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Concept of mathematics in convex analysis
specifically in convex analysis, the convex compactification is a compactification which is simultaneously a convex subset in a locally convex space in functional
Convex_compactification
Embedding a topological space into a compact space as a dense subset
In mathematics, in general topology, compactification is the process or result of making a topological space into a compact space. A compact space is a
Compactification (mathematics)
Compactification_(mathematics)
Type of mathematical functions
method for compactification of C n {\displaystyle \mathbb {C} ^{n}} , but not the only method like the Riemann sphere that was compactification of C {\displaystyle
Function of several complex variables
Function_of_several_complex_variables
Space of complex matrices with positive definite imaginary part
familiar compactification of modular curves by adding cusp points. A finer class of compactifications is given by toroidal compactifications, which depend
Siegel_upper_half-space
Space which has no holes through it
{\displaystyle \operatorname {SU} (n)} is simply connected. The one-point compactification of R {\displaystyle \mathbb {R} } is not simply connected (even though
Simply_connected_space
Measure in mathematical analysis
{\displaystyle \lim I={\frac {1}{\pi }}\int _{-1}^{+1}(1-y^{2})^{3/2}dy} . Convex compactification Young, L. C. (1942). "Generalized Surfaces in the Calculus of Variations"
Young_measure
List of concrete topologies and topological spaces
Projectively extended real line Stone–Čech compactification Stone topology Stone–Čech remainder Wallman compactification This lists named topologies of uniform
List_of_topologies
boundary, Tits boundary, Thurston boundary. See also projective space and compactification. Busemann function given a ray, γ : [0, ∞)→X, the Busemann function
Glossary of Riemannian and metric geometry
Glossary_of_Riemannian_and_metric_geometry
British mathematician (1905–2000)
various generalization is in Chapter 3 from the perspective of convex compactifications. White, Brian (1997), "The Mathematics of F. J. Almgren Jr.", Notices
Laurence_Chisholm_Young
Algebraic variety that is a moduli space for principally polarized abelian varieties
particular, a compactification of A2(2) is birationally equivalent to the Segre cubic which is in fact rational. Similarly, a compactification of A2(3) is
Siegel_modular_variety
Busemann functions by constants. Eberlein & O'Neill (1973) defined a compactification of a Hadamard manifold X which uses Busemann functions. Their construction
Busemann_function
Open convex self-dual cones
mathematics, symmetric cones, sometimes called domains of positivity, are open convex self-dual cones in Euclidean space which have a transitive group of symmetries
Symmetric_cone
"exterior completion", like completion of a locally convex space, or Stone–Čech compactification of a topological space. A dual construction is called
Envelope_(category_theory)
American mathematician
he published two papers setting out what is now called Stone–Čech compactification theory. This theory grew out of his attempts to understand more deeply
Marshall_H._Stone
Manifold with inversion symmetry
SU(1,1). It is a bounded domain in the complex plane C. The one-point compactification of C, the Riemann sphere, is the dual space, a homogeneous space for
Hermitian_symmetric_space
Duality for locally compact abelian groups
to characterize the Bohr compactification of an arbitrary abelian locally compact topological group. The Bohr compactification B ( G ) {\displaystyle B(G)}
Pontryagin_duality
American mathematician
(born 7 May 1938) is an American mathematician, specializing in locally convex spaces, harmonic analysis, and partial differential equations. After receiving
Bertram_John_Walsh
finite groups. A convex polyhedron C in hyperbolic space is called geometrically finite if its closure C in the conformal compactification of hyperbolic
Geometric_finiteness
Certain topology in mathematics
Stone–Čech compactification of ω1 is ω1+1, just as its one-point compactification (in sharp contrast to ω, whose Stone–Čech compactification is much larger
Order_topology
{\displaystyle M_{g,n}(J,\nu ,\beta )} has a natural compactification, called the Gromov-Uhlenbeck compactification. A stable map in the sense of symplectic geometry
Stable_map
Type of smooth complex surface of kodaira dimension 0
string, the Spin(32)/Z2 heterotic string, and M-theory are related by compactification on a K3 surface. For example, the Type IIA string compactified on a
K3_surface
Continuous deformation between two continuous functions
homology (which is, roughly speaking, the homology of the compactification, and compactification is not homotopy invariant). In order to define the fundamental
Homotopy
Complement of an open subset
{\displaystyle X} ; the "surrounding space" does not matter here. Stone–Čech compactification, a process that turns a completely regular Hausdorff space into a compact
Closed_set
compact type as a compactification of a finite-dimensional complex semisimple Jordan algebra. The automorphism group of the compactification becomes a complex
Mutation_(Jordan_algebra)
American mathematician
symmetries gives rise to the conformal group... However, conformal compactification introduces closed timelike curves so Segal portrayed the spatial part
Irving_Segal
geometry) Mumford vanishing theorem (algebraic geometry) Nagata's compactification theorem (algebraic geometry) Noether's theorem on rationality for surfaces
List_of_theorems
long as we have no article on Martin boundary, see Compactification (mathematics)#Other compactification theories. Bishop, C. (1991), "A characterization
List of probabilistic proofs of non-probabilistic theorems
List_of_probabilistic_proofs_of_non-probabilistic_theorems
Concept in functional analysis
Banach-Mazur compactum A note on the Banach-Mazur distance to the cube The Banach-Mazur compactum is the Alexandroff compactification of a Hilbert cube manifold
Banach–Mazur_compactum
Algebro-geometric stability condition
) {\displaystyle ({\mathcal {X}},{\mathcal {L}})} admits a natural compactification ( X ¯ , L ¯ ) {\displaystyle ({\bar {\mathcal {X}}},{\bar {\mathcal
K-stability
Skeletonized version of algebraic geometry
consensus trees based on tropical optimization. Tropical analysis Tropical compactification Hartnett, Kevin (5 September 2018). "Tinkertoy Models Produce New Geometric
Tropical_geometry
Polytope associated with combinatorial problems
manifold with corners by allowing the points to approach each other. This compactification can be factored as S 1 × W d + 1 {\displaystyle S^{1}\times W_{d+1}}
Cyclohedron
symmetric spaces of rank 1, this is the usual (one point) conformal compactification of V {\displaystyle V} . Tube type is important because bounded symmetric
Shilov_boundary
Algebraic surface defined by a cubic polynomial
singularities. In particular, this compactification is isomorphic to the K-moduli space of cubic surfaces. Further compactifications of the moduli have been considered
Cubic_surface
Compact Riemann surface of genus 3
is the modular curve X(7) and the projective Klein quartic is its compactification, just as the dodecahedron (with a cusp in the center of each face)
Klein_quartic
Differential equations involving stochastic processes
{\displaystyle {\widehat {M}}=M\cup \{\infty \}} be the one-point compactification and x 0 {\displaystyle x_{0}} be F 0 {\displaystyle {\mathcal {F}}_{0}}
Stochastic differential equation
Stochastic_differential_equation
Generalization of mass, length, area and volume
limits, the dual of L ∞ {\displaystyle L^{\infty }} and the Stone–Čech compactification. All these are linked in one way or another to the axiom of choice
Measure_(mathematics)
{\displaystyle P_{1},\ldots ,P_{m}} be a collection of (not necessarily convex) polygons in the Euclidean plane and suppose that for every side s i {\displaystyle
Translation_surface
Isomorphism of differentiable manifolds
classification by observing that the mapping class group acted naturally on a compactification of Teichmüller space; as this enlarged space was homeomorphic to a
Diffeomorphism
Cannon–Thurston map. Here "extends" means that the map between hyperbolic compactifications i ^ : H ∪ ∂ H → G ∪ ∂ G {\displaystyle {\hat {i}}:H\cup \partial H\to
Cannon–Thurston_map
Nonempty, upper-bounded, downward-closed subset
{\displaystyle B} can be chosen so that this gives the Stone-Čech compactification after modifying map to have the universal property. Here is a relation
Ideal_(order_theory)
{\displaystyle T_{1}} . Accumulation point See limit point. Alexandroff compactification Alexandrov topology The topology of a space X is an Alexandrov topology
Glossary_of_general_topology
*-algebra of bounded operators on a Hilbert space
correspond to exotic *-homomorphisms into C and describe the Stone–Čech compactification of Z. Examples: The predual of the von Neumann algebra L∞(R) of essentially
Von_Neumann_algebra
Set whose pairs have minima and maxima
spectra of completely distributive complete lattices, and Hausdorff compactifications. Continuous Lattices. Vol. 871. pp. 159–208. doi:10.1007/BFb0089907
Lattice_(order)
78-dimensional exceptional simple Lie group
of E6 up to dimension 78 are shown to the right. The E6 polytope is the convex hull of the roots of E6. It therefore exists in 6 dimensions; its symmetry
E6_(mathematics)
are known as K-moduli. The K-moduli space was first constructed as a compactification of the moduli of K-polystable Fano manifolds by Li–Wang–Xu and Odaka
K-stability_of_Fano_varieties
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CONVEX COMPACTIFICATION