On étudie le phénomène de retard à la bifurcation dans des systèmes dynamiques discrets du plan. La distinction d’une courbe invariante par le système permet de ramener l’étude de ce phénomène à l’étude d’un objet. On démontre la présence du retard dans les systèmes analytiques oscillants. On fait état d’un nouveau phénomène découvert expérimentalement qui apparaît dans les systèmes non inversibles: la courbe invariante présente une succession de pôles exponentiellement étroits. On démontre la présence...

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,...

Newton's method for computation of a square root yields a difference equation which can be solved using the hyperbolic cotangent function. For the computation of the third root Newton's sequence presents a harder problem, which already Cayley was trying to solve. In the present paper two mutually inverse functions are defined in order to solve the difference equation, instead of the hyperbolic cotangent and its inverse. Several coefficients in the expansion around the fixed points are obtained,...

Conditions under which the solutions of a partial difference equations system can be probability functions are examined.When the coefficients of the system are polynomials then the partial difference equations system satisfied by generating functions associated to these distributions are easily obtained; they give useful recurrence relations for the moments. Three examples are given as well.

Given $\{h_1,\cdots ,h_t\}$ a finite subset of ${ℝ}^d$, we study the continuous complex valued functions and the Schwartz complex valued distributions $f$ defined on ${ℝ}^d$ with the property that the forward differences ${\Delta }_{h_k}^{m_k}f$ are (in distributional sense) continuous exponential polynomials for some natural numbers $m_1,\cdots ,m_t$.

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...