In mathematics, specifically functional analysis, Mercer's theorem is a representation of a symmetric positive-definite function on a square as a sum of a convergent sequence of product functions. This theorem, presented in, is one of the most notable results of the work of James Mercer. It is an important theoretical tool in the theory of integral equations; it is used in the Hilbert space theory of stochastic processes, for example the Karhunen–Loève theorem; and it is also used to characterize a symmetric positive semi-definite kernel.
In mathematics, integral equations are equations in which an unknown function appears under an integral sign.
This article itemizes the various lists of mathematics topics. Some of these lists link to hundreds of articles; some link only to a few. The template to the right includes links to alphabetical lists of all mathematical articles. This article brings together the same content organized in a manner better suited for browsing.
Otto Toeplitz was a German mathematician working in functional analysis.
Alberto Pedro Calderón was an Argentinian mathematician. His name is associated with the University of Buenos Aires, but first and foremost with the University of Chicago, where Calderón and his mentor, the analyst Antoni Zygmund, developed the theory of singular integral operators. This created the "Chicago School of (hard) Analysis". Calderón's work ranged over a wide variety of topics: from singular integral operators to partial differential equations, from interpolation theory to Cauchy integrals on Lipschitz curves, from ergodic theory to inverse problems in electrical prospection. Calderón's work has also had a powerful impact on practical applications including signal processing, geophysics, and tomography.
In mathematics, generalized functions, or distributions, are objects extending the notion of functions. There is more than one recognized theory. Generalized functions are especially useful in making discontinuous functions more like smooth functions, and describing discrete physical phenomena such as point charges. They are applied extensively, especially in physics and engineering.
In mathematics, a vector measure is a function defined on a family of sets and taking vector values satisfying certain properties. It is a generalization of the concept of finite measure, which takes nonnegative real values only.
Percy John Daniell was a pure and applied mathematician. In a series of papers published between 1918 and 1928, he developed and expanded a generalized theory of integration and differentiation, which is today known as the Daniell integral. In the setting of integration, he also worked on results that lead to the Daniell-Kolmogorov extension theorem in the theory of stochastic processes, independently of Andrey Kolmogorov. He was an Invited Speaker of the ICM in 1920 at Strasbourg.
Oskar Bolza was a German mathematician, and student of Felix Klein. He was born in Bad Bergzabern, Palatinate, then a district of Bavaria, known for his research in the calculus of variations, particularly influenced by Karl Weierstrass' 1879 lectures on the subject.
In mathematics, the Fox derivative is an algebraic construction in the theory of free groups which bears many similarities to the conventional derivative of calculus. The Fox derivative and related concepts are often referred to as the Fox calculus, or the free differential calculus. The Fox derivative was developed in a series of five papers by mathematician Ralph Fox, published in Annals of Mathematics beginning in 1953.
Hubert Stanley Wall was an American mathematician who worked primarily in the field of continued fractions. He is also known as one of the leading proponents of the Moore method of teaching.
In mathematics, a Barnes integral or Mellin–Barnes integral is a contour integral involving a product of gamma functions. They were introduced by Ernest William Barnes. They are closely related to generalized hypergeometric series.
Special functions are particular mathematical functions which have more or less established names and notations due to their importance in mathematical analysis, functional analysis, physics, or other applications.
In physics, quantum dynamics is the quantum version of classical dynamics. Quantum dynamics deals with the motions, and energy and momentum exchanges of systems whose behavior is governed by the laws of quantum mechanics. Quantum dynamics is relevant for burgeoning fields, such as quantum computing and atomic optics.
In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the x-axis. The Lebesgue integral extends the integral to a larger class of functions. It also extends the domains on which these functions can be defined.
In probability theory and statistics, Campbell's theorem or the Campbell–Hardy theorem is either a particular equation or set of results relating to the expectation of a function summed over a point process to an integral involving the mean measure of the point process, which allows for the calculation of expected value and variance of the random sum. One version of the theorem, also known as Campbell's formula, entails an integral equation for the aforementioned sum over a general point process, and not necessarily a Poisson point process. There also exist equations involving moment measures and factorial moment measures that are considered versions of Campbell's formula. All these results are employed in probability and statistics with a particular importance in the theory of point processes and queueing theory as well as the related fields stochastic geometry, continuum percolation theory, and spatial statistics.
In computing, a Digital Object Identifier or DOI is a persistent identifier or handle used to uniquely identify objects, standardized by the International Organization for Standardization (ISO). An implementation of the Handle System, DOIs are in wide use mainly to identify academic, professional, and government information, such as journal articles, research reports and data sets, and official publications though they also have been used to identify other types of information resources, such as commercial videos.
Dover Publications, also known as Dover Books, is an American book publisher founded in 1941 by Hayward Cirker and his wife, Blanche. It primarily publishes reissues, books no longer published by their original publishers. These are often, but not always, books in the public domain. The original published editions may be scarce or historically significant. Dover republishes these books, making them available at a significantly reduced cost.
Mathematical Reviews is a journal published by the American Mathematical Society (AMS) that contains brief synopses, and in some cases evaluations, of many articles in mathematics, statistics, and theoretical computer science. The AMS also publishes an associated online bibliographic database called MathSciNet which contains an electronic version of Mathematical Reviews and additionally contains citation information for over 3.5 million items as of 2018.
