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	Dimension - (Site not responding. Last check: 2007-10-10)
		In mathematics, dimensions are the parameters required to describe the position and relevant characteristics of any object within a conceptual space —where the dimensions of a space are the total number of different parameters used for all possible objects considered in the model.
		The inductive dimension of a topological space may refer to the small inductive dimension or the large inductive dimension, and is based on the analogy that n+1-dimensional balls have n dimensional boundaries, permitting an inductive definition based on the dimension of the boundaries of open sets.
		The Krull dimension of a commutative ring, named after Wolfgang Krull (1899 - 1971), is defined to be the maximal number of strict inclusions in an increasing chain of prime ideals in the ring.
	en.wikipedia.2es.com.pl /wiki/Dimension (1871 words)

	Science Fair Projects - Isoperimetric dimension
		In mathematics, the isoperimetric dimension of a manifold is a notion of dimension that tries to capture how the large scale behavior of the manifold resembles that of a Euclidean space (unlike the topological dimension or the Hausdorff dimension which compare different local behaviors against those of the Euclidean space).
		The isoperimetric dimension of M is the supremum on all d-s such that M satisfies a d-dimensional isoperimetric inequality.
		In general, the isoperimetric dimension is preserved by quasi isometries, both by quasi-isometries between manifolds, between graphs, and even by quasi isometries carrying manifolds to graphs, with the respective definitions.
	www.all-science-fair-projects.com /science_fair_projects_encyclopedia/Isoperimetric_dimension (944 words)

	Dimension - Facts, Information, and Encyclopedia Reference article
		Dimension (from Latin "measured out") is, in essence, the number of degrees of freedom available for movement in a space.
		In common usage, the dimensions of an object are the measurements that define its shape and size.
		Science fiction texts often mention the concept of dimension, when really referring to parallel universes, alternate universes, or other planes of existence, or concepts that are beyond the reader.
	www.startsurfing.com /encyclopedia/d/i/m/Dimension.html (1161 words)

	PlanetMath: isoperimetric inequality
		The analogous statement is true in arbitrary dimension.
		The advantage of this formulation is that it does not depend on the notion of surface area, and so can be generalized to arbitrary measure spaces with a metric.
		This is version 9 of isoperimetric inequality, born on 2003-10-17, modified 2005-09-17.
	planetmath.org /encyclopedia/IsoperimetricInequality.html (341 words)

	Isoperimetric dimension - Wikipedia, the free encyclopedia
		The hyperbolic plane has topological dimension 2 and isoperimetric dimension infinity.
		The isoperimetric dimension of any finite graph is 0.
		The isoperimetric dimension of a d-dimensional grid is d.
	en.wikipedia.org /wiki/Isoperimetric_dimension (810 words)

	Isoperimetry - Wikipedia, the free encyclopedia
		The theorem is usually stated in the form of an inequality that relates the perimeter and area of a closed curve in the plane.
		Modern formulations of isoperimetric problems are sometimes given in terms of sub-Riemannian geometry; Dido's problem specifically finds expression in terms of the Heisenberg group: given an arc connecting two points, the "height" z of a point in the Heisenberg group corresponds to the area subtended by the arc.
		The isoperimetric theorem generalises to higher dimensional spaces: the domain with volume 1 with the minimal surface area is always a ball.
	en.wikipedia.org /wiki/Isoperimetric_theorem (585 words)

	Dimension Summary
		In simplest terms, the dimension of a mathematical object is the number of independent parameters required to describe that object.
		It is well known that the dimension of a line is 1, the dimension of a plane is 2, and the dimension of space is 3.
		Adding the three Euler angles, for a total 6 dimensions, allows the current degrees of freedom —orientation and trajectory —of the airplane to be known.
	www.bookrags.com /Dimension (3541 words)

	Dimension (Site not responding. Last check: 2007-10-10)
		It is conventional (and for most practical purposes entirely sensible) to consider this as three spatial dimensions and one of time.
		It is somewhat different to the three spatial dimensions in that there is only one of it, and movement seems to be possible in only one direction.
		Science fiction texts often mention the concept of dimension, when really refering to either parallel universes, or concepts that are beyond the reader.
	dimension.iqnaut.net (824 words)

	misspelledsearch.info: dimension cross stitch variant information page (Site not responding. Last check: 2007-10-10)
		Also, in science fiction, a "dimension" can also refer to a separate world or plane of existence, though this meaning is not discussed in this article.
		In physics, the dimension of a quality is the expression of that quality in basic units: the dimension of speed, for example, is length divided by time.
		This dimension cross stitch variant index site has been developed to help wayward users find the information they are looking for, no matter how they are mistakenly spelled or mistyped.
	www.misspelledsearch.info /dimension%20cross%20stitch.htm (792 words)

	Amazon.com: "isoperimetric dimension": Key Phrase page (Site not responding. Last check: 2007-10-10)
		The isoperimetric dimension of a graph In spectral geometry.
		Isoperimetric Inequalities: Differential Geometric and Analytic Perspectives (Cambridge Tracts in Mathematics) by Isaac Chavel
		The isoperimetric dimension of the Heisenberg group is n + 1, and it follows from this that capn8H,,, > 0 for I <...
	www.amazon.com /phrase/isoperimetric-dimension (600 words)

	2100 Chip Dimension Upgrade (Site not responding. Last check: 2007-10-10)
		Super Dimension Cavalry Southern Cross - Super Dimension Cavalry Southern Cross (è¶æç©ºé¨å£ãµã¶ã³ã¯ãã¹ ChÅjikÅ« Kidan Sazan Kurosu) was the third Japanese animated (anime) series released under the "Super Dimension" moniker by the sponsor Big West.
		Complex dimension - In mathematics, complex dimension usually refers to the dimension of a complex manifold M, or complex algebraic variety V. If the complex dimension is d, the real dimension will be 2d.
		Isoperimetric dimension - In mathematics, the isoperimetric dimension of a manifold is a notion of dimension that tries to capture how the large scale behavior of the manifold resembles that of a Euclidean space (unlike the topological dimension or the Hausdorff dimension which compare different local behaviors against those of the Euclidean space).
	www.morganmatroc-ecd.com /2100chipdimensionupgrade.html (771 words)

	Publications and Preprints (Site not responding. Last check: 2007-10-10)
		I have also studied the singularities of the Ricci flow in dimension 3.
		Abstract: In this paper, we study the singularities of the Ricci flow in dimension three.
		Dimension Reduction under the Ricci Flow on Manifolds with Positive Curvature Operator.
	www.math.columbia.edu /~cao/research.html (466 words)

	Springer Online Reference Works
		The principal subject of the theory of convex sets is the study of convex bodies, which are finite (i.e.
		is not stipulated, one speaks of degenerate convex bodies or of convex bodies of lower dimension.
		The points of the boundary of a convex body are classified by the minimal dimension of the faces to which they belong, and also by the dimension of the set of supporting hyperplanes at the point.
	eom.springer.de /c/c026340.htm (1126 words)

	Springer Online Reference Works
		There are several isoperimetric inequalities for the lowest eigenvalue of boundary value problems, similar to the Rayleigh–Faber–Krahn inequality.
		The lowest non-trivial Neumann eigenvalue also satisfies an isoperimetric inequality.
		Back in the membrane problem, if one goes beyond the lowest eigenvalue, there are several isoperimetric inequalities as well as a number of open problems.
	eom.springer.de /r/r130040.htm (826 words)

	Deming Dimension (Site not responding. Last check: 2007-10-10)
		Minkowski-Bouligand dimension - In fractal geometry, the Minkowski-Bouligand dimension or Minkowski dimension is a way of determining the fractal dimension of a set S in a Euclidean space R^n, or more generally of a metric space (X,d).
		This dimension is also known as the packing dimension or, less accurately, the box-...
		Super Dimension - ;Super Dimension: Term initially used in Big West's anime and science fiction trilogy that consisted of the unrelated stories of The Super Dimension Fortress Macross, The Super Dimension Century Orguss, and The Super Dimension Cavalry Southern Cross.
	de81.360mkt.info /demingdimension.html (1117 words)

	2005Song.html (Site not responding. Last check: 2007-10-10)
		This means that the volume needed to fill a cycle of volume r is bounded above by Cr^(k+1)/k for some constant C depending only on k and X. This extends a result of M. Gromov from the context of Riemannian manifolds to that of metric spaces.
		We furthermore show: If all asymptotic cones of X have 'dimension' strictly less than k+1 in the sense that images of Lipschitz maps from R^(k+1) have (k+1)-dimensional measure 0 then X admits an isoperimetric inequality of sub-Euclidean type for k-dimensional cycles.
		In particular, isoperimetric inequalities can be used to detect the Euclidean rank of proper cocompact Hadamard spaces.
	www.math.princeton.edu /~fhang/2006Wenger.html (196 words)

	Amazon.com: "isoperimetric constants": Key Phrase page (Site not responding. Last check: 2007-10-10)
		Our choice of isoperimetric constants will even be invariant with respect to the discretization of the Riemannian manifold.
		Then we discuss the Cheeger and isoperimetric constants and their effect on estimating eigenvalues and eigenfunctions.
		Chapter V is devoted to estimating the Cheeger and isoperimetric constants in...
	www.amazon.com /phrase/isoperimetric-constants (536 words)

	[No title] (Site not responding. Last check: 2007-10-10)
		the topological dimension does not reflect their geometry at infinity.
		However, one may introduce {\it an isoperimetric dimension} through isoperimetric inequalities.
		The dimension introduced in this way is not a number but a family of functions indexed by a parameter $p$, $1
	www.math.technion.ac.il /~techm/20040127141520040127lev (53 words)

	Dimension Free Weak Concentration Of Measure Phenomenon (ResearchIndex)
		Dimension Free Weak Concentration Of Measure Phenomenon (ResearchIndex)
		Dimension Free Weak Concentration Of Measure Phenomenon (1995)
		0.1: Discrete Isoperimetric Inequalities and the Probability of..
	citeseer.ist.psu.edu /bobkov95dimension.html (288 words)

	Isoperimetric Dimension (Site not responding. Last check: 2007-10-10)
		In mathematics, the isoperimetric dimension of a manifold is a notion of dimension that tries to capture how the large scale behavior of the manifold...
		Osada, Isoperimetric dimension and estimates of heat kernels of pre-- Sierpinski carpets.
		A Note on the Isoperimetric Dimension of a Manifold, Expositiones Mathematicae, Vol.
	www.isoperimetricdimension.info (364 words)

	UCI Differential Geometry
		We classify homogeneous order d solutions to (fully nonlinear) elliptic equations in dimension three, and in higher dimensions except d=2.
		A special case is the well-known result that any nonparametric minimal cone of dimension three must be flat.
		It is well known that a domain D in R^n satisfies the classical isoperimetric inequality (CII) and that equality holds in CII if and only if D is a ball.
	www.math.uci.edu /~scgas/geomsem/geomsem-spring2003.html (181 words)

	Isoperimetric dimension - Education - Information - Educational Resources - Encyclopedia - Music (Site not responding. Last check: 2007-10-10)
		Isoperimetric dimension - Education - Information - Educational Resources - Encyclopedia - Music
		\mathrm\,(\partial D)\geq C\mathrm\,(D) which obviously implies infinite isoperimetric dimension.
		In rough terms, this means that a graph "mimicking" a given manifold (as the grid mimicks the Euclidean space) would have the same isoperimetric dimension as the manifold.
	education.music.us /I/Isoperimetric-dimension.htm (1003 words)

	Robert Young's home page (Site not responding. Last check: 2007-10-10)
		In particular, if a nilpotent group satisfies the isoperimetric inequality $\delta(l)2$ then it satisfies the averaged isoperimetric inequality $\delta^{\text{avg}}(l)
		In the case of non-abelian free nilpotent groups, the bounds we give are asymptotically sharp.
		In this note, we use their methods to bound the number of hyperbolic manifolds with diameter less than $d$ and show that the number grows double-exponentially with volume.
	www.math.uchicago.edu /~rjyoung (451 words)

	Ort Braude Academic College
		the notion of dimension is more delicate, e.g.
		The dimension introduced in this way is not a number but a
		euclidean case using the notion of the isoperimetric dimension.
	braude.ort.org.il /news/msgshow.asp?tid=200401271547 (107 words)

	Amazon.com: "isoperimetric problems": Key Phrase page (Site not responding. Last check: 2007-10-10)
		See all pages with references to isoperimetric problems.
		This material evolves naturally from minimization and isoperimetric problems involving critical points of higher type.
		History One of the earliest problems in geometry was the isoperimetric problem, which was considered by the ancient Greeks.
	amazon.com /phrase/isoperimetric-problems (511 words)

	MAAA Seminar - abstracts
		The dimension of its global attractor will be esitmated and shown
		The Shattering dimension of a class is a real-valued version of the
		bounded class of functions are exponential in the shattering dimension of the class.
	www.wisdom.weizmann.ac.il /mathusers/vered/analsemabs.html (10965 words)

	Higher dimensional isoperimetric functions in hyperbolic groups. (Site not responding. Last check: 2007-10-10)
		For a finitely presented group G, the following statements are equivalent:
		$\ell_\infty$-cohomology of G vanishes for any normed vector space as coefficients and any dimension at least 2,
		G satisfies linear isoperimetric inequalities for filling real cycles in each positive dimension,
	www.math.uiuc.edu /~mineyev/math/art/abs_hd.html (51 words)

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