We provide a number of new construction techniques for cubical complexes and cubical polytopes, and thus for cubifications (hexahedral mesh generation). As an application we obtain an instance of a cubical $4$-polytope that has a nonorientable dual manifold (a Klein bottle). This confirms an existence conjecture of Hetyei (1995).

More systematically, we prove that every normal crossing codimension one immersion of a compact 2-manifold into {\small $\R^3$} is PL-equivalent to a dual manifold immersion of a cubical 4-polytope. As an instance we obtain a cubical 4-polytope with a cubification of Boy's surface as a dual manifold immersion, and with an odd number of facets. Our explicit example has 17,718 vertices and 16,533 facets. Thus we get a parity-changing operation for three-dimensional cubical complexes (hex meshes); this solves problems of Eppstein, Thurston, and others.

We estimate an integral on the Sierpinski gasket and justify a theorem in the paper [Lund et al. 98]. The integral relates to the Laplace transform of the Hausdorff measure. It is fundamental and useful in some other contexts [Dong and Lau 04].

We introduce primality tests using algebraic groups. Some previously known tests are naturally interpreted as special cases of these algebraic group tests. Moreover, in this framework we can generalize Lucas and {\small $n+1$} tests in a natural way.

We summarize the classification of all firm and residually connected geometries satisfying the conditions {\small $(IP)_2$} and {\small $(2T)_1$} and on which the Janko group {\small $J_3$} acts flag-transitively and residually weakly primitively. We state some facts regarding the results. The complete list of geometries is available as a supplement to this note [Leemans 03c].

Minimal factorisations of groups can be used in cryptography. It is not yet known if one exists for each finite group, although it has been shown that a minimal counterexample of a group without one must be simple. We prove existence of minimal factorisations for some sporadic groups.

We propose in this work a computational criterion to test if a free divisor {\small $D\subset {\bf C}^n$} verifies the Logarithmic Comparison Theorem (LCT); that is, whether the complex of logarithmic differential forms computes the cohomology of the complement of {\small $D$} in {\small ${\bf C}^n$}.

For Spencer free divisors {\small $D\equiv(f=0)$}, we solve a conjecture about the generators of the annihilating ideal of {\small $1/f$} and make a conjecture on the nature of Euler homogeneous free divisors which verify LCT. In addition, we provide examples of free divisors defined by weighted homogeneous polynomials that are not locally quasi-homogeneous.

A perfect form is a form {\small $f=mx^2+ny^2+kxy$} with integral coefficients {\small $(m,n,k)$} such that {\small $f(\mathbb{Z}^2)$} is a multiplicative semigroup. The growth rate of the number of perfect forms in cubes of increasing side {\small $L$} in the space of the coefficients is known for small cubes, where all perfect forms are known. A form is perfect if its coefficients belong to the image of a map, {\small $Q$}, from {\small $\mathbb{Z}^4$} to {\small $\mathbb{Z}^3$}. This property of perfect forms allows us to estimate from below the growth rate of their number for larger values of {\small $L$}. The conjecture that all perfect forms are generated by {\small $Q$} allows us to reformulate results and conjectures on the numbers of the images {\small $Q(\mathbb{Z}^4)$} in cubes of side {\small $L$} in terms of the numbers of perfect forms. In particular, the proportion of perfect elliptic forms in a ball of radius {\small $R$} should decrease faster than {\small $R^{-3/4}$} and the proportion of all perfect forms in a ball of radius {\small $R$} should decrease faster than {\small $2/\sqrt{R}$}.

We study the Veech group of an origami, i.e., of a translation surface, tessellated by parallelograms. We show that it is isomorphic to the image of a certain subgroup of {\small $\aut^+(F_2)$ in $\slzwei(\ZZ) \cong \out^+(F_2)$}. Based on this, we present an algorithm that determines the Veech group.

We describe a numerical experiment concerning the order of magnitude of {\small $% q(x):=M\left( x\right) /\sqrt{x}$}, where {\small $M(x)$} is the Mertens function (the summatory function of the Möbius function). It is known that, if the Riemann hypothesis is true and all nontrivial zeros of the Riemann zeta-function are simple, {\small $q(x)$} can be approximated by a series of trigonometric functions of {\small $\log x$}. We try to obtain an {\small $\Omega $}-estimate of the order of {\small $q(x)$} by searching for increasingly large extrema of the sum of the first {\small $10^{2}$, $10^{4}$}, and {\small $10^{6}$} terms of this series. Based on the extrema found in the range {\small $10^{4}\leq x\leq 10^{10^{10}}$} we conjecture that {\small $q(x)=\Omega _{\pm }(\sqrt{\log \log \log x})$}.

We present a conjecture for the power-law exponent in the asymptotic number of types of plane curves as the number of self-intersections goes to infinity. In view of the description of prime alternating links as flype equivalence classes of plane curves, a similar conjecture is made for the asymptotic number of prime alternating knots.

The rationale leading to these conjectures is given by quantum field theory. Plane curves are viewed as configurations of loops on random planar lattices, that are in turn interpreted as a model of two-dimensional quantum gravity with matter. The identification of the universality class of this model yields the conjecture.

Since approximate counting or sampling planar curves with more than a few dozens of intersections is an open problem, direct confrontation with numerical data yields no convincing indication on the correctness of our conjectures. However, our physical approach yields a more general conjecture about connected systems of curves. We take advantage of this to design an original and feasible numerical test, based on recent perfect samplers for large planar maps. The numerical data strongly support our identification with a conformal field theory recently described by Read and Saleur.

A group is termed parafree if it is residually nilpotent and has the same nilpotent quotients as a given free group. Since free groups are residually nilpotent, they are parafree. Nonfree parafree groups abound and they all have many properties in common with free groups. Finitely presented parafree groups have solvable word problems, but little is known about the conjugacy and isomorphism problems. The conjugacy problem plays an important part in determining whether an automorphism is inner, which we term the inner automorphism problem. We will attack these and other problems about parafree groups experimentally, in a series of papers, of which this is the first and which is concerned with the isomorphism problem. The approach that we take here is to distinguish some parafree groups by computing the number of epimorphisms onto selected finite groups. It turns out, rather unexpectedly, that an understanding of the quotients of certain groups leads to some new results about equations in free and relatively free groups. We touch on this only lightly here but will discuss this in more depth in a future paper.

We prove a very general identity, conjectured by Henri Cohen, involving the generating function of the family {\small $(\zeta(2r+4s+3))_{r,s\ge 0}$}: it unifies two identities, proved by Koecher in 1980 and Almkvist and Granville in 1999, for the generating functions of {\small $(\zeta(2r+3))_{r\ge 0}$} and {\small $(\zeta(4s+3))_{s\ge 0}$}, respectively. As a consequence, we obtain that, for any integer {\small $j \ge 0$}, there exists at least {\small $[j/2]+1$ } Apéry-like formulae for {\small $\zeta(2j+3)$}.