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Pfeffer Integral
In mathematics, the Pfeffer integral is an integration technique created by Washek Pfeffer as an attempt to extend the Henstock–Kurzweil integral to a multidimensional domain. This was to be done in such a way that the fundamental theorem of calculus would apply analogously to the theorem in one dimension, with as few preconditions on the function under consideration as possible. The integral also permits analogues of the chain rule and other theorems of the integral calculus for higher dimensions. Definition The construction is based on the Henstock or gauge integral, however Pfeffer proved that the integral, at least in the one dimensional case, is less general than the Henstock integral. It relies on what Pfeffer refers to as a set of bounded variation, this is equivalent to a Caccioppoli set. The Riemann sums of the Pfeffer integral are taken over partitions made up of such sets, rather than intervals as in the Riemann or Henstock integrals. A gauge is used, exactly as in the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Washek Pfeffer
Washek F. Pfeffer (November 14, 1936–January 3, 2021) was a Czech-born US mathematician and Emeritus Professor at the University of California, Davis. Pfeffer was one of the world's pre-eminent authorities on real integration and has authored several books on the topic of integration, and numerous papers on these topics and others related to many areas of real analysis and measure theory. Pfeffer gave his name to the Pfeffer integral, which extends a Riemann-type construction for the integral of a measurable function both to higher-dimensional domains and, in the case of one dimension, to a superset of the Lebesgue integrable In mathematics, the integral of a non-negative Function (mathematics), function of a single variable can be regarded, in the simplest case, as the area between the Graph of a function, graph of that function and the axis. The Lebesgue integral, ... functions. External linksUC Davis memorial 1936 births Living people 20th-century American mathem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Henstock–Kurzweil Integral
In mathematics, the Henstock–Kurzweil integral or generalized Riemann integral or gauge integral – also known as the (narrow) Denjoy integral (), Luzin integral or Perron integral, but not to be confused with the more general wide Denjoy integral – is one of a number of inequivalent definitions of the integral of a function. It is a generalization of the Riemann integral, and in some situations is more general than the Lebesgue integral. In particular, a function is Lebesgue integrable over a subset of \R^n if and only if the function and its absolute value are Henstock–Kurzweil integrable. This integral was first defined by Arnaud Denjoy (1912). Denjoy was interested in a definition that would allow one to integrate functions like: f(x) = \frac\sin\left(\frac\right). This function has a singularity at 0, and is not Lebesgue-integrable. However, it seems natural to calculate its integral except over the interval \varepsilon, \delta/math> and then let \varepsilon ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fundamental Theorem Of Calculus
The fundamental theorem of calculus is a theorem that links the concept of derivative, differentiating a function (mathematics), function (calculating its slopes, or rate of change at every point on its domain) with the concept of integral, integrating a function (calculating the area under its graph, or the cumulative effect of small contributions). Roughly speaking, the two operations can be thought of as inverses of each other. The first part of the theorem, the first fundamental theorem of calculus, states that for a continuous function , an antiderivative or indefinite integral can be obtained as the integral of over an interval with a variable upper bound. Conversely, the second part of the theorem, the second fundamental theorem of calculus, states that the integral of a function over a fixed Interval (mathematics), interval is equal to the change of any antiderivative between the ends of the interval. This greatly simplifies the calculation of a definite integral pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Caccioppoli Set
In mathematics, a Caccioppoli set is a subset of \R^n whose boundary is (in a suitable sense) measurable and has (at least locally) a ''finite measure''. A synonym is set of (locally) finite perimeter. Basically, a set is a Caccioppoli set if its characteristic function is a function of bounded variation, and its perimeter is the total variation of the characteristic function. History The basic concept of a Caccioppoli set was first introduced by the Italian mathematician Renato Caccioppoli in the paper : considering a plane set or a surface defined on an open set in the plane, he defined their measure or area as the total variation in the sense of Tonelli of their defining functions, i.e. of their parametric equations, ''provided this quantity was bounded''. The ''measure of the boundary of a set was defined as a functional'', precisely a set function, for the first time: also, being defined on open sets, it can be defined on all Borel sets and its value can be approxim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
McShane Integral
In the branch of mathematics known as integration theory, the McShane integral, created by Edward J. McShane, is a modification of the Henstock-Kurzweil integral. The McShane integral is equivalent to the Lebesgue integral. Definition Free tagged partition Given a closed interval of the real line, a ''free tagged partition'' P ''of'' ,b/math> is a set : \ where : a = a_0 , : \left , \int_a^bf - S(f, P) \ 0, let's choose the gauge \delta(t) such that \delta(a)=\delta(b)=\varepsilon/4 and \delta(t)=b-a if t\in]a,b[. Any free tagged partition P=\ of ,b can be decomposed into sequences like (a, _,x_, for j=1,...,\lambda, (b, _,x_, for k=1,...,\mu, and (t_, _,x_, where r=1,...,\nu, such that t_\in]a,b[ (\lambda+\mu+\nu=n). This way, we have the Riemann sum S(f, P) = \sum_^\nu \displaystyle(x_-x_) and by consequence , S(P,f)-(b-a), =\textstyle \sum_^\lambda \displaystyle(x_-x_)+\textstyle \sum_^\mu \displaystyle(x_-x_). Therefore if P is a free tagged \delta-fin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lebesgue Integral
In mathematics, the integral of a non-negative Function (mathematics), function of a single variable can be regarded, in the simplest case, as the area between the Graph of a function, graph of that function and the axis. The Lebesgue integral, named after france, French mathematician Henri Lebesgue, is one way to make this concept rigorous and to extend it to more general functions. The Lebesgue integral is more general than the Riemann integral, which it largely replaced in mathematical analysis since the first half of the 20th century. It can accommodate functions with discontinuities arising in many applications that are pathological from the perspective of the Riemann integral. The Lebesgue integral also has generally better analytical properties. For instance, under mild conditions, it is possible to exchange limits and Lebesgue integration, while the conditions for doing this with a Riemann integral are comparatively baroque. Furthermore, the Lebesgue integral can be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
