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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 cotto ... [...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 computati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Measure Theory
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many similarities and can often be treated together in a single mathematical context. Measures are foundational in probability theory, integration theory, and can be generalized to assume negative values, as with electrical charge. Far-reaching generalizations (such as spectral measures and projection-valued measures) of measure are widely used in quantum physics and physics in general. The intuition behind this concept dates back to ancient Greece, when Archimedes tried to calculate the area of a circle. But it was not until the late 19th and early 20th centuries that measure theory became a branch of mathematics. The foundations of modern measure theory were laid in the works of Émile Borel, Henri Lebesgue, Nikolai Luzin, Johann Radon, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Descriptive Set Theory
In mathematical logic, descriptive set theory (DST) is the study of certain classes of " well-behaved" subsets of the real line and other Polish spaces. As well as being one of the primary areas of research in set theory, it has applications to other areas of mathematics such as functional analysis, ergodic theory, the study of operator algebras and group actions, and mathematical logic. Polish spaces Descriptive set theory begins with the study of Polish spaces and their Borel sets. A Polish space is a second-countable topological space that is metrizable with a complete metric. Heuristically, it is a complete separable metric space whose metric has been "forgotten". Examples include the real line \mathbb, the Baire space \mathcal, the Cantor space \mathcal, and the Hilbert cube I^. Universality properties The class of Polish spaces has several universality properties, which show that there is no loss of generality in considering Polish spaces of certain restricted f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Set Theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concerned with those that are relevant to mathematics as a whole. The modern study of set theory was initiated by the German mathematicians Richard Dedekind and Georg Cantor in the 1870s. In particular, Georg Cantor is commonly considered the founder of set theory. The non-formalized systems investigated during this early stage go under the name of '' naive set theory''. After the discovery of paradoxes within naive set theory (such as Russell's paradox, Cantor's paradox and the Burali-Forti paradox) various axiomatic systems were proposed in the early twentieth century, of which Zermelo–Fraenkel set theory (with or without the axiom of choice) is still the best-known and most studied. Set theory is commonly employed as a foundational ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself. A topological space is a set endowed with a structure, called a '' topology'', which allows defining continuous deformation of subspaces, and, more generally, all kinds of continuity. Euclidean spaces, and, more generally, metric spaces are examples of a topological space, as any distance or metric defines a topology. The deformations that are considered in topology are homeomorphisms and homotopies. A property that is invariant under such deformations is a topological property. Basic examples of topological properties are: the dimension, which allows distinguishing between a line and a surface; compactness, which allows distinguishing between a line and a circle; ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History One of the earliest known mathematicians were Thales of Miletus (c. 624–c.546 BC); he has been hailed as the first true mathematician and the first known individual to whom a mathematical discovery has been attributed. He is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales' Theorem. The number of known mathematicians grew when Pythagoras of Samos (c. 582–c. 507 BC) established the Pythagorean School, whose doctrine it was that mathematics ruled the universe and whose motto was "All is number". It was the Pythagoreans who coined the term "mathematics", and with whom the study of mathematics for its own sake begins. The first woman mathematician recorded by history was Hyp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hausdorff–Young Inequality
The Hausdorff−Young inequality is a foundational result in the mathematical field of Fourier analysis. As a statement about Fourier series, it was discovered by and extended by . It is now typically understood as a rather direct corollary of the Plancherel theorem, found in 1910, in combination with the Riesz-Thorin theorem, originally discovered by Marcel Riesz in 1927. With this machinery, it readily admits several generalizations, including to multidimensional Fourier series and to the Fourier transform on the real line, Euclidean spaces, as well as more general spaces. With these extensions, it is one of the best-known results of Fourier analysis, appearing in nearly every introductory graduate-level textbook on the subject. The nature of the Hausdorff-Young inequality can be understood with only Riemann integration and infinite series as prerequisite. Given a continuous function , define its "Fourier coefficients" by :c_n=\int_0^1 e^f(x)\,dx for each integer . The Hausd ... [...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 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 (indeterminate moment problem) whereas a Hausdorff moment problem always has a unique solution if it is solvable (determinate moment problem). In the indeterminate moment problem case, there are infinite measures corresponding to the same prescribed moments a ... [...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 separ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hausdorff Paradox
The Hausdorff paradox is a paradox in mathematics named after Felix Hausdorff. It involves the sphere (a 3-dimensional sphere in ). It states that if a certain countable subset is removed from , then the remainder can be divided into three disjoint subsets and such that and are all congruent. In particular, it follows that on S^2 there is no finitely additive measure defined on all subsets such that the measure of congruent sets is equal (because this would imply that the measure of is simultaneously 1/3, 1/2, and 2/3 of the non-zero measure of the whole sphere). The paradox was published in ''Mathematische Annalen'' in 1914 and also in Hausdorff's book, ''Grundzüge der Mengenlehre'', the same year. The proof of the much more famous Banach–Tarski paradox uses Hausdorff's ideas. The proof of this paradox relies on the axiom of choice. This paradox shows that there is no finitely additive measure on a sphere defined on ''all'' subsets which is equal on congruent pieces. ( ... [...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 funda ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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