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Pál Turán
Pál Turán (; 18 August 1910 – 26 September 1976) also known as Paul Turán, was a Hungarian mathematician who worked primarily in extremal combinatorics. He had a long collaboration with fellow Hungarian mathematician Paul Erdős, lasting 46 years and resulting in 28 joint papers. Life and education Turán was born into a Jewish family in Budapest on 18 August 1910.At the same period of time, Turán and Erdős were famous answerers in the journal '' KöMaL''. He received a teaching degree at the University of Budapest in 1933 and the PhD degree under Lipót Fejér in 1935 at Eötvös Loránd University. As a Jew, he fell victim to numerus clausus, and could not get a university job for several years. He was sent to labour service at various times from 1940-44. He is said to have been recognized and perhaps protected by a fascist guard, who, as a mathematics student, had admired Turán's work. Turán became associate professor at the University of Budapest in 1945 and ful ... [...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] |
Numerus Clausus
''Numerus clausus'' ("closed number" in Latin) is one of many methods used to limit the number of students who may study at a university. In many cases, the goal of the ''numerus clausus'' is simply to limit the number of students to the maximum feasible in some particularly sought-after areas of studies with an intent to keep a constant supply of qualified workforce and thus limit competition. In historical terms however, in some countries, ''numerus clausus'' policies were religious or racial quotas, both in intent and function. Historical use Countries legislating limitations on the admission of Jewish students, at various times, have included: Austria, Canada, Hungary, Imperial Russia, Iraq, Latvia (from 1934 under Kārlis Ulmanis regime), Netherlands, Poland, Romania, United States, Vichy France, and Yugoslavia among others. Selected examples Germany A whole series of ''numerus clausus'' resolutions were adopted in 1929 on the grounds of race and place of origin, not re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probabilistic Number Theory
In mathematics, Probabilistic number theory is a subfield of number theory, which explicitly uses probability to answer questions about the integers and integer-valued functions. One basic idea underlying it is that different prime numbers are, in some serious sense, like independent random variables. This however is not an idea that has a unique useful formal expression. The founders of the theory were Paul Erdős, Aurel Wintner and Mark Kac during the 1930s, one of the periods of investigation in analytic number theory. Foundational results include the Erdős–Wintner theorem and the Erdős–Kac theorem on additive functions. See also *Number theory *Analytic number theory *Areas of mathematics *List of number theory topics *List of probability topics *Probabilistic method *Probable prime In number theory, a probable prime (PRP) is an integer that satisfies a specific condition that is satisfied by all prime numbers, but which is not satisfied by most composite numbers. Di ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gábor Halász (mathematician)
Gábor Halász (born 25 December 1941, in Budapest) is a Hungarian mathematician. He specialised in number theory and mathematical analysis, especially in analytic number theory. He is a member of the Hungarian Academy of Sciences. Since 1985, he is professor at the Faculty of Sciences of the Eötvös Loránd University, Budapest. With Pál Turán, Halász proved zero density results on the roots of the Riemann zeta function. He co-invented the Halász–Montgomery inequality with Hugh Montgomery. Awards, prizes *Alfréd Rényi Prize (1972) *Paul Erdős Prize (1976) *Tibor Szele Commemorative Medal of the János Bolyai Mathematical Society The János Bolyai Mathematical Society (Bolyai János Matematikai Társulat, BJMT) is the Hungarian mathematical society, named after János Bolyai, a 19th-century Hungarian mathematician, a co-discoverer of non-Euclidean geometry. It is the profes ... (1985) References Sources * Ki Kicsoda, 2006, MTI, Budapest. * Halász'homepageat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normal Order Of An Arithmetic Function
In number theory, a normal order of an arithmetic function is some simpler or better-understood function which "usually" takes the same or closely approximate values. Let ''f'' be a function on the natural numbers. We say that ''g'' is a normal order of ''f'' if for every ''ε'' > 0, the inequalities : (1-\varepsilon) g(n) \le f(n) \le (1+\varepsilon) g(n) hold for ''almost all'' ''n'': that is, if the proportion of ''n'' ≤ ''x'' for which this does not hold tends to 0 as ''x'' tends to infinity. It is conventional to assume that the approximating function ''g'' is continuous and monotone. Examples * The Hardy–Ramanujan theorem: the normal order of ω(''n''), the number of distinct prime factors of ''n'', is log(log(''n'')); * The normal order of Ω(''n''), the number of prime factors of ''n'' counted with multiplicity, is log(log(''n'')); * The normal order of log(''d''(''n'')), where ''d''(''n'') is the number of divisors of ''n'', is log(2)&nb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hardy–Ramanujan Theorem
In mathematics, the Hardy–Ramanujan theorem, proved by , states that the normal order of the number ω(''n'') of distinct prime factors of a number ''n'' is log(log(''n'')). Roughly speaking, this means that most numbers have about this number of distinct prime factors. Precise statement A more precise version states that for every real-valued function ''ψ''(''n'') that tends to infinity as ''n'' tends to infinity :, \omega(n)-\log\log n, <\psi(n)\sqrt or more traditionally : for '''' (all but an infinitesimal proportion of) integers. That is, let ''g''(''x'') be the number of positive integers ''n'' less than ''x'' for which the above inequality fails: then ''g''(''x'')/''x'' converges to zero as ''x'' goes to infinity. History A simple proof to the result was given by |
Turán Sieve
In number theory, the Turán sieve is a technique for estimating the size of "sifted sets" of positive integers which satisfy a set of conditions which are expressed by congruences. It was developed by Pál Turán in 1934. Description In terms of sieve theory the Turán sieve is of ''combinatorial type'': deriving from a rudimentary form of the inclusion–exclusion principle. The result gives an ''upper bound'' for the size of the sifted set. Let ''A'' be a set of positive integers ≤ ''x'' and let ''P'' be a set of primes. For each ''p'' in ''P'', let ''A''''p'' denote the set of elements of ''A'' divisible by ''p'' and extend this to let ''A''''d'' be the intersection of the ''A''''p'' for ''p'' dividing ''d'', when ''d'' is a product of distinct primes from ''P''. Further let ''A''1 denote ''A'' itself. Let ''z'' be a positive real number and ''P''(''z'') denote the product of the primes in ''P'' which are ≤ ''z''. The object of the sieve is to estimate :S(A,P,z) = ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Theory
In mathematics, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are connected by '' edges'' (also called ''links'' or ''lines''). A distinction is made between undirected graphs, where edges link two vertices symmetrically, and directed graphs, where edges link two vertices asymmetrically. Graphs are one of the principal objects of study in discrete mathematics. Definitions Definitions in graph theory vary. The following are some of the more basic ways of defining graphs and related mathematical structures. Graph In one restricted but very common sense of the term, a graph is an ordered pair G=(V,E) comprising: * V, a set of vertices (also called nodes or points); * E \subseteq \, a set of edges (also called links or lines), which are unordered pairs of vertices (that is, an edge is associated with t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (mathematics), series, and analytic functions. These theories are usually studied in the context of Real number, real and Complex number, complex numbers and Function (mathematics), 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 (mathematics), 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 i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Number Theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics."German original: "Die Mathematik ist die Königin der Wissenschaften, und die Arithmetik ist die Königin der Mathematik." Number theorists study prime numbers as well as the properties of mathematical objects made out of integers (for example, rational numbers) or defined as generalizations of the integers (for example, algebraic integers). Integers can be considered either in themselves or as solutions to equations (Diophantine geometry). Questions in number theory are often best understood through the study of Complex analysis, analytical objects (for example, the Riemann zeta function) that encode properties of the integers, primes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Leukemia
Leukemia ( also spelled leukaemia and pronounced ) is a group of blood cancers that usually begin in the bone marrow and result in high numbers of abnormal blood cells. These blood cells are not fully developed and are called ''blasts'' or ''leukemia cells''. Symptoms may include bleeding and bruising, bone pain, fatigue, fever, and an increased risk of infections. These symptoms occur due to a lack of normal blood cells. Diagnosis is typically made by blood tests or bone marrow biopsy. The exact cause of leukemia is unknown. A combination of genetic factors and environmental (non-inherited) factors are believed to play a role. Risk factors include smoking, ionizing radiation, petrochemicals (such as benzene), prior chemotherapy, and Down syndrome. People with a family history of leukemia are also at higher risk. There are four main types of leukemia— acute lymphoblastic leukemia (ALL), acute myeloid leukemia (AML), chronic lymphocytic leukemia (CLL) and chronic myeloi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
