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Affine Algebraic Set
In algebraic geometry, an affine variety or affine algebraic variety is a certain kind of algebraic variety that can be described as a subset of an affine space. More formally, an affine algebraic set is the set of the common zeros over an algebraically closed field of some family of polynomials in the polynomial ring k _1, \ldots,x_n An affine variety is an affine algebraic set which is not the union of two smaller algebraic sets; algebraically, this means that (the radical of) the ideal generated by the defining polynomials is prime. One-dimensional affine varieties are called affine algebraic curves, while two-dimensional ones are affine algebraic surfaces. Some texts use the term ''variety'' for any algebraic set, and ''irreducible variety'' an algebraic set whose defining ideal is prime (affine variety in the above sense). In some contexts (see, for example, Hilbert's Nullstellensatz), it is useful to distinguish the field in which the coefficients are considered, from ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cubic With Double Point
Cubic may refer to: Science and mathematics * Cube (algebra), "cubic" measurement * Cube, a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex ** Cubic crystal system, a crystal system where the unit cell is in the shape of a cube * Cubic function, a polynomial function of degree three * Cubic equation, a polynomial equation (reducible to ''ax''3 + ''bx''2 + ''cx'' + ''d'' = 0) * Cubic form, a homogeneous polynomial of degree 3 * Cubic graph (mathematics - graph theory), a graph where all vertices have degree 3 * Cubic plane curve (mathematics), a plane algebraic curve ''C'' defined by a cubic equation * Cubic reciprocity (mathematics - number theory), a theorem analogous to quadratic reciprocity * Cubic surface, an algebraic surface in three-dimensional space * Cubic zirconia, in geology, a mineral that is widely synthesized for use as a diamond simulacra * CUBIC, a histology method Computing * Cubic IDE, a modular d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quotient Ring
In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite similar to the quotient group in group theory and to the quotient space in linear algebra. It is a specific example of a quotient, as viewed from the general setting of universal algebra. Starting with a ring R and a two-sided ideal I in , a new ring, the quotient ring , is constructed, whose elements are the cosets of I in R subject to special + and \cdot operations. (Quotient ring notation almost always uses a fraction slash ""; stacking the ring over the ideal using a horizontal line as a separator is uncommon and generally avoided.) Quotient rings are distinct from the so-called "quotient field", or field of fractions, of an integral domain as well as from the more general "rings of quotients" obtained by localization. Formal quotient ring construction Given a ring R and a two-sided ideal I in , we may define an e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Doubling Oriented
Doubling may refer to: Mathematics * Arithmetical doubling of a count or a measure, expressed as: ** Multiplication by 2 ** Increase by 100%, i.e. one-hundred percent ** Doubling the cube (i. e., hypothetical geometric construction of a cube with twice the volume of a given cube) * Doubling time, the length of time required for a quantity to double in size or value * Doubling map, a particular infinite two-dimensional geometrical construction Music * The composition or performance of a melody with itself or itself transposed at a constant interval such as the octave, third, or sixth, Voicing (music)#Doubling * The assignment of a melody to two instruments in an arrangement * The playing of two (or more) instruments alternately by a single player, e.g. ''Flute, doubling piccolo'' ** Musicians who play more than one woodwind instrument are called woodwind doublers or reed players * Double tracking, a recording technique in which a musical part (or vocal) is recorded twice ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projective Variety
In algebraic geometry, a projective variety is an algebraic variety that is a closed subvariety of a projective space. That is, it is the zero-locus in \mathbb^n of some finite family of homogeneous polynomials that generate a prime ideal, the defining ideal of the variety. A projective variety is a projective curve if its dimension is one; it is a projective surface if its dimension is two; it is a projective hypersurface if its dimension is one less than the dimension of the containing projective space; in this case it is the set of zeros of a single homogeneous polynomial. If ''X'' is a projective variety defined by a homogeneous prime ideal ''I'', then the quotient ring :k _0, \ldots, x_nI is called the homogeneous coordinate ring of ''X''. Basic invariants of ''X'' such as the degree and the dimension can be read off the Hilbert polynomial of this graded ring. Projective varieties arise in many ways. They are complete, which roughly can be expressed by saying t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integral Closure
In commutative algebra, an element ''b'' of a commutative ring ''B'' is said to be integral over a subring ''A'' of ''B'' if ''b'' is a root of some monic polynomial over ''A''. If ''A'', ''B'' are fields, then the notions of "integral over" and of an "integral extension" are precisely " algebraic over" and " algebraic extensions" in field theory (since the root of any polynomial is the root of a monic polynomial). The case of greatest interest in number theory is that of complex numbers integral over Z (e.g., \sqrt or 1+i); in this context, the integral elements are usually called algebraic integers. The algebraic integers in a finite extension field ''k'' of the rationals Q form a subring of ''k'', called the ring of integers of ''k'', a central object of study in algebraic number theory. In this article, the term '' ring'' will be understood to mean ''commutative ring'' with a multiplicative identity. Definition Let B be a ring and let A \subset B be a subring of B. An el ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normal Scheme
In algebraic geometry, an algebraic variety or scheme ''X'' is normal if it is normal at every point, meaning that the local ring at the point is an integrally closed domain. An affine variety ''X'' (understood to be irreducible) is normal if and only if the ring ''O''(''X'') of regular functions on ''X'' is an integrally closed domain. A variety ''X'' over a field is normal if and only if every finite birational morphism from any variety ''Y'' to ''X'' is an isomorphism. Normal varieties were introduced by . Geometric and algebraic interpretations of normality A morphism of varieties is finite if the inverse image of every point is finite and the morphism is proper. A morphism of varieties is birational if it restricts to an isomorphism between dense open subsets. So, for example, the cuspidal cubic curve ''X'' in the affine plane ''A''2 defined by ''x''2 = ''y''3 is not normal, because there is a finite birational morphism ''A''1 → ''X'' (namely, ''t'' maps to (''t''3, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic geometry, number theory, analytic combinatorics, and applied mathematics, as well as in physics, including the branches of hydrodynamics, thermodynamics, quantum mechanics, and twistor theory. By extension, use of complex analysis also has applications in engineering fields such as nuclear, aerospace, mechanical and electrical engineering. As a differentiable function of a complex variable is equal to the sum function given by its Taylor series (that is, it is analytic), complex analysis is particularly concerned with analytic functions of a complex variable, that is, '' holomorphic functions''. The concept can be extended to functions of several complex variables. Complex analysis is contrasted with real analysis, which dea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hartogs' Extension Theorem
In the theory of functions of several complex variables, Hartogs's extension theorem is a statement about the singularities of holomorphic functions of several variables. Informally, it states that the support of the singularities of such functions cannot be compact, therefore the singular set of a function of several complex variables must (loosely speaking) 'go off to infinity' in some direction. More precisely, it shows that an isolated singularity is always a removable singularity for any analytic function of complex variables. A first version of this theorem was proved by Friedrich Hartogs,See the original paper of and its description in various historical surveys by , and . In particular, in this last reference on p. 132, the Author explicitly writes :-"''As it is pointed out in the title of , and as the reader shall soon see, the key tool in the proof is the Cauchy integral formula''". and as such it is known also as Hartogs's lemma and Hartogs's principle: in ear ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Localization Of A Ring
Localization or localisation may refer to: Biology * Localization of function, locating psychological functions in the brain or nervous system; see Linguistic intelligence * Localization of sensation, ability to tell what part of the body is affected by touch or other sensation; see Allochiria * Neurologic localization, in neurology, the process of deducing the location of injury based on symptoms and neurological examination * Nuclear localization signal, an amino acid sequence on the surface of a protein which acts like a 'tag' to localize the protein in the cell * Sound localization, a listener's ability to identify the location or origin of a detected sound * Subcellular localization, organization of cellular components into different regions of a cell Engineering and technology * GSM localization, determining the location of an active cell phone or wireless transceiver * Robot localization, figuring out robot's position in an environment * Indoor positioning system, a networ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Saturation (commutative Algebra)
Saturation, saturated, unsaturation or unsaturated may refer to: Chemistry *Saturated and unsaturated compounds, a classification of compounds related to their ability to resist addition reactions ** Degree of unsaturation ** Saturated fat or saturated fatty acid **Unsaturated fat or unsaturated fatty acid ** Non-susceptibility of an organometallic compound to oxidative addition * Saturation of protein binding sites * Saturation of enzymes with a substrate * Saturation of a solute in a solution, as related to the solute's maximum solubility at equilibrium ** Supersaturation, where the concentration of a solute exceeds its maximum solubility at equilibrium ** Undersaturation, where the concentration of a solute is less than its maximum solubility at equilibrium Biology * Oxygen saturation, a clinical measure of the amount of oxygen in a patient's blood * Saturation pollination, a pollination technique * Saturated mutagenesis, a form of site-directed mutagenesis * Saturatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hypersurface
In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface. A hypersurface is a manifold or an algebraic variety of dimension , which is embedded in an ambient space of dimension , generally a Euclidean space, an affine space or a projective space. Hypersurfaces share, with surfaces in a three-dimensional space, the property of being defined by a single implicit equation, at least locally (near every point), and sometimes globally. A hypersurface in a (Euclidean, affine, or projective) space of dimension two is a plane curve. In a space of dimension three, it is a surface. For example, the equation :x_1^2+x_2^2+\cdots+x_n^2-1=0 defines an algebraic hypersurface of dimension in the Euclidean space of dimension . This hypersurface is also a smooth manifold, and is called a hypersphere or an -sphere. Smooth hypersurface A hypersurface that is a smooth manifold is called a ''smooth hypersurface''. In , a smooth hypersurface is ori ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dimension Of An Algebraic Variety
In mathematics and specifically in algebraic geometry, the dimension of an algebraic variety may be defined in various equivalent ways. Some of these definitions are of geometric nature, while some other are purely algebraic and rely on commutative algebra. Some are restricted to algebraic varieties while others apply also to any algebraic set. Some are intrinsic, as independent of any embedding of the variety into an affine or projective space, while other are related to such an embedding. Dimension of an affine algebraic set Let be a field, and be an algebraically closed extension. An affine algebraic set is the set of the common zeros in of the elements of an ideal in a polynomial ring R=K _1, \ldots, x_n Let A=R/I be the ''K''-algebra of the polynomial functions over . The dimension of is any of the following integers. It does not change if is enlarged, if is replaced by another algebraically closed extension of and if is replaced by another ideal having ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
