Topic

# Infinite set

About: Infinite set is a research topic. Over the lifetime, 2501 publications have been published within this topic receiving 54613 citations.

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Yale University

^{1}TL;DR: In this article, the problem of identifying an unknown regular set from examples of its members and nonmembers is addressed, where the regular set is presented by a minimaMy adequate teacher, which can answer membership queries about the set and can also test a conjecture and indicate whether it is equal to the unknown set and provide a counterexample if not.

Abstract: The problem of identifying an unknown regular set from examples of its members and nonmembers is addressed. It is assumed that the regular set is presented by a minimaMy adequate Teacher, which can answer membership queries about the set and can also test a conjecture and indicate whether it is equal to the unknown set and provide a counterexample if not. (A counterexample is a string in the symmetric difference of the correct set and the conjectured set.) A learning algorithm L* is described that correctly learns any regular set from any minimally adequate Teacher in time polynomial in the number of states of the minimum dfa for the set and the maximum length of any counterexample provided by the Teacher. It is shown that in a stochastic setting the ability of the Teacher to test conjectures may be replaced by a random sampling oracle, EX( ). A polynomial-time learning algorithm is shown for a particular problem of context-free language identification.

2,157 citations

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TL;DR: This article introduces the formal notion of the family of α-shapes of a finite point set in R 3 .

Abstract: Frequently, data in scientific computing is in its abstract form a finite point set in space, and it is sometimes useful or required to compute what one might call the “shape” of the set. For that purpose, this article introduces the formal notion of the family of a-shapes of a finite point set in R3. Each shape is a well-defined polytope, derived from the Delaunay triangulation of the point set, with a parameter a e R controlling the desired level of detail. An algorithm is presented that constructs the entire family of shapes for a given set of size n in time 0(n2), worst case. A robust implementation of the algorithm is discussed, and several applications in the area of scientific computing are mentioned.

1,980 citations

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TL;DR: A vague set is a set of objects, each of which has a grade of membership whose value is a continuous subinterval of according to the inequality of the following type:

Abstract: A vague set is a set of objects, each of which has a grade of membership whose value is a continuous subinterval of

1,567 citations

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TL;DR: This paper will answer the question in the affirmative by determining the exact upper bound of T if T is a family of subsets of some infinite set S then either there exists to each number n a set A ⊂ S with |A| = n such that |T ∩ A| = 2n or there exists some number N such that •A| c for each A⩾ N and some constant c.

1,029 citations

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TL;DR: In this paper, it was shown that the Green9s functions for an elastic layered medium can be expressed as a double integral over frequency and horizontal wavenumber, which can be exactly represented by a discrete summation.

Abstract: Green9s functions for an elastic layered medium can be expressed as a double integral over frequency and horizontal wavenumber. We show that, for any time window, the wavenumber integral can be exactly represented by a discrete summation. This discretization is achieved by adding to the particular point source an infinite set of specified circular sources centered around the point source and distributed at equal radial interval. Choice of this interval is dependent on the length of time desired for the point source response and determines the discretized set of horizontal wavenumbers which contribute to the solution. Comparisons of the results obtained with those derived using the two-dimensional discretization method (Bouchon, 1979) are presented. They show the great accuracy of the two methods.

928 citations