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Topic: Topological dimension

  Fractal dimension
This constant is the fractal dimension of the Koch curve.
The fractal dimension of a plane region is 2.
Therefore, the fractal dimension serves as an interpolation of the topological dimension.
www.math.okstate.edu /mathdept/dynamics/lecnotes/node37.html   (767 words)

 3.2 Topological Dimension
The dimension on any other space will be defined as one greater that the dimension of the object that could be used to completely separate any part of the first space from the rest.
The dimension of a space should be the maximum of its local dimensions where the local dimension is defined as one more than the dimension of the lowest dimensional object with the capacity to separate any neighborhood of the space into two parts.
A topological property of an entity is one that remains invariant under continuous, one-to-one transformations or homeomorphisms.
hypertextbook.com /chaos/32.shtml   (858 words)

  Cartan's Corner (using Charlotte Technology)
The Pfaff topological dimension of the thermodynamic 1-form becomes equal to 2, and the resulting thermodynamic system is an isolated equilibrium system that does not exchange matter or radiation with its environment.
The technique furnishes a universal, topological foundation for the partial differential equations of hydrodynamics and electrodynamics; the topological technique does not depend upon a metric, connection or a variational principle.
The topological defects, in the otherwise flat surface of fluid density discontinuity, appear as a pair of zero mean curvature surfaces, with a conical (dimple) singularity at each end.
www22.pair.com /csdc/car/carhomep.htm   (2276 words)

  NationMaster - Encyclopedia: Quotient space
Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity.
In mathematics, a topological group G is a group that is also a topological space such that the group multiplication G × G → G and the inverse operation G → G are continuous maps.
The topological dimension of a quotient space can be more (as well as less) than the dimension of the original space; space-filling curves provide such examples.
www.nationmaster.com /encyclopedia/Quotient-space   (2427 words)

  Lebesgue covering dimension - Wikipedia, the free encyclopedia
In mathematics, the Lebesgue covering dimension or topological dimension of a topological space is defined to be the minimum value of n, such that any open cover has a refinement in which no point is included in more than n+1 elements.
The Lebesgue covering dimension coincides with the affine dimension of a finite simplicial complex; this is the Lebesgue covering theorem.
The idea of topological dimension first became a topic of considerable interest in the early 20th century.
en.wikipedia.org /wiki/Topological_dimension   (688 words)

 Hausdorff dimension - Wikipedia, the free encyclopedia
Intuitively, the dimension of a set (for example, a subset of Euclidean space) is the number of independent parameters needed to describe a point in the set.
One mathematical concept which closely models this naive idea is that of topological dimension of a set.
For example box-counting dimension, generalises the idea of counting the squares of graph paper in which a point of X can be found, as the size of the squares is made smaller and smaller.
en.wikipedia.org /wiki/Hausdorff_dimension   (1914 words)

 Springer Online Reference Works
Another, inductive, approach (see Inductive dimension) to the definition of the dimension of a topological space is possible, based on the separation of the space by subspaces of smaller dimension.
Dimension theory is most meaningful, first, for the class of metric spaces with a countable base, and, secondly, for the class of all metric spaces.
Several of the dimensions most used in algebra and ring theory may be defined on the lattice of submodules of some module, globalizing the definition by considering the supremum (or a similar invariant) of the dimension of all modules (perhaps restricting to a certain class of modules).
eom.springer.de /D/d032450.htm   (2461 words)

 Julia Sets
Topologically, connected Julia sets are either equivalent to a severely deformed circle or to a curve with an infinite series of branches and sub-branches called a dendrite (e.g., the Julia set for c=0+i)" (Elert 22.shtml).
We are all familiar with to the topological dimension in describing the dimensionality of an object.
However, the fractal (non-topological) dimension of fractals (such as the Cantor dust of points or the Sierpinski gasket) incorporates the concept that their infinite ramifications in effect cover more of their Euclidean space than their topological dimension would suggest.
www.mcgoodwin.net /julia/juliajewels.html   (4935 words)

 Fractal FAQs - WikiFAQ - Answers to Frequently Asked Questions (FAQ)
Topological dimension is the "normal" idea of dimension; a point has topological dimension 0, a line has topological dimension 1, a surface has topological dimension 2, etc.
A set S has topological dimension k if each point in S has arbitrarily small neighborhoods whose boundaries meet S in a set of dimension k-1, and k is the least nonnegative integer for which this holds.
A strange attractor is an attractor that is topologically distinct from a periodic orbit or a limit cycle.
www.wikifaq.com /index.php?title=Fractal_FAQs&redirect=no   (832 words)

 Fractal dimension
This is a figure the Euclidean dimension of which is 1 (it is a broken line) and the fractal dimension of which is greater than 1 and, moreover, is not a whole number.
This dimension is often said to be the dimension of Hausdorff-Besicovitch or, at least, it is suggested by the context.
Moreover there are other approaches of the notion of dimension which are not all equivalent, and the works of Kolmogorov and Tihomirov show a connection between the covering dimension and the notion of entropy (this may bring back some vague memories, for some of you!).
fractals.iut.u-bordeaux1.fr /jpl/dimension_a.html   (1246 words)

The notion of dimension was invented for the purpose of measuring this "qualitative" topological property.
Dimension is formalized in mathematics as the intrinsic dimension of a topological space.
In particular, the dimension of a subspace of R^n is equal to the number of linearly independent vectors needed to generate it (i.e., the number of vectors in its basis).
forum.myspace.com /index.cfm?fuseaction=messageboard.viewThread&entryID=7387051&categoryID=0&IsSticky=0&groupID=100000297   (1420 words)

 Fractal Curves and Dimension
The topological dimension of a smooth curve is, as one would expect, one and that of a sphere is two which may seem very intuitive.
To understand the notion of the similarity dimension, first observe that, if the initial line segment was 1 unit in length, then the second stage curve that consists of four segments each one third of the initial line, is 4/3 units in length.
By one of Brouwer's theorems this function preserves the topological dimension of the segment (which is, of course 1).
www.cut-the-knot.org /do_you_know/dimension.shtml   (1290 words)

 Fractal Dimension
The dimension of the union of finitely many sets is the largest dimension of any one of them, so if we ``grow hair'' on a plane, the result is still a two-dimensional set.
Not surprisingly, the box dimensions of ordinary Euclidean objects such as points, curves, surfaces, and solids coincide with their topological dimensions of 0, 1, 2, and 3-- this is, of course, what we would want to happen, and follows from the discussion at the beginning of §5.1.
Thus, the embedding dimension of a plane is 2, the embedding dimension of a sphere is 3, and the embedding dimension of a klein bottle is 4, even though they all have (topological) dimension two.
www.math.sunysb.edu /~scott/Book331/Fractal_Dimension.html   (1303 words)

 Computer Simulation of Fractal Structure of Flocs   (Site not responding. Last check: 2007-10-26)
The definition of topological dimension is based on the way in which an object must be cut for it to be divided into two parts.
A point is indivisible and has topological dimension 0, while to be severed a line requires removal of a point and is given a dimension 1.
Fractal dimension is a measurement of disorder, corresponding to the degree of irregularity and complexity of the space-filling capacity of the fractal object.
www.dekker.com /sdek/242672276-28938433/abstract~content=a713562224~db=enc   (781 words)

 R. Kiehn Abstract.   (Site not responding. Last check: 2007-10-26)
ABSTRACT: Methods of continuous topological evolution, expressed in terms of Cartan's theory of exterior differential forms, are used to construct a cosmological model of the present universe.
Such defects form long lived states of Pfaff dimension 3, and as such are states far from equilibrium which are of Pfaff topological dimension of 2 or less.
Hence, a cosmological model based on the assumption that the universe is a dilute, non-equilibrium turbulent gas near its critical point explains the the granularity of the night sky, the inverse square law of gravitation, and the expansion of the universe.
www.mindspring.com /~cerebroscopic/Kiehn.html   (296 words)

 [No title]   (Site not responding. Last check: 2007-10-26)
This dimension is a variant of the Hausdorff dimension, with the diameter of a set replaced by some gauge functions of the smallest return time of the set into itself (Poincar\'e recurrence of the set).
This dimension will be related to what we call {\it polynomial entropy}, which is the exponent of the algebraic growth of the complexity function, whenever the topological entropy of the subshift is zero.
A topological partition of a homogeneous space is a finite collection ${\cal V}$ of its open sets, such that for $V,V' \in {\cal V}$, $V \cap V' = \emptyset$, the (topological) dimension of $\overline{V} \cap \overline{V'}$ is strictly smaller than the dimension of $X$, and $\{\overline{V}: V \in {\cal V}\}$ covers $X$.
www.ma.utexas.edu /mp_arc/html/papers/00-231   (4031 words)

Further results are upper estimates of the box dimension for so-called weakly hyperbolic systems in terms of the topological pressure and the topological entropy.
Further the topological entropy of a (not necessarily uniformly) continuous mapping in a metric space is defined.
Further estimates of the box dimension are given which are formulated in terms of topological entropy and global inverse Lyapunov-exponents.
www.mpipks-dresden.mpg.de /~gelfert/short_diss.html   (844 words)

 Fractal dimension
Roughly, fractal dimension can be calculated by taking the limit of the quotient of the log change in object size and the log change in measurement scale, as the measurement scale approaches zero.
Since the dimension 1.261 is larger than the dimension 1 of the lines making up the curve, the snowflake curve is a fractal.
A set S has topological dimension k if each point in S has arbitrarily small neighborhoods whose boundaries meet S in a set of dimension k-1, and k is the least nonnegative integer for which this holds.
fractals.iut.u-bordeaux1.fr /sci-faq/dimension.html   (520 words)

 Model Simplification and Sample Decimation
Topological dimensions of shapes constitute an important feature of sample data.
We present a Voronoi based dimension detection algorithm that assigns a dimension to a sample point which is the topological dimension of the manifold it belongs to.
Based on this dimension detection, the shapes of arbitrary dimension can be reconstructed from their samples.
www.cse.ohio-state.edu /~tamaldey/dimensiondetection.htm   (62 words)

 Virtual Physics WWW Archive - Fractals, Wavelets, Percolation and Disorder
Like in the case of the topological dimension, if a set A is contained in a set B then a theorem says that Hausdorff dimension of A is less or equal to that of B. Thus every set containing a square (or, what is equivalent, a disk) has Hausdorff dimension greater or equal to 2.
The Hausdorff dimension of Sierpinski Carpet is not an integer.
Dimension theory is developed for sets in even more general spaces, but some statements valid in the euclidean case might fail in general.
www.swin.edu.au /chem/complex/vp/vp07/vp07.html   (4350 words)

 Fractals and Fractal Architecture - Dimension   (Site not responding. Last check: 2007-10-26)
A point in a three-dimensional space, with the three dimensions being the length, the width and the height, is described by three numbers, for example by the three ordinates.
The “topological dimension”, however, proceeds from the fact that each structure can be reduced to a set of points.
Fractal dimension is not an integer in contrast to the dimension in Euclidean geometry.
www.iemar.tuwien.ac.at /modul23/Fractals/subpages/04Dimension.html   (1650 words)

 sciforums.com - some basic questions
The definition of 'dimension' is a precise, rigorous mathematical concept.
to see how topological dimension leads to the notion of degrees of freedom is a bit harder, and requires some study.
The similarity method for calculating fractal dimension is great if you have a fractal composed of a certain number of identical versions of itself.
www.sciforums.com /showthread.php?t=13735   (2104 words)

 What is the Definition of a Fractal
What Mandelbrot means by topological dimension is that a curve is one dimensional, planar figures are 2-dimensional, and space figures are 3-dimensional.
This comes out to 2.726…, but, like the Sierpinski gasket, it's topological dimension is actually 2, as it could possibly be made out of lots and lots of paper cutouts.
Many fractals that require a 3-space to construct have a topological dimension of 2, and many of the curves that are considered to have a fractal dimension of 1 need to be drawn on a 2-dimensional paper.
members.fortunecity.com /bradstro/fractal.htm   (2028 words)

 Trading System Solutions - Publications - Fractal dimension – numerical characteristic of trend - What do most of ...
Formally as fractal can be termed set with Hausdorff dimension bigger than topological dimension.
If we measure their length on one scale (time frame), than on more detailed scale (time frame) the length would be bigger than if it would be usual line.
Popular measure of fractal dimension is Hirst parameter H (Edgar E. Peters, Fractal Market Analysis.
www.tsresearch.com /public/fractal   (933 words)

 Sierpinski Gasket
This family of objects will be discussed in dimensions 1, 2, 3, and an attempt will be made to visualise it in the 4th dimension.
The dimension of the gasket is log 3 / log 2 = 1.5849, ie: it lies dimensionally between a line and a plane.
All that we need is the equivalent of the pyramid in the 4th dimension (normally the name pyramid is given to the 3D square base pyramid, a 2D pyramid is an equilateral triangle, and a 1D pyramid is a line segment).
local.wasp.uwa.edu.au /~pbourke/fractals/gasket   (2172 words)

 Wolfram Research, Inc.
Fractal dimension is a subtle mathematical concept, and the interested reader is encouraged to consult the references in the bibliography.
The mathematician Hermann Weyl supposedly explained that "space is 3-dimensional because the walls of a prison are 2-dimensional." This description is sometimes called "topological dimension." In topological dimension, a point has dimension 0, a line has dimension 1, and a plane has dimension 2.
The use of a fourth "dimension" to measure time in physics also adds to the confusion, sometimes leading people to say "Time is the Fourth Dimension." Mathematicians consider spaces of four or higher dimensions that have nothing to do with time.
documents.wolfram.com /teachersedition/Teacher/FDB.html   (1825 words)

 Dimension | World of Mathematics
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.
For example, the has a fractal dimension of 1.26..., because when its size is tripled the number of disks required to cover it increases by a factor of four, or 3^(1.26...).
www.bookrags.com /research/dimension-wom   (516 words)

 Fractal Analysis of Trabecular Bone
These dimensions are known as topological dimensions, and have been used for many years to describe the shape and position of objects.
Benoit Mandelbrot, however, found that certain geometrical objects couldn't be described well with the usual topological dimensions, and formulated the idea of a fractional or fractal dimension, existing somewhere between the usual topological dimensions.
The amount of space filled by one of these objects is represented by the fractal dimension or index (D), which can be thought of as a "filling factor".
www.rad.washington.edu /exhibits/fractal.html   (1254 words)

 2.6 Self-Similarity Dimension
The self-similarity dimensions of the objects in fig.
As depicted by the Peano curve the Hausdorff-Besicovitch dimension of a fractal doesn't have to be noninteger.
The fractal dimension of the Koch curve in sect.
www.weihenstephan.de /ane/dimensions/subsection3_3_6.html   (333 words)

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