Askey–Gasper Inequality
In mathematics, the Askey–Gasper inequality is an inequality for Jacobi polynomials proved by and used in the proof of the Bieberbach conjecture. Statement It states that if \beta\geq 0, \alpha+\beta\geq -2, and -1\leq x\leq 1 then :\sum_^n \frac \ge 0 where :P_k^(x) is a Jacobi polynomial. The case when \beta=0 can also be written as :_3F_2 \left (-n,n+\alpha+2,\tfrac(\alpha+1);\tfrac(\alpha+3),\alpha+1;t \right)>0, \qquad 0\leq t-1. In this form, with a non-negative integer, the inequality was used by Louis de Branges in his proof of the Bieberbach conjecture. Proof gave a short proof of this inequality, by combining the identity :\begin &\frac\times _3F_2 \left (-n,n+\alpha+2,\tfrac(\alpha+1);\tfrac(\alpha+3),\alpha+1;t \right)\\ =&\sum_ \frac \times _3F_2\left (-n+2j,n-2j+\alpha+1,\tfrac(\alpha+1);\tfrac(\alpha+2),\alpha+1;t \right ) \end with the Clausen inequality. Generalizations give some generalizations of the Askey–Gasper inequality to basic hypergeometri ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Jacobi Polynomial
In mathematics, Jacobi polynomials (occasionally called hypergeometric polynomials) P_n^(x) are a class of classical orthogonal polynomials. They are orthogonal with respect to the weight (1-x)^\alpha(1+x)^\beta on the interval 1,1/math>. The Gegenbauer polynomials, and thus also the Legendre, Zernike and Chebyshev polynomials, are special cases of the Jacobi polynomials. The Jacobi polynomials were introduced by Carl Gustav Jacob Jacobi. Definitions Via the hypergeometric function The Jacobi polynomials are defined via the hypergeometric function as follows: :P_n^(z)=\frac\,_2F_1\left(-n,1+\alpha+\beta+n;\alpha+1;\tfrac(1-z)\right), where (\alpha+1)_n is Pochhammer's symbol (for the rising factorial). In this case, the series for the hypergeometric function is finite, therefore one obtains the following equivalent expression: :P_n^ (z) = \frac \sum_^n \frac \left(\frac\right)^m. Rodrigues' formula An equivalent definition is given by Rodrigues' formula: :P_n^(z) = ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Bieberbach Conjecture
In complex analysis, de Branges's theorem, or the Bieberbach conjecture, is a theorem that gives a necessary condition on a holomorphic function in order for it to map the open unit disk of the complex plane injectively to the complex plane. It was posed by and finally proven by . The statement concerns the Taylor coefficients a_n of a univalent function, i.e. a one-to-one holomorphic function that maps the unit disk into the complex plane, normalized as is always possible so that a_0=0 and a_1=1. That is, we consider a function defined on the open unit disk which is holomorphic and injective ('' univalent'') with Taylor series of the form :f(z)=z+\sum_ a_n z^n. Such functions are called ''schlicht''. The theorem then states that : , a_n, \leq n \quad \textn\geq 2. The Koebe function (see below) is a function for which a_n=n for all n, and it is schlicht, so we cannot find a stricter limit on the absolute value of the nth coefficient. Schlicht functions The normalization ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Louis De Branges
Louis may refer to: People * Louis (given name), origin and several individuals with this name * Louis (surname) * Louis (singer), Serbian singer Other uses * Louis (coin), a French coin * HMS ''Louis'', two ships of the Royal Navy See also * Derived terms * King Louis (other) * Saint Louis (other) * Louis Cruise Lines * Louis dressing, for salad * Louis Quinze, design style Associated terms * Lewis (other) * Louie (other) * Luis (other) * Louise (other) * Louisville (other) Associated names * * Chlodwig, the origin of the name Ludwig, which is translated to English as "Louis" * Ladislav and László - names sometimes erroneously associated with "Louis" * Ludovic, Ludwig, Ludwick, Ludwik Ludwik () is a Polish given name. Notable people with the name include: * Ludwik Czyżewski, Polish WWII general * Ludwik Fleck (1896–1961), Polish medical doctor and biologist * Ludwik Gintel (1899–1973), Polish- ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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De Branges's Theorem
In complex analysis, de Branges's theorem, or the Bieberbach conjecture, is a theorem that gives a necessary condition on a holomorphic function in order for it to map the open unit disk of the complex plane injectively to the complex plane. It was posed by and finally proven by . The statement concerns the Taylor coefficients a_n of a univalent function, i.e. a one-to-one holomorphic function that maps the unit disk into the complex plane, normalized as is always possible so that a_0=0 and a_1=1. That is, we consider a function defined on the open unit disk which is holomorphic and injective ('' univalent'') with Taylor series of the form :f(z)=z+\sum_ a_n z^n. Such functions are called ''schlicht''. The theorem then states that : , a_n, \leq n \quad \textn\geq 2. The Koebe function (see below) is a function for which a_n=n for all n, and it is schlicht, so we cannot find a stricter limit on the absolute value of the nth coefficient. Schlicht functions The normalizations ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Clausen Inequality
Clausen is a Danish language, Danish patronymic surname, literally meaning ''child of Claus'', Claus being a German language, German form of the Greek language, Greek Νικόλαος, Nikolaos, (cf. Nicholas), used in Denmark at least since the 16th century. The spelling variant Klausen has identical pronunciation (as does the often interchangeable Claussen). The two variants are number 34 and 85 on the top 100 of surnames in Denmark. Occurrences of Clausen/Klausen as a surname outside Denmark and Schleswig-Holstein are due to immigration. Immigrants to English-speaking countries sometimes changed the spelling to Clauson. People * Alden W. Clausen (1923–2013), American businessman and former President of the World Bank * Alf Clausen (1941–2025), American film and television musical composer * Andrea Clausen (born 1959), German stage actress (Burgtheater, Vienna) * Arne Clausen (born 1932), Argentine Methodist Pastor * Casey Clausen (born 1981), American football player *Christo ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Basic Hypergeometric Series
In mathematics, basic hypergeometric series, or ''q''-hypergeometric series, are q-analog, ''q''-analogue generalizations of generalized hypergeometric series, and are in turn generalized by elliptic hypergeometric series. A series ''x''''n'' is called hypergeometric if the ratio of successive terms ''x''''n''+1/''x''''n'' is a rational function of ''n''. If the ratio of successive terms is a rational function of ''q''''n'', then the series is called a basic hypergeometric series. The number ''q'' is called the base. The basic hypergeometric series _2\phi_1(q^,q^;q^;q,x) was first considered by . It becomes the hypergeometric series F(\alpha,\beta;\gamma;x) in the limit when base q =1. Definition There are two forms of basic hypergeometric series, the unilateral basic hypergeometric series φ, and the more general bilateral basic hypergeometric series ψ. The unilateral basic hypergeometric series is defined as :\;_\phi_k \left[\begin a_1 & a_2 & \ldots & a_ \\ b_1 & b_2 & ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Turán's Inequalities
In mathematics, Turán's inequalities are some inequalities for Legendre polynomials found by (and first published by ). There are many generalizations to other polynomials, often called Turán's inequalities, given by and other authors. If P_n is the nth Legendre polynomial, Turán's inequalities state that :\,\! P_n(x)^2 > P_(x)P_(x)\ \text\ -10 \ \text\ -1 See also * *Sturm Chain
In mathematics, the Sturm sequence of a univariate polynomial is a sequence of polynomials associated with and its derivative by a variant of Euclid's algorithm for polynomials. Sturm's theorem expresses the number of distinct real roots of lo ...
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American Journal Of Mathematics
The ''American Journal of Mathematics'' is a bimonthly mathematics journal published by the Johns Hopkins University Press. History The ''American Journal of Mathematics'' is the oldest continuously published mathematical journal in the United States, established in 1878 at the Johns Hopkins University by James Joseph Sylvester, an English-born mathematician who also served as the journal's editor-in-chief from its inception through early 1884. Initially W. E. Story was associate editor in charge; he was replaced by Thomas Craig (mathematician), Thomas Craig in 1880. For volume 7 Simon Newcomb became chief editor with Craig managing until 1894. Then with volume 16 it was "Edited by Thomas Craig with the Co-operation of Simon Newcomb" until 1898. Other notable mathematicians who have served as editors or editorial associates of the journal include Frank Morley, Oscar Zariski, Lars Ahlfors, Hermann Weyl, Wei-Liang Chow, S. S. Chern, André Weil, Harish-Chandra, Jean Dieudonné, Hen ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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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]   |
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Theoretical Computer Science (journal)
''Theoretical Computer Science'' (''TCS'') is a computer science journal published by Elsevier, started in 1975 and covering theoretical computer science. The journal publishes 52 issues a year. It is abstracted and indexed by Scopus and the Science Citation Index. According to the Journal Citation Reports, its 2020 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 ... is 0.827. References Computer science journals Elsevier academic journals Academic journals established in 1975 {{comp-sci-theory-stub ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Cambridge University Press
Cambridge University Press was the university press of the University of Cambridge. Granted a letters patent by King Henry VIII in 1534, it was the oldest university press in the world. Cambridge University Press merged with Cambridge Assessment to form Cambridge University Press and Assessment under Queen Elizabeth II's approval in August 2021. With a global sales presence, publishing hubs, and offices in more than 40 countries, it published over 50,000 titles by authors from over 100 countries. Its publications include more than 420 academic journals, monographs, reference works, school and university textbooks, and English language teaching and learning publications. It also published Bibles, runs a bookshop in Cambridge, sells through Amazon, and has a conference venues business in Cambridge at the Pitt Building and the Sir Geoffrey Cass Sports and Social Centre. It also served as the King's Printer. Cambridge University Press, as part of the University of Cambridge, was a ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Inequalities (mathematics)
Inequality may refer to: * Inequality (mathematics), a relation between two quantities when they are different. * Economic inequality, difference in economic well-being between population groups ** Income inequality, an unequal distribution of income ** Wealth inequality, an unequal distribution of wealth ** Spatial inequality, the unequal distribution of income and resources across geographical regions ** International inequality, economic differences between countries * Social inequality, unequal opportunities and rewards for different social positions or statuses within a group ** Gender inequality, unequal treatment or perceptions due to gender ** Racial inequality, social distinctions between racial and ethnic groups within a society * Health inequality, differences in the quality of health and healthcare across populations * Educational inequality, the unequal distribution of academic resources * Environmental inequality, unequal environmental harms between differe ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |