is hostility to, prejudice towards, or discrimination against Jews. A person who holds such positions is called an antisemite. Antisemitism is considered to be a form of racism. Antisemitism has historically been manifested in many ways, ranging from expressions of hatred of or discrimination against individual Jews to organized pogroms by mobs, police forces, or genocide. Although the term did not come into common usage until the 19th century, it is also applied to previous and later anti-Jewish incidents. Notable instances of persecution include the Rhineland massacres preceding the First Crusade in 1096, the Edict of Expulsion from England in 1290, the 1348–1351 persecution of Jews during the Black Death, the massacres of Spanish Jews in 1391, the persecutions of the Spanish Inquisition, the expulsion from Spain in 1492, the Cossack massacres in Ukraine from 1648 to 1657, various anti-Jewish pogroms in the Russ ... [...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] |
Classical Analysis
Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions. These theories are usually studied in the context of real and complex numbers and 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 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 implicitly present in the early days of ancient Greek mathematics. For instance, an infinite geometric sum is implicit in Zeno's paradox of the dichotomy. (St ... [...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 ''μ'' to the sequences 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, μ 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 μ is allowed to be the whole real line; the Stieltjes moment problem, for , +∞); and the Hausdorff moment problem for a bounded interval, which without loss of generality may be taken as , 1 Existence A sequence of numbers ''m''''n'' is the sequence of moments of a measure ''μ'' if an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Physics
Mathematical physics refers to 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 (also known as physical mathematics). Scope There are several distinct branches of mathematical physics, and these roughly correspond to particular historical periods. Classical mechanics The rigorous, abstract and advanced reformulation of Newtonian mechanics adopting the Lagrangian mechanics and the Hamiltonian mechanics even in the presence of constraints. Both formulations are embodied in analytical mechanics and lead to understanding the deep interplay of the notions of symmetry (physics), symmetry and conservation law, con ... [...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] |
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