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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. The generalized hypergeometric series is sometimes just called the hypergeometric series, though this term also sometimes just refers to the Gaussian hypergeometric series. Generalized hypergeometric functions include the (Gaussian) hypergeometric function and the confluent hypergeometric function as special cases, which in turn have many particular special functions as special cases, such as elementary functions, Bessel functions, and the classical orthogonal polynomials.
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https://en.wikipedia.org/wiki/Generalized_hypergeometric_series
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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 map is the homogeneity of one-to-one mappings in any dimensions, which simplifies definitions and algorithms comparing to combinatorial maps. For this reason, generalized maps are sometimes used instead of combinatorial maps, even to represent orientable closed partitions. Like combinatorial maps, generalized maps are used as efficient data structure in image representation and processing, in geometrical modeling, they are related to simplicial set and to combinatorial topology, and this is a boundary representation model (B-rep or BREP), i.e. it represents object by its boundaries.
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https://en.wikipedia.org/wiki/Generalized_maps
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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 entry can be any nonzero value. An example of a generalized permutation matrix is . {\displaystyle {\begin{bmatrix}0&0&3&0\\0&-7&0&0\\1&0&0&0\\0&0&0&{\sqrt {2}}\end{bmatrix}}.}
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https://en.wikipedia.org/wiki/Monomial_matrix
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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 that cannot be obtained in this way. Generalized polygons satisfying a technical condition known as the Moufang property have been completely classified by Tits and Weiss. Every generalized n-gon with n even is also a near polygon.
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https://en.wikipedia.org/wiki/Generalized_n-gon
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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 generating function is not actually regarded as a function, and the "variable" remains an indeterminate. Generating functions were first introduced by Abraham de Moivre in 1730, in order to solve the general linear recurrence problem.
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https://en.wikipedia.org/wiki/Exponential_generating_series
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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 series; definitions and examples are given below. Every sequence in principle has a generating function of each type (except that Lambert and Dirichlet series require indices to start at 1 rather than 0), but the ease with which they can be handled may differ considerably.
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https://en.wikipedia.org/wiki/Exponential_generating_series
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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 series. These expressions in terms of the indeterminate x may involve arithmetic operations, differentiation with respect to x and composition with (i.e., substitution into) other generating functions; since these operations are also defined for functions, the result looks like a function of x. Indeed, the closed form expression can often be interpreted as a function that can be evaluated at (sufficiently small) concrete values of x, and which has the formal series as its series expansion; this explains the designation "generating functions".
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https://en.wikipedia.org/wiki/Exponential_generating_series
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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 and fractional powers of x are examples of functions that do not have a corresponding formal power series. Generating functions are not functions in the formal sense of a mapping from a domain to a codomain. Generating functions are sometimes called generating series, in that a series of terms can be said to be the generator of its sequence of term coefficients.
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https://en.wikipedia.org/wiki/Exponential_generating_series
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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 by the identity element 1 as a left R-module over itself. If there is a finite generating set, then a module is said to be finitely generated. This applies to ideals, which are the submodules of the ring itself.
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https://en.wikipedia.org/wiki/Generating_set_of_a_module
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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 = r 1 g 1 + ⋯ + r m g m . {\displaystyle x=r_{1}g_{1}+\cdots +r_{m}g_{m}.}
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https://en.wikipedia.org/wiki/Generating_set_of_a_module
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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 quotient of a free module, a useful fact.) A generating set of a module is said to be minimal if no proper subset of the set generates the module.
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https://en.wikipedia.org/wiki/Generating_set_of_a_module
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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, a minimal generating set {2, 3}. What is uniquely determined by a module is the infimum of the numbers of the generators of the module.
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https://en.wikipedia.org/wiki/Generating_set_of_a_module
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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} .
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https://en.wikipedia.org/wiki/Generating_set_of_a_module
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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.
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https://en.wikipedia.org/wiki/Generating_set_of_a_module
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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.}
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https://en.wikipedia.org/wiki/Generic_polynomial
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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, such that the splitting field M of P has Galois group G over L, and such that every extension K/F with Galois group G can be obtained as the splitting field of a polynomial which is the specialization of P resulting from setting the n indeterminates to n elements of F. This is sometimes called F-generic or relative to the field F; a Q-generic polynomial, which is generic relative to the rational numbers is called simply generic. The existence, and especially the construction, of a generic polynomial for a given Galois group provides a complete solution to the inverse Galois problem for that group. However, not all Galois groups have generic polynomials, a counterexample being the cyclic group of order eight.
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https://en.wikipedia.org/wiki/Generic_polynomial
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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 genus g surface is a two-dimensional manifold. The classification theorem for surfaces states that every compact connected two-dimensional manifold is homeomorphic to either the sphere, the connected sum of tori, or the connected sum of real projective planes.
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https://en.wikipedia.org/wiki/Double_torus
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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 sequence of polynomials in characteristic classes that arise as coefficients in formal power series with good multiplicative properties.
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https://en.wikipedia.org/wiki/Hirzebruch_polynomial
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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 objects called multivectors. Compared to other formalisms for manipulating geometric objects, geometric algebra is noteworthy for supporting vector division and addition of objects of different dimensions.
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https://en.wikipedia.org/wiki/Geometric_algebra
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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 chose to call them "geometric algebras"). Clifford defined the Clifford algebra and its product as a unification of the Grassmann algebra and Hamilton's quaternion algebra.
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https://en.wikipedia.org/wiki/Geometric_algebra
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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 derived operations allow a correspondence of elements, subspaces and operations of the algebra with geometric interpretations. For several decades, geometric algebras went somewhat ignored, greatly eclipsed by the vector calculus then newly developed to describe electromagnetism.
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https://en.wikipedia.org/wiki/Geometric_algebra
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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 vector algebra such as oriented area, oriented angle of rotation, torque, angular momentum and the electromagnetic field.
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https://en.wikipedia.org/wiki/Geometric_algebra
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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.
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https://en.wikipedia.org/wiki/Geometric_algebra
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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) and the conformal geometric algebra.
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https://en.wikipedia.org/wiki/Geometric_algebra
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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 Chris Doran, as the preferred mathematical framework for physics. Proponents claim that it provides compact and intuitive descriptions in many areas including classical and quantum mechanics, electromagnetic theory and relativity. GA has also found use as a computational tool in computer graphics and robotics.
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https://en.wikipedia.org/wiki/Geometric_algebra
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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 ratio 3. Similarly 10, 5, 2.5, 1.25, ... is a geometric sequence with common ratio 1/2.
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https://en.wikipedia.org/wiki/Geometrical_progression
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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 start value. The sum of a geometric progression's terms is called a geometric series.
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https://en.wikipedia.org/wiki/Geometrical_progression
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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 term can be obtained by multiplying the previous term by 1 / 2 {\displaystyle 1/2} . In general, a geometric series is written as a + a r + a r 2 + a r 3 + . .
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https://en.wikipedia.org/wiki/Geometric_series
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. {\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 frequently used mathematical tools such as the Taylor series, the complex Fourier series, and the matrix exponential. The name geometric succession indicates each term is the geometric mean of its two neighboring terms, similar to how the name Arithmetic succession indicates each term is the arithmetic mean of its two neighboring terms.
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https://en.wikipedia.org/wiki/Geometric_series
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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} ^{3}} — such that the function is bijective so that its inverse exists. The study of geometry may be approached by the study of these transformations.
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https://en.wikipedia.org/wiki/Transformation_(geometry)
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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 stack of a group.
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https://en.wikipedia.org/wiki/Gerbe
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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 cohomology classes and additional structures attached to them. "Gerbe" is a French (and archaic English) word that literally means wheat sheaf.
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https://en.wikipedia.org/wiki/Gerbe
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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 finite field, equivalently, a finite extension of F q ( T ) {\displaystyle \mathbb {F} _{q}(T)} , the field of rational functions in one variable over the finite field with q = p n {\displaystyle q=p^{n}} elements.An axiomatic characterization of these fields via valuation theory was given by Emil Artin and George Whaples in the 1940s.
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https://en.wikipedia.org/wiki/Global_field
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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 with the structure of a graded Lie algebra. Any parabolic Lie algebra is also a graded Lie algebra.
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https://en.wikipedia.org/wiki/Graded_Lie_algebra
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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 the theory of Lie derivatives.
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https://en.wikipedia.org/wiki/Graded_Lie_algebra
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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-supersymmetric) setting, and then tensorizes to obtain the supersymmetric analog.Still greater generalizations are possible to Lie algebras over a class of braided monoidal categories equipped with a coproduct and some notion of a gradation compatible with the braiding in the category. For hints in this direction, see Lie superalgebra#Category-theoretic definition.
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https://en.wikipedia.org/wiki/Graded_Lie_algebra
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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 is widely used for graded algebras, which are graded vector spaces with additional structures.
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https://en.wikipedia.org/wiki/Graded_dimension
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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}\}} where A 1 < ⋯ < A m {\displaystyle A_{1}<\cdots
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https://en.wikipedia.org/wiki/Gradually_varied_surface
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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 result, graph C*-algebras provide a common framework for investigating many well-known classes of C*-algebras that were previously studied independently. Among other benefits, this provides a context in which one can formulate theorems that apply simultaneously to all of these subclasses and contain specific results for each subclass as special cases.
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https://en.wikipedia.org/wiki/Graph_C*-algebra
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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 describing and visualizing properties of the C*-algebra. This visual quality has led to graph C*-algebras being referred to as "operator algebras we can see." Another advantage of graph C*-algebras is that much of their structure and many of their invariants can be readily computed. Using data coming from the graph, one can determine whether the associated C*-algebra has particular properties, describe the lattice of ideals, and compute K-theoretic invariants.
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https://en.wikipedia.org/wiki/Graph_C*-algebra
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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 graph, then the partitioned graph may be better suited for analysis and problem-solving than the original. Finding a partition that simplifies graph analysis is a hard problem, but one that has applications to scientific computing, VLSI circuit design, and task scheduling in multiprocessor computers, among others.
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https://en.wikipedia.org/wiki/Graph_partitioning
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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 minimum cut and maximum cut problems.
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https://en.wikipedia.org/wiki/Graph_partitioning
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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.
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https://en.wikipedia.org/wiki/Graph_polynomial
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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-zero flows with integer flow amounts modulo the argument. The (inverse of the) Ihara zeta function, defined as a product of binomial terms corresponding to certain closed walks in a graph. The Martin polynomial, used by Pierre Martin to study Euler tours The matching polynomials, several different polynomials defined as the generating function of the matchings of a graph. The reliability polynomial, a polynomial that describes the probability of remaining connected after independent edge failures The Tutte polynomial, a polynomial in two variables that can be defined (after a small change of variables) as the generating function of the numbers of connected components of induced subgraphs of the given graph, parameterized by the number of vertices in the subgraph.
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https://en.wikipedia.org/wiki/Graph_polynomial
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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 distinct non-antipodal points on the sphere, there is a unique great circle passing through both. (Every great circle through any point also passes through its antipodal point, so there are infinitely many great circles through two antipodal points.) The shorter of the two great-circle arcs between two distinct points on the sphere is called the minor arc, and is the shortest surface-path between them.
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https://en.wikipedia.org/wiki/Great_circle
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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 coincides with a diameter of the sphere, and therefore every great circle is concentric with the sphere and shares the same radius.
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https://en.wikipedia.org/wiki/Great_circle
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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 great circle is called a great disk: it is the intersection of a ball and a plane passing through its center. In higher dimensions, the great circles on the n-sphere are the intersection of the n-sphere with 2-planes that pass through the origin in the Euclidean space Rn + 1.
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https://en.wikipedia.org/wiki/Great_circle
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In mathematics, a greatest common divisor matrix (sometimes abbreviated as GCD matrix) is a matrix.
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https://en.wikipedia.org/wiki/GCD_matrix
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In mathematics, a ground field is a field K fixed at the beginning of the discussion.
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https://en.wikipedia.org/wiki/Ground_field
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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 for more information. A group which can be expressed as a direct sum of non-trivial subgroups is called decomposable, and if a group cannot be expressed as such a direct sum then it is called indecomposable.
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https://en.wikipedia.org/wiki/Direct_sum_of_groups
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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: injectivity implies that only conjugation by the identity element is the identity automorphism, meaning the group is centerless, while surjectivity implies it has no outer automorphisms.
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https://en.wikipedia.org/wiki/Complete_group
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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 structure. If a group acts on a structure, it will usually also act on objects built from that structure.
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https://en.wikipedia.org/wiki/Group_action
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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.
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https://en.wikipedia.org/wiki/Group_action
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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 permuting the elements of the set. Although the group of all permutations of a set depends formally on the set, the concept of group action allows one to consider a single group for studying the permutations of all sets with the same cardinality.
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https://en.wikipedia.org/wiki/Group_action
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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 1 → N → ι G → π Q → 1. {\displaystyle 1\to N\;{\overset {\iota }{\to }}\;G\;{\overset {\pi }{\to }}\;Q\to 1.} If G {\displaystyle G} is an extension of Q {\displaystyle Q} by N {\displaystyle N} , then G {\displaystyle G} is a group, ι ( N ) {\displaystyle \iota (N)} is a normal subgroup of G {\displaystyle G} and the quotient group G / ι ( N ) {\displaystyle G/\iota (N)} is isomorphic to the group Q {\displaystyle Q} .
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https://en.wikipedia.org/wiki/Extension_(algebra)
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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 by some.Since any finite group G {\displaystyle G} possesses a maximal normal subgroup N {\displaystyle N} with simple factor group G / N {\displaystyle G/N} , all finite groups may be constructed as a series of extensions with finite simple groups. This fact was a motivation for completing the classification of finite simple groups. An extension is called a central extension if the subgroup N {\displaystyle N} lies in the center of G {\displaystyle G} .
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https://en.wikipedia.org/wiki/Extension_(algebra)
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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 theory of group schemes based on the notion of group functor instead of scheme theory. A formal group is usually defined as a particular kind of a group functor.
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https://en.wikipedia.org/wiki/Group_sheaf
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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 addition operation is an infinite group, which is generated by a single element called 1 (these properties characterize the integers in a unique way). The concept of a group was elaborated for handling, in a unified way, many mathematical structures such as numbers, geometric shapes and polynomial roots.
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https://en.wikipedia.org/wiki/Elementary_group_theory
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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, called the symmetry group of the object, and the transformations of a given type form a general group. Lie groups appear in symmetry groups in geometry, and also in the Standard Model of particle physics. The Poincaré group is a Lie group consisting of the symmetries of spacetime in special relativity.
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https://en.wikipedia.org/wiki/Elementary_group_theory
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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.
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https://en.wikipedia.org/wiki/Elementary_group_theory
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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 smaller, better-understandable pieces, such as subgroups, quotient groups and simple groups.
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https://en.wikipedia.org/wiki/Elementary_group_theory
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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, which culminated with the classification of finite simple groups, completed in 2004. Since the mid-1980s, geometric group theory, which studies finitely generated groups as geometric objects, has become an active area in group theory.
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https://en.wikipedia.org/wiki/Elementary_group_theory
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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).
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https://en.wikipedia.org/wiki/Iwasawa_group
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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 in general, but s ≥ 2 for p=2.In Berkovich & Janko (2008, p. 257), Iwasawa's proof was deemed to have essential gaps, which were filled by Franco Napolitani and Zvonimir Janko. Roland Schmidt (1994) has provided an alternative proof along different lines in his textbook.
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https://en.wikipedia.org/wiki/Iwasawa_group
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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 words, a finite p-group is an Iwasawa group if and only if it is a PT-group.
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https://en.wikipedia.org/wiki/Iwasawa_group
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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).
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https://en.wikipedia.org/wiki/Boundedly_generated_group
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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, the converse is not true. Formally, the class of elementary amenable groups is the smallest subclass of the class of all groups that satisfies the following conditions: it contains all finite and all abelian groups if G is in the subclass and H is isomorphic to G, then H is in the subclass it is closed under the operations of taking subgroups, forming quotients, and forming extensions it is closed under directed unions.The Tits alternative implies that any amenable linear group is locally virtually solvable; hence, for linear groups, amenability and elementary amenability coincide.
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https://en.wikipedia.org/wiki/Elementary_amenable_group
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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) simple group S such that S ≤ A ≤ Aut ( S ) . {\displaystyle S\leq A\leq \operatorname {Aut} (S).}
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https://en.wikipedia.org/wiki/Almost_simple_group
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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, provided the group is nontrivial, of type II1, i.e. it will possess a unique, faithful, tracial state.Examples of ICC groups are the group of permutations of an infinite set that leave all but a finite subset of elements fixed, and free groups on two generators.In abelian groups, every conjugacy class consists of only one element, so ICC groups are, in a way, as far from being abelian as possible. == References ==
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https://en.wikipedia.org/wiki/Infinite_conjugacy_class_property
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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.
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https://en.wikipedia.org/wiki/Supersoluble_group
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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, smooth, or defined over a field. This extra generality allows one to study richer infinitesimal structures, and this can help one to understand and answer questions of arithmetic significance. The category of group schemes is somewhat better behaved than that of group varieties, since all homomorphisms have kernels, and there is a well-behaved deformation theory. Group schemes that are not algebraic groups play a significant role in arithmetic geometry and algebraic topology, since they come up in contexts of Galois representations and moduli problems. The initial development of the theory of group schemes was due to Alexander Grothendieck, Michel Raynaud and Michel Demazure in the early 1960s.
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https://en.wikipedia.org/wiki/Multiplicative_group_scheme
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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 function (f(-x)=-f(x)). This allows the expansion of the function in a series solely of sines (odd) or cosines (even). The choice between odd and even is typically motivated by boundary conditions associated with a differential equation satisfied by f ( x ) {\displaystyle f(x)} . Example Calculate the half range Fourier sine series for the function f ( x ) = cos ( x ) {\displaystyle f(x)=\cos(x)} where 0 < x < π {\displaystyle 0
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https://en.wikipedia.org/wiki/Half_range_Fourier_series
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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} .
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https://en.wikipedia.org/wiki/Half-exponential_function
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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 accurate, but even though not literally true, "half integer" is the conventional term.
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https://en.wikipedia.org/wiki/Half-integer
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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 subset of the dyadic rationals (numbers produced by dividing an integer by a power of two).
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https://en.wikipedia.org/wiki/Half-integer
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In mathematics, a handle decomposition of a 3-manifold allows simplification of the original 3-manifold into pieces which are easier to study.
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https://en.wikipedia.org/wiki/Handle_decompositions_of_3-manifolds
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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 purpose of a handle decomposition is to have a language analogous to CW-complexes, but adapted to the world of smooth manifolds. Thus an i-handle is the smooth analogue of an i-cell. Handle decompositions of manifolds arise naturally via Morse theory. The modification of handle structures is closely linked to Cerf theory.
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https://en.wikipedia.org/wiki/Handle_decomposition
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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).
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https://en.wikipedia.org/wiki/Ore's_harmonic_number
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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 sequence of the form 1 a , 1 a + d , 1 a + 2 d , 1 a + 3 d , ⋯ , {\displaystyle {\frac {1}{a}},\ {\frac {1}{a+d}},\ {\frac {1}{a+2d}},\ {\frac {1}{a+3d}},\cdots ,} where a is not zero and −a/d is not a natural number, or a finite sequence of the form 1 a , 1 a + d , 1 a + 2 d , 1 a + 3 d , ⋯ , 1 a + k d , {\displaystyle {\frac {1}{a}},\ {\frac {1}{a+d}},\ {\frac {1}{a+2d}},\ {\frac {1}{a+3d}},\cdots ,\ {\frac {1}{a+kd}},} where a is not zero, k is a natural number and −a/d is not a natural number or is greater than k.
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https://en.wikipedia.org/wiki/Harmonic_progression_(mathematics)
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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 turned out to play an important role in the mathematical theory of quasicrystals. Some related concepts are model sets, Meyer sets, and cut-and-project sets.
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https://en.wikipedia.org/wiki/Harmonious_set
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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" comes from the Sanskrit harṣa (joy) + da (give), meaning joy-giver. The term "Niven number" arose from a paper delivered by Ivan M. Niven at a conference on number theory in 1977.
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https://en.wikipedia.org/wiki/Harshad_numbers
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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 (though its topology is not the quotient topology, but that defined by the metric below). Each unit interval is referred to as one of the hedgehog's spines. A κ {\displaystyle \kappa } -hedgehog space is sometimes called a hedgehog space of spininess κ {\displaystyle \kappa } .
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https://en.wikipedia.org/wiki/Hedgehog_space
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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} and y {\displaystyle y} lie in different spines. Although their disjoint union makes the origins of the intervals distinct, the metric makes them equivalent by assigning them 0 distance. Hedgehog spaces are examples of real trees.
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https://en.wikipedia.org/wiki/Hedgehog_space
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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 . {\displaystyle z(t)=t.\,} As the parameter t increases, the point (x(t),y(t),z(t)) traces a right-handed helix of pitch 2π (or slope 1) and radius 1 about the z-axis, in a right-handed coordinate system. In cylindrical coordinates (r, θ, h), the same helix is parametrised by: r ( t ) = 1 , {\displaystyle r(t)=1,\,} θ ( t ) = t , {\displaystyle \theta (t)=t,\,} h ( t ) = t .
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https://en.wikipedia.org/wiki/Helix
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{\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 mathematically constructing a helix is to plot the complex-valued function exi as a function of the real number x (see Euler's formula). The value of x and the real and imaginary parts of the function value give this plot three real dimensions. Except for rotations, translations, and changes of scale, all right-handed helices are equivalent to the helix defined above. The equivalent left-handed helix can be constructed in a number of ways, the simplest being to negate any one of the x, y or z components.
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https://en.wikipedia.org/wiki/Helix
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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.
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https://en.wikipedia.org/wiki/Induced-hereditary_property
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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 successive equilibria. In generic dynamical systems heteroclinic connections are of high co-dimension, that is, they will not persist if parameters are varied.
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https://en.wikipedia.org/wiki/Heteroclinic_cycle
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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 dynamics of trajectories near to heteroclinic networks is intermittent: trajectories spend a long time performing one type of behaviour (often, close to equilibrium), before switching rapidly to another type of behaviour. This type of intermittent switching behaviour has led to several different groups of researchers using them as a way to model and understand various type of neural dynamics.
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https://en.wikipedia.org/wiki/Heteroclinic_network
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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, suggesting that it does not have the square-symmetry of a homogeneous relation on a set where A = B . {\displaystyle A=B.} Commenting on the development of binary relations beyond homogeneous relations, researchers wrote, "...a variant of the theory has evolved that treats relations from the very beginning as heterogeneous or rectangular, i.e. as relations where the normal case is that they are relations between different sets."
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https://en.wikipedia.org/wiki/Binary_relations
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In mathematics, a hexadecagon (sometimes called a hexakaidecagon or 16-gon) is a sixteen-sided polygon.
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https://en.wikipedia.org/wiki/Hexadecagon
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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 synonymous with a mathematical hierarchy. The term hierarchy is used to stress a hierarchical relation among the elements.
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https://en.wikipedia.org/wiki/Hierarchy_(mathematics)
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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 ′ {\displaystyle n'} is bigger than n {\displaystyle n} only because we can get to n ′ {\displaystyle n'} from n {\displaystyle n} using m {\displaystyle m} .
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https://en.wikipedia.org/wiki/Hierarchy_(mathematics)
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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 pre-ordered set) should not be confused with the more general concept of a hierarchy in the social realm, particularly when one is constructing computational models that are used to describe real-world social, economic or political systems. These hierarchies, or complex networks, are much too rich to be described in the category Set of sets.
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https://en.wikipedia.org/wiki/Hierarchy_(mathematics)
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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, the preorder defining the hierarchy is the class-containment relation. Containment hierarchies are thus special cases of hierarchies.
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https://en.wikipedia.org/wiki/Hierarchy_(mathematics)
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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 a unique surjective discrete valuation (of rank 1) associated to a choice of a local parameter of the fields, unless they are archimedean local fields such as the real numbers and complex numbers. Similarly, there is a discrete valuation of rank n on almost all n-dimensional local fields, associated to a choice of n local parameters of the field.
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https://en.wikipedia.org/wiki/Higher_local_field
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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 completion of local rings of higher dimensional schemes. Higher local fields are an important part of the subject of higher dimensional number theory, forming the appropriate collection of objects for local considerations.
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https://en.wikipedia.org/wiki/Higher_local_field
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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 range −r to +r. When traversing any row or column of an ASM or HSASM, the partial sum of its entries must always be non-negative.High spin ASMs have found application in statistical mechanics and physics, where they have been found to represent symmetry groups in ice crystal formation. Some typical examples of HSASMs are shown below: ( 0 0 2 0 0 2 − 1 1 2 − 1 2 − 1 0 1 − 1 2 ) ; ( 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 ) ; ( 0 0 0 2 0 2 0 0 2 − 2 2 0 0 2 0 0 ) ; ( 0 2 0 0 0 0 0 2 2 0 0 0 0 0 2 0 ) .
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https://en.wikipedia.org/wiki/Higher_spin_alternating_sign_matrix
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{\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}}.} The set of HSASMs is a superset of the ASMs. The extreme points of the convex hull of the set of r-spin HSASMs are themselves integer multiples of the usual ASMs. == References ==
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https://en.wikipedia.org/wiki/Higher_spin_alternating_sign_matrix
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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 the subject was done by Alaoglu and Erdős (1944). Alaoglu and Erdős tabulated all highly abundant numbers up to 104, and showed that the number of highly abundant numbers less than any N is at least proportional to log2 N.
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https://en.wikipedia.org/wiki/Highly_abundant_number
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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. While originally motivated by questions of geometric topology and bundle theory, they are today most often used in stable homotopy theory.
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https://en.wikipedia.org/wiki/Highly_structured_ring_spectrum
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