Mathematika最新文献

Let $$ be integers. Using a fragmentation technique, we characterise $$-tuples $$ of non-empty families of partitions of $$ such that it suffices that an order-$$ tensor has bounded $$-rank for each $$ for it to have bounded $$-rank. On the way, we prove power lower bounds on suitable products of diagonal tensors, providing a qualitative answer to a question of Naslund.

Let $$ be a positive integer. Let $$ be a henselian local ring with residue field $$ of $$th level $$. We give some upper and lower bounds for the $$th Waring number $$ in terms of $$ and $$. In large number of cases, we are able to compute $$. Similar results for the $$th Waring number of the total ring of fractions of $$ are obtained. We then provide applications. In particular, we compute $$ and $$ for $$ and any prime $$.

We investigate polytopes inscribed into a sphere that are normally equivalent (or strongly isomorphic) to a given polytope $$. We show that the associated space of polytopes, called the inscribed cone of $$, is closed under Minkowski addition. Inscribed cones are interpreted as type cones of ideal hyperbolic polytopes and as deformation spaces of Delaunay subdivisions. In particular, testing if there is an inscribed polytope normally equivalent to $$ is polynomial time solvable. Normal equivalence is decided on the level of normal fans and we study the structure of inscribed cones for various classes of polytopes and fans, including simple, simplicial, and even. We classify (virtually) inscribable fans in dimension 2 as well as inscribable permutahedra and nestohedra. A second goal of the paper is to introduce inscribed virtual polytopes. Polytopes with a fixed normal fan $$ form a monoid with respect to Minkowski addition and the associated Grothendieck group is called the type space of $$. Elements of the type space correspond to formal Minkowski differences and are naturally equipped with vertices and hence with a notion of inscribability. We show that inscribed virtual polytopes form a subgroup, which can be nontrivial even if $$ does not have actual inscribed polytopes. We relate inscribed virtual polytopes to routed particle trajectories, that is, piecewise-linear trajectories of particles in a ball with restricted directions. The state spaces gives rise to connected groupoids generated by reflections, called reflection groupoids. The endomorphism groups of reflection groupoids can be thought of as discrete holonomy groups of the trajectories and we determine when they are reflection groups.

In this paper, we consider the Finsler $$-Laplacian torsion equation. The domain of the problem is bounded by a conical surface supporting a Neumann-type condition, and an unknown surface supporting both a Dirichlet and a Neumann condition. The case when the cone coincides with the punctured space is included. We show that the existence of a weak solution implies that the unknown surface lies on the boundary of a Finsler-ball. Incidentally, some properties of the Finsler–Minkowski norms are proved here under mild smoothness assumptions.

In this article, we will get nontrivial estimates for the central values of degree six Rankin–Selberg $$-functions $$ associated with a $$ form $$ and a $$ form $$ using the delta symbol approach in the hybrid $$ level and $$-aspect.

Let $$ be a prime number, $$ and $$ positive integers coprime with $$. We provide the explicit continued fraction expansion of the $$th root of $$ in the power series field $$. We determine its irrationality exponent.

Let $$ be any positive and non-square integer. We prove an upper bound for the first two moments of the length $$ of the period of the continued fraction expansion for $$. This allows to improve the existing results for the large deviations of $$.

We investigate strong divisibility sequences and produce lower and upper bounds for the density of integers in the sequence that only have (somewhat) large prime factors. We focus on the special cases of Fibonacci numbers and elliptic divisibility sequences, discussing the limitations of our methods. At the end of the paper, there is an appendix by Sandro Bettin on divisor closed sets that we use to study the density of prime terms that appear in strong divisibility sequences.

Assuming the Riemann hypothesis, we investigate the shifted moments of the zeta function

Let $$ be a convex body in $$ in which a ball rolls freely and which slides freely in a ball. Let $$ be the intersection of $$ i.i.d. random half-spaces containing $$ chosen according to a certain prescribed probability distribution. We prove an asymptotic upper bound on the variance of the mean width of $$ as $$. We achieve this result by first proving an asymptotic upper bound on the variance of the weighted volume of random polytopes generated by $$ i.i.d. random points selected according to certain probability distributions, then, using polarity, we transfer this to the circumscribed model.