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Dimension Doubling Theorem
In probability theory, the dimension doubling theorems are two results about the Hausdorff dimension of an image of a Brownian motion. In their core both statements say, that the dimension of a set A under a Brownian motion doubles almost surely In probability theory, an event is said to happen almost surely (sometimes abbreviated as a.s.) if it happens with probability 1 (with respect to the probability measure). In other words, the set of outcomes on which the event does not occur ha .... The first result is due to Henry P. McKean jr and hence called McKean's theorem (1955). The second theorem is a refinement of McKean's result and called Kaufman's theorem (1969) since it was proven by Robert Kaufman. Dimension doubling theorems Let (\Omega,\mathcal,P) be a probability space. For a d-dimensional Brownian motion W(t) and a set A\subset [0,\infty) we define the image of A under W, i.e. :W(A):=\\subset \R^d. McKean's theorem Let W(t) be a Brownian motion in dimension d\ge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Theory
Probability theory or probability calculus 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 of probability, axioms. Typically these axioms formalise probability in terms of a probability space, which assigns a measure (mathematics), 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 (probability theory), event. Central subjects in probability theory include discrete and continuous random variables, probability distributions, and stochastic processes (which provide mathematical abstractions of determinism, non-deterministic or uncertain processes or measured Quantity, quantities that may either be single occurrences or evolve over time in a random fashion). Although it is no ... [...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 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 computation of di ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Image (mathematics)
In mathematics, for a function f: X \to Y, the image of an input value x is the single output value produced by f when passed x. The preimage of an output value y is the set of input values that produce y. More generally, evaluating f at each Element (mathematics), element of a given subset A of its Domain of a function, domain X produces a set, called the "image of A under (or through) f". Similarly, the inverse image (or preimage) of a given subset B of the codomain Y is the set of all elements of X that map to a member of B. The image of the function f is the set of all output values it may produce, that is, the image of X. The preimage of f is the preimage of the codomain Y. Because it always equals X (the domain of f), it is rarely used. Image and inverse image may also be defined for general Binary relation#Operations, binary relations, not just functions. Definition The word "image" is used in three related ways. In these definitions, f : X \to Y is a Function (mat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brownian Motion
Brownian motion is the random motion of particles suspended in a medium (a liquid or a gas). The traditional mathematical formulation of Brownian motion is that of the Wiener process, which is often called Brownian motion, even in mathematical sources. This motion pattern typically consists of Randomness, random fluctuations in a particle's position inside a fluid sub-domain, followed by a relocation to another sub-domain. Each relocation is followed by more fluctuations within the new closed volume. This pattern describes a fluid at thermal equilibrium, defined by a given temperature. Within such a fluid, there exists no preferential direction of flow (as in transport phenomena). More specifically, the fluid's overall Linear momentum, linear and Angular momentum, angular momenta remain null over time. The Kinetic energy, kinetic energies of the molecular Brownian motions, together with those of molecular rotations and vibrations, sum up to the caloric component of a fluid's in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Almost Surely
In probability theory, an event is said to happen almost surely (sometimes abbreviated as a.s.) if it happens with probability 1 (with respect to the probability measure). In other words, the set of outcomes on which the event does not occur has probability 0, even though the set might not be empty. The concept is analogous to the concept of "almost everywhere" in measure theory. In probability experiments on a finite sample space with a non-zero probability for each outcome, there is no difference between ''almost surely'' and ''surely'' (since having a probability of 1 entails including all the sample points); however, this distinction becomes important when the sample space is an infinite set, because an infinite set can have non-empty subsets of probability 0. Some examples of the use of this concept include the strong and uniform versions of the law of large numbers, the continuity of the paths of Brownian motion, and the infinite monkey theorem. The terms almost certai ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Henry McKean
Henry P. McKean, Jr. (December 10, 1930 – April 20, 2024) was an American mathematician at the Courant Institute in New York University. He worked in various areas of analysis. He obtained his PhD in 1955 from Princeton University under William Feller. McKean was elected to the National Academy of Sciences in 1980. In 2007 he was awarded the Leroy P. Steele Prize for his life's work. In 1978 he was an invited speaker at the International Congress of Mathematicians in Helsinki (''Algebraic curves of infinite genus arising in the theory of nonlinear waves''). In 2012 he became a fellow of the American Mathematical Society. His doctoral students include Michael Arbib, Luigi Chierchia, Donald A. Dawson, Harry Dym, Daniel Stroock, Eugene Trubowitz, Victor Moll and Pierre van Moerbeke and Uri Keich. McKean died on April 20, 2024, at the age of 93. Works Selected articles * * * * * * * Books *with Kiyosi Itô: ''Diffusion processes and their sample paths.'' Springer 1965. * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rober Kaufman (mathematician)
Rober may refer to: * Röber, a list of people with the surname Röber, Rober or Roeber * Rober (footballer), Spanish footballer Roberto González Bayón (born 2001) * Roberto Rober Correa (born 1992), Spanish footballer * Rober Eryol (1930–2000), Turkish footballer * Roberto Robert Ibáñez (born 1993), Spanish footballer * Roberto Róber Pier Roberto "Róber" Suárez Pier (born 16 February 1995) is a Spanish professional footballer who plays as a centre-back for Sporting de Gijón. Club career Deportivo Born in Oleiros, Galicia, Pier was a youth product of Deportivo de La Coruña. ... (born 1995), Spanish footballer * Roberto Rober Sierra (born 1996), Spanish footballer {{nickname Hypocorisms ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Null Set
In mathematical analysis, a null set is a Lebesgue measurable set of real numbers that has measure zero. This can be characterized as a set that can be covered by a countable union of intervals of arbitrarily small total length. The notion of null set should not be confused with the empty set as defined in set theory. Although the empty set has Lebesgue measure zero, there are also non-empty sets which are null. For example, any non-empty countable set of real numbers has Lebesgue measure zero and therefore is null. More generally, on a given measure space M = (X, \Sigma, \mu) a null set is a set S \in \Sigma such that \mu(S) = 0. Examples Every finite or countably infinite subset of the real numbers is a null set. For example, the set of natural numbers , the set of rational numbers and the set of algebraic numbers are all countably infinite and therefore are null sets when considered as subsets of the real numbers. The Cantor set is an example of an uncountable ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wiener Process
In mathematics, the Wiener process (or Brownian motion, due to its historical connection with Brownian motion, the physical process of the same name) is a real-valued continuous-time stochastic process discovered by Norbert Wiener. It is one of the best known Lévy processes (càdlàg stochastic processes with stationary increments, stationary independent increments). It occurs frequently in pure and applied mathematics, economy, economics, quantitative finance, evolutionary biology, and physics. The Wiener process plays an important role in both pure and applied mathematics. In pure mathematics, the Wiener process gave rise to the study of continuous time martingale (probability theory), martingales. It is a key process in terms of which more complicated stochastic processes can be described. As such, it plays a vital role in stochastic calculus, diffusion processes and even potential theory. It is the driving process of Schramm–Loewner evolution. In applied mathematics, the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
