This study describes the program bTd, which was developed for the decomposition of any $n$-tangle with $1 < n < 10$} into base {\small$n$}-tangles using the Skein relation. The program enables us to compute HOMFLY polynomials of knots and links with a large number of crossing points within a matter of hours (see Examples 4.4 and 4.5). This contrasts with the results of attempting computations using Hecke algebras $H(q,n)$} with $18 \ge n$. Such a computation did not complete even after a period of thirty days in a recent examination by the first author and F. Kako [Imafuji and Ochiai 02, Murakami 89, Ochiai and Murakami 94, Ochiai and Kako 95]. In this paper, we first introduce two new concepts: an oriented ordered tangle and a subdivision of a tangle. We then present some examples of base-tangle decompositions achieved using the present program along with the corresponding computational times.

This paper describes a numerical method for finding good packings in Grassmannian manifolds equipped with various metrics. This investigation also encompasses packing in projective spaces. In each case, producing a good packing is equivalent to constructing a matrix that has certain structural and spectral properties. By alternately enforcing the structural condition and then the spectral condition, it is often possible to reach a matrix that satisfies both. One may then extract a packing from this matrix.

This approach is both powerful and versatile. In cases in which experiments have been performed, the alternating projection method yields packings that compete with the best packings recorded. It also extends to problems that have not been studied numerically. For example, it can be used to produce packings of subspaces in real and complex Grassmannian spaces equipped with the Fubini–Study distance; these packings are valuable in wireless communications. One can prove that some of the novel configurations constructed by the algorithm have packing diameters that are nearly optimal.

In this article we report on extensive calculations concerning the Gorenstein defect for Hecke algebras of spaces of modular forms of prime weight $p$ at maximal ideals of residue characteristic $p$ such that the attached mod-$p$ Galois representation is unramified at $p$ and the Frobenius at $p$ acts by scalars. The results lead us to ask the question whether the Gorenstein defect and the multiplicity of the attached Galois representation are always equal to $2$. We review the literature on the failure of the Gorenstein property and multiplicity one, discuss in some detail a very important practical improvement of the modular-symbols algorithm over finite fields, and include precise statements on the relationship between the Gorenstein defect and the multiplicity of Galois representations.

We have verified that the van der Waerden number $W(2, 6)$ is 1132, that is, 1132 is the smallest integer n = W(2, 6) such that whenever the set of integers {1, 2, . . . , $n$} is 2-colored, there exists a monochromatic arithmetic progression of length 6. This was accomplished by applying special preprocessing techniques that drastically reduced the required search space. The exhaustive search showing that $W(2, 6)$ = 1132 was carried out by formulating the problem as a satisfiability (SAT) question for a Boolean formula in conjunctive normal form (CNF), and then using a SAT solver specifically designed for the problem. The parallel backtracking computation was run over multiple Beowulf clusters, and in the last phase, field programmable gate arrays (FPGAs) were used to speed up the search. The fact that $W(2, 6)$ > 1131 was shown previously by the first author.

Let $f(2,3,4)$} denote the smallest integer $n$ such that there exists a l$K_4$}-free graph of order $n$ for which any $2$-coloring of its edges yields at least one monochromatic triangle. It is well known that such a number must exist. For a long time, the best known upper bound, provided by J. Spencer, was $f(2,3,4)$ < 3 \cdot 10^9. Recently, L. Lu announced that $f(2,3,4)$ < 10,000. In this note, we will give a computer-assisted proof showing that $f(2,3,4)$ < 1000. To prove this, we will generalize an idea of Goodman's, giving a necessary and sufficient condition for a graph $G$ to yield a monochromatic triangle for every edge coloring.

We analyze properties of the 2-adic valuations of $S(n,k)$, the Stirling numbers of the second kind. For fixed $k \in \mathbb{N}$, a conjectured pattern for the valuation is provided in terms of the dyadic format of $n$. This conjecture is established for $k=5$.

We prove that there exist infinitely many triples $a,b,c$ of nonzero rational numbers with the property that $a^2+1$, $b^2+1$, $c^2+1$, $ab+1$, $ac+1$, and $bc+1$ are perfect squares.

In this paper we give upper and lower bounds as well as a heuristic estimate on the number of vertices of the convex closure of the set $ G_n={((a,b) : a,b\in \Z,\; ab \equiv 1$ (mod $n$), $1\leq a,b\leq n-1}$. The heuristic is based on an asymptotic formula of Renyi and Sulanke. After describing two algorithms to determine the convex closure, we compare the numeric results with the heuristic estimate, and find that they do not agree--there are some interesting peculiarities, for which we provide a heuristic explanation. We then describe some numerical work on the convex closure of the graph of random quadratic and cubic polynomials over $\Z_n$. In this case the numeric results are in much closer agreement with the heuristic, which strongly suggests that the curve $xy=1$ (mod $n$) is ``atypical.''

We give some heuristics for counting elliptic curves with certain properties. In particular, we rederive the Brumer-McGuinness heuristic for the number of curves with positive/negative discriminant up to {\small$X$}, which is an application of lattice-point counting. We then introduce heuristics that allow us to predict how often we expect an elliptic curve $E$ with even parity to have $L(E,1)=0$. We find that we expect there to be about $c_1X^{19/24}(\log X)^{3/8}$ curves with $|\Delta|<X$ with even parity and positive (analytic) rank; since Brumer and McGuinness predict {\small$cX^{5/6}$} total curves, this implies that, asymptotically, almost all even-parity curves have rank $0$. We then derive similar estimates for ordering by conductor, and conclude by giving various data regarding our heuristics and related questions.