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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 In mathematics, a Caccioppoli set is a set whose boundary is measurable and has (at least locally) a ''finite measure''. A synonym is set of (locally) finite perimeter. Basically, a set ... [...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 In mathematics, the concept of a measure is a generalization and formalization of geometrical measures ( length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many simil .... 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 functions. External linksUC Davis memorial 1936 births Living people 20th-century American mat ... [...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 (pronounced ), 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 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 and then let . Trying to create a general theory, Denjoy used trans ... [...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 differentiating a function (calculating its slopes, or rate of change at each time) with the concept of integrating a function (calculating the area under its graph, or the cumulative effect of small contributions). The two operations are inverses of each other apart from a constant value which depends on where one starts to compute area. The first part of the theorem, the first fundamental theorem of calculus, states that for a function , an antiderivative or indefinite integral may be obtained as the integral of over an interval with a variable upper bound. This implies the existence of antiderivatives for continuous functions. Conversely, the second part of the theorem, the second fundamental theorem of calculus, states that the integral of a function over a fixed interval is equal to the change of any antiderivative between the ends of the interval. This greatly simplifies the calculation of a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Caccioppoli Set
In mathematics, a Caccioppoli set is a set whose boundary is 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. 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 approximated by the values it takes on an increasing net of subsets. Another clearly stated (and demonstrated) propert ... [...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 of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the -axis. The Lebesgue integral, named after French mathematician Henri Lebesgue, extends the integral to a larger class of functions. It also extends the domains on which these functions can be defined. Long before the 20th century, mathematicians already understood that for non-negative functions with a smooth enough graph—such as continuous functions on closed bounded intervals—the ''area under the curve'' could be defined as the integral, and computed using approximation techniques on the region by polygons. However, as the need to consider more irregular functions arose—e.g., as a result of the limiting processes of mathematical analysis and the mathematical theory of probability—it became clear that more careful approximation techniques were needed to define a suitable integral. Also, one might ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
