Showing papers in "American Mathematical Monthly in 1971"
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TL;DR: In this article, Markov Random Fields and Gibbs Ensembles are used for Markov random fields and Gibbs ensembles, and Gibbs ensemble is used for Gibbs Ensemble.
Abstract: (1971). Markov Random Fields and Gibbs Ensembles. The American Mathematical Monthly: Vol. 78, No. 2, pp. 142-154.
181 citations
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TL;DR: The Faber Polynomials and the Faber Series as discussed by the authors have been used extensively in the literature for the purpose of identifying the most appropriate nouns for a given class of polynomials.
Abstract: (1971). Faber Polynomials and the Faber Series. The American Mathematical Monthly: Vol. 78, No. 6, pp. 577-596.
133 citations
TL;DR: In this article, isometries in normed spaces are studied in the context of the American Mathematical Monthly: Vol. 78, No. 6, pp. 655-658.
Abstract: (1971). Isometries in Normed Spaces. The American Mathematical Monthly: Vol. 78, No. 6, pp. 655-658.
TL;DR: In this article, Fractional Derivatives and Leibniz Rule are discussed in the context of fractional derivatives and fractional leebniz rule. The American Mathematical Monthly: Vol. 78, No. 6, pp. 645-649.
Abstract: (1971). Fractional Derivatives and Leibniz Rule. The American Mathematical Monthly: Vol. 78, No. 6, pp. 645-649.
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TL;DR: A history of mathematics education in the United States and Canada that you really wait for now is coming as mentioned in this paper. But, when you feel hard to find it as yours, what to do? Borrow to your friends and don't know when to give back it to her or him.
Abstract: Interestingly, a history of mathematics education in the united states and canada that you really wait for now is coming. It's significant to wait for the representative and beneficial books to read. Every book that is provided in better way and utterance will be expected by many peoples. Even you are a good reader or not, feeling to read this book will always appear when you find it. But, when you feel hard to find it as yours, what to do? Borrow to your friends and don't know when to give back it to her or him.
TL;DR: In this paper, the generalized Fibonacci numbers were studied and discussed in the context of generalized finite numbers, and the American Mathematical Monthly: Vol. 78, No. 10, pp. 1108-1109.
Abstract: (1971). On Generalized Fibonacci Numbers. The American Mathematical Monthly: Vol. 78, No. 10, pp. 1108-1109.
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TL;DR: In this article, what is a Martingale? The American Mathematical Monthly: Vol. 78, No. 5, pp. 451-463, with a discussion of the meaning of Martingales.
Abstract: (1971). What is a Martingale? The American Mathematical Monthly: Vol. 78, No. 5, pp. 451-463.
TL;DR: The Integral Inequalities Involving a Function and its Derivative (IIN) as discussed by the authors is an example of a function and its relation to a function. The American Mathematical Monthly: Vol. 78, No. 7, pp. 705-741.
Abstract: (1971). Integral Inequalities Involving a Function and its Derivative. The American Mathematical Monthly: Vol. 78, No. 7, pp. 705-741.
TL;DR: In this paper, the three-bug problem is considered in the context of cyclic pursuit and the three bugs problem is defined as a three-dimensional combinatorial optimization problem.
Abstract: (1971). Cyclic Pursuit or “the Three Bugs Problem”. The American Mathematical Monthly: Vol. 78, No. 6, pp. 631-639.
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TL;DR: In this article, what is a convex set and how it relates to a set of convex sets are discussed. But they do not discuss the relation between convex and concave sets.
Abstract: (1971). What Is a Convex Set? The American Mathematical Monthly: Vol. 78, No. 6, pp. 616-631.
TL;DR: It has been known for some time that there exist sequences on 3 symbols which contain no 2 identically equal consecutive segments, and sequences on 2 symbols that contain no 3 identically identical consecutive segments as mentioned in this paper.
Abstract: It has long been known (see [1–3, 5, 6, 10–12, 15, 16]) that there exist sequences on 3 symbols which contain no 2 identically equal consecutive segments, and sequences on 2 symbols which contain no 3 identically equal consecutive segments. Indeed, Axel Thue obtained these results around 1906. See [6] for a brief account of the contexts of the various independent rediscoveries of these results, and see [7,14] for an account of other properties of these sequences. Let X be a set and let s = x1x2x3 be a sequence on X . Then for i+1 k, s[i+1;k] = xi+1xi+2 xk is a segment of s, and the segments s[i+ 1; j];s[ j + 1;k] are consecutive. The segments s[i+ 1; j] and s[p+ 1;q] are identically equal if k i = q p and xi+1 = xp+1;xi+2 = xp+2; : : : ;xk = xq or, in other words, if s[i+1;k] = s[p+1;q] in X , the free semigroup generated by the set X . An interesting situation arises when we allow the symbols within a segment to commute with each other. It will be convenient to use the following terminology. Given a set X and a sequence s on X , we regard segments of s as elements of X . (Thus the results mentioned above say that there exist sequences on 3 symbols without 2nd powers as segments, and sequences on 2 symbols without 3rd powers.) Now let X denote the free commutative semigroup generated by X , and let α : X 7! X be the natural homomorphism (α(x) = x for x 2 X). If s has k consecutive segments f1; : : : ; fk such that α( f1) = = α( fk), then we say that s has a kth power mod α . In this language, the question of the title is: Does there exist a sequence on four symbols without 2nd powers mod α? It is an easy matter to verify that every sequence on 3 symbols contains 2nd powers mod α , and that every sequence on 2 symbols has 3rd powers mod α . For example, if X = fx;yg, one can show by examining all cases that the longest elements of X which do not contain a 3rd power mod α are xxyyxyyxx;xxyyxyyxy; and a few others. Evdomikov [4] constructed a sequence on 25 symbols without 2nd powers mod α , and conjectured that perhaps 5 symbols would suffice. Justin [8], with a remarkable half-page proof, constructed a sequence on 2 symbols without 5th powers mod α . This sequence is obtained by successive iterations of the transformation x 7! xxxxy and y 7! xyyyy, starting with x. Thus the first few iterations give x;xxxxy;(xxxxy)4xyyyy; [(xxxxy)4xyyyy]4xxxxy(xyyyy)4. Then in 1970 a paper appeared [13] in which P. A. B. Pleasants gave a construction of a sequence on 5 symbols without 2nd powers mod α . Pleasants’ sequence, which extends to infinity in both directions, is constructed by
TL;DR: In this paper, the patterns of visible and nonvisible lattice points are described and analyzed in terms of visible points and visible points, respectively, in the context of lattice point diagrams.
Abstract: (1971). Patterns of Visible and Nonvisible Lattice Points. The American Mathematical Monthly: Vol. 78, No. 5, pp. 487-496.