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Quaternary Cubic
In mathematics, a quaternary cubic form is a degree 3 homogeneous polynomial in four variables. The zeros form a cubic surface in 3-dimensional projective space. Invariants and studied the ring of invariants of a quaternary cubic, which is a ring generated by invariants of degrees 8, 16, 24, 32, 40, 100. The generators of degrees 8, 16, 24, 32, 40 generate a polynomial ring. The generator of degree 100 is a skew invariant, whose square is a polynomial in the other generators given explicitly by Salmon. Salmon also gave an explicit formula for the discriminant as a polynomial in the generators, though pointed out that the formula has a widely copied misprint in it. Sylvester pentahedron A generic quaternary cubic can be written as a sum of 5 cubes of linear forms, unique up to multiplication by cube roots of unity. This was conjectured by Sylvester in 1851, and proven 10 years later by Clebsch. The union of the 5 planes where these 5 linear forms vanish is called the Sylvester ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homogeneous Polynomial
In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial of degree 5, in two variables; the sum of the exponents in each term is always 5. The polynomial x^3 + 3 x^2 y + z^7 is not homogeneous, because the sum of exponents does not match from term to term. The function defined by a homogeneous polynomial is always a homogeneous function. An algebraic form, or simply form, is a function defined by a homogeneous polynomial.However, as some authors do not make a clear distinction between a polynomial and its associated function, the terms ''homogeneous polynomial'' and ''form'' are sometimes considered as synonymous. A binary form is a form in two variables. A ''form'' is also a function defined on a vector space, which may be expressed as a homogeneous function of the coordinates over any basis. A polynomial of degree 0 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cubic Surface
In mathematics, a cubic surface is a surface in 3-dimensional space defined by one polynomial equation of degree 3. Cubic surfaces are fundamental examples in algebraic geometry. The theory is simplified by working in projective space rather than affine space, and so cubic surfaces are generally considered in projective 3-space \mathbf^3. The theory also becomes more uniform by focusing on surfaces over the complex numbers rather than the real numbers; note that a complex surface has real dimension 4. A simple example is the Fermat cubic surface :x^3+y^3+z^3+w^3=0 in \mathbf^3. Many properties of cubic surfaces hold more generally for del Pezzo surfaces. Rationality of cubic surfaces A central feature of smooth scheme, smooth cubic surfaces ''X'' over an algebraically closed field is that they are all rational variety, rational, as shown by Alfred Clebsch in 1866. That is, there is a one-to-one correspondence defined by rational functions between the projective plane \mathbf^2 min ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
James Joseph Sylvester
James Joseph Sylvester (3 September 1814 – 15 March 1897) was an English mathematician. He made fundamental contributions to matrix theory, invariant theory, number theory, partition theory, and combinatorics. He played a leadership role in American mathematics in the later half of the 19th century as a professor at the Johns Hopkins University and as founder of the '' American Journal of Mathematics''. At his death, he was a professor at Oxford University. Biography James Joseph was born in London on 3 September 1814, the son of Abraham Joseph, a Jewish merchant. James later adopted the surname ''Sylvester'' when his older brother did so upon emigration to the United States. At the age of 14, Sylvester was a student of Augustus De Morgan at the University of London (now University College London). His family withdrew him from the university after he was accused of stabbing a fellow student with a knife. Subsequently, he attended the Liverpool Royal Institutio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alfred Clebsch
Rudolf Friedrich Alfred Clebsch (19 January 1833 – 7 November 1872) was a German mathematician who made important contributions to algebraic geometry and invariant theory. He attended the University of Königsberg and was habilitated at Humboldt University of Berlin, Berlin. He subsequently taught in Berlin and University of Karlsruhe, Karlsruhe. His collaboration with Paul Gordan in University of Giessen, Giessen led to the introduction of Clebsch–Gordan coefficients for spherical harmonics, which are now widely used in quantum mechanics. Together with Carl Neumann at University of Göttingen, Göttingen, he founded the mathematical research journal ''Mathematische Annalen'' in 1868. In 1883, Adhémar Jean Claude Barré de Saint-Venant, Saint-Venant translated Clebsch's work on Elasticity (physics), elasticity into French and published it as ''Théorie de l'élasticité des Corps Solides''. Books by A. Clebsch Vorlesungen über Geometrie(Teubner, Leipzig, 1876-1891) edited b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ternary Cubic
In mathematics, a ternary cubic form is a homogeneous degree 3 polynomial in three variables. Invariant theory The ternary cubic is one of the few cases of a form of degree greater than 2 in more than 2 variables whose ring of invariants was calculated explicitly in the 19th century. The ring of invariants The algebra of invariants of a ternary cubic under SL3(C) is a polynomial algebra generated by two invariants ''S'' and ''T'' of degrees 4 and 6, called Aronhold invariants. The invariants are rather complicated when written as polynomials in the coefficients of the ternary cubic, and are given explicitly in The ring of covariants The ring of covariants is given as follows. The identity covariant ''U'' of a ternary cubic has degree 1 and order 3. The Hessian ''H'' is a covariant of ternary cubics of degree 3 and order 3. There is a covariant ''G'' of ternary cubics of degree 8 and order 6 that vanishes on points ''x'' lying on the Salmon conic of the polar of ''x'' with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ternary Quartic
In mathematics, a ternary quartic form is a degree 4 homogeneous polynomial in three variables. Hilbert's theorem showed that a positive semi-definite ternary quartic form over the reals can be written as a sum of three squares of quadratic forms. Invariant theory The ring of invariants is generated by 7 algebraically independent invariants of degrees 3, 6, 9, 12, 15, 18, 27 (discriminant) , together with 6 more invariants of degrees 9, 12, 15, 18, 21, 21, as conjectured by . discussed the invariants of order up to about 15. The Salmon invariant is a degree 60 invariant vanishing on ternary quartics with an inflection bitangent. Catalecticant The catalecticant of a ternary quartic is the resultant of its 6 second partial derivatives. It vanishes when the ternary quartic can be written as a sum of five 4th powers of linear forms. See also * Ternary cubic * Invariants of a binary form References * * * * *. * * *{{Citation , last1=Thomsen , first1=H. Ivah , title=Some ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Invariants Of A Binary Form
In mathematical invariant theory, an invariant of a binary form is a polynomial in the coefficients of a binary form in two variables ''x'' and ''y'' that remains invariant under the special linear group acting on the variables ''x'' and ''y''. Terminology A binary form (of degree ''n'') is a homogeneous polynomial \sum_^n \binom a_x^y^i = a_nx^n + \binom a_x^y + \cdots + a_0y^n. The group SL_2(\mathbb) acts on these forms by taking x to ax + by and y to cx + dy. This induces an action on the space spanned by a_0, \ldots, a_n and on the polynomials in these variables. An invariant is a polynomial in these n + 1 variables a_0, \ldots, a_n that is invariant under this action. More generally a covariant is a polynomial in a_0, \ldots, a_n, x, y that is invariant, so an invariant is a special case of a covariant where the variables x and y do not occur. More generally still, a simultaneous invariant is a polynomial in the coefficients of several different forms in x and y. In ter ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Für Die Reine Und Angewandte Mathematik
''Crelle's Journal'', or just ''Crelle'', is the common name for a mathematics journal, the ''Journal für die reine und angewandte Mathematik'' (in English: ''Journal for Pure and Applied Mathematics''). History The journal was founded by August Leopold Crelle (Berlin) in 1826 and edited by him until his death in 1855. It was one of the first major mathematical journals that was not a proceedings of an academy. It has published many notable papers, including works of Niels Henrik Abel, Georg Cantor, Gotthold Eisenstein, Carl Friedrich Gauss and Otto Hesse. It was edited by Carl Wilhelm Borchardt from 1856 to 1880, during which time it was known as ''Borchardt's Journal''. The current editor-in-chief is Daniel Huybrechts (Rheinische Friedrich-Wilhelms-Universität Bonn). Past editors * 1826–1856: August Leopold Crelle * 1856–1880: Carl Wilhelm Borchardt * 1881–1888: Leopold Kronecker, Karl Weierstrass Karl Theodor Wilhelm Weierstrass (; ; 31 October 1815 � ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Philosophical Transactions Of The Royal Society
''Philosophical Transactions of the Royal Society'' is a scientific journal published by the Royal Society. In its earliest days, it was a private venture of the Royal Society's secretary. It was established in 1665, making it the second journal in the world exclusively devoted to science, after the '' Journal des sçavans'', and therefore also the world's longest-running scientific journal. It became an official society publication in 1752. The use of the word ''philosophical'' in the title refers to natural philosophy, which was the equivalent of what would now be generally called ''science''. Current publication In 1887 the journal expanded and divided into two separate publications, one serving the physical sciences ('' Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences'') and the other focusing on the life sciences ('' Philosophical Transactions of the Royal Society B: Biological Sciences''). Both journals now publish theme ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Invariant Theory
Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties, such as vector spaces, from the point of view of their effect on functions. Classically, the theory dealt with the question of explicit description of polynomial functions that do not change, or are ''invariant'', under the transformations from a given linear group. For example, if we consider the action of the special linear group ''SLn'' on the space of ''n'' by ''n'' matrices by left multiplication, then the determinant is an invariant of this action because the determinant of ''A X'' equals the determinant of ''X'', when ''A'' is in ''SLn''. Introduction Let G be a group, and V a finite-dimensional vector space over a field k (which in classical invariant theory was usually assumed to be the complex numbers). A representation of G in V is a group homomorphism \pi:G \to GL(V), which induces a group action of G on V. If k /math> is the space of polynomial functions on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
