For many meta-Fibonacci sequences it is possible to identify a partition of the sequence into successive intervals (sometimes called blocks) with the property that the sequence behaves “similarly” in each block. This partition provides insights into the sequence properties. To date, for any given sequence, only ad hoc methods have been available to identify this partition. We apply a new concept—the spot-based generation sequence—to derive a general methodology for identifying this partition for a large class of meta-Fibonacci sequences. This class includes the Conolly and Conway sequences and many of their well-behaved variants, and even some highly chaotic sequences, such as Hofstadter’s famous Q-sequence.

In the online compendium, J. L. Loday, Encyclopedia of Types of Algebras, it is asked whether the alternative operad is Koszul. The purpose of this note is to demonstrate that the answer to this question is negative. In doing so, we have been aided by the programs Albert and PARI/GP.

We investigate experimentally the growth regimes of the number of LEGO structures that can be constructed contiguously from n blocks of equal shape and color.

We present a theoretical algorithm that given any finite presentation of a group as input, will terminate with answer yes if and only if the group is large. We then implement a practical version of this algorithm using MAGMA and apply it to a range of presentations. Our main focus is on two-generator one-relator presentations, for which we have a complete picture of largeness if the relator has exponent sum zero in one generator and word length at most 12, as well as if the relator is in the commutator subgroup and has word length at most 18. Indeed, all but a tiny number of presentations define large groups. Finally, we look at fundamental groups of closed hyperbolic 3-manifolds, for which the algorithm readily determines that at least a quarter of the groups in the SnapPea closed census are large.

We test a conjectural nonabelian refinement of the classical 2- adic Main Conjecture of Iwasawa theory. In the first part, we show how, in the special case that we study, the validity of this refinement is equivalent to a congruence condition on the coefficients of some power series. Then, in the second part, we explain how to compute the first coefficients of this power series and thus numerically check the conjecture in that setting.

We establish properties of and propose a conjecture concerning $\sum_m (S(x + m))2$, where $S$ is a piecewise polynomial cardinal spline in $L^2(\mathbb{R})$.

The integer Chebyshev problem is the problem of finding an integer polynomial of degree n such that the supremum norm on [0, 1] is minimized. The most common technique used to find upper bounds is by explicit construction of an example. This is often (although not always) done by heavy computational use of the LLL algorithm and simplex method. One of the first methods developed to find lower bounds employed a sequence of polynomials known as the Gorshkov–Wirsing polynomials.

This paper studies properties of the Gorshkov–Wirsing polynomials. It is shown how to construct generalized Gorshkov–Wirsing polynomials on any interval $[a, b]$, with $a, b ∈ \mathbb{Q}$. An extensive search for generalized Gorshkov–Wirsing polynomials is carried out for a large family of $[a, b]$. Using generalized Gorshkov– Wirsing polynomials, LLL, and the simplex method, upper and lower bounds for the integer Chebyshev constant on intervals other than $[0, 1]$ are calculated. These methods are compared with other existing methods.

The 4-dimensional abstract Kummer variety $K^4$ with 16 nodes leads to the $K^3$ surface by resolving the 16 singularities. Here we present a simplicial realization of this minimal resolution. Starting with a minimal 16-vertex triangulation of $K^4$, we resolve its 16 isolated singularities—step by step—by simplicial blowups. As a result we obtain a 17-vertex triangulation of the standard PL $K^3$ surface. A key step is the construction of a triangulated version of the mapping cylinder of the Hopf map from real projective 3-space onto the 2-sphere with the minimum number of vertices. Moreover, we study simplicial Morse functions and the changes of their levels between the critical points. In this way we obtain slicings through the $K^3$ surface of various topological types.

This paper classifies the maximal finite subgroups of ${\rm Sp}_{2n}(\mathbb{Q})$ for $1 ≤ n ≤ 11$ up to conjugacy in ${\rm GL}_{2n}(\mathbb{Q})$.