Showing papers in "American Mathematical Monthly in 2006"

The Prehistory of the Hardy Inequality

Abstract: (2006). The Prehistory of the Hardy Inequality. The American Mathematical Monthly: Vol. 113, No. 8, pp. 715-732.

Infinitely Divisible Matrices

Abstract: Assembling natural numbers in such nice patterns often has interesting consequences, and so it is in this case. Each of the aforementioned matrices is endowed with positive definiteness of a very high order: for every positive real number r the matrices with entries a\-, b\-, c\-, and d\are positive semidefinite. This special property is called infinite divisibility and is the subject of this paper. Positive semidefinite matrices arise in diverse contexts: calculus (Hessians at minima of functions), statistics (correlation matrices), vibrating systems (stiffness matrices), quantum mechanics (density matrices), harmonic analysis (positive definite functions), to name just a few. Many of the test matrices used by numerical analysts are positive definite. One of the interests of this paper might be the variety of examples that are provided in it. The general theorems and methods presented in the context of these examples are, in fact, powerful techniques that could be used elsewhere. In this introductory section we begin with the basic definitions and notions related to positive semidefinite matrices.

Prime Number Races

Abstract: This talk is a survey of “prime number races”. Chebyshev noticed in the first half of the nineteenth century that for any given value of x, there always seem to be more primes of the form 4n+3 less than x than there are of the form 4n+1. Similar observations have been made with primes of the form 3n+2 and 3n+1, primes of the form 10n+3, 10n+7 and 10n+1, 10n+9, and many others besides. More generally, one can consider primes of the form qn + a, qn + b, qn + c, . . . for our favorite constants q, a, b, c, . . . and try to figure out which forms are “preferred” over the others – not to mention figuring out what, precisely, being “preferred” means. We describe these phenomena in greater detail and explain the efforts that have been made at understanding them. This talk should be accessible to graduate students. 1272547773

Noncrossing Partitions in Surprising Locations

Abstract: 1. Introduction.Certain mathematical structures make a habit of reoccuring in the most diverselist of settings. Some obvious examples exhibiting this intrusive type of behaviorinclude the Fibonacci numbers, the Catalan numbers, the quaternions, and themodular group. In this article, the focus is on a lesser known example: the non-crossing partition lattice. The focus of the article is a gentle introduction to thelattice itself in three of its many guises: as a way to encode parking functions, asa key part of the foundations of noncommutative probability, and as a buildingblock for a contractible space acted on by a braid group. Since this article is aimedprimarily at nonspecialists, each area is briefly introduced along the way.The noncrossing partition lattice is a relative newcomer to the mathematicalworld. First defined and studied by Germain Kreweras in 1972 [33], it caught theimagination of combinatorialists beginning in the 1980s [20], [21], [22], [23], [29],[37], [39], [40], [45], and has come to be regarded as one of the standard objectsin the field. In recent years it has also played a role in areas as diverse as low-dimensional topology and geometric group theory [9], [12], [13], [31], [32] as wellas the noncommutative version of probability [2], [3], [35], [41], [42], [43], [49],[50]. Due no doubt to its recent vintage, it is less well-known to the mathematicalcommunity at large than perhaps it deserves to be, but hopefully this short paperwill help to remedy this state of affairs.2. A motivating example.Before launching into a discussion of the noncrossing partition lattice itself, wequickly consider a motivating example: the Catalan numbers. The Catalan num-bers are a favorite pastime of many amateur (and professional) mathematicians. Inaddition, they also have a connection with the noncrossing partition lattice (Theo-rem 3.1).Example 2.1(Catalan numbers). The Catalan numbers are the numbers C

On the Fundamental Theorem of Algebra

Abstract: One of the simplest proofs that every nontrivial polynomial P has a zero goes as follows. Observe that |P(z)| → ∞ as |z| → ∞, so we may find an R > 0 with |P(z)| > |P(0)| for all |z| ≥ R. Since any real-valued continuous function on a compact set attains a minimum, |P(z)| attains a minimum for |z| ≤ R at some point z1 and, by the previous sentence, this must be a global minimum. By translation, we may suppose z1 = 0 and, by multiplying P by a constant, we may suppose that a0 = P(0) is real and nonnegative. Now we know that P(z) = a0 − ∑N r=m ar zr with am = 0 and m ≥ 1. Choose ω so that b = amω is real and strictly positive. Then, if η is real and small,

Monotonicity Rules in Calculus

Abstract: 1. RULES FOR MONOTONICITY. In the first semester of calculus a student learns that if a function f is continuous on an interval [a, b] and has a positive (negative) derivative on (a, b), then f is increasing (decreasing) on [a, b]. This result is obtained easily by means of the Lagrange mean value theorem. The functions that the student proves monotone in this way are usually polynomials, rational functions, or other elementary functions. If one is attempting to establish the monotonicity of a quotient of two functions, one often finds that the derivative of the quotient is quite messy and the process tedious. Several authors have developed refinements of this method for proving monotonicity of quotients. The first such refinement of which we are aware is the following one by M. Gromov [11, p. 42], which appears in his work in differential geometry (Gromov’s proof uses only monotonicity and elementary properties of integrals):

Simple Norm Inequalities

Abstract: (2006). Simple Norm Inequalities. The American Mathematical Monthly: Vol. 113, No. 3, pp. 256-260.

Fastest mixing markov chain on a path

Abstract: (2006). Fastest Mixing Markov Chain on a Path. The American Mathematical Monthly: Vol. 113, No. 1, pp. 70-74.

Recovering a function from a dini derivative

Abstract: provides a clear answer if we can assume that F' is Riemann integrable. Students of analysis will learn that if F' is Lebesgue integrable the same formula can be used, interpreting the integral in this more general sense. A full resolution of the problem requires a more general integral still, that of Denjoy and Perron (known frequently now as the Henstock-Kurzweil integral). The main question of this paper is, as it was for Lebesgue, whether a function can be recovered as an indefinite integral of one of its Dini derivatives?that is, when does the formula

Ten Problems in Experimental Mathematics

Abstract: This article was stimulated by the recent SIAM ''100 DigitChallenge'' of Nick Trefethen, beautifully described in a recent book. Indeed, these ten numeric challenge problems are also listed in a recent book by two of present authors, where they are followed by the ten symbolic/numeric challenge problems that are discussed in this article. Our intent was to present ten problems that are characteristic of the sorts of problems that commonly arise in ''experimental mathematics''. The challenge in each case is to obtain a high precision numeric evaluation of the quantity, and then, if possible, to obtain a symbolic answer, ideally one with proof. Our goal in this article is to provide solutions to these ten problems, and in the process present a concise account of how one combines symbolic and numeric computation, which may be termed ''hybrid computation'', in the process of mathematical discovery.

A Short Proof of the Simple Continued Fraction Expansion of e

Abstract: (2006). A Short Proof of the Simple Continued Fraction Expansion of e. The American Mathematical Monthly: Vol. 113, No. 1, pp. 57-62.

The Pythagorean Theorem: What Is It About?

Abstract: (2006). The Pythagorean Theorem: What Is It About? The American Mathematical Monthly: Vol. 113, No. 3, pp. 261-265.

More Formulas for π

Abstract: Proof. If the weak topology of H were metrizable, then because of the Baire category theorem and Corollary 2, H couldn’t be complete in this metric. There would have to exist a weakly fundamental sequence {gk}k=1 in H not converging to any vector in H . Since the sequence is weakly fundamental, for every h in H the limit limk→∞〈h, gk〉 would exist. In particular, every such sequence would be bounded. Again by the uniform boundedness principle also the norms ‖gk‖ would have to be bounded. But then by the weak compactness of every set Bo,n the sequence would converge weakly to some vector in H , a contradiction to the initial assumption.

A New Proof of Euclid's Theorem

Abstract: (2006). A New Proof of Euclid's Theorem. The American Mathematical Monthly: Vol. 113, No. 10, pp. 937-938.

A $1 Problem

Abstract: Since the proof of [3] is based on the maximum principle, which is available in its full generality only for (cooperative systems of) second order equations, one would tend to believe that such type of result cannot hold for (1) or (2). Trying to get a negative answer to the question whether or not positive solutions are radially symmetric will necessarily lead to a strict p.d.e. approach and will hence be hard to obtain. The radially decreasing part of the claim however allows an o.d.e.-counterexample as we will show shortly. Let us fix this part in a conjecture:

A Geometric Proof That e Is Irrational and a New Measure of Its Irrationality

Abstract: The proof leads in section 3 to a new measure of irrationality for e, that is, a lower bound on the distance from e to a given rational number, as a function of its denominator. A connection with the greatest prime factor of a number is discussed in section 4. In section 5 we compare the new irrationality measure for e with a known one, and state a numbertheoretic conjecture that implies the known measure is almost always stronger. The new measure is applied in section 6 to prove a special case of a result from [24], leading to another conjecture. Finally, in section 7 we recall a theorem of G. Cantor that can be proved by a similar construction.

Asymmetric Rhythms and Tiling Canons

Abstract: rhythms, melodic structures, harmonies, and lyrics in the songs we hear. This article focuses on what may be the most important of these aspects: rhythm. We examine the mathematics of some rhythmic structures common in popular and folk music. Anyone who listens to rock music is familiar with the repeated drum beat?one, two, three, four?based on a 4/4 measure. Fifteen minutes listening to a Top 40 ra dio station offers evidence enough that most rock music has this basic beat (Audio Example l).1 But if we tune the radio to different frequencies, we may hear popular music (jazz, Latin, African) with different characteristic rhythms (Audio Example 2). Although much of this music is also based on the 4/4 measure, some instruments play repeated patterns that are not synchronized with the 4/4 beat, creating syncopation? an exciting tension between different components of the rhythm. This article is con cerned with classifying and counting rhythms that are maximally syncopated in the sense that, even when shifted, they cannot be synchronized with the division of a mea sure into two parts. In addition, we discuss rhythms that cannot be aligned with other even divisions of the measure. Our results have a surprising application to rhythmic canons.

Bernstein Polynomials and Brownian Motion

Abstract: One of the greatest pleasures in mathematics is the surprising connections that often appear between apparently disconnected ideas and theories. Some particularly striking instances exist in the interaction between probability theory and analysis. One of the simplest is the elegant proof of the Weierstrass approximation theorem by S. Bernstein [2]: on the surface, this states that if f : [0, 1] → C is a continuous function, then the sequence of Bernstein polynomials

A Geometric Interpretation of an Infinite Product for the Lemniscate Constant

Abstract: is known as the lemniscate constant. Here B(x, y) and V(z) are the beta and gamma functions, respectively, whose definitions we recall in the next section. The lemniscate constant gets its name from the fact that the arclength of the lemniscate with polar equation r2 = cos(2#) is given by 2L, just as the arclength of the unit circle is given by 2tt. In our work, the lemniscate constant will arise in the guise of an area. We will see that the area enclosed by the curve C4 defined by x4 + y4 ? 1 is La/2, and this will allow us to give a geometric meaning to the product formula (2). For more details on the remarkable lemniscate constant, we refer the reader to [11, pp. 420-423]. Equation (1) is classical. It was discovered in 1593 by Fran?ois Vi?te. It was the first exact analytic expression ever discovered for tt and constitutes the first known use of an infinite product in mathematics. Vi?te discovered the formula by considering the areas of regular 2"-gons inscribed in a unit circle. For Vi?te's original paper, in Latin or translated into English, see [5]. For a discussion of Vi?te's product and its place in the history of mathematics and tt see [4, pp. 92-96]. Equation (2) appeared in [17] as a consequence of a general method for constructing similar infinite product formulas. We will show that the similarity between equations (1) and (2) goes beyond mere typographical appearances. We will see that (2) is related to the curve C4 in much the same way that Vi?te's product is related to the circle.

Simpson's Rule Is Exact for Quintics

Abstract: (2006). Simpson's Rule Is Exact for Quintics. The American Mathematical Monthly: Vol. 113, No. 2, pp. 144-155.

Dynamical Systems Method and a Homeomorphism Theorem

Abstract: 1. R. B. Banks, Slicing Pizzas, Racing Turtles, and Further Adventures in Applied Mathematics, Princeton University Press, Princeton, 1999. 2. A. C. Doyle, “The Stone of Boxman’s Drift,” in Uncollected Stories: The Unknown Conan Doyle, Doubleday, New York, 1982. 3. G. Shortley and D. Williams, Elements of Physics, Prentice-Hall, Englewood Cliffs, NJ, 1971. 4. A. J. Simoson, Falling down a hole through the Earth, Math. Mag. 77 (2004) 171–189.

Waiting Times for Patterns and a Method of Gambling Teams

Abstract: We flip a fair coin five times. Which pattern is “more difficult” to get: HHHHH or HTHTH? If we posed this question to the typical man on the street, the most likely answer would be: the first one. Of course, we know that this answer is not correct, for both patterns have the same probability of occurring, namely, 1/32. However, there is a sense in which a street-smart person is, in fact, correct. If we flip the coin without stopping, then the average waiting time until the first occurrence of the pattern HHHHH is 62, whereas for the pattern HTHTH it is 42. From this perspective, the pattern HHHHH is indeed “more difficult” to get. Now, if we ask a person familiar with probability theory (but unfamiliar with this particular topic) to rank the average waiting times until the patterns HHHHH, HHHHT , HHHTH, and HTHTH occur, then most likely the first pattern would get rank 1 (the longest average waiting time), the second—2, the third—3, and the last one—4 (the shortest average waiting time). This ranking is based on the “intuitive” idea that long runs of the same outcomes, such as HHHH or HHHHH, require more time until they occur. In fact, the average waiting times are 62, 32, 34, and 42, respectively. All the foregoing waiting times are easily and elegantly obtained by using martingale theory and the “optional stopping theorem,” as shown in the classical paper by Li [16] and briefly described in section 2. Our focus in the present paper is the average time until we observe the first of several different patterns. Suppose, for instance, that Melanie flips a coin until she observes either HHHTH or HTHTH, while Kyle flips another coin until he observes either HHHHT or HHHTH. Since Kyle was assigned the two patterns with the shortest waiting times, 32 and 34 versus 34 and 42, one would expect him to have a shorter average waiting time. In fact, the averages are the same—22 for both Melanie and Kyle. Let us present another counterintuitive fact. Consider the two patterns HHHHT and HHHTH. What is the probability that in a stochastic sequence of heads and tails the pattern HHHHT appears earlier than HHHTH? Since the average waiting times (32 and 34, respectively) are close to each other, one might guess that the probability would be reasonably close to 1/2. However, the exact answer is 2/3! As we will see, this probability is determined by the relationship between patterns rather than by their individual average waiting times. Finally, consider two special patterns: a run of Hs of length r and a run of T s of length ρ . The expected waiting time until the run of Hs is 2 − 2, while for the run of T s it is 2 − 2. We can ask: what is the expected time until either of these two runs happens for the first time? Using results presented in this article

The Broken Spaghetti Noodle

Abstract: A popular question in recreational mathematics is the following: If we drop a spaghetti noodle and it breaks at two random places, what is the probability that we can form a triangle with the three resulting segments? See, for example, [2, chap. 1, sec. 6], [3, p. 6], [4, p. 31], or [7, pp. 30–36]. This is an elementary problem in geometric probability. Clearly the length of the noodle (or equivalently our choice of unit length) does not matter, so the problem amounts to choosing two numbers at random from the interval (0, 1), say a and b with a < b, and looking at the resulting intervals (0, a), (a, b), and (b, 1). We will be able to form a triangle when the positive numbers a, b − a, and 1 − b satisfy the triangle inequality (i.e., when no interval is longer than the combined lengths of the other two). Equivalently, this will be the case when all three intervals have length less than 1/2. Therefore, a triangle can be formed precisely when the following three inequalities hold: a < 1/2, b − a < 1/2, and b > 1/2. Figure 1 shows all possible outcomes 0 < a < b < 1, and the darker shaded region consists of all “favorable” outcomes, when a triangle can be formed. Comparing areas, we see that the probability of succeeding in getting a triangle is 1/4.

Reflections on the Arbelos

Abstract: 1. INTRODUCTION. Written in block capitals on lined paper, the letter bore a postmark from a Northeastern seaport. \" I have a problem that I would like to solve, \" the letter began, \" but unfortunately I cannot. I dropped out from 2nd year high school, and this problem is too tough for me. \" There followed the diagram shown in Figure 1 along with the statement of the problem: to prove that the line segment DE has the same length as the line segment AB.

The Constructive Mathematics of A. A. Markov

Abstract: Andrei Andreevich Markov Jr. (born September 22, 1903, in St. Petersburg; died Oc tober 11, 1979, in Moscow) was the late and only child of the great Russian mathe matician Andrei Andreevich Markov Sr. (born June 14, 1856, in Ryazan7; died July 20, 1922, in Petrograd), universally recognized, in particular, for his contributions to the theory of probability (e.g., Markov chains and Markov processes). At his father's suggestion, young Andrei entered the chemistry section of the School of Physics and Mathematics at the University of Petrograd (formerly St. Petersburg, then Leningrad, and now again St. Petersburg). The young man was fascinated by chemistry and al ready by 1920 had taken part in chemical research. The results of his and his coau thor's work were published in 1924. Thus Markov's first paper dealt with chemistry. In his sophomore year he became interested in theoretical physics, and he graduated in 1924 with a physics degree. Markov's publications in chemistry were followed by a series of papers on the three body problem and dynamical systems (1926-1937), and a paper on Schr?dinger's quantum mechanics. The latter was one of the first papers on quantum mechanics published in the U.S.S.R., appearing less than a year after Schr?dinger's own ground breaking series of publications. In this connection it should be noted that it was Markov who, in 1931, introduced the concept of an abstract (topological) dynamical system. In 1932 Markov published an intriguing paper (in German) on relativity titled "Deriving a World Metric from the Relation 'Earlier Than.' " His interest in abstract mathematics is represented by series of papers on topology, algebra, analysis, and geometry. After World War II Markov's interests turned to axiomatic set theory, mathematical logic, and the foundations of mathematics. He founded the Russian school of con structive mathematics in the late 1940s and early 1950s. But in private conversations Markov often said that he had nurtured constructivist convictions for a very long time, in fact, long before the war. The Moscow mathematical school had been interested in constructivism, especially intuitionism, since its inception in the 1920s. It is enough to mention the 1925 work of Kolmogorov on intuitionistic logic [10]. It may be that this interest was due, at

The Poincare conjecture

Abstract: On November 11, 2002, Grigoriĭ Yakovlevich Perelman, a geometer working in the St. Petersburg section of the Steklov Mathematical Institute at Fontanka 27, published on the internet a forty-page paper titled “An Entropy Formula for the Ricci Flow and Its Geometric Applications.” The fourth page of the dry introduction, full of technical terms, ends with the sentence: Finally, in Section 13, we give a brief sketch of a proof of the Geometrization Conjecture.

Congruent Numbers and Elliptic Curves

Abstract: (2006). Congruent Numbers and Elliptic Curves. The American Mathematical Monthly: Vol. 113, No. 4, pp. 308-317.

A Proof of the Continued Fraction Expansion of 1/M

Abstract: (2006). A Proof of the Continued Fraction Expansion of e1/M. The American Mathematical Monthly: Vol. 113, No. 1, pp. 62-66.

Another Short Proof of Descartes's Rule of Signs

Abstract: (2006). Another Short Proof of Descartes's Rule of Signs. The American Mathematical Monthly: Vol. 113, No. 9, pp. 829-830.

Solids Circumscribing Spheres

Abstract: (2006). Solids Circumscribing Spheres. The American Mathematical Monthly: Vol. 113, No. 6, pp. 521-540.