We show that the edge graph of a 6-dimensional polytope with 12 facets has diameter at most 6, thus verifying the $d$-step conjecture of Klee and Walkup in the case $d = 6$. This implies that for all pairs $(d, n)$ with $n − d ≤ 6$, the diameter of the edge graph of a $d$-polytope with $n$ facets is bounded by 6, which proves the Hirsch conjecture for all $n − d ≤ 6$. We prove this result by establishing this bound for a more general structure, so-called matroid polytopes, by reduction to a small number of satisfiability problems.

We study experimentally systems of orthogonal polynomialswith respect to self-similar measures. When the support of the measure is a Cantor set, we observe some interesting properties of the polynomials, both on the Cantor set and in the gaps of the Cantor set. We introduce an effective method to visualize the graph of a function on a Cantor set. We suggest a new perspective, based on the theory of dynamical systems, for studying families $P_n(x)$ of orthogonal functions as functions of $n$ for fixed values of $x$.

A numerical process to approximate optimal partitions in any dimension is reported. The key idea of the method is to relax the problem into a functional framework based on the famous result of Γ-convergence obtained by Modica and Mortolla.

The classical class invariants of Weber are introduced as quotients of Thetanullwerte, enabling the computation of these invariants more efficiently than as quotients of values of the Dedekind η-function. We show also how to compute the unit group of suitable ring class fields by proving the fact that most of the invariants introduced by Weber are actually units in the corresponding ring class fields.

We compute many dimensions of spaces of finite type invariants of virtual knots (of several kinds) and the dimensions of the corresponding spaces of “weight systems,” finding everything to be in agreement with the conjecture that “every weight system integrates.”

It is a long-standing questionwhether an arbitrary variety is desingularized by finitely many normalized Nash blowups. We consider this question in the case of a toric variety. We interpret the normalized Nash blowup in polyhedral terms, show how continued fractions can be used to give an affirmative answer for a toric surface, and report on a computer investigation in which over a thousand 3- and 4-dimensional toric varieties were successfully resolved.

This paper is devoted to presenting numerical simulations and a theoretical interpretation of results for determining the minimal $k$-partitions of a domain $\Omega$ as considered in Helffer et al., Nodal Domains and Spectral Minimal Partitions. More precisely, using the double-covering approach introduced by B. Helffer, M. and T. Hoffmann-Ostenhof, and M. Owen and further developed for questions of isospectrality by the authors in collaboration with T. Hoffmann-Ostenhof and S. Terracini in Helffer et al., and Bonnaillie-Noël et al. Aharonov–Bohm Hamiltonians, Isospectrality and Minimal Partitions.”, we analyze the variation of the eigenvalues of the one-pole Aharonov– Bohm Hamiltonian on the square and the nodal picture of the associated eigenfunctions as a function of the pole. This leads us to discover new candidates for minimal k-partitions of the square with a specific topological type and without any symmetric assumption, in contrast to our previous works. This illustrates also recent results of B. Noris and S. Terracini; see Noris and Terracini 10, Nodal Sets of Magnetic Schrödinger Operators of Aharonov–Bohm Type and Energy Minimizing Partitions”. This finally supports or disproves conjectures for the minimal 3- and 5-partitions on the square.

We construct a planar quartic system and demonstrate that it has at least 26 limit cycles. The vector field is symmetric and integrable, but non-Hamiltonian. The proof is based on a verified computation of zeros of pseudo-Abelian integrals, together with the symmetry properties.

An amicable pair for an elliptic curve $E/\mathbb{Q}$ is a pair of primes $(p, q)$ of good reduction for $E \sharp\tilde{E}_p(\mathbb{F}_p) = q$ and $\sharp\tilde{E}_q(\mathbb{F}q) = p$. In this paper we study elliptic amicable pairs and analogously defined longer elliptic aliquot cycles. We show that there exist elliptic curves with arbitrarily long aliquot cycles, but that CM elliptic curves (with $j \neq 0$) have no aliquot cycles of length greater than two.We give conjectural formulas for the frequency of amicable pairs. For CM curves, the derivation of precise conjectural formulas involves a detailed analysis of the values of the Grössencharacter evaluated at primes $\mathfrak{p}$ in $\operatorname{End}(E )$ having the property that $\sharp\tilde{E}_{\mathfrak{p}}(\mathbb{F}_{\mathfrak{p}})$ is prime. This is especially intricate for the family of curves with $j = 0$.

Methods from numerical algebraic geometry are applied in combination with techniques from classical representation theory to show that the variety of 3 × 3 × 4 tensors of border rank 4 is cut out by polynomials of degree 6 and 9. Combined with results of Landsberg and Manivel, this furnishes a computational solution of an open problem in algebraic statistics, namely, the set-theoretic version of Allman’s salmon conjecture for 4 × 4 × 4 tensors of border rank 4. A proof without numerical computation was given recently by Friedland and Gross.