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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. Turing was highly influential in the development of theoretical com ... 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 an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...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 one thousandth of a kilogram. Originally defined as of 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 4 °C, the temperature of maximum density of water. However, by the late 19th century, there was an effort to make the Base unit (measurement), base unit the kilogram and the gram a derived unit. In 1960, the new International System of Units defined a ''gram'' as one one-thousandth of a kilogram (i.e., one gram is Scientific notation, 1×10−3 kg). The kilogram, as of 2019, is defined by the International Bureau of Weights and Measures from the fixed numerical value of the Planck constant (), which is kg⋅m2⋅s−1. Official SI symbol The only unit symbol for gram that is recognised by the International System of ... [...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 \operatorname(s) > 1 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 is consid ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Academic Press
Academic Press (AP) is an academic book publisher founded in 1941. It was acquired by Harcourt, Brace & World in 1969. Reed Elsevier bought Harcourt in 2000, and 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 Well-known products include the ''Methods in Enzymology'' series and encyclopedias such as ''The International Encyclopedia of Public Health'' and the ''Encyclopedia of Neuroscience''. See also * Akademische Verlagsgesellschaft (AVG) — the German predecessor, founded in 1906 by Leo Jolowicz (1868–1940), the father of Walter Jolowicz Walter may refer to: People * Walter (name), both a surname and a given name * Little Walter, American blues harmonica player Marion Wa ... [...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. Turing 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. He is widely considered to be the father of theoretical computer science and artificial intelligence. Born in Maida Vale, London, Turing was raised in southern England. He graduated at King's College, Cambridge, with a degree in mathematics. Whilst he was a fellow at Cambridge, he published a proof demonstrating that some purely mathematical yes–no questions can never be answered by computation and defined a Turing machine, and went on to prove that the halting problem for Turing machines is undecidable. In 1938, he obtained his PhD from the Department of Mathemati ... [...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†... [...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 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 function is holomorphic in t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
1953 Introductions
Events January * January 6 РThe Asian Socialist Conference opens in Rangoon, Burma. * January 12 РEstonian ̩migr̩s found a government-in-exile in Oslo. * January 14 ** Marshal Josip Broz Tito is chosen President of Yugoslavia. ** The CIA-sponsored Robertson Panel first meets to discuss the UFO phenomenon. * January 15 РGeorg Dertinger, foreign minister of East Germany, is arrested for spying. * January 19 Р71.1% 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 has yet to be broken. * January 20 РDwight D. Eisenhower is sworn in as the 34th President of the United States. * January 24 ** Mau Mau Uprising: Rebels in Kenya kill the Ruck family (father, mother, and six-year-old son). ** Leader of East Germany Walter Ulbricht announces that agriculture will be co ... [...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 * April 25 – Francis Crick and James D. Watson of 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 1962, they will share the Nobel Prize in Medicine with Maurice Wilkins, who publishes X-ray crystallography results for DNA in the same issue of ''Nature'' in 1953. The third related article published at the same time is by Rosalind Franklin and Raymond Goslin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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