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A new proof is given of the connecting homomorphism.

Let $N$ be a nilpotent group with torsion subgroup $TN$, and let $\alpha: TN\rightarrow \tilde T$ be a surjective homomorphism such that $\operatorname{ker}\alpha$ is normal in $N$. Then $\alpha$ determines a nilpotent group $\tilde N$ such that $T\tilde N=\tilde T$ and a function $\alpha_*$ from the Mislin genus of $N$ to that of $\tilde N$ if $N$ (and hence $\tilde N$) is finitely generated. The association $\alpha\mapsto\alpha_*$ satisfies the usual functorial conditions. Moreover $[N,N]$ is finite if and only if $[\tilde N,\tilde N]$ is finite and $\alpha_*$ is then a homomorphism of abelian groups. If $\tilde N$ belongs to the special class studied by Casacuberta and Hilton (Comm. in Alg. 19(7) (1991), 2051--2069), then $\alpha_*$ is surjective. The construction $\alpha_*$ thus enables us to prove that the genus of $N$ is non-trivial in many cases in which $N$ itself is not in the special class; and to establish non-cancellation phenomena relating to such groups $N$.

I had the good fortune to work closely with Alan Turing and to know him well for the last 12 years of his short life. It is a rare experience to meet an authentic genius. Those of us privileged to inhabit the world of scholarship are familiar with the intellectual stimulation furnished by talented colleagues. We can admire the ideas they share with us and are usually able to understand their source; we may even often believe that we ourselves could have created such concepts and originated such thoughts. However, the experience of sharing the intellectual life of a genius is entirely different; one realizes that one is in the presence of an intelligence, a sensitivity of such profundity and originality that one is filled with wonder and excitement. Alan Turing was such a genius, and those, like myself, who had the astonishing and unexpected opportunity created by the strange exigencies of the Second World War to be able to count Turing as colleague and friend will never forget that experience, nor can we ever lose its immense benefit to us. Before the war, in 1935–36, Turing had done fundamental work in mathematical logic and had invented a concept that has come to be known as the ‘universal Turing machine’ (see Chapter 6). His purpose was to make precise the notion of a computable mathematical function, but he had in fact provided a blueprint for the most basic principles of computer design and for the foundations of computer science. I joined the distinguished team of mathematicians and first-class chess players working on the Enigma code in January 1942. Alan Turing was the acknowledged leading light of that team. However, I must emphasize that we were a team—this was no one-man show! Indeed, Turing’s contribution was somewhat different from that of the rest of the team, being more concerned with improving our methods, especially the machines we used to help us, and less concerned with our daily output of deciphered messages. It was due to the efforts of Turing and the entire team that Churchill was able to describe our work as ‘my secret weapon’.

SummaryThis paper describes how the authors used a set of magnetic toys to discover analogues in 3 dimensions of well known theorems about binomial coefficients. In particular, they looked at the Star of David theorem involving the six nearest neighbors to a binomial coefficient . If one labels the vertices of the bounding hexagon with the numbers 1, 2, 3, 4, 5, 6, consecutively (in either direction), then the product of the coefficients with even labels is the same as the product as the coefficients with odd labels. Furthermore the two figures formed by connecting the odd and even vertices are both equilateral triangles arranged so that a rotation of ±60° exchanges the triangles. There is a generalized Star of David theorem concerning a semi-regular hexagon with similar results. The paper describes analogous results for trinomial coefficients involving, sometimes but not always, tetrahedra instead of triangles.

Dedicated to the memory of Russell Towle, a remarkable man who contributed so much to geometry and to other aspects of the quality of life.We introduce an expanded notation where r + s = n, for the binomial coefficient , and then use this expanded notation to develop theorems involving 8 binomial coefficients, analogous to the Star of David Theorem, which. in its original form, involved the 6 neighbours of a given binomial coefficient in the Pascal Triangle (see Section 3), that appeared in [1,2,3,4,5,6,7,8,9].

A general study is undertaken of product-wedge-diagonal (= PWD ) structures on a space. In part this concept may be viewed as arising from G.W. Whitehead's fat-wedge characterization of Lusternik–Schnirelmann category. From another viewpoint PWD -structures occupy a distinguished position among those structures that provide data allowing Hopf invariants to be defined. Indeed the Hopf invariant associated with a PWD -structure is a crucial component of the structure. Our overall theme addresses the basic question of existence of compatible structures on X and Y with regard to a map X → Y . A principal result of the paper uses Hopf invariants to formulate a Berstein–Hilton type result when the space involved is a double mapping cylinder (or homotopy pushout). A decomposition formula for the Hopf invariant (extending previous work of Marcum) is provided in case the space is a topological join U * V that has PWD -structure defined canonically via the join structure in terms of diagonal maps on U and V .

In [1, Chapter 3, Section 2], we collected together results we had previously obtained relating to the question of which positive integers m were Lucasian, that is, factors of some Lucas number Ln. We pointed out that the behaviors of odd and even numbers m were quite different. Thus, for example, 2 and 4 are both Lucasian but 8 is not; for the sequence of residue classes mod 8 of the Lucas numbers, n ⩾ 0, readsand thus does not contain the residue class 0*. On the other hand, it is a striking fact that, if the odd number s is Lucasian, then so are all of its positive powers.

We describe in detail Serre's application of spectral sequence theory to the study of the relations between the homology of total space, base space and fibre in a Serre fibration; and we apply the results to establish that a 1-connected space X has homology groups (in positive dimension) in a Serre class C if and only if its homotopy groups are in C . We include in this paper some personal reflections on the contact the author had with Serre during the decade of the 1950's when Serre's revolutionary work in homotopy theory was completely changing the face of algebraic topology.

We give a sustained example, drawn largely from earlier publications, of how we may freely pursue a line of mathematical enquiry if we are not constrained, unnaturally, to confine ourselves to a single mathematical subdiscipline; and we draw conclusions from the study of this example which are relevant at many levels of mathematical instruction. We also include the statement and proof of a new result (Theorem 4.1) in linear algebra which is obviously fundamental to the geometrical investigation which constitutes the leit-motif of the paper.

A rational number is said to be expressed in Egyptian form (or Egyptian presentation) if it is written as a sum of fractions where each fraction has the numerator 1 and all of the denominators are distinct. The individual terms of die sum are often referred to as unit fractions. The problem of finding such expressions, and many related questions, has a long history, apparently beginning with Ahmes in the Rhind Papyrus. It is for this reason that we talk of ‘Egyptian’ form.

(1999). Relating Geometry and Algebra in the Pascal Triangle, Hexagon, Tetrahedron, and Cuboctahedron Part II: Geometry and Algebra in Higher Dimensions: Identifying the Pascal Cuboctahedron. The College Mathematics Journal: Vol. 30, No. 4, pp. 279-292.

We prove that there is a bijection between the Mislin genus of a circle bundle over a certain nilpotent base space M M , which is constructed from a nilpotent group N N of a certain specified type, and the Mislin genus of N N itself.

Over the years, research in mathematical problem-solving has examined expert/novice problem-solving performance on various types of problems and subjects. In particular, DeFranco examined two groups of Ph.D. mathematicians as they solved four mathematics problems and found that although all were content experts , only one group were problem-solving experts . Based on this study, this article posits the notion that one distinguishing feature between experts and novices is that experts tend to look for special features of a problem and use algorithms only as a “fail-safe” system while novices act like “machines” relying on algorithms to solve the problems. Why? The article explores the idea that novice problem solvers learned to solve problems the way they learned proof, that is, in a formal, abstract and mechanizable way. Beliefs about proof and the culture in which it is practiced help frame a mathematician's view of the discipline and ultimately impacts classroom practice. The authors believe that current classroom instruction tends to create a culture that fosters algorithmic proficiency and a “machine-like” approach to the learning of mathematics and problem-solving. Further, they argue that mathematicians need to be aware of the distinction between knowing a proof is true and explaining why it is true. When these distinctions are appreciated and practiced during classroom instruction, then and only then will students begin to acquire the mathematical knowledge to become better problem solvers.

(1997). The Faces of the Tri-Hexaflexagon. Mathematics Magazine: Vol. 70, No. 4, pp. 243-251.

We will discuss in this article a fractal-like structure made from a flat piece of paper. What will motivate most people to want to make the structure is that it is pretty. In fact, the exercise has two rather obvious uses. First, greeting card companies may want to use the idea to manufacture interesting 3-dimensional cards which fit conveniently into envelopes; and, second, teachers may wish to teach students how to make it (and this process will also involve teaching them some beautiful mathematics). The structure itself is a model for a stage in a self-similarity process leading to a fractal. Building the model involves scoring a flat piece of paper in a prescribed manner, cutting along some of the score lines, and then folding some lines as ‘mountain’ folds and others as ‘valley’ folds.

We use methods of algebraic topology to study when a connected topological space admits ann-mean map.

and which is not primarily erected for this purpose. The applications bear witness to the power of mathematics, but are not its real motivation. The springs of 686 [October MATHEMATICS: QUESTIONS AND ANSWERS This content downloaded from 207.46.13.120 on Wed, 14 Sep 2016 04:15:20 UTC All use subject to http://about.jstor.org/terms

The Mislin genus $\Cal G(N)$ of a finitely generated nilpotent group $N$ with finite commutator subgroup admits an abelian group structure. If $N$ satisfies some additional conditions ---we say that $N$ belongs to $\Cal N_1$--- we know exactly the structure of $\Cal G(N)$. Considering a direct product $ N_1 \times \cdots \times N_k$ of groups in $\Cal N_1$ takes us virtually always out of $\Cal N_1$. We here calculate the Mislin genus of such a direct product.

In their stimulating paper [1], to which we here make a friendly and constructive response the authors, Branko Grunbaum and Geoffrey Shephard, introduce the interesting geometric concept of polyhedral set, generalizing the familiar notion of polyhedron, but confining themselves, for their present purposes, to subsets of 1R3. They discuss dissections of such sets, especially relatively open convex dissections of bounded polyhedral sets and show, by easily accessible arguments, the nice properties of the Euler characteristic X relative to such dissections. They are thereby led to a formula for X(P), namely, Theorem 4 of [1]

In three earlier articles we drew attention to an astonishing error which appears in several texts in the United States, even some of otherwise very high quality. The error relates to a fallacious procedure for the calculation of the Euler characteristic of a polyhedron, producing, in particular, the extraordinary statement that the Euler characteristic of a torus is 2.

Let N be a P-local nilpotent group. We say that N is finitely generated (fg) as P-local group if there is a finite subset S of N such that the smallest P-local subgroup of N containing S is N itself; we could also say that S generates N as P-local group and write N=〈S〉 p. If G is nilpotent, we say that G is finitely generated at every prime (fgp) if Gp is fg as p-local group for all primes p. Such groups share with fg nilpotent groups many important properties (e.g., they are Hopfian). We show that they satisfy the axioms for a Serre class, as extended by Hilton and Roitberg to nilpotent groups.

Disclaimer This article is full of generalizations, many about student attitudes. Such generalizations, unlike generalizations in mathematics, are not invalidated by the existence of counterexamples. All of us know the joy of teaching those exceptional undergraduates who really want to learn and are stimulated by the beauty and power of the mathematics they are learning. It is these students who give us the conviction that we are, as teachers, doing something thoroughly worthwhile, despite our many failures. 1. INTRODUCTION. I have been distressed for many years by the defects in our methods of evaluating students-at all levels from 3rd grade upwards to graduate students-and have railed especially against the unfortunate effects of over-frequent testing and inappropriate tests, especially multiple-choice and standardized tests. I have not changed my point of view on these issues; but I have come to realize that the ubiquity of intrusive testing, contaminating the learning environment, distorting the curriculum, and undermining the fabric of the teacher-student relationship, springs from a deep malaise in our whole approach to education, and is not to be remedied simply by attempting to improve the quality and relevance of the tests administered and reducing their frequency. Such a superficial approach may be thought of as treating a symptom and leaving the cause of the disease untouched. In this article I would like to develop this thought, and then raise the question of whether any change-any beneficial change, that is-is possible. I will largely confine myself to undergraduate education-principally, of course, in mathematics. 2. THE ROOT CAUSE OF THE PROBLEM. I have often remarked that, in this country, there is a fundamental confusion of education with training. Thus, whereas drivers of lethal automobiles stand in evident need of first-class training, we provide 'driver education'; and whereas teachers stand in evident need of first-class education, we provide 'teacher training.' Indeed, I would now go further and say that, in general, our students-and their parents-only feel comfortable with the presence of a curricular item if it has an evident value for the students'

Should you always follow instructions? All self-respecting human beings, and therefore all our readers, must answer this question with a resounding NO! In the next paragraph we describe two aspects of our paper-folding, and building, instructions where we do advise rather rigid adherence to our specifications. However, we are very far from recommending that you fold all your regular polygons and construct all your polyhedra exactly as described. What we have done is to give you algorithms for the relevant constructions. Machines follow algorithms with relentless fervor, while human beings look for special ways of doing particular, convenient things. Always feel free to use your ingenuity to avoid an algorithm that is not working for you, or seems to you to be unduly complex. A word to the wise We've done a lot of field-testing of the “hands-on” material in this book. Our instructions seem to be, on the whole, quite comprehensible to most readers. However, there are two basic types of error that people seem prone to make in carrying out our instructions. Material error In doing mathematics, it is absurd to specify the quality of paper on which the mathematics should be done. However, when we describe to you how to make mathematical models, we must insist that the choice of material is not arbitrary. Instructions for making models that are easily constructed using gummed mailing tape are unlikely to be effective if a strip of paper taken from an exercise book is used instead.

This easy-to-read 2010 book demonstrates how a simple geometric idea reveals fascinating connections and results in number theory, the mathematics of polyhedra, combinatorial geometry, and group theory. Using a systematic paper-folding procedure it is possible to construct a regular polygon with any number of sides. This remarkable algorithm has led to interesting proofs of certain results in number theory, has been used to answer combinatorial questions involving partitions of space, and has enabled the authors to obtain the formula for the volume of a regular tetrahedron in around three steps, using nothing more complicated than basic arithmetic and the most elementary plane geometry. All of these ideas, and more, reveal the beauty of mathematics and the interconnectedness of its various branches. Detailed instructions, including clear illustrations, enable the reader to gain hands-on experience constructing these models and to discover for themselves the patterns and relationships they unearth.

Peter Hilton is Distinguished Professor of Mathematics at the State University of New York at Binghamton. He is the author of numerous books and research articles, mainly in the areas of algebraic topology, homological algebra, and group theory. He has a long-standing interest in mathematics education?he was chairman of the United States Commission on Mathematical Instruction and chairman of the National Research Council Com? mittee on Applied Mathematics Training?and has been much concerned, with Jean Pedersen, to revive the study of geometry at the pre-college and college levels and to develop an inte? grated mathematics curriculum.