Showing papers in "American Mathematical Monthly in 2008"

Sudoku, Gerechte Designs, Resolutions, Affine Space, Spreads, Reguli, and Hamming Codes

Abstract: Solving a Sudoku puzzle involves putting the symbols 1, . . . , 9 into the cells of a 9 × 9 grid partitioned into 3 × 3 subsquares, in such a way that each symbol occurs just once in each row, column, or subsquare. Such a solution is a special case of a gerechte design, in which an n×n grid is partitioned into n regions with n squares in each, and each of the symbols 1, . . . , n occurs once in each row, column, or region. Gerechte designs originated in statistical design of agricultural experiments, where they ensure that treatments are fairly exposed to localised variations in the field containing the experimental plots. In this paper we consider several related topics. In the first section, we define gerechte designs and some generalizations, and explain a computational technique for finding and classifying them. The second section looks at the statistical background, explaining how such designs are used for designing agricultural experiments, and what additional properties statisticians would like them to have. In the third section, we focus on a special class of Sudoku solutions which we call “symmetric”. They turn out to be related to some important topics in finite geometry over the 3-element field, and to ∗This research partially supported by NSF Grant Number DMS-0510625.

The Hermite—Hadamard Inequality on Simplices

Abstract: (2008). The Hermite—Hadamard Inequality on Simplices. The American Mathematical Monthly: Vol. 115, No. 4, pp. 339-345.

An Elementary Proof of the Hitting Time Theorem

Abstract: In this note, we give an elementary proof of the random walk hitting time theorem, which states that, for a left-continuous random walk on Z starting at a nonnegative integer k, the conditional probability that the walk hits the origin for the first time at time n, given that it does hit zero at time n, is equal to k/n. Here, a walk is called left-continuous when its steps are bounded from below by −1. We start by introducing some notation. Let Pk denote the law of a random walk starting at k ≥ 0, let {Yi } ∞=1 be the independent and identically distributed (i.i.d.) steps of the random walk, let Sn = k + Y1 +···+ Yn be the position of the random walk starting at k after n steps, and let T0 = inf{n : Sn = 0}

Lost (and Found) in Translation: André's Actual Method and Its Application to the Generalized Ballot Problem

Abstract: (2008). Lost (and Found) in Translation: Andre's Actual Method and Its Application to the Generalized Ballot Problem. The American Mathematical Monthly: Vol. 115, No. 4, pp. 358-363.

An Elementary Proof of Marden's Theorem

Abstract: (2008). An Elementary Proof of Marden's Theorem. The American Mathematical Monthly: Vol. 115, No. 4, pp. 330-338.

Mathematical people : profiles and interviews

Abstract: This unique collection contains extensive and in-depth interviews with mathematicians who have shaped the field of mathematics in the twentieth century. Collected by two mathematicians respected in the community for their skill in communicating mathematical topics to a broader audience, the book is also rich with photographs and includes an introduction by Philip J. Davis.

From Mobius To Gyrogroups

Abstract: Abraham A. UngarAmer. Math. Monthly, Vol. 115(2), (2008), pp. 138–144.1. IntroductionThe evolution from Mo¨bius to gyrogroups began in 1988 [12], and is still ongoingin [14, 15]. Gyrogroups, a natural generalization of groups, lay a fruitful bridgebetween nonassociativealgebraand hyperbolicgeometry, just asgroupslayafruitfulbridge between associative algebra and Euclidean geometry. More than 150 yearshave passed since the German mathematician August Ferdinand Mo¨bius (1790–1868) first studied the transformations that now bear his name [9, p. 71]. Yet, therich structure he thereby exposed is still far from being exhausted, as the evolutionfrom Mo¨bius to gyrogroups demonstrates.2. FromM¨obiusMo¨bius transformations of the complex open unit disc D= {z ∈ C: |z| < 1} ofthe complex plane Care studied in most books on function theory of one complexvariable. According to [10, p. 176], these are important for at least two reasons:(i) they play a central role in non-Euclidean geometry [15], and (ii) they are theonly automorphisms of the disc. Ahlfors’ book [1],

Surprising Sinc Sums and Integrals

Abstract: We show that a variety of trigonometric sums have unexpected closed forms by relating them to cognate integrals.

Qualitative Tools for Studying Periodic Solutions and Bifurcations as Applied to the Periodically Harvested Logistic Equation

Abstract: (2008). Qualitative Tools for Studying Periodic Solutions and Bifurcations as Applied to the Periodically Harvested Logistic Equation. The American Mathematical Monthly: Vol. 115, No. 3, pp. 202-219.

Fair Majority Voting (or How to Eliminate Gerrymandering)

Abstract: 1. THE PROBLEM. Something is rotten in the electoral state of the United States. Mathematics is involved. Advances in computer technology—hardware and software— have permitted a great leap “forward” in the fine art of political gerrymandering—“the practice of dividing a geographical area into electoral districts, often of highly irregular shape, to give one political party an unfair advantage by diluting the opposition’s voting strength” (according to Black’s Law Dictionary). It is generally acknowledged that some four hundred of the 435 seats in the House of Representatives are “safe,” and many claim that districting determines elections, not votes. Recent congressional elections (especially those of 2002 and 2004)— summarized in Table 1—show the shocking impact of gerrymandering. Incumbent candidates, in tailored districts, are almost certain of reelection (over 98% in 2002 and 2004, over 94% in 2006). If an election is deemed “competitive” when the spread in votes between the winner and the runner-up is 6% or less, then 5.5% of the elections were competitive in 2002, 2.3% in 2004 and 9.0% in 2006. Many candidates ran unopposed by a candidate from one of the two major parties in all three elections. In Michigan, the Democratic candidates together out-polled the Republican candidates by some 35,000 votes in 2002, yet elected only six representatives to the Republican’s nine. In the 2002 Maryland elections, Republican representatives needed an average of 376,455 votes to be elected, the Democratic representatives only 150,708. In the 2004 Connecticut elections, the Democratic candidates as a group out-polled the Republican candidates by over 156,000 votes; nevertheless, only two were elected to the Republican’s three. In all three elections Massachusetts elected only Democrats: in 2002 six of the ten were elected without Republican opposition, in 2004 five and in 2006 seven. Ohio elected eleven Republican and seven Democratic representatives in 2006, and yet the Democratic candidates received 211,347 more votes than did the

Hilbert's inequality and Witten's zeta-function

Abstract: 1. INTRODUCTION. In this article we explore a variety of pleasing connections between analysis, number theory, and operator theory, while revisiting a number of beautiful inequalities originating with Hilbert, Hardy and others. We shall first discuss the aforementioned Hilbert inequality [14], [18] and then apply it to various multiple zeta values. In consequence we obtain the norm of the classical Hilbert matrix, in the process illustrating the interplay of numerical and symbolic computation with classical mathematics. 2. HILBERT'S (EASIER) INEQUALITY. The inequality in question is:

Primitive Juggling Sequences

Abstract: (2008). Primitive Juggling Sequences. The American Mathematical Monthly: Vol. 115, No. 3, pp. 185-194.

Reciprocity Relations for Bernoulli Numbers

Abstract: (2008). Reciprocity Relations for Bernoulli Numbers. The American Mathematical Monthly: Vol. 115, No. 3, pp. 237-244.

The Carathéodory and Kobayashi Metrics and Applications in Complex Analysis

Abstract: The Caratheodory and Kobayashi metrics have proved to be important tools in the function theory of several complex variables But they are less familiar in the context of one complex variable Our purpose here is to gather in one place the basic ideas about these important invariant metrics for domains in the plane and to provide some illuminating examples and applications

Optimal Congressional Apportionment

Abstract: The purpose of this article is to expand the list of divisor methods to include those associated with the logarithmic and identric means. We show that these two methods stem from mirror-image objective functions in terms of optimization, in the same sense that the Hill and Webster methods have mirror-image objective functions. We then explore the connections of the various objective functions to information theory and statistics, concluding that the identric mean and arithmetic mean (Webster) objective functions are the most natural, and moreover that the former has certain theoretical advantages. Finally, we compare optimal congressional allocations associated with the four means: geometric, logarithmic, identric, and arithmetic. Our principal reference on congressional apportionment is the comprehensive monograph by Balinski and Young [1], which provides rich historical context, empiri cal support for the Webster method, and Appendix A which lays out the mathematical structure, including optimization criteria. We adopt their notation and avoidance of positive lower bounds, which are easily added to divisor calculations; we also ignore the upper bounds which are currently inactive and which could easily be added to the calculations if necessary. Two recent articles in this Monthly have dealt with the apportionment problem. Grimmett [4] proposed a randomized (lottery) scheme which guarantees quota in an expected value sense in the absence of lower bounds. Balinski [2] presented a general theory of coherence for apportionment problems, including the congressional problem.

Summing a Curious, Slowly Convergent Series

Abstract: (2008). Summing a Curious, Slowly Convergent Series. The American Mathematical Monthly: Vol. 115, No. 6, pp. 525-540.

Triangles, Ellipses, and Cubic Polynomials

Abstract: (2008). Triangles, Ellipses, and Cubic Polynomials. The American Mathematical Monthly: Vol. 115, No. 8, pp. 679-689.

How to compute $\sum 1/ n^2$ by solving triangles

Abstract: A new elementary proof is given for Euler's classical formula for the sum of squared reciprocals of natural numbers.

On Sums of Positive Integers That Are Not of the Form ax + by

Abstract: (2008). On Sums of Positive Integers That Are Not of the Form ax + by. The American Mathematical Monthly: Vol. 115, No. 4, pp. 363-364.

A Peculiar Connection Between the Axiom of Choice and Predicting the Future

Abstract: 1. INTRODUCTION. We often model systems that change over time as functions from the real numbers R (or a subinterval of R) into some set S of states, and it is often our goal to predict the behavior of these systems. Generally, this requires rules governing their behavior, such as a set of differential equations or the assumption that the system (as a function) is analytic. With no such assumptions, the system could be an arbitrary function, and the values of arbitrary functions are notoriously hard to predict. After all, if someone proposed a strategy for predicting the values of an arbitrary function based on its past values, a reasonable response might be, "That is impossible. Given any strategy for predicting the values of an arbitrary function, one could just define a function that diagonalizes against it: whatever the strategy predicts, define the function to be something else." This argument, however, makes an appeal to induction: to diagonalize against the proposed strategy at a point t, we must have already defined our function for all s 0 such that the prediction is valid on [t, t + s). Noting that any countable set of reals has measure 0, we can restate this informally: at almost every instant t, the strategy predicts some "?-glimpse" of the future. We should emphasize that these results do not give a practical means of predicting

A Probabilistic Proof of Wallis's Formula for π

Abstract: (2008). A Probabilistic Proof of Wallis's Formula for π. The American Mathematical Monthly: Vol. 115, No. 8, pp. 740-745.

Periodic Orbits for Billiards on an Equilateral Triangle

Abstract: 1. INTRODUCTION. The trajectory of a billiard ball in motion on a frictionless billiards table is completely determined by its initial position, direction, and speed. When the ball strikes a bumper, we assume that the angle of incidence equals the angle of reflection. Once released, the ball continues indefinitely along its trajectory with constant speed unless it strikes a vertex, at which point it stops. If the ball returns to its initial position with its initial velocity direction, it retraces its trajectory and continues to do so repeatedly; we call such trajectories periodic. Nonperiodic trajectories are either infinite or singular; in the later case the trajectory terminates at a vertex.

When Is a Periodic Function the Curvature of a Closed Plane Curve

Abstract: (2008). When Is a Periodic Function the Curvature of a Closed Plane Curve? The American Mathematical Monthly: Vol. 115, No. 5, pp. 405-414.

Derivatives and Eulerian numbers

Abstract: The aim of the present note is to give a closed form formula for the nth deriva-tive of a function which satis es Riccati’s di erential equation with constantcoecients. The paper is organized as follows. In section 2 we recall the de -nition and basic properties of Eulerian numbers. We introduce and prove thementioned formula in section 3. Some examples where the formula can be ap-plied are included in section 4. In one of them, using a formula derived byHamilton, we arrive at an elegant integral expression for Eulerian numbers.

Edge Detection Using Fourier Coefficients

Abstract: numerical method used to solve nonlinear partial differential equations (PDEs), is an example of such a method [12] The method approximates the Fourier coefficients of the solution of a PDE The Fourier coefficients are then used to calculate an approx imation to the solution The accurate reconstruction of the solution requires that the positions of the discontinuities of the solution be known [5] In this paper we discuss techniques for using a function's Fourier coefficients to determine the locations and sizes of the jump discontinuities of the function At first glance the spectral representation of the signal?the Fourier series or trans form associated with the signal?does not seem to be the ideal place to look for information about discontinuities in the signal When a signal is discontinuous the con vergence of the Fourier series or transform associated with the signal is not uniform; in such cases the Gibbs phenomenon [11] appears and truncating the series after any finite number of terms always leads to 0(1) oscillations in the reconstructed signal (For a nice, detailed treatment of the Gibbs phenomenon, see [6]) Considering the question again, however, one realizes that if a discontinuity is characterized by a "phenomenon," then the existence of the discontinuity is indeed encoded in the coefficients The question becomes how to effectively "decode" the discontinuity One does not do this by directly summing the series?one uses the spec tral representation in a somewhat different way to "concentrate" the function about the discontinuity In what follows, we explain how this is done We restrict ourselves to pe riodic (or compactly supported) functions and only consider Fourier series (Those in terested in seeing a more general theory of concentration factors are referred to [3,4]) Much of the information in this article is well known [3, 4] The use of the Euler

Arithmetic in the Ring of Formal Power Series with Integer Coefficients

Abstract: of polynomials R[x] over R, namely the ring ^[[x]] of formal powers series in one variable over R, is hardly ever mentioned in such a course. In most cases, it is relegated to the homework problems (or to the exercises in the textbooks), and one learns that, like R[x], R[[x]] is an integral domain provided that R is an integral domain. More surprising is to learn that, in contrast to the situation of polynomials, in R[[x]] there are many invertible elements: while the only units in R[x] are the units of R, a necessary and sufficient condition for a power series to be invertible is that its constant term be invertible in R. This fact makes the study of arithmetic in /?[[*]] simple when R is a field: the only prime element is the variable x. As might be expected, the study of prime factorization in Z[[x]] is much more interesting (and complicated), but to the best of our knowledge it is not treated in detail in the available literature. After some basic considerations, it is apparent that the question of deciding whether or not an integral power series is prime is a difficult one, and it seems worthwhile to develop criteria to determine irreducibility in Z[[x]] similar to Eisenstein's criterion for polynomials. In this note we propose an easy argument that provides us with an infinite class of irreducible power series over Z. As in the case of Eisenstein's criterion in 7L\x\, our criteria give only sufficient conditions, and the question of whether or not a given power series is irreducible remains open in a vast array of cases, including quadratic polynomials. It is important to note that irreducibility in Z[x] and in Z[[x]] are, in general, un related. For instance, 6 + x + x2 is irreducible in Z[x] but can be factored in Z[[x]], while 2 + Ix + 3x2 is irreducible in Z[[x]] but equals (2 + x)(l +3x) as a polyno mial (observe that this is not a proper factorization in Z[[x]] since 1 + 3x is invert

Venn Symmetry and Prime Numbers: A Seductive Proof Revisited

Abstract: An n-Venn diagram is a Venn diagram on n sets, which is defined to be a collection of n simple closed curves (Jordan curves) C1,C2, ,Cn in the plane such that any two intersect in finitely many points and each of the 2n sets of the form ∩C i i is nonempty and connected, where i is one of “interior” or “exterior” Thus the Venn regions are all bounded except for the region exterior to all curves; each bounded region is the interior of a Jordan curve See [6] for much more information on Venn diagrams An n-Venn diagram is symmetric if each curve Ci is ρ i (C1), where ρ is a rotation of order n about some center (we use O for the fixed point of rotation ρ) We use Boolean notation for combinations of sets, with the 0-1 string e1e2 en representing ∩C i i , where i is interior (respectively, exterior) if ei = 1 (respectively, 0) Thus 111 1 represents F , the full intersection of all the interiors, 000 0 is the intersection of all the exteriors (the unbounded region), and 100 0 represents the set of points interior to C1 and exterior to the others In a symmetric Venn diagram, rotation of a region by ρ corresponds to a rightward cyclic shift of the Boolean string The universally familiar three-circle Venn diagram is symmetric, as is the one on two sets using two circles For about 40 years a major open question was whether symmetric n-Venn diagrams exist for all prime n Henderson found one for n = 5 and also (unpublished) for n = 7 Much later, Hamburger [3] settled the case of 11, which was quite complicated, and then in 2004 Griggs, Killian, and Savage [1] found an approach that works for all primes So we now have the strikingly beautiful theorem that a symmetric n-Venn diagram exists if and only if n is prime But there is a small problem: Henderson’s proof, which appears to be very simple, has a gap Here is the proof from [4] Suppose 1 ≤ k ≤ n − 1 Since a symmetric n-Venn diagram is symmetric with respect to a rotation of 2π/n, the regions corresponding to the Boolean strings with k 1s must come in groups of size n, each group consisting of one such region and its images under repeated rotation by 2π/n Therefore n divides (n k ) This concludes the proof because the only n for which this is true for the specified k-values are the primes (an easy-to-prove fact of number theory; see [5]) This is a very seductive argument The primeness arises in such a cute way that one wants it to be true Thus the proof has been repeated in many papers in the decades since it was first published Yet there are problems The proof does not call upon the connectedness of the Venn regions Without connectedness the result is false; see Figure 1 (due to Grunbaum [2]), which shows a diagram satisfying all of the conditions

Easy Proofs of Some Borwein Algorithms for π

Abstract: 1. E. Artin, Einf?hrung in die Theorie der Gammafunktion, Teubner, Leipzig, 1931. (English translation by M. Butler, The Gamma Function, Holt, Rinehart, and Winston, San Francisco, CA, 1964.) 2. G. D. Birkhoff, Note on the gamma function, Bull. Amer. Math. Soc. 20 (1914) 1-10. 3. H. Bohr and J. Mollerup, L rebog i matematisk Analyse, vol. 3, Jul. Gjellerups Forlag, Copenhagen, 1922. 4. A. Pringsheim, Zur Theorie der Gamma-Functionen, Math. Annalen 31 (1888) 455-481. 5. R. Remmert, Wielandt's theorem about the r-function, this MONTHLY 103 (1996) 214-220. 6. -, Classical Topics in Complex Function Theory, Springer-Verlag, New York, 1998. 7. W. Rudin, Principles of Mathematical Analysis, 3rd ed., McGraw-Hill, New York, 1976. 8. G. K. Srinivasan, The gamma function: An eclectic tour, this Monthly 114 (2007) 297-315. 9. D. V. Widder, Advanced Calculus, Prentice-Hall, Englewood Cliffs, NJ, 1961.

Prime Number Patterns

Abstract: (2008) Prime Number Patterns The American Mathematical Monthly: Vol 115, No 4, pp 279-296

Inflating the Cube Without Stretching

Abstract: Here by isometric we mean that the geodesic distance between pairs of points on the non-convex polyhedron is always equal to the geodesic distance between of the corresponding pairs of points on a cube. Alternatively, it means that two surfaces can be triangulated in such a way that they now consist of congruent triangles which are glued according to the same combinatorial rules. For example, if we push a vertex v of a cube C inside as shown in the Figure below, we obtain a polyhedron P whose surface ∂P is isometric to S = ∂C. Of course, vol(P ) < vol(C) in this case.