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Rational Normal Curve
In mathematics, the rational normal curve is a smooth, rational curve of degree in projective n-space . It is a simple example of a projective variety; formally, it is the Veronese variety when the domain is the projective line. For it is the plane conic and for it is the twisted cubic. The term "normal" refers to projective normality, not normal schemes. The intersection of the rational normal curve with an affine space is called the moment curve. Definition The rational normal curve may be given parametrically as the image of the map :\nu:\mathbf^1\to\mathbf^n which assigns to the homogeneous coordinates the value :\nu: :T\mapsto \left ^n:S^T:S^T^2:\cdots:T^n \right In the affine coordinates of the chart the map is simply :\nu:x \mapsto \left (x, x^2, \ldots, x^n \right ). That is, the rational normal curve is the closure by a single point at infinity of the affine curve :\left (x, x^2, \ldots, x^n \right ). Equivalently, rational normal curve may be understoo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...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] |
Scheme (mathematics)
In mathematics, specifically algebraic geometry, a scheme is a structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations and define the same algebraic variety but different schemes) and allowing "varieties" defined over any commutative ring (for example, Fermat curves are defined over the integers). Scheme theory was introduced by Alexander Grothendieck in 1960 in his treatise '' Éléments de géométrie algébrique'' (EGA); one of its aims was developing the formalism needed to solve deep problems of algebraic geometry, such as the Weil conjectures (the last of which was proved by Pierre Deligne). Strongly based on commutative algebra, scheme theory allows a systematic use of methods of topology and homological algebra. Scheme theory also unifies algebraic geometry with much of number theory, which eventually led to Wiles's proof of Fermat's Last Theorem. Schemes elaborate the fundamental idea that an a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complete Intersection
In mathematics, an algebraic variety ''V'' in projective space is a complete intersection if the ideal of ''V'' is generated by exactly ''codim V'' elements. That is, if ''V'' has dimension ''m'' and lies in projective space ''P''''n'', there should exist ''n'' − ''m'' homogeneous polynomials: :F_i(X_0,\cdots,X_n), 1\leq i\leq n - m, in the homogeneous coordinates ''X''''j'', which generate all other homogeneous polynomials that vanish on ''V''. Geometrically, each ''F''''i'' defines a hypersurface; the intersection of these hypersurfaces should be ''V''. The intersection of hypersurfaces will always have dimension at least ''m'', assuming that the field of scalars is an algebraically closed field such as the complex numbers. The question is essentially, can we get the dimension down to ''m'', with no extra points in the intersection? This condition is fairly hard to check as soon as the codimension . When then ''V'' is automatically a hypersurface and there is nothin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ideal (ring Theory)
In mathematics, and more specifically in ring theory, an ideal of a ring is a special subset of its elements. Ideals generalize certain subsets of the integers, such as the even numbers or the multiples of 3. Addition and subtraction of even numbers preserves evenness, and multiplying an even number by any integer (even or odd) results in an even number; these closure and absorption properties are the defining properties of an ideal. An ideal can be used to construct a quotient ring in a way similar to how, in group theory, a normal subgroup can be used to construct a quotient group. Among the integers, the ideals correspond one-for-one with the non-negative integers: in this ring, every ideal is a principal ideal consisting of the multiples of a single non-negative number. However, in other rings, the ideals may not correspond directly to the ring elements, and certain properties of integers, when generalized to rings, attach more naturally to the ideals than to the elem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Hyperplane
In geometry, a hyperplane is a generalization of a two-dimensional plane in three-dimensional space to mathematical spaces of arbitrary dimension. Like a plane in space, a hyperplane is a flat hypersurface, a subspace whose dimension is one less than that of the ambient space. Two lower-dimensional examples of hyperplanes are one-dimensional lines in a plane and zero-dimensional points on a line. Most commonly, the ambient space is -dimensional Euclidean space, in which case the hyperplanes are the -dimensional "flats", each of which separates the space into two half spaces. A reflection across a hyperplane is a kind of motion ( geometric transformation preserving distance between points), and the group of all motions is generated by the reflections. A convex polytope is the intersection of half-spaces. In non-Euclidean geometry, the ambient space might be the -dimensional sphere or hyperbolic space, or more generally a pseudo-Riemannian space form, and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
General Position
In algebraic geometry and computational geometry, general position is a notion of genericity for a set of points, or other geometric objects. It means the ''general case'' situation, as opposed to some more special or coincidental cases that are possible, which is referred to as special position. Its precise meaning differs in different settings. For example, generically, two lines in the plane intersect in a single point (they are not parallel or coincident). One also says "two generic lines intersect in a point", which is formalized by the notion of a ''generic point''. Similarly, three generic points in the plane are not collinear; if three points are collinear (even stronger, if two coincide), this is a degenerate case. This notion is important in mathematics and its applications, because degenerate cases may require an exceptional treatment; for example, when stating general theorems or giving precise statements thereof, and when writing computer programs (see '' generic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Field (mathematics)
In mathematics, a field is a set (mathematics), set on which addition, subtraction, multiplication, and division (mathematics), division are defined and behave as the corresponding operations on rational number, rational and real numbers. A field is thus a fundamental algebraic structure which is widely used in algebra, number theory, and many other areas of mathematics. The best known fields are the field of rational numbers, the field of real numbers and the field of complex numbers. Many other fields, such as field of rational functions, fields of rational functions, algebraic function fields, algebraic number fields, and p-adic number, ''p''-adic fields are commonly used and studied in mathematics, particularly in number theory and algebraic geometry. Most cryptographic protocols rely on finite fields, i.e., fields with finitely many element (set), elements. The theory of fields proves that angle trisection and squaring the circle cannot be done with a compass and straighte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projective Linear Group
In mathematics, especially in the group theoretic area of algebra, the projective linear group (also known as the projective general linear group or PGL) is the induced action of the general linear group of a vector space ''V'' on the associated projective space P(''V''). Explicitly, the projective linear group is the quotient group : PGL(''V'') = GL(''V'')/Z(''V'') where GL(''V'') is the general linear group of ''V'' and Z(''V'') is the subgroup of all nonzero scalar transformations of ''V''; these are quotiented out because they act trivially on the projective space and they form the kernel of the action, and the notation "Z" reflects that the scalar transformations form the center of the general linear group. The projective special linear group, PSL, is defined analogously, as the induced action of the special linear group on the associated projective space. Explicitly: : PSL(''V'') = SL(''V'')/SZ(''V'') where SL(''V'') is the special linear group over ''V'' and SZ('' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Congruence Relation
In abstract algebra, a congruence relation (or simply congruence) is an equivalence relation on an algebraic structure (such as a group (mathematics), group, ring (mathematics), ring, or vector space) that is compatible with the structure in the sense that algebraic operations done with equivalent elements will yield equivalent elements. Every congruence relation has a corresponding Equivalence class, quotient structure, whose elements are the equivalence classes (or congruence classes) for the relation. Definition The definition of a congruence depends on the type of algebraic structure under consideration. Particular definitions of congruence can be made for group (mathematics), groups, ring (mathematics), rings, vector spaces, module (mathematics), modules, semigroups, lattice (order), lattices, and so forth. The common theme is that a congruence is an equivalence relation on an algebraic object that is compatible with the algebraic structure, in the sense that the operat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Base (topology)
In mathematics, a base (or basis; : bases) for the topology of a topological space is a family \mathcal of open subsets of such that every open set of the topology is equal to the union of some sub-family of \mathcal. For example, the set of all open intervals in the real number line \R is a basis for the Euclidean topology on \R because every open interval is an open set, and also every open subset of \R can be written as a union of some family of open intervals. Bases are ubiquitous throughout topology. The sets in a base for a topology, which are called , are often easier to describe and use than arbitrary open sets. Many important topological definitions such as continuity and convergence can be checked using only basic open sets instead of arbitrary open sets. Some topologies have a base of open sets with specific useful properties that may make checking such topological definitions easier. Not all families of subsets of a set X form a base for a topology on X. Under ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
