text
stringlengths 9
3.55k
| source
stringlengths 31
280
|
|---|---|
(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 in terms of irreducible representations, even though it is unitary and completely reducible. An example where this happens is the infinite symmetric group, where the von Neumann group algebra is the hyperfinite type II1 factor. The further theory divides up the Plancherel measure into a discrete and a continuous part. For semisimple groups, and classes of solvable Lie groups, a very detailed theory is available.
|
https://en.wikipedia.org/wiki/Noncommutative_harmonic_analysis
|
In mathematics, noncommutative projective geometry is a noncommutative analog of projective geometry in the setting of noncommutative algebraic geometry.
|
https://en.wikipedia.org/wiki/Noncommutative_projective_geometry
|
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 earlier by M. Adler (1978) and Y. Manin (1978) in the context of one-dimensional integrable systems.
|
https://en.wikipedia.org/wiki/Noncommutative_residue
|
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 topology is related to analytic noncommutative geometry.
|
https://en.wikipedia.org/wiki/Noncommutative_topology
|
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 traditional modelling methods, such as linear regression and basic statistical methods, nonlinear modelling can be utilized efficiently in a vast number of situations where traditional modelling is impractical or impossible. The newer nonlinear modelling approaches include non-parametric methods, such as feedforward neural networks, kernel regression, multivariate splines, etc., which do not require a priori knowledge of the nonlinearities in the relations.
|
https://en.wikipedia.org/wiki/Nonlinear_model
|
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 phenomenological modelling, nonlinear modelling can be utilized in processes and systems where the theory is deficient or there is a lack of fundamental understanding on the root causes of most crucial factors on system. Phenomenological modelling describes a system in terms of laws of nature. Nonlinear modelling can be utilized in situations where the phenomena are not well understood or expressed in mathematical terms. Thus nonlinear modelling can be an efficient way to model new and complex situations where relationships of different variables are not known.
|
https://en.wikipedia.org/wiki/Nonlinear_model
|
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 variables and conditional to the satisfaction of a system of equalities and inequalities, collectively termed constraints. It is the sub-field of mathematical optimization that deals with problems that are not linear.
|
https://en.wikipedia.org/wiki/Non-linear_programming
|
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 widely used before Karl Weierstrass sought to replace them with the (ε, δ)-definition of limit starting in the 1870s.
|
https://en.wikipedia.org/wiki/Nonstandard_calculus
|
(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 Jerzy Łoś. According to Howard Keisler, "Robinson solved a three hundred year old problem by giving a precise treatment of infinitesimals. Robinson's achievement will probably rank as one of the major mathematical advances of the twentieth century."
|
https://en.wikipedia.org/wiki/Nonstandard_calculus
|
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).
|
https://en.wikipedia.org/wiki/Nuclear_map
|
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. In Banach spaces this is no longer possible for general nuclear operators, it is however possible for 2 3 {\displaystyle {\tfrac {2}{3}}} -nuclear operator via the Grothendieck trace theorem. The general definition for Banach spaces was given by Grothendieck. This article presents both cases but concentrates on the general case of nuclear operators on Banach spaces.
|
https://en.wikipedia.org/wiki/Nuclear_operators_between_Banach_spaces
|
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 were introduced by Alexander Grothendieck. The topology on nuclear spaces can be defined by a family of seminorms whose unit balls decrease rapidly in size.
|
https://en.wikipedia.org/wiki/Nuclear_spaces
|
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 a sort of converse to this is often true: if a "naturally occurring" topological vector space is not a Banach space, then there is a good chance that it is nuclear.
|
https://en.wikipedia.org/wiki/Nuclear_spaces
|
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 per subdomain is used to further coordinate the solution between the subdomains globally. The problems on the subdomains are independent, which makes domain decomposition methods suitable for parallel computing. Domain decomposition methods are typically used as preconditioners for Krylov space iterative methods, such as the conjugate gradient method, GMRES, and LOBPCG.
|
https://en.wikipedia.org/wiki/Domain_decomposition
|
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 Schwarz method.
|
https://en.wikipedia.org/wiki/Domain_decomposition
|
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 methods, such as FETI, the continuity of the solution across the subdomain interface is enforced by Lagrange multipliers.
|
https://en.wikipedia.org/wiki/Domain_decomposition
|
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.
|
https://en.wikipedia.org/wiki/Domain_decomposition
|
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 by multiple-point constraints. Finite element simulations of moderate size models require solving linear systems with millions of unknowns. Several hours per time step is an average sequential run time, therefore, parallel computing is a necessity. Domain decomposition methods embody large potential for a parallelization of the finite element methods, and serve a basis for distributed, parallel computations.
|
https://en.wikipedia.org/wiki/Domain_decomposition
|
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 turns out to be an application of cohomology theory to the problem of constructing a cross-section of a bundle.
|
https://en.wikipedia.org/wiki/Obstruction_theory
|
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 usually taken to be f ( λ x ) = λ Δ f ( x ) {\displaystyle f(\lambda x)=\lambda ^{\Delta }f(x)} for some choice of exponent Δ, and for all dilations λ. This is equivalent to f being a homogeneous function of degree Δ. Examples of scale-invariant functions are the monomials f ( x ) = x n {\displaystyle f(x)=x^{n}} , for which Δ = n, in that clearly f ( λ x ) = ( λ x ) n = λ n f ( x ) . {\displaystyle f(\lambda x)=(\lambda x)^{n}=\lambda ^{n}f(x)~.} An example of a scale-invariant curve is the logarithmic spiral, a kind of curve that often appears in nature.
|
https://en.wikipedia.org/wiki/Scale_invariance
|
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).
|
https://en.wikipedia.org/wiki/Scale_invariance
|
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 defined.
|
https://en.wikipedia.org/wiki/Product_of_group_subsets
|
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 only if ST = TS.
|
https://en.wikipedia.org/wiki/Product_of_group_subsets
|
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. Examples are the product of sets, groups (described below), rings, and other algebraic structures. The product of topological spaces is another instance.There is also the direct sum – in some areas this is used interchangeably, while in others it is a different concept.
|
https://en.wikipedia.org/wiki/Directly_indecomposable
|
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 continuous embedding theorems.
|
https://en.wikipedia.org/wiki/Continuous_embedding
|
In mathematics, operator K-theory is a noncommutative analogue of topological K-theory for Banach algebras with most applications used for C*-algebras.
|
https://en.wikipedia.org/wiki/Operator_K-theory
|
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 operators. The study, which depends heavily on the topology of function spaces, is a branch of functional analysis. If a collection of operators forms an algebra over a field, then it is an operator algebra. The description of operator algebras is part of operator theory.
|
https://en.wikipedia.org/wiki/Operator_Theory
|
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 so-called, regular p-groups. The relationship between power and commutator structure forms a central theme in the modern study of p-groups, as exemplified in the work on uniformly powerful p-groups. The word "agemo" is just "omega" spelled backwards, and the agemo subgroup is denoted by an upside-down omega.
|
https://en.wikipedia.org/wiki/Omega_and_agemo_subgroup
|
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 1912.
|
https://en.wikipedia.org/wiki/Riesz_projector
|
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 particular, this condition is always satisfied if M is compact),then it is a locally trivial fibration. This is a foundational result in differential topology due to Charles Ehresmann, and has many variants.
|
https://en.wikipedia.org/wiki/Ehresmann's_lemma
|
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.
|
https://en.wikipedia.org/wiki/Orbit_capacity
|
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 from some form of incompleteness, Turing focused on a method so that a complete system of logic may be constructed from a given system of logic. By repeating the process a sequence L1, L2, … of logic is obtained, each more complete than the previous one. A logic L can then be constructed in which the provable theorems are the totality of theorems provable with the help of the L1, L2, … etc. Thus Turing showed how one can associate logic with any constructive ordinal. == References ==
|
https://en.wikipedia.org/wiki/Ordinal_logic
|
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 possible definitions, and a choice between them is an orientation of the space. Real vector spaces, Euclidean spaces, and spheres are orientable.
|
https://en.wikipedia.org/wiki/Orientation_reversing
|
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-orientable space.
|
https://en.wikipedia.org/wiki/Orientation_reversing
|
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 of differential forms. A generalization of the notion of orientability of a space is that of orientability of a family of spaces parameterized by some other space (a fiber bundle) for which an orientation must be selected in each of the spaces which varies continuously with respect to changes in the parameter values.
|
https://en.wikipedia.org/wiki/Orientation_reversing
|
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 coordinate qk is the curve, surface, or hypersurface on which qk is a constant. For example, the three-dimensional Cartesian coordinates (x, y, z) is an orthogonal coordinate system, since its coordinate surfaces x = constant, y = constant, and z = constant are planes that meet at right angles to one another, i.e., are perpendicular. Orthogonal coordinates are a special but extremely common case of curvilinear coordinates.
|
https://en.wikipedia.org/wiki/Orthogonal_coordinate_system
|
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,g\rangle =\int {\overline {f(x)}}g(x)\,dx.} The functions f {\displaystyle f} and g {\displaystyle g} are orthogonal when this integral is zero, i.e. ⟨ f , g ⟩ = 0 {\displaystyle \langle f,\,g\rangle =0} whenever f ≠ g {\displaystyle f\neq g} .
|
https://en.wikipedia.org/wiki/Orthogonal_functions
|
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 , … } {\displaystyle \{f_{0},f_{1},\ldots \}} is a sequence of orthogonal functions of nonzero L2-norms ‖ f n ‖ 2 = ⟨ f n , f n ⟩ = ( ∫ f n 2 d x ) 1 2 {\textstyle \left\|f_{n}\right\|_{2}={\sqrt {\langle f_{n},f_{n}\rangle }}=\left(\int f_{n}^{2}\ dx\right)^{\frac {1}{2}}} . It follows that the sequence { f n / ‖ f n ‖ 2 } {\displaystyle \left\{f_{n}/\left\|f_{n}\right\|_{2}\right\}} is of functions of L2-norm one, forming an orthonormal sequence. To have a defined L2-norm, the integral must be bounded, which restricts the functions to being square-integrable.
|
https://en.wikipedia.org/wiki/Orthogonal_functions
|
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).
|
https://en.wikipedia.org/wiki/Orthogonal_polynomials_on_the_unit_circle
|
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 vectors. In the case of function spaces, families of orthogonal functions are used to form a basis. The concept has been used in the context of orthogonal functions, orthogonal polynomials, and combinatorics.
|
https://en.wikipedia.org/wiki/Orthogonality_(mathematics)
|
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.
|
https://en.wikipedia.org/wiki/Orthogonal_subspace
|
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 _{j})=2\delta _{ij},} where δ i j {\displaystyle \delta _{ij}} is the Kronecker delta. This is so the embedded Pauli matrices corresponding to the three embedded subalgebras of SU(2) are conventionally normalized.
|
https://en.wikipedia.org/wiki/Gell-Mann_matrices
|
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 } {\displaystyle \{\lambda _{1},\lambda _{2},\lambda _{3}\}} { λ 4 , λ 5 , x } , {\displaystyle \{\lambda _{4},\lambda _{5},x\},} and { λ 6 , λ 7 , y } , {\displaystyle \{\lambda _{6},\lambda _{7},y\},} where the x and y are linear combinations of λ 3 {\displaystyle \lambda _{3}} and λ 8 {\displaystyle \lambda _{8}} .
|
https://en.wikipedia.org/wiki/Gell-Mann_matrices
|
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.
|
https://en.wikipedia.org/wiki/Gell-Mann_matrices
|
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.
|
https://en.wikipedia.org/wiki/Overconvergent_modular_form
|
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 notion of Hodge–Tate representation. Hodge–Tate representations are related to certain decompositions of p-adic cohomology theories analogous to the Hodge decomposition, hence the name p-adic Hodge theory. Further developments were inspired by properties of p-adic Galois representations arising from the étale cohomology of varieties. Jean-Marc Fontaine introduced many of the basic concepts of the field.
|
https://en.wikipedia.org/wiki/P-adic_Hodge_theory
|
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 reformulate the Fuchsian uniformization of a complex Riemann surface (an isomorphism from the upper half plane to a universal covering space of the surface) in a way that makes sense for p-adic curves. The existence of a Fuchsian uniformization is equivalent to the existence of a canonical indigenous bundle over the Riemann surface: the unique indigenous bundle that is invariant under complex conjugation and whose monodromy representation is quasi-Fuchsian. For p-adic curves the analogue of complex conjugation is the Frobenius endomorphism, and the analogue of the quasi-Fuchsian condition is an integrality condition on the indigenous line bundle. So p-adic Teichmüller theory, the p-adic analogue the Fuchsian uniformization of Teichmüller theory, is the study of integral Frobenius invariant indigenous bundles.
|
https://en.wikipedia.org/wiki/P-adic_Teichmuller_theory
|
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-adic-valued functions on spaces of interest. Applications of p-adic analysis have mainly been in number theory, where it has a significant role in diophantine geometry and diophantine approximation.
|
https://en.wikipedia.org/wiki/P-adic_analysis
|
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 over p-adic fields show distinctive features; for example aspects relating to convexity and the Hahn–Banach theorem are different.
|
https://en.wikipedia.org/wiki/P-adic_analysis
|
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 cohomology
|
https://en.wikipedia.org/wiki/P-adic_cohomology
|
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 coordinates have found many applications, e.g., the potential theory of edges.
|
https://en.wikipedia.org/wiki/Parabolic_cylindrical_coordinates
|
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 taking a representation of M A {\displaystyle MA} , extending it to P by letting N act trivially, and inducing the result from P to G. There are some generalizations of parabolic induction using cohomology, such as cohomological parabolic induction and Deligne–Lusztig theory.
|
https://en.wikipedia.org/wiki/Philosophy_of_cusp_forms
|
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.
|
https://en.wikipedia.org/wiki/Even_number
|
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.g., 22 (even number) and 13 (odd number) have opposite parities.
|
https://en.wikipedia.org/wiki/Even_number
|
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.
|
https://en.wikipedia.org/wiki/Even_number
|
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. In an odd base, the number is even according to the sum of its digits—it is even if and only if the sum of its digits is even.
|
https://en.wikipedia.org/wiki/Even_number
|
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 , . . .
|
https://en.wikipedia.org/wiki/Stable_range_condition
|
, 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\leq n} also generate the unit ideal. If R {\displaystyle R} is a commutative Noetherian ring of Krull dimension d {\displaystyle d} , then the stable range of R {\displaystyle R} is at most d + 1 {\displaystyle d+1} (a theorem of Bass).
|
https://en.wikipedia.org/wiki/Stable_range_condition
|
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 binary operation) is flexible if the binary operation with which it is equipped is flexible. Similarly, a nonassociative algebra is flexible if its multiplication operator is flexible. Every commutative or associative operation is flexible, so flexibility becomes important for binary operations that are neither commutative nor associative, e.g. for the multiplication of sedenions, which are not even alternative. In 1954, Richard D. Schafer examined the algebras generated by the Cayley–Dickson process over a field and showed that they satisfy the flexible identity.
|
https://en.wikipedia.org/wiki/Flexible_identity
|
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 closure of a field K is unique up to an isomorphism that fixes every member of K. Because of this essential uniqueness, we often speak of the algebraic closure of K, rather than an algebraic closure of K. The algebraic closure of a field K can be thought of as the largest algebraic extension of K. To see this, note that if L is any algebraic extension of K, then the algebraic closure of L is also an algebraic closure of K, and so L is contained within the algebraic closure of K. The algebraic closure of K is also the smallest algebraically closed field containing K, because if M is any algebraically closed field containing K, then the elements of M that are algebraic over K form an algebraic closure of K. The algebraic closure of a field K has the same cardinality as K if K is infinite, and is countably infinite if K is finite.
|
https://en.wikipedia.org/wiki/Separably_closed_field
|
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 method consists of splitting a space into subspaces, for which the homology or cohomology groups may be easier to compute. The sequence relates the (co)homology groups of the space to the (co)homology groups of the subspaces.
|
https://en.wikipedia.org/wiki/Mayer–Vietoris_sequence
|
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 simplicial homology and singular cohomology. In general, the sequence holds for those theories satisfying the Eilenberg–Steenrod axioms, and it has variations for both reduced and relative (co)homology.
|
https://en.wikipedia.org/wiki/Mayer–Vietoris_sequence
|
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 subspaces so that, together with their intersection, they have simpler (co)homology than that of the whole space may allow a complete deduction of the (co)homology of the space. In that respect, the Mayer–Vietoris sequence is analogous to the Seifert–van Kampen theorem for the fundamental group, and a precise relation exists for homology of dimension one.
|
https://en.wikipedia.org/wiki/Mayer–Vietoris_sequence
|
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.
|
https://en.wikipedia.org/wiki/Cohomotopy_group
|
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 group G {\displaystyle G} might then be regarded as a good approximation to the space X {\displaystyle X} , and consequently the theorem is sometimes interpreted to mean that homotopy theory can be viewed as part of group theory. More precisely, the theorem states that every path-connected topological space is homology-equivalent to the classifying space K ( G , 1 ) {\displaystyle K(G,1)} of a discrete group G {\displaystyle G} , where homology-equivalent means there is a map K ( G , 1 ) → X {\displaystyle K(G,1)\rightarrow X} inducing an isomorphism on homology. The theorem is attributed to Daniel Kan and William Thurston who published their result in 1976.
|
https://en.wikipedia.org/wiki/Kan-Thurston_theorem
|
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.
|
https://en.wikipedia.org/wiki/Vertical_tangent
|
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 the category of sets in other settings. From another point of view, representable functors for a category C are the functors given with C. Their theory is a vast generalisation of upper sets in posets, and of Cayley's theorem in group theory.
|
https://en.wikipedia.org/wiki/Universal_element
|
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 be invertible if it is locally a free module of rank 1. In other words, M P ≅ R P {\displaystyle M_{P}\cong R_{P}} for all primes P of R. Now, if M is an invertible R-module, then its dual M* = Hom(M,R) is its inverse with respect to the tensor product, i.e. M ⊗ R M ∗ ≅ R {\displaystyle M\otimes _{R}M^{*}\cong R} . The theory of invertible modules is closely related to the theory of codimension one varieties including the theory of divisors.
|
https://en.wikipedia.org/wiki/Invertible_module
|
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 algorithm for solving the problem until the Cantor–Zassenhaus algorithm of 1981. It is currently implemented in many well-known computer algebra systems.
|
https://en.wikipedia.org/wiki/Berlekamp's_algorithm
|
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 . {\displaystyle L(\gamma )=\int _{a}^{b}F\left(\gamma (t),{\dot {\gamma }}(t)\right)\,\mathrm {d} t.} Finsler manifolds are more general than Riemannian manifolds since the tangent norms need not be induced by inner products. Every Finsler manifold becomes an intrinsic quasimetric space when the distance between two points is defined as the infimum length of the curves that join them. Élie Cartan (1933) named Finsler manifolds after Paul Finsler, who studied this geometry in his dissertation (Finsler 1918).
|
https://en.wikipedia.org/wiki/Finsler_geometry
|
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., πTTM: TTM → TM. Some authors index these maps by their ranges instead, so for them, that map would be written πTM. The second tangent bundle arises in the study of connections and second order ordinary differential equations, i.e., (semi)spray structures on smooth manifolds, and it is not to be confused with the second order jet bundle.
|
https://en.wikipedia.org/wiki/Double_tangent_bundle
|
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 gives rise to a double vector bundle structure (TE,E,TM,M). In the special case (E, p, M) = (TM, πTM, M), where TE = TTM is the double tangent bundle, the secondary vector bundle (TTM, (πTM)∗, TM) is isomorphic to the tangent bundle (TTM, πTTM, TM) of TM through the canonical flip.
|
https://en.wikipedia.org/wiki/Secondary_vector_bundle_structure
|
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 version of the theorem states that a weakly closed subset C {\displaystyle C} of a Banach space X {\displaystyle X} is weakly compact if and only if the dual norm each continuous linear functional on X {\displaystyle X} attains a maximum on C . {\displaystyle C.} The hypothesis of completeness in the theorem cannot be dropped.
|
https://en.wikipedia.org/wiki/James'_theorem
|
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 between normed spaces, whereas this article discusses topologies on such spaces in the more general setting of topological vector spaces (TVSs).
|
https://en.wikipedia.org/wiki/Topology_of_uniform_convergence
|
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.
|
https://en.wikipedia.org/wiki/Dunford–Schwartz_theorem
|
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 vertices to be at distance one, they may also be called strict unit distance graphs or faithful unit distance graphs. As a hereditary family of graphs, they can be characterized by forbidden induced subgraphs. The unit distance graphs include the cactus graphs, the matchstick graphs and penny graphs, and the hypercube graphs.
|
https://en.wikipedia.org/wiki/Unit_distance_graph
|
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 {\displaystyle n^{4/3}} .
|
https://en.wikipedia.org/wiki/Unit_distance_graph
|
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 apart. According to the Beckman–Quarles theorem, the only plane transformations that preserve all unit distance graphs are the isometries.
|
https://en.wikipedia.org/wiki/Unit_distance_graph
|
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, and more specifically complete for the existential theory of the reals.
|
https://en.wikipedia.org/wiki/Unit_distance_graph
|
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 loop. A directed graph is a DAG if and only if it can be topologically ordered, by arranging the vertices as a linear ordering that is consistent with all edge directions. DAGs have numerous scientific and computational applications, ranging from biology (evolution, family trees, epidemiology) to information science (citation networks) to computation (scheduling). Directed acyclic graphs are sometimes instead called acyclic directed graphs or acyclic digraphs.
|
https://en.wikipedia.org/wiki/Directed_Acyclic_Graph
|
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 fail to be exact, but in ways that can still be controlled.
|
https://en.wikipedia.org/wiki/Left_exact_functor
|
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.
|
https://en.wikipedia.org/wiki/Zig-zag_lemma
|
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 stably finite ring has IBN. Commutative rings, noetherian rings and artinian rings are stably finite. Subrings of stably finite rings and matrix rings over stably finite rings are stably finite. A ring satisfying Klein's nilpotence condition is stably finite.
|
https://en.wikipedia.org/wiki/Stably_finite_ring
|
In mathematics, particularly in abstract algebra, a semigroup with involution or a *-semigroup is a semigroup equipped with an involutive anti-automorphism, which—roughly speaking—brings it closer to a group because this involution, considered as unary operator, exhibits certain fundamental properties of the operation of taking the inverse in a group: uniqueness, double application "cancelling itself out", and the same interaction law with the binary operation as in the case of the group inverse. It is thus not a surprise that any group is a semigroup with involution. However, there are significant natural examples of semigroups with involution that are not groups.
|
https://en.wikipedia.org/wiki/Free_semigroup_with_involution
|
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 multiplication as taking inverses has in the general linear group (which is a subgroup of the full linear monoid). However, for an arbitrary matrix, AAT does not equal the identity element (namely the diagonal matrix).
|
https://en.wikipedia.org/wiki/Free_semigroup_with_involution
|
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 binary relations between a set and itself, with the involution being the converse relation, and the multiplication given by the usual composition of relations. Semigroups with involution appeared explicitly named in a 1953 paper of Viktor Wagner (in Russian) as result of his attempt to bridge the theory of semigroups with that of semiheaps.
|
https://en.wikipedia.org/wiki/Free_semigroup_with_involution
|
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 complex numbers are an extension field of the real numbers; the real numbers are a subfield of the complex numbers. Field extensions are fundamental in algebraic number theory, and in the study of polynomial roots through Galois theory, and are widely used in algebraic geometry.
|
https://en.wikipedia.org/wiki/Adjunction_(field_theory)
|
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.
|
https://en.wikipedia.org/wiki/Indeterminate_equations
|
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 multiple solutions for the variable x {\displaystyle x} in the complex plane—unless it can be rewritten in the form a n ( x − b ) n = 0 {\displaystyle a_{n}(x-b)^{n}=0} . Non-degenerate conic equation: A x 2 + B x y + C y 2 + D x + E y + F = 0 , {\displaystyle Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0,} where at least one of the given parameters A {\displaystyle A} , B {\displaystyle B} , and C {\displaystyle C} is non-zero, and x {\displaystyle x} and y {\displaystyle y} are real variables.
|
https://en.wikipedia.org/wiki/Indeterminate_equations
|
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^{2}=z^{2},} in which the variables x {\displaystyle x} , y {\displaystyle y} , and z {\displaystyle z} are required to be positive integers. The equation of the Fermat–Catalan conjecture: a m + b n = c k , {\displaystyle a^{m}+b^{n}=c^{k},} in which the variables a {\displaystyle a} , b {\displaystyle b} , c {\displaystyle c} are required to be coprime positive integers, and the variables m {\displaystyle m} , n {\displaystyle n} , and k {\displaystyle k} are required to be positive integers satisfying the following equation: 1 m + 1 n + 1 k < 1. {\displaystyle {\frac {1}{m}}+{\frac {1}{n}}+{\frac {1}{k}}<1.}
|
https://en.wikipedia.org/wiki/Indeterminate_equations
|
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 solution would imply an infinite number of solutions (since the system would be describable in terms of at least one free variable), but that property does not extend to nonlinear systems (e.g., the system with the equation x 2 = 1 {\displaystyle x^{2}=1} ). An indeterminate system by definition is consistent, in the sense of having at least one solution. For a system of linear equations, the number of equations in an indeterminate system could be the same as the number of unknowns, less than the number of unknowns (an underdetermined system), or greater than the number of unknowns (an overdetermined system). Conversely, any of those three cases may or may not be indeterminate.
|
https://en.wikipedia.org/wiki/Indeterminate_system
|
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 module, but the converse fails to hold over some rings, such as Dedekind rings that are not principal ideal domains. However, every projective module is a free module if the ring is a principal ideal domain such as the integers, or a polynomial ring (this is the Quillen–Suslin theorem). Projective modules were first introduced in 1956 in the influential book Homological Algebra by Henri Cartan and Samuel Eilenberg.
|
https://en.wikipedia.org/wiki/Locally_free_module
|
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).
|
https://en.wikipedia.org/wiki/Injective_envelope
|
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 algebraic geometry and indispensable tools for much research on other topics in algebraic geometry and number theory. An abelian variety can be defined by equations having coefficients in any field; the variety is then said to be defined over that field. Historically the first abelian varieties to be studied were those defined over the field of complex numbers.
|
https://en.wikipedia.org/wiki/Principal_polarization
|
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.
|
https://en.wikipedia.org/wiki/Principal_polarization
|
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 the Dedekind domain by the prime, which is a finite field for all finite primes. This induces a map from the fraction field to any such finite field.
|
https://en.wikipedia.org/wiki/Principal_polarization
|
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 zero in Picard varieties) and Albanese varieties of other algebraic varieties. The group law of an abelian variety is necessarily commutative and the variety is non-singular. An elliptic curve is an abelian variety of dimension 1. Abelian varieties have Kodaira dimension 0.
|
https://en.wikipedia.org/wiki/Principal_polarization
|
In mathematics, particularly in algebraic topology, Alexander–Spanier cohomology is a cohomology theory for topological spaces.
|
https://en.wikipedia.org/wiki/Alexander–Spanier_cohomology
|
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.
|
https://en.wikipedia.org/wiki/Tautness_(topology)
|
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-skeleton is obtained by stopping at the n-th step.
|
https://en.wikipedia.org/wiki/Skeleton_(topology)
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.