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Montgomery's Pair Correlation Conjecture
In mathematics, Montgomery's pair correlation conjecture is a conjecture made by that the pair correlation between pairs of zeros of the Riemann zeta function (normalized to have unit average spacing) is :1-\left(\frac\right)^ + \delta(u), which, as Freeman Dyson pointed out to him, is the same as the pair correlation function of random Hermitian matrices. Conjecture ''Under the assumption that the Riemann Hypothesis is true.'' Let \alpha\leq \beta be fixed, as T\to \infty'' : N(T;\alpha,\beta)\, := \, \sum_ 1 \, \sim \, \left( \int\limits_\alpha^\beta \left(1-\left(\frac\right)^2 \right) \mathrmu +\delta_( alpha,\beta\right)\frac\log\frac and we count over A = \, where each \gamma, \gamma' is the imaginary part of the non-trivial zeros of zeta function, that is \tfrac+i\gamma. Also \delta_0 denotes the delta measure supported at 0. Explanation Informally, this means that the chance of finding a zero in a very short interval of length 2π''L''/log(''T'') at a distanc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hugh Montgomery At Oberwolfach 2008
Hugh may refer to: *Hugh (given name) Noblemen and clergy French * Hugh the Great (died 956), Duke of the Franks * Hugh Magnus of France (1007–1025), co-King of France under his father, Robert II * Hugh, Duke of Alsace (died 895), modern-day France * Hugh of Austrasia (7th century), Mayor of the Palace of Austrasia * Hugh I, Count of Angoulême (1183–1249) * Hugh II, Count of Angoulême (1221–1250) * Hugh III, Count of Angoulême (13th century) * Hugh IV, Count of Angoulême (1259–1303) * Hugh, Bishop of Avranches (11th century), France * Hugh I, Count of Blois (died 1248) * Hugh II, Count of Blois (died 1307) * Hugh of Brienne (1240–1296), Count of the medieval French County of Brienne * Hugh, Duke of Burgundy (d. 952) * Hugh I, Duke of Burgundy (1057–1093) * Hugh II, Duke of Burgundy (1084–1143) * Hugh III, Duke of Burgundy (1142–1192) * Hugh IV, Duke of Burgundy (1213–1272) * Hugh V, Duke of Burgundy (1294–1315) * Hugh Capet (939–996), King of France * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Operator
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a Map (mathematics), mapping V \to W between two vector spaces that preserves the operations of vector addition and scalar multiplication. The same names and the same definition are also used for the more general case of module (mathematics), modules over a ring (mathematics), ring; see Module homomorphism. If a linear map is a bijection then it is called a . In the case where V = W, a linear map is called a (linear) ''endomorphism''. Sometimes the term refers to this case, but the term "linear operator" can have different meanings for different conventions: for example, it can be used to emphasize that V and W are Real number, real vector spaces (not necessarily with V = W), or it can be used to emphasize that V is a function space, which is a common convention in functional analysis. Some ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conjectures
In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 1995 by Andrew Wiles), have shaped much of mathematical history as new areas of mathematics are developed in order to prove them. Important examples Fermat's Last Theorem In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers a, ''b'', and ''c'' can satisfy the equation ''a^n + b^n = c^n'' for any integer value of ''n'' greater than two. This theorem was first conjectured by Pierre de Fermat in 1637 in the margin of a copy of '' Arithmetica'', where he claimed that he had a proof that was too large to fit in the margin. The first successful proof was released in 1994 by Andrew Wiles, and formally published in 1995, after 358 years of effort by mat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zeta And L-functions
Zeta (, ; uppercase Ζ, lowercase ζ; grc, ζῆτα, el, ζήτα, label= Demotic Greek, classical or ''zē̂ta''; ''zíta'') is the sixth letter of the Greek alphabet. In the system of Greek numerals, it has a value of 7. It was derived from the Phoenician letter zayin . Letters that arose from zeta include the Roman Z and Cyrillic З. Name Unlike the other Greek letters, this letter did not take its name from the Phoenician letter from which it was derived; it was given a new name on the pattern of beta, eta and theta. The word ''zeta'' is the ancestor of ''zed'', the name of the Latin letter Z in Commonwealth English. Swedish and many Romanic languages (such as Italian and Spanish) do not distinguish between the Greek and Roman forms of the letter; "''zeta''" is used to refer to the Roman letter Z as well as the Greek letter. Uses Letter The letter ζ represents the voiced alveolar fricative in Modern Greek. The sound represented by zeta in Greek before ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Duke Mathematical Journal
''Duke Mathematical Journal'' is a peer-reviewed mathematics journal published by Duke University Press. It was established in 1935. The founding editors-in-chief were David Widder, Arthur Coble, and Joseph Miller Thomas. The first issue included a paper by Solomon Lefschetz. Leonard Carlitz served on the editorial board for 35 years, from 1938 to 1973. The current managing editor is Richard Hain (Duke University). Impact According to the journal homepage, the journal has a 2018 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 ... of 2.194, ranking it in the top ten mathematics journals in the world. References External links * Mathematics journals Mathematical Journal Publications established in 1935 Multilingual journals English-language journals French- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics Of Computation
''Mathematics of Computation'' is a bimonthly mathematics journal focused on computational mathematics. It was established in 1943 as ''Mathematical Tables and other Aids to Computation'', obtaining its current name in 1960. Articles older than five years are available electronically free of charge. Abstracting and indexing The journal is abstracted and indexed in Mathematical Reviews, Zentralblatt MATH, Science Citation Index, CompuMath Citation Index, and Current Contents/Physical, Chemical & Earth Sciences. According to the ''Journal Citation Reports'', the journal has a 2020 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 ... of 2.417. References External links * Delayed open access journals English-language journals Mathematics journals Publications ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs. The society is one of the four parts of the Joint Policy Board for Mathematics and a member of the Conference Board of the Mathematical Sciences. History The AMS was founded in 1888 as the New York Mathematical Society, the brainchild of Thomas Fiske, who was impressed by the London Mathematical Society on a visit to England. John Howard Van Amringe was the first president and Fiske became secretary. The society soon decided to publish a journal, but ran into some resistance, due to concerns about competing with the American Journal of Mathematics. The result was the '' Bulletin of the American Mathematical Society'', with Fiske as editor-in-chief. The de facto journal, as intended, was influential i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lehmer Pair
In the study of the Riemann hypothesis, a Lehmer pair is a pair of zeros of the Riemann zeta function that are unusually close to each other. They are named after Derrick Henry Lehmer, who discovered the pair of zeros : \begin & \tfrac 1 2 + i\,7005.06266\dots \\ pt& \tfrac 1 2 + i\,7005.10056\dots \end (the 6709th and 6710th zeros of the zeta function). More precisely, a Lehmer pair can be defined as having the property that their complex coordinates \gamma_n and \gamma_ obey the inequality :\frac \ge C\sum_ \left(\frac+\frac\right) for a constant C>5/4. It is an unsolved problem whether there exist infinitely many Lehmer pairs. If so, it would imply that the De Bruijn–Newman constant is non-negative, a fact that has been proven unconditionally by Brad Rodgers and Terence Tao. See also *Montgomery's pair correlation conjecture References {{reflist, refs= {{citation , last1 = Csordas , first1 = George , last2 = Smith , first2 = Wayne , last3 = Varga , firs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cray X-MP
The Cray X-MP was a supercomputer designed, built and sold by Cray Research. It was announced in 1982 as the "cleaned up" successor to the 1975 Cray-1, and was the world's fastest computer from 1983 to 1985 with a quad-processor system performance of 800 MFLOPS. The principal designer was Steve Chen. Description The X-MP's main improvement over the Cray-1 was that it was a shared-memory parallel vector processor, the first such computer from Cray Research. It housed up to four CPUs in a mainframe that was nearly identical in outside appearance to the Cray-1. The X-MP CPU had a faster 9.5 nanosecond clock cycle (105 MHz), compared to 12.5 ns for the Cray-1A. It was built from bipolar gate-array integrated circuits containing 16 emitter-coupled logic gates each. The CPU was very similar to the Cray-1 CPU in architecture, but had better memory bandwidth (with two read ports and one write port to the main memory instead of only one read/write port) and improved ch ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Random Matrix
In probability theory and mathematical physics, a random matrix is a matrix-valued random variable—that is, a matrix in which some or all elements are random variables. Many important properties of physical systems can be represented mathematically as matrix problems. For example, the thermal conductivity of a lattice can be computed from the dynamical matrix of the particle-particle interactions within the lattice. Applications Physics In nuclear physics, random matrices were introduced by Eugene Wigner to model the nuclei of heavy atoms. Wigner postulated that the spacings between the lines in the spectrum of a heavy atom nucleus should resemble the spacings between the eigenvalues of a random matrix, and should depend only on the symmetry class of the underlying evolution. In solid-state physics, random matrices model the behaviour of large disordered Hamiltonians in the mean-field approximation. In quantum chaos, the Bohigas–Giannoni–Schmit (BGS) conjecture asse ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
