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Regular Digon In Spherical Geometry-2
Regular may refer to: Arts, entertainment, and media Music * "Regular" (Badfinger song) * Regular tunings of stringed instruments, tunings with equal intervals between the paired notes of successive open strings Other uses * Regular character, a main character who appears more frequently and/or prominently than a recurring character * Regular division of the plane, a series of drawings by the Dutch artist M. C. Escher which began in 1936 Language * Regular inflection, the formation of derived forms such as plurals in ways that are typical for the language ** Regular verb * Regular script, the newest of the Chinese script styles Mathematics Algebra and number theory * Regular category, a kind of category that has similarities to both Abelian categories and to the category of sets * Regular chains in computer algebra * Regular element (other), certain kinds of elements of an algebraic structure * Regular extension of fields * Regular ideal (multiple defini ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular (Badfinger Song)
Regular may refer to: Arts, entertainment, and media Music * Regular (Badfinger song), "Regular" (Badfinger song) * Regular tunings of stringed instruments, tunings with equal intervals between the paired notes of successive open strings Other uses * Regular character, a main character who appears more frequently and/or prominently than a recurring character * Regular division of the plane, a series of drawings by the Dutch artist M. C. Escher which began in 1936 Language * Regular inflection, the formation of derived forms such as plurals in ways that are typical for the language ** Regular verb * Regular script, the newest of the Chinese script styles Mathematics Algebra and number theory * Regular category, a kind of category that has similarities to both Abelian categories and to the category of sets * Regular chains in computer algebra * Regular element (other), certain kinds of elements of an algebraic structure * Regular extension of fields * Regular ideal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular P-group
In mathematical finite group theory, the concept of regular ''p''-group captures some of the more important properties of abelian ''p''-groups, but is general enough to include most "small" ''p''-groups. Regular ''p''-groups were introduced by . Definition A finite ''p''-group ''G'' is said to be regular if any of the following equivalent , conditions are satisfied: * For every ''a'', ''b'' in ''G'', there is a ''c'' in the derived subgroup ' of the subgroup ''H'' of ''G'' generated by ''a'' and ''b'', such that ''a''''p'' · ''b''''p'' = (''ab'')''p'' · ''c''''p''. * For every ''a'', ''b'' in ''G'', there are elements ''c''''i'' in the derived subgroup of the subgroup generated by ''a'' and ''b'', such that ''a''''p'' · ''b''''p'' = (''ab'')''p'' · ''c''1''p'' ⋯ ''c''k''p''. * For every ''a'', ''b'' in ''G'' and every positive integer ''n'', there are elements ''c''''i'' in the derived subgroup of the subgroup generated by ''a'' and ''b'' such that ''a''''q'' · ''b''''q' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Singular Point
In mathematics, in the theory of ordinary differential equations in the complex plane \Complex, the points of \Complex are classified into ''ordinary points'', at which the equation's coefficients are analytic functions, and ''singular points'', at which some coefficient has a singularity. Then amongst singular points, an important distinction is made between a regular singular point, where the growth of solutions is bounded (in any small sector) by an algebraic function, and an irregular singular point, where the full solution set requires functions with higher growth rates. This distinction occurs, for example, between the hypergeometric equation, with three regular singular points, and the Bessel equation which is in a sense a limiting case, but where the analytic properties are substantially different. Formal definitions More precisely, consider an ordinary linear differential equation of -th order f^(z) + \sum_^ p_i(z) f^ (z) = 0 with meromorphic functions. The equa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Part
In mathematics, the regular part of a Laurent series In mathematics, the Laurent series of a complex function f(z) is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where a Taylor series expansio ... consists of the series of terms with positive powers.. That is, if :f(z) = \sum_^ a_n (z - c)^n, then the regular part of this Laurent series is :\sum_^ a_n (z - c)^n. In contrast, the series of terms with negative powers is the principal part. References Complex analysis {{mathanalysis-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Measure
In mathematics, a regular measure on a topological space is a measure for which every measurable set can be approximated from above by open measurable sets and from below by compact measurable sets. Definition Let (''X'', ''T'') be a topological space and let Σ be a σ-algebra on ''X''. Let ''μ'' be a measure on (''X'', Σ). A measurable subset ''A'' of ''X'' is said to be inner regular if :\mu (A) = \sup \ This property is sometimes referred to in words as "approximation from within by compact sets." Some authors use the term tight as a synonym for inner regular. This use of the term is closely related to tightness of a family of measures, since a finite measure ''μ'' is inner regular if and only if, for all ''ε'' > 0, there is some compact subset ''K'' of ''X'' such that ''μ''(''X'' \ ''K'') < ''ε''. This is precisely the condition that the [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Function
In algebraic geometry, a morphism between algebraic varieties is a function between the varieties that is given locally by polynomials. It is also called a regular map. A morphism from an algebraic variety to the affine line is also called a regular function. A regular map whose inverse is also regular is called biregular, and the biregular maps are the isomorphisms of algebraic varieties. Because regular and biregular are very restrictive conditions – there are no non-constant regular functions on projective varieties – the concepts of rational and birational maps are widely used as well; they are partial functions that are defined locally by rational fractions instead of polynomials. An algebraic variety has naturally the structure of a locally ringed space; a morphism between algebraic varieties is precisely a morphism of the underlying locally ringed spaces. Definition If ''X'' and ''Y'' are closed subvarieties of \mathbb^n and \mathbb^m (so they are affine vari ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cauchy-continuous Function
In mathematics, a Cauchy-continuous, or Cauchy-regular, function is a special kind of continuous function between metric spaces (or more general spaces). Cauchy-continuous functions have the useful property that they can always be (uniquely) extended to the Cauchy completion of their domain. Definition Let X and Y be metric spaces, and let f : X \to Y be a function from X to Y. Then f is Cauchy-continuous if and only if, given any Cauchy sequence \left(x_1, x_2, \ldots\right) in X, the sequence \left(f\left(x_1\right), f\left(x_2\right), \ldots\right) is a Cauchy sequence in Y. Properties Every uniformly continuous function is also Cauchy-continuous. Conversely, if the domain X is totally bounded, then every Cauchy-continuous function is uniformly continuous. More generally, even if X is not totally bounded, a function on X is Cauchy-continuous if and only if it is uniformly continuous on every totally bounded subset of X. Every Cauchy-continuous function is continuous. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Borel Regular Measure
In mathematics, an outer measure ''μ'' on ''n''-dimensional Euclidean space R''n'' is called a Borel regular measure if the following two conditions hold: * Every Borel set ''B'' ⊆ R''n'' is ''μ''-measurable in the sense of Carathéodory's criterion: for every ''A'' ⊆ R''n'', ::\mu (A) = \mu (A \cap B) + \mu (A \setminus B). * For every set ''A'' ⊆ R''n'' there exists a Borel set ''B'' ⊆ R''n'' such that ''A'' ⊆ ''B'' and ''μ''(''A'') = ''μ''(''B''). Notice that the set ''A'' need not be ''μ''-measurable: ''μ''(''A'') is however well defined as ''μ'' is an outer measure. An outer measure satisfying only the first of these two requirements is called a ''Borel measure'', while an outer measure satisfying only the second requirement (with the Borel set B replaced by a measurable set B) is called a ''regular measure''. The Lebesgue outer measure on R''n'' is an example of a Borel regular measure. It can ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semigroup With Involution
In mathematics, particularly in abstract algebra, a semigroup with involution or a *-semigroup is a semigroup equipped with an involutive anti-automorphism, which—roughly speaking—brings it closer to a group because this involution, considered as unary operator, exhibits certain fundamental properties of the operation of taking the inverse in a group: * Uniqueness * Double application "cancelling itself out". * The same interaction law with the binary operation as in the case of the group inverse. It is thus not a surprise that any group is a semigroup with involution. However, there are significant natural examples of semigroups with involution that are not groups. An example from linear algebra is a set of real-valued n-by-n square matrices with the matrix-transpose as the involution. The map which sends a matrix to its transpose is an involution because the transpose is well defined for any matrix and obeys the law , which has the same form of interaction with multi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Semigroup
In mathematics, a regular semigroup is a semigroup ''S'' in which every element is regular, i.e., for each element ''a'' in ''S'' there exists an element ''x'' in ''S'' such that . Regular semigroups are one of the most-studied classes of semigroups, and their structure is particularly amenable to study via Green's relations. History Regular semigroups were introduced by J. A. Green in his influential 1951 paper "On the structure of semigroups"; this was also the paper in which Green's relations were introduced. The concept of ''regularity'' in a semigroup was adapted from an analogous condition for rings, already considered by John von Neumann. It was Green's study of regular semigroups which led him to define his celebrated relations. According to a footnote in Green 1951, the suggestion that the notion of regularity be applied to semigroups was first made by David Rees. The term inversive semigroup (French: demi-groupe inversif) was historically used as synonym in the p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Semi-algebraic System
In computer algebra, a regular semi-algebraic system is a particular kind of triangular system of multivariate polynomials over a real closed field. Introduction Regular chains and triangular decompositions are fundamental and well-developed tools for describing the complex solutions of polynomial systems. The notion of a regular semi-algebraic system is an adaptation of the concept of a regular chain focusing on solutions of the real analogue: semi-algebraic systems. Any semi-algebraic system S can be decomposed into finitely many regular semi-algebraic systems S_1, \ldots, S_e such that a point (with real coordinates) is a solution of S if and only if it is a solution of one of the systems S_1, \ldots, S_e.Changbo Chen, James H. Davenport, John P. May, Marc Moreno-Maza, Bican Xia, Rong XiaoTriangular decomposition of semi-algebraic systems Proceedings of 2010 International Symposium on Symbolic and Algebraic Computation (ISSAC 2010), ACM Press, pp. 187–194, 2010. Forma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Von Neumann Regular Ring
In mathematics, a von Neumann regular ring is a ring ''R'' (associative, with 1, not necessarily commutative) such that for every element ''a'' in ''R'' there exists an ''x'' in ''R'' with . One may think of ''x'' as a "weak inverse" of the element ''a''; in general ''x'' is not uniquely determined by ''a''. Von Neumann regular rings are also called absolutely flat rings, because these rings are characterized by the fact that every left ''R''-module is flat. Von Neumann regular rings were introduced by under the name of "regular rings", in the course of his study of von Neumann algebras and continuous geometry. Von Neumann regular rings should not be confused with the unrelated regular rings and regular local rings of commutative algebra. An element ''a'' of a ring is called a von Neumann regular element if there exists an ''x'' such that . An ideal \mathfrak is called a (von Neumann) regular ideal if for every element ''a'' in \mathfrak there exists an element ''x'' in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |