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Hausdorff Method
Hausdorff may refer to: * A Hausdorff space, when used as an adjective, as in "the real line is Hausdorff" * Felix Hausdorff (1868–1942), the German mathematician after whom Hausdorff spaces are named * Hausdorff dimension, a measure theoretic concept of dimension * Hausdorff distance or Hausdorff metric, which measures how far two compact non-empty subsets of a metric space are from each other * Hausdorff density * Hausdorff maximal principle * Hausdorff measure * Hausdorff moment problem * Hausdorff paradox {{disambig, surname ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hausdorff Space
In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" (T2) is the most frequently used and discussed. It implies the uniqueness of limits of sequences, nets, and filters. Hausdorff spaces are named after Felix Hausdorff, one of the founders of topology. Hausdorff's original definition of a topological space (in 1914) included the Hausdorff condition as an axiom. Definitions Points x and y in a topological space X can be '' separated by neighbourhoods'' if there exists a neighbourhood U of x and a neighbourhood V of y such that U and V are disjoint (U\cap V=\varnothing). X is a Hausdorff space if any two distinct points in X are separated by neighbourhoods. This condition is the third separation axiom ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Felix Hausdorff
Felix Hausdorff ( , ; November 8, 1868 – January 26, 1942) was a German mathematician who is considered to be one of the founders of modern topology and who contributed significantly to set theory, descriptive set theory, measure theory, and functional analysis. Life became difficult for Hausdorff and his family after Kristallnacht in 1938. The next year he initiated efforts to emigrate to the United States, but was unable to make arrangements to receive a research fellowship. On 26 January 1942, Felix Hausdorff, along with his wife and his sister-in-law, died by suicide by taking an overdose of veronal, rather than comply with German orders to move to the Endenich camp, and there suffer the likely implications, about which he held no illusions. Life Childhood and youth Hausdorff's father, the Jewish merchant Louis Hausdorff (1843–1896), moved with his young family to Leipzig in the autumn of 1870, and over time worked at various companies, including a linen-and cotton goo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hausdorff Dimension
In mathematics, Hausdorff dimension is a measure of ''roughness'', or more specifically, fractal dimension, that was first introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff dimension of a single point is zero, of a line segment is 1, of a square is 2, and of a cube is 3. That is, for sets of points that define a smooth shape or a shape that has a small number of corners—the shapes of traditional geometry and science—the Hausdorff dimension is an integer agreeing with the usual sense of dimension, also known as the topological dimension. However, formulas have also been developed that allow calculation of the dimension of other less simple objects, where, solely on the basis of their properties of scaling and self-similarity, one is led to the conclusion that particular objects—including fractals—have non-integer Hausdorff dimensions. Because of the significant technical advances made by Abram Samoilovitch Besicovitch allowing computation of di ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hausdorff Distance
In mathematics, the Hausdorff distance, or Hausdorff metric, also called Pompeiu–Hausdorff distance, measures how far two subsets of a metric space are from each other. It turns the set of non-empty compact subsets of a metric space into a metric space in its own right. It is named after Felix Hausdorff and Dimitrie Pompeiu. Informally, two sets are close in the Hausdorff distance if every point of either set is close to some point of the other set. The Hausdorff distance is the longest distance you can be forced to travel by an adversary who chooses a point in one of the two sets, from where you then must travel to the other set. In other words, it is the greatest of all the distances from a point in one set to the closest point in the other set. This distance was first introduced by Hausdorff in his book ''Grundzüge der Mengenlehre'', first published in 1914, although a very close relative appeared in the doctoral thesis of Maurice Fréchet in 1906, in his study of the space of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hausdorff Density
In measure theory, a field of mathematics, the Hausdorff density measures how concentrated a Radon measure is at some point. Definition Let \mu be a Radon measure and a\in\mathbb^ some point in Euclidean space. The ''s''-dimensional upper and lower Hausdorff densities are defined to be, respectively, : \Theta^(\mu,a)=\limsup_\frac and : \Theta_^(\mu,a)=\liminf_\frac where B_(a) is the ball of radius ''r'' > 0 centered at ''a''. Clearly, \Theta_^(\mu,a)\leq \Theta^(\mu,a) for all a\in\mathbb^. In the event that the two are equal, we call their common value the s-density of \mu at ''a'' and denote it \Theta^(\mu,a). Marstrand's theorem The following theorem states that the times when the ''s''-density exists are rather seldom. : Marstrand's theorem: Let \mu be a Radon measure on \mathbb^. Suppose that the ''s''-density \Theta^(\mu,a) exists and is positive and finite for ''a'' in a set of positive \mu measure. Then ''s'' is an integer. Preiss' theorem In 1987 David Prei ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hausdorff Maximal Principle
In mathematics, the Hausdorff maximal principle is an alternate and earlier formulation of Zorn's lemma proved by Felix Hausdorff in 1914 (Moore 1982:168). It states that in any partially ordered set, every totally ordered subset is contained in a maximal totally ordered subset. The Hausdorff maximal principle is one of many statements equivalent to the axiom of choice over ZF (Zermelo–Fraenkel set theory without the axiom of choice). The principle is also called the Hausdorff maximality theorem or the Kuratowski lemma (Kelley 1955:33). Statement The Hausdorff maximal principle states that, in any partially ordered set, every totally ordered subset is contained in a maximal totally ordered subset (a totally ordered subset that, if enlarged in any way, does not remain totally ordered). In general, there may be many maximal totally ordered subsets containing a given totally ordered subset. An equivalent form of the Hausdorff maximal principle is that in every partially ordered ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hausdorff Measure
In mathematics, Hausdorff measure is a generalization of the traditional notions of area and volume to non-integer dimensions, specifically fractals and their Hausdorff dimensions. It is a type of outer measure, named for Felix Hausdorff, that assigns a number in ,∞to each set in \R^n or, more generally, in any metric space. The zero-dimensional Hausdorff measure is the number of points in the set (if the set is finite) or ∞ if the set is infinite. Likewise, the one-dimensional Hausdorff measure of a simple curve in \R^n is equal to the length of the curve, and the two-dimensional Hausdorff measure of a Lebesgue-measurable subset of \R^2 is proportional to the area of the set. Thus, the concept of the Hausdorff measure generalizes the Lebesgue measure and its notions of counting, length, and area. It also generalizes volume. In fact, there are ''d''-dimensional Hausdorff measures for any ''d'' ≥ 0, which is not necessarily an integer. These measures are fundamenta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hausdorff Moment Problem
In mathematics, the Hausdorff moment problem, named after Felix Hausdorff, asks for necessary and sufficient conditions that a given sequence be the sequence of moments :m_n = \int_0^1 x^n\,d\mu(x) of some Borel measure supported on the closed unit interval . In the case , this is equivalent to the existence of a random variable supported on , such that . The essential difference between this and other well-known moment problems is that this is on a bounded interval, whereas in the Stieltjes moment problem In mathematics, the Stieltjes moment problem, named after Thomas Joannes Stieltjes, seeks necessary and sufficient conditions for a sequence (''m''0, ''m''1, ''m''2, ...) to be of the form :m_n = \int_0^\infty x^n\,d\mu(x) for some measure ''μ ... one considers a half-line , and in the Hamburger moment problem one considers the whole line . The Stieltjes moment problems and the Hamburger moment problems, if they are solvable, may have infinitely many solutions ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
