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Israeli Inventions And Discoveries
This is a list of inventions and discoveries by Israeli scientists and researchers, working locally or overseas. There are over 6,000 startups currently in Israel. There are currently more than 30 technology companies valued over US$1 billion (unicorn startups) in Israel, more than all of Europe combined. Mathematics * Johnson–Lindenstrauss lemma, a mathematical result concerning low-distortion embeddings of points from high-dimensional into low-dimensional Euclidean space contributed by Joram Lindenstrauss. * Development of the measurement of rigidity by Elon Lindenstrauss in ergodic theory, and their applications to number theory''. * Proof of Szemerédi's theorem solved by mathematician Hill Furstenberg. * Expansion of axiomatic set theory and the ZF set theory by Abraham Fraenkel. * Development of the area of automorphic forms and L-functions by Ilya Piatetski-Shapiro. * Development of Sauer–Shelah lemma and Shelah cardinal. * Development of the first proof of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dianthus Caryophyllus Colori Joy (p) 2005-12-04
''Dianthus'' () is a genus of about 340 species of flowering plants in the family (biology), family Caryophyllaceae, native plant, native mainly to Europe and Asia, with a few species in north Africa and in southern Africa, and one species (''D. repens'') in arctic North America. Common names include carnation (''D. caryophyllus''), pink (''D. plumarius'' and related species) and sweet william (''D. barbatus''). Description The species are mostly herbaceous plant, herbaceous perennial plant, perennials, a few are annual plant, annual or biennial plant, biennial, and some are low subshrubs with woody basal stems. The leaf, leaves are opposite, simple, mostly linear and often strongly glaucous grey green to blue green. The flowers have five petals, typically with a Serration, frilled or Pinking shears, pinked margin, and are (in almost all species) pale to dark pink. One species, ''D. knappii'', has yellow flowers with a purple centre. Some species, particularly the perennial pin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sauer–Shelah Lemma
In combinatorial mathematics and extremal set theory, the Sauer–Shelah lemma states that every family of sets with small VC dimension consists of a small number of sets. It is named after Norbert Sauer and Saharon Shelah, who published it independently of each other in 1972. The same result was also published slightly earlier and again independently, by Vladimir Vapnik and Alexey Chervonenkis, after whom the VC dimension is named. In his paper containing the lemma, Shelah gives credit also to Micha Perles, and for this reason the lemma has also been called the Perles–Sauer–Shelah lemma.. Buzaglo et al. call this lemma "one of the most fundamental results on VC-dimension", and it has applications in many areas. Sauer's motivation was in the combinatorics of set systems, while Shelah's was in model theory and that of Vapnik and Chervonenkis was in statistics. It has also been applied in discrete geometry. and graph theory.. Definitions and statement If \textstyle \mathcal=\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shimshon Amitsur
Shimshon Avraham Amitsur (born Kaplan; he, שמשון אברהם עמיצור; August 26, 1921 – September 5, 1994) was an Israeli mathematician. He is best known for his work in ring theory, in particular PI rings, an area of abstract algebra. Biography Amitsur was born in Jerusalem and studied at the Hebrew University under the supervision of Jacob Levitzki. His studies were repeatedly interrupted, first by World War II and then by the 1948 Arab–Israeli War. He received his M.Sc. degree in 1946, and his Ph.D. in 1950. Later, for his joint work with Levitzki, he received the first Israel Prize in Exact Sciences. He worked at the Hebrew University until his retirement in 1989. Amitsur was a visiting scholar at the Institute for Advanced Study from 1952 to 1954. He was an Invited Speaker at the ICM in 1970 in Nice. He was a member of the Israel Academy of Sciences, where he was the Head for Experimental Science Section. He was one of the founding editors of the '' Isra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Amitsur–Levitzki Theorem
In algebra, the Amitsur–Levitzki theorem states that the algebra of ''n'' × ''n'' matrices over a commutative ring satisfies a certain identity of degree 2''n''. It was proved by . In particular matrix rings are polynomial identity rings such that the smallest identity they satisfy has degree exactly 2''n''. Statement The standard polynomial of degree ''n'' is :S_n(x_1,\dots,x_n) = \sum_\text(\sigma)x_ \cdots x_ in non-commuting variables ''x''1, ..., ''x''''n'', where the sum is taken over all ''n''! elements of the symmetric group ''S''''n''. The Amitsur–Levitzki theorem states that for ''n'' × ''n'' matrices ''A''1, ..., ''A''2''n'' whose entries are taken from a commutative ring then :S_(A_1,\dots,A_) = 0. Proofs gave the first proof. deduced the Amitsur–Levitzki theorem from the Koszul–Samelson theorem about primitive cohomology of Lie algebras. and gave a simple combinatorial proof as follows. By linearity it is enough to pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ron Aharoni
Ron Aharoni ( he, רון אהרוני ) (born 1952) is an Israeli mathematician, working in finite and infinite combinatorics. Aharoni is a professor at the Technion – Israel Institute of Technology, where he received his Ph.D. in mathematics in 1979. With Nash-Williams and Shelah he generalized Hall's marriage theorem by obtaining the right transfinite conditions for infinite bipartite graphs. He subsequently proved the appropriate versions of the Kőnig theorem and the Menger theorem for infinite graphs (the latter with Eli Berger). Aharoni is the author of several nonspecialist books; the most successful is '' Arithmetic for Parents'', a book helping parents and elementary school teachers in teaching basic mathematics. He also wrote a book on the connections between ''Mathematics, poetry and beauty'' and on philosophy, ''The Cat That is not There''. His book, "Man detaches meaning", is on a mechanism common to jokes and poetry. His last to date book iCircularity: A Common Se ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Menger Theorem
In the mathematical discipline of graph theory, Menger's theorem says that in a finite graph, the size of a minimum cut set is equal to the maximum number of disjoint paths that can be found between any pair of vertices. Proved by Karl Menger in 1927, it characterizes the connectivity of a graph. It is generalized by the max-flow min-cut theorem, which is a weighted, edge version, and which in turn is a special case of the strong duality theorem for linear programs. Edge connectivity The edge-connectivity version of Menger's theorem is as follows: :Let ''G'' be a finite undirected graph and ''x'' and ''y'' two distinct vertices. Then the size of the minimum edge cut for ''x'' and ''y'' (the minimum number of edges whose removal disconnects ''x'' and ''y'') is equal to the maximum number of pairwise edge-independent paths from ''x'' to ''y''. :Extended to all pairs: a graph is ''k''-edge-connected (it remains connected after removing fewer than ''k'' edges) if and only if e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kőnig's Theorem (graph Theory)
In the mathematical area of graph theory, Kőnig's theorem, proved by , describes an equivalence between the maximum matching problem and the minimum vertex cover problem in bipartite graphs. It was discovered independently, also in 1931, by Jenő Egerváry in the more general case of weighted graphs. Setting A vertex cover in a graph is a set of vertices that includes at least one endpoint of every edge, and a vertex cover is ''minimum'' if no other vertex cover has fewer vertices. A matching in a graph is a set of edges no two of which share an endpoint, and a matching is ''maximum'' if no other matching has more edges. It is obvious from the definition that any vertex-cover set must be at least as large as any matching set (since for every edge in the matching, at least one vertex is needed in the cover). In particular, the minimum vertex cover set is at least as large as the maximum matching set. Kőnig's theorem states that, in any bipartite graph, the minimum vertex c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hall's Marriage Theorem
In mathematics, Hall's marriage theorem, proved by , is a theorem with two equivalent formulations: * The combinatorial formulation deals with a collection of finite sets. It gives a necessary and sufficient condition for being able to select a distinct element from each set. * The graph theoretic formulation deals with a bipartite graph. It gives a necessary and sufficient condition for finding a matching that covers at least one side of the graph. Combinatorial formulation Statement Let \mathcal F be a family of finite sets. Here, \mathcal F is itself allowed to be infinite (although the sets in it are not) and to contain the same set multiple times. Let X be the union of all the sets in \mathcal F, the set of elements that belong to at least one of its sets. A transversal for F is a subset of X that can be obtained by choosing a distinct element from each set in \mathcal F. This concept can be formalized by defining a transversal to be the image of an injective function f: ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Joseph Bernstein
Joseph Bernstein (sometimes spelled I. N. Bernshtein; he, יוס(י)ף נאומוביץ ברנשטיין; russian: Иосиф Наумович Бернштейн; born 18 April 1945) is a Soviet-born Israeli mathematician working at Tel Aviv University. He works in algebraic geometry, representation theory, and number theory. Biography Bernstein received his Ph.D. in 1972 under Israel Gelfand at Moscow State University. In 1981, he emigrated to the United States due to growing anti-semitism in the Soviet Union. Bernstein was a professor at Harvard during 1983-1993. He was a visiting scholar at the Institute for Advanced Study in 1985-86 and again in 1997-98. In 1993, he moved to Israel to take a professorship at Tel Aviv University (emeritus since 2014). Awards and honors Bernstein received a gold medal at the 1962 International Mathematical Olympiad. He was elected to the Israel Academy of Sciences and Humanities in 2002 and was elected to the United States National Academ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bernstein–Sato Polynomial
In mathematics, the Bernstein–Sato polynomial is a polynomial related to differential operators, introduced independently by and , . It is also known as the b-function, the b-polynomial, and the Bernstein polynomial, though it is not related to the Bernstein polynomials used in approximation theory. It has applications to singularity theory, monodromy theory, and quantum field theory. gives an elementary introduction, while and give more advanced accounts. Definition and properties If f(x) is a polynomial in several variables, then there is a non-zero polynomial b(s) and a differential operator P(s) with polynomial coefficients such that :P(s)f(x)^ = b(s)f(x)^s. The Bernstein–Sato polynomial is the monic polynomial of smallest degree amongst such polynomials b(s). Its existence can be shown using the notion of holonomic D-modules. proved that all roots of the Bernstein–Sato polynomial are negative rational numbers. The Bernstein–Sato polynomial can also be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Avi Wigderson
Avi Wigderson ( he, אבי ויגדרזון; born 9 September 1956) is an Israeli mathematician and computer scientist. He is the Herbert H. Maass Professor in the school of mathematics at the Institute for Advanced Study in Princeton, New Jersey, United States of America. His research interests include complexity theory, parallel algorithms, graph theory, cryptography, distributed computing, and neural networks. Wigderson received the Abel Prize in 2021 for his work in theoretical computer science. Biography Avi Wigderson was born in Haifa, Israel, to Holocaust survivors. Wigderson is a graduate of the Hebrew Reali School in Haifa, and did his undergraduate studies at the Technion in Haifa, Israel, graduating in 1980, and went on to graduate study at Princeton University. He received his PhD in computer science in 1983 after completing a doctoral dissertation, titled "Studies in computational complexity", under the supervision of Richard Lipton. After short-term positions at t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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