Showing papers in "American Mathematical Monthly in 2004"

The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities

Abstract: This lively, problem-oriented text, first published in 2004, is designed to coach readers toward mastery of the most fundamental mathematical inequalities With the Cauchy-Schwarz inequality as the initial guide, the reader is led through a sequence of fascinating problems whose solutions are presented as they might have been discovered - either by one of history's famous mathematicians or by the reader The problems emphasize beauty and surprise, but along the way readers will find systematic coverage of the geometry of squares, convexity, the ladder of power means, majorization, Schur convexity, exponential sums, and the inequalities of Holder, Hilbert, and Hardy The text is accessible to anyone who knows calculus and who cares about solving problems It is well suited to self-study, directed study, or as a supplement to courses in analysis, probability, and combinatorics

Cauchy's Interlace Theorem for Eigenvalues of Hermitian Matrices

Abstract: Hermitian matrices have real eigenvalues. The Cauchy interlace theorem states that the eigenvalues of a Hermitian matrix A of order n are interlaced with those of any principal submatrix of order n − 1. Theorem 1 (Cauchy Interlace Theorem). Let A be a Hermitian matrix of order n, and let B be a principal submatrix of A of order n − 1 .I fλn ≤ λn−1 ≤ · ·· ≤λ2 ≤ λ1 lists the eigenvalues of A and µn ≤ µn−1 ≤ · ·· ≤µ3 ≤ µ2 the eigenvalues of B, then λn ≤ µn ≤ λn−1 ≤ µn−1 ≤ · ·· ≤λ2 ≤ µ2 ≤ λ1.

Surprises from mathematics education research: student (mis)use of mathematical definitions

Abstract: 1. INTRODUCTION. The authors of this paper met at a summer institute sponsored by the Oregon Collaborative for Excellence in the Preparation of Teachers (OCEPT). Edwards is a researcher in undergraduate mathematics education. Ward, a pure mathematician teaching at an undergraduate institution, had had little exposure to mathematics education research prior to the OCEPT program. At the institute, Edwards described to Ward the results of her Ph.D. dissertation [5] on student understanding and use of definitions in undergraduate real analysis. In that study, tasks involving the definitions of “limit” and “continuity,” for example, were problematic for some of the students. Ward’s intuitive reaction was that those words were “loaded” with connotations from their nonmathematical use and from their less than completely rigorous use in elementary calculus. He said, “I’ll bet students have less difficulty or, at least, different difficulties with definitions in abstract algebra. The words there, like ‘group’ and ‘coset,’ are not so loaded.” Eventually, with OCEPT support, the authors studied student understanding and use of definitions in an introductory abstract algebra course populated by undergraduate mathematics majors and taught by Ward. The “surprises” in the title are outcomes that surprised Ward, among others. He was surprised to see his algebra students having difficulties very similar to those of Edwards’s analysis students. (So he lost his bet.) In particular, he was surprised to see difficulties arising from the students’ understanding of the very nature of mathematical definitions, not just from the content of the definitions. Upon hearing of Edwards’s dissertation work, some other mathematicians who teach undergraduates found those difficulties surprising even when restricted to real analysis. Hereafter, we present a simple two-part theoretical framework borrowed from philosophy and from mathematics education literature. Although it is not our intent to give an extensive report of either study, we next indicate the methodology used in Edwards’s dissertation and in our joint abstract algebra study so that the reader may know the context from which our observations are drawn. We then list the “surprising” difficulties of the two groups of students, documenting them with examples from the studies and using the framework to provide a possible explanation for them. We conclude with what we see as the implications for undergraduate teaching, along with some specific classroom activities that the studies and our experience as teachers suggest might be of value. 2. FRAMEWORK. It is commonly noted in mathematics departments that undergraduate mathematics majors often experience difficulties when trying to write mathematical proofs in their introductory abstract algebra, real analysis, or number theory courses. Some researchers have investigated certain aspects of students’ understanding or success in proof-writing [8], [16], [11]. In particular, Moore [11] notes that, while attempting to write formal proofs, students do not necessarily understand the content of relevant definitions or how to use these definitions in proof-writing. Edwards’s study

A Simple Proof of Descartes's Rule of Signs

Abstract: (2004). A Simple Proof of Descartes's Rule of Signs. The American Mathematical Monthly: Vol. 111, No. 6, pp. 525-526.

Hypergraphs, Entropy, and Inequalities

Abstract: (2004). Hypergraphs, Entropy, and Inequalities. The American Mathematical Monthly: Vol. 111, No. 9, pp. 749-760.

L'Hospital Rules for Monotonicity and the Wilker-Anglesio Inequality

Abstract: (2004). L'Hospital Rules for Monotonicity and the Wilker-Anglesio Inequality. The American Mathematical Monthly: Vol. 111, No. 10, pp. 905-909.

Asymptotic Behaviour of Nonlinear Systems

Abstract: (2004). Asymptotic Behaviour of Nonlinear Systems. The American Mathematical Monthly: Vol. 111, No. 10, pp. 864-889.

The Early History of the Ham Sandwich Theorem

Abstract: (2004). The Early History of the Ham Sandwich Theorem. The American Mathematical Monthly: Vol. 111, No. 1, pp. 58-61.

Lost in a Forest

Abstract: Call a path an escape path if it eventually leads out of the forest no matter what the initial starting point or the relative orientations of the path and forest. To solve the "lost in a forest" problem we must find the "best" escape path. Bellman proposed two different interpretations of "best," one in which the maximum time to escape is minimized, and one in which the expected time to escape is minimized. A third interpretation (see

Projective Geometry over F 1 and the Gaussian Binomial coefficients

Abstract: There is no field with only one element, yet there is a well-defined notion of what projective geometry over such a field means. This notion is familiar to experts and plays an interesting role behind the scenes in combinatorics and algebra, but it is rarely discussed as such. The purpose of this article is to bring it to the attention of a broader audience, as the solution to a puzzle about Gaussian binomial coefficients.

History or Heritage? An Important Distinction in Mathematics and for Mathematics Education

Abstract: only about its significance but even concerning legitimacy-that is, whether or not an historical interpretation counts as history at all. In this paper I consider the latter issue, and also note some consequences for education. The disagreements are general, in that they may arise for any branch of mathematics in any period or culture; so they need a general resolution. I offer one in the form of a distinction in the ways of interpreting a piece of mathematics of the past. Take such a mathematical notion N; it could be anything from one notation through a definition, proof, proof-method or algorithm to a theorem, a wide-ranging theory, a whole branch of mathematics, and ways of teaching it. By its 'history', which becomes a technical term, one considers the development of N during a particular period: its launch and early forms, its impact, and applications in and/or outside mathematics, and so on. It addresses the question 'What happened in the past?' by offering descriptions. Maybe some kinds of explanation will also be attempted, to answer the companion question 'Why did it happen?'. History should also regard as important two companion questions, namely 'What did not happen in the past?' and 'Why not?'. The reasons may involve the other side of this distinction, which I call 'heritage'. There one is largely concerned with the effect of N upon later work, during any relevant period including that of its launch. Some modernised versions of N are likely to be taken, for heritage is largely concerned with the question 'How did we get here?', that is, to some current version of the context in question.

A Simple Proof of Sharkovsky's Theorem

Abstract: (2004). A Simple Proof of Sharkovsky's Theorem. The American Mathematical Monthly: Vol. 111, No. 7, pp. 595-599.

An Elementary Proof of Euler's Formula for ζ(2m)

Abstract: (2004). An Elementary Proof of Euler's Formula for ζ(2m) The American Mathematical Monthly: Vol. 111, No. 5, pp. 430-431.

A note on the Effros theorem

Abstract: such that ex = x for every x in X and g(hx) = (gh)x for g and h in G and x in X (here e denotes the neutral element of G). It is easily seen that for each g in G the map x gx is a homeomorphism of X whose inverse is the map x g-1x. If x belongs to X and U is a subset of G, then Ux = {gx : g E U). The action of G on X is transitive if Gx = X for every x in X. It is micro-transitive if for every x in X and every neighborhood U of e in G the set Ux is a neighborhood of x in X. A metric on a space X is admissible if it generates the topology on X. A space is Polish if it has an admissible complete metric.

A New Proof of Darboux's Theorem

Abstract: Darboux's Theorem. Let I be an open interval, and let f : I -> R be a differentiable function. If a and b are points of I with a < b and if y lies between f' (a) and f' (b), then there exists a number x in [a, b] such that f'(x) = y.

From Hilbert's Superposition Problem to Dynamical Systems

Abstract: (2004). From Hilbert's Superposition Problem to Dynamical Systems. The American Mathematical Monthly: Vol. 111, No. 7, pp. 608-624.

Polynomials in the Nation's Service: Using Algebra to Design the Advanced Encryption Standard

Abstract: 1. INTRODUCTION. Cryptography, the science of transforming communications so that only the intended recipient can understand them, should be a mathematician’s playground. Certain aspects of cryptography are indeed quite mathematical. Publickey cryptography, in which the encryption key is public but only the intended recipient holds the decryption key, is an excellent demonstration of this. Both Diffie-Hellman key exchange and the RSA encryption algorithm rely on elementary number theory, while elliptic curves power more advanced public-key systems [21], [4]. But while public key has captured mathematicians’ attention, such cryptography is in fact a show horse, far too slow for most needs. Public key is typically used only for key exchange. Once a key is established, the workhorses of encryption, privateor symmetric-key cryptosystems, take over. While Boolean functions are the mainstay of private-key cryptosystems, until recently most private-key cryptosystems were an odd collection of tricks, lacking an overarching mathematical theory. That changed in 2001, with the U.S. government’s choice of Rijndael 1 as the Advanced Encryption Standard. Polynomials provide Rijndael’s structure and yield proofs of security. Cryptographic design may not yet fully be a science, but Rijndael’s polynomials brought to cryptographic design “more matter, with less art” (Hamlet, act 2, scene 2, 97). Rijndael is a “block-structured cryptosystem,” encrypting 128-bit blocks of data using a 128-, 192-, or 256-bit key. Rijndael variously uses x −1 , x 7 + x 6 + x 2 + x, x 7 + x 6 + x 5 + x 4 + 1, x 4 + 1, 3x 3 + x 2 + x + 2, and x 8 + 1 to provide cryptographic security. (Of course, x −1 is not strictly a polynomial, but in the finite field GF(2 8 ) x −1 = x 254 and so we will consider it one.) In this paper I will show how polynomials came to play a critical role in what may become the most widely-used algorithm of the new century. To set the stage, I will begin with a discussion of a decidedly nonalgebraic algorithm, the 1975 U.S. Data Encryption Standard (DES), which, aside from RC4 in web browsers and relatively insecure cable-TV signal encryption, is the most widely-used cryptosystem in the world. 2 I will concentrate on attacks on DES, showing how they shaped future ciphers, and explain the reasoning that led to Rijndael, and explain the role that each of Rijndael’s polynomials play. I will end by discussing how the algebraic structure that promises security may also introduce vulnerabilities. Cryptosystems consist of two pieces: the algorithm, or method, for encryption, and a secret piece of information, called the key. In the nineteenth century, Auguste Kerckhoffs observed that any cryptosystem used by more than a very small group of people will eventually leak the encryption technique. Thus the secrecy of a system must reside in the key.

An Old Friend Revisited

Abstract: (2004). An Old Friend Revisited. The American Mathematical Monthly: Vol. 111, No. 5, pp. 437-439.

Heesch's Tiling Problem

Abstract: 1. INTRODUCTION. Let T be a tile in the plane. By calling T a tile, we mean that T is a topological disk whose boundary is a simple closed curve. But also implicit in the word “tile” is our intent to use congruent or reflected copies of T to cover the plane without gaps or overlapping; that is, we want to tessellate the plane with copies of T . In a minor abuse of language, one often speaks of T (as opposed to copies of it) as tiling or tessellating the plane, in the sense that T generates a tessellation. A tessellation by T may or may not be possible, so in order to learn something of T ’s abilities with regard to tessellating the plane, we perform the following procedure: around a centrally placed copy of T , we attempt to form a full layer, or corona ,o f congruent copies of T . We require as part of the definition that no point of T should be visible from the exterior of a corona to a Flatland creature in this plane. Also, we should form the corona without allowing gaps or overlapping, just as if we were building a tessellation. If a corona can be formed, then we attempt to surround this corona with yet another corona, and then another, and so on; if we get stuck, we go back and change a previously placed tile and try again. If T tessellates the plane, then this procedure will never end. On the other hand, if T does not tessellate the plane, and if we check all of the possible ways of forming a first corona, a second corona, and so forth, we will find that there is a maximum number of coronas that can be formed. This maximum number of layers that can be formed around a single centrally placed copy of T is called the Heesch number of T and is denoted by H (T ). We consider a few examples before proceeding. Consider first a regular hexagon. All bees know that a regular hexagon tessellates the plane, so H =∞ for a regular

Interesting Dynamics and Inverse Limits in a Family of One-dimensional Maps

Abstract: 1. INTRODUCTION. In the past thirty or forty years inverse limits have been used extensively in dynamical systems as well as in continuum theory as a means to attack a myriad of unsolved problems. A study of one-dimensional branched manifolds by R. F. Williams [16] provided an early demonstration of the utility of the inverse limit construction in dynamical systems. In continuum theory, many surprising and complicated examples have been constructed using inverse limits. For example, a continuum admitting only constant maps and the identity as continuous transformations of the continuum into itself was constructed by Howard Cook [4] in the 1960s via inverse limits. The inverse limit technique is particularly useful in that it allows one to build complicated structures out of simpler ones. For instance, in Example 2 of this paper (see Figure 2) we demonstrate how a continuum containing a sin(1/x)-curve results from an inverse limit on the interval [0, 1] using a single simple piecewise-linear continuous function on the interval. Example 3 (see Figure 6) shows that even more complicated continua (indecomposable continua) may show up in inverse limits on intervals with equally simple looking bonding maps. This particular example actually consists of two copies of a continuum naturally associated with the \" Smale horseshoe \" glued together at a common end-point. Interest in inverse limits is increased by the realization that through the inverse limit construction one is able to turn the study of a dynamical system consisting of a space and a continuous function on that space into the study of another space with a homeomorphism on that space (Theorem 3.9). After reading with interest a paper by Susan Bassein that appeared in this MONTHLY in 1998 [2], we decided that the simple two-parameter family of

An Intuitive Derivation of Heron's Formula

Abstract: From elementary geometry we learn that two triangles are congruent if their edges have the same lengths, so it should come as no surprise that the edge lengths of a triangle determine the area of that triangle. On the other hand, the explicit formula for the area of a triangle in terms of its edge lengths, named for Heron of Alexandria (although attributed to Archimedes [4]), seems to be less commonly remembered (as compared with, say, the formulas for the volume of a sphere or the area of a rectangle). One reason why Heron's formula is so easily forgotten may be that proofs are usually presented as unwieldy verifications of an already known formula, rather than as expositions that derive a formula from scratch in a constructive and intuitive manner. Perhaps the derivation that follows, while not truly elementary, will render Heron's formula more memorable for its symmetric and intuitive factorization. The first step of this derivation is to recall that the square of the area of a triangle is a polynomial in the edge lengths. More generally, suppose that T is a simplex in RT with vertices xo, xi, ... , x,, where x0 = 0, the origin. Let A denote the n x n matrix whose columns are given by the vectors xl, ..., xn, and suppose that the xi are ordered so that A has positive determinant. The volume of T is then given by det(A) = n! V(T), implying that

The Origin and Early Impact of the Moore Method

Abstract: (2004). The Origin and Early Impact of the Moore Method. The American Mathematical Monthly: Vol. 111, No. 6, pp. 465-486.

Lattices in ℂ and Finite Subsets of a Circle

Abstract: 2. SPACES OF FINITE SUBSETS. For a topological space X, let expk X be the set of all nonempty finite subsets of X of cardinality at most k. There is a map from the Cartesian product of k copies of X with itself to expk X that sends (xI, ..., xk) to {x } U ... U {xk }. The quotient topology gives expk X the structure of a topological space. Notice that when m < k the space expm X is canonically embedded in expk X. Clearly, exp, X = X. The simplest nontrivial example is provided by the space exp2 S1, which is homeomorphic to the Mdbius band. One way to see this is as follows [3]. Identify S' with the boundary of an open disk D in the projective plane. Notice that AM = RP2\D is a M~ibius band. For each point x of M there exist at most two lines that pass through x and are tangent to S1. Let T (x) in exp2 S1 be the corresponding set of tangency points. Then

How Conics Govern Möbius Transformations

Abstract: (2004). How Conics Govern Mobius Transformations. The American Mathematical Monthly: Vol. 111, No. 9, pp. 779-790.

Alice through Looking Glass after Looking Glass: The Mathematics of Mirrors and Kaleidoscopes

Abstract: 1. ALICE AND THE MIRRORS. Let us imagine that Lewis Carroll stopped condensing determinants long enough to write a third Alice book called Alice Through Looking Glass After Looking Glass. The book opens with Alice in her chamber in front of a peculiar cone-shaped arrangement of three looking glasses. She steps through one of the looking glasses and finds herself in a new virtual chamber that looks almost like her own. On closer examination she discovers that she is now left-handed and her books are all written backward. There are also virtual mirrors in this chamber. Stepping through one of them, she continues her exploration and passes through many virtual chambers until, to her great relief, she suddenly finds herself back in her own real chamber, just in time for tea. Eager to have new adventures, Alice wonders how many different ways the mirrors could be arranged so that she could have other trips through the looking glasses and still return the same day for tea. Alice’s problem was solved (for all dimensions) by H. S. M. Coxeter [4], who classified all possible systems of n mirrors in n-dimensional Euclidean space whose reflections generate a finite group of orthogonal matrices. In this paper we describe Coxeter’s results, emphasizing the connection with kaleidoscopes. The mathematical tools involved are some linear algebra (including determinants), basic group theory, and a bit of graph theory. We also give plans for three-dimensional kaleidoscopes that exhibit the symmetries of the three types of Platonic solids. The mathematics of kaleidoscopes in n dimensions is the study of those finite groups of orthogonal n × n real matrices that are generated by reflection matrices. These groups appeared in many parts of mathematics in the late nineteenth and early twentieth centuries, in connection with geometry, invariant theory, and Lie groups, especially in the work of W. Killing, E. Cartan, and H. Weyl [8]. As abstract groups, almost all of them turn out to be very familiar: dihedral groups, the symmetric group of all permutations, the group of all signed permutations, and the group of all evenlysigned permutations. There are also six exceptional groups that occur in dimensions three to eight.

Newton, Maclaurin, and the Authority of Mathematics

Abstract: (2004). Newton, Maclaurin, and the Authority of Mathematics. The American Mathematical Monthly: Vol. 111, No. 10, pp. 841-852.

The Spiral of Theodorus

Abstract: (2004). The Spiral of Theodorus. The American Mathematical Monthly: Vol. 111, No. 3, pp. 230-237.

Finding ζ(2p) From a Product of Sines

Abstract: 1. K. P. S. Bhaskara Rao, The Theory of Generalized Inverses over Commutative Rings, Taylor and Francis, London, 2002. 2. D. Carlson, C. R. Johnson, D. C. Lay, and A. D. Porter, Linear Algebra Gems. Assets for Undergraduate Mathematics, Mathematical Association of America, Washington, D.C., 2002. 3. D. A. Harville, Matrix Algebra from a Statistician's Perspective, Springer-Verlag, New York, 1997 4. M. Machover, Matrices which take a given vector into a given vector, this MONTHLY 74 (1967) 851-852. 5. C. R. Rao and S. K. Mitra, Generalized Inverse of Matrices and its Applications, Wiley, New York, 1971.