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Twisted Cubic
In mathematics, a twisted cubic is a smooth, rational curve ''C'' of degree three in projective 3-space P3. It is a fundamental example of a skew curve. It is essentially unique, up to projective transformation (''the'' twisted cubic, therefore). In algebraic geometry, the twisted cubic is a simple example of a projective variety that is not linear or a hypersurface, in fact not a complete intersection. It is the three-dimensional case of the rational normal curve, and is the image of a Veronese map of degree three on the projective line. Definition The twisted cubic is most easily given parametrically as the image of the map :\nu:\mathbf^1\to\mathbf^3 which assigns to the homogeneous coordinate :T/math> the value :\nu: :T\mapsto ^3:S^2T:ST^2:T^3 In one coordinate patch of projective space, the map is simply the moment curve :\nu:x \mapsto (x,x^2,x^3) That is, it is the closure by a single point at infinity of the affine curve (x,x^2,x^3). The twisted cubic is a ... [...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] |
Coordinate Patch
In mathematics, particularly topology, an atlas is a concept used to describe a manifold. An atlas consists of individual ''charts'' that, roughly speaking, describe individual regions of the manifold. In general, the notion of atlas underlies the formal definition of a manifold and related structures such as vector bundles and other fiber bundles. Charts The definition of an atlas depends on the notion of a ''chart''. A chart for a topological space ''M'' is a homeomorphism \varphi from an open subset ''U'' of ''M'' to an open subset of a Euclidean space. The chart is traditionally recorded as the ordered pair (U, \varphi). When a coordinate system is chosen in the Euclidean space, this defines coordinates on U: the coordinates of a point P of U are defined as the coordinates of \varphi(P). The pair formed by a chart and such a coordinate system is called a local coordinate system, coordinate chart, coordinate patch, coordinate map, or local frame. Formal definition of at ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Secant Variety
In algebraic geometry, the secant variety \operatorname(V), or the variety of chords, of a projective variety V \subset \mathbb^r is the Zariski closure of the union of all secant lines (chords) to ''V'' in \mathbb^r: :\operatorname(V) = \bigcup_ \overline (for x = y, the line \overline is the tangent line.) It is also the image under the projection p_3: (\mathbb^r)^3 \to \mathbb^r of the closure ''Z'' of the incidence variety :\. Note that ''Z'' has dimension 2 \dim V + 1 and so \operatorname(V) has dimension at most 2 \dim V + 1. More generally, the k^ secant variety is the Zariski closure of the union of the linear spaces spanned by collections of k+1 points on V. It may be denoted by \Sigma_k. The above secant variety is the first secant variety. Unless \Sigma_k=\mathbb^r, it is always singular along \Sigma_, but may have other singular points. If V has dimension ''d'', the dimension of \Sigma_k is at most kd+d+k. A useful tool for computing the dimension of a secant variety is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Secant Line
In geometry, a secant is a line (geometry), line that intersects a curve at a minimum of two distinct Point (geometry), points.. The word ''secant'' comes from the Latin word ''secare'', meaning ''to cut''. In the case of a circle, a secant intersects the circle at exactly two points. A Chord (geometry), chord is the line segment determined by the two points, that is, the interval (mathematics), interval on the secant whose ends are the two points. Circles A straight line can intersect a circle at zero, one, or two points. A line with intersections at two points is called a ''secant line'', at one point a ''tangent line'' and at no points an ''exterior line''. A ''chord'' is the line segment that joins two distinct points of a circle. A chord is therefore contained in a unique secant line and each secant line determines a unique chord. In rigorous modern treatments of plane geometry, results that seem obvious and were assumed (without statement) by Euclid in Euclid's Elements, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tangent
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is, intuitively, the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More precisely, a straight line is tangent to the curve at a point if the line passes through the point on the curve and has slope , where ''f'' is the derivative of ''f''. A similar definition applies to space curves and curves in ''n''-dimensional Euclidean space. The point where the tangent line and the curve meet or intersect is called the ''point of tangency''. The tangent line is said to be "going in the same direction" as the curve, and is thus the best straight-line approximation to the curve at that point. The tangent line to a point on a differentiable curve can also be thought of as a '' tangent line approximation'', the graph of the affine function that best approximates the original function at the given point ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Union (set Theory)
In set theory, the union (denoted by ∪) of a collection of Set (mathematics), sets is the set of all element (set theory), elements in the collection. It is one of the fundamental operations through which sets can be combined and related to each other. A refers to a union of Zero, zero () sets and it is by definition equal to the empty set. For explanation of the symbols used in this article, refer to the List of mathematical symbols, table of mathematical symbols. Binary union The union of two sets ''A'' and ''B'' is the set of elements which are in ''A'', in ''B'', or in both ''A'' and ''B''. In set-builder notation, : A \cup B = \. For example, if ''A'' = and ''B'' = then ''A'' ∪ ''B'' = . A more elaborate example (involving two infinite sets) is: : ''A'' = : ''B'' = : A \cup B = \ As another example, the number 9 is ''not'' contained in the union of the set of prime numbers and the set of even numbers , because 9 is neither prime nor even. Sets cannot ha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Radical Ideal
Radical (from Latin: ', root) may refer to: Politics and ideology Politics *Classical radicalism, the Radical Movement that began in late 18th century Britain and spread to continental Europe and Latin America in the 19th century *Radical politics, the political intent of fundamental societal change * Radical Party (other), several political parties *Radicals (UK), a British and Irish grouping in the early to mid-19th century *Radicalization *Politicians from the Radical Civic Union Ideologies * Radical chic, a term coined by Tom Wolfe to describe the pretentious adoption of radical causes *Radical feminism, a perspective within feminism that focuses on patriarchy * Radical Islam, or Islamic extremism * Radical Christianity * Radical veganism, a radical interpretation of veganism, usually combined with anarchism *Radical Reformation, an Anabaptist movement concurrent with the Protestant Reformation Science and mathematics Science *Radical (chemistry), an atom, molecule, o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homogeneous Ideal
In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups R_i such that . The index set is usually the set of nonnegative integers or the set of integers, but can be any monoid. The direct sum decomposition is usually referred to as gradation or grading. A graded module is defined similarly (see below for the precise definition). It generalizes graded vector spaces. A graded module that is also a graded ring is called a graded algebra. A graded ring could also be viewed as a graded -algebra. The associativity is not important (in fact not used at all) in the definition of a graded ring; hence, the notion applies to non-associative algebras as well; e.g., one can consider a graded Lie algebra. First properties Generally, the index set of a graded ring is assumed to be the set of nonnegative integers, unless otherwise explicitly specified. This is the case in this article. A graded ring is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quadratic Form
In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example, 4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong to a fixed field , such as the real or complex numbers, and one speaks of a quadratic form ''over'' . Over the reals, a quadratic form is said to be '' definite'' if it takes the value zero only when all its variables are simultaneously zero; otherwise it is '' isotropic''. Quadratic forms occupy a central place in various branches of mathematics, including number theory, linear algebra, group theory ( orthogonal groups), differential geometry (the Riemannian metric, the second fundamental form), differential topology ( intersection forms of manifolds, especially four-manifolds), Lie theory (the Killing form), and statistics (where the exponent of a zero-mean multivariate normal distribution has the quadratic form -\mathbf^\math ... [...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] |
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] |
