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Probabilistic Proofs Of Non-probabilistic Theorems
Probability theory routinely uses results from other fields of mathematics (mostly, analysis). The opposite cases, collected below, are relatively rare; however, probability theory is used systematically in combinatorics via the probabilistic method. They are particularly used for non-constructive proofs. Analysis * Normal numbers exist. Moreover, computable normal numbers exist. These non-probabilistic existence theorems follow from probabilistic results: (a) a number chosen at random (uniformly on (0,1)) is normal almost surely (which follows easily from the strong law of large numbers); (b) some probabilistic inequalities behind the strong law. The existence of a normal number follows from (a) immediately. The proof of the existence of computable normal numbers, based on (b), involves additional arguments. All known proofs use probabilistic arguments. * Dvoretzky's theorem which states that high-dimensional convex bodies have ball-like slices is proved probabilistically. No determ ... [...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] |
The Doctrine Of Chances
''The Doctrine of Chances'' was the first textbook on probability theory, written by 18th-century French mathematician Abraham de Moivre and first published in 1718.. De Moivre wrote in English because he resided in England at the time, having fled France to escape the persecution of Huguenots. The book's title came to be synonymous with ''probability theory'', and accordingly the phrase was used in Thomas Bayes' famous posthumous paper ''An Essay towards solving a Problem in the Doctrine of Chances'', wherein a version of Bayes' theorem was first introduced. Editions The full title of the first edition was ''The doctrine of chances: or, a method for calculating the probabilities of events in play''; it was published in 1718, by W. Pearson, and ran for 175 pages. Published in 1738 by Woodfall and running for 258 pages, the second edition of de Moivre's book introduced the concept of normal distributions as approximations to binomial distributions. In effect de Moivre proved a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maximum-minimums Identity
In mathematics, the maximum-minimums identity is a relation between the maximum element of a set ''S'' of ''n'' numbers and the minima of the 2''n'' − 1 non-empty subsets of ''S''. Let ''S'' = . The identity states that :\begin \max\ & = \sum_^n x_i - \sum_\min\ +\sum_\min\ - \cdots \\ & \qquad \cdots + \left(-1\right)^\min\,\end or conversely :\begin \min\ & = \sum_^n x_i - \sum_\max\ +\sum_\max\ - \cdots \\ & \qquad \cdots + \left(-1\right)^\max\. \end For a probabilistic proof, see the reference. See also * Inclusion–exclusion principle In combinatorics, a branch of mathematics, the inclusion–exclusion principle is a counting technique which generalizes the familiar method of obtaining the number of elements in the union of two finite sets; symbolically expressed as : , A \cup ... * Maxima and minima#In relation to sets References * {{cite book , last = Ross , first = Sheldon , title = A First Course in Probability , publisher = Prentice ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Martingale Convergence
In mathematicsspecifically, in the theory of stochastic processesDoob's martingale convergence theorems are a collection of results on the limits of supermartingales, named after the American mathematician Joseph L. Doob. Informally, the martingale convergence theorem typically refers to the result that any supermartingale satisfying a certain boundedness condition must converge. One may think of supermartingales as the random variable analogues of non-increasing sequences; from this perspective, the martingale convergence theorem is a random variable analogue of the monotone convergence theorem, which states that any bounded monotone sequence converges. There are symmetric results for submartingales, which are analogous to non-decreasing sequences. Statement for discrete-time martingales A common formulation of the martingale convergence theorem for discrete-time martingales is the following. Let X_1, X_2, X_3, \dots be a supermartingale. Suppose that the supermartingale is bound ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lipschitz Function
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] |
Picard Theorem
In complex analysis, Picard's great theorem and Picard's little theorem are related theorems about the range of an analytic function. They are named after Émile Picard. The theorems Little Picard Theorem: If a function f: \mathbb \to\mathbb is entire and non-constant, then the set of values that f(z) assumes is either the whole complex plane or the plane minus a single point. Sketch of Proof: Picard's original proof was based on properties of the modular lambda function, usually denoted by λ, and which performs, using modern terminology, the holomorphic universal covering of the twice punctured plane by the unit disc. This function is explicitly constructed in the theory of elliptic functions. If ''f'' omits two values, then the composition of ''f'' with the inverse of the modular function maps the plane into the unit disc which implies that ''f'' is constant by Liouville's theorem. This theorem is a significant strengthening of Liouville's theorem which states that the i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Basel Problem
The Basel problem is a problem in mathematical analysis with relevance to number theory, concerning an infinite sum of inverse squares. It was first posed by Pietro Mengoli in 1650 and solved by Leonhard Euler in 1734, and read on 5 December 1735 in ''The Saint Petersburg Academy of Sciences''. Since the problem had withstood the attacks of the leading mathematicians of the day, Euler's solution brought him immediate fame when he was twenty-eight. Euler generalised the problem considerably, and his ideas were taken up years later by Bernhard Riemann in his seminal 1859 paper "On the Number of Primes Less Than a Given Magnitude", in which he defined his zeta function and proved its basic properties. The problem is named after Basel, hometown of Euler as well as of the Bernoulli family who unsuccessfully attacked the problem. The Basel problem asks for the precise summation of the reciprocals of the squares of the natural numbers, i.e. the precise sum of the infinite series: \sum_ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boundary Harnack Principle
Boundary or Boundaries may refer to: * Border, in political geography Entertainment * ''Boundaries'' (2016 film), a 2016 Canadian film * ''Boundaries'' (2018 film), a 2018 American-Canadian road trip film *Boundary (cricket), the edge of the playing field, or a scoring shot where the ball is hit to or beyond that point *Boundary (sports), the sidelines of a field Mathematics and physics *Boundary (topology), the closure minus the interior of a subset of a topological space; an edge in the topology of manifolds, as in the case of a 'manifold with boundary' *Boundary (graph theory), the vertices of edges between a subgraph and the rest of a graph *Boundary (chain complex), its abstractization in chain complexes *Boundary value problem, a differential equation together with a set of additional restraints called the boundary conditions * Boundary (thermodynamics), the edge of a thermodynamic system across which heat, mass, or work can flow Psychology and sociology *Personal boundari ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Doob's Martingale Convergence Theorems
In mathematicsspecifically, in the theory of stochastic processesDoob's martingale convergence theorems are a collection of results on the limits of supermartingales, named after the American mathematician Joseph L. Doob. Informally, the martingale convergence theorem typically refers to the result that any supermartingale satisfying a certain boundedness condition must converge. One may think of supermartingales as the random variable analogues of non-increasing sequences; from this perspective, the martingale convergence theorem is a random variable analogue of the monotone convergence theorem, which states that any bounded monotone sequence converges. There are symmetric results for submartingales, which are analogous to non-decreasing sequences. Statement for discrete-time martingales A common formulation of the martingale convergence theorem for discrete-time martingales is the following. Let X_1, X_2, X_3, \dots be a supermartingale. Suppose that the supermartingale is b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ivan Privalov
Ivan Ivanovich Privalov (russian: Ива́н Ива́нович Привáлов; 11 February 1891 – 13 July 1941) was a Russian mathematician best known for his work on analytic functions. Biography Privalov graduated from Moscow State University (MSU) in 1913 studying under Dimitri Egorov and Nikolai Lusin. He obtained his master's degree from MSU in 1916 and became professor at Imperial Saratov University (1917—1922). In 1922 he was appointed as Professor at MSU and worked there for the rest of his life. Corresponding member of the USSR Academy of Sciences (since 1939). Member of the French Mathematical Society (Société Mathématique de France) and the Mathematical Circle of Palermo (Circolo Matematico di Palermo The Circolo Matematico di Palermo (Mathematical Circle of Palermo) is an Italian List of mathematical societies, mathematical society, founded in Palermo by Sicilian geometer Giovanni Battista Guccia, Giovanni B. Guccia in 1884. [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harmonic Function
In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f: U \to \mathbb R, where is an open subset of that satisfies Laplace's equation, that is, : \frac + \frac + \cdots + \frac = 0 everywhere on . This is usually written as : \nabla^2 f = 0 or :\Delta f = 0 Etymology of the term "harmonic" The descriptor "harmonic" in the name harmonic function originates from a point on a taut string which is undergoing harmonic motion. The solution to the differential equation for this type of motion can be written in terms of sines and cosines, functions which are thus referred to as ''harmonics''. Fourier analysis involves expanding functions on the unit circle in terms of a series of these harmonics. Considering higher dimensional analogues of the harmonics on the unit ''n''-sphere, one arrives at the spherical harmonics. These functions satisfy Laplace's equation and over time "harmonic" ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Holomorphic Function
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivative in a neighbourhood is a very strong condition: it implies that a holomorphic function is infinitely differentiable and locally equal to its own Taylor series (''analytic''). Holomorphic functions are the central objects of study in complex analysis. Though the term ''analytic function'' is often used interchangeably with "holomorphic function", the word "analytic" is defined in a broader sense to denote any function (real, complex, or of more general type) that can be written as a convergent power series in a neighbourhood of each point in its domain. That all holomorphic functions are complex analytic functions, and vice versa, is a major theorem in complex analysis. Holomorphic functions are also sometimes referred to as ''regular fu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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