In this paper we discuss planar quadrilateral (PQ) nets as discrete models for convex affine surfaces. As a main result, we prove a necessary and sufficient condition for a PQ net to admit a Lelieuvre co-normal vector field. Particular attention is given to the class of surfaces with discrete harmonic co-normals, which we call discrete affine minimal surfaces, and the subclass of surfaces with co-planar discrete harmonic co-normals, which we call discrete improper affine spheres. Within this classes,...

We use the concept of intrinsic metrics to give a new definition for an isoperimetric constant of a graph. We use this novel isoperimetric constant to prove a Cheeger-type estimate for the bottom of the spectrum which is nontrivial even if the vertex degrees are unbounded.

The authors consider the difference equation $${\Delta }^m[y_n-p_n{y}_{n-k}]+\delta q_n{y}_{\sigma (n+m-1)}=0(*)$$ where $m\ge 2$, $\delta =\pm 1$, $k\in N_0=\{0,1,2,\cdots \}$, $\Delta y_n=y_{n+1}-y_n$, $q_n>0$, and $\left\{\sigma \right(n\left)\right\}$ is a sequence of integers with $\sigma \left(n\right)\le n$ and ${lim}_{n\to \infty }\sigma \left(n\right)=\infty$. They obtain results on the classification of the set of nonoscillatory solutions of ($*$) and use a fixed point method to show the existence of solutions having certain types of asymptotic behavior. Examples illustrating the results are included.

En choisissant des “caractères” et des “logarithmes”, méromorphes sur $ℂ$, construits à l’aide de la fonction Gamma d’Euler, et en utilisant des séries de factorielles convergentes, nous sommes en mesure, dans une première partie, de donner une “forme normale” pour les solutions d’un système aux différences singulier régulier. Nous pouvons alors définir une matrice de connexion d’un tel système. Nous étudions ensuite, suivant une idée de G.D. Birkhoff, le lien de celles-ci avec le problème de la classification...

We establish conditions which guarantee that the second order difference equation $${\Delta }^2{x}_k+p_k{x}_{k+1}=0$$ possesses a nontrivial solution with at least two generalized zero points in a given discrete interval

Using Fan’s Min-Max Theorem we investigate existence of solutions and their dependence on parameters for some second order discrete boundary value problem. The approach is based on variational methods and solutions are obtained as saddle points to the relevant Euler action functional.

We investigate the criticality of the one term $2n$-order difference operators $l{\left(y\right)}_k={\Delta }^n\left(r_k{\Delta }^n{y}_k\right)$. We explicitly determine the recessive and the dominant system of solutions of the equation $l{\left(y\right)}_k=0$. Using their structure we prove a criticality criterion.