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Gaussian Isoperimetric Inequality
In mathematics, the Gaussian isoperimetric inequality, proved by Boris Tsirelson and Vladimir Sudakov, and later independently by Christer Borell, states that among all sets of given Gaussian measure in the ''n''-dimensional Euclidean space, half-spaces have the minimal Gaussian boundary measure. Mathematical formulation Let \scriptstyle A be a measurable subset of \scriptstyle\mathbf^n endowed with the standard Gaussian measure \gamma^n with the density /(2\pi)^. Denote by : A_\varepsilon = \left\ the ε-extension of ''A''. Then the ''Gaussian isoperimetric inequality'' states that : \liminf_ \varepsilon^ \left\ \geq \varphi(\Phi^(\gamma^n(A))), where : \varphi(t) = \frac\quad\quad\Phi(t) = \int_^t \varphi(s)\, ds. Proofs and generalizations The original proofs by Sudakov, Tsirelson and Borell were based on Paul Lévy's spherical isoperimetric inequality. Sergey Bobkov proved a functional generalization of the Gaussian isoperimetric inequality, from a cert ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boris Tsirelson
Boris Semyonovich Tsirelson (May 4, 1950 – January 21, 2020) ( he, בוריס סמיונוביץ' צירלסון, russian: Борис Семёнович Цирельсон) was a Russian–Israeli mathematician and Professor of Mathematics at Tel Aviv University in Israel, as well as a Wikipedian, Wikipedia editor. Biography Tsirelson was born in Saint Petersburg, Leningrad to a History of the Jews in Russia, Russian Jewish family. From his father Simeon's side, he was the great-nephew of rabbi Yehuda Leib Tsirelson, chief rabbi of Bessarabia from 1918 to 1941, and a prominent posek and Jewish leader. He obtained his Master of Science from the Saint Petersburg State University, University of Leningrad and remained there to pursue graduate studies. He obtained his Doctor of Philosophy, Ph.D. in 1975, with thesis "General properties of bounded Gaussian processes and related questions" written under the direction of Ildar Abdulovich Ibragimov. Later, he participated in the Re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vladimir Sudakov
Vladimir may refer to: Names * Vladimir (name) for the Bulgarian, Croatian, Czech, Macedonian, Romanian, Russian, Serbian, Slovak and Slovenian spellings of a Slavic name * Uladzimir for the Belarusian version of the name * Volodymyr for the Ukrainian version of the name * Włodzimierz (given name) for the Polish version of the name * Valdemar for the Germanic version of the name * Wladimir for an alternative spelling of the name Places * Vladimir, Russia, a city in Russia * Vladimir Oblast, a federal subject of Russia * Vladimir-Suzdal, a medieval principality * Vladimir, Ulcinj, a village in Ulcinj Municipality, Montenegro * Vladimir, Gorj, a commune in Gorj County, Romania * Vladimir, a village in Goiești Commune, Dolj County, Romania * Vladimir (river), a tributary of the Gilort in Gorj County, Romania * Volodymyr (city), a city in Ukraine Religious leaders * Metropolitan Vladimir (other), multiple * Jovan Vladimir (d. 1016), ruler of Doclea and a saint of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Christer Borell
Christer or Krister are varieties of the masculine given name Kristian, derived from the Latin name ''Christianus'', which in turn comes from the Greek word ''khristianós'', which means "follower of Christ". The name, written in its two variants Christer and Krister, is quite common in the Nordic countries. Notable people with the name include: *Catherine Christer Hennix (born 1948), Swedish-American composer, philosopher, scientist and visual artist associated with drone minimal music * Christer Abris (formerly Abrahamsson, born 1947), Swedish former ice hockey goaltender * Christer Adelsbo, born 1962, is a Swedish social democratic politician who has been a member of the Riksdag since 2002 *Christer Basma (born 1972), Norwegian football coach and defender * Christer Björkman (born 1957), Swedish singer * Christer Boucht (1911–2009), Finnish-Swedish lawyer, adventure traveller and writer *Christer Boustedt (1939–1986), Swedish musician and actor *Christer Dahl (born 1940), ... [...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] |
Euclidean Space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any positive integer dimension (mathematics), dimension, including the three-dimensional space and the ''Euclidean plane'' (dimension two). The qualifier "Euclidean" is used to distinguish Euclidean spaces from other spaces that were later considered in physics and modern mathematics. Ancient History of geometry#Greek geometry, Greek geometers introduced Euclidean space for modeling the physical space. Their work was collected by the Greek mathematics, ancient Greek mathematician Euclid in his ''Elements'', with the great innovation of ''mathematical proof, proving'' all properties of the space as theorems, by starting from a few fundamental properties, called ''postulates'', which either were considered as eviden ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Half-space (geometry)
In geometry, a half-space is either of the two parts into which a plane divides the three-dimensional Euclidean space. If the space is two-dimensional, then a half-space is called a half-plane (open or closed). A half-space in a one-dimensional space is called a ''half-line'' or '' ray''. More generally, a half-space is either of the two parts into which a hyperplane divides an affine space. That is, the points that are not incident to the hyperplane are partitioned into two convex sets (i.e., half-spaces), such that any subspace connecting a point in one set to a point in the other must intersect the hyperplane. A half-space can be either ''open'' or ''closed''. An open half-space is either of the two open sets produced by the subtraction of a hyperplane from the affine space. A closed half-space is the union of an open half-space and the hyperplane that defines it. A half-space may be specified by a linear inequality, derived from the linear equation that specifies the defin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minkowski Content
The Minkowski content (named after Hermann Minkowski), or the boundary measure, of a set is a basic concept that uses concepts from geometry and measure theory to generalize the notions of length of a smooth curve in the plane, and area of a smooth surface in space, to arbitrary measurable sets. It is typically applied to fractal boundaries of domains in the Euclidean space, but it can also be used in the context of general metric measure spaces. It is related to, although different from, the Hausdorff measure. Definition For A \subset \mathbb^, and each integer ''m'' with 0 \leq m \leq n, the ''m''-dimensional upper Minkowski content is :M^(A) = \limsup_ \frac and the ''m''-dimensional lower Minkowski content is defined as :M_*^m(A) = \liminf_ \frac where \alpha(n-m)r^ is the volume of the (''n''−''m'')-ball of radius r and \mu is an n-dimensional Lebesgue measure. If the upper and lower ''m''-dimensional Minkowski content of ''A'' are equal, then their common v ... [...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] |
Paul Lévy (mathematician)
Paul Pierre Lévy (15 September 1886 – 15 December 1971) was a French mathematician who was active especially in probability theory, introducing fundamental concepts such as local time, stable distributions and characteristic functions. Lévy processes, Lévy flights, Lévy measures, Lévy's constant, the Lévy distribution, the Lévy area, the Lévy arcsine law, and the fractal Lévy C curve are named after him. Biography Lévy was born in Paris to a Jewish family which already included several mathematicians. His father Lucien Lévy was an examiner at the École Polytechnique. Lévy attended the École Polytechnique and published his first paper in 1905, at the age of nineteen, while still an undergraduate, in which he introduced the Lévy–Steinitz theorem. His teacher and advisor was Jacques Hadamard. After graduation, he spent a year in military service and then studied for three years at the École des Mines, where he became a professor in 1913. During Worl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spherical Isoperimetric Inequality
In mathematics, the isoperimetric inequality is a geometric inequality involving the perimeter of a set and its volume. In n-dimensional space \R^n the inequality lower bounds the surface area or perimeter \operatorname(S) of a set S\subset\R^n by its volume \operatorname(S), :\operatorname(S)\geq n \operatorname(S)^ \, \operatorname(B_1)^, where B_1\subset\R^n is a unit sphere. The equality holds only when S is a sphere in \R^n. On a plane, i.e. when n=2, the isoperimetric inequality relates the square of the circumference of a closed curve and the area of a plane region it encloses. ''Isoperimetric'' literally means "having the same perimeter". Specifically in \R ^2, the isoperimetric inequality states, for the length ''L'' of a closed curve and the area ''A'' of the planar region that it encloses, that : L^2 \ge 4\pi A, and that equality holds if and only if the curve is a circle. The isoperimetric problem is to determine a plane figure of the largest possible area whose ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sergey Bobkov
Sergey Bobkov (Russian: Cергей Германович Бобков; born March 15, 1961) is a mathematician. Currently Bobkov is a professor at the University of Minnesota, Twin Cities. He was born in Vorkuta ( Komi Republic, Russia) and graduated from the Department of Mathematics and Mechanics in Leningrad State University. In 1988 he earned PhD in Mathematics and Physics (under direction of Vladimir N. Sudakov, Steklov Institute of Mathematics) and in 1997 earned his Doctor of Science. During 1998–2000 Bobkov held positions at Syktyvkar State University, Russia. From 1995 to 1996 he was an Alexander von Humboldt Fellow at Bielefeld University, Germany. He spent the summers of 2001 and 2002 as an EPSRC Fellow at Imperial College London, UK. Bobkov was awarded a Simons Fellowship (2012) and Humboldt Research Award (2014). Bobkov is known for research in mathematics on the border of probability theory, analysis, convex geometry and information theory. He has achieved im ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semigroup
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplicatively: ''x''·''y'', or simply ''xy'', denotes the result of applying the semigroup operation to the ordered pair . Associativity is formally expressed as that for all ''x'', ''y'' and ''z'' in the semigroup. Semigroups may be considered a special case of magmas, where the operation is associative, or as a generalization of groups, without requiring the existence of an identity element or inverses. The closure axiom is implied by the definition of a binary operation on a set. Some authors thus omit it and specify three axioms for a group and only one axiom (associativity) for a semigroup. As in the case of groups or magmas, the semigroup operation need not be commutative, so ''x''·''y'' is not necessarily equal to ''y''·''x''; a well-known example of an operation that is as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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