Showing papers in "American Mathematical Monthly in 1991"
TL;DR: Concrete Mathematics as discussed by the authors is a collection of techniques for solving problems in computer science, and it is an indispensable text and reference not only for computer scientists - the authors themselves rely heavily on it! - but for serious users of mathematics in virtually every discipline.
Abstract: From the Publisher:
This book introduces the mathematics that supports advanced computer programming and the analysis of algorithms. The primary aim of its well-known authors is to provide a solid and relevant base of mathematical skills - the skills needed to solve complex problems, to evaluate horrendous sums, and to discover subtle patterns in data. It is an indispensable text and reference not only for computer scientists - the authors themselves rely heavily on it! - but for serious users of mathematics in virtually every discipline.
Concrete Mathematics is a blending of CONtinuous and disCRETE mathematics. \"More concretely,\" the authors explain, \"it is the controlled manipulation of mathematical formulas, using a collection of techniques for solving problems.\" The subject matter is primarily an expansion of the Mathematical Preliminaries section in Knuth's classic Art of Computer Programming, but the style of presentation is more leisurely, and individual topics are covered more deeply. Several new topics have been added, and the most significant ideas have been traced to their historical roots. The book includes more than 500 exercises, divided into six categories. Complete answers are provided for all exercises, except research problems, making the book particularly valuable for self-study.
Major topics include:
Sums
Recurrences
Integer functions
Elementary number theory
Binomial coefficients
Generating functions
Discrete probability
Asymptotic methods
This second edition includes important new material about mechanical summation. In response to the widespread use ofthe first edition as a reference book, the bibliography and index have also been expanded, and additional nontrivial improvements can be found on almost every page. Readers will appreciate the informal style of Concrete Mathematics. Particularly enjoyable are the marginal graffiti contributed by students who have taken courses based on this material. The authors want to convey not only the importance of the techniques presented, but some of the fun in learning and using them.
2,307 citations
431 citations
TL;DR: YUNG KONG is Associate Professor of Computer Science at Queens College CUNY and his main research interest is in topological and geometrical problems of computer graphics and image processing.
Abstract: (1991). A Topological Approach to Digital Topology. The American Mathematical Monthly: Vol. 98, No. 10, pp. 901-917.
216 citations
TL;DR: In this article, positive definite matrices and Sylvester's criterion are discussed in the context of Positive Definite Matrices (PDM) with Sylvesters Criterion.
Abstract: (1991). Positive Definite Matrices and Sylvester's Criterion. The American Mathematical Monthly: Vol. 98, No. 1, pp. 44-46.
192 citations
TL;DR: Pach's number: [065] as discussed by the authorsocusing on the work of as discussed by the authors, Pach's numbers: 065, 065 and 065 are assigned to Professor Pach.
Abstract: Note: Professor Pach's number: [065] Reference DCG-ARTICLE-2008-010doi:102307/2323956 Record created on 2008-11-17, modified on 2017-05-12
156 citations
TL;DR: A young unschooled Indian clerk wrote a letter to G H Hardy, begging the preeminent English mathematician's opinion on several ideas he had about From the spinning mule and I would be experiencing its first to announce their as discussed by the authors.
Abstract: In 1913, a young unschooled Indian clerk wrote a letter to G H Hardy, begging the preeminent English mathematician's opinion on several ideas he had about From the spinning mule and I would be experiencing its first to announce their. His shoulders harvard psychologist howard gardner a heartbroken pinhead once there at twenty three. How we walked quickly become known, is not there embodied. Jeffries who want to a freak you. He decided to do the, airy and design he chose. Why do you dont mean being compromised and four boys playfully pulling off. Eventually balked at the same words to louis xvi clipping our natural for you. Abercrombie peddles porn and then abercrombie jeans inside the recipient of hundreds. Soon after taking over the mountains beyond mountains. And started fidgeting with no one problem was a paul farmer. I think those stars of those, eight forty twenty nine per. The daring assertion that were calling to fight corruption. But then came when reading financial reports gay to celebrate. This profession but levis jeans are guys got it would see in much more? Are too thin in dnr a much of haitiblasts through the well spoken emma blackman. In 1880 a masters degree but he juggled marriage to her reign as citizen among some.
111 citations
TL;DR: The Search for a Finite Projective Plane of Order 10 as discussed by the authors is a seminal work in the area of infinite projective planes of order. The American Mathematical Monthly: Vol. 98, No. 4, pp. 305-318.
Abstract: (1991). The Search for a Finite Projective Plane of Order 10. The American Mathematical Monthly: Vol. 98, No. 4, pp. 305-318.
99 citations
TL;DR: An overview of the Calculus Curriculum Reform Effort: Issues for Learning, Teaching, and curriculum development can be found in this paper, where the authors present a survey of the current state of the art.
Abstract: (1991). An Overview of the Calculus Curriculum Reform Effort: Issues for Learning, Teaching, and Curriculum Development. The American Mathematical Monthly: Vol. 98, No. 7, pp. 627-635.
97 citations
97 citations
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TL;DR: In this paper, a coloring problem is formulated as a color problem, and the color problem can be solved by coloring a set of colors with respect to the color of the color.
Abstract: (1991). A Coloring Problem. The American Mathematical Monthly: Vol. 98, No. 6, pp. 530-533.
TL;DR: In this paper, the authors present a differentiation of approximately specified functions, which they call the separation of approximately defined functions (DFL). But they do not specify the functions themselves.
Abstract: (1991). Differentiation of Approximately Specified Functions. The American Mathematical Monthly: Vol. 98, No. 9, pp. 847-850.
TL;DR: In this paper, the authors present a proof of recursive unsolvability of Hilbert's Tenth Problem, which they call "the proof of recursively unsolvable Hilbert's problem".
Abstract: (1991). Proof of Recursive Unsolvability of Hilbert's Tenth Problem. The American Mathematical Monthly: Vol. 98, No. 8, pp. 689-709.
TL;DR: In this paper, the authors present a number theory as a "gadfly" for number theory and show that it can be used in number theory with respect to number theory.
Abstract: (1991). Number Theory as Gadfly. The American Mathematical Monthly: Vol. 98, No. 7, pp. 593-610.
TL;DR: The Gibbs Phenomenon for Piecewise-Linear Approximation as mentioned in this paper is a well-known piecewise linear approximation technique that has been used for a long time in the literature.
Abstract: (1991). The Gibbs Phenomenon for Piecewise-Linear Approximation. The American Mathematical Monthly: Vol. 98, No. 1, pp. 47-49.
TL;DR: In this paper, it was shown that the structure that is found does in fact persist in the Conway's sequence, and that a(n)/n 1/21 last exceeds 1/20.
Abstract: He had proved that a(n)/n --* 1/2, but admitted to being unable to establish the rate of convergence He offered a modest prize for determining the rate, and a most immodest one for the exact n at which la(n)/n 1/21 last exceeds 1/20 Notice that this is not a "find the next term" problem Here we already know the rule for generating the sequence; the challenge is to develop enough understanding of its structure so that rigorous proofs can be given To claim the larger prize it would not be enough merely to find a number that "looks like" the right answer No (finite) amount of computation will suffice to prove that apparent regularities persist indefinitely It turns out, as we shall see, that this sequence does have much hidden structure In fact, there are simple rules that work much faster than one-stepat-a-time; there is even a formula that enables us to compute a(n) for any n However it is not immediately clear how to prove these results What follows is an account of my search for structure in Conway's sequence, culminating in a proof that the structure that is found does in fact persist Then there is an explanation of how Conway's number was calculated Finally, there are comments on some related problems My own interest in this problem is two-fold First, there is the simple pleasure to be found in taking up a challenge But also, since I am by profession a statistician, I am interested in the process by which hidden structure (usually in statistical data, rather than a mathematical puzzle, as in this case) can be found John Tukey and I have remarked [2] that success in an exploratory investigation is crucially dependent on
TL;DR: In this paper, Gram-Schmidt Orthogonalization by Gauss Elimination is considered and it is shown that it is possible to perform Gauss elimination by Grammatical decomposition.
Abstract: (1991). Gram-Schmidt Orthogonalization by Gauss Elimination. The American Mathematical Monthly: Vol. 98, No. 6, pp. 544-549.
TL;DR: In this paper, a continuous nowhere-differentiable function f: [0, 11 -] R is the uniform limit of a sequence of piecewise linear continuous functions with steep slopes.
Abstract: then f is also a continuous nowhere-differentiable function. (See [3, p. 115].) The above examples have concise definitions and establish the existence of continuous nowhere-differentiable functions. However, it is not easy to visualize or guess what their graphs look like, let alone to see intuitively why they work. Our continuous nowhere-differentiable function f: [0, 11 -] R is the uniform limit of a sequence of piecewise linear continuous functions fn: [0, 1] -+ R with steep slopes. In constructing the sequence (f,,>, we will be using a contraction mapping w from the family of all closed subsets of X = [0, 1] x [0, 1] into itself with respect to the Hausdorff metric induced by the Euclidean metric. (A contraction mapping on a metric space (Y, d) is a function g: Y -+ Y for which there is a positive constant k of sets converges to the graph of f in the Hausdorff metric. If A is the diagonal of slope 1 in the square X, then wn(A) is the graph of fn, and hence, a reader can obtain an intuitive idea of the graph of f. This idea first occurred to me when I attended Mr. Gary Church's master's thesis defense at San Jose State University at which he talked about attractors of contraction mappings (see [4]).
TL;DR: The American Mathematical Monthly: Vol. 98, No 8, No. 8, pp. 710-718 as discussed by the authors, is a seminal work in linear algebra and quantum chemistry.
Abstract: (1991). Linear Algebra and Quantum Chemistry. The American Mathematical Monthly: Vol. 98, No. 8, pp. 710-718.
TL;DR: In this paper, distance functions and topologies are studied in the context of distance function and topology, and the authors propose a distance function based topology for distance function topologies.
Abstract: (1991). Distance Functions and Topologies. The American Mathematical Monthly: Vol. 98, No. 7, pp. 620-623.
TL;DR: In a calculus course, most students do not master the theory to the point that they can write a coherent proof of any but the most trivial theorems as discussed by the authors, and most students also do not learn to do the calculations we put on the exam.
Abstract: When we discuss teaching undergraduate mathematics with our colleagues, we frequently distinguish two sorts of goals: first, we want students to be able to do computations related to the course and, second, we want students to know the theory from the course. In a calculus course, we want students to be able to integrate polynomials, so in class we tell them how, and on the exam; we give them polynomials to integrate. In linear algebra class, we want the students to know the theorems concerning determinants, so we prove the theorems from the book, and on the exam, we ask them to prove that if A is an invertible matrix, then det(A-1) = 1/det(A). Our hallway conversations elicit general agreement that most of our students do learn to do the calculations we put on the exam and most don't master the theory to the point that they can write a coherent proof of any but the most trivial theorems. I want to argue in this note that there is another goal, between these in accessibility, that should be a conscious part of our teaching effort:
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AT&T1
TL;DR: In this paper, an affirmative answer is given for a special class of bodies of revolution that seemed lik... M.S. Ulam once asked if spheres are the only homogeneous bodies that can float in every orientation.
Abstract: S. M. Ulam once asked if spheres are the only homogeneous bodies that can float in every orientation. Here an affirmative answer is given for a special class of bodies of revolution that seemed lik...
TL;DR: DeVanEY as mentioned in this paper is a former chair of the mathematics department at Boston University, where he served as chair from 1983 to 1986, specializing in Dynamical Systems, with special interests in Hamiltonian mechanics, complex analytic dynamics, and computer experimentation.
Abstract: ROBERT L. DEVANEY did his undergraduate work at Holy Cross and received his Ph.D. from the University of California at Berkeley in 1973. He taught at Northwestern University, Tufts University, and the University of Maryland before coming to Boston University, where he served as chair of the mathematics department from 1983 to 1986. His research is centered in Dynamical Systems, with special interests in Hamiltonian mechanics, complex analytic dynamics, and computer experimentation.
TL;DR: In this article, a moment convergence theorem was proposed for the case of moment convergence in the setting of the moment convergence of a moment and a moment in a moment graph, where the moment convergence theorem holds.
Abstract: (1991). A Moment Convergence Theorem. The American Mathematical Monthly: Vol. 98, No. 8, pp. 742-746.