### A matrix inequality for Möbius functions.

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We present in this paper a stability study concerning finite volume schemes applied to the two-dimensional Maxwell system, using rectangular or triangular meshes. A stability condition is proved for the first-order upwind scheme on a rectangular mesh. Stability comparisons between the Yee scheme and the finite volume formulation are proposed. We also compare the stability domains obtained when considering the Maxwell system and the convection equation.

We study the interplay between recurrences for zeta related functions at integer values, 'Minor Corner Lattice' Toeplitz determinants and integer composition based sums. Our investigations touch on functional identities due to Ramanujan and Grosswald, the transcendence of the zeta function at odd integer values, the Li Criterion for the Riemann Hypothesis and pseudo-characteristic polynomials for zeta related functions. We begin with a recent result for ζ(2s) and some seemingly new Bernoulli relations,...

Let $h$ be a complex valued multiplicative function. For any $N\in \mathbb{N}$, we compute the value of the determinant ${D}_{N}:={det}_{i|N,j|N}\left(\frac{h\left(\left(i,j\right)\right)}{ij}\right)$ where $\left(i,j\right)$ denotes the greatest common divisor of $i$ and $j$, which appear in increasing order in rows and columns. Precisely we prove that $${D}_{N}=\prod _{p{}^{l}\parallel N}{\left(\frac{1}{{p}^{l\left(l+1\right)}}\stackrel{l}{\prod _{i=1}}\left(h\left({p}^{i}\right)-h\left({p}^{i-1}\right)\right)\right)}^{\tau \left(N/{p}^{l}\right)}.$$ This means that ${D}_{N}^{1/\tau \left(N\right)}$ is a multiplicative function of $N$. The algebraic apparatus associated with this result allows us to prove the following two results. The first one is the characterization of real multiplicative functions $f\left(n\right)$, with $0\le f\left(p\right)<1$, as minimal values of certain...

Consider the group ${SL}_{2}\left({\mathbf{O}}_{K}\right)$ over the ring of algebraic integers of a number field $K$. Define the height of a matrix to be the maximum over all the conjugates of its entries in absolute value. Let ${SL}_{2}({\mathbf{O}}_{K},t)$ be the number of matrices in ${SL}_{2}\left({\mathbf{O}}_{K}\right)$ with height bounded by $t$. We determine the asymptotic behaviour of ${SL}_{2}({\mathbf{O}}_{K},t)$ as $t$ goes to infinity including an error term,$${SL}_{2}({\mathbf{O}}_{K},t)=C{t}^{2n}+O\left({t}^{2n-\eta}\right)$$with $n$ being the degree of $K$. The constant $C$ involves the discriminant of $K$, an integral depending only on the signature of $K$, and the value of the Dedekind zeta function...

This is a survey of recent results concerning (integer) matrices whose leading principal minors are well-known sequences such as Fibonacci, Lucas, Jacobsthal and Pell (sub)sequences. There are different ways for constructing such matrices. Some of these matrices are constructed by homogeneous or nonhomogeneous recurrence relations, and others are constructed by convolution of two sequences. In this article, we will illustrate the idea of these methods by constructing some integer matrices of this...