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Topic: Hausdorff measure

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Hausdorff dimension

Outer measure

Null set

Lebesgue measure

Lebesgue integration

Henri Lebesgue

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	Felix Hausdorff - Wikipedia, the free encyclopedia
		Hausdorff studied at the University of Leipzig, obtaining his Ph.D. in 1891.
		Hausdorff was the first to state a generalization of Cantor's Continuum Hypothesis; his Aleph Hypothesis, which appears in his 1908 article Grundzüge einer Theorie der geordneten Mengen, is equivalent to what is now called the Generalized Continuum Hypothesis.
		He introduced the concepts now called Hausdorff measure and Hausdorff dimension, which have been useful in the theory of fractals.
	en.wikipedia.org /wiki/Felix_Hausdorff (449 words)

	Hausdorff biography
		Hausdorff studied at Leipzig University under Heinrich Bruns and Adolph Mayer, graduating in 1891 with a doctorate in applications of mathematics to astronomy.
		Hausdorff returned to Bonn in 1921, by this time an emminent mathematician, and he worked there until 1935 when he was forced to retire by the Nazi regime.
		Hausdorff's paper includes a proof that the dimension of the middle-third Cantor set is log 2/log 3.
	www-history.mcs.st-andrews.ac.uk /Biographies/Hausdorff.html (1823 words)

	Hausdorff dimension Summary
		This number, D(A), is called the Hausdorff dimension of A. For many fractals, such as the snowflake curve, the Menger curve, Sierpinski's triangle and carpet, the Hausdorff dimension is equal to the self-similarity dimension (which is also equal to the capacity dimension).
		To carry this construction of this measure, we use a theory of measure which is appropriate for metric spaces.
		Define a family of metric outer measures on X using the Method II construction of outer measures due to Munroe and described in the article outer measure.
	www.bookrags.com /Hausdorff_dimension (2287 words)

	Hausdorff Measure and Dimension (Site not responding. Last check: 2007-10-16)
		is thus the measure of a d-dimensional unit ball, and rulers of
		The 1-dimensional measure, or length, of the rectangle is thus infinite.
		is equal to the Hausdorff dimension of that object.
	www.ph.ed.ac.uk /~ted/thesis/node72.html (277 words)

	PlanetMath: Hausdorff dimension
		Hausdorff dimension is easy to calculate for simple objects like the Sierpinski gasket or a Koch curve.
		In fact, in the case of the Sierpinski gasket, one can take the individual triangles in each approximation as balls in the covering.
		This is version 12 of Hausdorff dimension, born on 2002-05-31, modified 2006-08-02.
	planetmath.org /encyclopedia/HausdorffDimension.html (160 words)

	Ars Mathematica » Blog Archive » Hausdorff Surprises
		The Hausdorff dimension is used to define the dimension of fractals, for example, the dimension of the Sierpinski triangle is log(3)/log(2).
		For integer dimensions, the Hausdorff measure is equivalent to the Lebesgue measure.
		The Hausdorff dimension of a set is the point where the d-dimensional Hausdorff measure changes from infinity to zero, i.e.
	www.arsmathematica.net /archives/2005/05/16/hausdorff-surprises (285 words)

	Felix Hausdorff (Site not responding. Last check: 2007-10-16)
		Hausdorff's main work was in topology and set theory.
		He also introduced the Hausdorff measure and the term "metric space" is due to him.
		Hausdorff worked at Bonn until 1935 when he was forced to retire by the Nazi regime.
	www.stetson.edu /~efriedma/periodictable/html/Hf.html (256 words)

	Hausdorff Distance Image Comparison
		The function h(A,B) is called the directed Hausdorff `distance' from A to B (this function is not symmetric and thus is not a true distance).
		Thus the Hausdorff distance, H(A,B), measures the degree of mismatch between two sets, as it reflects the distance of the point of A that is farthest from any point of B and vice versa.
		Therefore this measure generalizes the directed Hausdorff measure, by replacing the maximum with a quantile.
	www.cs.cornell.edu /Info/People/dph/hausdorff/hausdorff.html (830 words)

	Hausdorff-Based Matching (Site not responding. Last check: 2007-10-16)
		The generalized Hausdorff measure provides a means of determining the resemblance of one point set to another, by examining the fraction of points in one set that lie near points in the other set (and perhaps vice versa).
		A C implementation of Hausdorff matching (for matching with translation or with translation and scaling) is available as a tar file via ftp.
		We have used the generalized Hausdorff measure to search for a two-dimensional model (a point set represented as a bitmap) in a bitmap image (usually the intensity edges from some image) under various transformations.
	www.cs.cornell.edu /vision/hausdorff/hausmatch.html (410 words)

	Box Counting Dimension (Site not responding. Last check: 2007-10-16)
		Though elegant, the Hausdorff measure and dimension are difficult to calculate in practice.
		For this reason, a single arrangement of rulers is used to measure the coverage, and this measure is assumed to be satisfactory.
		used to measure the curve is reduced by a factor of 3, the number required to cover it increases by a factor of 4.
	www.ph.ed.ac.uk /~ted/thesis/node73.html (965 words)

	BioMed Central \| Full text \| Structural similarity of loops in protein families: toward the understanding of protein ...
		In our previous work we showed that a measure derived from the loop regions can distinguish homologous from analogous proteins with the same or higher accuracy compared to the conventional measures which are based on comparing proteins in structurally aligned regions only [7].
		This new measure is based on the concept of the Hausdorff metric, which is used in mathematical topology to define a distance between two point sets of a metric space.
		Hausdorff distance can be calculated under the assumption that the Cα atoms for both structures are in a common coordinate frame which is defined by the structural alignment between two domains.
	www.biomedcentral.com /1471-2148/5/10 (3431 words)

	[No title] (Site not responding. Last check: 2007-10-16)
		Assume that the measures $\lmu$ and $\omu$ are Gibbs measures.
		Moreover, the Hausdorff measure $m_H(s, \cdot)$ is equivalent to the measure $m(\cdot)$.
		Thus for the measure $\nu = \chi^* \mu$, $$ \underline d_{\nu}(x) = \overline d_{\nu}(x) = \frac{2 h_{\mu}(\sigma)}{\log (\alpha \beta)} \qquad \text{if $x \in A$} $$ and $$ \underline d_{\nu}(x) = \overline d_{\nu}(x) = \frac{h_{\mu}(\sigma)}{\log \alpha} \qquad \text{if $x \in B$}.
	www.ma.utexas.edu /mp_arc/papers/93-332 (8047 words)

	Hausdorff dimension
		A way to accurately measure the dimension of complicated sets such as fractals.
		The Hausdorff dimension, named after Felix Hausdorff, coincides with the more familiar notion of dimension in the case of well-behaved sets.
		But a Hausdorff dimension is not always a natural number.
	www.daviddarling.info /encyclopedia/H/Hausdorff_dimension.html (238 words)

	Springer Online Reference Works
		For a slightly different slant, [a9] discusses applications of some of the ideas of geometric measure theory in the theory of Sobolev spaces and functions of bounded variation.
		It is common in analysis to construct measures as solutions to equations, and one would like to be able to deduce something about the structure of these measures (for example, that they are rectifiable).
		The utility of the notion lies in the fact that tangent measures often possess more regularity than the original measure, thus allowing a wider range of analytical techniques to be used upon them.
	eom.springer.de /G/g130040.htm (1568 words)

	Budapest University of Technology
		A simple consequence of this the theorem is the statement 1.5.:There is existing the invariant, non-negative, additive set function, explained all on the subsets; on which the measure of the group is 1.
		verifies in the case of a compact, metrizable group the existence of a measure, that is not identically zero and is in consideriton of metric is invariant.
		In the remaining part of this Chapter, we examine the characteristics of that topological group on which there is existing a non-identically zero, left-invariant measure.
	www.math.bme.hu /~arpi/doktori.htm (700 words)

	Hausdorff Distance
		Felix Hausdorff (1868 -1942) devised a metric function between subsets of a metric space.
		two sets are within Hausdorff distance r from each other iff any point of one set is within distance r from some point of the other set.
		The Hausdorff metric h(A,B) is defined in terms of the neighborhoods.
	www.cut-the-knot.org /do_you_know/Hausdorff.shtml (687 words)

	Mesh Approximation Review (Site not responding. Last check: 2007-10-16)
		Definition 1a : The one sided Hausdorff distance between two meshes M1 and M2 is defined H(M1, M2) = max(min(d(a,b))), for 'a' on the surface of M1, b on the surface of M2, and where d(a,b) is the L2 distance from a to b.
		The Hausdorff distance is a good approximation for comparing meshes that are very similar, guaranteeing a specific error bound between two meshes, but as the Hausdorff measure grows larger, it becomes less meaningfull.
		The error measures the deviation in pixel units that a surface point (with associated colour and normal) may shift in the image space.
	www.cs.sfu.ca /~aclement/personal/LitReview/litreview.html (1668 words)

	Chaos and Fractals in Financial Markets, Part 3, by J. Orlin Grabbe
		But, once we measure all the holes in the carpet, we discovered that what we are left with is carpet that has been entirely consumed by holes.
		Consider: one way to measure the length of something is to take a measuring rod of length m, lay it alongside what you are measuring, mark the end point of the measuring rod, and repeat the process until you have the number N of measuring rod lengths.
		To calculate a Hausdorff dimension, we note that at each stage of the construction, we replace each line segment with N = 4 segments, after dividing the original line segment by a scale factor r = 3.
	www.aci.net /kalliste/Chaos3.htm (4169 words)

	Hausdorff dimension - Wikipedia, the free encyclopedia
		For example, the Cantor set has topological dimension zero, but in some sense it behaves as a higher dimensional space.
		Some authors adopt a slightly different definition of Hausdorff measure than the one chosen here, the difference being that it is normalized in such a way that Hausdorff m-dimensional measure in the case of Euclidean space coincides exactly with the Lebesgue measure λ.
		Hausdorff, Dimension und äusseres Mass, Mathematische Annalen 79 (1919).
	en.wikipedia.org /wiki/Hausdorff_dimension (1928 words)

	Math Seminars.
		Hausdorff and packing measure for some hyperbolic attractors
		We study their Hausdorff dimension and appropriate dimensional Hausdorff and packing measure.
		We prove that the Hausdorff measure is zero, the packing measure is positive and finite, and the ``natural'' (Sinai-Bowen-Ruelle) measure on the attractor is equivalent to the packing measure.
	www.math.psu.edu /dynsys/abstracts-2001/simonk.html (68 words)

	PlanetMath: Hausdorff measure
		is nothing else than the Lebesgue measure of
		Cross-references: limit, contains, sequences, function, cover, enumerable, infimum, gamma function, constant, diameter, metric space, fractals, integer, Hausdorff dimension, dimension, infinite, positive, finite, fixed, Lebesgue measure, subset, surface, regular, volume, area, length, measure, real number
		This is version 5 of Hausdorff measure, born on 2004-06-30, modified 2006-10-02.
	planetmath.org /encyclopedia/HausdorffMeasure.html (228 words)

	UW Rainwater Seminar
		Let $C_1$ and $C_2$ be two affine Cantor sets in the real line and let $d_1$ and $d_2$ be their Hausdorff dimensions.
		It is natural to ask whether the corresponding Hausdorff measure of $C_1 + C_2$ is positive.
		As a corollary, we obtain the result that if $C_1$ and $C_2$ are homogeneous affine Cantor sets of the same diameter, then the $(d_1+d_2)$-dimensional Hausdorff measure of $C_1 + C_2$ is always zero.
	www.math.washington.edu /~solomyak/Seminar/04/ilgar.html (177 words)

	Citebase - The Hausdorff measure of stable trees
		In particular we derive the exact Hausdorff measure function for Aldous' continuum random tree and for its level sets.
		It follows that both the uniform measure on the tree and the local time measure on a level set coincide with certain Hausdorff measures.
		Slightly less precise results are obtained for the Hausdorff measure of general stable trees.
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/0509690 (140 words)

	UNT Department of Mathematics: Dynamics and Analysis Seminar
		It is known that the Hausdorff dimension of $J(f_\lambda)$ is equal to $2$, while the $2$-dimensional Hausdorff measure of $J(f_\lambda)$ is zero.
		We also prove that $h_\lambda$-dimensional Hausdorff measure of $J_r(f_\lambda)$ is non-zero and finite on each horizontal strip, while the $h_\lambda$-dimensional packing measure is locally infinite at each point of $J_r(f_\lambda)$.
		Disintegrations of these measures in terms of ergodic limits are investigated in terms of linkage and separation.
	www.math.unt.edu /seminars/dynamic.shtml (4675 words)

	The Farey Room
		The measure is shown as a color, with fl=zero, blue=small, green=larger, yellow=large, red=larger still.
		When we say 'Hausdorff Measure' above, what we really mean is this: Take the real number line between 0 and 1, and divide it into N equal bins.
		When properly normalized, the resulting 'density' on the real-number line converges to a stable limit as N is increased, and as the number of samples is increased.
	linas.org /art-gallery/farey/fthumb.html (1305 words)

	The Strange World of the Hausdorff Metric Geometry
		A metric is a function that provides us a way to measure the distance between two objects.
		The Hausdorff metric allows us to measure the distance between certain types of sets.
		B) varies as we alter the position and radii of two disks A, B in the plane.
	faculty.gvsu.edu /schlicks/HausdorffGeometry/H2.htm (373 words)

	Stein, E.M. and Shakarchi, R.: Real Analysis: Measure Theory, Integration, and Hilbert Spaces.
		Here the focus is on the development of measure and integration theory, differentiation and integration, Hilbert spaces, and Hausdorff measure and fractals.
		This book reflects the objective of the series as a whole: to make plain the organic unity that exists between the various parts of the subject, and to illustrate the wide applicability of ideas of analysis to other fields of mathematics and science.
		After setting forth the basic facts of measure theory, Lebesgue integration, and differentiation on Euclidian spaces, the authors move to the elements of Hilbert space, via the L2 theory.
	press.princeton.edu /titles/8008.html (477 words)

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