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Godfrey Harold Hardy
Godfrey Harold Hardy (7 February 1877 – 1 December 1947) was an English mathematician, known for his achievements in number theory and mathematical analysis. In biology, he is known for the Hardy–Weinberg principle, a basic principle of population genetics. G. H. Hardy is usually known by those outside the field of mathematics for his 1940 essay ''A Mathematician's Apology'', often considered one of the best insights into the mind of a working mathematician written for the layperson. Starting in 1914, Hardy was the mentor of the Indian mathematician Srinivasa Ramanujan, a relationship that has become celebrated.THE MAN WHO KNEW INFINITY: A Life of the Genius Ramanujan . Retrieved 2 December 2010. Hardy almost immediately rec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cranleigh
Cranleigh is a village and civil parish, about southeast of Guildford in Surrey, England. It lies on a minor road east of the A281, which links Guildford with Horsham. It is in the north-west corner of the Weald, a large remnant forest, the main local remnant being Winterfold Forest directly north-west on the northern Greensand Ridge. Etymology Until the mid-1860s, the place was usually spelt Cranley. The Post Office persuaded the vestry to use "''-leigh''" to avoid misdirections to nearby Crawley in West Sussex. The older spelling is publicly visible in the ''Cranley Hotel''. The name is recorded in the '' Pipe Rolls'' as ''Cranlea'' in 1166 and ''Cranelega'' in 1167. A little later in the '' Feet of Fines'' of 1198 the name is written as ''Cranele''. Etymologists consider all these versions to be the fusion of the Old English words "Cran", meaning " crane", and "Lēoh" that together mean 'a woodland clearing visited by cranes'. The name is popularly believed to come from imp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hardy–Weinberg Principle
In population genetics, the Hardy–Weinberg principle, also known as the Hardy–Weinberg equilibrium, model, theorem, or law, states that allele and genotype frequencies in a population will remain constant from generation to generation in the absence of other evolutionary influences. These influences include ''genetic drift'', '' mate choice'', ''assortative mating'', ''natural selection'', ''sexual selection'', ''mutation'', '' gene flow'', ''meiotic drive'', ''genetic hitchhiking'', '' population bottleneck'', '' founder effect,'' ''inbreeding and outbreeding depression''. In the simplest case of a single locus with two alleles denoted ''A'' and ''a'' with frequencies and , respectively, the expected genotype frequencies under random mating are for the AA homozygotes, for the aa homozygotes, and for the heterozygotes. In the absence of selection, mutation, genetic drift, or other forces, allele frequencies ''p'' and ''q'' are constant between generations, so equilibri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Srinivasa Ramanujan
Srinivasa Ramanujan (; born Srinivasa Ramanujan Aiyangar, ; 22 December 188726 April 1920) was an Indian mathematician. Though he had almost no formal training in pure mathematics, he made substantial contributions to mathematical analysis, number theory, infinite series, and continued fractions, including solutions to mathematical problems then considered unsolvable. Ramanujan initially developed his own mathematical research in isolation: according to Hans Eysenck: "He tried to interest the leading professional mathematicians in his work, but failed for the most part. What he had to show them was too novel, too unfamiliar, and additionally presented in unusual ways; they could not be bothered". Seeking mathematicians who could better understand his work, in 1913 he began a postal correspondence with the English mathematician G. H. Hardy at the University of Cambridge, England. Recognising Ramanujan's work as extraordinary, Hardy arranged for him to travel to Cambridge. I ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Camille Jordan
Marie Ennemond Camille Jordan (; 5 January 1838 – 22 January 1922) was a French mathematician, known both for his foundational work in group theory and for his influential ''Cours d'analyse''. Biography Jordan was born in Lyon and educated at the École polytechnique. He was an engineer by profession; later in life he taught at the École polytechnique and the Collège de France, where he had a reputation for eccentric choices of notation. He is remembered now by name in a number of results: * The Jordan curve theorem, a topological result required in complex analysis * The Jordan normal form and the Jordan matrix in linear algebra * In mathematical analysis, Jordan measure (or ''Jordan content'') is an area measure that predates measure theory * In group theory, the Jordan–Hölder theorem on composition series is a basic result. * Jordan's theorem on finite linear groups Jordan's work did much to bring Galois theory into the mainstream. He also investigated the Mathie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hardy–Littlewood Zeta-function Conjectures
In mathematics, the Hardy–Littlewood zeta-function conjectures, named after Godfrey Harold Hardy and John Edensor Littlewood, are two conjectures concerning the distances between zeros and the density of zeros of the Riemann zeta function. Conjectures In 1914, Godfrey Harold Hardy proved that the Riemann zeta function \zeta\bigl(\tfrac+it\bigr) has infinitely many real zeros. Let N(T) be the total number of real zeros, N_0(T) be the total number of zeros of odd order of the function \zeta\bigl(\tfrac+it\bigr), lying on the interval (0,T]. Hardy and Littlewood claimed two conjectures. These conjectures – on the distance between real zeros of \zeta\bigl(\tfrac+it\bigr) and on the density of zeros of \zeta\bigl(\tfrac+it\bigr) on intervals (T,T+H] for sufficiently great T > 0, H = T^ and with as less as possible value of a > 0, where \varepsilon > 0 is an arbitrarily small number – open two new directions in the investigation of the Riemann zeta function. 1. For any \varep ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hardy Field
In mathematics, a Hardy field is a field consisting of germs of real-valued functions at infinity that are closed under differentiation. They are named after the English mathematician G. H. Hardy. Definition Initially at least, Hardy fields were defined in terms of germs of real functions at infinity. Specifically we consider a collection ''H'' of functions that are defined for all large real numbers, that is functions ''f'' that map (''u'',∞) to the real numbers R, forsome real number ''u'' depending on ''f''. Here and in the rest of the article we say a function has a property " eventually" if it has the property for all sufficiently large ''x'', so for example we say a function ''f'' in ''H'' is ''eventually zero'' if there is some real number ''U'' such that ''f''(''x'') = 0 for all ''x'' ≥ ''U''. We can form an equivalence relation on ''H'' by saying ''f'' is equivalent to ''g'' if and only if ''f'' − ''g'' is eventually zero. The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hardy–Littlewood Circle Method
In mathematics, the Hardy–Littlewood circle method is a technique of analytic number theory. It is named for G. H. Hardy and J. E. Littlewood, who developed it in a series of papers on Waring's problem. History The initial idea is usually attributed to the work of Hardy with Srinivasa Ramanujan a few years earlier, in 1916 and 1917, on the asymptotics of the partition function. It was taken up by many other researchers, including Harold Davenport and I. M. Vinogradov, who modified the formulation slightly (moving from complex analysis to exponential sums), without changing the broad lines. Hundreds of papers followed, and the method still yields results. The method is the subject of a monograph by R. C. Vaughan. Outline The goal is to prove asymptotic behavior of a series: to show that for some function. This is done by taking the generating function of the series, then computing the residues about zero (essentially the Fourier coefficients). Technically, the genera ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hardy's Theorem
In mathematics, Hardy's theorem is a result in complex analysis describing the behavior of holomorphic functions. Let f be a holomorphic function on the open ball centered at zero and radius R in the complex plane, and assume that f is not a constant function. If one defines :I(r) = \frac \int_0^\! \left, f(r e^) \ \,d\theta for 0< r < R, then this function is strictly and is a convex function of . See also * *Hadamard three-circle theorem In complex analysis, a branch ...
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Hardy's Inequality
Hardy's inequality is an inequality in mathematics, named after G. H. Hardy. It states that if a_1, a_2, a_3, \dots is a sequence of non-negative real numbers, then for every real number ''p'' > 1 one has :\sum_^\infty \left (\frac\right )^p\leq\left (\frac\right )^p\sum_^\infty a_n^p. If the right-hand side is finite, equality holds if and only if a_n = 0 for all ''n''. An integral version of Hardy's inequality states the following: if ''f'' is a measurable function with non-negative values, then :\int_0^\infty \left (\frac\int_0^x f(t)\, dt\right)^p\, dx\le\left (\frac\right )^p\int_0^\infty f(x)^p\, dx. If the right-hand side is finite, equality holds if and only if ''f''(''x'') = 0 almost everywhere. Hardy's inequality was first published and proved (at least the discrete version with a worse constant) in 1920 in a note by Hardy. The original formulation was in an integral form slightly different from the above. General one-dimensional version The general weighted one ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hardy–Littlewood Inequality
In mathematical analysis, the Hardy–Littlewood inequality, named after G. H. Hardy and John Edensor Littlewood, states that if f and g are nonnegative measurable real functions vanishing at infinity that are defined on n-dimensional Euclidean space \mathbb R^n, then :\int_ f(x)g(x) \, dx \leq \int_ f^*(x)g^*(x) \, dx where f^* and g^* are the symmetric decreasing rearrangements of f and g, respectively. The decreasing rearrangement f^* of f is defined via the property that for all r >0 the two super-level sets :E_f(r)=\left\ \quad and \quad E_(r)=\left\ have the same volume (n-dimensional Lebesgue measure) and E_(r) is a ball in \mathbb R^n centered at x=0, i.e. it has maximal symmetry. Proof The layer cake representation allows us to write the general functions f and g in the form f(x)= \int_0^\infty \chi_ \, dr \quad and \quad g(x)= \int_0^\infty \chi_ \, ds where r \mapsto \chi_ equals 1 for rs the indicator functions x \mapsto \chi_(x) and x \map ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hardy Notation
Hardy may refer to: People * Hardy (surname) * Hardy (given name) * Hardy (singer), American singer-songwriter Places Antarctica * Mount Hardy, Enderby Land * Hardy Cove, Greenwich Island * Hardy Rocks, Biscoe Islands Australia * Hardy, South Australia, a locality * Cape Hardy, a headland in South Australia * Hardy Inlet, Western Australia Canada * Hardy Township, Ontario, Canada, administered by the Loring, Port Loring and District, Ontario, services board * Port Hardy, British Columbia * Hardy, Saskatchewan, Canada, a hamlet United States * Hardy, Arkansas, a city * Hardy, California, an unincorporated community * Hardy, Iowa, a city * Hardy, Kentucky, an unincorporated community * Hardy, Mississippi, an unincorporated community * Hardy, Montana, an unincorporated community * Hardy, Nebraska, a village * Hardy, Virginia, an unincorporated community * Hardy County, West Virginia * Hardy Dam, Michigan * Hardy Lake, Indiana, a state reservoir * Hardy Pond, Massachusetts * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hardy Space
In complex analysis, the Hardy spaces (or Hardy classes) ''Hp'' are certain spaces of holomorphic functions on the unit disk or upper half plane. They were introduced by Frigyes Riesz , who named them after G. H. Hardy, because of the paper . In real analysis Hardy spaces are certain spaces of distributions on the real line, which are (in the sense of distributions) boundary values of the holomorphic functions of the complex Hardy spaces, and are related to the ''Lp'' spaces of functional analysis. For 1 ≤ ''p'' < ∞ these real Hardy spaces ''Hp'' are certain s of ''Lp'', while for ''p'' < 1 the ''Lp'' spaces have some undesirable properties, and the Hardy spaces are much better behaved. There are also higher-dimensional generalizations, consisting of certain holomorphic functions on |
