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0.1. How it all began. When the second author translated a result on algebraic sur faces into the language of lattice polygons using toric geometry, he got a simple inequality for lattice polygons. This inequality had originally been discovered by Scott [12]. The first author then found a third proof. Subsequently, both authors went through a phase of polygon addiction. Once you get started drawing lattice polygons on graph paper and discovering relations between their numerical invariants, it is not so easy to stop! (The gentle reader has been warned.) Thus, it was just unavoidable that the authors came up with new inequalities: Scott's inequality can be sharpened if one takes into account another invariant, which is de fined by peeling off the skins of the polygons like an onion (see Section 3).

A large part of mathematical demography is built upon one fundamental theorem, the "strong ergodic theorem" of demography. If the fertility and mortality age-schedules of a population remain unchanged over time, its age distribution, no matter what its initial shape, will converge in time to a fixed and stable form. In brief, when demographic behavior remains unchanged, the population, it is said, converges to stability. This short paper presents a new argument for the convergence of the age structure, one that is self-contained, and that brings the mechanism behind convergence into full view. The idea is simple. Looked at directly, the dynamics of the age-distribution say little to our normal intuition. Looked at from a slightly different angle though, population dynamics define a smoothing or averaging process over the generations -- a process comfortable to our intuition. This smoothing and resmoothing turns out to be the mechanism that forces the age structure toward a fixed and final form.

When we pause to reflect on Ramanujan’s life, we see that there were certain events that seemingly were necessary in order that Ramanujan and his mathematics be brought to posterity. One of these was V. Ramaswamy Aiyer’s founding of the Indian Mathematical Society on 4 April 1907, for had he not launched the Indian Mathematical Society, then the next necessary episode, namely, Ramanujan’s meeting with Ramaswamy Aiyer at his office in Tirtukkoilur in 1910, would also have not taken place. Ramanujan had carried with him one of his notebooks, and Ramaswamy Aiyer not only recognized the creative spirit that produced its contents, but he also had the wisdom to contact others, such as R. Ramachandra Rao, in order to bring Ramanujan’s mathematics to others for appreciation and support. The large mathematical community that has thrived on Ramanujan’s discoveries for nearly a century owes a huge debt to V. Ramaswamy Aiyer. 1. THE BEGINNING. Toward the end of the first paper [57], [58 ,p . 36] that Ramanujan published in England, at the beginning of Section 13, he writes, “I shall conclude this paper by giving a few series for 1/π.” (In fact, Ramanujan concluded his paper a couple of pages later with another topic: formulas and approximations for the perimeter of an ellipse.) After sketching his ideas, which we examine in detail in Sections 3 and 9, Ramanujan records three series representations for 1/π .A s is customary, set

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

We show that a "simple" differential equation modeling a garden-variety damped forced pendulum can exhibit extraordinarily complicated and unstable behavior. While instability and control might at first glance appear contradictory, we can use the pendulum's instability to control it. Such results are vital in robotics: the forced pendulum is a basic subsystem of any robot. Most of the mathematical methods used in this paper were initially developed in celestial mechanics, largely by Poincare. The literature of the field tends to be quite advanced indeed (see [1] and [11]); one object of this paper is to show that computer programs, properly used, can make these advanced topics transparent. All the computer-generated pictures in this paper were produced by the programs Planar Systems and Planar Iterations [6], both written by Ben Hinkle (now at Maple). 1. SOME PARALLELS IN CELESTIAL MECHANICS. When I was a graduate

have some nice, relevant properties. At the same time another group of people combinatorialists are busy trying to produce many different types of arrays of numbers with special features. We have the feeling that the two groups are not well acquwainted with each other's work, although recently some mathematical journals have published papers reporting results obtained by the playing community, and we hope that this paper will contribute to increasing the mathematicians' interest in these problems. The problems are natural and mathematically easy to formulate, but are highly nontrivial and can be attacked using powerful combinatorial machinery.

The Cauchy-Davenport theorem states that if A and B are nonempty sets of congruence classes modulo a prime p, and if |A| = k and |B| = l, then the sumset A + B contains at least min(p, k + l − 1) congruence classes. It follows that the sumset 2A contains at least min(p, 2k − 1) congruence classes. Erdős and Heilbronn conjectured 30 years ago that there are at least min(p, 2k − 3) congruence classes that can be written as the sum of two distinct elements of A. Erdős has frequently mentioned this problem in his lectures and papers (for example, Erdős-Graham [4, p. 95]). The conjecture was recently proven by Dias da Silva and Hamidoune [3], using linear algebra and the representation theory of the symmetric group. The purpose of this paper is to give a simple proof of the Erdős-Heilbronn conjecture that uses only the most elementary properties of polynomials. The method, in fact, yields generalizations of both the Erdős-Heilbronn conjecture and the Cauchy-Davenport theorem.

We agree. And we think there's a fairly wide consensus on this among experienced abstract algebra instructors, and an even wider one among experienced students. Statement: There's little the conscientious math professor can do about it. The stuff is simply too hard for most students. Students are not well-prepared and they are unwilling to make the effort to learn this very difficult material. We disagree. But we suspect that many experienced abstract algebra instructors hold such beliefs. This is especially true for some excellent instructors: Their lectures are truly masterpieces, surely you can't improve much on that; so if the students still fail, that's too bad, but it can't really be helped. We claim that, far from being an immutable fact of nature resulting from inadequacies of the student, this failure is, at least in part, an artifact of a too narrowly conceived view of instruction. In fact, replacing the lecture method with constructive, interactive methods involving computer activities and cooperative learn- ing, can change radically the amount of meaningful learning achieved by average students. In this paper we would like to paint a picture of such an alternative approach, which we and others have been developing and using in our classes over the last several years. We are painfully aware of the limitations inherent in any attempt to give such a description by means of the written text only. It would have been much better if you could actually visit our classes and observe the dynamics of the students' interactions with both the computer and their peers. By way of compro- mise, we will try to simulate such a visit by organizing our paper around several classroom "scenarios" and some commentary on the events depicted in each scenario. As a matter of principle, we have tried to make the scenarios as realistic as space limitation permits.

1. INTRODUCTION. The study of knots and links begins with simple intuitive problems but quickly leads to sophisticated mathematics. This paper will provide the reader with an accessible route that begins with basic knot theory and leads into interesting realms of modern research. The specific topic of the paper is the enhanced linking number, X, a new invariant of links that provides a simple tool to address some of the fundamental problems in the study of linking. The linked rings illustrated on the left in Figure 1 are clearly linked; when a magician pulls such a link apart we know we have been tricked. A mathematical proof that this link is nontrivial depends on showing that its linking number is 1, where the linking number is the basic invariant of link theory, first described by Gauss two hundred years ago. With the link on the right the situation is less clear. If this link is built from rope or beads, then one quickly finds that unlinking the two is impossible. However, formulating a mathematical proof of this is more difficult; for instance, the linking number turns out to be of no help. The enhanced linking number reduces the proof that this link is nontrivial to a simple calculation. This article describes the enhanced linking number and shows how it can be applied to this basic problem of distinguishing links. It will also discuss the role of X in a variety of advanced topics, such as periodicity, symmetry, Brunnian linking, and the modern theory of finite type invariants.

In this article we shall give a modern interpretation of transformation geometry. This subject has recently become of great interest to mathematics educators for use in kindergarten to high school, but has been paid too little attention at the college level. The usual approach to transformation geometry [5] or [12] consists of giving the classical geometries and then presenting their transformation groups. This certainly runs contrary to the ideas of the founder of transformation geometry, Felix Klein (1849-1929), who believed very strongly in a unified approach to mathematics whenever possible. I shall adopt the view that the proper approach to transformation geometry is through the geometry of homogeneous spaces and that this affords a modern interpretation of Klein's program (the Erlanger Programm) as outlined in his original paper of 1872 [7]. Roughly speaking, Klein's program says that a geometry on a space determines a group of transformations of the space and that a group of transformations on the space determines a geometry. While most modern mathematicians have been using the first idea (that very valuable information can be gained by looking at the group of transformations which leave the geometry invariant) we have neglected the second one. I claim that the theory of homogeneous spaces affords us a modern interpretation of the second point of Klein's program. I am not claiming that Klein foresaw the theory of homogeneous spaces but rather that with the help of the works of Riemann and E. Cartan (1869-1951) we can make precise (via the theory of homogeneous spaces) the notion that an arbitrarily chosen group will determine a geometry. If we are to adopt Klein's approach to transformations, what definition are we to take for geometry? There is certainly no easy answer to this question but the approach of G. F. B. Riemann (1826-1866) is certainly the most modern in spirit and is the one I shall use here. Riemann's inaugural address [111 begins: "As is well known, geometry presupposes the concept of space, as well as assuming the basic principles for constructions in space. It gives only nominal definitions of these things, while their essential specifications appear in the form of axioms. The relationship between these presuppositions [the concept of space, and the basic properties of space] is left in the dark; we do not see whether, or to what extent, any connection between them is necessary, or a priori whether any connection between them is even possible." This means that we must first decide what "space" should be. In section 2 we define space to be a homogeneous space; that is, the quotient space of a topological group G by a subgroup L so that M = GIL. In section 3 we assume that G is a subgroup of the group of nonsingular matrices (that is, G and L are Lie groups). In this section we add some geometric structure to that of space, which we call a geometry on the homogeneous space, and so obtain the notion of "lines." Throughout the last two sections of the paper we stress the fact that this definition of geometry includes in it, as special cases, Euclidean, Spherical and Hyperbolic geometries. We treat these three special cases in detail. (I do not claim, however, that this is the most general definition of geometry.) The first section of this paper gives a brief historical background before studying homogeneous spaces and their geometry. Where do homogeneous spaces belong in the realm of mathematics? They are not just used as an interpretation of Klein's Erlanger program; they are also used to do function theory (harmonic analysis [6]), to serve as models in differential geometry [8], and are used in mathematical physics [2]. It is as Klein remarks in the notes he added (in 1893) to the original Erlangen address ([7], p. 244): "A model, whether constructed and observed or only vividly imagined, is for this geometry not a means to an end, but the subject itself."

As the speaker representing the Mathematical Association of America at this session I propose this morning to call your attention to one of the more important duties of a mathematician, namely, the duty of explaining as clearly as possible mathematical truths and discoveries both to his fellow mathematicians and to students of mathematics in general. The American Mathematical Society is especially devoted to the encouragement of research and the secretary of that society has called to your attention during this meeting the remarkable increase in the number of papers presented annually to the Society during the past five years. This increase has been such as to make the problem of publishing the results of mathematical research in America a very acute and difficult one. At the same time it has become more desirable than ever before that the papers published in our American journals should be as clear and as easily readible as possible. The Mathematical Association of America is especially concerned with the teaching and exposition of mathematical truth and it is pretty generally agreed that it is highly desirable that care should be taken to make this teaching and exposition to students as good as possible. I fear, however, that some of my friends who are particularly interested in research agree to this in a rather condescending manner; their tone implying that, whilst nothing should be done to discourage anyone beginning the study of mathematics, such care is not so necessary nor even so desirable when writing for fellow mathematicians. The underlying idea is that a competent mathematician usually prefers to glance at a paper, see the results arrived at, and then derive these results in his own individual manner. I believe that this opinion is not justified by the facts and in support of this belief I shall mention two instances which have recently come to my attention where mathematicians of great competence failed, through a lack of detail or of clarity in available expositions of known results, to arrive at immediate and important corollaries of these results. I shall be happy if my talk this morning tends to make the editors of our mathematical journals insist more definitely on clearness of exposition when considering papers submitted for publication. One can surmise that the writers of at least some of our papers set down their results with the referee of the paper more in mind than the prospective readers. They neglect, therefore, to state or. emphasize points which they think will be familiar to the referee. Of course this is unfortunate since there is us'ually a one-to-one

on the interval I. However, it has also long been known that for n functions which are only (n 1)-times differentiable (so that their Wronskian is defined) the sufficiency part of the above statement no longer holds. Peano [12] seems to have been the first to point this out, and Bocher [3] has given an example which shows that even if the functions involved are infinitely differentiable on I, the identical vanishing of the Wronskian is still not sufficient to imply their linear dependence on I. It was then recognized that the functions involved must satisfy other conditions, supplementary to the vanishing of their Wronskian, in order to guarantee their linear dependence. Peano [13] and Bocher [5] have given such conditions (for example, Lemma 3 of Sec. 2 of this paper) and Bocher has shown that his conditions include those of Peano. In this paper we look at this problem from a slightly different point of view. Namely, instead of placing linear dependence in the forefront and looking for a condition to supplement the identical vanishing of the Wronskian, rather, we shall put the vanishing of the Wronskian in the forefront and ask for a kind of generalized dependence (necessarily weaker than linear dependence) which is equivalent to it. This point of view has led the author to define a new type of dependencef relation for functions of a real variable which he has called "local linear dependence." It is shown that local linear dependence and the identical

Let D be the ring of integers in a finite extension of the rationals. The classic examination of the factorization properties of algebraic integers usually begins with the study of norms. In this paper, we show using the ideal class group, C(D), of D that a deeper examination of such properties is possible. Using the class group, we construct an object known as a block monoid, which allows us to offer proofs of three major results from the theory of nonunique factorizations: Geroldinger's theorem, Carlitz's theorem, and Valenza's theorem. The combinatorial properties of block monoids offer a glimpse into two widely studied constants from additive number theory, the Davenport constant and the cross number. Moreover, block monoids allow us to extend these results to the more general classes of Dedekind domains and Krull domains.

For a fixed $n\ge2$, consider an $n\times n$ matrix $M$ whose entries are random integers bounded by $k$ in absolute value. In this paper, we examine the probability that $M$ is singular (hence has eigenvalue 0), and the probability that $M$ has at least one rational eigenvalue. We show that both of these probabilities tend to 0 as $k$ increases. More precisely, we establish an upper bound of size $k^{-2+\epsilon}$ for the probability that $M$ is singular, and size $k^{-1+\epsilon}$ for the probability that $M$ has a rational eigenvalue. These results generalize earlier work by Kowalsky for the case $n=2$ and answer a question posed by Hetzel, Liew, and Morrison., Comment: 9 pages, 1 figure

The paper begins with an elementary treatment of a standard trammel (trammel of Archimedes), a line segment of fixed length whose ends slide along two perpendicular axes. During the motion, points on the trammel trace ellipses, and the trammel produces an astroid as an envelope that is also the envelope of the family of traced ellipses. Two generalizations are introduced: a zigzag trammel, obtained by dividing a standard trammel into several hinged pieces, and a flexible trammel whose length may vary during the motion. All properties regarding traces and envelopes of a standard trammel are extended to these more general trammels. Applications of zigzag trammels are given to problems involving folding doors. Flexible trammels provide not only a deeper understanding of the standard trammel but also a new solution of a classical problem of determining the envelope of a family of straight lines. They also reveal unexpected connections between various classical curves; for example, the cycloid and the quadratrix of Hippias, curves known from antiquity.

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

1. THE MAIN RESULT. Liouville's theorem states that every bounded holomor phic function on C is constant. Let us recall that holomorphic functions / on open subsets U of the complex plane have the mean value property, that is, for every closed disk B(z,r) in U, the value of / at its center z is equal to the average of the values of f on the circle S(z,r) = dB(z,r). In this paper, we shall present an elementary proof for the following stronger result (where, as usual, we do not distinguish between C and R2)

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

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.

Some years ago W. Plesken told the first author of a simple but interesting construction of a Lie algebra from a finite group. The authors posed themselves the question as to what the structure of this Lie algebra might be. In particular, for which groups does the construction produce a simple Lie algebra? The answer is given in the present paper; it uses some textbook results on representations of finite groups, which we explain along the way. Little knowledge of the theory of Lie algebras is required beyond the dfinition of a Lie algebra itself and the definitions of simple and semisimple Lie algebras. Thus this exposition may serve as the basis for some entertaining examples er exercises in a graduate course on the representation theory of finite groups.

Observe that Aac ? b2 > 0 is required for the convergence of (1). The goal of this paper is to present a new proof of (2). We illustrate a technique that will apply to any rational integrand. Providing new proofs of an elementary result, such as (2), is usually an effective tool to introduce students to more interesting math ematics. The method discussed here has a rich history that we describe in section 2. It is an unfortunate fact that, despite our best efforts, evaluating definite integrals is not very much in fashion today. Thus we rephrase the previous evaluation as a question in dynamical systems: replace the parameters a, b, and c in (1) with new ones given by the rules

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

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

The first widely accepted proof of the Fundamental Theorem of Algebra was published by Gaus in 1799 in his Ph.D. thesis, although to current standards this proof has gaps. Argand gave a proof (with only small gaps) in 1814 which was based on a flawed proof of d’Alembert of 1746. Many more proofs followed, including three more proofs by Gaus. For a more about the history of the Fundamental Theorem of Algebra, see [5, 6]. Proofs roughly can be divided up in three categories (see [3] for a collection of proofs). First there are the topological proofs (see [1, 8]). These proofs are based on topological considerations such as the winding number of a curve in R around 0. Gaus’ original proof might fit in this category as well. Then there are analytical proofs (see [9]) which are related to Liouville’s result that an entire non-constant function on C is unbounded. Finally there are the algebraic proofs (see [4, 10]). These proofs only use the facts that every odd polynomial with real coefficients has a real root, and that every complex number has a square root. The deeper reasons why these proves work can be understood in terms of Galois Theory. For a linear algebra course, the Fundamental Theorem of Algebra is needed, so it is therefore desireable to have a proof of it in terms of linear algebra. In this paper we will prove that every square matrix with complex coefficients has an eigenvector. This is equivalent to the Fundamental Theorem of Algebra. In fact we will prove the slightly stronger result that any number of commuting square matrices with complex entries will have a common eigenvector. The proof is entirely within the framework of linear algebra, and unlike most other algebraic proves of the Fundamental Theorem of Algebra, it does not require Galois Theory or splitting fields. Another (but longer) proof using linear algebra can be found in [7].

Despite the fact that almost all real numbers are absolutely normal---that is, the digits in their expansions to any base occur in all possible configurations with the expected frequency---not one specific example of an absolutely normal number is known. In this note we investigate the opposite extreme, numbers that are normal to no base whatsoever, and we succeed in writing down explicitly such a number., 9 pages. The details of the main construction have changed somewhat, though the method of proof is the same. The paper has been expanded a bit for clarity and completeness

The purpose of this paper is to present two elementary (and perhaps somewhat novel) solutions of the Dirichlet problem for ellipsoids in R"'1. One of these is based on an elegant result of Ernst Fischer-of Riesz-Fischer fame. By the Dirichlet problem (for the Laplacian) we mean the following: Given a bounded region (nonempty, open, connected set) fQ in RD, n > 2, and given a continuous function f: d{l -> R (called the bounda;y data), find a continuous function u: Q -> R such that u is C2 on Q

This paper arose from a course I gave on algebraic structures, where some of the results of Sections 1 and 2 and some examples from Section 4 were presented as exercises and then discussed in the classroom. In addition, the students were asked to calculate B(a) and C(a) for certain specific examples, sometimes with the aid of a software package. Generally B(a) is not a subgroup of G, and it may be empty. However, E(a) = B(a) U C(a) is a group, which is called the reversing symmetry group of a. In dynamical systems theory, the group element a represents the time evolution operator of a dynamical system. We present some results familiar to people working in time reversing dynamical systems, but our presentation is given in an

This paper contains an exposition of two theorems on Banach spaces. Let X and Y be real Banach spaces and let f be a map from X to Y such that f(0)=0. The Mazur-Ulam Theorem says that if such a map is isometric (distance-preserving) and surjective, then it is linear. In general, it is necessary to assume that f is surjective. However, for a very large class of spaces this assumption is not necessary for the conclusion of the theorem. Let epsilon be a given positive number. An epsilon-isometry is a map from X to Y that preserves distances to within epsilon. It is known that if f is such a map between real Banach spaces, if f(0) = 0, and if f is surjective, then there exists a linear isometry g such that f and g are uniformly close. This was proved in 1945 by Hyers and Ulam for the special case of Hilbert spaces, and then extended to Banach spaces over the years by several authors. Here the assumption that f be surjective is necessary even when X and Y are Euclidean spaces.

. In 1947 Fine [Fin] introduced and solved a problem of maximizing the distance a jeep can travel into the desert using n drums of fuel. Subsequently, Phipps [Phi], Alway [Alw], and Gale [Gal] gave other solutions to the original problem or considered related problems. As mentioned in [Fin], the original problem is similar to one which arose in air transport operations in the China theater during World War II, and it has been suggested that there may be applications to Arctic expeditions and interplanetary travel. Near the end of [Gal], the author states, "An apparently simple question is the round trip problem in which fuel is available at both ends of the desert, but I must confess . . . that I have not been able to find the solution. It is not hard to see that one can do at least as well in this case as in the case of two jeeps making one-way trips, but it may be possible to do better. The difficulty here as with many optimization problems is that there does not appear to be any simple way to determine whether or not a given solution is optimal." Gale's problem can be interpreted in two equivalent ways. (i) Given unlimited fuel at each end of a desert of given length, find a round trip across the desert which uses as little fuel as possible. (ii) Given a fixed amount of fuel which can be distributed between the two ends of a desert, find the maximum length desert which can be crossed in a round trip using the available fuel. We find it convenient to consider (ii) and give an optimal solution for it. We also describe a solution for the analogous round trip problem where the two allowed depots may be placed anywhere in the desert. In each of the above problems the jeep can carry exactly 1 drum. It is implicit that the jeep can store whatever fraction of a drum is desired at any point in the desert. (Perhaps the driver carries large plastic bags for fuel storage.) In [Dew], Dewdney proposed an interesting variation of the one-way problem. Although Dewdney's problem was given in terms of drums, gallons, and miles, it can be rephrased as follows: Find the maximum distance a jeep can travel into the desert using n drums of fuel where the jeep can carry 1 drum plus 1/5 of a drum in its tank, but only drums can be stored. That is the jeep can dump at most 5/6 of its fuel capacity in the desert. It is interesting to note that Dewdney's problem has been solved as a linear programming problem; an optimal algorithm for Dewdney's problem appears in [Jac]. But the problems solved in this paper apparently are not easily posed as either linear or dynamic programming problems. In [Gal], Gale also points out that "there is a feeling among many people that the original jeep problem can be solved by the functional equation method of dynamic programming. . . I know of no way of solving the problem by this method." 1. A BRIEF DESCRIPTION OF THE SOLUTION TO GALE'S PROBLEM. In solving Gale's problem we will start by considering the longest desert which can be crossed in a round trip if there are m drums of fuel at the start S and k drums of

Little has been published about this problem but a good account of it can be found in [6], where it is shown that S is infinite and has density zero. Moreover, [6] contains a table giving all the elements of S less than 73. In the present paper, we extend this table to include all the elements of S less than 1000. But our main purpose is first to give necessary conditions that k must satisiFy in order to belong to S. Then we will describe a general method to find the squares which are sums of k consecutive squares, for any given k, this method being an application of the theory of Pell's equation. Furthermore, we will show that if k belongs to S, then there exist infinitely many squares that can be written as the sum of k consecutive squares if and only if k itself is not a square. As usual, we will write Pallk iffpalk but p+1+ k; cr alwayswill denote a strictly positive integer.

Stimulating problems are at the heart of many great advances in mathematics. In fact, whole subjects owe their existence to a single problem which resisted solution. Nevertheless, we tend to present only polished theories, devoid of both the motivating problems and the long road to their solution. As a consequence, we deprive our students of both an example of the process by which mathematics is created and of the central problems which fueled its development. A more motivating approach could, for example, begin a discussion of infinite sets with Galileo's observation that there are as many integers as there are perfect squares. This observation seems as paradoxical to today's students as it did to Galileo. Its ingeniously simple resolution (through a better definition of "size") is a tremendous educational experience, an example of the kind of education which the German logician Heinrich Scholz characterized as "that which remains after we have forgotten everything we learned". We have designed a lower division honors course aimed at giving students the "big picture". In the course we examine the evolution of selected great problems from five mathematical subjects. Crucial to achieving this goal is the use of original sources to demonstrate the fundamental ideas developed for solving these problems. Studying original sources allows students truly to appreciate the progress achieved through time in the clarity and sophistication of concepts and techniques, and also reveals how progress is repeatedly stifled by certain ways of thinking until some quantum leap ushers in a new era. In addition to allowing a firsthand look at the mathematical mindscape of the time, no other method would show so clearly the evolution of mathematical rigor and the conception of what constitutes an acceptable proof. Thus most homework assignments focus on gaps and difficult points in the original texts. Since mathematics is not created in a social vacuum, we supplement the mathematical content with cultural, biographical, and mathematical history, as well as a variety of prose readings, ranging from Plato's dialogue Socrates and the Slave Boy to modern writings such as an excerpt on "Mathematics and the End of the World" from [8]. They form the basis of regular class discussions. Two good sources for such readings are [11, 18]. To encourage student involvement, the discussions are led by one or two students, and everybody is expected to contribute. As the finale, each student gives a short presentation of a research paper written on a topic of his or her choice. Our course serves as an "Introduction to Mathematics," drawing good students to the subject. It attracts students from remarkably diverse disciplines, serving as a general education course for some while acting as a springboard to further mathematics for others. Here are our mathematical themes and original sources.

is claimed to settle a problem of Stein [2] by possessing various properties, one of which is that every integer is included in at least one arithmetic progression in the system. Actually it is not hard to see that none of the arithmetic progressions in (1) contain 0 or any power of 3. In the present paper the system (1) will be modified so as to provide a correct example of the type requested by Stein. A system {(mi: ai): i E S} is called a cover in case its union is the set of all integers, and exact if its sets are pairwise disjoint. The system is incongruent if mi 0 mi whenever i 0 j, and infinite if S is infinite. Stein's problem is to find an infinite incongruent exact cover {(mi: ai): i E S} such that

The usual proofs of Bocher's Theorem rely either on the theory of superharmonic functions ([4], Theorem 5.4) or series expansions using spherical harmonics ([5], Chapter X, Theorem XII). (The referee has called our attention to the proof given by G. E. Raynor [7]. Raynor points out that the original proof of Maxime Bocher [2] implicitly uses some non-obvious properties of the level surfaces of a harmonic function.) In this, paper we offer a different and simpler approach to this theorem. The only results about harmonic functions needed are the minimum principle, Harnack's Inequality, and the solvability of the Dirichlet problem in Bn. We will investigate a harmonic function by studying its dilates. For u a function defined on Bn \ {0} and r E (0, 1), the dilate ur is the function defined on (l/r)Bn \ {0} by

f(z) = z2 _ A evaluated at the point z = zo should remain bounded as n tends to oo. We have only to mention the book [14] by Mandelbrot which has rendered this subject accessible to a much wider audience with its fine illustrations and graphical descriptions. A recent wave of activity in this area has generated, among others, the papers by Barnsley et al. [1, 2, 3, 4] and by Mandelbrot [15, 16]. A problem which is closely related to the one just described is to analyze the behaviour of the n-th iterate Fn(xO) of the function

The method of Lagrange for finding extrema of functions subject to equality constraints was published in 1788 in the famous book Mecanique Analytique. The works of Karush, John, Kuhn and Tucker concerning optimization subject to inequality constraints appeared more than 150 years after that. The purpose of this paper is to call attention to important papers, published as contributions to mechanics, containing fundamental ideas concerning optimization theory. The most important works in this respect were done primarily by Fourier, Cournot, Farkas and further by Gauss, Ostrogradsky and Hamel. (Author)

for an appropriate choice of xq. There are two standard and equivalent ways to derive it: algebraically (via the Taylor expansion) and geometrically (via the tangent line). We show initially how it can be derived in yet another way, via a constrained opti mization problem, by considering the special case of solving p(x) ? 0 for its smallest zero, where p(x) is a polynomial with all real zeros. We then do the same for several more classical zerofinders and construct some apparently new ones using the same optimization framework. While some of the resulting methods are applicable only to polynomial equations, others, like Newton's method, can be used for general nonlinear equations. The usefulness of this approach does not limit itself to the mere construction of zerofinders. As we will see later, it also allows us to carry out a controlled enhancement of those zerofinders. The paper is organized as follows: in section 2 we introduce our notation and state some preliminaries, in section 3 we derive Newton's and other methods, and in section 4 we show how these methods can be enhanced.

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.

The answer is yes, and follows from a version of Gauss's lemma applied to number fields. Gauss's lemma plays an important role in the study of unique factorization, and it was a failure of unique factorization that led to the development of the theory of algebraic integers. These developments were the basis of algebraic number theory, and also of much of ring and module theory. We take the opportunity afforded by this problem to discuss some of these historical developments, on the (at times flimsy) excuse of introducing the necessary notions for the proofs. It will take us some time to get to the answer to Question 1.1, so we ask for the reader's patience. To make for easier reading, we give names to many of the results. While some of these are standard, others are the inventions of the authors. The paper is organized as follows: First we discuss some of the history of unique factorization. In sections 3 and 4 we discuss how Euler and Lam6 ran afoul of unique factorization when dealing with Fermat's Last Theorem. Section 5 relates Kummer's own struggle with the failure of unique factorization, and his solution for cyclotomic fields. Section 6 deals with Kronecker's and Dedekind's extensions of Kummer's work, setting up the stage for our treatment of Question 1.1. In section 7 we recall the basic notions associated with number fields that we need, and proceed in section 8 to prove Gauss's lemma and a version of its corollaries for number fields, providing an answer to Question 1.1. In section 9 we discuss some related ring theoretic notions and provide an alternative approach to answering our question. Finally, in section 10 we consider the question in the setting of function fields (which are closely related to number fields) and give an example to show that Question 1.1 has a negative answer there.

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.

In two famous and popular puzzles a participant is required to compare two numbers of which she is shown only one. In the first one there are two envelopes with money in them. The sum of money in one of the envelopes is twice as large as the other sum. An envelope is selected at random and handed to you. If the sum in this envelope is x, then the sum in the other one is (1/2)(2x) + (1/2)(0.5x) = 1.25x. Hence, you are better off switching to the other envelope no matter what sum you see, which is paradoxical. In the second puzzle two distinct numbers are written on two slips of paper. One of them is selected at random and you observe it. How can you guess, with probability greater than 1/2 of being correct, whether this number is the larger or the smaller? We show that there is one principle behind the two puzzles: The ranking of n random variables X1, ... , Xn cannot be independent of each of them, unless the ranking is fixed. Thus, unless there is nothing to be learned about the ranking, there must be at least one variable the observation of which conveys information about it.

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.

The case r = 1 is fairly well-known: if A = [al, . . ., an] and B = [bl, . . ., b"]T are such that AB [1], then al, . . ., an are relatively prime. In this case, there is an n X n integral matrix that has an integral inverse and whose first row is al, . . ., an; see [10, Thm. II.1, p. 13]. Hajja's Problem suggests a natural generalization that has not received much attention and will be a main topic of this article: what are the properties that two or more given rows must have so they can serve as the first rows of an invertible matrix? The property central to our paper is the following. A matrix A with entries in a communtative ring R with unity is left good if, for every vector x, the ideal (xA) generated by the entries in the vector xA is the same as the ideal (x) generated by the entries in the vector x. In the context of matrices with integral entries, this is equivalent to requiring that the greatest common divisor of the entries in xA is the same as the greatest common divisor of the entries in x. Since, for any matrix A and any vector x, it is obvious that (xA) c (x), the content of left goodness is in the reverse containment. Our goal is to prove the following.

1. REMODELING CALCULUS THE INSTITUTION. Surely the renewal of Calculus is a good idea, one good enough to attract the attention and energy of many good people. But this is Calculus the Institution that peculiarly American academic event and all its supporting structures and expectations. Professor Knisely, however, barely hints at matters of institutional implementation, so I conclude that he is addressing Calculus, the System of Knowledge and Technique. As such, his paper is, perhaps, a warm-up exercise to a deep and long overdue reconsideration of the appropriate intellectual content of Calculus, one that has been postponed while we attempt to remodel Calculus the Institution. This remodeling has proven to be an arduous task for two reasons: (1) the renovation is taking place whilst the owners and stakeholders continue to inhabit the institution (a constraint applying to most educational reform); and relatedly (2) we have left all the larger structural features of the institution intact, including those features that connect it to the outside world, e.g., to the rapidly changing K-12 education. The basic architecture and its place in the larger world are untouched. I suggest that we embark on the more fundamentaJ rebuilding towards which Knisely points. In so doing we need to come to terms with the relations, existing and possible, between Calculus the Institution (C-INST) and Calculus the System of Knowledge and Technique (C-KNOWL). And we need to look more deeply and critically at the assumptions, largely tacit, that hold the status quo in place and provide some concrete, implementable alternatives.

This paper should be considered as expository classroom notes for the instructor. Existence of partitions of unity for metric spaces is usually proved via some rather exacting set-theoretical and topological arguments using the (equivalent) concept of paracompactness. Although the standard proof of paracompactness of metric spaces is the one given by M. E. Rudin [Ru], in his 1965 PhD thesis, Michael Mather showed that it is easier, for metric spaces, to show directly the existence of locally finite partitions of unity for arbitrary covers [Ma]. This proof does not seem to be so well-known although it has been reproduced in [Bo] (see also the appendix of [Do]). It is astonishing that Mather's arguments do not appear in recent textbooks. We hope that the following exposition for countable covers will popularise it. An advantage of the method is that the same proof can be used in the smooth category. Although it is formally covered by the countable case, we will first explain the method in the case of finite covers. T/his will show how easy it is to obtain partitions of unity for compact metric spaces. For the sake of completeness, let us recall a few definitions. If X is a topological space, the support of a continuous function : X > R is the closed set supp(f ) = f xl ( x) + o} . A partition of unity on X is a family (

Leonard Adleman recently found the solution to a problem in graph theory using DNA technology. (See [Adleman].) However, once the solution was obtained, Adleman checked that the solution was correct by hand. As a result, DNA evidence served no justificatory function in this case. Perhaps this is fortunate since "the mathematical establishment has often expressed its displeasure with certain types of 'proof' visual, mechanical, experimental, probabilistic" [Davis 1995, p. 212]. At the risk of displeasing the mathematical establishment, I will suggest in this paper how DNA evidence might legitimately stand in for a mathematical proof. THE DIRECTED HAMILTONIAN PATH PROBLEM. The problem in graph theory that Adleman solved is an instance of the directed Hamiltonian path problem (DHPP). A directed graph is a group of towns connected in various ways by one-way roads. One of the towns is designated the start town and another town is designated the destination town. A path is a sequence of towns such that each town in the sequence is connected by a one-way road to the next town in the sequence. A Hamiltonian path (HP) is a path from the start town to the destination town that visits every other town exactly one time. The DHPP asks if a given directed graph contains an HP and if so what the HP is. The DHPP is an example of an NP-complete problem. A distinguishing feature of NP-complete problems is that while it is easy to check that a solution is correct, it is difficult to find a solution in the first place. In the case of the DHPP, it is easy to check whether or not any particular path is an HP, but it is difficult to find a path that is an HP.

1. INTRODUCTION. Given initial position and direction, the flight-path of a ship in Euclidean space is completely determined by how much it turns and how much it twists at each odometer reading. This is an intuitive interpretation of the Fundamental Theorem for Space Curves, which states that curvature K and torsion , as functions of arclength s, determine a space curve uniquely up to rigid motion. This statement of the Fundamental Theorem ([14], §1-8) should be tempered with the reservations expressed by Nomizu [12] and Wong & Lai [15]. Given a parametric space curve, there are well-known formulae for the arclength, curvature, and torsion (as functions of the parameter). Given two functions of one parameter (potentially curvature and torsion parametrized by arc-length) one might like to find a parametrized space curve for which the two functions are the curvature and torsion. This activity, called "solving natural equations" ([14], §1-10), is generally achieved by solving Riccati equations like dw/ds = -iz/2 iKW + i7W /2. Although the solution generally exists, it usually cannot be obtained explicitly. Euler [6] found explicit integral formulae for plane curves (where z - O) through direct geometric analysis. Hoppe [9] developed a general method for solving the natural equations for space curves by solving Riccati equations through a complicated sequence of integral transformations. He digressed to obtain formulae for the tangent, normal, and binormal indicatrices for general helices and essentially for curves of constant precession. Enneper [5] obtained explicit closed-form solutions for helices on revolved conic sections through direct geometric analysis. A curve of constant precession is defined by the property that as the curve is traversed with unit speed, its centrode revolves about a fixed axis with constant angle and constant speed. In this paper we obtain an arclength-parametrized closed-form solution of the natural equations for curves of constant precession through direct geometric analysis. As part of this analysis, we obtain a new theorem for curves of constant precession analogous with Lancret's Theorem for general helices. We provide the first rendering of a curve of constant precession. We also note for the first time that curves of constant precession lie on circular hyperboloids of one sheet and have closure conditions that are simply related to their arclength, curvature, and torsion. These are 3-type curves, except one family of closed 2-type curves (when Z = 4,u; see [2], [3], and [1]). Given a closed C3 curve in space, it is rather obvious that the curvature and torsion functions will be periodic functions of the arclength, with period equal the total arclength. This is a necessary condition but, as the circular helices (K and z both constant) show, not a sufficient condition that integral curves be closed. Efimov [4] and Fenchel [7] independently formulated The Closed Curve Problem. Find (explicit) necessary and sufficient conditions that determine when, given two periodic functions K(S) and z(s) with the same period L, the integral curve is closed.

giving a way to calculate as many of the B's as desired. Formula (1.2), stated in tha way or in the form (1.3) or in various other forms, has been known a long time. Thus, in 1748 Euler gave (1.3) (obscurely expressed) in section 76 of his Introductio [4], [5]. Adams and Hippisley [1] gave the formula in the form (1.2) (their formula 6.361). Formula (1.3) is given as formula (0.314) in any of the several editions (Russian original, German, or English) of Ryshik and Gradstein [16], where, however, p is restricted to be a natural number. Thinking to remove this restriction of Ryshik and Gradstein, von Holdt [17] published still another derivation using properties of double sums to establish that (1.2) holds true for rational real p. His derivation avoids use of differentiation of the series in (1.1). We mention this because differentiation of formal power series affords a quick proof of (1.2) and has been used before. The basic recurrence relation is not as widely known as it should be, and has been rediscovered repeatedly. Thus Barrucand [2] found (1.2) again and made applications of it. Cappellucci [18] found it in the form (1.3) but attributed it to Hansted [19]. Hindenburg [12], in his treatise on the multinomial theorem (p. 291) gave (1.3) in a perfectly obscure notation. Many other references could be cited. Actually, the recurrence is implicit in still another class of formulas widespread in the literature, stemming from the early work of Hindenburg's student Rothe [15]. I myself have written a number of papers, e.g., [6]-[11] having to do with a formula of Rothe and its consequences for combinatorics, special functions and number theory. What we shall do here is to derive the formulas again and put them in a

For example, when f(x) =9x 2+x+ 1 = 3(x) and a =2, we find from Theorem 1 that a necessary condition for 4r + 2r + 1 to be prime is that r be a power of 3. A similar result, which has been applied by Liu, Reed, and Truong [1] based on f(x) = x2x+1= 16(x), and a = 2, is that a necessary condition for 4r 2r + 1 to be prime is that r > 1 be a multiple of 4 with no prime factor greater than three, i.e., that r = 2a3ft with a > 2, ,B > 0. When r is of this form, it can be shown (Theorem 4) that every prime q which divides 4r 2r + 1 is of the form 6rk + 1. Thus, in attempting to factor 412-212+1 =16,773,121, it suffices to look only for prime factors of the form q=72k+ 1. (In fact, the factors are q1 = 433 = 72 x 6 + 1 and q2= 38,737 = 72 x 538 + 1, where 433 is only the second prime number in the sequence 72k + 1.) The techniques introduced in this paper involve the factorization of Dn(x r) over the rational field, the factorization of tDn(a) over the integers, and, for comparison and completeness, the factorization of tDn(x) and Dn (Xr) over the integers modulo q.

Geometric problems involving a finite set of points in the plane have intrigued mathematicians for many years. As far back as 1893, J. J. Sylvester [4] posed the following problem: If S is a set of points in the plane, not all collinear, must there exist a line that contains exactly two points of S? This problem remained unsolved for forty years until it was rediscovered by P. Erdos in 1933 and solved a few days later by T. Gallai. This problem and a myriad of other famous problems in plane geometry are discussed in the fascinating paper by V. Klee [2]. In this note, we are concerned with results about maximum distances between pairs of points in the plane. An early problem of this type was posed by Hopf and Pannwitz [1] in 1934: What is the maximum number of pairs of points that can realize the diameter, the maximum distance between points of a set? The answer to this question appears in [3] and is the same as the number of points in the set. This is achieved by placing one point at the center of a circle of unit radius and the remaining points on a part of the boundary of the circle so that the two most distant points also are at distance 1 apart. We shall state and solve a variation of this problem. Consider a set N of n points in the Eucidean plane indexed by the integers 1, 2,.. ., n. A point j is called a furthest neighbour of a point i if d(i, j) = maxl,k n d(i,k), where d is the Euclidean distance function. Each point has at least one, and possibly several, furthest neighbours. Let n, denote the number of furthest neighbours of point i. Finally let