Showing papers in "American Mathematical Monthly in 1959"
7,408 citations
4,135 citations
TL;DR: The theory of generalised functions and their Fourier transforms is discussed in this paper. But the analysis of Fourier transform is limited to the case of generalized functions, and it is not suitable for generalised function analysis.
Abstract: 1. Introduction 2. The theory of generalised functions and their Fourier transforms 3. Definitions, properties and Fourier transforms of particular generalised functions 4. The asymptotic estimation of Fourier transforms 5. Fourier series Index.
1,327 citations
996 citations
TL;DR: The Integrability of Certain Functions and Related Summability Methods as discussed by the authors is a seminal work in the field of algebraic geometry, and it has been used extensively in many applications.
Abstract: (1959). The Integrability of Certain Functions and Related Summability Methods. The American Mathematical Monthly: Vol. 66, No. 5, pp. 361-775.
881 citations
485 citations
TL;DR: In this article, the authors present a set of sets of acquaintances and strangers at any party, where the set of friends and strangers is defined as a set, i.e., sets of associates and strangers.
Abstract: (1959). On Sets of Acquaintances and Strangers at any Party. The American Mathematical Monthly: Vol. 66, No. 9, pp. 778-783.
326 citations
316 citations
305 citations
202 citations
TL;DR: In this article, the problem of reducing an arbitrary truth-functional formula to a shortest equivalent in normal form was studied, and a theorem was established that a normal formula is uniliterally redundant if and only if not all its clauses are prime implicants of it.
Abstract: What is called a truth-functional formula in (alternational) normal form is built up of sentence letters 'p', 'q', etc., or their negations 'i', 'a', etc., or both, by using only the notations of conjunction and alternation, and in such a way as to subject alternations never to conjunction but only vice versa: thus 'pq v firs v f'. This paper is concerned with the problem of reducing an arbitrary truth-functional formula to a shortest equivalent in normal form. Part of the content of my last previous paper on the subject* will be presented anew in an improved way, and a further theorem will be established. I shall not assume familiarity with my previous papers. Let us sharpen our terminology. Sentence letters and their negations are called literals. Literals and conjunctions of them are called fundamentalformulas, provided that none contains the same letter twice. Fundamental formulas and alternations of them are called normal, and are said to have those fundamental formulas as their clauses. On these definitions, a formula is convertible to nornal form only if it is not self-contradictory; but there is no serious loss in setting aside the self-contradictory cases. A prime implicant of a formula 4) is a fundamental formula that logically implies 4 but ceases to when deprived of any one literal. A normal formula will be called uniliterally redundant if it is equivalent to what remains of itself on dropping some one occurrence of a literal. Obviously then a normal formula is uniliterally redundant if and only if not all its clauses are prime implicants of it. Consequently, in particular,
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TL;DR: In this paper, the voting problem is formulated as a voting problem, and the voting rules are described in terms of a voting set-up and a set of voting rules, respectively.
Abstract: (1959). The Voting Problem. The American Mathematical Monthly: Vol. 66, No. 9, pp. 761-763.
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TL;DR: In this paper, an improved solution is presented for the problem of finding the smallest number of direct pairwise comparisons which will always suffice to rank n objects according to some transitive characteristic.
Abstract: : An improved solution is presented for the problem of finding the smallest number of direct pairwise comparisons which will always suffice to rank n objects according to some transitive characteristic. In his book, Mathematical Snapshots, Steinhaus discusses the problem of ranking n objects according to some transitive characteristic, by means of successive pairwise comparisons. In this paper, the terminology was adopted of a tennis tournament by n players. The problem may be briefly stated: 'What is the smallest number of matches which will always suffice to rank all n players.'
TL;DR: A Connected Topology for Integers as mentioned in this paper is a topology for the Integers, which is based on the topology of the integers of the Integer's graph.
Abstract: (1959). A Connected Topology for the Integers. The American Mathematical Monthly: Vol. 66, No. 8, pp. 663-665.
TL;DR: The second edition of the paper this article incorporates recent developments in functional analysis to make the selection of topics more appropriate for current courses in functional analytically oriented courses in this article.
Abstract: Analyzes the theory of normed linear spaces and of linear mappings between such spaces, providing the necessary foundation for further study in many areas of analysis. Strives to generate an appreciation for the unifying power of the abstract linear-space point of view in surveying the problems of linear algebra, classical analysis, and differential and integral equations. This second edition incorporates recent developments in functional analysis to make the selection of topics more appropriate for current courses in functional analysis. Additions to this new edition include: a chapter on Banach algebras, and material on weak topologies and duality, equicontinuity, the Krein-Milman theorem, and the theory of Fredholm operators. Greater emphasis is also placed on closed unbounded linear operators, with more illustrations drawn from ordinary differential equations.
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TL;DR: In this paper, an optimal search procedure was proposed to find the optimal solution for a given problem in the optimal search space, and the algorithm was shown to work well in most cases.
Abstract: (1959). An Optimal Search Procedure. The American Mathematical Monthly: Vol. 66, No. 8, pp. 690-693.
TL;DR: In this paper, heuristic reasoning in the theory of numbers has been studied in the context of heuristics in the field of number theory, and heuristic reasoning has been applied to the case of numbers.
Abstract: (1959). Heuristic Reasoning in the Theory of Numbers. The American Mathematical Monthly: Vol. 66, No. 5, pp. 375-384.
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