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Dvoretzky's Theorem
In mathematics, Dvoretzky's theorem is an important structural theorem about normed vector spaces proved by Aryeh Dvoretzky in the early 1960s, answering a question of Alexander Grothendieck. In essence, it says that every sufficiently high-dimensional normed vector space will have low-dimensional subspaces that are approximately Euclidean. Equivalently, every high-dimensional bounded symmetric convex set has low-dimensional sections that are approximately ellipsoids. A new proof found by Vitali Milman in the 1970s was one of the starting points for the development of asymptotic geometric analysis (also called ''asymptotic functional analysis'' or the ''local theory of Banach spaces''). Original formulations For every natural number ''k'' ∈ N and every ''ε'' > 0 there exists a natural number ''N''(''k'', ''ε'') ∈ N such that if (''X'', ‖·‖) is any normed space of dimension ''N''(''k'', ''ε''), ther ... [...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] |
Concentration Of Measure
In mathematics, concentration of measure (about a median) is a principle that is applied in measure theory, probability and combinatorics, and has consequences for other fields such as Banach space theory. Informally, it states that "A random variable that depends in a Lipschitz way on many independent variables (but not too much on any of them) is essentially constant". The concentration of measure phenomenon was put forth in the early 1970s by Vitali Milman in his works on the local theory of Banach spaces, extending an idea going back to the work of Paul Lévy. It was further developed in the works of Milman and Gromov, Maurey, Pisier, Schechtman, Talagrand, Ledoux, and others. The general setting Let (X, d) be a metric space with a measure \mu on the Borel sets with \mu(X) = 1. Let :\alpha(\epsilon) = \sup \left\, where :A_\epsilon = \left\ is the \epsilon-''extension'' (also called \epsilon-fattening in the context of the Hausdorff distance) of a set A. The funct ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Banach Spaces
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well-defined limit that is within the space. Banach spaces are named after the Polish mathematician Stefan Banach, who introduced this concept and studied it systematically in 1920–1922 along with Hans Hahn and Eduard Helly. Maurice René Fréchet was the first to use the term "Banach space" and Banach in turn then coined the term "Fréchet space." Banach spaces originally grew out of the study of function spaces by Hilbert, Fréchet, and Riesz earlier in the century. Banach spaces play a central role in functional analysis. In other areas of analysis, the spaces under study are often Banach spaces. Definition A Banach space is a complete norme ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Joram Lindenstrauss
Joram Lindenstrauss ( he, יורם לינדנשטראוס) (October 28, 1936 – April 29, 2012) was an Israeli mathematician working in functional analysis. He was a professor of mathematics at the Einstein Institute of Mathematics. Biography Joram Lindenstrauss was born in Tel Aviv. He was the only child of a pair of lawyers who immigrated to Israel from Berlin. He began to study mathematics at the Hebrew University of Jerusalem in 1954 while serving in the army. He became a full-time student in 1956 and received his master's degree in 1959. In 1962 Lindenstrauss earned his Ph.D. from the Hebrew University (dissertation: ''Extension of Compact Operators'', advisors: Aryeh Dvoretzky, Branko Grünbaum). He worked as a postdoc at Yale University and the University of Washington in Seattle from 1962 - 1965. He was appointed senior lecturer at the Hebrew University in 1965, associate professor on 1967 and full professor in 1969. He became the Leon H. and Ada G. Miller Memorial ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tadeusz Figiel
Tadeusz Figiel (born 2 July 1948 in Gdańsk) is a Polish mathematician specializing in functional analysis. Biography In 1970 Figiel graduated in mathematics at the University of Warsaw. He received his doctorate in 1972 under the supervision of Aleksander Pełczyński and then habilitated in 1975 with habilitation thesis ''O modułach wypukłości i gładkości'' (On modules of convexity and smoothness) at the (Instytut Matematyczny PAN). There Figiel was appointed in 1983 an associate professor and in 1990 a full professor. He is the head of the Gdańsk Branch of the Polish Academy of Sciences and the editor-in-chief of the journal ''Studia Mathematica''.''Złota księga nauk ekonomicznych, prawnych i ścisłych 2005'' (The Golden Book of Economics, Law and Science 2005), wyd. Mastermedia sp. z o.o. i wyd. Helion, Gliwice 2005, p. 67 Figiel received in 1976 the Stefan Banach Award, in 1988 the of First Degree (together with Zbigniew Ciesielski), in 1989 the , and in 2004 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chebyshev Distance
In mathematics, Chebyshev distance (or Tchebychev distance), maximum metric, or L∞ metric is a metric defined on a vector space where the distance between two vectors is the greatest of their differences along any coordinate dimension. It is named after Pafnuty Chebyshev. It is also known as chessboard distance, since in the game of chess the minimum number of moves needed by a king to go from one square on a chessboard to another equals the Chebyshev distance between the centers of the squares, if the squares have side length one, as represented in 2-D spatial coordinates with axes aligned to the edges of the board. For example, the Chebyshev distance between f6 and e2 equals 4. Definition The Chebyshev distance between two vectors or points ''x'' and ''y'', with standard coordinates x_i and y_i, respectively, is :D_(x,y) := \max_i(, x_i -y_i, ).\ This equals the limit of the L''p'' metrics: :\lim_ \bigg( \sum_^n \left, x_i - y_i \^p \bigg)^, hence it is also known as th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Noga Alon
Noga Alon ( he, נוגה אלון; born 17 February 1956) is an Israeli mathematician and a professor of mathematics at Princeton University noted for his contributions to combinatorics and theoretical computer science, having authored hundreds of papers. Academic background Alon is a Professor of Mathematics at Princeton University and a Baumritter Professor Emeritus of Mathematics and Computer Science at Tel Aviv University, Israel. He graduated from the Hebrew Reali School in 1974 and received his Ph.D. in Mathematics at the Hebrew University of Jerusalem in 1983 and had visiting positions in various research institutes including MIT, The Institute for Advanced Study in Princeton, IBM Almaden Research Center, Bell Labs, Bellcore and Microsoft Research. He serves on the editorial boards of more than a dozen international journals; since 2008 he is the editor-in-chief of ''Random Structures and Algorithms''. He has given lectures in many conferences, including plenary addresses ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gideon Schechtman
Gideon Schechtman (; born 14 February 1947) is an Israeli mathematician and professor of mathematics at the Weizmann Institute of Science. Academic career Schechtman received his Ph.D. in mathematics from the Hebrew University of Jerusalem in 1976 and was a postdoctoral fellow at Ohio State University. Since 1980 he has been affiliated with the Weizmann Institute, where he became emeritus professor in 2017. His research focuses predominantly on functional analysis and the geometry of Banach spaces. Schechtman is an editor of the Israel Journal of Mathematics '' Israel Journal of Mathematics'' is a peer-reviewed mathematics journal published by the Hebrew University of Jerusalem (Magnes Press). Founded in 1963, as a continuation of the ''Bulletin of the Research Council of Israel'' (Section F), the jou .... References {{DEFAULTSORT:Schechtman, Gideon 1947 births Functional analysts Einstein Institute of Mathematics alumni Israeli Jews Israeli mathematicians Wei ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Israel Journal Of Mathematics
'' Israel Journal of Mathematics'' is a peer-reviewed mathematics journal published by the Hebrew University of Jerusalem (Magnes Press). Founded in 1963, as a continuation of the ''Bulletin of the Research Council of Israel'' (Section F), the journal publishes articles on all areas of mathematics. The journal is indexed by ''Mathematical Reviews'' and Zentralblatt MATH. Its 2009 MCQ was 0.70, and its 2009 impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a scientometric index calculated by Clarivate that reflects the yearly mean number of citations of articles published in the last two years in a given journal, as i ... was 0.754. External links * Mathematics journals Publications established in 1963 English-language journals Bimonthly journals Hebrew University of Jerusalem {{math-journal-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Yehoram Gordon
Jehoram (meaning "Jehovah is exalted" in Biblical Hebrew) was the name of several individuals in the Tanakh. The female version of this name is Athaliah. *The son of Toi, King of Hamath who was sent by his father to congratulate David on the occasion of his victory over Hadadezer (2 Samuel 8:10) * Jehoram of Israel or Joram, King of Israel (ruled c. 852/49–842/41) * Jehoram of Judah or Joram, King of Judah (ruled c. 849/48–842/41) *A Levite of the family of Gershom (1 Chronicles 26:25) *A priest sent by Jehoshaphat to instruct the people in Judah (2 Chronicles The Book of Chronicles ( he, דִּבְרֵי־הַיָּמִים ) is a book in the Hebrew Bible, found as two books (1–2 Chronicles) in the Christian Old Testament. Chronicles is the final book of the Hebrew Bible, concluding the third sect ... 17:8) In modern days, it is also the name of: * Yehoram Gaon (born 1939), Israeli singer and actor {{disambiguation, hndis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dvoretzky Dimension
Dvoretzky is a surname. Notable people with the surname include: * Aryeh Dvoretzky (1916–2008), Russian-born Israeli mathematician, eighth president of the Weizmann Institute of Science * Moshe Dvoretzky Moshe Dvoretzky (alternative spelling "Dvoretzki", bg, Моше Дворецки; born 1922 in Haskovo, Bulgaria) was a Bulgarian actor. Moshe Dvoretzky was one of the founders of the Dimitrovgrad Theatre. He played dozens of leading parts ther ... (1922–1988), Bulgarian actor See also * Dvoretzky's theorem {{Surname ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unit Sphere
In mathematics, a unit sphere is simply a sphere of radius one around a given center. More generally, it is the set of points of distance 1 from a fixed central point, where different norms can be used as general notions of "distance". A unit ball is the closed set of points of distance less than or equal to 1 from a fixed central point. Usually the center is at the origin of the space, so one speaks of "the unit ball" or "the unit sphere". Special cases are the unit circle and the unit disk. The importance of the unit sphere is that any sphere can be transformed to a unit sphere by a combination of translation and scaling. In this way the properties of spheres in general can be reduced to the study of the unit sphere. Unit spheres and balls in Euclidean space In Euclidean space of ''n'' dimensions, the -dimensional unit sphere is the set of all points (x_1, \ldots, x_n) which satisfy the equation : x_1^2 + x_2^2 + \cdots + x_n ^2 = 1. The ''n''-dimensional open unit ball ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
