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In mathematics, a generalized space is a generalization of a topological space. Impetuses for such a generalization comes at least in two forms: A desire
Generalized_space
In mathematics, specifically in category theory, a generalized metric space is a metric space but without the symmetry property and some other properties
Generalized_metric_space
Objects extending the notion of functions
theory of generalized functions in order to define weak solutions of partial differential equations (i.e. solutions which are generalized functions,
Generalized_function
Construction for adding objects to a Hilbert space
the strict confines of the Hilbert space theory. This was supplied by the apparatus of distributions, and a generalized eigenfunction theory was developed
Rigged_Hilbert_space
Area of mathematics using condensed sets
whereas condensed mathematics can be developed strictly within ZFC. Generalized space Pyknotic set "Math 205 – Topics in Number Theory – Lecture 1 video"
Condensed_mathematics
Type of mathematical space
for spaces of functions rather than just numbers or geometrical points. The idea of regarding functions as themselves points of a generalized space dates
Compact_space
Model for representing text documents
as WordNet. Models based on and extending the vector space model include: Generalized vector space model Latent semantic analysis Rocchio Classification
Vector_space_model
Technique to solve partial differential equations
(NNs) as a regularization agent that limits the space of admissible solutions, increasing the generalizability of the function approximation. This way, embedding
Physics-informed neural networks
Physics-informed_neural_networks
System configuration relative to another
configuration. The generalized velocities are the time derivatives of the generalized coordinates of the system. The adjective "generalized" distinguishes
Generalized_coordinates
Type of vector space in math
vector spaces, examples of Hilbert spaces include spaces of square-integrable functions, spaces of sequences, Sobolev spaces consisting of generalized functions
Hilbert_space
Mathematical category
themselves for displaying the space-like aspect of a topos, because a non-trivial topos may fail to have any. Generalized points are geometric morphisms
Topos
Algebraic element satisfying some of the criteria of an inverse
}A=A.} A generalized inverse exists for an arbitrary matrix, and when a matrix has a regular inverse, this inverse is its unique generalized inverse.
Generalized_inverse
Type of mathematical space
mathematics, a generalized flag variety (or simply flag variety) is a homogeneous space whose points are flags in a finite-dimensional vector space V over a
Generalized_flag_variety
Vector space with generalized dot product
orthogonality (zero inner product) of vectors. Inner product spaces generalize Euclidean vector spaces, in which the inner product is the dot product or scalar
Inner_product_space
Space of possible positions for all objects in a physical system
of a system are called generalized coordinates, and the space defined by these coordinates is called the configuration space of the physical system.
Configuration_space_(physics)
Mathematical concept
thick. The notion can be compared to other approaches of introducing generalized spaces for the purpose of homological algebra such as Clausen and Scholze's
Pyknotic_set
Type of incidence structure
containing many quadrangles. A generalized quadrangle is by definition a polar space of rank two. They are the generalized n-gons with n = 4 and near 2n-gons
Generalized_quadrangle
Generalization of the notion of convergence that is found in general topology
In mathematics, a convergence space, also called a generalized convergence, is a set together with a relation called a convergence that satisfies certain
Convergence_space
Field of mathematics which studies incidence structures
geometries are generalized quadrangles. If α = s + 1 these are called Steiner systems. For n > 2, a generalized n-gon is a partial linear space whose incidence
Incidence_geometry
Branch of mathematics
algebras are treated as analogues of algebras of functions on generalized, or "noncommutative", spaces. The subject is not a single formalism. It includes operator-algebraic
Noncommutative_geometry
Set on which a generalization of volumes and integrals is defined
A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes. It contains an underlying set
Measure_space
Epilepsy syndrome that is characterised by generalised seizures with no apparent cause
Generalized epilepsy is a form of epilepsy characterized by generalized seizures that occur with no obvious cause. Generalized seizures, as opposed to
Generalized_epilepsy
Sets of coordinates on phase space which can be used to describe a physical system
details. As Hamiltonian mechanics are generalized by symplectic geometry and canonical transformations are generalized by contact transformations, so the
Canonical_coordinates
Result in integral geometry
ISBN 0-201-13500-0 Ueno, Seitarô (1955), "On the densities in a two-dimensional generalized space", Memoirs of the Faculty of Science, 9: 65–77, doi:10.2206/kyushumfs
Crofton_formula
Integral transform type in mathematics
transform that generalizes the spherical mean operator to homogeneous spaces. Instead of integrating over spheres, one integrates over generalized spheres:
Orbital_integral
Concept in geometry
l. Finite polar spaces (where P is a finite set) are also studied as combinatorial objects. A polar space of rank two is a generalized quadrangle; in this
Polar_space
Mathematical vector components
T_{p}M/T_{p}N} is a generalized space of normal vectors. If M is a Riemannian manifold, the above sequence splits, and the tangent space of M at p decomposes
Tangential and normal components
Tangential_and_normal_components
All even-degree subgraphs of a graph
homology theory, the binary cycle space may be generalized to cycle spaces over arbitrary rings. The cycle space of a graph can be described with increasing
Cycle_space
Algebraic structure
generalized topology in the Boolean algebra. Given an interior algebra its open elements form a generalized topology. Conversely given a generalized topological
Interior_algebra
Vector satisfying some of the criteria of an eigenvector
for each of the generalized eigenspaces of A {\displaystyle A} . Together the two chains of generalized eigenvectors span the space of all 5-dimensional
Generalized_eigenvector
Study of triangles in other spaces than the Euclidean plane
symmetric spaces Schläfli orthoschemes - right simplexes (right triangles generalized to n dimensions) - studied by Schoute who called the generalized trigonometry
Generalized_trigonometry
Algebraic structure in linear algebra
Scalars can also be, more generally, elements of any field. Vector spaces generalize Euclidean vectors, which allow modeling of physical quantities (such
Vector_space
Euclidean space surface
of generalized helicoids are ruled generalized helicoids. Their profile curves are lines and the surfaces are ruled surfaces. circular generalized helicoids
Generalized_helicoid
Nonspecific long-lasting anxiety
Generalized anxiety disorder (GAD) is an anxiety disorder characterized by excessive, uncontrollable, and often irrational worry about events or activities
Generalized_anxiety_disorder
Operation in calculus
real-valued functions on a set X, generalized by Nicolas Bourbaki to functions with values in a locally compact topological vector space. See Hildebrandt 1953 for
Integral
Overview of mechanics based on the least action principle
are Lagrangian mechanics (using generalized coordinates and corresponding generalized velocities in configuration space) and Hamiltonian mechanics (using
Analytical_mechanics
Topological vector spaces
Methods of the theory of generalized functions. Taylor & Francis. ISBN 0-415-27356-0 Vladimirov, V.S. (2001) [1994], "Generalized function", Encyclopedia
Spaces of test functions and distributions
Spaces_of_test_functions_and_distributions
Modification using the principle of template matching
The generalized Hough transform (GHT), introduced by Dana H. Ballard in 1981, is the modification of the Hough transform using the principle of template
Generalised_Hough_transform
Formula in calculus
polynomial remainder theorem (the little Bézout theorem, or factor theorem), generalized to an appropriate class of functions.[citation needed] The full generalization
Chain_rule
Property of a differential manifold that includes complex structures
i is the interior product. A generalized complex structure is a generalized almost complex structure such that the space of smooth sections of L is closed
Generalized_complex_structure
Matrix of partial derivatives of a vector-valued function
function in several variables generalizes the gradient of a scalar-valued function in several variables, which in turn generalizes the derivative of a scalar-valued
Jacobian matrix and determinant
Jacobian_matrix_and_determinant
Statement relating differentiable symmetries to conserved quantities
the target space is the cotangent bundle of space of generalized positions. In field theory, M is the spacetime manifold and the target space is the set
Noether's_theorem
Method of mathematical integration
Furthermore, the Lebesgue integral can be generalized in a straightforward way to more general spaces, measure spaces, such as those that arise in probability
Lebesgue_integral
Integral of sin(x)/x from 0 to infinity
however, integrable in the sense of the improper Riemann integral or the generalized Riemann or Henstock–Kurzweil integral. This can be seen by using Dirichlet's
Dirichlet_integral
Differential calculus on function spaces
the points. However, if the curve is constrained to lie on a surface in space, then the solution is less obvious, and possibly many solutions may exist
Calculus_of_variations
Study of Lie groups, Lie algebras and differential equations
actually created them, for example through his theories of symmetric and generalized spaces, including all the attendant apparatus (moving frames, exterior differential
Lie_theory
object by its boundaries. The definition of generalized map in any dimension is given in and: A nD generalized map (or nG-map) is an (n + 2)-tuple G = (D
Generalized_map
Statistical method
problem in linear regression. the partial least squares (PLS) approach to generalized inverses". SIAM Journal on Scientific and Statistical Computing. 5 (3):
Partial least squares regression
Partial_least_squares_regression
analysis, a Gelfand–Shilov space S α β {\displaystyle S_{\alpha }^{\beta }} is a space of test functions for the theory of generalized functions, introduced
Gelfand–Shilov_space
Instantaneous rate of change (mathematics)
velocity, and the second derivative is its acceleration. Derivatives can be generalized to functions of several real variables. In this case, the derivative
Derivative
Generalized sphere of dimension n (mathematics)
1-dimensional circle is in 2-dimensional space, the 2-dimensional sphere is usually depicted embedded in 3-dimensional space, and a general n {\displaystyle
N-sphere
Differential operator in mathematics
Laplacian can be generalized in certain ways to non-Euclidean spaces, where it may be elliptic, hyperbolic, or ultrahyperbolic. In Minkowski space the Laplace–Beltrami
Laplace_operator
Generalized topological space
Chu spaces generalize the notion of topological space by dropping the requirements that the set of open sets be closed under union and finite intersection
Chu_space
Perspective on spatial and temporal proesses
assessment in mental health. Benjamin Bach and colleagues have generalized the space-time cube into a framework for temporal data visualization that
Time_geography
Technique in integral evaluation
in 1769. Although generalized to triple integrals by Lagrange in 1773, and used by Legendre, Laplace, and Gauss, and first generalized to n variables by
Integration_by_substitution
Framework of superstring theory
with only two space dimensions and one time dimension, but it can be generalized to any number of dimensions. Indeed, hyperbolic space can have more than
M-theory
Boundary condition for generalized functions
restriction of a function to the boundary of its domain to "generalized" functions in a Sobolev space. This is particularly important for the study of partial
Trace_operator
Formula for the derivative of a product
{du}{dx}}\cdot v+u\cdot {\frac {dv}{dx}}.} The rule may be extended or generalized to products of three or more functions, to a rule for higher-order derivatives
Product_rule
Theorem in mathematics
complex variable. It generalizes to functions from n-tuples (of real or complex numbers) to n-tuples, and to functions between vector spaces of the same finite
Inverse_function_theorem
Matrix of second derivatives
m=1.} In the context of several complex variables, the Hessian may be generalized. Suppose f : C n → C , {\displaystyle f\colon \mathbb {C} ^{n}\to \mathbb
Hessian_matrix
Adjunction between a category of co/presheaf under the co/Yoneda embedding
Rutten, J.J.M.M. (1998), "Weighted colimits and formal balls in generalized metric spaces", Topology and Its Applications, 89 (1–2): 179–202, doi:10
Isbell_duality
Study of rates of change
It was also during this period that the differentiation was generalized to Euclidean space and the complex plane. The 20th century brought two major steps
Differential_calculus
Differentiation under the integral sign formula
after integrating over Ω ( t ) {\displaystyle \Omega (t)} and using generalized Stokes' theorem on the second term, reduces to the three desired terms
Leibniz_integral_rule
Theorem
result in the theory of generalized functions, published by Laurent Schwartz in 1952. It states, in broad terms, that the generalized functions introduced
Schwartz_kernel_theorem
Circulation density in a vector field
field v. Grad and div generalize to all oriented pseudo-Riemannian manifolds, with the same geometric interpretation, because the spaces of 0-forms and n-forms
Curl_(mathematics)
Multivariate derivative (mathematics)
increase. The gradient transforms like a vector under change of basis of the space of variables of f {\displaystyle f} . If the gradient of a function is non-zero
Gradient
Field of study in physics
In physics, generalized hydrodynamics (GHD) is an extension of ordinary hydrodynamics to non-equilibrium integrable systems. Such systems have a large
Generalized_hydrodynamics
Generalization of Sobolev spaces
space when 1 ≤ p, q ≤ ∞. These spaces, as well as the similarly defined Triebel–Lizorkin spaces, serve to generalize more elementary function spaces such
Besov_space
Generalised concept of incidence structure of polygons
In mathematics, a generalized polygon is an incidence structure introduced by Jacques Tits in 1959. Generalized n-gons encompass as special cases projective
Generalized_polygon
Musical keyboards with regular arrangements of keys
A generalized keyboard is a musical keyboard, a type of isomorphic keyboard, with regular, tile-like arrangements usually with rectangular or hexagonal
Generalized_keyboard
Branch of mathematical analysis
{\displaystyle ^{[1]}} The Coimbra derivative can be generalized to any order, leading to the Coimbra Generalized Order Differintegration Operator (GODO) For q
Fractional_calculus
Theorem in vector calculus
the curl theorem, or the rotor theorem. It is a special case of the generalized Stokes theorem. In the language of differential forms, the vector field
Stokes'_theorem
Method of detecting shapes within images
was invented by Richard Duda and Peter Hart in 1972, who called it a "generalized Hough transform" after the related 1962 patent of Paul Hough. The transform
Hough_transform
Physical spaces representing position and momentum, Fourier-transform duals
the exchange of differentials in the generalized coordinates and velocities for the differentials in generalized momenta and their time derivatives, p
Position_and_momentum_spaces
Type of derivative in mathematics
line approximation. In multivariable calculus, the same property is generalized to define the derivative of a vector-valued function or function of a
Derivative (multivariable calculus)
Derivative_(multivariable_calculus)
On converting relations to functions of several real variables
rigorous form of the implicit function theorem. Ulisse Dini (1845–1918) generalized the real-variable version of the implicit function theorem to the context
Implicit_function_theorem
Class of statistical models
In statistics, a generalized linear model (GLM) is a flexible generalization of ordinary linear regression. The GLM generalizes linear regression by allowing
Generalized_linear_model
formulated in generalized coordinates of motion. Note that "generalized coordinates of motion" are related to—but distinct from—generalized coordinates
Generalized_filtering
Vector operator in vector calculus
there are more of the field vectors exiting from an infinitesimal region of space than entering it. A point at which the flux is outgoing has positive divergence
Divergence
Fundamental space of geometry
was not applied in spaces of dimension more than three until the 19th century. Ludwig Schläfli generalized Euclidean geometry to spaces of dimension n, using
Euclidean_space
Matrix factorisation in mathematics
T are upper triangular. The generalized Schur decomposition is also sometimes called the QZ decomposition. The generalized eigenvalues λ {\displaystyle
Schur_decomposition
Mathematical notion of infinitesimal difference
dimension, any inner product space is a Hilbert space, any normed vector space is a Banach space and any topological vector space is complete. As a result
Differential_(mathematics)
Notation of differential calculus
\mathbf {A} \end{aligned}}} Many symbolic operations of derivatives can be generalized in a straightforward manner by the gradient operator in Cartesian coordinates
Notation_for_differentiation
Relationship between derivatives and integrals
by James Gregory (1638–1675). Isaac Barrow (1630–1677) proved a more generalized version of the theorem, while his student Isaac Newton (1642–1727) completed
Fundamental theorem of calculus
Fundamental_theorem_of_calculus
Quantum mechanical model based on mathematical matrices
are coordinates on some more general notion of "space" and proves theorems about these generalized spaces by exploiting the analogy with ordinary geometry
Matrix_theory_(physics)
Feature detection algorithm in computer vision
Hessian, or more generally considering a more general family of generalized scale-space interest points. Recently, a slight variation of the descriptor
Scale-invariant feature transform
Scale-invariant_feature_transform
Generalized Euclidean space in mathematics
mechanics, the projective Hilbert space or ray space P ( H ) {\displaystyle \mathbf {P} (H)} of a complex Hilbert space H {\displaystyle H} is the set of
Projective_Hilbert_space
Estimation procedure for correlated data
In statistics, a generalized estimating equation (GEE) is used to estimate the parameters of a generalized linear model with a possible unmeasured correlation
Generalized estimating equation
Generalized_estimating_equation
Formulation of classical mechanics
{\displaystyle N} generalized coordinates q 1 , q 2 , … , q N {\displaystyle q_{1},\,q_{2},\dots ,q_{N}} and the time t {\displaystyle t} . The generalized momenta
Hamilton–Jacobi_equation
Function spaces generalizing finite-dimensional p norm spaces
same normed space and so they may both be called " L p {\displaystyle L^{p}} space". The above definitions generalize to Bochner spaces. In general,
Lp_space
Mathematical space with a notion of distance
topological space is the Sierpiński space. Sets equipped with an extended pseudoquasimetric were studied by William Lawvere as "generalized metric spaces". From
Metric_space
Statement about integration on manifolds
In vector calculus and differential geometry the generalized Stokes theorem (sometimes with apostrophe as Stokes' theorem or Stokes's theorem), also called
Generalized_Stokes_theorem
Theorem in mathematics
coordinate-free manner; hence, they generalize to the case when G {\displaystyle G} is a subset of a Banach space. There is no exact analog of the mean
Mean_value_theorem
Computational problem
winning strategy in a generalized geography game is PSPACE-complete. Let GG = { ⟨G, b⟩ | P1 has a winning strategy for the generalized geography game played
Generalized_geography
listed. A generalized suffix array can be generated for a generalized suffix tree. When compared to a generalized suffix tree, while the generalized suffix
Generalized_suffix_array
Mathematical method in calculus
_{M}d(u\wedge v)-(-1)^{k}\int \limits _{M}u\wedge dv.} An application of generalized Stokes' theorem gives the result: ∫ M d u ∧ v = ∮ ∂ M u ∧ v − ( − 1 )
Integration_by_parts
Conditions for switching order of integration in calculus
distributions, that is, generalized functions Kuratowski–Ulam theorem – analog of Fubini's theorem for arbitrary second countable Baire spaces Symmetry of second
Fubini's_theorem
Definite integral of a scalar or vector field along a path
is defined over a plane (n = 2), its graph is a surface z = f(x, y) in space, and the line integral gives the (signed) cross-sectional area bounded by
Line_integral
Physics generalization
The generalized uncertainty principle (GUP) is a proposed extension of the Heisenberg uncertainty principle that incorporates potential effects of gravitational
Generalized uncertainty principle
Generalized_uncertainty_principle
correspond to algorithms. Topos Logic: Internal logics of topoi (generalized spaces) are intuitionistic. Constructivism (philosophy of mathematics) Brouwer
Constructive_logic
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