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c_ns3opyijad54 | (For Pontryagin duality the Plancherel measure is some Haar measure on the dual group to G, the only issue therefore being its normalization.) For general locally compact groups, or even countable discrete groups, the von Neumann group algebra need not be of type I and the regular representation of G cannot be written ... | Noncommutative harmonic analysis |
c_qnpcmmid6yho | In mathematics, noncommutative projective geometry is a noncommutative analog of projective geometry in the setting of noncommutative algebraic geometry. | Noncommutative projective geometry |
c_zpoq1x65pyol | In mathematics, noncommutative residue, defined independently by M. Wodzicki (1984) and Guillemin (1985), is a certain trace on the algebra of pseudodifferential operators on a compact differentiable manifold that is expressed via a local density. In the case of the circle, the noncommutative residue had been studied e... | Noncommutative residue |
c_y0n8uqqhv8z8 | In mathematics, noncommutative topology is a term used for the relationship between topological and C*-algebraic concepts. The term has its origins in the Gelfand–Naimark theorem, which implies the duality of the category of locally compact Hausdorff spaces and the category of commutative C*-algebras. Noncommutative to... | Noncommutative topology |
c_sgsfth332ggn | In mathematics, nonlinear modelling is empirical or semi-empirical modelling which takes at least some nonlinearities into account. Nonlinear modelling in practice therefore means modelling of phenomena in which independent variables affecting the system can show complex and synergetic nonlinear effects. Contrary to tr... | Nonlinear model |
c_x2sji4q4av4a | Thus the nonlinear modelling can utilize production data or experimental results while taking into account complex nonlinear behaviours of modelled phenomena which are in most cases practically impossible to be modelled by means of traditional mathematical approaches, such as phenomenological modelling. Contrary to phe... | Nonlinear model |
c_uw50vx1jyzwa | In mathematics, nonlinear programming (NLP) is the process of solving an optimization problem where some of the constraints or the objective function are nonlinear. An optimization problem is one of calculation of the extrema (maxima, minima or stationary points) of an objective function over a set of unknown real vari... | Non-linear programming |
c_0qioo4sm0xde | In mathematics, nonstandard calculus is the modern application of infinitesimals, in the sense of nonstandard analysis, to infinitesimal calculus. It provides a rigorous justification for some arguments in calculus that were previously considered merely heuristic. Non-rigorous calculations with infinitesimals were wide... | Nonstandard calculus |
c_31g7mu32lj9q | (See history of calculus.) For almost one hundred years thereafter, mathematicians such as Richard Courant viewed infinitesimals as being naive and vague or meaningless.Contrary to such views, Abraham Robinson showed in 1960 that infinitesimals are precise, clear, and meaningful, building upon work by Edwin Hewitt and ... | Nonstandard calculus |
c_7bpr0okvwscl | In mathematics, nuclear operators are an important class of linear operators introduced by Alexander Grothendieck in his doctoral dissertation. Nuclear operators are intimately tied to the projective tensor product of two topological vector spaces (TVSs). | Nuclear map |
c_7cxfio9d05ti | In mathematics, nuclear operators between Banach spaces are a linear operators between Banach spaces in infinite dimensions that share some of the properties of their counter-part in finite dimension. In Hilbert spaces such operators are usually called trace class operators and one can define such things as the trace. ... | Nuclear operators between Banach spaces |
c_rj8rbolmvb57 | In mathematics, nuclear spaces are topological vector spaces that can be viewed as a generalization of finite dimensional Euclidean spaces and share many of their desirable properties. Nuclear spaces are however quite different from Hilbert spaces, another generalization of finite dimensional Euclidean spaces. They wer... | Nuclear spaces |
c_tmthrjuw6bjv | Vector spaces whose elements are "smooth" in some sense tend to be nuclear spaces; a typical example of a nuclear space is the set of smooth functions on a compact manifold. All finite-dimensional vector spaces are nuclear. There are no Banach spaces that are nuclear, except for the finite-dimensional ones. In practice... | Nuclear spaces |
c_qrtf2dhthwrs | In mathematics, numerical analysis, and numerical partial differential equations, domain decomposition methods solve a boundary value problem by splitting it into smaller boundary value problems on subdomains and iterating to coordinate the solution between adjacent subdomains. A coarse problem with one or few unknowns... | Domain decomposition |
c_8vcyctmrrk19 | In overlapping domain decomposition methods, the subdomains overlap by more than the interface. Overlapping domain decomposition methods include the Schwarz alternating method and the additive Schwarz method. Many domain decomposition methods can be written and analyzed as a special case of the abstract additive Schwar... | Domain decomposition |
c_gu8yn2so9ukg | In non-overlapping methods, the subdomains intersect only on their interface. In primal methods, such as Balancing domain decomposition and BDDC, the continuity of the solution across subdomain interface is enforced by representing the value of the solution on all neighboring subdomains by the same unknown. In dual met... | Domain decomposition |
c_92ze4xb5j6h3 | The FETI-DP method is hybrid between a dual and a primal method. Non-overlapping domain decomposition methods are also called iterative substructuring methods. Mortar methods are discretization methods for partial differential equations, which use separate discretization on nonoverlapping subdomains. | Domain decomposition |
c_tfpvhfsa8p13 | The meshes on the subdomains do not match on the interface, and the equality of the solution is enforced by Lagrange multipliers, judiciously chosen to preserve the accuracy of the solution. In the engineering practice in the finite element method, continuity of solutions between non-matching subdomains is implemented ... | Domain decomposition |
c_vpbna5vrmbqn | In mathematics, obstruction theory is a name given to two different mathematical theories, both of which yield cohomological invariants. In the original work of Stiefel and Whitney, characteristic classes were defined as obstructions to the existence of certain fields of linear independent vectors. Obstruction theory t... | Obstruction theory |
c_mnxanupzu6uh | In mathematics, one can consider the scaling properties of a function or curve f (x) under rescalings of the variable x. That is, one is interested in the shape of f (λx) for some scale factor λ, which can be taken to be a length or size rescaling. The requirement for f (x) to be invariant under all rescalings is usual... | Scale invariance |
c_641kb3typ5me | In polar coordinates (r, θ), the spiral can be written as θ = 1 b ln ( r / a ) . {\displaystyle \theta ={\frac {1}{b}}\ln(r/a)~.} Allowing for rotations of the curve, it is invariant under all rescalings λ; that is, θ(λr) is identical to a rotated version of θ(r). | Scale invariance |
c_49etuiik185b | In mathematics, one can define a product of group subsets in a natural way. If S and T are subsets of a group G, then their product is the subset of G defined by S T = { s t: s ∈ S and t ∈ T } . {\displaystyle ST=\{st:s\in S{\text{ and }}t\in T\}.} The subsets S and T need not be subgroups for this product to be well d... | Product of group subsets |
c_9gniu5n7f4da | The associativity of this product follows from that of the group product. The product of group subsets therefore defines a natural monoid structure on the power set of G. A lot more can be said in the case where S and T are subgroups. The product of two subgroups S and T of a group G is itself a subgroup of G if and on... | Product of group subsets |
c_0qgjm13z07vw | In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one talks about the product in category theory, which formalizes these notions.... | Directly indecomposable |
c_acytsx3px6nk | In mathematics, one normed vector space is said to be continuously embedded in another normed vector space if the inclusion function between them is continuous. In some sense, the two norms are "almost equivalent", even though they are not both defined on the same space. Several of the Sobolev embedding theorems are co... | Continuous embedding |
c_y8eq82xxizk0 | In mathematics, operator K-theory is a noncommutative analogue of topological K-theory for Banach algebras with most applications used for C*-algebras. | Operator K-theory |
c_tve5bwi3mmyv | In mathematics, operator theory is the study of linear operators on function spaces, beginning with differential operators and integral operators. The operators may be presented abstractly by their characteristics, such as bounded linear operators or closed operators, and consideration may be given to nonlinear operato... | Operator Theory |
c_l9u6lrg5a3qk | In mathematics, or more specifically group theory, the omega and agemo subgroups described the so-called "power structure" of a finite p-group. They were introduced in (Hall 1933) where they were used to describe a class of finite p-groups whose structure was sufficiently similar to that of finite abelian p-groups, the... | Omega and agemo subgroup |
c_uk5btas6u0wo | In mathematics, or more specifically in spectral theory, the Riesz projector is the projector onto the eigenspace corresponding to a particular eigenvalue of an operator (or, more generally, a projector onto an invariant subspace corresponding to an isolated part of the spectrum). It was introduced by Frigyes Riesz in ... | Riesz projector |
c_hh0kd5n9u1ua | In mathematics, or specifically, in differential topology, Ehresmann's lemma or Ehresmann's fibration theorem states that if a smooth mapping f: M → N {\displaystyle f\colon M\rightarrow N} , where M {\displaystyle M} and N {\displaystyle N} are smooth manifolds, is a surjective submersion, and a proper map (in particu... | Ehresmann's lemma |
c_9vf4tp1dfh6w | In mathematics, orbit capacity of a subset of a topological dynamical system may be thought of heuristically as a “topological dynamical probability measure” of the subset. More precisely, its value for a set is a tight upper bound for the normalized number of visits of orbits in this set. | Orbit capacity |
c_2bs9tqha2fp4 | In mathematics, ordinal logic is a logic associated with an ordinal number by recursively adding elements to a sequence of previous logics. The concept was introduced in 1938 by Alan Turing in his PhD dissertation at Princeton in view of Gödel's incompleteness theorems.While Gödel showed that every logic system suffers... | Ordinal logic |
c_px5v9z6sxqcb | In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space is orientable if such a consistent definition exists. In this case, there are two... | Orientation reversing |
c_3x1j74zfcxq3 | A space is non-orientable if "clockwise" is changed into "counterclockwise" after running through some loops in it, and coming back to the starting point. This means that a geometric shape, such as , that moves continuously along such a loop is changed into its own mirror image . A Möbius strip is an example of a non-o... | Orientation reversing |
c_qaqnzmrfxh7x | Various equivalent formulations of orientability can be given, depending on the desired application and level of generality. Formulations applicable to general topological manifolds often employ methods of homology theory, whereas for differentiable manifolds more structure is present, allowing a formulation in terms o... | Orientation reversing |
c_osrvjycczaxv | In mathematics, orthogonal coordinates are defined as a set of d coordinates q = ( q 1 , q 2 , … , q d ) {\displaystyle \mathbf {q} =(q^{1},q^{2},\dots ,q^{d})} in which the coordinate hypersurfaces all meet at right angles (note that superscripts are indices, not exponents). A coordinate surface for a particular coord... | Orthogonal coordinate system |
c_5tfa4dg53byl | In mathematics, orthogonal functions belong to a function space that is a vector space equipped with a bilinear form. When the function space has an interval as the domain, the bilinear form may be the integral of the product of functions over the interval: ⟨ f , g ⟩ = ∫ f ( x ) ¯ g ( x ) d x . {\displaystyle \langle f... | Orthogonal functions |
c_c7bmkh4l0vi7 | As with a basis of vectors in a finite-dimensional space, orthogonal functions can form an infinite basis for a function space. Conceptually, the above integral is the equivalent of a vector dot product; two vectors are mutually independent (orthogonal) if their dot-product is zero. Suppose { f 0 , f 1 , … } {\displays... | Orthogonal functions |
c_vgecovulkucp | In mathematics, orthogonal polynomials on the unit circle are families of polynomials that are orthogonal with respect to integration over the unit circle in the complex plane, for some probability measure on the unit circle. They were introduced by Szegő (1920, 1921, 1939). | Orthogonal polynomials on the unit circle |
c_n9ro27ns0c9g | In mathematics, orthogonality is the generalization of the geometric notion of perpendicularity to the linear algebra of bilinear forms. Two elements u and v of a vector space with bilinear form B are orthogonal when B(u, v) = 0. Depending on the bilinear form, the vector space may contain nonzero self-orthogonal vecto... | Orthogonality (mathematics) |
c_57ya8ycikdnd | In mathematics, orthogonality is the generalization of the geometric notion of perpendicularity. Orthogonality is also used with various meanings that are often weakly related or not related at all with the mathematical meanings. | Orthogonal subspace |
c_7xhwbpem8ew7 | In mathematics, orthonormality typically implies a norm which has a value of unity (1). Gell-Mann matrices, however, are normalized to a value of 2. Thus, the trace of the pairwise product results in the ortho-normalization condition tr ( λ i λ j ) = 2 δ i j , {\displaystyle \operatorname {tr} (\lambda _{i}\lambda _{... | Gell-Mann matrices |
c_h03cv5u6dgvz | In this three-dimensional matrix representation, the Cartan subalgebra is the set of linear combinations (with real coefficients) of the two matrices λ 3 {\displaystyle \lambda _{3}} and λ 8 {\displaystyle \lambda _{8}} , which commute with each other. There are three significant SU(2) subalgebras: { λ 1 , λ 2 , λ 3 } ... | Gell-Mann matrices |
c_68tpsegbng1t | The SU(2) Casimirs of these subalgebras mutually commute. However, any unitary similarity transformation of these subalgebras will yield SU(2) subalgebras. There is an uncountable number of such transformations. | Gell-Mann matrices |
c_egajnzzs11vu | In mathematics, overconvergent modular forms are special p-adic modular forms that are elements of certain p-adic Banach spaces (usually infinite dimensional) containing classical spaces of modular forms as subspaces. They were introduced by Nicholas M. Katz in 1972. | Overconvergent modular form |
c_93tvz9eojw64 | In mathematics, p-adic Hodge theory is a theory that provides a way to classify and study p-adic Galois representations of characteristic 0 local fields with residual characteristic p (such as Qp). The theory has its beginnings in Jean-Pierre Serre and John Tate's study of Tate modules of abelian varieties and the noti... | P-adic Hodge theory |
c_zwaj8qerar03 | In mathematics, p-adic Teichmüller theory describes the "uniformization" of p-adic curves and their moduli, generalizing the usual Teichmüller theory that describes the uniformization of Riemann surfaces and their moduli. It was introduced and developed by Shinichi Mochizuki (1996, 1999). The first problem is to reform... | P-adic Teichmuller theory |
c_2jb9c7quu47c | In mathematics, p-adic analysis is a branch of number theory that deals with the mathematical analysis of functions of p-adic numbers. The theory of complex-valued numerical functions on the p-adic numbers is part of the theory of locally compact groups. The usual meaning taken for p-adic analysis is the theory of p-ad... | P-adic analysis |
c_esqab3n9rz79 | Some applications have required the development of p-adic functional analysis and spectral theory. In many ways p-adic analysis is less subtle than classical analysis, since the ultrametric inequality means, for example, that convergence of infinite series of p-adic numbers is much simpler. Topological vector spaces ov... | P-adic analysis |
c_34u9pgdd6fk8 | In mathematics, p-adic cohomology means a cohomology theory for varieties of characteristic p whose values are modules over a ring of p-adic integers. Examples (in roughly historical order) include: Serre's Witt vector cohomology Monsky–Washnitzer cohomology Infinitesimal cohomology Crystalline cohomology Rigid cohomol... | P-adic cohomology |
c_w1zv5dy8w90c | In mathematics, parabolic cylindrical coordinates are a three-dimensional orthogonal coordinate system that results from projecting the two-dimensional parabolic coordinate system in the perpendicular z {\displaystyle z} -direction. Hence, the coordinate surfaces are confocal parabolic cylinders. Parabolic cylindrical ... | Parabolic cylindrical coordinates |
c_zd1z1yq9desb | In mathematics, parabolic induction is a method of constructing representations of a reductive group from representations of its parabolic subgroups. If G is a reductive algebraic group and P = M A N {\displaystyle P=MAN} is the Langlands decomposition of a parabolic subgroup P, then parabolic induction consists of tak... | Philosophy of cusp forms |
c_26yoh530xdzo | In mathematics, parity is the property of an integer of whether it is even or odd. An integer is even if it is a multiple of two, and odd if it is not. For example, −4, 0, 82 are even because By contrast, −3, 5, 7, 21 are odd numbers. | Even number |
c_nt1ut9t2wi5h | The above definition of parity applies only to integer numbers, hence it cannot be applied to numbers like 1/2 or 4.201. See the section "Higher mathematics" below for some extensions of the notion of parity to a larger class of "numbers" or in other more general settings. Even and odd numbers have opposite parities, e... | Even number |
c_8380173nq72e | In particular, the parity of zero is even. Any two consecutive integers have opposite parity. A number (i.e., integer) expressed in the decimal numeral system is even or odd according to whether its last digit is even or odd. | Even number |
c_fu0ez5x03ko9 | That is, if the last digit is 1, 3, 5, 7, or 9, then it is odd; otherwise it is even—as the last digit of any even number is 0, 2, 4, 6, or 8. The same idea will work using any even base. In particular, a number expressed in the binary numeral system is odd if its last digit is 1; and it is even if its last digit is 0.... | Even number |
c_9vupfqckapob | In mathematics, particular in abstract algebra and algebraic K-theory, the stable range of a ring R {\displaystyle R} is the smallest integer n {\displaystyle n} such that whenever v 0 , v 1 , . . . | Stable range condition |
c_b7ywayqk2kpd | , v n {\displaystyle v_{0},v_{1},...,v_{n}} in R {\displaystyle R} generate the unit ideal (they form a unimodular row), there exist some t 1 , . . . , t n {\displaystyle t_{1},...,t_{n}} in R {\displaystyle R} such that the elements v i − v 0 t i {\displaystyle v_{i}-v_{0}t_{i}} for 1 ≤ i ≤ n {\displaystyle 1\leq i\le... | Stable range condition |
c_j65t3h3vsc8b | In mathematics, particularly abstract algebra, a binary operation • on a set is flexible if it satisfies the flexible identity: a ∙ ( b ∙ a ) = ( a ∙ b ) ∙ a {\displaystyle a\bullet \left(b\bullet a\right)=\left(a\bullet b\right)\bullet a} for any two elements a and b of the set. A magma (that is, a set equipped with a... | Flexible identity |
c_imlq2ykvq0y7 | In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed. It is one of many closures in mathematics. Using Zorn's lemma or the weaker ultrafilter lemma, it can be shown that every field has an algebraic closure, and that the algebraic c... | Separably closed field |
c_a4ehxw3948bt | In mathematics, particularly algebraic topology and homology theory, the Mayer–Vietoris sequence is an algebraic tool to help compute algebraic invariants of topological spaces, known as their homology and cohomology groups. The result is due to two Austrian mathematicians, Walther Mayer and Leopold Vietoris. The metho... | Mayer–Vietoris sequence |
c_uepdo82n55wo | It is a natural long exact sequence, whose entries are the (co)homology groups of the whole space, the direct sum of the (co)homology groups of the subspaces, and the (co)homology groups of the intersection of the subspaces. The Mayer–Vietoris sequence holds for a variety of cohomology and homology theories, including ... | Mayer–Vietoris sequence |
c_0pdlhqehk3wb | Because the (co)homology of most spaces cannot be computed directly from their definitions, one uses tools such as the Mayer–Vietoris sequence in the hope of obtaining partial information. Many spaces encountered in topology are constructed by piecing together very simple patches. Carefully choosing the two covering su... | Mayer–Vietoris sequence |
c_dtf2fg6bgt4b | In mathematics, particularly algebraic topology, cohomotopy sets are particular contravariant functors from the category of pointed topological spaces and basepoint-preserving continuous maps to the category of sets and functions. They are dual to the homotopy groups, but less studied. | Cohomotopy group |
c_hw7yexiaewfm | In mathematics, particularly algebraic topology, the Kan-Thurston theorem associates a discrete group G {\displaystyle G} to every path-connected topological space X {\displaystyle X} in such a way that the group cohomology of G {\displaystyle G} is the same as the cohomology of the space X {\displaystyle X} . The grou... | Kan-Thurston theorem |
c_6evihkhaobyz | In mathematics, particularly calculus, a vertical tangent is a tangent line that is vertical. Because a vertical line has infinite slope, a function whose graph has a vertical tangent is not differentiable at the point of tangency. | Vertical tangent |
c_4g9ugmzz0a30 | In mathematics, particularly category theory, a representable functor is a certain functor from an arbitrary category into the category of sets. Such functors give representations of an abstract category in terms of known structures (i.e. sets and functions) allowing one to utilize, as much as possible, knowledge about... | Universal element |
c_8gyjm1qtvmqi | In mathematics, particularly commutative algebra, an invertible module is intuitively a module that has an inverse with respect to the tensor product. Invertible modules form the foundation for the definition of invertible sheaves in algebraic geometry. Formally, a finitely generated module M over a ring R is said to b... | Invertible module |
c_bepm0w11lsy9 | In mathematics, particularly computational algebra, Berlekamp's algorithm is a well-known method for factoring polynomials over finite fields (also known as Galois fields). The algorithm consists mainly of matrix reduction and polynomial GCD computations. It was invented by Elwyn Berlekamp in 1967. It was the dominant ... | Berlekamp's algorithm |
c_8305vchu7gfu | In mathematics, particularly differential geometry, a Finsler manifold is a differentiable manifold M where a (possibly asymmetric) Minkowski functional F(x, −) is provided on each tangent space TxM, that enables one to define the length of any smooth curve γ: → M as L ( γ ) = ∫ a b F ( γ ( t ) , γ ˙ ( t ) ) d t . {\d... | Finsler geometry |
c_3h98ftlj2ksm | In mathematics, particularly differential topology, the double tangent bundle or the second tangent bundle refers to the tangent bundle (TTM,πTTM,TM) of the total space TM of the tangent bundle (TM,πTM,M) of a smooth manifold M . A note on notation: in this article, we denote projection maps by their domains, e.g., πTT... | Double tangent bundle |
c_r5rk4lycs825 | In mathematics, particularly differential topology, the secondary vector bundle structure refers to the natural vector bundle structure (TE, p∗, TM) on the total space TE of the tangent bundle of a smooth vector bundle (E, p, M), induced by the push-forward p∗: TE → TM of the original projection map p: E → M. This give... | Secondary vector bundle structure |
c_dl5ecrp5tx1c | In mathematics, particularly functional analysis, James' theorem, named for Robert C. James, states that a Banach space X {\displaystyle X} is reflexive if and only if every continuous linear functional's norm on X {\displaystyle X} attains its supremum on the closed unit ball in X . {\displaystyle X.} A stronger versi... | James' theorem |
c_99hw8dxan2pe | In mathematics, particularly functional analysis, spaces of linear maps between two vector spaces can be endowed with a variety of topologies. Studying space of linear maps and these topologies can give insight into the spaces themselves. The article operator topologies discusses topologies on spaces of linear maps bet... | Topology of uniform convergence |
c_w716quuw11v7 | In mathematics, particularly functional analysis, the Dunford–Schwartz theorem, named after Nelson Dunford and Jacob T. Schwartz, states that the averages of powers of certain norm-bounded operators on L1 converge in a suitable sense. | Dunford–Schwartz theorem |
c_64h4hofdmzm4 | In mathematics, particularly geometric graph theory, a unit distance graph is a graph formed from a collection of points in the Euclidean plane by connecting two points whenever the distance between them is exactly one. To distinguish these graphs from a broader definition that allows some non-adjacent pairs of vertice... | Unit distance graph |
c_etnhce7b0nfi | The generalized Petersen graphs are non-strict unit distance graphs. An unsolved problem of Paul Erdős asks how many edges a unit distance graph on n {\displaystyle n} vertices can have. The best known lower bound is slightly above linear in n {\displaystyle n} —far from the upper bound, proportional to n 4 / 3 {\displ... | Unit distance graph |
c_vod6g1erjd22 | The number of colors required to color unit distance graphs is also unknown (the Hadwiger–Nelson problem): some unit distance graphs require five colors, and every unit distance graph can be colored with seven colors. For every algebraic number there is a unit distance graph with two vertices that must be that distance... | Unit distance graph |
c_xths50thqa59 | It is possible to construct a unit distance graph efficiently, given its points. Finding all unit distances has applications in pattern matching, where it can be a first step in finding congruent copies of larger patterns. However, determining whether a given graph can be represented as a unit distance graph is NP-hard... | Unit distance graph |
c_jnu37lhvz7ma | In mathematics, particularly graph theory, and computer science, a directed acyclic graph (DAG) is a directed graph with no directed cycles. That is, it consists of vertices and edges (also called arcs), with each edge directed from one vertex to another, such that following those directions will never form a closed lo... | Directed Acyclic Graph |
c_s4mkqz4supwr | In mathematics, particularly homological algebra, an exact functor is a functor that preserves short exact sequences. Exact functors are convenient for algebraic calculations because they can be directly applied to presentations of objects. Much of the work in homological algebra is designed to cope with functors that ... | Left exact functor |
c_wqep51khvp1b | In mathematics, particularly homological algebra, the zig-zag lemma asserts the existence of a particular long exact sequence in the homology groups of certain chain complexes. The result is valid in every abelian category. | Zig-zag lemma |
c_hk4h2g6ls8oq | In mathematics, particularly in abstract algebra, a ring R is said to be stably finite (or weakly finite) if, for all square matrices A and B of the same size with entries in R, AB = 1 implies BA = 1. This is a stronger property for a ring than having the invariant basis number (IBN) property. Namely, any nontrivial st... | Stably finite ring |
c_da1jm7m427jc | In mathematics, particularly in abstract algebra, a semigroup with involution or a *-semigroup is a semigroup equipped with an involutive anti-automorphism, which—roughly speaking—brings it closer to a group because this involution, considered as unary operator, exhibits certain fundamental properties of the operation ... | Free semigroup with involution |
c_bbuebdcaro2w | An example from linear algebra is the multiplicative monoid of real square matrices of order n (called the full linear monoid). The map which sends a matrix to its transpose is an involution because the transpose is well defined for any matrix and obeys the law (AB)T = BTAT, which has the same form of interaction with ... | Free semigroup with involution |
c_jlt79zgram1o | Another example, coming from formal language theory, is the free semigroup generated by a nonempty set (an alphabet), with string concatenation as the binary operation, and the involution being the map which reverses the linear order of the letters in a string. A third example, from basic set theory, is the set of all ... | Free semigroup with involution |
c_ehxbcfrun8vb | In mathematics, particularly in algebra, a field extension is a pair of fields K ⊆ L , {\displaystyle K\subseteq L,} such that the operations of K are those of L restricted to K. In this case, L is an extension field of K and K is a subfield of L. For example, under the usual notions of addition and multiplication, the... | Adjunction (field theory) |
c_3gy7u2m0vghj | In mathematics, particularly in algebra, an indeterminate equation is an equation for which there is more than one solution. For example, the equation a x + b y = c {\displaystyle ax+by=c} is a simple indeterminate equation, as is x 2 = 1 {\displaystyle x^{2}=1} . Indeterminate equations cannot be solved uniquely. | Indeterminate equations |
c_gwwwcb9g0mrr | In fact, in some cases it might even have infinitely many solutions. Some of the prominent examples of indeterminate equations include: Univariate polynomial equation: a n x n + a n − 1 x n − 1 + ⋯ + a 2 x 2 + a 1 x + a 0 = 0 , {\displaystyle a_{n}x^{n}+a_{n-1}x^{n-1}+\dots +a_{2}x^{2}+a_{1}x+a_{0}=0,} which has multip... | Indeterminate equations |
c_phfmv82hths4 | Pell's equation: x 2 − P y 2 = 1 , {\displaystyle \ x^{2}-Py^{2}=1,} where P {\displaystyle P} is a given integer that is not a square number, and in which the variables x {\displaystyle x} and y {\displaystyle y} are required to be integers. The equation of Pythagorean triples: x 2 + y 2 = z 2 , {\displaystyle x^{2}+y... | Indeterminate equations |
c_6nr9hlvowrr1 | In mathematics, particularly in algebra, an indeterminate system is a system of simultaneous equations (e.g., linear equations) which has more than one solution (sometimes infinitely many solutions). In the case of a linear system, the system may be said to be underspecified, in which case the presence of more than one... | Indeterminate system |
c_7jojejycuduh | In mathematics, particularly in algebra, the class of projective modules enlarges the class of free modules (that is, modules with basis vectors) over a ring, by keeping some of the main properties of free modules. Various equivalent characterizations of these modules appear below. Every free module is a projective mod... | Locally free module |
c_bdow1nrdxmcf | In mathematics, particularly in algebra, the injective hull (or injective envelope) of a module is both the smallest injective module containing it and the largest essential extension of it. Injective hulls were first described in (Eckmann & Schopf 1953). | Injective envelope |
c_ofbgqpafse10 | In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions. Abelian varieties are at the same time among the most studied objects in... | Principal polarization |
c_ykrkeaxlrrt5 | Such abelian varieties turn out to be exactly those complex tori that can be embedded into a complex projective space. Abelian varieties defined over algebraic number fields are a special case, which is important also from the viewpoint of number theory. | Principal polarization |
c_xqgh9wxr6nlh | Localization techniques lead naturally from abelian varieties defined over number fields to ones defined over finite fields and various local fields. Since a number field is the fraction field of a Dedekind domain, for any nonzero prime of your Dedekind domain, there is a map from the Dedekind domain to the quotient of... | Principal polarization |
c_z8azaxrye5o0 | Given a curve with equation defined over the number field, we can apply this map to the coefficients to get a curve defined over some finite field, where the choices of finite field correspond to the finite primes of the number field. Abelian varieties appear naturally as Jacobian varieties (the connected components of... | Principal polarization |
c_g4v4q16eagfl | In mathematics, particularly in algebraic topology, Alexander–Spanier cohomology is a cohomology theory for topological spaces. | Alexander–Spanier cohomology |
c_aaxoxvpm4ksi | In mathematics, particularly in algebraic topology, a taut pair is a topological pair whose direct limit of cohomology module of open neighborhood of that pair which is directed downward by inclusion is isomorphic to the cohomology module of original pair. | Tautness (topology) |
c_3hfa9cyibshi | In mathematics, particularly in algebraic topology, the n-skeleton of a topological space X presented as a simplicial complex (resp. CW complex) refers to the subspace Xn that is the union of the simplices of X (resp. cells of X) of dimensions m ≤ n. In other words, given an inductive definition of a complex, the n-ske... | Skeleton (topology) |
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