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Topic: Compactification

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  Springer Online Reference Works   (Site not responding. Last check: 2007-09-17)
One of the fundamental methods in compactification theory is Aleksandrov's method of centred systems of open sets [7], which was initially used for the construction of the maximal compactification, and was subsequently extensively utilized by many mathematicians.
The concept of a compactification is useful in the study of the dimension of the remainder.
Moreover, a consequence of the existence of a compactification with a remainder of dimension
eom.springer.de /c/c023550.htm   (1691 words)

 Compactification (physics) - Wikipedia, the free encyclopedia
The mechanism behind this type of compactification is described by the Kaluza-Klein theory, under the name compact dimension.
It assumes that the shape of the internal manifold is a Calabi-Yau manifold or generalized Calabi-Yau manifold which is equipped with non-zero values of fluxes, i.e.
The hypothetical concept of the anthropic landscape in string theory follows from a large number of possibilities in which the integers that characterize the fluxes can be chosen without violating rules of string theory.
en.wikipedia.org /wiki/Compactification_(physics)   (238 words)

 Looking for extra dimensions
The most studied superstring compactification is heterotic string theory compactified on a Calabi-Yau space in six-dimensions (or three complex dimensions).
This avoids the situation in the Kaluza-Klein compactification where the Planck mass in four spacetime dimensions depends on the volume of the compactified space, which is hard to control dynamically.
In superstring theory with Kaluza-Klein compactification, there are several different energy scales that come into play in going from a string theory to a low energy effective particle theory that is consistent with observed particle physics and cosmology.
superstringtheory.com /experm/exper5a1.html   (1201 words)

 Encyclopedia :: encyclopedia : Compactification (physics)   (Site not responding. Last check: 2007-09-17)
In string theory, compactification refers to "curling up" the extra dimensions (six in the superstring theory), usually on Calabi-Yau spaces or on orbifolds.
The mechanism behind this type of compactification is described by the Kaluza-Klein theory.
The formulation of more precise versions of the meaning of compactification in this context has been promoted by discoveries such as the mysterious duality.
www.hallencyclopedia.com /topic/Compactification_(physics).html   (101 words)

 compactification   (Site not responding. Last check: 2007-09-17)
The resolution to this comes with the idea of compactification: 6 out of the 9 spatial dimensions need to be taken small and compact.
A particularly important class of 6 dimensional manifolds used in string compactification is known as Calabi-Yau manifolds.
However, it was not found so far a suitable compactification scheme which results are in complete agreement with experiment (eg, masses of elementary particles).
www.mit.edu /~nleonard/x/xdiv/string/node4.html   (128 words)

 [No title]
B. Bordbar and J. Pym, The set of idempotents in the weakly almost periodic compactification of the integers is not closed, Trans.
R. Butcher, The Stone-Cech compactification of a semigroup and its algebra of measures, Ph.D Dissertation (1975), University of Sheffield.
A. Maleki and D. Strauss, Homomorphisms, ideals and commutativity in the Stone-Cech compactification of a discrete semigroup, Topology and its Applications 71 (1996), 47-61.
members.aol.com /nhindman/bibliogr.html   (3776 words)

 Looking for extra dimensions
In string theory, Kaluza-Klein compactification of the extra dimensions has one important difference from the particle theory version.
Supersymmetry breaking and compactification of higher dimensions have to work together to give the low energy physics we observe in accelerator detectors.
Braneworld models in general are very different from superstring Kaluza-Klein compactification models because they don't require there to be so many steps between the Planck scale and the electroweak scale.
superstringtheory.com /experm/exper51.html   (989 words)

 The Stone-Cech Compactification
The Stone-Cech compactification is, in a suitable sense, the largest.
However, there are a few examples where the one-point compactification and the Stone-Cech compactification are the same.
We construct the Stone-Cech compactification using a weak topology on part of the dual of
www.math.unl.edu /~s-bbockel1/929/node16.html   (215 words)

 Superstring Theory
Since it is via compactification, which yields local non-Abelian gauge symmetries, and other symmetry groups (to describe all the known particles and forces, plus some unknow ones), the mathematical treatment will be presented in the followings for a better understanding of the subject.
This is similar to the Kaluza-Klein compactification when a dimension was compactified on a circle with components of the 26-dimensional metric tensor.
Orbifold compactification is the simplest and it preserves the equation of string in its simple form.
universe-review.ca /R15-18-string.htm   (8365 words)

 David C. Murphy: Research   (Site not responding. Last check: 2007-09-17)
Presently I am working on the classification of compactifications of algebraic groups - varieties containing an open subset isomorphic to an algebraic group G such that the action of G on itself by left translations extends to an action on the whole variety.
The so-called "wonderful compactification" of a reductive group is a special case.
One-parameter subgroups play a critical role in the affine case, which is similar to their application to instability problems in geometric invariant theory.
max.cs.kzoo.edu /~dmurphy/researchindex.html   (1030 words)

 Calabi-Yau manifold - Wikipedia, the free encyclopedia
Compactification on Calabi-Yau n-folds are important because they leave some of the original supersymmetry unbroken.
More precisely, in the absence of fluxes, compactification on a Calabi-Yau 3-fold (real dimension 6) leaves one quarter of the original supersymmetry unbroken if the holonomy is the full SU(3).
supercharges in a compactification of type I. When fluxes are included the supersymmetry condition instead implies that the compactification manifold be a generalized Calabi-Yau, a notion introduced in 2002 by Nigel Hitchin.
en.wikipedia.org /wiki/Calabi-Yau_manifold   (822 words)

This process, developed by Aleksandrov (biography), (also spelled Alexandroff), is called compactification.
The sphere is compact, hence it is the "compactification" of the plane.
The base for the compactification of s consists of the open balls of s, and the open balls centered at ω, which correspond to regions beyond b, for values of b that approach infinity.
www.mathreference.com /top-cs,cfc.html   (749 words)

 Citebase - Moduli Stabilization in String Gas Compactification   (Site not responding. Last check: 2007-09-17)
compactification in the context of massless string gas cosmology.
We found that the volume moduli, the shape moduli, and the flux moduli are stabilized at the self dual point in the moduli space.
Thus, it is proved that this simple compactification model is stable.
citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:hep-th/0509074   (203 words)

 Springer Online Reference Works   (Site not responding. Last check: 2007-09-17)
-space; for a normal space it coincides with the Stone–Čech compactification.
Not every Hausdorff compactification of a Tikhonov space is a compactification of Wallman type.
Compactifications that are not Wallman compactifications were constructed by V.M. Ul'yanov [a1].
eom.springer.de /w/w097050.htm   (132 words)

 Deligne-Mumford compactification
The Deligne-Mumford compactification is obtained as a moduli space for stable genus g curves with marked points, (instead of considering just smooth curves, as in
This translates to: every genus 0 irreducible component has at least three marked or nodal points and every genus 1 component has at least one marked or nodal point.
This space is important because it is a smooth compactification (as a stack, at least) with easy-to-understand boundary components that give an inductive structure to all the moduli spaces of curves.
www.aimath.org /WWN/modspacecurves/aim/glossary/node6.html   (111 words)

 PlanetMath: Alexandrov one-point compactification
The Alexandrov one-point compactification of a non-compact topological space
one-point compactification, Alexandroff one-point compactification, Aleksandrov one-point compactification, Alexandrov compactification, Aleksandrov compactification, Alexandroff compactification
This is version 6 of Alexandrov one-point compactification, born on 2003-07-27, modified 2005-02-06.
planetmath.org /encyclopedia/AlexandrovOnePointCompactification.html   (73 words)

 Giovanni Curi
The construction of Stone-Cech compactification in [2] enlightens aspects of this compactification that may be of interest also for the topologist or the locale theorist not concerned with foundational issues.
A constructive version of Alexandroff compactification is also obtained.
It is also shown how Stone-Cech compactification can itself be used to prove that certain hom-sets are small.
www.math.unipd.it /~gcuri   (833 words)

 Compactification - Wikipedia, the free encyclopedia
The term compactification is used in two different fields:
Compactification (mathematics), the enlarging of a topological space to make it compact
Compactification (physics), the "curling up" of extra dimensions in string theory
en.wikipedia.org /wiki/Compactification   (95 words)

 ACTA MATHEMATICA UNIVERSITATIS COMENIANAE   (Site not responding. Last check: 2007-09-17)
Abstract.  In this paper we introduce GF-compactifications, which are compactifications of GF-spaces (a new notion introduced by the authors).
We study properties of this new kind of compactification and prove that every GF-compactification is of Wallman type.
We also prove that every metrizable compactification of a metric space $X$ is a GF-compactification and, as a corollary, that every metric compactification is of Wallman type, giving a new proof of a result that dates back to Aarts.
www.univie.ac.at /EMIS/journals/AMUC/_vol-73/_no_1/_arenas/arenas.html   (76 words)

 [No title]
By the universal property of the Stone-Cech compactification, for the inclusion map f: X \subset I, and since Y is compact Hausdorff, there exists a map q:BX \to Y such that the composition of the natural map X \to BX followed by q coincides with f.
In this case (at least) BX is called the Stone-Cech compactification of X.
The solenoid group (which can be defined in various ways, for example as the inverse limit of the family of coverings of the circle, ordered by divisibility) is maybe the most interesting example.
www.lehigh.edu /dmd1/public/www-data/yr1013.txt   (1401 words)

 Voyage vers l'infiniment petit
Un exemple simple de compactification dans un espace bidimensionnel est le tuyau d'arrosage : une dimension est enroulée sur elle-même pour former un cercle (on l'observe en sectionnant le tuyau), alors que l'autre s'étend sur une certaine distance et a deux bouts.
Compactification sur une sphère dans les théories de Kaluza-Klein.
La compactification peut être réalisée à l'aide d'objets mathématique plus compliqués, par exemple les orbifolds.
www.diffusion.ens.fr /vip/tableJ02.html   (849 words)

 Stone-Cech Compactification - NoiseFactory Science Archives (http://noisefactory.co.uk)
This means that the circle is a compactification of (0,1).
Consequently a space has a compactification (and so is Tychonov) if and only if it can be identified with a subspace of a compact Hausdorff space.
X is the "largest" possible compactification X can have, and we can classify all the compactifications of the space X by constructing the quotients of
noisefactory.co.uk /maths/stone-cech.html   (1649 words)

 Abstracts   (Site not responding. Last check: 2007-09-17)
In particular I will go over the construction of the Stone-Cech compactification of a discrete space as a set of ultrafilters.
Abstract: I will show how to extend the operation on a discrete semigroup (such as (N,+), which is the granddaddy of all semigroups) to its Stone-Cech compactification so that the compactification becomes a right topological semigroup.
This will include results whose first proofs were elementary but complicated, other results first proved using the algebra of beta N for which elementary proofs have subsequently been found, and results that appear unlikely to succumb to elementary proofs in the foreseeable future.
www.theoryofnumbers.com /CANT/2006/abstracts1.htm   (1658 words)

 IngentaConnect Manifold-theoretic compactifications of configuration spaces   (Site not responding. Last check: 2007-09-17)
We present new definitions for and give a comprehensive treatment of the canonical compactification of configuration spaces due to Fulton–MacPherson and Axelrod–Singer in the setting of smooth manifolds, as well as a simplicial variant of this compactification initiated by Kontsevich.
We stratify the canonical compactification, identifying the difieomorphism types of the strata in terms of spaces of configurations in the tangent bundle, and give completely explicit local coordinates around the strata as needed to define a manifold with corners.
We analyze the quotient map from the canonical to the simplicial compactification, showing it is a homotopy equivalence.
www.ingentaconnect.com /content/klu/29/2004/00000010/00000003/art00006   (215 words)

 Van Tan: On the compactification problem for Stein surfaces
, On the compactification of a Stein surface.
, (a) On the compactification of strongly pseudoconvex surfaces.
; (b) On the compactification of strongly pseudoconvex surfaces II.
www.numdam.org /numdam-bin/item?id=CM_1989__71_1_1_0   (98 words)

 Atlas: Exact approximations to Stone-Cech compactification by Giovanni Curi   (Site not responding. Last check: 2007-09-17)
Stone-Cech compactification is obtained as a particular case of this construction in those settings (such as ordinary set theory, or more generally, topos theory) in which the class Hom(L, [0, 1]) of [0, 1]
Together with the described compactification, this allows us to characterize the class of locales for which Stone-Cech compactification can be defined constructively, and yields an effective (type-theoretic) construction of Stone-Cech compactification of locally compact locales.
The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # carg-11.
atlas-conferences.com /cgi-bin/abstract/carg-11   (221 words)

 AMCA: The parabolic compactification and its application to the approximation of unbounded functions by H. Gingold   (Site not responding. Last check: 2007-09-17)
AMCA: The parabolic compactification and its application to the approximation of unbounded functions by H. Gingold
The properties of a compactification that maps the n dimensional Euclidean space onto a "parabolic bowl" are studied.
The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Mathematical Conference Abstracts.
at.yorku.ca /c/a/q/i/57.htm   (223 words)

 the Bohr compactification of the Reals
One use made of Pontryagin duality is to give a general definition of an almost-periodic function on a non-compact group G in LCA.
For that, we define the Bohr compactification B(G) of G as H', where H is as a group G', but given the discrete topology.
I am still having difficulty imagining what the bohr compactification of the real line is like.
www.physicsforums.com /showthread.php?t=14590   (2864 words)

 J.S. Pym, May/03
ABSTRACT: A semigroup compactification of a (Hausdorff) topological group G
compactifications of locally compact groups leading to our collaborations.
given and the structure of the compactification lattice is determined.
www.math.uwo.ca /~milnes/JSPMay03.htm   (2206 words)

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