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Lebedev–Milin Inequality
In mathematics, the Lebedev–Milin inequality is any of several inequalities for the coefficients of the exponential of a power series, found by and . It was used in the proof of the Bieberbach conjecture, as it shows that the Milin conjecture implies the Robertson conjecture. They state that if :\sum_ \beta_kz^k = \exp\left(\sum_ \alpha_kz^k\right) for complex numbers \beta_k and \alpha_k, and n is a positive integer, then :\sum_^, \beta_k, ^2 \le \exp\left(\sum_^\infty k, \alpha_k, ^2\right), :\sum_^, \beta_k, ^2 \le (n+1)\exp\left(\frac\sum_^\sum_^m(k, \alpha_k, ^2 - 1/k)\right), :, \beta_n, ^2 \le \exp\left(\sum_^n(k, \alpha_k, ^2 -1/k)\right). See also exponential formula In combinatorial mathematics, the exponential formula (called the polymer expansion in physics) states that the exponential generating function for structures on finite sets is the exponential of the exponential generating function for connected str ... (on exponentiation of power series). References ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 in 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] |
Milin Conjecture
Milin may refer to: Places *Milin, Lower Silesian Voivodeship (south-west Poland) *Milin, Greater Poland Voivodeship (west-central Poland) *Milín, a municipality and village in the Czech Republic *Mainling County, in Tibet People Surname * Ferdo Milin (born 1977), Croatian football manager and former player * Gabriel Milin (1822–1895), French Breton-language poet * Isaak Moiseevich Milin Isaak Moiseevich Milin, (Исаак Моисеевич Милин); * February 16, 1919, Oster, Ukrainian Soviet Socialist Republic – † November 17, 1992 Saint-Petersburg (former Leningrad), Russian Federation) was a prominent Soviet/Russian ma ... (1919–1992), Soviet/Russian mathematician * Petar Milin (born 1979), Croatian rower Given name * Milin Dokthian (born 1996), Thai singer {{disambiguation, geo, given name, surname ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Robertson 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 in 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] |
Exponential Formula
In combinatorial mathematics, the exponential formula (called the polymer expansion in physics) states that the exponential generating function for structures on finite sets is the exponential of the exponential generating function for connected structures. The exponential formula is a power-series version of a special case of Faà di Bruno's formula. Statement For any formal power series of the form f(x)=a_1 x+x^2+x^3+\cdots+x^n+\cdots\, we have \exp f(x)=e^=\sum_^\infty x^n,\, where b_n = \sum_ a_\cdots a_, and the index \pi runs through the list of all partitions \ of the set \. (When k = 0, the product is empty and by definition equals 1.) One can write the formula in the following form: b_n = B_n(a_1,a_2,\dots,a_n), and thus \exp\left(\sum_^\infty x^n \right) = \sum_^\infty x^n, where B_n(a_1,\ldots,a_n) is the nth complete Bell polynomial. Alternatively, the exponential formula can also be written using the cycle index of the symmetric group, as follows:\exp\left(\sum ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationall ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
North-Holland Publishing Company
Elsevier () is a Dutch academic publishing company specializing in scientific, technical, and medical content. Its products include journals such as ''The Lancet'', '' Cell'', the ScienceDirect collection of electronic journals, '' Trends'', the '' Current Opinion'' series, the online citation database Scopus, the SciVal tool for measuring research performance, the ClinicalKey search engine for clinicians, and the ClinicalPath evidence-based cancer care service. Elsevier's products and services also include digital tools for data management, instruction, research analytics and assessment. Elsevier is part of the RELX Group (known until 2015 as Reed Elsevier), a publicly traded company. According to RELX reports, in 2021 Elsevier published more than 600,000 articles annually in over 2,700 journals; as of 2018 its archives contained over 17 million documents and 40,000 e-books, with over one billion annual downloads. Researchers have criticized Elsevier for its high profit margi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Amsterdam
Amsterdam ( , , , lit. ''The Dam on the River Amstel'') is the Capital of the Netherlands, capital and Municipalities of the Netherlands, most populous city of the Netherlands, with The Hague being the seat of government. It has a population of 907,976 within the city proper, 1,558,755 in the City Region of Amsterdam, urban area and 2,480,394 in the Amsterdam metropolitan area, metropolitan area. Located in the Provinces of the Netherlands, Dutch province of North Holland, Amsterdam is colloquially referred to as the "Venice of the North", for its large number of canals, now designated a World Heritage Site, UNESCO World Heritage Site. Amsterdam was founded at the mouth of the Amstel River that was dammed to control flooding; the city's name derives from the Amstel dam. Originally a small fishing village in the late 12th century, Amsterdam became a major world port during the Dutch Golden Age of the 17th century, when the Netherlands was an economic powerhouse. Amsterdam is th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Louis De Branges
Louis de Branges de Bourcia (born August 21, 1932) is a French-American mathematician. He is the Edward C. Elliott Distinguished Professor of Mathematics at Purdue University in West Lafayette, Indiana. He is best known for proving the long-standing Bieberbach conjecture in 1984, now called de Branges's theorem. He claims to have proved several important conjectures in mathematics, including the generalized Riemann hypothesis. Born to American parents who lived in Paris, de Branges moved to the US in 1941 with his mother and sisters. His native language is French. He did his undergraduate studies at the Massachusetts Institute of Technology (1949–53), and received a PhD in mathematics from Cornell University (1953–57). His advisors were Wolfgang Fuchs and then-future Purdue colleague Harry Pollard. He spent two years (1959–60) at the Institute for Advanced Study and another two (1961–62) at the Courant Institute of Mathematical Sciences. He was appointed to Purdue in 19 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Mathematical Monthly
''The American Mathematical Monthly'' is a mathematical journal founded by Benjamin Finkel in 1894. It is published ten times each year by Taylor & Francis for the Mathematical Association of America. The ''American Mathematical Monthly'' is an expository journal intended for a wide audience of mathematicians, from undergraduate students to research professionals. Articles are chosen on the basis of their broad interest and reviewed and edited for quality of exposition as well as content. In this the ''American Mathematical Monthly'' fulfills a different role from that of typical mathematical research journals. The ''American Mathematical Monthly'' is the most widely read mathematics journal in the world according to records on JSTOR. Tables of contents with article abstracts from 1997–2010 are availablonline The MAA gives the Lester R. Ford Awards annually to "authors of articles of expository excellence" published in the ''American Mathematical Monthly''. Editors *2022� ... [...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 was the first president and Fiske became secretary. The society soon decided to publish a journal, but ran into some resistance, due to 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 influential ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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