Showing papers in "American Mathematical Monthly in 2005"
TL;DR: Virgil tells us that Aeneas, on his quest to found Rome, is shipwrecked and blown ashore at Carthage and Dido falls in love with him, but he does not return her love, and she kills herself.
Abstract: This refers to the legend of Dido. Virgil's version has it that Dido, daughter of the king of Tyre, fled her home after her brother had killed her husband. Then she ended up on the north coast of Africa, where she bargained to buy as much land as she could enclose with an oxhide. So she cut the hide into thin strips, and then she faced, and presumably solved, the problem of enclosing the largest possible area within a given perimeterthe isoperimetric problem. But earthly factors mar the purity of the problem, for surely the clever Dido would have chosen an area by the coast so as to exploit the shore as part of the perimeter. This is essential for the mathematics as well as for the progress of the story. Virgil tells us that Aeneas, on his quest to found Rome, is shipwrecked and blown ashore at Carthage. Dido falls in love with him, but he does not return her love. He sails away and Dido kills herself. Kline concludes [23, p. 135]:
232 citations
TL;DR: Earlier work unnoticed in these papers is described that anticipate many of the results attributed to others on the series expansion of composite functions.
Abstract: poraries on the series expansion of composite functions. Here, earlier work unnoticed in these papers is described. These anticipate many of the results attributed to others. This earlier work originates with Arbogast, with later reworkings by Knight, West, De Morgan, and others. The series expansion in powers of x of a suitably-differentiable composite function g(f(x)) has the form
78 citations
TL;DR: In this article, double integral for Euler's Constant and In and an Analog of Hadjicostas's Formula are discussed, and the double integral is shown to be equivalent to a double integral.
Abstract: (2005). Double Integrals for Euler's Constant and In and an Analog of Hadjicostas's Formula. The American Mathematical Monthly: Vol. 112, No. 1, pp. 61-65.
61 citations
TL;DR: A faster product for π and a new integral for In is described in this paper. But the authors do not discuss the relation between the product and the Integral Integral for In.
Abstract: (2005). A Faster Product for π and a New Integral for In . The American Mathematical Monthly: Vol. 112, No. 8, pp. 729-734.
44 citations
TL;DR: The dynamics of algebraically defined transformations on the p-adics are studied and it is seen that there is a strong connection between the topological property of minimality, which is easy to check for such transformations, and the measure-theoretic property of ergodicity.
Abstract: (2005). Measurable Dynamics of Simple p-adic Polynomials. The American Mathematical Monthly: Vol. 112, No. 3, pp. 212-232.
34 citations
31 citations
TL;DR: It is a classic theorem of Arens that if a Hausdorff space X is noncompact, locally compact, and locally connected, then H(X) is a topological group because the compact-open topology coincides with the topology that H( X) inherits from H(αX), where αX is the Alexandroff one-point compactification of X.
Abstract: If X is a topological space, then we let H(X) denote the group of autohomeomorphisms of X equipped with the compact-open topology. For subsets A and B of X we define [A, B] = {h ∈ H(X) : h(A) ⊂ B}, and we recall that the topology on H(X) is generated by the subbasis SX = {[K , O] : K compact, O open in X}. If X is a compact Hausdorff space, then H(X) is a topological group, that is, composition and taking inverses are continuous operations (see Arens [2]). It is well known that even for locally compact separable metric spaces the inverse operation on H(X) may not be continuous, thus H(X) is, in general, not a topological group (see the example to follow). However, it is a classic theorem of Arens [2] that if a Hausdorff space X is noncompact, locally compact, and locally connected, then H(X) is a topological group because the compact-open topology coincides with the topology that H(X) inherits from H(αX), where αX is the Alexandroff one-point compactification of X . We improve on this result as follows. (Recall that a continuum is a compact connected space. If A is a subset of X , then int A and ∂A denote the interior and boundary of A in X , respectively.)
30 citations
TL;DR: A Simple Proof of Sion's Minimax Theorem as mentioned in this paper is a simple proof of the Sion minimax theorem, and it can be found in Section 2.1.
Abstract: (2005). A Simple Proof of Sion's Minimax Theorem. The American Mathematical Monthly: Vol. 112, No. 4, pp. 356-358.
30 citations
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TL;DR: It is my belief that the authors' collective acceptance of proportionality as the ultimate measure of justice is due to a more fundamental concept that lurks in the background, but whose implications are rich for their understanding of what is equitable: I call it coherence.
Abstract: Why? Aristotle's arguments are tautological: "The just... is a species of the proportionate.... For proportion is equality of ratios, and involves four terms at least...; and the just, too, involves at least four terms, and the ratio between one pair is the same as that between the other pair; for there is a similar distinction between the persons and between the things. As the term A, then, is to B, so will C be to D, and therefore, alternatively, as A is to C, B will be to D.... The conjunction, then, of the term A with C and of B with D is what is just in distribution, and this species of the just is intermediate, and the unjust is what violates the proportion; for the proportional is intermediate, and the just is proportional." No further logical reasons are advanced. What then is the source of the well nigh universal belief that when something is to be shared, what is fair is what is proportional? Is it simply a question of tradition? Should one agree with Blaise Pascal when he says, "Custom makes equity for the sole reason that it is received; it is the mysterious foundation of its authority"? It is my belief that our collective acceptance of proportionality as the ultimate measure of justice is due to a more fundamental concept that lurks in the background, but whose implications are rich for our understanding of what is equitable: I call it coherence.
30 citations
28 citations
TL;DR: In this article no Sylow-theory, no theory of p-groups, no relatively free groups (generators and relations), no group actions, not even the structure of finite Abelian groups will be invoked.
Abstract: 1. INTRODUCTION. There are many introductory texts on group theory that, in particular, classify all groups of order at most fifteen. What about order sixteen? Well, there are fourteen of them, and they do of course make their appearance in higher level texts such as [1] or [7]. However, to the author's best knowledge, the classification of the groups of order sixteen is always obtained as a special case of a sophisticated theory of p-groups, and many details are left to the reader to verify. This is annoying since it, for example, obstructs the teaching of an advanced undergraduate course devoted to the classification of all groups of order no larger than 23 or even 31. Here we give a complete classification of the groups of order sixteen that is based on elementary facts. More specifically, we look at extensions of order eight groups N by C 2 , the cyclic group of order two. A crucial role is played by Lemma 2, which allows us to discard three of the five groups of order eight. We also deal a lot with automorphisms of groups. In this article no Sylow-theory, no theory of p-groups, no relatively free groups (generators and relations), no group actions, not even the structure of finite Abelian groups will be invoked. The most \" difficult \" prerequisite we take for granted is Fact 5 about conjugacy classes and centralizers. Semidirect products appear as an additional means to describe the fourteen groups of order sixteen. Yet no knowledge of semidirect products is required; they do not appear in the proofs.
TL;DR: Some of the most basic results on the geometry of unit discs in two-dimensional normed spaces, while adding a few results and some new proofs of their own.
Abstract: longer holds. These geometries are homogeneous but not isotropic. In this article we survey some of the most basic results on the geometry of unit discs in two-dimensional normed spaces, while adding a few results and some new proofs of our own. These results answer simple questions about the perimeter of the unit disc, its area, and the relationships between these two quantities.
TL;DR: Theorems of Fibonacci, Chebyshev, and Orthogonal Polynomials: Foundations of Mathematical Monthly .
Abstract: (2005). Fibonacci, Chebyshev, and Orthogonal Polynomials. The American Mathematical Monthly: Vol. 112, No. 7, pp. 612-630.
TL;DR: The purpose of this article is to present a triangular array of numbers similar to Pascal’s triangle and to prove a corresponding criterion for the twin prime pairs and to relate it to some recent work of John D’Angelo.
Abstract: 1. INTRODUCTION. Two of the most ubiquitous objects in mathematics are the sequence of prime numbers and the binomial coefficients (and thus Pascal’s triangle). A connection between the two is given by a well-known characterization of the prime numbers: Consider the entries in the kth row of Pascal’s triangle, without the initial and final entries. They are all divisible by k if and only if k is a prime. It is the purpose of this article to present a triangular array of numbers similar to Pascal’s triangle and to prove a corresponding criterion for the twin prime pairs .A further goal is to place all this in the context of some classical orthogonal polynomials and to relate it to some recent work of John D’Angelo. To begin, and for the sake of completeness, we present a short proof of the Pascal triangle criterion. First suppose that k = p is prime. Then we see that in p j = p! j! ( p − j)! (1 ≤ j ≤ p − 1)
TL;DR: This paper will generalize the idea of factor rings from the integers to the Gaussian integers and discuss what new objects can be found in this manner.
Abstract: 1. INTRODUCTION AND HISTORY. The Gaussian integers are defined to be the set Z[i] = {a + bi : a, b ∈ Z, i = √ −1}. These sit inside the complex numbers C and thus obey the usual rules of addition and multiplication; indeed, despite the presence of the imaginary i, they are quite similar to the " traditional " integers. In fact, in the set Z[i] one can define (Gaussian integer) primes, construct analogues of the Euclidean division algorithm and the Euler φ-function, discuss Pythagorean triples, generalize the twin-prime problem, and much more. In this paper, we will generalize the idea of factor rings from the integers to the Gaussian integers and discuss what new objects can be found in this manner. (Recall that integer factor rings are the familiar objects Z/n, where n signifies the ideal in Z generated by n. These rings are also written as Z/nZ or Z n .) If the Gaussian integers were no more than a practice area for generalizing concepts from the standard integers, they would still be of great interest to students and researchers alike. Yet these numbers have been (and still are) far more than just a convenient teaching tool. They have played roles in the development of two of the great theorems of mathematics of the last two centuries (the reciprocity theorems and Fer-mat's Last Theorem) and have helped inspire the creation of algebraic number theory. Taking a minute to provide a brief review of this fascinating history will be time well spent. (Much of what follows can be found in [9]; see also [7].) Our story begins over two hundred years ago, long before either abstract algebra or modern number theory came into existence. In the late 1700s, Leonhard Euler noticed some intriguing patterns that arose in his study of the equation x
TL;DR: This work can establish a correspondence of a Pythagorean triple in which m and n are relatively prime with a matrix belonging to SL2(Z) (the group of two-by-two integral matrices of determinant one) whose entries depend onm and n.
Abstract: The Pythagorean triples that are relatively prime (called the primitive triples) have the elementary and beautiful characterization as integers x = m2 − n2, y = 2mn, z = m2 + n2 (when y is even) for relatively prime integers m and n of opposite parity. One can think of this as replacing the parameter t for the circle with the fraction m/n and then scaling. Our motivation for understanding the triples stems from the realization that one can enumerate the rational numbers on the line by using the modular group, in a sense reversing the Euclidean algorithm [2]. Now the line can be transformed by a linear fractional transformation to the circle. This transformation changes fractions to rational points on the circle, and after scaling this process gives rise to Pythagorean triples. Roughly speaking, we can establish a correspondence of a Pythagorean triple [m2 − n2, 2mn,m2 + n2] in which m and n are relatively prime with a matrix belonging to SL2(Z) (the group of two-by-two integral matrices of determinant one) whose entries depend on m and n. Since the modular group = PSL2(Z) = SL2(Z)/{±I } is essentially a free group, it follows that there is an underlying tree structure to Pythagorean triples. Making this tree structure and its connection to the modular group explicit is a bit delicate, but the payoff is worth the effort. Our main results can be summarized as follows:
TL;DR: A key role in this story is played by group characters, where a character X of a finite Abelian group G is a homomorphism from G into C*, the multiplicative group of nonzero complex numbers.
Abstract: (2005). Quadratic Reciprocity in a Finite Group. The American Mathematical Monthly: Vol. 112, No. 3, pp. 251-256.
TL;DR: An elementary proof of a theorem first proved by J. A. Erdos, which states that every noninvertible n × n matrix is a finite product of matrices M with the property that M = M .
Abstract: In this note we give an elementary proof of a theorem first proved by J. A. Erdos [3]. This theorem, which is the main result of [3], states that every noninvertible n × n matrix is a finite product of matrices M with the property that M = M . (These are known as idempotent matrices. Noninvertible matrices are also called singular matrices.) An alternative formulation of this result reads: every noninvertible linear mapping of a finite dimensional vector space is a finite product of idempotent linear mappings α, linear mappings that satisfy α = α. This result was motivated by a result of J. M. Howie asserting that each selfmapping α of a nonempty finite set X with image size at most |X| − 1 (which occurs precisely when α is noninvertible) is a product of idempotent mappings. We shall see that Erdos’s theorem is a consequence of Howie’s result. Together the papers [3] and [4] are cited in over one hundred articles, dealing with subjects including universal algebra, ring theory, topology, and combinatorics. Since its publication, various proofs of the result in [3] have appeared. For example, a semigroup theoretic proof appears in [1, pp. 121-131] and linear operator theory is used to prove the theorem in [2]. Here we give a new proof using a basic combinatorial argument. Unlike the previous proofs our argument involves only elementary results from linear algebra and one basic result concerning permutations. On the way to proving the main result of this note we provide a short proof of Howie’s result. Throughout this paper X signifies an arbitrary nonempty finite set. If α : A → X, where A is a subset of X, then A is the domain of α; we denote this set by dom(α). Naturally, the set α(A) is called the image of α and is denoted by im(α). Recall that a mapping α is injective (or one-to-one) if α(x) 6= α(y) for all x and y in dom(α) with x 6= y. Let TX denote the set of all mappings from X to X with domain X. We note that this set is closed under composition of mappings and that this composition is associative. We now define one of the most important notions we require in the proofs in this note. For a mapping α : dom(α) → X we say that α is a restriction of an element β of TX if β and α agree on the domain of α. In other words, β(x) = α(x) for all x in dom(α). For x and y in X we denote the transposition that fixes every point of X other than x or y and that maps x to y, and vice versa, by (x y).
TL;DR: This note derives a closed-form expression for Rk (x) when x > 2, together with a related nested radical, via an application of a hyperbolic cosine identity in contrast with the use of the half-angle cosine formula required to evaluate R(bk,..., bl) in [1].
Abstract: (2005). More Nested Square Roots of 2. The American Mathematical Monthly: Vol. 112, No. 9, pp. 822-825.
TL;DR: The Zeroes of the Nth Partial Sum of the Exponential Series is defined as the sum of the zeros of the exponential series divided by the number of zeroes.
Abstract: (2005). On the Zeroes of the Nth Partial Sum of the Exponential Series. The American Mathematical Monthly: Vol. 112, No. 10, pp. 891-909.
Journal Article•
TL;DR: Theorem 1 of [1] proves the irreducibility of f(x) if one takes n = 15 or 16, because f(15) = 4 .
Abstract: (2005). On an Irreducibility Criterion of M. Ram Murty. The American Mathematical Monthly: Vol. 112, No. 3, pp. 269-270.
TL;DR: It is seen that all eigenvalues of L(p) are positive and, on the basis of (5), that II P II| furnishes a lower bound for them.
Abstract: (L(p)[Q], Q)k = (PQ, PQ)k+m = +IIell2m |> IIPII\QII, (5) showing that L(p) is positive. Therefore, we see that all eigenvalues of L(p) are positive and, on the basis of (5), that II P II| furnishes a lower bound for them. Furthermore, equality holds in (4) if and only if either P 0 or II P II| is the smallest eigenvalue of L(p) and Q is an eigenvector corresponding to it (unless Q = 0). A particular case in which equality holds in (4) occurs when P = P(y) belongs to Nm and Q = Q(z) to Nk, where y E RP, z E Rq, and IR = RP x Rq. Added in proof. Professor Luo Xuebo, who was one of his coauthor's Ph.D. supervisors, died in March 2004. Zhu-Jun Zheng expresses his deep respect for and everlasting memory of his deceased colleague and mentor.
TL;DR: It follows that N is finitely generated, which contradicts the definition of N, therefore p is a prime ideal, and p = Ann(Rij) for some j.
Abstract: Since the other inclusion is trivial, we get N = Y-=I Rni + aL. It follows that N is finitely generated, which contradicts the definition of N. Therefore p is a prime ideal. Since M is finitely generated, we have M/N = Rxj + ... + Rx, for some x,..., x, in M, where x signifies the equivalence class of x in M/N, hence p = n=IAnn(R ). Because p is a prime ideal, p = Ann(Rij) for some j. Suppose that the set {yi + rixj }, generates N + Rxj, where yi is in N and ri in R. By an argument similar to the earlier one, we have N = Y=l Ryi + pxj. Since pM is contained in N, we obtain
TL;DR: A classical result due to Cauchy on the distribution of zeros of a polynomial may be stated as follows: all zeros z of P satisfy |z| ≤ r, where r is the positive solution of the equation z − |an−1|zn−1 −|an−2 |zn−2 − · · · − |a0|.
Abstract: A classical result due to Cauchy (see [8, p. 122]) on the distribution of zeros of a polynomial may be stated as follows: Theorem 1. If P(z) = zn + an−1zn−1 + an−2zn−2 + · · · + a0 is a polynomial with complex coefficients, then all zeros z of P satisfy |z| ≤ r , where r is the positive solution of the equation z − |an−1|zn−1 − |an−2|zn−2 − · · · − |a0| = 0. Dı́az-Barrero [4], [5] recently improved this estimate by identifying an annulus containing all the zeros of a polynomial, where the inner and outer radii are expressed in terms of binomial coefficients and Fibonacci numbers. In this note, we use the wellknown identity
TL;DR: 6. , Counting subgroups of given index in Hecke groups, Contemp.
Abstract: (2005). An Elementary Proof of Lyapunov's Theorem. The American Mathematical Monthly: Vol. 112, No. 7, pp. 651-653.
TL;DR: The French Revolution exhibits such characteristics in generous measure in the years following its occurrence in 1789, with the creation of a new school in Paris, called the Ecole Centrale des Travaux Publics, conceived as the one and only institution to train engineers, both civilian and military.
Abstract: 1. WHAT KIND OF NEW SCHOOL? After revolution, confusion. It may be clear who has lost, but not who has won. The winners fight the winners, and large power vacua open up unpredictably. The French Revolution exhibits such characteristics in generous measure in the years following its occurrence in 1789. For the engineering profession three dangers arose from the ensuing chaos. First, especially following the social chaos of 1790–1793, there was considerable haemorrhage of officers from the army and navy, some going abroad, others returning to their families in the countryside; thus the higher ranks needed rapid replenishment [47]. Second, all institutions of higher education were closed in 1793; while the engineering schools were soon open again with few changes, their own roles needed reappraisal. Third, the transport situation of the country required maintenance and improvement following years of neglect: not only roads but also the new technology of canals, and the coastal harbours and sea travel. Thus ‘the republic needed academics’ [39, title]. The accumulation of these dangers, especially the first one with the fragility of the borders of The Hexagon (as France is known familiarly), led to an emergency council being set up in Paris. Its main decision was the creation of a new school in Paris, called the Ecole Centrale des Travaux Publics and launched in December 1794. Under supporters such as the chemist François Fourcroy (1755–1809), it seems to have been conceived as the one and only institution to train engineers, both civilian and military. To this end four hundred students were rapidly enrolled, and ‘revolutionary courses’ in mathematics and chemistry were taught, with the help of dozens of rapidly recruited officers (listed in [26, pp. 389–390]). It sounds like nonsense, similar to a contemporary new school that closed after a few months: namely, the Ecole Normale, which was intended to provide similar forced feeding for future teachers and administrators [20]. But the impracticality of the vision must soon have been recognized. For example, most of the military schools were outside Paris, and no naval school could possibly run in the city. Thus the role of the school was changed to that of a preparatory institution for the other schools, with a three-year curriculum. The change was reflected in a change of name, made in September 1795, to Ecole Polytechnique; the adjective first appeared in a pamphlet published that year by an involved politician, Claude Prieur de la Côte d’Or (1763–1832), on ‘l’enseignement polytechnique de l’Ecole Centrale des Travaux Publics’ [53]. The adjective was presumably intended to convey the idea of a plurality of techniques;1 at all events, very soon the name became definitive.