Showing papers in "American Mathematical Monthly in 1998"
TL;DR: In this paper, unique developments in non-integer bases have been discussed in the context of non-negative integer bases, and the authors present an overview of the most recent developments.
Abstract: (1998). Unique Developments in Non-Integer Bases. The American Mathematical Monthly: Vol. 105, No. 7, pp. 636-639.
174 citations
TL;DR: It is shown that eigenvalues play a central role when it comes to ensuring existence and uniqueness of Krylov solutions; they are not merely an artifact of convergence analyses.
Abstract: 1. INTRODUCTION. We explain why Krylov methods make sense, and why it is natural to represent a solution to a linear system as a member of a Krylov space. In particular we show that the solution to a nonsingular linear system Ax = b lies in a Krylov space whose dimension is the degree of the minimal polynomial of A. Therefore, if the minimal polynomial of A has low degree then the space in which a Krylov method searches for the solution can be small. In this case a Krylov method has the opportunity to converge fast. When the matrix is singular, however, Krylov methods can fail. Even if the linear system does have a solution, it may not lie in a Krylov space. In this case we describe a class of right-hand sides for which a solution lies in a Krylov space. As it happens, there is only a single solution that lies in a Krylov space, and it can be obtained from the Drazin inverse. Our discussion demonstrates that eigenvalues play a central role when it comes to ensuring existence and uniqueness of Krylov solutions; they are not merely an artifact of convergence analyses.
161 citations
TL;DR: In this article, Descartes' Rule of Signs Revisited was revisited and the American Mathematical Monthly: Vol. 105, No. 5, pp. 447-451.
Abstract: (1998). Descartes' Rule of Signs Revisited. The American Mathematical Monthly: Vol. 105, No. 5, pp. 447-451.
126 citations
TL;DR: In this paper, the authors propose quelques observations concernant l'ouvrage de Stanislas Dehaene The number sense, and examine ici en particulier les questions portant sur la relation entre les nombres and le langage dans a perspective cognitive.
Abstract: L'A. propose quelques observations concernant l'ouvrage de Stanislas Dehaene The number sense. How the mind creates mathematics (1997) qui explore tous les aspects de la relation entre les hommes et les nombres : la numerosite chez les autres animaux, la numerosite et le calcul simple chez les bebes, l'histoire de l'expression du nombre dans le langage, l'histoire de la notation du nombre, le circuit neuronal necessaire pour faire de l'arithmetique et du calcul, la localisation dans le cerveau, l'ordre mathematique de l'univers, etc ... L'A. examine ici en particulier les questions portant sur la relation entre les nombres et le langage dans une perspective cognitive, puis explique ce que Dehaene entend par le sens du nombre en caracterisant les mathematiques comme une formalisation progressive de nos intuitions sur les ensembles, le nombre, l'espace, le temps et la logique
119 citations
TL;DR: Two dual assertions: the first on learning and the second on teaching (or vice versa Versa) as mentioned in this paper were made by the authors of this paper. But they did not discuss the relationship between learning and teaching.
Abstract: (1998). Two Dual Assertions: The First on Learning and the Second on Teaching (or Vice Versa) The American Mathematical Monthly: Vol. 105, No. 6, pp. 497-507.
115 citations
TL;DR: In this article, the authors present a Polynomial Equations and Convex Polytopes (PE) for convex polytopes with polynomial equations.
Abstract: (1998). Polynomial Equations and Convex Polytopes. The American Mathematical Monthly: Vol. 105, No. 10, pp. 907-922.
111 citations
TL;DR: In this article, the authors separate hyperplanes and the Authorship of the Disputed Federalist Papers, and present an alternative approach to the same problem in the context of hyperplanes in geometry.
Abstract: (1998). Separating Hyperplanes and the Authorship of the Disputed Federalist Papers. The American Mathematical Monthly: Vol. 105, No. 7, pp. 601-608.
79 citations
75 citations
TL;DR: The surprise examination or Unexpected hanging paradox as mentioned in this paper is a classic example of a paradox in the setting of algebraic geometry, and it has been studied extensively in the literature.
Abstract: (1998). The Surprise Examination or Unexpected Hanging Paradox. The American Mathematical Monthly: Vol. 105, No. 1, pp. 41-51.
57 citations
TL;DR: The main issue is interpreted as how the teaching of algorithms should be handled in the schools, and questions of how the Standards should address aspects of algorithms will be relatively easy.
Abstract: (1998). Doing and Proving: The Place of Algorithms and Proofs in School Mathematics. The American Mathematical Monthly: Vol. 105, No. 3, pp. 252-255.
TL;DR: In this article, Lanczos' generalized derived derived is used to describe the generalized derivative of the generalized derivative of Lanczos's generalized derivative, which is a generalization of the generalized derived.
Abstract: (1998). Lanczos' Generalized Derivative. The American Mathematical Monthly: Vol. 105, No. 4, pp. 320-326.
TL;DR: The Newton and Halley Methods for Complex Roots as mentioned in this paper have been used for complex root discovery for the first time in 1998, and they are described in detail in the American Mathematical Monthly: Vol 105, No. 9, pp. 806-818.
Abstract: (1998). The Newton and Halley Methods for Complex Roots. The American Mathematical Monthly: Vol. 105, No. 9, pp. 806-818.
TL;DR: The number of homomorphisms from Zm into Z into Z, where i 2 + 1 = 0 and p 2 + p+1 = 0, was studied by Gallian and Jungreis as discussed by the authors.
Abstract: I. J. A. Gallian and J. Van Buskirk, The number of homomorphisms from Zm into Z,, Amer. Math. Momhly 91 (1984) 196-197. 2. J. A. Gallian and D. S. Jungreis, Homomorphisms from Zm[i) into Z,[i) and Zm[ p) into Z,[ p), where i 2 + 1 = 0 and p 2 + p + 1 = 0, Amer. Math. Monthly 95 (1988) 247-249. 3. D. Poulakis, On the homomorphisms of a family of finite rings, Bull. Greek Math. Soc. 30 (1989) 43-47.
TL;DR: A Stroll Through the Gaussian Primes as discussed by the authors is a recent work that walks through the Gaussians and shows that it is possible to walk through a Gaussian prime.
Abstract: (1998). A Stroll Through the Gaussian Primes. The American Mathematical Monthly: Vol. 105, No. 4, pp. 327-337.
TL;DR: In this article, the Fourier-Motzkinetic elimination method is used to integrate over a polyhedron in an integration-over-a-polyhedron setting.
Abstract: (1998). Integration Over a Polyhedron: An Application of the Fourier-Motzkin Elimination Method. The American Mathematical Monthly: Vol. 105, No. 3, pp. 246-251.
TL;DR: In this article, a generalization of the Fermat-Torricelli problem is presented, which is based on generalizations of conics and on a generalized version of the problem.
Abstract: (1998). On Generalizations of Conics and on a Generalization of the Fermat-Torricelli Problem. The American Mathematical Monthly: Vol. 105, No. 8, pp. 732-743.
TL;DR: In this article, Univalent Polynomials and Non-Negative Trigonometric Sums (NNTS) are used to represent trigonometric sums in the context of univalent polynomials.
Abstract: (1998). Univalent Polynomials and Non-Negative Trigonometric Sums. The American Mathematical Monthly: Vol. 105, No. 6, pp. 508-522.
TL;DR: In this paper, product-free subsets of groups are defined and discussed in the context of product free subgroups. The American Mathematical Monthly: Vol. 105, No. 10, pp. 900-906
Abstract: (1998). Product-Free Subsets of Groups. The American Mathematical Monthly: Vol. 105, No. 10, pp. 900-906.
TL;DR: Reflections on Reflection in a Spherical Mirror as discussed by the authors is an example of a reflection in a mirror that can be viewed as a reflection of the reflection in the mirror itself, but not the reflection itself.
Abstract: (1998). Reflections on Reflection in a Spherical Mirror. The American Mathematical Monthly: Vol. 105, No. 6, pp. 523-528.
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TL;DR: The American Mathematical Monthly (AMM) Vol. 105, No. 3, pp. 263-270 as mentioned in this paper, is the first publication of the Function: Part II.
Abstract: (1998). Function: Part II. The American Mathematical Monthly: Vol. 105, No. 3, pp. 263-270.
TL;DR: Theorem of Burnside on Matrix Rings as discussed by the authors is a theorem of theorems of a theorem prover for theorem of theorem 651-653 of the matrix ring theorem.
Abstract: (1998). A Theorem of Burnside on Matrix Rings. The American Mathematical Monthly: Vol. 105, No. 7, pp. 651-653.
TL;DR: In this article, it is shown that any interesting dynamics of any map from [0, 1] to [0, 1] whose graph consists of two, joined line segments ends up (after scaling and reflecting as necessary) in a square region in which the graph meets the left edge, has its peak on the top edge, and descends to the lower right corner.
Abstract: It is not hard to see that any interesting dynamics of any map from [0, 1] to [0, 1] whose graph consists of two, joined line segments ends up (after scaling and reflecting as necessary) in a square region in which the graph meets the left edge, has its peak on the top edge, and descends to the lower right corner, as illustrated in Figure 1. Despite the simplicity of its maps, this family exhibits a surprising variety of dynamics. In Section 2 we see non-chaotic maps with attracting and non-attracting periodic behavior. In Section 3 we show that every map in the family with a
TL;DR: The TIMSS video study as mentioned in this paper found that the proportion of concrete settings was significantly lower in the United States than in Japan or Germany, and that mathematics instruction is oriented toward "problem solving" and also that routine skills are emphasized.
Abstract: situations involve only abstract concepts such as numbers and geometric shapes-even specific examples such as a rectangle three inches wide and five inches long-in contrast to concrete situations that involve physical or social representations of abstract concepts-such as a 3-by-5 inch index card. One reason reviewers were interested in this dichotomy was the call from some advocates of mathematics education reform in the United States for far more mathematics instruction to be done in "real world" contexts. Reviewers wanted to determine whether the proportion of concrete settings was significantly lower in the United States than in Japan or Germany. The locus of control of the solution of a problem-whether the task in the context of the class essentially included its own method of solution-is another aspect of the extent to which instruction was consistent with some current calls for reform of mathematics education in the United States. An important component of problem solving is the solver's control of the process of solving a problem [3]. A comparison of the extent to which exercises appear to require solvers to find solution methods should provide one indication of the extent to which mathematics instruction is oriented toward "problem solving" and also, perhaps, of the extent to which routine skills are emphasized. Similarly, the complexity of exercises is another indicator of their sophistication and mathematical richness, especially from a problem solving perspective. Determining whether a problem requires a "single-step" or "multi-step" solution depends upon not only the statement of the problem but also the presumed sophistication of the solver. For example, solving a simple linear equation such as 5x + 4 = 19 would be considered a two-step process for students who are first learning how to solve such equations but would constitute one very small step for students who had developed proficiency at solving systems of linear equations. As a result, reviewers had to consider the problem in the context of the class when judging its complexity. The reviewers classified the subject and described the subject matter of each class. Each of the 90 classes was classified according to its likely position in a traditional (1980s) United States college preparatory mathematics curriculum. The three classifications used were "Before Algebra," "Algebra," and "Geometry." Approximately half the classes were classified as Geometry, 30% as Algebra, and the remainder as Before Algebra. The reviewers also rated each class on the basis of their judgment of its potential for helping students understand mathematics. The five categories used 1998] EIGHTH GRADE MATHEMATICS CLASSES 795 This content downloaded from 157.55.39.178 on Sun, 24 Jul 2016 06:17:36 UTC All use subject to http://about.jstor.org/terms were "Weak," "Almost Good," "Good," "Better," and "Strong." While this is quite subjective, the four reviewers were able to reach complete agreement in all 90 cases. Readers of this paper who would like to make their own judgments may send email to timss@ed.gov to inquire about obtaining copies of the tables and other data from the TIMSS video study. The table for each class consisted of a few pages, typically about three, that included a summary description of each segment of the class. Segments were defined and identified by the members of the Stigler laboratory. A new segment started when there was a significant change in activity or in the content being presented. Reviewers categorized each segment by the nature of its mathematical activity and by its role in the structure of the class. The segments became the nodes of the directed graphs used to represent graphically the structure of the
TL;DR: The Second Number of Plutarch as mentioned in this paper is the second number of the first number of a column in the Second Book of the Roman Republic (Second Number of the Second Column).
Abstract: (1998). On the Second Number of Plutarch. The American Mathematical Monthly: Vol. 105, No. 5, pp. 446-446.
TL;DR: In this paper, the Archimedes' Cattle problem is studied in the context of the problem of solving the Cattle Problem in the American Mathematical Monthly: Vol. 105, No. 4, pp. 305-319.
Abstract: (1998). Archimedes' Cattle Problem. The American Mathematical Monthly: Vol. 105, No. 4, pp. 305-319.
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TL;DR: The American Mathematical Monthly (AMM) Vol. 105, No. 5, pp. 430-445 as mentioned in this paper is the most cited publication in the history of graph and marriage.
Abstract: (1998). Graphs and Marriages. The American Mathematical Monthly: Vol. 105, No. 5, pp. 430-445.
TL;DR: In this article, the authors present Mathematics on a Distant Planet (MDS), a novel approach to mathematics on a distant planet (MDP) which is an extension of the MDS.
Abstract: (1998). Mathematics on a Distant Planet. The American Mathematical Monthly: Vol. 105, No. 7, pp. 640-650.
TL;DR: The modern formulation of Descartes' Rule of Signs provides a list of possible numbers of both positive and negative roots for a given sign sequence as discussed by the authors, assuming none of the signs are zero.
Abstract: The modern formulation of Descartes' Rule of Signs provides a list of possible numbers of both positive and negative roots for a given sign sequence. We have shown (assuming none of the signs are zero) that polynomials exist with any of the possible numbers of positive roots. By replacing x by -x, we can provide a polynomial with the given sign sequence that contains any given number of negative roots allowable by Descartes. We have not addressed trying to accommodate both the positive and negative roots simultaneously. The following question is unanswered here and seems to be open: Given a sign sequence (which may include some zeros), do there exist polynomials containing positive and negative roots numbering each of the possible combinations allowed by Descartes' Rule of Signs?