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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)^, 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, then the conjecture states : \lim_ \frac= \int\limits_\alpha^\beta 1-\left(\frac\right)^2 \mathrmu and where each \gamma, \gamma' is the imaginary part of the non-trivial zeros of Riemann zeta function, that is \tfrac+i\gamma. Explanation Informally, this means that the chance of finding a zero in a very short interval of length 2π''L''/log(''T'') at a distance 2π''u''/log(''T'') from a zero 1/2+''iT'' is about ''L'' times the expression above. (The factor 2π/log(''T'') is a normalization factor that can be thought of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hugh Montgomery At Oberwolfach 2008
Hugh is the English-language variant of the masculine given name , itself the Old French variant of '' Hugo (name)">Hugo'', a short form of Continental Germanic Germanic name">given names beginning in the element "mind, spirit" (Old English ). The Germanic name is on record beginning in the 8th century, in variants ''Chugo, Hugo, Huc, Ucho, Ugu, Uogo, Ogo, Ougo,'' etc. The name's popularity in the Middle Ages ultimately derives from its use by Frankish nobility, beginning with Duke of the Franks and Count of Paris Hugh the Great (898–956). The Old French form was adopted into English from the Norman period (e.g. Hugh of Montgomery, 2nd Earl of Shrewsbury d. 1098; Hugh d'Avranches, 1st Earl of Chester, d. 1101). The spelling ''Hugh'' in English is from the Picard variant spelling '' Hughes'', where the orthography ''-gh-'' takes the role of ''-gu-'' in standard French, i.e. to express the phoneme /g/ as opposed to the affricate /ʒ/ taken by the grapheme ''g'' before front ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fourier Transform
In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the transform is a complex-valued function of frequency. The term ''Fourier transform'' refers to both this complex-valued function and the mathematical operation. When a distinction needs to be made, the output of the operation is sometimes called the frequency domain representation of the original function. The Fourier transform is analogous to decomposing the sound of a musical chord into the intensities of its constituent pitches. Functions that are localized in the time domain have Fourier transforms that are spread out across the frequency domain and vice versa, a phenomenon known as the uncertainty principle. The critical case for this principle is the Gaussian function, of substantial importance in probability theory and statist ... [...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 or Fermat's conjecture (now a theorem, proven in 1995 by Andrew Wiles), have shaped much of mathematical history as new areas of mathematics are developed in order to prove them. Resolution of conjectures Proof Formal mathematics is based on ''provable'' truth. In mathematics, any number of cases supporting a universally quantified conjecture, no matter how large, is insufficient for establishing the conjecture's veracity, since a single counterexample could immediately bring down the conjecture. Mathematical journals sometimes publish the minor results of research teams having extended the search for a counterexample farther than previously done. For instance, the Collatz conjecture, which concerns whether or not certain sequences of integers terminate, has been tested for all integers up to 1.2 × 1012 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zeta And L-functions
Zeta (, ; uppercase Ζ, lowercase ζ; , , 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 Romance 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 400 BC is disputed. See Ancient Greek phonolog ... [...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 type of journal ranking. Journals with higher impact factor values are considered more prestigious or important within their field. The Impact Factor of a journa ... of 2.194, ranking it in the top ten mathematics journals in the world. References External links * Mathematics journals Mathematical Journal Academic journals established in 1935 Multilingual journals English-language journals ... [...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 of 2.417. References External links * Delayed open access journals English-language journals Mathematics journals Academic journals established in 1943 American Mathematical Society academic journals Bimonthly journals {{math-journal-stub ... [...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 became the first president while Fiske became secretary. The society soon decided to publish a journal, but ran into some resistance over 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 influentia ... [...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 The de Bruijn–Newman constant, denoted by \Lambda and named after Nicolaas Govert de Bruijn and Charles Michael Newman, is a mathematical constant defined via the zeros of a certain function H(\lambda,z), where \lambda is a real parameter ... is non-negative, a fact that has be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arnold Schönhage
Arnold Schönhage (born 1 December 1934 in Lockhausen, now Bad Salzuflen) is a German mathematician and computer scientist. Schönhage was professor at the Rheinische Friedrich-Wilhelms-Universität, Bonn, and also in Tübingen and Konstanz. Together with Volker Strassen, he developed the Schönhage–Strassen algorithm for the multiplication of large numbers that has a runtime of '' O''(''N'' log ''N'' log log ''N''). For many years, this was the fastest way to multiply large integers, although Schönhage and Strassen predicted that an algorithm with a run-time of N(logN) should exist. In 2019, Joris van der Hoeven and David Harvey finally developed an algorithm with this runtime, proving that Schönhage's and Strassen's prediction had been correct. Schönhage designed and implemented together with Andreas F. W. Grotefeld and Ekkehart Vetter a multitape Turing machine, called TP, in software. The machine is programmed in TPAL, an assembler lan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cray X-MP
The Cray X-MP was a supercomputer designed, built and sold by Cray, 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 (computer engineer), Steve Chen. Description The X-MP's main improvement over the Cray-1 was that it was a shared-memory Parallel computing, 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 junction transistor, bipolar gate-array integrated circuits containing 16 emitter-coupled logic Logic gate, gates each. The CPU was very similar to the Cray-1 CPU in architecture, but had better memory bandwidth (with two read por ... [...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 of its entries are sampled randomly from a probability distribution. Random matrix theory (RMT) is the study of properties of random matrices, often as they become large. RMT provides techniques like mean-field theory, diagrammatic methods, the cavity method, or the replica method to compute quantities like traces, spectral densities, or scalar products between eigenvectors. Many physical phenomena, such as the spectrum of nuclei of heavy atoms, the thermal conductivity of a lattice, or the emergence of quantum chaos, can be modeled mathematically as problems concerning large, random matrices. 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
