In mathematics, a regular surface may refer to A smooth surface , in differential geometry A surface with vanishing Irregularity of a surface irregularity , in algebraic geometry mathdab ... more details
language theory , a regular language is a formal language that can be expressed using a regular expression . Note that the regular expression features provided with many programming languages are Regular expression Patterns for non regular languages augmented with features that make them capable of recognizing languages that can not be expressed by the formal regular expressions as formally defined below . In the Chomsky hierarchy , regular languages are defined to be the languages that are generated by Type 3 grammars regular grammar s . Regular languages are very useful in input parsing and programming language design. Formal definition The collection of regular languages over an alphabet is defined recursively as follows The empty language is a regular language. For each a a belongs to , the Singleton mathematics singleton language a is a regular language. If A and B are regular languages, then A B union , A B concatenation , and A Kleene star are regular languages. No other languages over are regular. See Regular expression Formal language theory regular expression for its syntax and semantics. Note that the above cases are in effect the defining rules of regular expression. Examples All finite languages are regular in particular the empty string language is regular. Other typical examples include the language consisting of all strings over the alphabet ... several a s followed by several b s. A simple example of a language that is not regular is the set of strings ... to prove this fact rigorously are given below. Equivalence to other formalisms A regular language ... , or the more general alternating finite automaton it can be generated by a regular grammar it can ... on its alphabet The above properties are sometimes used as alternative definition of regular languages. Closure properties The regular languages are closure mathematics closed under the various operations, that is, if the languages K and L are regular, so is the result of the following operations the set ... more details
Graph families defined by their automorphisms In graph theory , a regular graph is a graph mathematics ... theory degree or valency. A regular directed graph must also satisfy the stronger condition that the indegree ... 29 isbn 9789810218591 ref A regular graph with vertices of degree span class texhtml var k var span is called a span class texhtml var k var span regular graph or regular graph of degree span class texhtml var k var span . Regular graphs of degree at most 2 are easy to classify A 0 regular graph consists of disconnected vertices, a 1 regular graph consists of disconnected edges, and a 2 regular graph consists of disconnected cycle graph theory cycle s and infinite chains. A 3 regular graph is known as a cubic graph . A strongly regular graph is a regular graph where every adjacent pair of vertices ... number n of neighbors in common. The smallest graphs that are regular but not strongly regular are the cycle ... regular for any math m math . A theorem by Crispin St. J. A. Nash Williams Nash Williams says that every span class texhtml var k var span regular graph on span class texhtml 2 var k var 1 span vertices has a Hamiltonian cycle . gallery Image 0 regular graph.svg 0 regular graph Image 1 regular graph.svg 1 regular graph Image 2 regular graph.svg 2 regular graph Image 3 regular graph.svg 3 regular graph gallery Algebraic properties Let A be the adjacency matrix of a graph. Then the graph is regular if and only if math textbf j 1, dots ,1 math is an eigenvector of A . ref name Cvetkovic ... math v v 1, dots,v n math , we have math sum i 1 n v i 0 math . A regular graph of degree k is connected ... for regular and connected graphs a graph is connected and regular if and only if the matrix J ... of powers of A . citation needed date February 2009 Let G be a k regular graph with diameter D and eigenvalues ... . citation needed date March 2009 Generation Regular graphs may be generated by the GenReg program. ref cite journal last Meringer first Markus year 1999 title Fast generation of regular graphs and construction ... more details
In field theory mathematics field theory , a branch of algebra, a field extension math L k math is said to be regular if k is algebraically closed in L and L is separable extension separable over k , or equivalently, math L otimes k overline k math is an integral domain when math overline k math is the algebraic closure of math k math that is, to say, math L, overline k math are linearly disjoint over k . In particular, any field extension of an algebraically closed field is regular. Also, a purely transcendental extension of a field is regular. There is also a similar notion a field extension math L k math is said to be self regular if math L otimes k L math is an integral domain. A self regular extension is algebraically closed in k . However, a self regular extension is not necessarily regular. Citation needed date February 2010 References M. Nagata 1985 . Commutative field theory new edition, Shokado. Japanese http www.shokabo.co.jp mybooks ISBN978 4 7853 1309 8.htm P.M. Cohn 2003 . Basic algebra. A. Weil, Foundations of algebraic geometry . Category Field theory Abstract algebra stub ... more details
In astronomy, a regular moon is a natural satellite following a relatively close and generally prograde orbit with little orbital inclination or orbital eccentricity eccentricity . They are believed to have formed in orbit about their primary astronomy primary , as opposed to irregular moon s, which were captured. There are at least 55 regular satellites of the eight planets one at Earth, eight at Jupiter, 22 named regular moons at Saturn not counting hundreds or thousands of moonlet s , 18 known at Uranus, and 6 small regular moons at Neptune. Large Triton appears to have been captured. It is thought that Pluto s four moons and Haumea s two were formed in orbit about those dwarf planet s out of Collisional family debris created in giant collisions . See also Irregular moon Inner moon Category Moons astronomy stub ca Sat l lit regular eo Regula satelito zh ... more details
In category theory , a regular category is a category with limit category theory finite limits and coequalizer ..., regular categories recapture many properties of abelian categories , like the existence of images , without requiring additivity. At the same time, regular categories provide a foundation for the study of a fragment of first order logic , known as regular logic. Definition A category C is called regular if it satisfies the following three properties C is finitely complete category finitely complete . If f X Y is a morphism in C , and div style text align center Image Regular category ... center Image Regular category 2.png div is a pullback, and if f is a regular epimorphism , then g is a regular epimorphism as well. A regular epimorphism is an epimorphism which appears as a coequalizer of some pair of morphisms. Examples Examples of regular categories include Category of sets Set ... are not regular Top , the category of topological space s and Continuous function topology continuous ... In a regular category, the regular epimorphism s and the monomorphism s form a factorization system . Every morphism f X Y can be factorized into a regular epimorphism e X E followed by a monomorphism m E Y , so that f me . The factorization is unique in the sense that if e X E is another regular ... the image of f . Exact sequences and regular functors In a regular category, a diagram of the form ... sense. A functor between regular categories is called regular , if it preserves finite limits and coequalizers of kernel pairs. A functor is regular if and only if it preserves finite limits and exact sequences. For this reason, regular functors are sometimes called exact functors . Functors that preserve finite limits are often said to be left exact . Regular logic and regular categories Regular logic is the fragment of first order logic that can express statements of the form center math forall x phi x to psi x math , center where math phi math and math psi math are regular Formula ... more details
class wikitable align right width 320 Regular polytope examples valign top File Regular pentagon.svg 160px BR A regular pentagon is a polygon , a two dimensional polytope with 5 Edge geometry edges , represented by Schl fli symbol 5 . Image POV Ray Dodecahedron.svg 160px BR A regular dodecahedron is a polyhedron ... symbol 5,3 . valign top File Schlegel wireframe 120 cell.png 160px BR A regular dodecaplex ... by Schl fli symbol 5,3,3 . shown here as a Schlegel diagram File Cubic honeycomb.png 160px BR A regular ... edges of an 8 cube can be shown in this orthogonal projection Petrie polygon In mathematics , a regular ... are also transitive on the symmetries of the polytope, and are regular polytopes of dimension n . Regular polytopes are the generalized analog in any number of dimensions of regular polygon s for example, the square geometry square or the regular pentagon and regular polyhedra for example, the cube . The strong symmetry of the regular polytopes gives them an aesthetics aesthetic quality that interests both non mathematicians and mathematicians. Classically, a regular polytope in n dimensions may be defined as having regular Facet geometry facets n &minus 1 faces and regular ... are alike. Note, however, that this definition does not work for abstract polytope s. A regular ... z , with regular facets as a, b, c, ..., y , and regular vertex figures as b, c, ..., y, z . Classification and description Regular polytopes are classified ... . For example the cube and the regular octahedron share the same symmetry, as do the regular dodecahedron and icosahedron . Indeed, symmetry groups are sometimes named after regular polytopes, for example the tetrahedral and icosahedral symmetries. Three special classes of regular polytope exist in every dimensionality Simplex Regular simplex Measure polytope Hypercube Cross polytope Orthoplex In two dimensions there are infinitely many regular polygon s. In three and four dimensions there are several ... more details
A regular semigroup is a semigroup S in which every element is regular , i.e., for each element a , there exists an element x such that axa a . ref Howie 1995 54. ref Regular semigroups are one of the most ... s relations . ref Howie 2002. ref Origins Regular semigroups were introduced by J. A. Green in his influential ... study of regular semigroups which led Green to define his celebrated Green s relations relations ... ways in which to define a regular semigroup S 1 for each a in S , there is an x in S , which ... way of expressing definition 2 above is to say that in a regular semigroup, V a is nonempty, for every ... aba a . ref Clifford and Preston 1961 p. 26. ref A regular semigroup in which idempotent s commute is an inverse semigroup , that is, every element has a unique inverse. To see this, let S be a regular ... of a is shown to be unique. Conversely, it can be shown that any inverse semigroup is a regular ... 6391 ref Theorem. The homomorphic image of a regular semigroup is regular. ref Howie 1995 Lemma 2.4.4. ref Examples of regular semigroups Every group mathematics group is regular. Every inverse semigroup is regular. Every band mathematics band idempotent semigroup is regular in the sense of this article, though this is not what is meant by a band mathematics Regular bands regular band . The bicyclic semigroup is regular. Any transformation semigroup full transformation semigroup is regular. A Rees matrix semigroup is regular. Green s relations Recall that the principal ideal s of a semigroup .... In a regular semigroup S , however, an element a axa automatically belongs to these ideals, without recourse to adjoining an identity. Green s relations can therefore be redefined for regular ... In a regular semigroup S , every math mathcal L math and math mathcal R math class contains at least .... ref Theorem. Let S be a regular semigroup, and let a and b be elements of S . Then math a , mathcal ... mathcal R math class is unique. ref name Howie 1995 Theorem 5.1.1 Special classes of regular semigroups ... more details
Regular semantics is a computing term which describes the guarantees provided by a data register shared by several processors in a parallel machine or in a Computer network network of computers working together. Regular semantics are defined for a variable computer science variable with a single writer but multiple readers. These semantics are stronger than safe semantics but weaker than atomic semantics they guarantee that there is a total order to the write operations which is consistent with real time computing real time and that read operations return either the value of the last completed write or that of one of the writes which are concurrent with the read. See also Atomic semantics Safe semantics References Lamport, Leslie On Interprocess Communication http research.microsoft.com en us um people lamport pubs interprocess.pdf 1986 DEFAULTSORT Regular Semantics Category Concurrency control Comp sci stub ... more details
A regular polyhedron is a polyhedron whose faces are Congruence geometry congruent regular polygons which are assembled in the same way around each vertex geometry vertex . A regular polyhedron is highly ... on its Flag geometry flag s. This last alone is a sufficient definition. A regular polyhedron ... and m the number of faces meeting at each vertex. There are 5 finite regular polyhedra, which are called ... pair dodecahedron icosahedron 5,3 . The regular polyhedra There are five Convex polygon convex regular polyhedra, known as the Platonic solid s , and four regular star polyhedron star polyhedra , the Kepler ... are regular polygon s. All the solid angle s of the polyhedron are congruent. Cromwell, 1997 Concentric spheres A regular polyhedron has all of three related spheres other polyhedra lack at least one ... to all edges. A circumsphere , tangent to all vertices. Symmetry The regular polyhedra are the most ... solids have an Euler characteristic of 2. Some of the regular stars have a Kepler Poinsot polyhedra ... in the interior of a regular polyhedron to the sides is independent of the location of the point. This is an extension ... Journal 37 5 , 2006, pp. 390 391. ref Duality of the regular polyhedra The regular polyhedra come in natural ... is linked from Polyhedron See also Regular polytope History of discovery Regular polytope History ... preceded the Greeks in their awareness of at least some of the regular polyhedra, as evidenced ... though not quite regular and may be found in northern Italy Citation needed date February 2007 . Greeks The earliest known written records of the regular convex solids originated from Classical ... characterise the Greek definition as follows A regular polygon is a Convex polygon convex planar figure with all edges equal and all corners equal A regular polyhedron is a solid convex figure with all faces being congruent regular polygons, the same number arranged all alike around each vertex. This definition rules out, for example, the square pyramid since although all the faces are regular, the square ... more details
In complex analysis , see holomorphic function . In mathematics , a regular function is a function that is analytic function analytic in a given region. ref http mathworld.wolfram.com RegularFunction.html Wolfram Mathworld Regular Function ref In complex analysis, any complex regular function is known as a holomorphic function . In algebraic geometry the term takes up a more specific definition, referring to an everywhere defined, polynomial function on an algebraic variety V with values in the field mathematics field K over which V is defined. For example, if V is the affine line over K , the regular functions on V make up a commutative ring , under pointwise multiplication of functions, isomorphic with the polynomial ring in one variable over K . In other words, the regular functions are just polynomials in some natural parameter on the affine line. More generally, for any affine variety V , the regular functions make up the coordinate ring of V , often written K V . This can be expressed in other ways. A regular function is the same as a morphism to the affine line, or in the language of scheme theory a global section of the structure sheaf . The reason for looking at regular functions becomes more apparent when one allows V to be a projective variety . Then regular functions on V become rare. For example morphisms from a projective space to the affine line must be constant regular functions on a projective space are constant functions. The same is true for any connected projective variety this can be viewed as an algebraic analogue of Liouville s theorem complex analysis Liouville s theorem in complex analysis . In fact taking the function field of an algebraic variety function field K V of an irreducible variety irreducible algebraic curve V , the functions F in the function field may all be realised as morphisms from V to the projective line over K . The image will either ... DEFAULTSORT Regular Function Category Algebraic varieties Category Types of functions nl Regelmatige ... more details
In set theory , a regular cardinal is a cardinal number that is equal to its own cofinality . So, crudely speaking, a regular cardinal is one which cannot be broken into a smaller collection of smaller parts. If the axiom of choice holds so that any cardinal number can be well ordered , an infinite cardinal math kappa math is regular if and only if it cannot be expressed as the cardinal sum of a set of cardinality less than math kappa math , the elements of which are cardinals less than math kappa math . The situation is slightly more complicated in contexts where the axiom of choice might fail ... math is regular if and only if it is a limit ordinal which is not the limit of a set of smaller ordinals which set has order type less than math alpha math . A regular ordinal is always an initial ordinal , though some initial ordinals are not regular. Infinite well ordered cardinals which are not regular are called singular cardinals . Finite cardinal numbers are typically not called regular or singular ... less than math omega math whose elements are ordinals less than math omega math , and is therefore a regular ordinal. math aleph 0 math aleph null is a regular cardinal because its initial ordinal, math omega math , is regular. It can also be seen directly to be regular, as the cardinal sum of a finite ... cardinal numbers, and is regular. math aleph omega math is the next cardinal number after the sequence ... regular are known as weakly inaccessible cardinals . They cannot be proved to exist within ... s of the aleph number aleph function , though not all fixed points are regular. For instance, the first ... cardinal is regular. Thus the regularity or singularity of most aleph numbers can be checked depending ... which is regular. Without the axiom of choice, there would be cardinal numbers which were not well ... only the aleph number s can meaningfully be called regular or singular cardinals. Furthermore, a successor aleph need not be regular. For instance, the union of a countable set of countable sets need ... more details
In mathematics , a regular measure on a topological space is a measure mathematics measure for which every measurable set is approximately open and approximately closed . Definition Let X , T be a topological space and let be a sigma algebra &sigma algebra on X that contains the topology T so that all open set open and closed set s are measurable set s, and is at least as fine as the Borel sigma algebra Borel &sigma algebra on X . Let be a measure on X , . A measurable subset A of X is said to be &mu regular if math mu A sup mu F F subseteq A, F mbox closed math and math mu A inf mu G G supseteq A, G mbox open . math Alternatively, A is a &mu regular set if and only if , for every > 0, there exists a closed set F and an open set G such that math F subseteq A subseteq G math and math mu G setminus F delta. math The two definitions are equivalent if math mu A math is finite otherwise, the second definition is stronger . If every measurable set is regular, then the measure &mu is said to be a regular measure . Some authors require the set F to be compact not just closed . ref harvnb Dudley 1989 loc Sect. 7.1 ref Examples Lebesgue measure on the real line is a regular measure see the regularity theorem for Lebesgue measure . Any Borel probability measure on any metric space is a regular measure. The trivial measure , which assigns measure zero to every measurable subset, is a regular measure. A trivial example of a non regular measure on the real line with its usual topology is the measure where math mu emptyset 0 math , math mu left 1 right 0 , , math , and math mu A infty , , math for any other set math A math . Notes references References cite book last Billingsley first Patrick title Convergence of Probability Measures publisher John Wiley & Sons, Inc. location New York year 1999 isbn 0 471 19745 9 cite book last Parthasarathy first K. R. title Probability ... regular measure Radon measure Regularity theorem for Lebesgue measure Category Measures measure ... more details
In the mathematics mathematical subject of knot theory , a regular isotopy of a Knot diagram Knot diagrams link diagram is the equivalence relation generated by using the 2nd and 3rd Reidemeister move s only. The notion of regular isotopy was introduced by Louis Kauffman Kauffman 1990 . It can be thought of as an isotopy of a ribbon pressed flat against the plane which keeps the ribbon flat. For diagrams in the plane this is a finer equivalence relation than ambient isotopy of a framed link , since the 2nd and 3rd Reidemeister moves preserve the winding number of the diagram Kauffman 1990, pp. 450ff. . However, for diagrams in the sphere considered as the plane plus infinity , the two notions are equivalent, due to the extra freedom of passing a strand through infinity. See also ambient isotopy knot polynomial Notes Empty section date July 2010 References L. H. Kauffman An invariant of regular isotopy , Transactions of the American Mathematical Society 318 2 , 1990, pp. 417 471 Category Knot theory knottheory stub ... more details
for regular Cauchy sequence Cauchy sequence In constructive mathematics In commutative algebra , if R is a commutative ring and M an R Module mathematics module , a nonzero element r in R is called M regular if r is not a zerodivisor on M , and M rM is nonzero. An R regular sequence on M is a d tuple r sub 1 sub , ..., r sub d sub in R such that for each i &le d , r sub i sub is M sub i 1 sub regular, where M sub i 1 sub is the quotient R module M r sub 1 sub , ..., r sub i 1 sub M . Such a sequence is also called an M sequence. An R regular sequence is usually called simply a regular sequence . It may be that r sub 1 sub , ..., r sub d sub is an M sequence, and yet some permutation of the sequence is not. It is, however, a theorem that if R is a local ring or if R is a graded ring and the r sub i sub are all homogeneous, then a sequence is an R sequence only if every permutation of it is an R sequence. The depth ring theory depth of R is defined as the maximum length of a regular R sequence on R . More generally, the depth of an R module M is the maximum length of an M regular sequence on M . The concept is inherently module theoretic and so there is no harm in approaching it from this point of view. The depth of a module is always at least 0 and no greater than the Krull dimension of the module. Examples If k is a field, it possesses no non zero non unit elements so its depth as a k module is 0 . If k is a field and X is an indeterminate, then X is a nonzerodivisor on the formal power series ring R k nowiki nowiki X nowiki nowiki , but R XR is a field and has no further nonzerodivisors. Therefore R has depth 1. If k is a field and X sub 1 sub , X sub 2 sub , ..., X sub d sub are indeterminates, then X sub 1 sub , X sub 2 sub , ..., X sub d sub form a regular sequence of length d on the polynomial ring k X sub 1 sub , X sub 2 sub , ..., X sub d sub and there are no longer R sequences, so R has depth d , as does the formal power series ring in d indeterminates over any ... more details
A regular economy is an economy characterized by an excess demand function which has the property that its slope at any equilibrium price vector is non zero. In other words, if we graph the excess demand function against prices, then the excess demand function cuts the x axis assuring that each equilibrium is locally unique. Local uniqueness in turn permits the use of comparative statics an analysis of how the economy responds to external shocks as long as these shocks are not too large. An important result due to G rard Debreu Debreu 1970 states that almost any economy, defined by an initial distribution of consumer s endowments, is regular. In technical terms, the set of nonregular economies is of Lebesgue measure zero. Combined with the Hopf index theorem index theorem this result implies that almost any economy will have a finite and odd number of equilibria. Image Regular economy.JPG thumb 300px Regular and nonregular economies References Debreu, G. 1970 Economies with a finite set of equilibria , Econometrica , 38 . Andreu Mas Colell Mas Colell, A. , Whinston, M. and Green, J. 1995 . Microeconomic Theory , Oxford University Press Category General equilibrium and disequilibrium ... more details
In the mathematics mathematical field of topology , a regular homotopy refers to a special kind of homotopy between immersion mathematics immersion s of one manifold in another. The homotopy must be a 1 parameter family of immersions. Similar to homotopy class es, one defines two immersions to be in the same regular homotopy class if there exists a regular homotopy between them. Regular homotopy for immersions is similar to isotopy of embeddings they are both restricted types of homotopies. Stated another way, two continuous functions math f,g M to N math are homotopic if they represent points in the same path components of the mapping space math C M,N math , given the compact open topology . The space of immersions is the subspace of math C M,N math consisting of immersions, denote it by math Imm M,N math . Two immersions math f,g M to N math are regularly homotopic if they represent points in the same path component of math Imm M,N math . Examples File Winding Number Around Point.svg thumb 300px This curve has total curvature 6 , and turning number 3. The Whitney Graustein theorem anchor Whitney Graustein theorem classifies the regular homotopy classes of a circle into the plane two immersions are regularly homotopic if and only if they have the same turning number equivalently, total curvature equivalently, if and only if their Gauss map s have the same degree winding number . File MorinSurfaceFromTheTop.PNG thumb Smale s classification of immersions of spheres shows that sphere eversion s exist, which can be realized via this Morin surface . Stephen Smale classified the regular homotopy classes of a k sphere immersed in math mathbb R n math they are classified by homotopy ... derivatives not vanishing. A corollary of his work is that there is only one regular homotopy class ..., i.e. one can turn the 2 sphere inside out . Both of these examples consist of reducing regular ... 4 276 0 On regular closed curves in the plane . Compositio Mathematica , 4 1937 , p. 276 284 Stephen ... more details
for regular irreducible representations of a finite group Gelfand Graev representation In mathematics , and in particular the theory of group representation s, the regular representation of a group G is the linear ... translation . One distinguishes the left regular representation given by left translation and the right regular representation given by the inverse of right translation. Finite groups For a finite group G , the left regular representation over a field mathematics field K is a linear representation ... by g , i.e. math lambda g h mapsto gh, text for all h in G. math For the right regular representation ... that the regular representation is generalized to topological group s such as Lie group ... g G , math lambda g f x f g 1 x math and math rho g f x f xg . math Significance of the regular ... a single orbit group theory orbit and Group action stabilizer the identity subgroup e of G . The regular ... the permutation representation doesn t decompose it is group action transitive the regular ... and K is the complex number field, the regular representation decomposes as a direct sum of representations ... to the number of conjugacy class es of G . The article on group algebra s articulates the regular representation for finite group s, as well as showing how the regular representation can be taken to be a module ... sum decomposition of the regular representation contains a representative of every isomorphism class of irreducible linear representations of G over K . You can say that the regular representation ... an algebraically closed field K such as the complex number s the regular representation of G is completely ... group G , the regular representation in the above sense should be replaced by a suitable space ... G module is the regular representation. This is the content of the normal basis theorem , a normal basis ... general algebras The regular representation of a group ring is such that the left hand and right hand regular representations give isomorphic modules and we often need not distinguish the cases . Given ... more details
In computer algebra , a regular chain is a particular kind of triangular set in a multivariate polynomial ring over a field. It enhances the notion of characteristic set . Introduction Given a linear system, one can convert it to a triangular system via Gaussian elimination . For the non linear case ... of regular chain was introduced, independently by Kalkbrener 1993 , Yang and Zhang 1994 . Regular chains also appear in Chou and Gao 1992 . Regular chains are special triangular sets which are used in different ... by regular chains. Examples Denote Q the rational number field. In Q x sub 1 sub , x sub 2 sub ... 3 x 1 math is a triangular set and also a regular chain. Two generic points given by T are a, a, a and a, a, a where ... of each component is 1, the number of free variables in the regular chain. Formal definitions The variables ... set is finite, and has cardinality at most n . Regular chain Let T t sub 1 sub , ..., t sub s sub ... of t sub i sub and h be the product of h sub i sub s. Then T is a regular chain if math mathrm resultant ... of a regular chain The quasi component W T described by the regular chain T is math W T V T setminus ... of a regular chain is its saturated ideal math mathrm sat T T h infty math . A classic result is that the Zariski ... algorithms available for triangular decompositions in either sense. Properties Let T be a regular chain in the polynomial ring R . The saturated ideal sat T is an unmixed ideal with dimension n T . A regular ... for sat T is algorithmic. Given a prime ideal P , there exists a regular chain C such that P sat C . If the first element of a regular chain C is an irreducible polynomial and the others are linear ... almost all linear changes of variables, there exists a regular chain C of the preceding shape such that P sat C . A triangular set is a regular chain if and only if it is a Ritt characteristic set of its saturated ideal. See also Characteristic set Groebner basis RegularChains Regular semi algebraic .... J. Symb. Comput. 15 2 143&ndash 167 1993 . D. Wang. Computing Triangular Systems and Regular Systems ... more details
In computing, a regular expression provides a concise and flexible means to match specify and recognize .... Common abbreviations for regular expression include regex and regexp . The concept of regular ... ed text editor ed and the filter grep . Citation needed date September 2011 A regular expression is written in a Formal language Programming languages formal language that can be interpreted by a regular ... specification . Historically, the concept of regular expressions is associated with Stephen Cole Kleene Kleene s formalism of regular sets , introduced in the 1950s. The following are examples of specifications which can be expressed as a regular expression the sequence of characters car appearing ... preceded by the word motor and separated by a named delimiter, or multiple. Regular expressions are used ... , integrate regular expressions into the syntax of the core language itself. Other programming ... language Python instead provide regular expressions through standard libraries. For yet other languages ... however, version C 11 provides regular expressions in its Standard Libraries . As an example of the syntax, the regular expression code bex code can be used to search for all instances of the string ... globbing . Wildcards differ from regular expressions in generally expressing only limited forms of patterns. Basic concepts A regular expression, often called a pattern, is an expression that specifies ... number of such expressions. Most formalisms provide the following operations to construct regular .... For example, code DON T CHANGE THIS TO colo?r REGULAR EXPRESSIONS DON T WORK LIKE WILDCARDS colou ... strings as the earlier example, code H ae? ndel code . The precise syntax for regular expressions ... See Pattern matching History The origins of regular expressions lie in automata theory and formal language ... regular sets . ref harvtxt Kleene 1956 ref The SNOBOL language was an early implementation of pattern matching , but not identical to regular expressions. Ken Thompson built Kleene s notation into the editor ... more details
Unreferenced date April 2007 A regular verb is any verb whose Grammatical conjugation conjugation follows the typical grammatical inflections of the language to which it belongs. A verb that cannot be conjugated like this is called an irregular verb . All natural language s, to different extents, have a number of irregular verbs. Auxiliary language s usually have a single regular pattern for all verbs as well as other part of speech parts of speech as a matter of design. Other constructed language s need not show such regularity, especially if they are designed to look similar to natural ones. The most simple form of regularity involves a single class of verbs, a single principal part the root linguistics root or a conjugated form in a given person, number, tense, aspect, mood, etc. , and a set of unique rules to produce each form in the verb paradigm . More complex regular patterns may have several verb classes e. g. distinguished by their infinitive ending , more than one principal part e. g. the infinitive and the first person singular, present tense, indicative mood , and more than one type of rule e. g. rules that add suffixes and other rules that change the vowel in the root . Sometimes it is highly subjective to state whether a verb is regular or not. For example, if a language has ten different conjugation patterns and two of them only comprise five or six verbs each while the rest are much more populated, it is a matter of choice to call the verbs in the smaller groups irregular . The concept of regular and irregular verbs belongs mainly in the context of second language acquisition, where the defining of rules and listing of exceptions is an important part of foreign language learning. The concepts can also be useful in psycholinguistics, where the ways in which ... verbs as though they were regular. This is regarded as evidence that we learn and process our native ... in their earliest utterances but then switch to incorrect regular forms for a time when they begin ... more details
, but the mapping from indexes to vertex coordinates is less uniform than in a regular grid. An example of a rectilinear grid that is not regular appears on logarithmic scale graph paper . A curvilinear grid or structured grid is a grid with the same combinatorial structure as a regular grid, in which ... stub DEFAULTSORT Regular Grid Category Tessellation Category Lattice points Category Mesh generation ... more details
Miles Devine was recruited. Regular John are currently working on their second album which Goman claims will be a motherfucker . During early 2011, Regular John appeared at the Big Day Out and shortly ... music albumreviews regular john the peaceful atom is a bomb.aspx Regular John The Peaceful ... 45 3Aalbums&id 2464 3Aregular john&tmpl component&print 1&page &option com content&Itemid 43 Regular ... regularjohn Regular John MySpace page DEFAULTSORT Regular John Category New South Wales musical ... more details
Image Regular divisibility lattice.svg thumb 360px A Hasse diagram of divisibility relationships among the regular numbers up to 400. The vertical scale is Logarithmic scale logarithmic . ref Inspired ..., and divisor lattices . ref Regular numbers are numbers that evenly divide powers of 60. As an example ... of a power of 60. Thus, they are also regular numbers . The numbers that evenly divide the powers ... of Babylonian mathematics , the divisors of powers of 60 are called regular numbers or regular .... In music theory , regular numbers occur in the ratios of tones in just intonation , also called Limit music 5 limit tuning for this reason. In computer science , regular numbers are often ... algorithm s for generating these numbers in order. Number theory Formally, a regular number is an integer ... is a divisor of math scriptstyle 60 max lceil i , 2 rceil,j,k math . The regular numbers ... few regular numbers are 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 25, 27, 30, 32 ... involving 5 smoothness . ref Although the regular numbers appear dense within the range from 1 to 60, they are quite sparse among the larger integers. A regular number n 2 sup i sup 3 sup j ... N . Therefore, the number of regular numbers that are at most N can be estimated as the volume ... big O notation , the number of regular numbers up to N is math frac left ln N sqrt 30 right 3 6 ... inverse reciprocal of a regular number has a finite representation, thus being easy to divide ... by the regular number 54 2 sup 1 sup 3 sup 3 sup . 54 is a divisor of 60 sup 3 sup , and 60 sup ... three sexagesimal places. The Babylonians used tables of reciprocals of regular numbers, some of which .... ref name aa harvtxt Aaboe 1965 . ref Although the primary reason for preferring regular numbers ... than reciprocals also involved regular numbers. For instance, tables of regular squares have been found ... 2 q 2 math generated by p , q both regular and less than 60. ref See harvtxt Conway Guy 1996 for a popular ... more details
Regular clergy , or just regulars , is applied in the Roman Catholic Church to clerics who follow a rule Latin regula in their life. Strictly, it means those members of religious order s who have made solemn profession . It contrasts with secular clergy . Terminology and history The observance of the Rule of St. Benedict procured for Benedictine monks at an early period the name of regulars . The Council of Verneuil 755 so refers to them in its third canon, and in its eleventh canon speaks of the ordo regularis as opposed to the ordo canonicus , formed by the canons who lived under the bishop according to the canonical regulations. There was question also of a regula canonicorum , or regula canonica , especially after the extension of the rule which Chrodegang , Bishop of Metz , had drawn up from the sacred canons 766 . ref cf. capitularies n. 69 circa 810, n. 138 of 818, 819, ed. Alf. Boretii ref . And when the canons were divided into two classes in the eleventh century, it was natural to call those who added religious poverty to their common life regulars, and those who gave up the common life, seculars. Before this we find mention of s culares canonici in the Chronicle of St. Bertin 821 ref Mart ne , Anecdot., III, 505. ref In fact as the monks were said to leave the world ref Augustine of Hippo , Serm. 40 de div. ref , sometimes those persons who were neither clerics nor monks were called seculars, as at times were clerics not bound by the rule. Sometimes also the name regulars was applied to the canons regular to distinguish them from monks. Thus the collection of Gratian jurist Gratian about 1139 ref C. xix, q. 2, c. 2 and q. 3, c. 1. ref speaks of canons regular, who make canonical profession, and live in a regular canonicate, in opposition to monks who wear the monastic ..., while the word religious is more generally used, the word regular is reserved for members of religious ... law fr Clerg r gulier pt Clero regular ... more details
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