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Superelliptic Curve
In mathematics, a superelliptic curve is an algebraic curve defined by an equation of the form :y^m = f(x), where m \geq 2 is an integer and ''f'' is a polynomial of degree d\geq 3 with coefficients in a field k; more precisely, it is the smooth projective curve whose function field defined by this equation. The case m=2 and d=3, 4 is an ''elliptic curve'', the case m=2 and d\ge 5 is a ''hyperelliptic curve'', and the case m=3 and d\geq 4 is an example of a ''trigonal curve''. Some authors impose additional restrictions, for example, that the integer m should not be divisible by the characteristic of k, that the polynomial f should be square free, that the integers ''m'' and ''d'' should be coprime, or some combination of these. Definition More generally, a ''superelliptic curve'' is a cyclic branched covering :C \to \mathbb^1 of the projective line of degree m \geq 2 coprime to the characteristic of the field of definition. The degree m of the covering map is also referred to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Algebraic Curve
In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane curve can be completed in a projective algebraic plane curve by homogenization of a polynomial, homogenizing its defining polynomial. Conversely, a projective algebraic plane curve of homogeneous equation can be restricted to the affine algebraic plane curve of equation . These two operations are each inverse function, inverse to the other; therefore, the phrase algebraic plane curve is often used without specifying explicitly whether it is the affine or the projective case that is considered. If the defining polynomial of a plane algebraic curve is irreducible polynomial, irreducible, then one has an ''irreducible plane algebraic curve''. Otherwise, the algebraic curve is the union of one or several irreducible curves, called its ''Irreduc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Kummer Theory
Kummer is a German surname. Notable people with the surname include: * Bernhard Kummer (1897–1962), German Germanist * Clare Kummer (1873–1958), American composer, lyricist and playwright * Clarence Kummer (1899–1930), American jockey * Christopher Kummer (born 1975), German economist * Corby Kummer (born 1957), American journalist * Dirk Kummer (born 1966), German actor, director, and screenwriter * Eberhard Kummer (1940–2019), Austrian concert singer, lawyer, and medieval music expert * Eduard Kummer, also known as the following Ernst Kummer * Eloise Kummer (1916–2008), American actress * Ernst Kummer (1810–1893), German mathematician ** Kummer configuration, a mathematical structure discovered by Ernst Kummer ** Kummer surface, a related geometrical structure discovered by Ernst Kummer * Ferdinand von Kummer (1816–1900), German general * Frederic Arnold Kummer (1873–1943), American author, playwright, and screenwriter * Friedrich August Kummer (1797–1879), Germ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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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 internationally, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Graduate Texts In Mathematics
Graduate Texts in Mathematics (GTM) () is a series of graduate-level textbooks in mathematics published by Springer-Verlag. The books in this series, like the other Springer-Verlag mathematics series, are yellow books of a standard size (with variable numbers of pages). The GTM series is easily identified by a white band at the top of the book. The books in this series tend to be written at a more advanced level than the similar Undergraduate Texts in Mathematics series, although there is a fair amount of overlap between the two series in terms of material covered and difficulty level. List of books #''Introduction to Axiomatic Set Theory'', Gaisi Takeuti, Wilson M. Zaring (1982, 2nd ed., ) #''Measure and Category – A Survey of the Analogies between Topological and Measure Spaces'', John C. Oxtoby (1980, 2nd ed., ) #''Topological Vector Spaces'', H. H. Schaefer, M. P. Wolff (1999, 2nd ed., ) #''A Course in Homological Algebra'', Peter Hilton, Urs Stammbach (1997, 2 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Superellipse
A superellipse, also known as a Lamé curve after Gabriel Lamé, is a closed curve resembling the ellipse, retaining the geometric features of semi-major axis and semi-minor axis, and symmetry about them, but defined by an equation that allows for various shapes between a rectangle and an ellipse. In two dimensional Cartesian coordinate system, a superellipse is defined as the set of all points (x,y) on the curve that satisfy the equation\left, \frac\^n\!\! + \left, \frac\^n\! = 1,where a and b are positive numbers referred to as Semidiameter, semi-diameters or Semi-major and semi-minor axes, semi-axes of the superellipse, and n is a positive parameter that defines the shape. When n=2, the superellipse is an ordinary ellipse. For n>2, the shape is more rectangular with rounded corners, and for 00, where m either equals to or differs from ''n''. If m=n, it is the Lamé's superellipses. If m\neq n, the curve possesses more flexibility of behavior, and is better possible fit to des ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Branched Covering
In mathematics, a branched covering is a map that is almost a covering map, except on a small set. In topology In topology, a map is a ''branched covering'' if it is a covering map everywhere except for a nowhere dense set known as the branch set. Examples include the map from a wedge of circles to a single circle, where the map is a homeomorphism on each circle. In algebraic geometry In algebraic geometry, the term branched covering is used to describe morphisms f from an algebraic variety V to another one W, the two dimensions being the same, and the typical fibre of f being of dimension 0. In that case, there will be an open set W' of W (for the Zariski topology) that is dense in W, such that the restriction of f to W' (from V' = f^(W') to W', that is) is unramified. Depending on the context, we can take this as local homeomorphism for the strong topology, over the complex numbers, or as an étale morphism in general (under some slightly stronger hypotheses, on flatness ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Hyperelliptic Curve
In algebraic geometry, a hyperelliptic curve is an algebraic curve of genus ''g'' > 1, given by an equation of the form y^2 + h(x)y = f(x) where ''f''(''x'') is a polynomial of degree ''n'' = 2''g'' + 1 > 4 or ''n'' = 2''g'' + 2 > 4 with ''n'' distinct roots, and ''h''(''x'') is a polynomial of degree 3. Therefore, in giving such an equation to specify a non-singular curve, it is almost always assumed that a non-singular model (also called a smooth completion), equivalent in the sense of birational geometry, is meant. To be more precise, the equation defines a quadratic extension of C(''x''), and it is that function field that is meant. The singular point at infinity can be removed (since this is a curve) by the normalization (integral closure) process. It turns out that after doing this, there is an open cover of the curve by two affine charts: the one already given by y^2 = f(x) and another one given by w^2 = v^f(1/v) . The glueing maps between the two charts are given by ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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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] [Amazon] |
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Thue Equation
In mathematics, a Thue equation is a Diophantine equation of the form f(x,y) = r, where f is an irreducible bivariate form of degree at least 3 over the rational numbers, and r is a nonzero rational number. It is named after Axel Thue, who in 1909 proved that a Thue equation can have only finitely many solutions in integers x and y, a result known as Thue's theorem. The Thue equation is solvable effectively: there is an explicit bound on the solutions x, y of the form (C_1 r)^ where constants C_1 and C_2 depend only on the form f. A stronger result holds: if K is the field generated by the roots of f, then the equation has only finitely many solutions with x and y integers of K, and again these may be effectively determined. Finiteness of solutions and diophantine approximation Thue's original proof that the equation named in his honour has finitely many solutions is through the proof of what is now known as Thue's theorem: it asserts that for any algebraic number \alpha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Siegel Identity
In mathematics, Siegel's identity refers to one of two formulae that are used in the resolution of Diophantine equations. Statement The first formula is : \frac + \frac = 1 . The second is : \frac \cdot\frac + \frac \cdot \frac = 1 . Application The identities are used in translating Diophantine problems connected with integral points on hyperelliptic curves into S-unit equations. See also * Siegel formula References * * * * * {{cite book , title=The Algorithmic Resolution of Diophantine Equations , volume=41 , series=London Mathematical Society Student Texts , first=N. P. , last=Smart , authorlink=Nigel Smart (cryptographer) , publisher=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 Assessme ... , year=1998 , isbn=0-521-64633-2 , page36–37, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Diophantine Problem
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''Diophantine'' means pertaining to the ancient Greek mathematician Diophantus. A number of concepts bear this name: *Diophantine approximation *Diophantine equation *Diophantine quintuple *Diophantine set In mathematics, a Diophantine equation is an equation of the form ''P''(''x''1, ..., ''x'j'', ''y''1, ..., ''y'k'') = 0 (usually abbreviated ''P''(', ') = 0) where ''P''(', ') is a polynomial with integer coefficients, where ''x''1, ..., ''x' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |