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Turing's Method
In mathematics, Turing's method is used to verify that for any given Gram point there lie ''m'' + 1 zeros of , in the region , where is the Riemann zeta function. It was discovered by Alan Turing Alan Mathison Turing (; 23 June 1912 – 7 June 1954) was an English mathematician, computer scientist, logician, cryptanalyst, philosopher and theoretical biologist. He was highly influential in the development of theoretical computer ... and published in 1953, although that proof contained errors and a correction was published in 1970 by R. Sherman Lehman. For every integer ''i'' with we find a list of Gram points \ and a complementary list \ , where is the smallest number such that : (-1)^i Z(g_i + h_i) > 0, where ''Z''(''t'') is the Hardy Z function. Note that may be negative or zero. Assuming that h_m = 0 and there exists some integer ''k'' such that h_k = 0 , then if : 1 + \frac -2, Then the bound is achieved and we have that there are ex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gram Point
The gram (originally gramme; SI unit symbol g) is a unit of mass in the International System of Units (SI) equal to one thousandth of a kilogram. Originally defined in 1795 as "the absolute weight of a volume of pure water equal to the cube of the hundredth part of a metre melting ice", the defining temperature (0 °C) was later changed to the temperature of maximum density of water (approximately 4 °C). Subsequent redefinitions agree with this original definition to within 30 parts per million (0.003%), with the maximum density of water remaining very close to 1 g/cm3, as shown by modern measurements. By the late 19th century, there was an effort to make the base unit the kilogram and the gram a derived unit. In 1960, the new International System of Units defined a ''gram'' as one thousandth of a kilogram (i.e., one gram is ). The kilogram, as of 2019, is defined by the International Bureau of Weights and Measures from the metre, the second, and from the f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemann Zeta Function
The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for and its analytic continuation elsewhere. The Riemann zeta function plays a pivotal role in analytic number theory and has applications in physics, probability theory, and applied statistics. Leonhard Euler first introduced and studied the function over the reals in the first half of the eighteenth century. Bernhard Riemann's 1859 article "On the Number of Primes Less Than a Given Magnitude" extended the Euler definition to a complex variable, proved its meromorphic continuation and functional equation, and established a relation between its zeros and the distribution of prime numbers. This paper also contained the Riemann hypothesis, a conjecture about the distribution of complex zeros of the Riemann zeta function that many mathematicians consider th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Academic Press
Academic Press (AP) is an academic book publisher founded in 1941. It launched a British division in the 1950s. Academic Press was acquired by Harcourt, Brace & World in 1969. Reed Elsevier said in 2000 it would buy Harcourt, a deal completed the next year, after a regulatory review. Thus, Academic Press is now an imprint of Elsevier. Academic Press publishes reference books, serials and online products in the subject areas of: * Communications engineering * Economics * Environmental science * Finance * Food science and nutrition * Geophysics * Life sciences * Mathematics and statistics * Neuroscience * Physical sciences * Psychology Psychology is the scientific study of mind and behavior. Its subject matter includes the behavior of humans and nonhumans, both consciousness, conscious and Unconscious mind, unconscious phenomena, and mental processes such as thoughts, feel ... Well-known products include the '' Methods in Enzymology'' series and encyclopedias such ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alan Turing
Alan Mathison Turing (; 23 June 1912 – 7 June 1954) was an English mathematician, computer scientist, logician, cryptanalyst, philosopher and theoretical biologist. He was highly influential in the development of theoretical computer science, providing a formalisation of the concepts of algorithm and computation with the Turing machine, which can be considered a model of a general-purpose computer. Turing is widely considered to be the father of theoretical computer science. Born in London, Turing was raised in southern England. He graduated from University of Cambridge, King's College, Cambridge, and in 1938, earned a doctorate degree from Princeton University. During World War II, Turing worked for the Government Code and Cypher School at Bletchley Park, Britain's codebreaking centre that produced Ultra (cryptography), Ultra intelligence. He led Hut 8, the section responsible for German naval cryptanalysis. Turing devised techniques for speeding the breaking of Germ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proceedings Of The London Mathematical Society
The London Mathematical Society (LMS) is one of the United Kingdom's learned societies for mathematics (the others being the Royal Statistical Society (RSS), the Institute of Mathematics and its Applications (IMA), the Edinburgh Mathematical Society and the Operational Research Society (ORS). History The Society was established on 16 January 1865, the first president being Augustus De Morgan. The earliest meetings were held in University College, but the Society soon moved into Burlington House, Piccadilly. The initial activities of the Society included talks and publication of a journal. The LMS was used as a model for the establishment of the American Mathematical Society in 1888. Mary Cartwright was the first woman to be President of the LMS (in 1961–62). The Society was granted a royal charter in 1965, a century after its foundation. In 1998 the Society moved from rooms in Burlington House into De Morgan House (named after the society's first president), at 57–5 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Z Function
In mathematics, the Z function is a function (mathematics), function used for studying the Riemann zeta function along the Riemann hypothesis, critical line where the argument is one-half. It is also called the Riemann–Siegel Z function, the Riemann–Siegel zeta function, the G._H._Hardy, Hardy function, the Hardy Z function and the Hardy–Littlewood zeta function conjectures, Hardy zeta function. It can be defined in terms of the Riemann–Siegel theta function and the Riemann zeta function by : Z(t) = e^ \zeta\left(\frac 1 2 + it\right). It follows from the functional equation of the Riemann zeta function that the Z function is real for real values of ''t''. It is an Even and odd functions, even function, and real analytic function, real analytic for real values. It follows from the fact that the Riemann–Siegel theta function and the Riemann zeta function are both holomorphic in the critical strip, where the imaginary part of ''t'' is between −1/2 and 1/2, that the Z ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
1953 Introductions
Events January * January 6 – The Asian Socialist Conference opens in Rangoon, Burma. * January 12 – Estonian émigrés found a Estonian government-in-exile, government-in-exile in Oslo. * January 14 ** Marshal Josip Broz Tito is chosen President of Socialist Federal Republic of Yugoslavia, Yugoslavia. ** The Central Intelligence Agency, CIA-sponsored Robertson Panel first meets to discuss the Unidentified flying object, UFO phenomenon. * January 15 ** Georg Dertinger, foreign minister of East Germany, is arrested for spying. ** British security forces in West Germany arrest 7 members of the Naumann Circle, a clandestine Neo-Nazi organization. * January 19 – 71.1% of all television sets in the United States are tuned into ''I Love Lucy'', to watch Lucy give birth to Little Ricky, which is more people than those who tune into Dwight Eisenhower's inauguration the next day. This record is never broken. * January 24 ** Mau Mau Uprising: Rebels in Kenya kill th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
1953 In Science
The year 1953 involved numerous significant events in science and technology, including the first description of the DNA double helix, the discovery of neutrinos, and the release of the first polio vaccine. Biology * February 15 – Linus Pauling proposes a DNA triple helix structure, which is rapidly shown to be incorrect. * February 28 – Francis Crick and James Watson enter The Eagle, Cambridge, for a pub lunch announcing "We have discovered the secret of life." * April 25 – Francis Crick and James D. Watson of the U.K. Medical Research Council's Unit for Research on the Molecular Structure of Biological Systems at the Cavendish Laboratory in the University of Cambridge publish " Molecular Structure of Nucleic Acids: A Structure for Deoxyribose Nucleic Acid" in the British journal ''Nature''. Their work is often ranked as one of the most dramatic biological discoveries of the 20th century, because of the structural beauty and functional logic of the DNA double helix. In ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
