Related papers: Algebraic independence and difference equations ov…

In a previous article [N. Delice, F.W. Nijhoff and S. Yoo-Kong, J. Phys. A: Math. Theor. 48(3) (2015), 035206] a novel class of elliptic Lax pairs for integrable lattice equations was introduced. The present article proposes a…

We study Popa's notion of rigidity for equivalence relations induced by actions on homogeneous spaces. For any lattices $\Gamma,\Lambda$ in a semisimple Lie group $G$ with finite center and no compact factors we prove that the action…

We derive exact formulas for the proportions of derangements and of derangements of $p$-power order in the affine classical groups $AU_m(q)$, $ASp_{2m}(q)$, $AO_{2m+1}(q)$ and $AO^{\pm}_{2m}(q)$, where $p$ denotes the characteristic of the…

The aim of this note is to give a direct proof for the following result proved by Fountain and Lewin: {\em Let $\alg$ be an independence algebra of finite rank and let $a$ be a singular endomorphism of $\alg $. Then $a=e_1... e_n$ where…

In our previous papers \cite{doz1,doz2} we studied Laguerre functions and polynomials on symmetric cones $\Omega=H/L$. The Laguerre functions $\ell^{\nu}_{\mathbf{n}}$, $\mathbf{n}\in\mathbf{\Lambda}$, form an orthogonal basis in…

The grouplike elements of a coalgebra over a field are known to be linearly independent over said field. Here we prove three variants of this result. One is a generalization to coalgebras over a commutative ring (in which case the linear…

Let $\phi: \R^d \longrightarrow \C$ be a compactly supported function which satisfies a refinement equation of the form $\phi(x) = \sum_{k\in\Lambda} c_k \phi(Ax - k),\quad c_k\in\C$, where $\Gamma\subset\R^d$ is a lattice, $\Lambda$ is a…

We prove upper bounds for Hecke-Laplace eigenfunctions on certain Riemannian manifolds of arithmetic type. The manifolds under consideration are d-fold products of 2-spheres or 3-spheres, realized as adelic quotients of quaternion algebras…

The well-known analogies between number fields and function fields have led to the transposition of many problems from one domain to the other. In this paper, we will discuss traffic of this sort, in both directions, in the theory of…

Let $\Lambda$ be a finite dimensional algebra over an algebraically closed field $k$. We survey some results on algebras of finite global dimension and address some open problems.

Let $G$ be a Lie group, and $H\subset G$ a closed subgroup. Let $\pi$ be an irreducible unitary representation of $G$. In this paper, we briefly discuss the orbit method and its application to the branching problem $\pi|_{H}$. We use the…

We study periodic points for endomorphisms $\sigma$ of abelian varieties $A$ over algebraically closed fields of positive characteristic $p$. We show that the dynamical zeta function $\zeta_\sigma$ of $\sigma$ is either rational or…

Given any non-polynomial $G$-function $F(z)=\sum\_{k=0}^\infty A\_k z^k$ of radius of convergence $R$, we consider the $G$-functions $F\_n^{[s]}(z)=\sum\_{k=0}^\infty \frac{A\_k}{(k+n)^s}z^k$ for any integers $s\geq 0$ and $n\geq 1$. For…

Let $p$ be an odd prime number, $E$ an elliptic curve defined over a number field. Suppose that $E$ has good reduction at any prime lying above $p$, and has supersingular reduction at some prime lying above $p$. In this paper, we construct…

Recently, we have established and used the generalized Littlewood theorem concerning contour integrals of the logarithm of analytical function to obtain new criteria equivalent to the Riemann hypothesis. Later, the same theorem was applied…

We extract an exact formula relating the number of lattice points in an expanding region of a complex semi-simple symmetric space and the automorphic spectrum from a spectral identity, which is obtained by producing two expressions for the…

We consider elliptic differential operators on either the entire Euclidean space $\mathbb{R}^d$ or on subsets consisting of a cube $\Lambda_L$ of integer length $L$. For eigenfunctions of the operator, and more general solutions of elliptic…

We show that orbit equivalence relations arising from essentially free ergodic probability measure preserving actions of Zariski dense discrete subgroups of simple algebraic groups are strongly prime. As a consequence, we prove the…

Consider some non-zero complex numbers $a_i, b_i, c_i, d_i$ with $1 \leq i \leq n$ and the associated classical Lotka-Volterra systems \[ \begin{cases} x' = a_i xy + b_i y \newline y' = c_i xy + d_i y \text{ .} \end{cases} \] We show that…

In recent years, various notions of algebraic independence have emerged as a central and unifying theme in a number of areas of applied mathematics, including algebraic statistics and the rigidity theory of bar-and-joint frameworks. In each…