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Bobkov's Inequality
In probability theory, Bobkov's inequality is a functional isoperimetric inequality for the canonical Gaussian measure. It generalizes the Gaussian isoperimetric inequality. The equation was proven in 1997 by the Russian mathematician Sergey Bobkov. Bobkov's inequality Notation: Let *\gamma^n(dx)=(2\pi)^e^d^nx be the canonical Gaussian measure on \R^n with respect to the Lebesgue measure In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides wit ..., *\phi(x)=(2\pi)^e^ be the one dimensional canonical Gaussian density *\Phi(t)=\gamma^1 \infty,t/math> the cumulative distribution function *I(t):=\phi(\Phi^(t)) be a function I(t): ,1to ,1/math> that vanishes at the end points \lim\limits_ I(t)=\lim\limits_ I(t)=0. Statement For every locally Lipschitz continuous (or smooth) function ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms. Typically these axioms formalise probability in terms of a probability space, which assigns a measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the sample space. Any specified subset of the sample space is called an event. Central subjects in probability theory include discrete and continuous random variables, probability distributions, and stochastic processes (which provide mathematical abstractions of non-deterministic or uncertain processes or measured quantities that may either be single occurrences or evolve over time in a random fashion). Although it is not possible to perfectly predict random events, much can be said about their behavior. Two major results in probability ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Functional (mathematics)
In mathematics, a functional (as a noun) is a certain type of function. The exact definition of the term varies depending on the subfield (and sometimes even the author). * In linear algebra, it is synonymous with linear forms, which are linear mapping from a vector space V into its Field (mathematics), field of scalars (that is, an element of the dual space V^*) "Let ''E'' be a free module over a commutative ring ''A''. We view ''A'' as a free module of rank 1 over itself. By the dual module ''E''∨ of ''E'' we shall mean the module Hom(''E'', ''A''). Its elements will be called functionals. Thus a functional on ''E'' is an ''A''-linear map ''f'' : ''E'' → ''A''." * In functional analysis and related fields, it refers more generally to a mapping from a space X into the field of Real numbers, real or complex numbers. "A numerical function ''f''(''x'') defined on a normed linear space ''R'' will be called a ''functional''. A functional ''f''(''x'') is said to be ''linear'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 ... [...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] |
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] |
Russia
Russia (, , ), or the Russian Federation, is a List of transcontinental countries, transcontinental country spanning Eastern Europe and North Asia, Northern Asia. It is the List of countries and dependencies by area, largest country in the world, with its internationally recognised territory covering , and encompassing one-eighth of Earth's inhabitable landmass. Russia extends across Time in Russia, eleven time zones and shares Borders of Russia, land boundaries with fourteen countries, more than List of countries and territories by land borders, any other country but China. It is the List of countries and dependencies by population, world's ninth-most populous country and List of European countries by population, Europe's most populous country, with a population of 146 million people. The country's capital and List of cities and towns in Russia by population, largest city is Moscow, the List of European cities by population within city limits, largest city entirely within E ... [...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 Informati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lebesgue Measure
In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides with the standard measure of length, area, or volume. In general, it is also called ''n''-dimensional volume, ''n''-volume, or simply volume. It is used throughout real analysis, in particular to define Lebesgue integration. Sets that can be assigned a Lebesgue measure are called Lebesgue-measurable; the measure of the Lebesgue-measurable set ''A'' is here denoted by ''λ''(''A''). Henri Lebesgue described this measure in the year 1901, followed the next year by his description of the Lebesgue integral. Both were published as part of his dissertation in 1902. Definition For any interval I = ,b/math>, or I = (a, b), in the set \mathbb of real numbers, let \ell(I)= b - a denote its length. For any subset E\subseteq\mathbb, the Lebesgue oute ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lipschitz Continuity
In mathematical analysis, Lipschitz continuity, named after German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there exists a real number such that, for every pair of points on the graph of this function, the absolute value of the slope of the line connecting them is not greater than this real number; the smallest such bound is called the ''Lipschitz constant'' of the function (or '' modulus of uniform continuity''). For instance, every function that has bounded first derivatives is Lipschitz continuous. In the theory of differential equations, Lipschitz continuity is the central condition of the Picard–Lindelöf theorem which guarantees the existence and uniqueness of the solution to an initial value problem. A special type of Lipschitz continuity, called contraction, is used in the Banach fixed-point theorem. We have the following chain of strict inclus ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Smooth Function
In mathematical analysis, the smoothness of a function (mathematics), function is a property measured by the number of Continuous function, continuous Derivative (mathematics), derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if it is differentiable everywhere (hence continuous). At the other end, it might also possess derivatives of all Order of derivation, orders in its Domain of a function, domain, in which case it is said to be infinitely differentiable and referred to as a C-infinity function (or C^ function). Differentiability classes Differentiability class is a classification of functions according to the properties of their derivatives. It is a measure of the highest order of derivative that exists and is continuous for a function. Consider an open set U on the real line and a function f defined on U with real values. Let ''k'' be a non-negative integer. The function f is said to be of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dominique Bakry
Roger-Dominique Bakry (born 12 December 1954), known as Dominique Bakry, is a French mathematician, a professor at the Université Paul-Sabatier in Toulouse, and a senior member of Institut Universitaire de France. Bakry graduated from , and prepared his PhD under the advisory of Paul-André Meyer and Marc Yor. Before coming to Toulouse, he was chargé de recherches at CNRS in Université Louis Pasteur of Strasbourg. His scientific work is at the interface of Analysis, Probability, and Geometry. His most influential works concern Riesz transforms and Markov semigroups. He gave his name to the Bakry-Émery criterion, developed in collaboration with Michel Émery Michel may refer to: * Michel (name), a given name or surname of French origin (and list of people with the name) * Míchel (nickname), a nickname (a list of people with the nickname, mainly Spanish footballers) * Míchel (footballer, born 1963), ... and published in 1984, and linked more generally to the curvature-dimens ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Michel Ledoux
Michel Ledoux (born 1958) is a French mathematician, specializing in probability theory. He is a professor at the University of Toulouse. Ledoux received in 1985 his PhD from the University of Strasbourg with thesis ''Propriétés limites des variables aléatoires vectorielles'' which was made under the supervision of Xavier Fernique. He has done important research on the isoperimetric inequality in analysis and probability theory. In 2010 he received the Servant Prize of the French Academy of Sciences. In 2014 he was an invited speaker at the International Congress of Mathematicians This is a list of International Congresses of Mathematicians Plenary and Invited Speakers. Being invited to talk at an International Congress of Mathematicians has been called "the equivalent, in this community, of an induction to a hall of fame." ... in Seoul and gave a talk ''Heat flows, geometric and functional inequalities''. Selected publications * * 2nd edition 2002 * * * References E ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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