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Locally Nilpotent Derivation
In mathematics, a Derivation (differential algebra), derivation \partial of a commutative ring A is called a locally nilpotent derivation (LND) if every element of A is annihilated by some power of \partial. One motivation for the study of locally nilpotent derivations comes from the fact that some of the counterexamples to Hilbert's 14th problem are obtained as the kernels of a derivation on a polynomial ring. Over a field k of characteristic zero, to give a locally nilpotent derivation on the integral domain A, finitely generated over the field, is equivalent to giving an action of the additive group (k,+) to the affine variety X = \operatorname(A). Roughly speaking, an affine variety admitting "plenty" of actions of the additive group is considered similar to an affine space. Definition Let A be a Ring (mathematics), ring. Recall that a Derivation (differential algebra), derivation of A is a map \partial\colon\, A\to A satisfying the product rule, Leibniz rule \partial (ab) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Derivation (differential Algebra)
In mathematics, a derivation is a function on an algebra which generalizes certain features of the derivative operator. Specifically, given an algebra ''A'' over a ring or a field ''K'', a ''K''-derivation is a ''K''-linear map that satisfies Leibniz's law: : D(ab) = a D(b) + D(a) b. More generally, if ''M'' is an ''A''-bimodule, a ''K''-linear map that satisfies the Leibniz law is also called a derivation. The collection of all ''K''-derivations of ''A'' to itself is denoted by Der''K''(''A''). The collection of ''K''-derivations of ''A'' into an ''A''-module ''M'' is denoted by . Derivations occur in many different contexts in diverse areas of mathematics. The partial derivative with respect to a variable is an R-derivation on the algebra of real-valued differentiable functions on R''n''. The Lie derivative with respect to a vector field is an R-derivation on the algebra of differentiable functions on a differentiable manifold; more generally it is a derivation on the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Localization Of A Ring
In commutative algebra and algebraic geometry, localization is a formal way to introduce the "denominators" to a given ring or module. That is, it introduces a new ring/module out of an existing ring/module ''R'', so that it consists of fractions \frac, such that the denominator ''s'' belongs to a given subset ''S'' of ''R''. If ''S'' is the set of the non-zero elements of an integral domain, then the localization is the field of fractions: this case generalizes the construction of the field \Q of rational numbers from the ring \Z of integers. The technique has become fundamental, particularly in algebraic geometry, as it provides a natural link to sheaf theory. In fact, the term ''localization'' originated in algebraic geometry: if ''R'' is a ring of functions defined on some geometric object ( algebraic variety) ''V'', and one wants to study this variety "locally" near a point ''p'', then one considers the set ''S'' of all functions that are not zero at ''p'' and localiz ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Masayoshi Miyanishi
Masayoshi is a masculine Japanese given name. Possible writings Masayoshi can be written using different kanji characters and can mean: *, "correct, justice, righteous; wherefore, a reason" *, "correct, justice, righteous; righteousness, justice, morality, honor, loyalty, meaning" *, "correct, justice, righteous; rejoice, take pleasure in" *, "correct, justice, righteous; intimate, friendly, harmonious" *, "correct, justice, righteous; graceful, gentle, pure" *, "correct, justice, righteous; good luck, joy, congratulations" *, "correct, justice, righteous; perfume, balmy, favorable, fragrant" *, "prosperous, bright, clear; good, pleasing, skilled" *, "prosperous, bright, clear; righteousness, justice, morality, honor, loyalty, meaning" *, "prosperous, bright, clear; good, bribe, servant" *, "gracious, elegant, graceful, refined; best regards, good" *, "gracious, elegant, graceful, refined; auspicious, happiness, blessedness, good omen, good fortune" *, "gracious, elegant, graceful, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tame Abstract Elementary Class
In model theory, a discipline within the field of mathematical logic, a tame abstract elementary class is an abstract elementary class (AEC) which satisfies a locality property for types called tameness. Even though it appears implicitly in earlier work of Shelah, tameness as a property of AEC was first isolated by Grossberg and VanDieren, who observed that tame AECs were much easier to handle than general AECs. Definition Let ''K'' be an AEC with joint embedding, amalgamation, and no maximal models. Just like in first-order model theory, this implies ''K'' has a universal model-homogeneous monster model \mathfrak. Working inside \mathfrak, we can define a semantic notion of types by specifying that two elements ''a'' and ''b'' have the same type over some base model M if there is an automorphism of the monster model sending ''a'' to ''b'' fixing M pointwise (note that types can be defined in a similar manner without using a monster model). Such types are called Galois types. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Affine Plane
In geometry, an affine plane is a two-dimensional affine space. Examples Typical examples of affine planes are *Euclidean planes, which are affine planes over the reals equipped with a metric, the Euclidean distance. In other words, an affine plane over the reals is a Euclidean plane in which one has "forgotten" the metric (that is, one does not talk of lengths nor of angle measures). * Vector spaces of dimension two, in which the zero vector is not considered as different from the other elements * For every field or division ring ''F'', the set ''F''2 of the pairs of elements of ''F'' * The result of removing any single line (and all the points on this line) from any projective plane Coordinates and isomorphism All the affine planes defined over a field are isomorphic. More precisely, the choice of an affine coordinate system (or, in the real case, a Cartesian coordinate system A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Automorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms of an object forms a group, called the automorphism group. It is, loosely speaking, the symmetry group of the object. Definition In the context of abstract algebra, a mathematical object is an algebraic structure such as a group, ring, or vector space. An automorphism is simply a bijective homomorphism of an object with itself. (The definition of a homomorphism depends on the type of algebraic structure; see, for example, group homomorphism, ring homomorphism, and linear operator.) The identity morphism ( identity mapping) is called the trivial automorphism in some contexts. Respectively, other (non-identity) automorphisms are called nontrivial automorphisms. The exact definition of an automorphism depends on the type of " ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Canad
{{Disambiguation, geo ...
Canad may refer to: * Sanjak of Çanad, an Ottoman-era district * Magyarcsanád, known in Serbian as Čanad, a village in Hungary * Cenad, known in Serbian as Čanad, a commune in Romania See also * Canad Inns, a chain of hotels * Canard (other) * Csanád (other) * Kanad, a town in India * Canada Canada is a country in North America. Its ten provinces and three territories extend from the Atlantic Ocean to the Pacific Ocean and northward into the Arctic Ocean, covering over , making it the world's second-largest country by tota ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangular Derivations
A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non-collinear, determine a unique triangle and simultaneously, a unique plane (i.e. a two-dimensional Euclidean space). In other words, there is only one plane that contains that triangle, and every triangle is contained in some plane. If the entire geometry is only the Euclidean plane, there is only one plane and all triangles are contained in it; however, in higher-dimensional Euclidean spaces, this is no longer true. This article is about triangles in Euclidean geometry, and in particular, the Euclidean plane, except where otherwise noted. Types of triangle The terminology for categorizing triangles is more than two thousand years old, having been defined on the very first page of Euclid's Elements. The names used for modern classification are eithe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Examples
Example may refer to: * '' exempli gratia'' (e.g.), usually read out in English as "for example" * .example, reserved as a domain name that may not be installed as a top-level domain of the Internet ** example.com, example.net, example.org, example.edu, second-level domain names reserved for use in documentation as examples * HMS ''Example'' (P165), an Archer-class patrol and training vessel of the Royal Navy Arts * ''The Example'', a 1634 play by James Shirley * ''The Example'' (comics), a 2009 graphic novel by Tom Taylor and Colin Wilson * Example (musician), the British dance musician Elliot John Gleave (born 1982) * ''Example'' (album), a 1995 album by American rock band For Squirrels See also * * Exemplar (other), a prototype or model which others can use to understand a topic better * Exemplum, medieval collections of short stories to be told in sermons * Eixample The Eixample (; ) is a district of Barcelona between the old city (Ciutat Vella) and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normalization Of An Algebraic Variety
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, ''t''2) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Contractible
In mathematics, a topological space ''X'' is contractible if the identity map on ''X'' is null-homotopic, i.e. if it is homotopic to some constant map. Intuitively, a contractible space is one that can be continuously shrunk to a point within that space. Properties A contractible space is precisely one with the homotopy type of a point. It follows that all the homotopy groups of a contractible space are trivial. Therefore any space with a nontrivial homotopy group cannot be contractible. Similarly, since singular homology is a homotopy invariant, the reduced homology groups of a contractible space are all trivial. For a topological space ''X'' the following are all equivalent: *''X'' is contractible (i.e. the identity map is null-homotopic). *''X'' is homotopy equivalent to a one-point space. *''X'' deformation retracts onto a point. (However, there exist contractible spaces which do not ''strongly'' deformation retract to a point.) *For any space ''Y'', any two maps ''f'',' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
