We consider a class of singular Riemannian manifolds, the deformed spheres S k , defined as the classical spheres with a one parameter family g[k] of singular Riemannian structures, that reduces for k = 1 to the classical metric. After giving explicit formulas for the eigenvalues and eigenfunctions of the metric Laplacian ∆SN k , we study the associated zeta functions ζ(s,∆SN k ). We introduce ...

Suppose $p$ is a prime, $t$ is a positive integer, and $f\!\in\!\mathbb{Z}[x]$ is a univariate polynomial of degree $d$ with coefficients of absolute value $<\!p^t$. We show that for any fixed $t$, we can compute the number of roots in $\mathbb{Z}/(p^t)$ of $f$ in deterministic time $(d+\log p)^{O(1)}$. This fixed parameter tractability appears to be new for $t\!\geq\!3$. A consequence for arit...

LetM be a tensor category with coefficients in a fieldK of characteristic 0, that is, a K-linear pseudo-abelian symmetric monoidal category such that the tensor product ⊗ of M is bilinear. Then symmetric and exterior powers of an object M ∈ M make sense, by using appropriate projectors relative to the action of the symmetric groups on tensor powers of M . One may therefore introduce the zeta fu...

If there are a positive (and b negative) coefficients among the λj’s, then we say that (1.1) has badge (a, b). Two linear forms are distinct if they (and their 2s-th powers) are not proportional; a representation is honest if its summands are pairwise distinct. Let B(p) denote the set of possible badges of honest representations of p. Badges are ordered componentwise: (a, b) (c, d) if a ≤ c and...

New results about certain sums Sn(k) of products of the Lucas numbers are derived. These sums are related to the generating function of the k-th powers of the Fibonacci numbers. The sums for Sn(k) are expressed by the binomial and the Fibonomial coefficients. Proofs of these formulas are based on a special inverse formula.

The assurance of power quality depends on factors which influence the performances of the availability: the reliability and its support – the maintenance. This paper deals with the interactions between the indicators of reliability, maintenance, availability and with the methods for the calculation of the reliability indicators. For an overhead line of 110 kV the characteristics of the reliabil...

We establish the existence of the asymptotic expansion of the Bergman kernel associated to the spin Dirac operators acting on high tensor powers of line bundles with non-degenerate mixed curvature (negative and positive eigenvalues) by extending [15]. We compute the second coefficient b1 in the asymptotic expansion using the method of [24].

The second Appell’s hypergeometric function F2(a, b, b ′, c, c′;x, y) has a Mellin convolution integral representation in the region (x + y) < 1 and a > 0. We apply a recently introduced asymptotic method for Mellin convolution integrals to derive three asymptotic expansions of F2(a, b, b ′, c, c′;x, y) in decreasing powers of x and y with x/y bounded. For certain values of the real parameters ...

We prove divisibility properties for sums of powers of binomial coefficients and of q-binomial coefficients. Dedicated to the memory of Paul Erdős