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Cubic Hypersurface
In mathematics, a cubic form is a homogeneous polynomial of degree 3, and a cubic hypersurface is the zero set of a cubic form. In the case of a cubic form in three variables, the zero set is a cubic plane curve. In , Boris Delone and Dmitry Faddeev showed that binary cubic forms with integer coefficients can be used to parametrize order (ring theory), orders in cubic fields. Their work was generalized in to include all cubic rings (a is a ring (mathematics), ring that is isomorphic to Z3 as a Module (mathematics), Z-module),In fact, Pierre Deligne pointed out that the correspondence works over an arbitrary Scheme (mathematics), scheme. giving a discriminant-preserving bijection between Orbit (group theory), orbits of a GL(2, Z)-Group action (mathematics), action on the space of integral binary cubic forms and cubic rings up to isomorphism. The classification of real cubic forms a x^3 + 3 b x^2 y + 3 c x y^2 + d y^3 is linked to the classification of umbilical points of surf ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Umbilical Point
In the differential geometry of surfaces in three dimensions, umbilics or umbilical points are points on a surface that are locally spherical. At such points the normal curvatures in all directions are equal, hence, both principal curvatures are equal, and every tangent vector is a ''principal direction''. The name "umbilic" comes from the Latin ''umbilicus'' (navel). Umbilic points generally occur as isolated points in the elliptical region of the surface; that is, where the Gaussian curvature is positive. The sphere is the only surface with non-zero curvature where every point is umbilic. A flat umbilic is an umbilic with zero Gaussian curvature. The monkey saddle is an example of a surface with a flat umbilic and on the plane every point is a flat umbilic. A torus can have no umbilics, but every closed surface of nonzero Euler characteristic, embedded smoothly into Euclidean space, has at least one umbilic. An unproven conjecture of Constantin Carathéodory states that every s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multilinear Algebra
Multilinear algebra is a subfield of mathematics that extends the methods of linear algebra. Just as linear algebra is built on the concept of a vector and develops the theory of vector spaces, multilinear algebra builds on the concepts of ''p''-vectors and multivectors with Grassmann algebras. Origin In a vector space of dimension ''n'', normally only vectors are used. However, according to Hermann Grassmann and others, this presumption misses the complexity of considering the structures of pairs, triplets, and general multi-vectors. With several combinatorial possibilities, the space of multi-vectors has 2''n'' dimensions. The abstract formulation of the determinant is the most immediate application. Multilinear algebra also has applications in the mechanical study of material response to stress and strain with various moduli of elasticity. This practical reference led to the use of the word tensor, to describe the elements of the multilinear space. The extra structure in a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homogeneous Polynomials
In mathematics, a homogeneous polynomial, sometimes called wikt:quantic, quantic in older texts, is a polynomial whose nonzero terms all have the same Degree of a polynomial, 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 (mathematics), function defined by a homogeneous polynomial. 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 (linear algebra), basis. A polynomial of degree 0 is always homogeneous; it is simply an element of the field (mathematics), field or ring (mathematics), ring of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Segre Cubic
In algebraic geometry, the Segre cubic is a cubic threefold embedded in 4 (or sometimes 5) dimensional projective space, studied by . Definition The Segre cubic is the set of points (''x''0:''x''1:''x''2:''x''3:''x''4:''x''5) of ''P''5 satisfying the equations :\displaystyle x_0+x_1+x_2+x_3+x_4+x_5= 0 :\displaystyle x_0^3+x_1^3+x_2^3+x_3^3+x_4^3+x_5^3 = 0. Properties The intersection of the Segre cubic with any hyperplane ''x''''i'' = 0 is the Clebsch cubic surface. Its intersection with any hyperplane ''x''''i'' = ''x''''j'' is Cayley's nodal cubic surface. Its dual is the Igusa quartic 3-fold in P4. Its Hessian is the Barth–Nieto quintic. A cubic hypersurface in ''P''4 has at most 10 nodes, and up to isomorphism the Segre cubic is the unique one with 10 nodes. Its nodes are the points conjugate to (1:1:1:−1:−1:−1) under permutations of coordinates. The Segre cubic is rational and furthermore birationally equivalent to a compactification of the Siegel modular v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Klein Cubic Threefold
In algebraic geometry, the Klein cubic threefold is the non-singular cubic threefold in 4-dimensional projective space given by the equation :V^2W+W^2X+X^2Y+Y^2Z+Z^2V =0 \, studied by . Its automorphism group is the group PSL2(11) of order 660 . It is unirational but not a rational variety. showed that it is birational to the moduli space of (1,11)-polarized abelian surfaces. References * * *{{Citation , authorlink=Felix Klein , last1=Klein , first1=Felix , title=Ueber die Transformation elfter Ordnung der elliptischen Functionen , doi=10.1007/BF02086276 , year=1879 , journal=Mathematische Annalen ''Mathematische Annalen'' (abbreviated as ''Math. Ann.'' or, formerly, ''Math. Annal.'') is a German mathematical research journal founded in 1868 by Alfred Clebsch and Carl Neumann. Subsequent managing editors were Felix Klein, David Hilbert, ... , issn=0025-5831 , volume=15 , issue=3 , pages=533–555, url=https://zenodo.org/record/1642598 3-folds ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Koras–Russell Cubic Threefold
In algebraic geometry, the Koras–Russell cubic threefolds are smooth affine complex threefolds diffeomorphic to \mathbf^3studied by . They have a hyperbolic action of a one-dimensional torus \mathbf^*with a unique fixed point, such that the quotients of the threefold and the tangent space of the fixed point by this action are isomorphic. They were discovered in the process of proving the Linearization Conjecture in dimension 3. A linear action of \mathbf^* on the affine space \mathbf^n is one of the form t*(x_1,\ldots,x_n)=(t^x_1,t^x_2,\ldots,t^x_n), where a_1,\ldots,a_n\in \mathbf and t\in\mathbf^*. The Linearization Conjecture in dimension n says that every algebraic action of \mathbf^* on the complex affine space \mathbf^n is linear in some algebraic coordinates on \mathbf^n. M. Koras and P. Russell made a key step towards the solution in dimension 3, providing a list of threefolds (now called Koras-Russell threefolds) and proving that the Linearization Conjecture for n=3 ho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cubic 3-fold
In algebraic geometry, a cubic threefold is a hypersurface of degree 3 in 4-dimensional projective space. Cubic threefolds are all unirational, but used intermediate Jacobians to show that non-singular cubic threefolds are not rational. The space of lines on a non-singular cubic 3-fold is a Fano surface. Examples *Koras–Russell cubic threefold *Klein cubic threefold *Segre cubic In algebraic geometry, the Segre cubic is a cubic threefold embedded in 4 (or sometimes 5) dimensional projective space, studied by . Definition The Segre cubic is the set of points (''x''0:''x''1:''x''2:''x''3:''x''4:''x''5) of ''P''5 satisfying t ... References * * *{{Citation , last1=Murre , first1=J. P. , author-link1=Jaap Murre , title=Algebraic equivalence modulo rational equivalence on a cubic threefold , url=http://www.numdam.org/item?id=CM_1972__25_2_161_0 , mr=0352088 , year=1972 , journal=Compositio Mathematica , issn=0010-437X , volume=25 , pages=161–206 Algebraic varieties ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fermat Cubic
In geometry, the Fermat cubic, named after Pierre de Fermat, is a surface defined by : x^3 + y^3 + z^3 = 1. \ Methods of algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ... provide the following parameterization of Fermat's cubic: : x(s,t) = : y(s,t) = : z(s,t) = . In projective space the Fermat cubic is given by :w^3+x^3+y^3+z^3=0. The 27 lines lying on the Fermat cubic are easy to describe explicitly: they are the 9 lines of the form (''w'' : ''aw'' : ''y'' : ''by'') where ''a'' and ''b'' are fixed numbers with cube −1, and their 18 conjugates under permutations of coordinates. ::::''Real points of Fermat cubic surface.'' References * * Algebraic surfaces {{algebraic-geometry-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elliptic Curve
In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If the field's characteristic is different from 2 and 3, then the curve can be described as a plane algebraic curve which consists of solutions for: :y^2 = x^3 + ax + b for some coefficients and in . The curve is required to be non-singular, which means that the curve has no cusps or self-intersections. (This is equivalent to the condition , that is, being square-free in .) It is always understood that the curve is really sitting in the projective plane, with the point being the unique point at infinity. Many sources define an elliptic curve to be simply a curve given by an equation of this form. (When the coefficient field has characteristic 2 or 3, the above equation is not quite general enough to include all non-singular cubic cu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cubic Plane Curve
In mathematics, a cubic plane curve is a plane algebraic curve defined by a cubic equation : applied to homogeneous coordinates for the projective plane; or the inhomogeneous version for the affine space determined by setting in such an equation. Here is a non-zero linear combination of the third-degree monomials : These are ten in number; therefore the cubic curves form a projective space of dimension 9, over any given field . Each point imposes a single linear condition on , if we ask that pass through . Therefore, we can find some cubic curve through any nine given points, which may be degenerate, and may not be unique, but will be unique and non-degenerate if the points are in general position; compare to two points determining a line and how five points determine a conic. If two cubics pass through a given set of nine points, then in fact a pencil of cubics does, and the points satisfy additional properties; see Cayley–Bacharach theorem. A cubic curve may ha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Umbilic Torus
The umbilic torus or umbilic bracelet is a single-edged 3-dimensional shape. The lone edge goes three times around the ring before returning to the starting point. The shape also has a single external face. A cross section of the surface forms a deltoid. The umbilic torus occurs in the mathematical subject of singularity theory, in particular in the classification of umbilical points which are determined by real cubic forms a x^3 + 3 b x^2 y + 3 c x y^2 + d y^3. The equivalence classes of such cubics form a three-dimensional real projective space and the subset of parabolic forms define a surface – the umbilic torus. Christopher Zeeman named this set the umbilic bracelet in 1976. The torus is defined by the following set of parametric equations. :x = \sin u \left(7+\cos\left( - 2v\right) + 2\cos\left( + v\right)\right) :y = \cos u \left(7 + \cos\left( - 2v\right) + 2\cos\left( + v\right)\right) :z = \sin\left( - 2v\right) + 2\sin \left( + v\right) :::\text-\pi \le u \le \pi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
