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Young's Inequality For Products
In mathematics, Young's inequality for products is a mathematical inequality about the product of two numbers. The inequality is named after William Henry Young and should not be confused with Young's convolution inequality. Young's inequality for products can be used to prove Hölder's inequality. It is also widely used to estimate the norm of nonlinear terms in PDE theory, since it allows one to estimate a product of two terms by a sum of the same terms raised to a power and scaled. Standard version for conjugate Hölder exponents The standard form of the inequality is the following, which can be used to prove Hölder's inequality. A second proof is via Jensen's inequality. Yet another proof is to first prove it with b = 1 an then apply the resulting inequality to \tfrac . The proof below illustrates also why Hölder conjugate exponent is the only possible parameter that makes Young's inequality hold for all non-negative values. The details follow: Young's inequa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Young
Young may refer to: * Offspring, the product of reproduction of a new organism produced by one or more parents * Youth, the time of life when one's age is low, often meaning the time between childhood and adulthood Music * The Young, an American rock band * ''Young'', an EP by Charlotte Lawrence, 2018 Songs * "Young" (Baekhyun and Loco song), 2018 * "Young" (The Chainsmokers song), 2017 * "Young" (Hollywood Undead song), 2009 * "Young" (Kenny Chesney song), 2002 * "Young" (Place on Earth song), 2018 * "Young" (Tulisa song), 2012 * "Young", by Ella Henderson, 2019 * "Young", by Lil Wayne from '' Dedication 6'', 2017 * "Young", by Nickel Creek from ''This Side'', 2002 * "Young", by Sam Smith from '' Love Goes'', 2020 * "Young", by Silkworm from '' Italian Platinum'', 2002 * "Young", by Vacations (band), 2016 * "Young", by Vallis Alps, 2015 * "Young", by Pixey, 2016 People Surname * Young (surname) Given name * Young (Korean name), Korean unisex given name and name elem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Loewner Order
In mathematics, Loewner order is the partial order defined by the convex cone of positive semi-definite matrices. This order is usually employed to generalize the definitions of monotone and concave/convex scalar functions to monotone and concave/convex Hermitian valued functions. These functions arise naturally in matrix and operator theory and have applications in many areas of physics and engineering. Definition Let ''A'' and ''B'' be two Hermitian matrices of order ''n''. We say that ''A ≥ B'' if ''A'' − ''B'' is positive semi-definite. Similarly, we say that ''A > B'' if ''A'' − ''B'' is positive definite. Although it is commonly discussed on matrices (as a finite-dimensional case), the Loewner order is also well-defined on operators (an infinite-dimensional case) in the analogous way. Properties When ''A'' and ''B'' are real scalars (i.e. ''n'' = 1), the Loewner order reduces to the usual ordering of R. Although some familiar proper ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
PlanetMath
PlanetMath is a free content, free, collaborative, mathematics online encyclopedia. Intended to be comprehensive, the project is currently hosted by the University of Waterloo. The site is owned by a US-based nonprofit corporation, "PlanetMath.org, Ltd". PlanetMath was started when the popular free online mathematics encyclopedia MathWorld was temporarily taken offline for 12 months by a court injunction as a result of the CRC Press lawsuit against the Wolfram Research company and its employee (and MathWorld's author) Eric Weisstein. Materials The main PlanetMath focus is on encyclopedia, encyclopedic entries. It formerly operated a self-hosted forum, but now encourages discussion via Gitter. An all-inclusive PlanetMath ''Free Encyclopedia'' book of 2,300 pages is available for the encyclopedia contents up to 2006 as a free download PDF file. Content development model PlanetMath implements a specific content creation system called ''authority model''. An author who starts a ne ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sanov's Theorem
In mathematics and information theory, Sanov's theorem gives a bound on the probability of observing an atypical sequence of samples from a given probability distribution. In the language of large deviations theory, Sanov's theorem identifies the rate function for large deviations of the empirical measure of a sequence of i.i.d. random variables. Let ''A'' be a set of probability distributions over an alphabet ''X'', and let ''q'' be an arbitrary distribution over ''X'' (where ''q'' may or may not be in ''A''). Suppose we draw ''n'' i.i.d. samples from ''q'', represented by the vector x^n = (x_1, x_2, \ldots, x_n). Then, we have the following bound on the probability that the empirical measure \hat_ of the samples falls within the set ''A'': :q^n(\hat_\in A) \le (n+1)^ 2^, where * q^n is the joint probability distribution on X^n, and * p^* is the information projection of ''q'' onto ''A''. * D_(P \, Q), the KL divergence, is given by: D_(P \, Q) = \sum_ P(x) \log \frac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rate Function
In mathematics — specifically, in large deviations theory — a rate function is a function used to quantify the probabilities of rare events. Such functions are used to formulate large deviation principles. A large deviation principle quantifies the asymptotic probability of rare events for a sequence of probabilities. A rate function is also called a Cramér function, after the Swedish probabilist Harald Cramér. Definitions Rate function An extended real-valued function I: X \to , +\infty/math> defined on a Hausdorff topological space X is said to be a rate function if it is not identically +\infty and is lower semi-continuous ''i.e.'' all the sub-level sets :\ \mbox c \geq 0 are closed in X. If, furthermore, they are compact, then I is said to be a good rate function. A family of probability measures (\mu_)_ on X is said to satisfy the large deviation principle with rate function I: X \to , +\infty) (and rate 1/\delta) if, for every closed set F \subseteq X a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Relative Entropy
Relative may refer to: General use *Kinship and family, the principle binding the most basic social units of society. If two people are connected by circumstances of birth, they are said to be ''relatives''. Philosophy *Relativism, the concept that points of view have no absolute truth or validity, having only relative, subjective value according to differences in perception and consideration, or relatively, as in the relative value of an object to a person * Relative value (philosophy) Economics * Relative value (economics) Popular culture Film and television * ''Relatively Speaking'' (1965 play), 1965 British play * ''Relatively Speaking'' (game show), late 1980s television game show * ''Everything's Relative'' (episode)#Yu-Gi-Oh! (Yu-Gi-Oh! Duel Monsters), 2000 Japanese anime ''Yu-Gi-Oh! Duel Monsters'' episode *'' Relative Values'', 2000 film based on the play of the same name. *'' It's All Relative'', 2003-4 comedy television series *''Intelligence is Relative'', tag lin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Large Deviations Theory
In probability theory, the theory of large deviations concerns the asymptotic behaviour of remote tails of sequences of probability distributions. While some basic ideas of the theory can be traced to Laplace, the formalization started with insurance mathematics, namely ruin theory with Cramér and Lundberg. A unified formalization of large deviation theory was developed in 1966, in a paper by Varadhan. Large deviations theory formalizes the heuristic ideas of ''concentration of measures'' and widely generalizes the notion of convergence of probability measures. Roughly speaking, large deviations theory concerns itself with the exponential decline of the probability measures of certain kinds of extreme or ''tail'' events. Introductory examples An elementary example Consider a sequence of independent tosses of a fair coin. The possible outcomes could be heads or tails. Let us denote the possible outcome of the i-th trial by where we encode head as 1 and tail as 0. Now let ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dual Pair
Dual or Duals may refer to: Paired/two things * Dual (mathematics), a notion of paired concepts that mirror one another ** Dual (category theory), a formalization of mathematical duality *** see more cases in :Duality theories * Dual number, a number system used in automatic differentiation * Dual (grammatical number), a grammatical category used in some languages * Dual county, a Gaelic games county which competes in both Gaelic football and hurling * Dual diagnosis, a psychiatric diagnosis of co-occurrence of substance abuse and a mental problem * Dual fertilization, simultaneous application of a P-type and N-type fertilizer * Dual impedance, electrical circuits that are the dual of each other * Dual SIM cellphone supporting use of two SIMs * Aerochute International Dual a two-seat Australian powered parachute design Acronyms and other uses * Dual (brand), a manufacturer of Hifi equipment * DUAL (cognitive architecture), an artificial intelligence design model * DUA ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dual Space
In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V,'' together with the vector space structure of pointwise addition and scalar multiplication by constants. The dual space as defined above is defined for all vector spaces, and to avoid ambiguity may also be called the . When defined for a topological vector space, there is a subspace of the dual space, corresponding to continuous linear functionals, called the continuous dual space. Dual vector spaces find application in many branches of mathematics that use vector spaces, such as in tensor analysis with finite-dimensional vector spaces. When applied to vector spaces of functions (which are typically infinite-dimensional), dual spaces are used to describe measures, distributions, and Hilbert spaces. Consequently, the dual space is an important concept in functional analysis. Early terms for ''dual'' include ''polarer Raum'' ahn 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Legendre Transformation
In mathematics, the Legendre transformation (or Legendre transform), first introduced by Adrien-Marie Legendre in 1787 when studying the minimal surface problem, is an involutive transformation on real-valued functions that are convex on a real variable. Specifically, if a real-valued multivariable function is convex on one of its independent real variables, then the Legendre transform with respect to this variable is applicable to the function. In physical problems, the Legendre transform is used to convert functions of one quantity (such as position, pressure, or temperature) into functions of the conjugate quantity (momentum, volume, and entropy, respectively). In this way, it is commonly used in classical mechanics to derive the Hamiltonian formalism out of the Lagrangian formalism (or vice versa) and in thermodynamics to derive the thermodynamic potentials, as well as in the solution of differential equations of several variables. For sufficiently smooth functions on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Conjugate
In mathematics and mathematical optimization, the convex conjugate of a function is a generalization of the Legendre transformation which applies to non-convex functions. It is also known as Legendre–Fenchel transformation, Fenchel transformation, or Fenchel conjugate (after Adrien-Marie Legendre and Werner Fenchel). The convex conjugate is widely used for constructing the dual problem in optimization theory, thus generalizing Lagrangian duality. Definition Let X be a real topological vector space and let X^ be the dual space to X. Denote by :\langle \cdot , \cdot \rangle : X^ \times X \to \mathbb the canonical dual pairing, which is defined by \left\langle x^*, x \right\rangle \mapsto x^* (x). For a function f : X \to \mathbb \cup \ taking values on the extended real number line, its is the function :f^ : X^ \to \mathbb \cup \ whose value at x^* \in X^ is defined to be the supremum: :f^ \left( x^ \right) := \sup \left\, or, equivalently, in terms of the in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inverse Function
In mathematics, the inverse function of a function (also called the inverse of ) is a function that undoes the operation of . The inverse of exists if and only if is bijective, and if it exists, is denoted by f^ . For a function f\colon X\to Y, its inverse f^\colon Y\to X admits an explicit description: it sends each element y\in Y to the unique element x\in X such that . As an example, consider the real-valued function of a real variable given by . One can think of as the function which multiplies its input by 5 then subtracts 7 from the result. To undo this, one adds 7 to the input, then divides the result by 5. Therefore, the inverse of is the function f^\colon \R\to\R defined by f^(y) = \frac . Definitions Let be a function whose domain is the set , and whose codomain is the set . Then is ''invertible'' if there exists a function from to such that g(f(x))=x for all x\in X and f(g(y))=y for all y\in Y. If is invertible, then there is exactly one functi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
