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Vitali Covering Theorem
In mathematics, the Vitali covering lemma is a combinatorial and geometric result commonly used in measure theory of Euclidean spaces. This lemma is an intermediate step, of independent interest, in the proof of the Vitali covering theorem. The covering theorem is credited to the Italian mathematician Giuseppe Vitali.. The theorem states that it is possible to cover, up to a Lebesgue-negligible set, a given subset ''E'' of R''d'' by a disjoint family extracted from a ''Vitali covering'' of ''E''. Vitali covering lemma There are two basic version of the lemma, a finite version and an infinite version. Both lemmas can be proved in the general setting of a metric space, typically these results are applied to the special case of the Euclidean space \mathbb^d. In both theorems we will use the following notation: if B = B(x,r) is a ball and c \in \mathbb, we will write cB for the ball B(x,cr). Finite version Theorem (Finite Covering Lemma). Let B_, \dots, B_ be any finite col ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
CRC Press
The CRC Press, LLC is an American publishing group that specializes in producing technical books. Many of their books relate to engineering, science and mathematics. Their scope also includes books on business, forensics and information technology. CRC Press is now a division of Taylor & Francis, itself a subsidiary of Informa. History The CRC Press was founded as the Chemical Rubber Company (CRC) in 1903 by brothers Arthur, Leo and Emanuel Friedman in Cleveland, Ohio, based on an earlier enterprise by Arthur, who had begun selling rubber laboratory aprons in 1900. The company gradually expanded to include sales of laboratory equipment to chemists. In 1913 the CRC offered a short (116-page) manual called the ''Rubber Handbook'' as an incentive for any purchase of a dozen aprons. Since then the ''Rubber Handbook'' has evolved into the CRC's flagship book, the '' CRC Handbook of Chemistry and Physics''. In 1964, Chemical Rubber decided to focus on its publishing ventures ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boca Raton, FL
Boca Raton ( ; es, Boca Ratón, link=no, ) is a city in Palm Beach County, Florida, United States. It was first incorporated on August 2, 1924, as "Bocaratone," and then incorporated as "Boca Raton" in 1925. The population was 97,422 in the 2020 census, and it was ranked as the 344th largest city in America in 2022. However, approximately 200,000 additional people with a Boca Raton postal address live outside of municipal boundaries, such as in West Boca Raton. As a business center, the city experiences significant daytime population increases. Boca Raton is north of Miami and is a principal city of the Miami metropolitan area, which had a population of 6,012,331 as of 2015. Boca Raton is home to the main campus of Florida Atlantic University and the corporate headquarters of Office Depot. It is also home to the Evert Tennis Academy, owned by former professional tennis player Chris Evert. Boca Town Center, an upscale shopping center in central Boca Raton, is one of th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert Space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturally and frequently in mathematics and physics, typically as function spaces. Formally, a Hilbert space is a vector space equipped with an inner product that defines a distance function for which the space is a complete metric space. The earliest Hilbert spaces were studied from this point of view in the first decade of the 20th century by David Hilbert, Erhard Schmidt, and Frigyes Riesz. They are indispensable tools in the theories of partial differential equations, quantum mechanics, Fourier analysis (which includes applications to signal processing and heat transfer), and ergodic theory (which forms the mathematical underpinning of thermodynamics). John von Neumann coined the term ''Hilbert space'' for the abstract concept that under ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Separable Space
In mathematics, a topological space is called separable if it contains a countable, dense subset; that is, there exists a sequence \_^ of elements of the space such that every nonempty open subset of the space contains at least one element of the sequence. Like the other axioms of countability, separability is a "limitation on size", not necessarily in terms of cardinality (though, in the presence of the Hausdorff axiom, this does turn out to be the case; see below) but in a more subtle topological sense. In particular, every continuous function on a separable space whose image is a subset of a Hausdorff space is determined by its values on the countable dense subset. Contrast separability with the related notion of second countability, which is in general stronger but equivalent on the class of metrizable spaces. First examples Any topological space that is itself finite or countably infinite is separable, for the whole space is a countable dense subset of itself. An importa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gaussian Measure
In mathematics, Gaussian measure is a Borel measure on finite-dimensional Euclidean space R''n'', closely related to the normal distribution in statistics. There is also a generalization to infinite-dimensional spaces. Gaussian measures are named after the Germany, German mathematician Carl Friedrich Gauss. One reason why Gaussian measures are so ubiquitous in probability theory is the central limit theorem. Loosely speaking, it states that if a random variable ''X'' is obtained by summing a large number ''N'' of independent random variables of order 1, then ''X'' is of order \sqrt and its law is approximately Gaussian. Definitions Let ''n'' ∈ N and let ''B''0(R''n'') denote the complete measure, completion of the Borel sigma algebra, Borel ''σ''-algebra on R''n''. Let ''λ''''n'' : ''B''0(R''n'') → [0, +∞] denote the usual ''n''-dimensional Lebesgue measure. Then the standard Gaussian measure ''γ''''n'' : ''B''0(R''n'') → [0, 1] is defined by :\gamma^ (A) = \frac \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
David Preiss
David Preiss FRS (born January 21, 1947) is a Czech and British mathematician, specializing in mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m .... He is a professor of mathematics at the University of Warwick Preiss is a recipient of the Ostrowski Prize (2011) and the winner of the 2008 London Mathematical Society Pólya Prize (LMS), Pólya Prize for his 1987 result on ''Geometry of Measures'', where he solved the remaining problem in the geometric theoretic structure of sets and measure (mathematics), measures in Euclidean space. He was an invited speaker at the ICM 1990 in Kyoto. He is a Fellow of the Royal Society (2004) and a Foreign Fellow of the Learned Society of the Czech Republic (2003). He is associate editor of the mathematical journal ''Real An ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Measurable
In mathematics, the concept of a measure is a generalization and formalization of Geometry#Length, area, and volume, 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, integral, integration theory, and can be generalized to assume signed measure, 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 Bo ... [...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] |
Besicovitch Covering Theorem
In mathematical analysis, a Besicovitch cover, named after Abram Samoilovitch Besicovitch, is an open cover of a subset ''E'' of the Euclidean space R''N'' by balls such that each point of ''E'' is the center of some ball in the cover. The Besicovitch covering theorem asserts that there exists a constant ''c''N depending only on the dimension ''N'' with the following property: * Given any Besicovitch cover F of a bounded set ''E'', there are ''c''''N'' subcollections of balls ''A''1 = , …, ''A''''c''''N'' = contained in F such that each collection ''A''i consists of disjoint balls, and : E \subseteq \bigcup_^ \bigcup_ B. Let G denote the subcollection of F consisting of all balls from the ''c''''N'' disjoint families ''A''1,...,''A''''c''''N''. The less precise following statement is clearly true: every point ''x'' ∈ R''N'' belongs to at most ''c''''N'' different balls from the subcollection G, and G remains a cover for ''E'' (every p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Henri Lebesgue
Henri Léon Lebesgue (; June 28, 1875 – July 26, 1941) was a French mathematician known for his theory of integration, which was a generalization of the 17th-century concept of integration—summing the area between an axis and the curve of a function defined for that axis. His theory was published originally in his dissertation ''Intégrale, longueur, aire'' ("Integral, length, area") at the University of Nancy during 1902. Personal life Henri Lebesgue was born on 28 June 1875 in Beauvais, Oise. Lebesgue's father was a typesetter and his mother was a school teacher. His parents assembled at home a library that the young Henri was able to use. His father died of tuberculosis when Lebesgue was still very young and his mother had to support him by herself. As he showed a remarkable talent for mathematics in primary school, one of his instructors arranged for community support to continue his education at the Collège de Beauvais and then at Lycée Saint-Louis and Lycée Louis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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