Showing papers in "American Mathematical Monthly in 2014"
TL;DR: In this article, the authors give two proofs of a folklore result relating numerical semigroups of embedding dimension two and binary cyclotomic polynomials and explore some consequences.
Abstract: We give two proofs of a folklore result relating numerical semigroups of embedding dimension two and binary cyclotomic polynomials and explore some consequences. In particular, we give a more conce...
42 citations
TL;DR: The topics discussed here are basic tropical algebra, tropical plane curves, some tropical intersections, and Viro's patchworking.
Abstract: This friendly introduction to tropical geometry is meant to be accessible to first year students in mathematics. The topics discussed here are basic tropical algebra, tropical plane curves, some tropical intersections, and Viro's patchworking. Each definition is explained with concrete examples and illustrations. To a great exten, this text is an updated of a translation from a french text by the first author. There is also a newly added section highlighting new developments and perspectives on tropical geometry. In addition, the final section provides an extensive list of references on the subject.
32 citations
TL;DR: Current mathematical trends in the field are discussed, including mathematical approaches that are able to address critical questions associated with tumor initiation; angiogenesis and vascular tumor growth; and the new frontier of computer-aided, patient-specific cancer evaluation and treatment.
Abstract: Mathematical and computational modeling approaches have been applied to every aspect of tumor growth from mutation acquisition and tumorigenesis to metastasis and treatment response. In thi...
28 citations
TL;DR: In this paper, asymptotic formulas for central extended binomial coefficients were derived using the distribution of the sum of independent discrete uniform rando(s) of a binomial coefficient.
Abstract: We derive asymptotic formulas for central extended binomial coefficients, which are generalizations of binomial coefficients, using the distribution of the sum of independent discrete uniform rando...
22 citations
TL;DR: It is shown how positive unital linear maps can be used to derive many of the inequalities in matrix analysis that render this principle more precise.
Abstract: The farther a normal matrix is from being a scalar, the more dispersed its eigenvalues should be. There are several inequalities in matrix analysis that render this principle more precise. Here it ...
22 citations
TL;DR: Harimsri joined the University of Victoria faculty there in 1969 and has held other faculty positions and visiting positions at many universities and research institutes around the world.
Abstract: faculty there in 1969. He has held other faculty positions and visiting positions at many universities and research institutes around the world. He has written more than 1000 papers and has collaborated with over 400 co-authors. His other publications include (for example) 21 books, monographs, and edited volumes. See http://www.math.uvic.ca/faculty/harimsri/. Department of Mathematics and Statistics, University of Victoria, British Columbia V8W 3R4, Canada harimsri@math.uvic.ca
21 citations
TL;DR: The axioms of set theory are often referred to as 'the' axiomatic of set theories as mentioned in this paper. But few of us could accurately quote what are referred as 'axioms' in set theory.
Abstract: Mathematicians manipulate sets with confidence almost every day, rarely making mistakes. Few of us, however, could accurately quote what are often referred to as ‘the' axioms of set theory. This su...
19 citations
TL;DR: An explicit analytical solution for the problem of minimization of the function i.e., the coordinates of the stationary point and the corresponding critical value as functions of {mj, xj, yj }3j are presented.
Abstract: We present an explicit analytical solution for the problem of minimization of the function i.e., we find the coordinates of the stationary point and the corresponding critical value as functions of {mj, xj, yj }3j. In addition, we also discuss the inverse problem of finding such values for m1, m2, and m3 for which the corresponding function F possesses a prescribed position of stationary point.
19 citations
TL;DR: This paper presented four remarkable facts about quotient sets, and these observations seem to have been overlooked by the Monthly, despite its intense coverage of quotient set over the years, despite the fact that they have been well-known in the literature.
Abstract: Our aim in this note is to present four remarkable facts about quotient sets. These observations seem to have been overlooked by the Monthly, despite its intense coverage of quotient sets over the years.
19 citations
TL;DR: The equivalence of the three approaches to knot coloring is presented and the simple combinatorial invariant suggested is equivalent to the well-known Fox n-coloring of arcs and lesser-known Dehn n-Coloring of regions.
Abstract: The 1926 paper of J. W. Alexander and G. B. Briggs suggests a simple combinato- rial invariant by coloring the crossings of a knot diagram. It is equivalent to the well-known Fox n-coloring of arcs and lesser-known Dehn n-coloring of regions. The equivalence of the three approaches to knot coloring is presented.
17 citations
TL;DR: The mathematical framework that governs the interaction of a forcegenerating microorganism with a surrounding viscous fluid is presented and the role of a dinoflagellate transverse flagellum as well as the flow structures near a choanoflagingllate is investigated.
Abstract: We present the mathematical framework that governs the interaction of a forcegenerating microorganism with a surrounding viscous fluid. We review slender-body theories that have been used to study flagellar motility, along with the method of regularized Stokeslets. We investigate the role of a dinoflagellate transverse flagellum as well as the flow structures near a choanoflagellate.
TL;DR: It is shown that the main geometric features of classical conics can be retrieved from more general equidistant sets, which can be thought of as a natural generalization for conics.
Abstract: This article is devoted to the study of classical and new results concerning equidistant sets, both from the topological and metric point of view. We include a review of the most interesting known ...
TL;DR: The corrected forms are shown to be connected with the problem of closed-form evaluation of series involving the Zeta functions, which happens to be an extensively-investigated subject since the time of Euler as in the classical three-century-old Goldbach theorem.
Abstract: We first present the corrected expression for a certain widely-recorded generalized Goldbach-Euler series The corrected forms are then shown to be connected with the prob- lem of closed-form evaluation of series involving the Zeta functions, which happens to be an extensively-investigated subject since the time of Euler as (for example) in the classical three-century old-Goldbach theorem
TL;DR: Her research is mainly in symbolic and algebraic dynamics and knot theory, and she has known her co-authors since they were graduate students together.
Abstract: SUSAN G. WILLIAMS is yet another Professor of mathematics at the University of South Alabama with a doctorate from Yale. Her advisor was Shizuo Kakutani. She has known her co-authors since they were graduate students together. Susan’s early interest in mathematics was encouraged by the Ross Program at Ohio State and her undergraduate experience at the University of Chicago. Her research is mainly in symbolic and algebraic dynamics and knot theory. Mathematical origami is one of her hobbies. Department of Mathematics and Statistics, University of South Alabama, Mobile, AL 36688 swilliam@southalabama.edu
TL;DR: In this paper, the Euclidean version of the Heron problem was revisited from a numerical perspective and a fast algorithm for solving it was proposed by exploiting the majorization minimization principle of computational statistics and rudimentary techniques from differential calculus.
Abstract: In a recent issue of this MONTHLY, Mordukhovich, Nam, and Salinas pose and solve an interesting non-differentiable generalization of the Heron problem in the framework of modern convex analysis. In the generalized Heron problem, we are given kC 1 closed con- vex sets in R d equipped with its Euclidean norm and asked to find the point in the last set such that the sum of the distances to the first k sets is minimal. In later work, the authors gen- eralize the Heron problem even further, relax its convexity assumptions, study its theoretical properties, and pursue subgradient algorithms for solving the convex case. Here, we revisit the original problem solely from the numerical perspective. By exploiting the majorization- minimization (MM) principle of computational statistics and rudimentary techniques from differential calculus, we are able to construct a very fast algorithm for solving the Euclidean version of the generalized Heron problem.
TL;DR: In this paper, a proof of Polya's random walk theorem using classical methods from special function theory and asymptotic analysis is presented, and the proof is used to prove polya's theorem.
Abstract: This note presents a proof of Polya's random walk theorem using classical methods from special function theory and asymptotic analysis.
TL;DR: An algebra of functions A enjoying the following properties is constructed, which complements those made by Cater and by Kim and Kwon, and published in the American Mathematical Monthly in 1984 and 2000, respectively.
Abstract: The Identity Theorem states that an analytic function (real or complex) on a connected domain is uniquely determined by its values on a sequence of distinct points that converge to a point of its domain. This result is not true in general in the real setting, if we relax the analytic hypothesis on the function to infinitely many times differentiable. In fact, we construct an algebra of functions A enjoying the following properties: (i) A is uncountably infinitely generated (that is, the cardinality of a minimal system of generators of A is uncountable); (ii) every nonzero element of A is nowhere analytic; (iii) A subset of C-infinity (R); (iv) every element of A has infinitely many zeros in R; and (v) for every f is an element of A\ {0} and n is an element of N, f((n)) (the nth derivative of f) enjoys the same properties as the elements in A\ {0}.
This construction complements those made by Cater and by Kim and Kwon, and published in the American Mathematical Monthly in 1984 and 2000, respectively.
TL;DR: Families A = (Ai)iεI of sets of nonnegative integers, each set containing 0, such that every nonnegative integer can be written uniquely in the form, with ai ε Ai for all i, and ai ≠ for only finitely many i.
Abstract: This paper proves a theorem of de Bruijn that classifies additive systems for the nonnegative integers, that is, families A = (Ai)ieI of sets of nonnegative integers, each set containing 0, such th...
TL;DR: Abstract Let g be a complex simple Lie algebra of rank ℓ, h the Coxeter number, m1, m2, …, mℓ the exponents of g, and C the Cartan matrix.
Abstract: Let g be a complex simple Lie algebra of rank l, h the Coxeter number, m1, m2, …, ml the exponents of g, and C the Cartan matrix Then
TL;DR: Two higher-dimensional generalizations of this invariant ratio of the catenary curve are developed and it is found that each invariant ratios identifies a class of hypersurfaces connected to classical objects from differential geometry.
Abstract: A well-known property of the catenary curve is that the ratio of the area under the curve to the arc length of the curve is independent of the interval over which these quanti- ties are concurrently measured. We develop two higher-dimensional generalizations of this invariant ratio, and find that each invariant ratio identifies a class of hypersurfaces connected to classical objects from differential geometry. 1. INTRODUCTION. The largest man-made monument in the United States is an inverted weighted catenary. 1 Standing 630 feet tall, the Gateway Arch in Saint Louis sits aside the Mississippi river as a symbolic passageway to the west and commem- orates the westward expansion of the United States, realized in part by the Louisiana purchase of 1803. Although this arch supports only its own weight, catenary-shaped arches have been used for centuries to provide passageways through weight-bearing walls. To see why a catenary arch is so stable, we observe that an idealized chain suspended between two points will always try to form a catenary. Such a chain expe- riences only tension forces; inverting the shape into an arch reverses these into pure compression forces which act along the curve and never at right angles to it. The 17th- century English scientist Robert Hooke phrased it best: "as hangs the flexible chain, so but inverted will stand the rigid arch." (See Figure 1.)
TL;DR: A student-friendly proof that the weighted Hermite polynomials form a complete orthonormal system (a basis) for the collection L2(ℝ) of real-valued functions is presented.
Abstract: We present a student-friendly proof that the weighted Hermite polynomials form a complete orthonormal system (a basis) for the collection L2(ℝ) of real-valued functions.
TL;DR: Marden's theorem characterizes the critical points of complex polynomials of degree 3 in a nice geometrical way and the proof of the theorem is based directly on the defining property of ellipses.
Abstract: Marden's theorem characterizes the critical points of complex polynomials of degree 3 in a nice geometrical way Our proof of the theorem is based directly on the defining property of ellipses
TL;DR: It is shown how logical circuits and piecewise linear equations are being used to meet the challenge of successfully model gene control in complex organisms.
Abstract: Living organisms contain thousands of genes. These genes contain the genetic code for protein molecules required for the functioning of the organism. During the development of the organisms, genes ...
TL;DR: A proof for a Perron-type theorem on the principal eigenvalue of nonsymmetric elliptic operators based on the strong maximum principle is provided.
Abstract: We provide a proof for a Perron-type theorem on the principal eigenvalue of nonsymmetric elliptic operators based on the strong maximum principle. This proof is modeled after a variational proof of...
TL;DR: A completely elementary proof of Booker's result, suitable for presentation in a first course in number theory, is given.
Abstract: Let q1 = 2. Supposing that we have defined qj for all 1 j k, let qk+1 be a prime factor of 1 + Q k j=1 qj. As was shown by Euclid over two thousand years ago, q1,q2,q3,... is then an infinite sequence of distinct primes. The sequence {qi} is not unique, since there is flexibility in the choice of the prime qk+1 dividing 1 + Q k j=1 qj. Mullin suggested studying the two sequences formed by (1) always taking qk+1 as small as possible, and (2) always taking qk+1 as large as possible. For each of these sequences, he asked whether every prime eventually appears. Recently, Booker showed that the second sequence omits infinitely many primes. We give a completely elementary proof of Booker's result, suitable for presentation in a first course in number theory.
TL;DR: It is shown that the exponential bound on the probability of a large deviation for sampling with replacement applies also to sampling without replacement, which includes as a special case the relationship between the binomial and hypergeometric distributions.
Abstract: We give a simple argument, based on drawing balls from urns, showing that the ex- ponential bound on the probability of a large deviation for sampling with replacement applies also to sampling without replacement. This result includes as a special case the relationship between the binomial and hypergeometric distributions. 1. INTRODUCTION. Two distributions that are encountered early in any course on probability are the binomial and the hypergeometric. For the binomial distribu- tion, we typically imagine a possibly biased coin that comes up heads with probabil- ity p and tails with probability 1 p. If we flip the coin independently n times, the number of times K that heads comes up has the binomial distribution: PrTKD kUD n k p k .1 p/ nk . One of the great virtues of the binomial distribution is that almost everything about it can be easily calculated or estimated by direct manipulation of simple expressions. Consider, for example, estimating the probability of a large devi- ation; that is, the probability that K exceeds its expectation pn by at least qn, where 0 < q < 1 p. To obtain such an estimate, we consider the probability generating function gK.u/D ExTu K U
TL;DR: The main purpose of this paper is to discuss an interpretation of the scheme for evaluating Wallis-type infinite products by means of Pólya urn models.
Abstract: A famous “curious identity” of Wallis gives a representation of the constant π in terms of a simply structured infinite product of fractions. Sondow and Yi [Amer. Math. Monthly 117 (2010) 9...
TL;DR: A combinatorial proof of the identity for the alternating convolution of the central binomial coefficients is given by applying an involution to certain colored permutations and showing that only permutations containing cycles of even length remain.
Abstract: We give a combinatorial proof of the identity for the alternating convolution of the central binomial coefficients. Our proof entails applying an involution to certain colored permutations and showing that only permutations containing cycles of even length remain. The combinatorial identity n X
TL;DR: Bailey and Borwein this article, 2014, American Mathematical Monthly, vol. 121, p. 191 and 206, reported that pi day is upon us again and we still do not know if pi is normal.
Abstract: Paper 23: David H. Bailey and Jonathan Borwein, “Pi day is upon us again and we still do not know if pi is normal,” American Mathematical Monthly, vol. 121 (2014), p. 191–206. Copyright 2014 Mathematical Association of America. All Rights Reserved.
TL;DR: Some of the central concepts and basic results in phylogenetics, which benefit from several branches of mathematics, including combinatorics, probability, and algebra, are explained.
Abstract: The idea that all life on earth traces back to a common beginning dates back at least to Charles Darwin's Origin of Species. Ever since, biologists have tried to piece together parts of this ‘tree ...