Showing papers in "American Mathematical Monthly in 2012"
TL;DR: In this paper, the classical Heron problem was extended to the nonempty closed convex subsets of a convex subset of the plane, where the objective is to find a point C such that the sum of the distances from C to the given points A and B is minimal.
Abstract: The classical Heron problem states: on a given straight line in the plane, find a point C such that the sum of the distances from C to the given points A and B is minimal This problem can be solved using standard geometry or differential calculus In the light of modern convex analysis, we are able to investigate more general versions of this problem In this paper we propose and solve the following problem: on a given nonempty closed convex subset of ℝs, find a point such that the sum of the distances from that point to n given nonempty closed convex subsets of ℝs is minimal
44 citations
TL;DR: Higher-dimensional generalizations of the SVD are discussed, which have become increasingly crucial with the newfound wealth of multidimensional data, and have launched new research initiatives in both theoretical and applied mathematics.
Abstract: The singular value decomposition (SVD) is a popular matrix factorization that has been used widely in applications ever since an efficient algorithm for its computation was developed in the 1970s. In recent years, the SVD has become even more prominent due to a surge in applications and increased computational memory and speed. To illustrate the vitality of the SVD in data analysis, we highlight three of its lesser-known yet fascinating applications. The SVD can be used to characterize political positions of congressmen, measure the growth rate of crystals in igneous rock, and examine entanglement in quantum computation. We also discuss higher-dimensional generalizations of the SVD, which have become increasingly crucial with the newfound wealth of multidimensional data, and have launched new research initiatives in both theoretical and applied mathematics. With its bountiful theory and applications, the SVD is truly extraordinary.
43 citations
TL;DR: These new indices accurately capture realistic data on income distribution, and give a better picture of how income data is shifting over time.
Abstract: Income distribution is described by a two-parameter model for the Lorenz curve. This model interpolates between self-similar behavior at the low and high ends of the income spectrum, and naturally ...
34 citations
TL;DR: In this paper, the authors formalized Gauss' geometric notion of winding number in the real-algebraic setting, from which they derived a real algebraic proof of the Funda-mental Theorem of Algebra.
Abstract: Sturm's theorem (1829/35) provides an elegant algorithm to count and locate the real roots of any real polynomial. In his residue calculus (1831/37), Cauchy extended Sturm's method to count and locate the complex roots of any complex polynomial. For holomorphic functions Cauchy's index is based on contour integration, but in the special case of polyno- mials it can effectively be calculated via Sturm chains using euclidean division as in the real case. In this way we provide an algebraic proof of Cauchy's theorem for polynomials over any real closed field. As our main tool, we formalize Gauss' geometric notion of winding number (1799) in the real-algebraic setting, from which we derive a real-algebraic proof of the Funda- mental Theorem of Algebra. The proof is elementary inasmuch as it uses only the intermediate value theorem and arithmetic of real polynomials. It can thus be formulated in the first-order language of real closed fields. Moreover, the proof is constructive and immediately translates to an algebraic root-finding algorithm. L'algest g´ enelle donne souvent plus qu'on lui demande. (Jean le Rond d'Alembert) 1
24 citations
TL;DR: A result frequently attributed to Napoleon Bonaparte is the topic of this note; it has an interesting history, and there are a considerable number of papers devoted to it.
Abstract: A result frequently attributed to Napoleon Bonaparte is the topic of this note; it has an interesting history, and there are a considerable number of papers devoted to it. Several relevant articles...
23 citations
TL;DR: It is proved that if n ≥ 4, then none of the elementary symmetric functions of 1, 1⁄2, …, 1¬1⁄n are integers.
Abstract: In 1946, P. Erdős and I. Niven proved that there are only finitely many positive integers n for which one or more of the elementary symmetric functions of 1, 1⁄2, …, 1⁄n are integers. In th...
20 citations
TL;DR: An elementary formula for the volume of arbitrary hyperplane sections of the n-dimensional cube is deduced and its application in various dimensions is shown.
Abstract: We deduce an elementary formula for the volume of arbitrary hyperplane sections of the n-dimensional cube and show its application in various dimensions.
17 citations
TL;DR: A family of conformal (angle preserving) projections of the sphere onto the plane is considered, referred to as the Lambert conic conformal projections, which include the Mercator map and the stereographic projection.
Abstract: We consider a family of conformal (angle preserving) projections of the sphere onto the plane. The family is referred to as the Lambert conic conformal projections. Special cases include the Mercat...
15 citations
TL;DR: The computation of square roots in ancient India in the context of the discovery of positional decimal arithmetic is examined.
Abstract: This article examines the computation of square roots in ancient India in the context of the discovery of positional decimal arithmetic.
15 citations
TL;DR: The minimal cardinalities of coverings and irredundant coverings of a vector space over an arbitrary field by proper linear subspaces are computed.
Abstract: We compute the minimal cardinalities of coverings and irredundant coverings of a vector space over an arbitrary field by proper linear subspaces. Analogues for affine linear subspaces are also given. Notation: The cardinality of a set S will be denoted by #S. For a vector space V over a field K , we denote its dimension by dim K. 1. LINEAR COVERINGS. Let V be a vector space over a field K. A linear cov- ering of V is a collectionfWigi2I of proper K -subspaces such that V D S i2I Wi . A linear covering is irredundant if for all J ( I , S i2J Wi6D V. Linear coverings exist if and only if dim V 2. The linear covering number LC.V/ of a vector space V of dimension at least 2 is the least cardinality #I of a linear coveringfWigi2I of V. The irredundant linear covering number ILC.V/ is the least cardinality of an irredundant linear covering of V. Thus LC.V/ ILC.V/. The main result of this note is a computation of LC.V/ and ILC.V/. Main Theorem. Let V be a vector space over a field K , with dim V 2. (a) If dim V and #K are not both infinite, then LC.V/D #KC 1.
15 citations
TL;DR: A family of magic squares is introduced, called linear magic squares, and it is shown that any parallel linear sudoku solution of sufficiently large order can be relabeled so that all of its subsquares are linear magic.
Abstract: We introduce a family of magic squares, called linear magic squares, and show that any parallel linear sudoku solution of sufficiently large order can be relabeled so that all of its subsquares are linear magic. As a consequence, we show that if n has prime factoriza- tion p k 1 1 p kt t and qD minf p k j j j 1 j tg, then there is a family of q.q 1/ mutually orthogonal magic sudoku solutions of order n 2 whenever q > 3; such an orthogonal family is complete if n is a prime power.
TL;DR: For example, Ethiopian Dinner is a game in which two players take turns eating morsels from a commo... as discussed by the authors, where the goal is to make sure that each player gets her favorite mouthful.
Abstract: If you are sharing a meal with a companion, then how is it best to make sure you get your favourite mouthfuls? Ethiopian Dinner is a game in which two players take turns eating morsels from a commo...
TL;DR: The mean value theorem of integral calculus states that for any continuous realvalued function f on an interval I, and for any two distinct real numbers a, b∈ I, there exists a value V(a, b) in the open interval between a and b that can be viewed as a two-variable mean on the interval I.
Abstract: The mean value theorem of integral calculus states that for any continuous realvalued function f on an interval I, and for any two distinct real numbers a, b∈ I, there exists a value V(a, b) in the...
TL;DR: A useful upper bound for the measure of spherical caps is proved and it is shown that spherical caps are ellipsoidal caps and the standard deviation of these caps is zero.
Abstract: We prove an useful upper bound for the measure of spherical caps. Consider the uniformly distributed measure σn−1 on the Euclidean unit sphere Sn−1 ⊂ R. On the sphere, as among only a handful other spaces, the isoperimetric problem is completely solved. This goes back to Levy [Le] and Schmidt [Sch] and states that caps have the minimal measure of a boundary among all sets with a fixed mass. For e ∈ [0, 1) and θ ∈ Sn−1 the cap C(e, θ), or shortly C( ), is a set of points x ∈ Sn−1 for which x · θ ≥ e, where · stands for the standard scalar product in R. See figure 1. Figure 1: A cap C(e, θ). A few striking properties of the high-dimensional sphere are presented in [Ba, Lecture 1, 8]. In such considerations, we often need a good estimation of the measure of a cap. Following the method used in [Ba, Lemma 2.2], we extend its proof to the skipped case of large e and get in an elementary way the desired bound. Theorem. For any e ∈ [0, 1) σn−1 (C(e)) ≤ e−ne 2/2. Figure 2: Small e. Proof. In the case of small e, for convenience, we repeat a beautiful argument used by Ball. Namely, for e ∈ [0, 1/ √ 2] we have (see Figure 2) σn−1 (C(e)) = voln (Cone ∩B(0, 1)) voln (Bn(0, 1)) ≤ voln ( B(P, √ 1− e2) ) voln (Bn(0, 1)) = √ 1− e2 n ≤ e−ne 2/2. For e ∈ [1/ √ 2, 1), it is enough to consider a different auxiliary ball which includes the set Cone ∩B(0, 1), see Figure 3. We obtain σn−1 (C(e)) ≤ voln (B (Q, r))) voln (Bn(0, 1)) = r = ( 1 2e )n ≤ e−ne 2/2, where the last inequality follows from the estimate Figure 3: Large e. By the congruence 1/2 r = e 1 . ∗Institute of Mathematics, University of Warsaw, Banacha 2, 02-097 Warszawa, Poland. t.tkocz@students.mimuw.edu.pl
TL;DR: A geometrical interpretation of submajorization is provided in which multisets are represented by points in an orbifold and bijections between multiset are representedBy paths between these points, which shows that sub majorization is closely related to the geometric principle that the shortest path between two points is a straight line.
Abstract: In this paper, we use submajorization to compare distances between either multisets of real numbers or multisets modulo translation on the real line. We provide a geometrical in- terpretation in which multisets are represented by points in an orbifold and bijections between multisets are represented by paths between these points. This interpretation shows that sub- majorization is closely related to the geometrical principle that the shortest path between two points is a straight line. Our results have applications to diverse problems from economics to music theory; moreover, they suggest generalizations of statistical measures of the center and spread of a distribution.
TL;DR: In this article, the probability that there are ever the same number of black and white balls in the polya urn, where b > w, was shown to be 1.
Abstract: We consider a Polya urn, started with b black and w white balls, where b > w. We compute the probability that there are ever the same number of black and white balls in the urn, and show that it is...
TL;DR: With two key identities the obstacle is overcome, proving the desired result and who discovered the requisite identities?
Abstract: It is tempting to try to reprove Euler's famous result that using power series methods of the sort taught in calculus 2. This leads to , the evaluation of which presents an obstacle. With two key i...
TL;DR: In this paper, the limits of a class of periodic continued radicals are derived and a connection between them and the fixed points of the Chebycheff polynomials is established.
Abstract: We compute the limits of a class of periodic continued radicals and we establish a connection between them and the fixed points of the Chebycheff polynomials.
TL;DR: In this article, the decomposition theory was introduced in the context of algebraic geometry and deals with the configuration of curves on an algebraic graph. But it is not related to our work.
Abstract: In a 1962 paper, Zariski introduced the decomposition theory that now bears his name. Although it arose in the context of algebraic geometry and deals with the configuration of curves on an algebra...
TL;DR: The sum of the reciprocals of all monic polynomials of a given degree over a finite field each raised to the power of k has a surprisingly simple result due to mysterious cancellations that occur in the sum.
Abstract: We consider the sum of the reciprocals of all monic polynomials of a given degree over a finite field 픽q each raised to the power of k. When k ≤ q, the sum has a surprisingly simple result due to mysterious cancellations that occur in the sum. We discuss this interesting phenomenon and provide a new inductive proof.
TL;DR: A new technique for proving and discovering some inequalities is introduced and it is shown that some inequalities can be proved and discovered using this technique.
Abstract: The aim of this note is to introduce a new technique for proving and discovering some inequalities.
TL;DR: The technique of parametrization of plane algebraic curves is presented from a number theorist's point of view and Kapferer's simple and beautiful proof that nonsingular curves of degree > 2 cannot be parametrized by rational functions is presented.
Abstract: We present the technique of parametrization of plane algebraic curves from a number theorist's point of view and present Kapferer's simple and beautiful (but little known) proof that nonsingular cu...
TL;DR: It is shown that Szegö's distribution formula for the eigenvalues of a family of Toeplitz matrices and g is real-valued and continuous on [-π, π].
Abstract: Suppose that -∞ < a < b < ∞, a ≤u1n ≤ u2n ≤… ≤unn ≤ b, and a ≤ v1n ≤ v2n ≤ … ≤ vnn ≤ b for n ≥ 1.We simplify and strengthen Weyl's definition of equal distribution of by showing that the fo...
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TL;DR: A general evaluation of the sinc integral evaluation is given, which is entirely explicit in the case of the product of three sinc functions.
Abstract: We resolve and further study a sinc integral evaluation, first posed in this Monthly in [1967, p. 1015], which was solved in [1968, p. 914] and withdrawn in [1970, p. 657]. After a short introducti...
TL;DR: It is shown how to reduce the periodic or antiperiodic Sturm-Liouville problems to an analysis of the Prüfer angle, providing a simple and flexible alternative to the usual approaches via operator theory or the Hill discriminant.
Abstract: It is shown how to reduce the periodic or antiperiodic Sturm-Liouville problems to an analysis of the Prufer angle This provides a simple and flexible alternative to the usual approaches via operator theory or the Hill discriminant
TL;DR: This paper shows how it is shown that for each fixed k, there are only finitely many odd perfect numbers with at most k distinct prime factors, and how this result, and many like it, follow from embedding the natural numbers in the supernatural numbers and imposing an appropriate topology on the latter.
Abstract: Since ancient times, a natural number has been called perfect if it equals the sum of its proper divisors; e.g., 6 = 1 + 2 + 3 is a perfect number. In 1913, Dickson showed that for each fixed k, th...
TL;DR: A class of functions satisfying what is called the Almost-Near property, or briefly the (AN) property is introduced and several examples, such as polynomial functions, complex exponential and covering maps are developed.
Abstract: We introduce a class of functions satisfying what we call the Almost-Near property, or briefly the (AN) property. The motivation of this investigation is provided by the phenomenon “almost implies ...
TL;DR: The following problem: does there exist a curve such that one can walk around it so that, at all moments, the two tangent segments to the curve have unequal lengths?
Abstract: There are two tangent segments to a strictly convex closed plane curve from every point in its exterior. We discuss the following problem: does there exist a curve such that one can walk around it so that, at all moments, the two tangent segments to the curve have unequal lengths?
TL;DR: It is proved that in this case the mathematical expectation of the length of the game is finite, and in principle it is equivalent to the graph of theGame of war, which has edges corresponding to all acceptable transitions, having the following property.
Abstract: The game of war is a popular international children's card game. In the beginning of the game, the deck is split into two parts, then each player reveals their top card. The player having the highest card collects both and returns them to the bottom of their hand. The player left with no cards loses. It is often wrongly assumed that this game is deterministic and the result is set once the cards have been dealt. However, this is not so; the rules of the game do not prescribe the order in which the winning player will place their cards on the bottom of the hand. First, we provide an example of a cycling game with fixed rules and then assume that each player can seldom but regularly change the returning order. We have proved that in this case the mathematical expectation of the length of the game is finite. In principle it is equivalent to the graph of the game, which has edges corresponding to all acceptable transitions, having the following property: from each initial configuration there is at least one path to the end of the game.
TL;DR: The total angular distance traversed by the spiral of Theodorus is governed by the Schneckenkonstante K, which is expressed as a sum of Riemann zeta-values at the half-integers and computed to 100 decimal places.
Abstract: The total angular distance traversed by the spiral of Theodorus is governed by the Schneckenkonstante K introduced by Hlawka. The only published estimate of K is the bound K 0:75. We express K as a sum of Riemann zeta-values at the half-integers and compute it to 100 decimal places. We find similar formulas involving the Hurwitz zeta-function for the analytic Theodorus spiral and the Theodorus constant introduced by Davis. 1. INTRODUCTION. Theodorus of Cyrene (ca. 460-399 B.C.) taught Plato math- ematics and was himself a pupil of Protagoras. Plato's dialogue Theaetetus tells that Theodorus was distinguished in the subjects of the quadrivium and also contains the following intriguing passage on irrational square-roots, quoted here from (12): (Theodorus) was proving to us a certain thing about square roots, I mean of three square feet and of five square feet, namely that these roots are not commensu- rable in length with the foot-length, and he went on in this way, taking all the separate cases up to the root of 17 square feet, at which point, for some reason, he stopped. It was discussed already in antiquity why Theodorus stopped at seventeen and what his method of proof was. There are at least four fundamentally different theories—not in- cluding the suggestion of Hardy and Wright that Theodorus simply became tired!—cf. (11, 12, 16). One of these theories is due to the German amateur mathematician J. Anderhub, cf. (4, 14). It involves the so-called square-root spiral of Theodorus or Quadratwurzel- schnecke. This spiral consists of a sequence of points P1, P2, P3;::: in the plane cir- culating anti-clockwise around a centre P0 such thatjP0 PnjD p n andjPn PnC1jD 1