Topic
Real line
About: Real line is a research topic. Over the lifetime, 3918 publications have been published within this topic receiving 51137 citations. The topic is also known as: infinite interval.
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06 Jun 2005
TL;DR: The persistence diagram of a real-valued function on a topological space is a multiset of points in the extended plane and it is proved that under mild assumptions on the function, the persistence diagram is stable.
Abstract: The persistence diagram of a real-valued function on a topological space is a multiset of points in the extended plane. We prove that under mild assumptions on the function, the persistence diagram is stable: small changes in the function imply only small changes in the diagram. We apply this result to estimating the homology of sets in a metric space and to comparing and classifying geometric shapes.
828 citations
TL;DR: In this article, necessary and sufficient conditions are given for a set of numbers to be the mutual distances of real points in Euclidean space, and matrices are found whose ranks determine the dimension of the smallest space containing such points, and methods for determining the configuration of these points and for approximating to them by points in a space of lower dimensionality.
Abstract: Necessary and sufficient conditions are given for a set of numbers to be the mutual distances of a set of real points in Euclidean space, and matrices are found whose ranks determine the dimension of the smallest Euclidean space containing such points. Methods are indicated for determining the configuration of these points, and for approximating to them by points in a space of lower dimensionality.
762 citations
TL;DR: In this paper, it was shown that for any continuous mapping of the real line into itself, the existence of a cycle of order n2 follows from the existence (1 − n 2 ) of cycle n 1.
Abstract: The basic result of this investigation may be formulated as follows. Consider the set of natural numbers in which the following relationship is introduced: n1 precedes n2 (n1 ≼ n2) if for any continuous mapping of the real line into itself the existence of a cycle of order n2 follows from the existence of a cycle of order n1. The following theorem holds.
567 citations
Book•
03 Oct 2003
TL;DR: In this paper, the authors present a list of well-known distributions notations and conventions, as well as prerequisites from probability and analysis list of distributions notation and conventions for stochastic processes.
Abstract: infinitely divisible distributions on the nonnegative integers infinitely divisible distributions on the nonnegative reals infinitely divisible distributions on the real line self-decomposability and stability infinite divisibility and mixtures infinite divisibility in stochastic processes. Appendices: prerequisites from probability and analysis list of well-known distributions notations and conventions.
488 citations
TL;DR: It is shown that far from being a curiosity, the trapezoidal rule is linked with computational methods all across scientific computing, including algorithms related to inverse Laplace transforms, special functions, complex analysis, rational approximation, integral equations, and the computation of functions and eigenvalues of matrices and operators.
Abstract: It is well known that the trapezoidal rule converges geometrically when applied to analytic functions on periodic intervals or the real line. The mathematics and history of this phenomenon are reviewed, and it is shown that far from being a curiosity, it is linked with computational methods all across scientific computing, including algorithms related to inverse Laplace transforms, special functions, complex analysis, rational approximation, integral equations, and the computation of functions and eigenvalues of matrices and operators.
481 citations