|
Miklós Ajtai
Miklós Ajtai (born 2 July 1946) is a computer scientist at the IBM Almaden Research Center, United States. In 2003, he received the Knuth Prize for his numerous contributions to the field, including a classic sorting network algorithm (developed jointly with J. Komlós and Endre Szemerédi), exponential lower bounds, superlinear time-space tradeoffs for branching programs, and other "unique and spectacular" results. He is a member of the U.S. National Academy of Sciences. Selected results One of Ajtai's results states that the length of proofs in propositional logic of the pigeonhole principle for ''n'' items grows faster than any polynomial in ''n''. He also proved that the statement "any two countable structures that are second-order equivalent are also isomorphic" is both consistent with and independent of ZFC. Ajtai and Szemerédi proved the corners theorem, an important step toward higher-dimensional generalizations of the Szemerédi theorem. With Komlós and Szemerà ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Budapest
Budapest (, ; ) is the capital and most populous city of Hungary. It is the ninth-largest city in the European Union by population within city limits and the second-largest city on the Danube river; the city has an estimated population of 1,752,286 over a land area of about . Budapest, which is both a city and county, forms the centre of the Budapest metropolitan area, which has an area of and a population of 3,303,786; it is a primate city, constituting 33% of the population of Hungary. The history of Budapest began when an early Celtic settlement transformed into the Roman town of Aquincum, the capital of Lower Pannonia. The Hungarians arrived in the territory in the late 9th century, but the area was pillaged by the Mongols in 1241–42. Re-established Buda became one of the centres of Renaissance humanist culture by the 15th century. The Battle of Mohács, in 1526, was followed by nearly 150 years of Ottoman rule. After the reconquest of Buda in 1686, the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Structure (mathematical Logic)
In universal algebra and in model theory, a structure consists of a set along with a collection of finitary operations and relations that are defined on it. Universal algebra studies structures that generalize the algebraic structures such as groups, rings, fields and vector spaces. The term universal algebra is used for structures with no relation symbols. Model theory has a different scope that encompasses more arbitrary theories, including foundational structures such as models of set theory. From the model-theoretic point of view, structures are the objects used to define the semantics of first-order logic. For a given theory in model theory, a structure is called a model if it satisfies the defining axioms of that theory, although it is sometimes disambiguated as a ''semantic model'' when one discusses the notion in the more general setting of mathematical models. Logicians sometimes refer to structures as " interpretations", whereas the term "interpretation" generally has ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cynthia Dwork
Cynthia Dwork (born June 27, 1958) is an American computer scientist at Harvard University, where she is Gordon McKay Professor of Computer Science, Radcliffe Alumnae Professor at the Radcliffe Institute for Advanced Study, and Affiliated Professor, Harvard Law School and Harvard's Department of Statistics. Dwork was elected a member of the National Academy of Engineering in 2008 for fundamental contributions to distributed algorithms and the security of cryptosystems. She is a distinguished scientist at Microsoft Research. Early life and education Dwork received her B.S.E. from Princeton University in 1979, graduating Cum Laude, and receiving the Charles Ira Young Award for Excellence in Independent Research. Dwork received her Ph.D. from Cornell University in 1983 for research supervised by John Hopcroft. Career and research Dwork is known for her research placing privacy-preserving data analysis on a mathematically rigorous foundation, including the co-invention of differenti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Crossing Number (graph Theory)
In graph theory, the crossing number of a graph is the lowest number of edge crossings of a plane drawing of the graph . For instance, a graph is planar if and only if its crossing number is zero. Determining the crossing number continues to be of great importance in graph drawing, as user studies have shown that drawing graphs with few crossings makes it easier for people to understand the drawing. The study of crossing numbers originated in Turán's brick factory problem, in which Pál Turán asked for a factory plan that minimized the number of crossings between tracks connecting brick kilns to storage sites. Mathematically, this problem can be formalized as asking for the crossing number of a complete bipartite graph. The same problem arose independently in sociology at approximately the same time, in connection with the construction of sociograms. Turán's conjectured formula for the crossing numbers of complete bipartite graphs remains unproven, as does an analogous formu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Crossing Number Inequality
In the mathematics of graph drawing, the crossing number inequality or crossing lemma gives a lower bound on the Crossing number (graph theory), minimum number of crossings of a given Graph (discrete mathematics), graph, as a function of the number of Edge (graph theory), edges and Vertex (graph theory), vertices of the graph. It states that, for graphs where the number of edges is sufficiently larger than the number of vertices, the crossing number is at least Proportionality (mathematics), proportional to . It has applications in VLSI design and combinatorial geometry, and was discovered independently by Miklós Ajtai, Ajtai, Václav Chvátal, Chvátal, Monty Newborn, Newborn, and Endre Szemerédi, Szemerédi and by F. Thomson Leighton, Leighton.. Statement and history The crossing number inequality states that, for an undirected simple graph with vertices and edges such that , the Crossing number (graph theory), crossing number obeys the Inequality (mathematics), inequality ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monty Newborn
Monroe "Monty" Newborn (born May 21, 1938), former chairman of the Computer Chess Committee of the Association for Computing Machinery, is a professor emeritus of computer science at McGill University in Montreal (formerly professor of electrical engineering at Columbia University). He briefly served as president of the International Computer Chess Association and co-wrote a computer chess program named Ostrich In the 1970's. Biography Monty Newborn received his Ph.D. in Electrical Engineering from The Ohio State University The Ohio State University, commonly called Ohio State or OSU, is a public land-grant research university in Columbus, Ohio. A member of the University System of Ohio, it has been ranked by major institutional rankings among the best publ ... in 1967. He was an assistant professor and associate professor at Columbia University in the Department of Electrical Engineering and Computer Science from 1967 to 1975. In 1975, he joined the School of Comput ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Václav Chvátal
Václav (VaÅ¡ek) Chvátal () is a Professor Emeritus in the Department of Computer Science and Software Engineering at Concordia University in Montreal, Quebec, Canada and a Visiting Professor at Charles University in Prague. He has published extensively on topics in graph theory, combinatorics, and combinatorial optimization. Biography Chvátal was born in 1946 in Prague and educated in mathematics at Charles University in Prague, where he studied under the supervision of ZdenÄ›k HedrlÃn. He fled Czechoslovakia in 1968, three days after the Soviet invasion, and completed his Ph.D. in Mathematics at the University of Waterloo, under the supervision of Crispin St. J. A. Nash-Williams, in the fall of 1970. Subsequently, he took positions at McGill University (1971 and 1978-1986), Stanford University (1972 and 1974-1977), the Université de Montréal (1972-1974 and 1977-1978), and Rutgers University (1986-2004) before returning to Montreal for the Canada Research Chair in Combi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fulkerson Prize
The Fulkerson Prize for outstanding papers in the area of discrete mathematics is sponsored jointly by the Mathematical Optimization Society (MOS) and the American Mathematical Society (AMS). Up to three awards of $1,500 each are presented at each (triennial) International Symposium of the MOS. Originally, the prizes were paid out of a memorial fund administered by the AMS that was established by friends of the late Delbert Ray Fulkerson to encourage mathematical excellence in the fields of research exemplified by his work. The prizes are now funded by an endowment administered by MPS. Winners SourceMathematical Optimization Society* 1979: ** Richard M. Karp for classifying many important NP-complete problems. ** Kenneth Appel and Wolfgang Haken for the four color theorem. ** Paul Seymour for generalizing the max-flow min-cut theorem to matroids. * 1982: ** D.B. Judin, Arkadi Nemirovski, Leonid Khachiyan, Martin Grötschel, László Lovász and Alexander Schrijver for the e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jeong Han Kim
Jeong Han Kim (; born July 20, 1962) is a South Korean mathematician. He studied physics and mathematical physics at Yonsei University, and earned his Ph.D. in mathematics at Rutgers University. He was a researcher at AT&T Bell Labs and at Microsoft Research, and was Underwood Chair Professor of Mathematics at Yonsei University. He is currently a Professor of the School of Computational Sciences at the Korea Institute for Advanced Study. His main research fields are combinatorics and computational mathematics. His best known contribution to the field is his proof that the Ramsey number R(3,t) has asymptotic order of magnitude t2/log t. He received the Fulkerson Prize in 1997 for his contributions to Ramsey theory. In 2008, he became president of the National Institute for Mathematical Sciences of South Korea and was also awarded the Kyung-Ahm Prize The Kyung-Ahm Prize is a series of awards presented annually from the Kyung-Ahm Education & Cultural Foundation. Founded in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ramsey's Theorem
In combinatorics, Ramsey's theorem, in one of its graph-theoretic forms, states that one will find monochromatic cliques in any edge labelling (with colours) of a sufficiently large complete graph. To demonstrate the theorem for two colours (say, blue and red), let and be any two positive integers. Ramsey's theorem states that there exists a least positive integer for which every blue-red edge colouring of the complete graph on vertices contains a blue clique on vertices or a red clique on vertices. (Here signifies an integer that depends on both and .) Ramsey's theorem is a foundational result in combinatorics. The first version of this result was proved by F. P. Ramsey. This initiated the combinatorial theory now called Ramsey theory, that seeks regularity amid disorder: general conditions for the existence of substructures with regular properties. In this application it is a question of the existence of ''monochromatic subsets'', that is, subsets of connected edges of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Szemerédi's Theorem
In arithmetic combinatorics, Szemerédi's theorem is a result concerning arithmetic progressions in subsets of the integers. In 1936, Erdős and Turán conjectured that every set of integers ''A'' with positive natural density contains a ''k''-term arithmetic progression for every ''k''. Endre Szemerédi proved the conjecture in 1975. Statement A subset ''A'' of the natural numbers is said to have positive upper density if :\limsup_\frac > 0. Szemerédi's theorem asserts that a subset of the natural numbers with positive upper density contains infinitely many arithmetic progressions of length ''k'' for all positive integers ''k''. An often-used equivalent finitary version of the theorem states that for every positive integer ''k'' and real number \delta \in (0, 1], there exists a positive integer :N = N(k,\delta) such that every subset of of size at least δ''N'' contains an arithmetic progression of length ''k''. Another formulation uses the function ''r''''k''(''N''), the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Corners Theorem
In arithmetic combinatorics, the corners theorem states that for every \varepsilon>0, for large enough N, any set of at least \varepsilon N^2 points in the N\times N grid \^2 contains a corner, i.e., a triple of points of the form \ with h\ne 0. It was first proved by Miklós Ajtai and Endre Szemerédi in 1974 using Szemerédi's theorem.. In 2003, József Solymosi gave a short proof using the triangle removal lemma. Statement Define a corner to be a subset of \mathbb^2 of the form \, where x,y,h\in \mathbb and h\ne 0. For every \varepsilon>0, there exists a positive integer N(\varepsilon) such that for any N\ge N(\varepsilon), any subset A\subseteq\^2 with size at least \varepsilon N^2 contains a corner. The condition h\ne 0 can be relaxed to h>0 by showing that if A is dense, then it has some dense subset that is centrally symmetric. Proof overview What follows is a sketch of Solymosi's argument. Suppose A\subset\^2 is corner-free. Construct an auxiliary tripartite graph G w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
