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c_i2m6alucv1kl | In mathematics, a generalized hypergeometric series is a power series in which the ratio of successive coefficients indexed by n is a rational function of n. The series, if convergent, defines a generalized hypergeometric function, which may then be defined over a wider domain of the argument by analytic continuation. ... | Generalized hypergeometric series |
c_0g4s3w77i9ne | In mathematics, a generalized map is a topological model which allows one to represent and to handle subdivided objects. This model was defined starting from combinatorial maps in order to represent non-orientable and open subdivisions, which is not possible with combinatorial maps. The main advantage of generalized ma... | Generalized maps |
c_1vutn4njr577 | In mathematics, a generalized permutation matrix (or monomial matrix) is a matrix with the same nonzero pattern as a permutation matrix, i.e. there is exactly one nonzero entry in each row and each column. Unlike a permutation matrix, where the nonzero entry must be 1, in a generalized permutation matrix the nonzero en... | Monomial matrix |
c_2xs4r9hlg8dq | In mathematics, a generalized polygon is an incidence structure introduced by Jacques Tits in 1959. Generalized n-gons encompass as special cases projective planes (generalized triangles, n = 3) and generalized quadrangles (n = 4). Many generalized polygons arise from groups of Lie type, but there are also exotic ones ... | Generalized n-gon |
c_6agyi2e6lhz5 | In mathematics, a generating function is a way of encoding an infinite sequence of numbers (an) by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinary series, the formal power series is not required to converge: in fact, the gener... | Exponential generating series |
c_mh00yn3r15xv | One can generalize to formal power series in more than one indeterminate, to encode information about infinite multi-dimensional arrays of numbers. There are various types of generating functions, including ordinary generating functions, exponential generating functions, Lambert series, Bell series, and Dirichlet serie... | Exponential generating series |
c_wor195xoa6wv | The particular generating function, if any, that is most useful in a given context will depend upon the nature of the sequence and the details of the problem being addressed. Generating functions are often expressed in closed form (rather than as a series), by some expression involving operations defined for formal ser... | Exponential generating series |
c_djm8gh64rre6 | However such interpretation is not required to be possible, because formal series are not required to give a convergent series when a nonzero numeric value is substituted for x. Also, not all expressions that are meaningful as functions of x are meaningful as expressions designating formal series; for example, negative... | Exponential generating series |
c_2vbfu6l9z03g | In mathematics, a generating set Γ of a module M over a ring R is a subset of M such that the smallest submodule of M containing Γ is M itself (the smallest submodule containing a subset is the intersection of all submodules containing the set). The set Γ is then said to generate M. For example, the ring R is generated... | Generating set of a module |
c_cbpmsj1j08gu | In particular, a principal ideal is an ideal that has a generating set consisting of a single element. Explicitly, if Γ is a generating set of a module M, then every element of M is a (finite) R-linear combination of some elements of Γ; i.e., for each x in M, there are r1, ..., rm in R and g1, ..., gm in Γ such that x ... | Generating set of a module |
c_8ao5z173oil8 | Put in another way, there is a surjection ⨁ g ∈ Γ R → M , r g ↦ r g g , {\displaystyle \bigoplus _{g\in \Gamma }R\to M,\,r_{g}\mapsto r_{g}g,} where we wrote rg for an element in the g-th component of the direct sum. (Coincidentally, since a generating set always exists, e.g. M itself, this shows that a module is a quo... | Generating set of a module |
c_hobvgc3yh3yl | If R is a field, then a minimal generating set is the same thing as a basis. Unless the module is finitely generated, there may exist no minimal generating set.The cardinality of a minimal generating set need not be an invariant of the module; Z is generated as a principal ideal by 1, but it is also generated by, say, ... | Generating set of a module |
c_n1n21jbgyb7j | Let R be a local ring with maximal ideal m and residue field k and M finitely generated module. Then Nakayama's lemma says that M has a minimal generating set whose cardinality is dim k M / m M = dim k M ⊗ R k {\displaystyle \dim _{k}M/mM=\dim _{k}M\otimes _{R}k} . | Generating set of a module |
c_o695x4z9xsen | If M is flat, then this minimal generating set is linearly independent (so M is free). See also: Minimal resolution. A more refined information is obtained if one considers the relations between the generators; see Free presentation of a module. | Generating set of a module |
c_pvesocrcg62o | In mathematics, a generic polynomial refers usually to a polynomial whose coefficients are indeterminates. For example, if a, b, and c are indeterminates, the generic polynomial of degree two in x is a x 2 + b x + c . {\displaystyle ax^{2}+bx+c.} | Generic polynomial |
c_l7o8zetz8t9m | However in Galois theory, a branch of algebra, and in this article, the term generic polynomial has a different, although related, meaning: a generic polynomial for a finite group G and a field F is a monic polynomial P with coefficients in the field of rational functions L = F(t1, ..., tn) in n indeterminates over F, ... | Generic polynomial |
c_81l3e4xxnst8 | In mathematics, a genus g surface (also known as a g-torus or g-holed torus) is a surface formed by the connected sum of g many tori: the interior of a disk is removed from each of g many tori and the boundaries of the g many disks are identified (glued together), forming a g-torus. The genus of such a surface is g. A ... | Double torus |
c_gbkp1ee7voqk | In mathematics, a genus of a multiplicative sequence is a ring homomorphism from the ring of smooth compact manifolds up to the equivalence of bounding a smooth manifold with boundary (i.e., up to suitable cobordism) to another ring, usually the rational numbers, having the property that they are constructed from a seq... | Hirzebruch polynomial |
c_7n9q5lamiwoa | In mathematics, a geometric algebra (also known as a real Clifford algebra) is an extension of elementary algebra to work with geometrical objects such as vectors. Geometric algebra is built out of two fundamental operations, addition and the geometric product. Multiplication of vectors results in higher-dimensional ob... | Geometric algebra |
c_qvqkm3krqnvl | The geometric product was first briefly mentioned by Hermann Grassmann, who was chiefly interested in developing the closely related exterior algebra. In 1878, William Kingdon Clifford greatly expanded on Grassmann's work to form what are now usually called Clifford algebras in his honor (although Clifford himself chos... | Geometric algebra |
c_87srvtsgbbsn | Adding the dual of the Grassmann exterior product (the "meet") allows the use of the Grassmann–Cayley algebra, and a conformal version of the latter together with a conformal Clifford algebra yields a conformal geometric algebra (CGA) providing a framework for classical geometries. In practice, these and several derive... | Geometric algebra |
c_rdnvmftxxljq | The term "geometric algebra" was repopularized in the 1960s by Hestenes, who advocated its importance to relativistic physics.The scalars and vectors have their usual interpretation, and make up distinct subspaces of a geometric algebra. Bivectors provide a more natural representation of the pseudovector quantities in ... | Geometric algebra |
c_rxyrbogrwl4f | A trivector can represent an oriented volume, and so on. An element called a blade may be used to represent a subspace of V {\displaystyle V} and orthogonal projections onto that subspace. | Geometric algebra |
c_acgr294vpepx | Rotations and reflections are represented as elements. Unlike a vector algebra, a geometric algebra naturally accommodates any number of dimensions and any quadratic form such as in relativity. Examples of geometric algebras applied in physics include the spacetime algebra (and the less common algebra of physical space... | Geometric algebra |
c_6g27jieposod | Geometric calculus, an extension of GA that incorporates differentiation and integration, can be used to formulate other theories such as complex analysis and differential geometry, e.g. by using the Clifford algebra instead of differential forms. Geometric algebra has been advocated, most notably by David Hestenes and... | Geometric algebra |
c_rsmkqvd9pk83 | In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of non-zero numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. For example, the sequence 2, 6, 18, 54, ... is a geometric progression with common ... | Geometrical progression |
c_8uvpb6ok442w | Examples of a geometric sequence are powers rk of a fixed non-zero number r, such as 2k and 3k. The general form of a geometric sequence is a , a r , a r 2 , a r 3 , a r 4 , … {\displaystyle a,\ ar,\ ar^{2},\ ar^{3},\ ar^{4},\ \ldots } where r ≠ 0 is the common ratio and a ≠ 0 is a scale factor, equal to the sequence's... | Geometrical progression |
c_ttuq13dmz5p3 | In mathematics, a geometric series is the sum of an infinite number of terms that have a constant ratio between successive terms. For example, the series 1 2 + 1 4 + 1 8 + 1 16 + ⋯ {\displaystyle {\frac {1}{2}}\,+\,{\frac {1}{4}}\,+\,{\frac {1}{8}}\,+\,{\frac {1}{16}}\,+\,\cdots } is geometric, because each successive ... | Geometric series |
c_9sol94lysocu | . {\displaystyle a+ar+ar^{2}+ar^{3}+...} , where a {\displaystyle a} is the coefficient of each term and r {\displaystyle r} is the common ratio between adjacent terms. The geometric series had an important role in the early development of calculus, is used throughout mathematics, and can serve as an introduction to fr... | Geometric series |
c_q0vm5ckolyuz | In mathematics, a geometric transformation is any bijection of a set to itself (or to another such set) with some salient geometrical underpinning. More specifically, it is a function whose domain and range are sets of points — most often both R 2 {\displaystyle \mathbb {R} ^{2}} or both R 3 {\displaystyle \mathbb {R} ... | Transformation (geometry) |
c_sfbvos75lbw8 | In mathematics, a gerbe (; French: ) is a construct in homological algebra and topology. Gerbes were introduced by Jean Giraud (Giraud 1971) following ideas of Alexandre Grothendieck as a tool for non-commutative cohomology in degree 2. They can be seen as an analogue of fibre bundles where the fibre is the classifying... | Gerbe |
c_q3bd7psvkn2e | Gerbes provide a convenient, if highly abstract, language for dealing with many types of deformation questions especially in modern algebraic geometry. In addition, special cases of gerbes have been used more recently in differential topology and differential geometry to give alternative descriptions to certain cohomol... | Gerbe |
c_g4rlthoqfxyg | In mathematics, a global field is one of two types of fields (the other one is local field) which are characterized using valuations. There are two kinds of global fields: Algebraic number field: A finite extension of Q {\displaystyle \mathbb {Q} } Global function field: The function field of an algebraic curve over a ... | Global field |
c_h0zs4fo0kbnl | In mathematics, a graded Lie algebra is a Lie algebra endowed with a gradation which is compatible with the Lie bracket. In other words, a graded Lie algebra is a Lie algebra which is also a nonassociative graded algebra under the bracket operation. A choice of Cartan decomposition endows any semisimple Lie algebra wit... | Graded Lie algebra |
c_95kreol0c9zc | A graded Lie superalgebra extends the notion of a graded Lie algebra in such a way that the Lie bracket is no longer assumed to be necessarily anticommutative. These arise in the study of derivations on graded algebras, in the deformation theory of Murray Gerstenhaber, Kunihiko Kodaira, and Donald C. Spencer, and in th... | Graded Lie algebra |
c_3nzyzc47hkam | A supergraded Lie superalgebra is a further generalization of this notion to the category of superalgebras in which a graded Lie superalgebra is endowed with an additional super Z / 2 Z {\displaystyle \mathbb {Z} /2\mathbb {Z} } -gradation. These arise when one forms a graded Lie superalgebra in a classical (non-supers... | Graded Lie algebra |
c_5c7lsuv6mb4i | In mathematics, a graded vector space is a vector space that has the extra structure of a grading or gradation, which is a decomposition of the vector space into a direct sum of vector subspaces, generally indexed by the integers. For "pure" vector spaces, the concept has been introduced in homological algebra, and it ... | Graded dimension |
c_aktiyb3ys79d | In mathematics, a gradually varied surface is a special type of digital surfaces. It is a function from a 2D digital space (see digital geometry) to an ordered set or a chain. A gradually varied function is a function from a digital space Σ {\displaystyle \Sigma } to { A 1 , … , A m } {\displaystyle \{A_{1},\dots ,A_{m... | Gradually varied surface |
c_efryf8m06zpl | In mathematics, a graph C*-algebra is a universal C*-algebra constructed from a directed graph. Graph C*-algebras are direct generalizations of the Cuntz algebras and Cuntz-Krieger algebras, but the class of graph C*-algebras has been shown to also include several other widely studied classes of C*-algebras. As a resul... | Graph C*-algebra |
c_35e6reiu2yr8 | Although graph C*-algebras include numerous examples, they provide a class of C*-algebras that are surprisingly amenable to study and much more manageable than general C*-algebras. The graph not only determines the associated C*-algebra by specifying relations for generators, it also provides a useful tool for describi... | Graph C*-algebra |
c_9ktumfo2ytcy | In mathematics, a graph partition is the reduction of a graph to a smaller graph by partitioning its set of nodes into mutually exclusive groups. Edges of the original graph that cross between the groups will produce edges in the partitioned graph. If the number of resulting edges is small compared to the original grap... | Graph partitioning |
c_35lxsyuo6ug5 | Recently, the graph partition problem has gained importance due to its application for clustering and detection of cliques in social, pathological and biological networks. For a survey on recent trends in computational methods and applications see Buluc et al. (2013). Two common examples of graph partitioning are minim... | Graph partitioning |
c_e82cu7o1mskc | In mathematics, a graph polynomial is a graph invariant whose values are polynomials. Invariants of this type are studied in algebraic graph theory. Important graph polynomials include: The characteristic polynomial, based on the graph's adjacency matrix. | Graph polynomial |
c_igrr3damresu | The chromatic polynomial, a polynomial whose values at integer arguments give the number of colorings of the graph with that many colors. The dichromatic polynomial, a 2-variable generalization of the chromatic polynomial The flow polynomial, a polynomial whose values at integer arguments give the number of nowhere-zer... | Graph polynomial |
c_978c7vyageq1 | In mathematics, a great circle or orthodrome is the circular intersection of a sphere and a plane passing through the sphere's center point.Any arc of a great circle is a geodesic of the sphere, so that great circles in spherical geometry are the natural analog of straight lines in Euclidean space. For any pair of dist... | Great circle |
c_pqa8ckpkl662 | Its arc length is the great-circle distance between the points (the intrinsic distance on a sphere), and is proportional to the measure of the central angle formed by the two points and the center of the sphere. A great circle is the largest circle that can be drawn on any given sphere. Any diameter of any great circle... | Great circle |
c_dcv8b7if61zp | Any other circle of the sphere is called a small circle, and is the intersection of the sphere with a plane not passing through its center. Small circles are the spherical-geometry analog of circles in Euclidean space. Every circle in Euclidean 3-space is a great circle of exactly one sphere. The disk bounded by a grea... | Great circle |
c_abwrxjz504pr | In mathematics, a greatest common divisor matrix (sometimes abbreviated as GCD matrix) is a matrix. | GCD matrix |
c_6y02hg3su4j8 | In mathematics, a ground field is a field K fixed at the beginning of the discussion. | Ground field |
c_4hi4hhb2q8kz | In mathematics, a group G is called the direct sum of two normal subgroups with trivial intersection if it is generated by the subgroups. In abstract algebra, this method of construction of groups can be generalized to direct sums of vector spaces, modules, and other structures; see the article direct sum of modules fo... | Direct sum of groups |
c_le2tv2p6geo9 | In mathematics, a group G is said to be complete if every automorphism of G is inner, and it is centerless; that is, it has a trivial outer automorphism group and trivial center. Equivalently, a group is complete if the conjugation map, G → Aut(G) (sending an element g to conjugation by g), is an isomorphism: injectivi... | Complete group |
c_y1zd7s8z20e4 | In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism group of the structure. It is said that the group acts on the space or structu... | Group action |
c_eg03mv4lgu83 | For example, the group of Euclidean isometries acts on Euclidean space and also on the figures drawn in it. For example, it acts on the set of all triangles. Similarly, the group of symmetries of a polyhedron acts on the vertices, the edges, and the faces of the polyhedron. | Group action |
c_s2rx5byurny1 | A group action on a vector space is called a representation of the group. In the case of a finite-dimensional vector space, it allows one to identify many groups with subgroups of GL(n, K), the group of the invertible matrices of dimension n over a field K. The symmetric group Sn acts on any set with n elements by perm... | Group action |
c_x6zw2seumg0h | In mathematics, a group extension is a general means of describing a group in terms of a particular normal subgroup and quotient group. If Q {\displaystyle Q} and N {\displaystyle N} are two groups, then G {\displaystyle G} is an extension of Q {\displaystyle Q} by N {\displaystyle N} if there is a short exact sequence... | Extension (algebra) |
c_y00fvdgh2dzz | Group extensions arise in the context of the extension problem, where the groups Q {\displaystyle Q} and N {\displaystyle N} are known and the properties of G {\displaystyle G} are to be determined. Note that the phrasing " G {\displaystyle G} is an extension of N {\displaystyle N} by Q {\displaystyle Q} " is also used... | Extension (algebra) |
c_lx2huybofk97 | In mathematics, a group functor is a group-valued functor on the category of commutative rings. Although it is typically viewed as a generalization of a group scheme, the notion itself involves no scheme theory. Because of this feature, some authors, notably Waterhouse and Milne (who followed Waterhouse), develop the t... | Group sheaf |
c_rk7rq8cq338y | In mathematics, a group is a non-empty set with an operation that satisfies the following constraints: the operation is associative, has an identity element, and every element of the set has an inverse element. Many mathematical structures are groups endowed with other properties. For example, the integers with the add... | Elementary group theory |
c_vm5xm1g3nnfc | Because the concept of groups is ubiquitous in numerous areas both within and outside mathematics, some authors consider it as a central organizing principle of contemporary mathematics.In geometry, groups arise naturally in the study of symmetries and geometric transformations: The symmetries of an object form a group... | Elementary group theory |
c_1065f2b2b5zb | Point groups describe symmetry in molecular chemistry. The concept of a group arose in the study of polynomial equations, starting with Évariste Galois in the 1830s, who introduced the term group (French: groupe) for the symmetry group of the roots of an equation, now called a Galois group. | Elementary group theory |
c_d5g1f4jgji7i | After contributions from other fields such as number theory and geometry, the group notion was generalized and firmly established around 1870. Modern group theory—an active mathematical discipline—studies groups in their own right. To explore groups, mathematicians have devised various notions to break groups into smal... | Elementary group theory |
c_h7oj8tou533e | In addition to their abstract properties, group theorists also study the different ways in which a group can be expressed concretely, both from a point of view of representation theory (that is, through the representations of the group) and of computational group theory. A theory has been developed for finite groups, w... | Elementary group theory |
c_aunb51x7fss3 | In mathematics, a group is called an Iwasawa group, M-group or modular group if its lattice of subgroups is modular. Alternatively, a group G is called an Iwasawa group when every subgroup of G is permutable in G (Ballester-Bolinches, Esteban-Romero & Asaad 2010, pp. 24–25). | Iwasawa group |
c_shti5cr54ao7 | Kenkichi Iwasawa (1941) proved that a p-group G is an Iwasawa group if and only if one of the following cases happens: G is a Dedekind group, or G contains an abelian normal subgroup N such that the quotient group G/N is a cyclic group and if q denotes a generator of G/N, then for all n ∈ N, q−1nq = n1+ps where s ≥ 1 i... | Iwasawa group |
c_a9kfxhaclput | As part of Schmidt's proof, he proves that a finite p-group is a modular group if and only if every subgroup is permutable, by (Schmidt 1994, Lemma 2.3.2, p. 55). Every subgroup of a finite p-group is subnormal, and those finite groups in which subnormality and permutability coincide are called PT-groups. In other word... | Iwasawa group |
c_umx1qhxcghfw | In mathematics, a group is called boundedly generated if it can be expressed as a finite product of cyclic subgroups. The property of bounded generation is also closely related with the congruence subgroup problem (see Lubotzky & Segal 2003). | Boundedly generated group |
c_lw230i8bvkbl | In mathematics, a group is called elementary amenable if it can be built up from finite groups and abelian groups by a sequence of simple operations that result in amenable groups when applied to amenable groups. Since finite groups and abelian groups are amenable, every elementary amenable group is amenable - however,... | Elementary amenable group |
c_zmo54n3rbmxh | In mathematics, a group is said to be almost simple if it contains a non-abelian simple group and is contained within the automorphism group of that simple group – that is, if it fits between a (non-abelian) simple group and its automorphism group. In symbols, a group A is almost simple if there is a (non-abelian) simp... | Almost simple group |
c_s69mzchinf83 | In mathematics, a group is said to have the infinite conjugacy class property, or to be an ICC group, if the conjugacy class of every group element but the identity is infinite.The von Neumann group algebra of a group is a factor if and only if the group has the infinite conjugacy class property. It will then be, provi... | Infinite conjugacy class property |
c_nh5deyebrwj0 | In mathematics, a group is supersolvable (or supersoluble) if it has an invariant normal series where all the factors are cyclic groups. Supersolvability is stronger than the notion of solvability. | Supersoluble group |
c_eae0vzq16z3s | In mathematics, a group scheme is a type of object from algebraic geometry equipped with a composition law. Group schemes arise naturally as symmetries of schemes, and they generalize algebraic groups, in the sense that all algebraic groups have group scheme structure, but group schemes are not necessarily connected, s... | Multiplicative group scheme |
c_94rcyty92cs4 | In mathematics, a half range Fourier series is a Fourier series defined on an interval {\displaystyle } instead of the more common {\displaystyle } , with the implication that the analyzed function f ( x ) , x ∈ {\displaystyle f(x),x\in } should be extended to {\displaystyle } as either an even (f(-x)=f(x)) or odd ... | Half range Fourier series |
c_bis3gfqb3jfh | In mathematics, a half-exponential function is a functional square root of an exponential function. That is, a function f {\displaystyle f} such that f {\displaystyle f} composed with itself results in an exponential function: for some constants a {\displaystyle a} and b {\displaystyle b} . | Half-exponential function |
c_kgy19te5aqoc | In mathematics, a half-integer is a number of the form where n {\displaystyle n} is a whole number. For example, are all half-integers. The name "half-integer" is perhaps misleading, as the set may be misunderstood to include numbers such as 1 (being half the integer 2). A name such as "integer-plus-half" may be more a... | Half-integer |
c_28nvuzrqlsdh | Half-integers occur frequently enough in mathematics and in quantum mechanics that a distinct term is convenient. Note that halving an integer does not always produce a half-integer; this is only true for odd integers. For this reason, half-integers are also sometimes called half-odd-integers. Half-integers are a subse... | Half-integer |
c_gacvvpk2akks | In mathematics, a handle decomposition of a 3-manifold allows simplification of the original 3-manifold into pieces which are easier to study. | Handle decompositions of 3-manifolds |
c_fgf7o172pjqc | In mathematics, a handle decomposition of an m-manifold M is a union where each M i {\displaystyle M_{i}} is obtained from M i − 1 {\displaystyle M_{i-1}} by the attaching of i {\displaystyle i} -handles. A handle decomposition is to a manifold what a CW-decomposition is to a topological space—in many regards the purpo... | Handle decomposition |
c_h8cecpcag3ey | In mathematics, a harmonic divisor number, or Ore number (named after Øystein Ore who defined it in 1948), is a positive integer whose divisors have a harmonic mean that is an integer. The first few harmonic divisor numbers are: 1, 6, 28, 140, 270, 496, 672, 1638, 2970, 6200, 8128, 8190 (sequence A001599 in the OEIS). | Ore's harmonic number |
c_033b2e0mrso8 | In mathematics, a harmonic progression (or harmonic sequence) is a progression formed by taking the reciprocals of an arithmetic progression. Equivalently, a sequence is a harmonic progression when each term is the harmonic mean of the neighboring terms. As a third equivalent characterization, it is an infinite sequenc... | Harmonic progression (mathematics) |
c_ux1iez3wf7zs | In mathematics, a harmonious set is a subset of a locally compact abelian group on which every weak character may be uniformly approximated by strong characters. Equivalently, a suitably defined dual set is relatively dense in the Pontryagin dual of the group. This notion was introduced by Yves Meyer in 1970 and later ... | Harmonious set |
c_cuc34sc6zk5y | In mathematics, a harshad number (or Niven number) in a given number base is an integer that is divisible by the sum of its digits when written in that base. Harshad numbers in base n are also known as n-harshad (or n-Niven) numbers. Harshad numbers were defined by D. R. Kaprekar, a mathematician from India. The word "... | Harshad numbers |
c_73k3egjb62lf | In mathematics, a hedgehog space is a topological space consisting of a set of spines joined at a point. For any cardinal number κ {\displaystyle \kappa } , the κ {\displaystyle \kappa } -hedgehog space is formed by taking the disjoint union of κ {\displaystyle \kappa } real unit intervals identified at the origin (tho... | Hedgehog space |
c_nxzrqweogt6b | The hedgehog space is a metric space, when endowed with the hedgehog metric d ( x , y ) = | x − y | {\displaystyle d(x,y)=\left|x-y\right|} if x {\displaystyle x} and y {\displaystyle y} lie in the same spine, and by d ( x , y ) = | x | + | y | {\displaystyle d(x,y)=\left|x\right|+\left|y\right|} if x {\displaystyle x}... | Hedgehog space |
c_rmgf1r7vv9tm | In mathematics, a helix is a curve in 3-dimensional space. The following parametrisation in Cartesian coordinates defines a particular helix; perhaps the simplest equations for one is x ( t ) = cos ( t ) , {\displaystyle x(t)=\cos(t),\,} y ( t ) = sin ( t ) , {\displaystyle y(t)=\sin(t),\,} z ( t ) = t . {\displays... | Helix |
c_l5jg660nuc1i | {\displaystyle h(t)=t.\,} A circular helix of radius a and slope a/b (or pitch 2πb) is described by the following parametrisation: x ( t ) = a cos ( t ) , {\displaystyle x(t)=a\cos(t),\,} y ( t ) = a sin ( t ) , {\displaystyle y(t)=a\sin(t),\,} z ( t ) = b t . {\displaystyle z(t)=bt.\,} Another way of mathematicall... | Helix |
c_9pi6v4z7xhgl | In mathematics, a hereditary property is a property of an object that is inherited by all of its subobjects, where the meaning of subobject depends on the context. These properties are particularly considered in topology and graph theory, but also in set theory. | Induced-hereditary property |
c_ox155aqdv2lf | In mathematics, a heteroclinic cycle is an invariant set in the phase space of a dynamical system. It is a topological circle of equilibrium points and connecting heteroclinic orbits. If a heteroclinic cycle is asymptotically stable, approaching trajectories spend longer and longer periods of time in a neighbourhood of... | Heteroclinic cycle |
c_pmyf5qav0ijp | In mathematics, a heteroclinic network is an invariant set in the phase space of a dynamical system. It can be thought of loosely as the union of more than one heteroclinic cycle. Heteroclinic networks arise naturally in a number of different types of applications, including fluid dynamics and populations dynamics. The... | Heteroclinic network |
c_t8fkeoj13o7o | In mathematics, a heterogeneous relation is a binary relation, a subset of a Cartesian product A × B , {\displaystyle A\times B,} where A and B are possibly distinct sets. The prefix hetero is from the Greek ἕτερος (heteros, "other, another, different"). A heterogeneous relation has been called a rectangular relation, ... | Binary relations |
c_q5oggvpg9enn | In mathematics, a hexadecagon (sometimes called a hexakaidecagon or 16-gon) is a sixteen-sided polygon. | Hexadecagon |
c_ln3uvgsi23tr | In mathematics, a hierarchy is a set-theoretical object, consisting of a preorder defined on a set. This is often referred to as an ordered set, though that is an ambiguous term that many authors reserve for partially ordered sets or totally ordered sets. The term pre-ordered set is unambiguous, and is always synonymou... | Hierarchy (mathematics) |
c_mzeo8f91qphy | Sometimes, a set comes equipped with a natural hierarchical structure. For example, the set of natural numbers N is equipped with a natural pre-order structure, where n ≤ n ′ {\displaystyle n\leq n'} whenever we can find some other number m {\displaystyle m} so that n + m = n ′ {\displaystyle n+m=n'} . That is, n ′ {\d... | Hierarchy (mathematics) |
c_xawpokj6ekmi | This idea can be applied to any commutative monoid. On the other hand, the set of integers Z requires a more sophisticated argument for its hierarchical structure, since we can always solve the equation n + m = n ′ {\displaystyle n+m=n'} by writing m = ( n ′ − n ) {\displaystyle m=(n'-n)} .A mathematical hierarchy (a p... | Hierarchy (mathematics) |
c_tklgmytqgpk3 | This is not just a pedantic claim; there are also mathematical hierarchies, in the general sense, that are not describable using set theory.Other natural hierarchies arise in computer science, where the word refers to partially ordered sets whose elements are classes of objects of increasing complexity. In that case, t... | Hierarchy (mathematics) |
c_7xjojdv420py | In mathematics, a higher (-dimensional) local field is an important example of a complete discrete valuation field. Such fields are also sometimes called multi-dimensional local fields. On the usual local fields (typically completions of number fields or the quotient fields of local rings of algebraic curves) there is ... | Higher local field |
c_grvlb1vrevoj | In contrast to one-dimensional local fields, higher local fields have a sequence of residue fields. There are different integral structures on higher local fields, depending how many residue fields information one wants to take into account.Geometrically, higher local fields appear via a process of localization and com... | Higher local field |
c_br79hacpb9iq | In mathematics, a higher spin alternating sign matrix is a generalisation of the alternating sign matrix (ASM), where the columns and rows sum to an integer r (the spin) rather than simply summing to 1 as in the usual alternating sign matrix definition. HSASMs are square matrices whose elements may be integers in the r... | Higher spin alternating sign matrix |
c_armu1yt7icga | {\displaystyle {\begin{pmatrix}0&0&2&0\\0&2&-1&1\\2&-1&2&-1\\0&1&-1&2\end{pmatrix}};\quad {\begin{pmatrix}0&0&2&0&0\\0&1&-1&2&0\\2&-1&-1&0&2\\0&0&2&0&0\\0&2&0&0&0\end{pmatrix}};\quad {\begin{pmatrix}0&0&0&2\\0&2&0&0\\2&-2&2&0\\0&2&0&0\end{pmatrix}};\quad {\begin{pmatrix}0&2&0&0\\0&0&0&2\\2&0&0&0\\0&0&2&0\end{pmatrix}}.... | Higher spin alternating sign matrix |
c_3t55n9lgg3kx | In mathematics, a highly abundant number is a natural number with the property that the sum of its divisors (including itself) is greater than the sum of the divisors of any smaller natural number. Highly abundant numbers and several similar classes of numbers were first introduced by Pillai (1943), and early work on t... | Highly abundant number |
c_2vbl57lfa27q | In mathematics, a highly structured ring spectrum or A ∞ {\displaystyle A_{\infty }} -ring is an object in homotopy theory encoding a refinement of a multiplicative structure on a cohomology theory. A commutative version of an A ∞ {\displaystyle A_{\infty }} -ring is called an E ∞ {\displaystyle E_{\infty }} -ring. Whi... | Highly structured ring spectrum |
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