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Quadric (algebraic Geometry)
In mathematics, a quadric or quadric hypersurface is the subspace of ''N''-dimensional space defined by a polynomial equation of degree 2 over a field (mathematics), field. Quadrics are fundamental examples in algebraic geometry. The theory is simplified by working in projective space rather than affine space. An example is the quadric surface :xy=zw in projective space ^3 over the complex numbers C. A quadric has a natural action of the orthogonal group, and so the study of quadrics can be considered as a descendant of Euclidean geometry. Many properties of quadrics hold more generally for projective homogeneous varieties. Another generalization of quadrics is provided by Fano varieties. By definition, a quadric ''X'' of dimension ''n'' over a field ''k'' is the subspace of \mathbf^ defined by ''q'' = 0, where ''q'' is a nonzero homogeneous polynomial of degree 2 over ''k'' in variables x_0,\ldots,x_. (A homogeneous polynomial is also called a form, and so ''q'' may be called a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Algebraic Geometry
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; the modern approach generalizes this in a few different aspects. The fundamental objects of study in algebraic geometry are algebraic variety, algebraic varieties, which are geometric manifestations of solution set, solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are line (geometry), lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves, and quartic curves like lemniscate of Bernoulli, lemniscates and Cassini ovals. These are plane algebraic curves. A point of the plane lies on an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of points of special interest like singular point of a curve, singular p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Algebraic Varieties
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Modern definitions generalize this concept in several different ways, while attempting to preserve the geometric intuition behind the original definition. Conventions regarding the definition of an algebraic variety differ slightly. For example, some definitions require an algebraic variety to be irreducible, which means that it is not the union of two smaller sets that are closed in the Zariski topology In algebraic geometry and commutative algebra, the Zariski topology is a topology defined on geometric objects called varieties. It is very different from topologies that are commonly used in real or complex analysis; in particular, it is not .... Under this definition, non-irreducible algebraic varieties are called algebraic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Rational Point
In number theory and algebraic geometry, a rational point of an algebraic variety is a point whose coordinates belong to a given field. If the field is not mentioned, the field of rational numbers is generally understood. If the field is the field of real numbers, a rational point is more commonly called a real point. Understanding rational points is a central goal of number theory and Diophantine geometry. For example, Fermat's Last Theorem may be restated as: for , the Fermat curve of equation x^n+y^n=1 has no other rational points than , , and, if is even, and . Definition Given a field , and an algebraically closed extension of , an affine variety over is the set of common zeros in of a collection of polynomials with coefficients in : :\begin & f_1(x_1,\ldots,x_n)=0, \\ & \qquad \quad \vdots \\ & f_r(x_1,\dots,x_n)=0. \end These common zeros are called the ''points'' of . A -rational point (or -point) of is a point of that belongs to , that is, a sequence (a_1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Rational Variety
In mathematics, a rational variety is an algebraic variety, over a given field ''K'', which is birationally equivalent to a projective space of some dimension over ''K''. This means that its function field is isomorphic to :K(U_1, \dots , U_d), the field of all rational functions for some set \ of indeterminates, where ''d'' is the dimension of the variety. Rationality and parameterization Let ''V'' be an affine algebraic variety of dimension ''d'' defined by a prime ideal ''I'' = ⟨''f''1, ..., ''f''''k''⟩ in K _1, \dots , X_n/math>. If ''V'' is rational, then there are ''n'' + 1 polynomials ''g''0, ..., ''g''''n'' in K(U_1, \dots , U_d) such that f_i(g_1/g_0, \ldots, g_n/g_0)=0. In other words, we have a x_i=\frac(u_1,\ldots,u_d) of the variety. Conversely, such a rational parameterization induces a field homomorphism of the field of functions of ''V'' into K(U_1, \dots , U_d). But this homomorphism is not necessarily onto. If such a parameteriza ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Rank (linear Algebra)
In linear algebra, the rank of a matrix is the dimension of the vector space generated (or spanned) by its columns. p. 48, § 1.16 This corresponds to the maximal number of linearly independent columns of . This, in turn, is identical to the dimension of the vector space spanned by its rows. Rank is thus a measure of the " nondegenerateness" of the system of linear equations and linear transformation encoded by . There are multiple equivalent definitions of rank. A matrix's rank is one of its most fundamental characteristics. The rank is commonly denoted by or ; sometimes the parentheses are not written, as in .Alternative notation includes \rho (\Phi) from and . Main definitions In this section, we give some definitions of the rank of a matrix. Many definitions are possible; see Alternative definitions for several of these. The column rank of is the dimension of the column space of , while the row rank of is the dimension of the row space of . A fundamental resul ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Nondegenerate Bilinear Form
In mathematics, specifically linear algebra, a degenerate bilinear form on a vector space ''V'' is a bilinear form such that the map from ''V'' to ''V''∗ (the dual space of ''V'') given by is not an isomorphism. An equivalent definition when ''V'' is finite-dimensional is that it has a non-trivial kernel: there exist some non-zero ''x'' in ''V'' such that :f(x,y)=0\, for all \,y \in V. Nondegenerate forms A nondegenerate or nonsingular form is a bilinear form that is not degenerate, meaning that v \mapsto (x \mapsto f(x,v)) is an isomorphism, or equivalently in finite dimensions, if and only if :f(x,y)=0 for all y \in V implies that x = 0. Using the determinant If ''V'' is finite-dimensional then, relative to some basis for ''V'', a bilinear form is degenerate if and only if the determinant of the associated matrix is zero – if and only if the matrix is ''singular'', and accordingly degenerate forms are also called singular forms. Likewise, a nondegenerate form is one for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Determinant
In mathematics, the determinant is a Scalar (mathematics), scalar-valued function (mathematics), function of the entries of a square matrix. The determinant of a matrix is commonly denoted , , or . Its value characterizes some properties of the matrix and the linear map represented, on a given basis (linear algebra), basis, by the matrix. In particular, the determinant is nonzero if and only if the matrix is invertible matrix, invertible and the corresponding linear map is an linear isomorphism, isomorphism. However, if the determinant is zero, the matrix is referred to as singular, meaning it does not have an inverse. The determinant is completely determined by the two following properties: the determinant of a product of matrices is the product of their determinants, and the determinant of a triangular matrix is the product of its diagonal entries. The determinant of a matrix is :\begin a & b\\c & d \end=ad-bc, and the determinant of a matrix is : \begin a & b & c \\ d & e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Hessian Matrix
In mathematics, the Hessian matrix, Hessian or (less commonly) Hesse matrix is a square matrix of second-order partial derivatives of a scalar-valued Function (mathematics), function, or scalar field. It describes the local curvature of a function of many variables. The Hessian matrix was developed in the 19th century by the German mathematician Otto Hesse, Ludwig Otto Hesse and later named after him. Hesse originally used the term "functional determinants". The Hessian is sometimes denoted by H or \nabla\nabla or \nabla^2 or \nabla\otimes\nabla or D^2. Definitions and properties Suppose f : \R^n \to \R is a function taking as input a vector \mathbf \in \R^n and outputting a scalar f(\mathbf) \in \R. If all second-order partial derivatives of f exist, then the Hessian matrix \mathbf of f is a square n \times n matrix, usually defined and arranged as \mathbf H_f= \begin \dfrac & \dfrac & \cdots & \dfrac \\[2.2ex] \dfrac & \dfrac & \cdots & \dfrac \\[2.2ex] \vdots & \vdot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Smooth Scheme
In algebraic geometry, a smooth scheme over a field is a scheme which is well approximated by affine space near any point. Smoothness is one way of making precise the notion of a scheme with no singular points. A special case is the notion of a smooth variety over a field. Smooth schemes play the role in algebraic geometry of manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...s in topology. Definition First, let ''X'' be an affine scheme of Glossary of scheme theory#finite, finite type over a field ''k''. Equivalently, ''X'' has a closed immersion into affine space ''An'' over ''k'' for some natural number ''n''. Then ''X'' is the closed subscheme defined by some equations ''g''1 = 0, ..., ''g''''r'' = 0, where each ''gi'' is in the polynomial ring ''k''[''x''1,..., ''x' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Characteristic (algebra)
In mathematics, the characteristic of a ring , often denoted , is defined to be the smallest positive number of copies of the ring's multiplicative identity () that will sum to the additive identity (). If no such number exists, the ring is said to have characteristic zero. That is, is the smallest positive number such that: : \underbrace_ = 0 if such a number exists, and otherwise. Motivation The special definition of the characteristic zero is motivated by the equivalent definitions characterized in the next section, where the characteristic zero is not required to be considered separately. The characteristic may also be taken to be the exponent of the ring's additive group, that is, the smallest positive integer such that: : \underbrace_ = 0 for every element of the ring (again, if exists; otherwise zero). This definition applies in the more general class of rngs (see '); for (unital) rings the two definitions are equivalent due to their distributive law. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Projective Cone
A projective cone (or just cone) in projective geometry is the union of all lines that intersect a projective subspace ''R'' (the apex of the cone) and an arbitrary subset ''A'' (the basis) of some other subspace ''S'', disjoint from ''R''. In the special case that ''R'' is a single point, ''S'' is a plane, and ''A'' is a conic section on ''S'', the projective cone is a conical surface; hence the name. Definition Let ''X'' be a projective space over some field ''K'', and ''R'', ''S'' be disjoint subspaces of ''X''. Let ''A'' be an arbitrary subset of ''S''. Then we define ''RA'', the cone with top ''R'' and basis ''A'', as follows : * When ''A'' is empty, ''RA'' = ''A''. * When ''A'' is not empty, ''RA'' consists of all those points on a line connecting a point on ''R'' and a point on ''A''. Properties * As ''R'' and ''S'' are disjoint, one may deduce from linear algebra and the definition of a projective space that every point on ''RA'' not in ''R'' or ''A'' is on exactly ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Quadric Cone
In mathematics, a quadric or quadric surface is a generalization of conic sections (ellipses, parabolas, and hyperbolas). In three-dimensional space, quadrics include ellipsoids, paraboloids, and hyperboloids. More generally, a quadric hypersurface (of dimension ''D'') embedded in a higher dimensional space (of dimension ) is defined as the zero set of an irreducible polynomial of degree two in variables; for example, ''D''1 is the case of conic sections (plane curves). When the defining polynomial is not absolutely irreducible, the zero set is generally not considered a quadric, although it is often called a ''degenerate quadric'' or a ''reducible quadric''. A quadric is an affine algebraic variety, or, if it is reducible, an affine algebraic set. Quadrics may also be defined in projective spaces; see , below. Formulation In coordinates , the general quadric is thus defined by the algebraic equationSilvio LevQuadricsin "Geometry Formulas and Facts", excerpted from 30th Edit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
