We first define the curvature indices of vertices of digital objects. Second, using these indices, we define the principal normal vectors of digital curves and surfaces. These definitions allow us to derive the Gauss-Bonnet theorem for digital objects. Third, we introduce curvature flow for isothetic polytopes defined in a digital space.

We consider the implicit discretization of Nagumo equation on finite lattices and show that its variational formulation corresponds in various parameter settings to convex, mountain-pass or saddle-point geometries. Consequently, we are able to derive conditions under which the implicit discretization yields multiple solutions. Interestingly, for certain parameters we show nonuniqueness for arbitrarily small discretization steps. Finally, we provide a simple example showing that the nonuniqueness...

The main objective of this paper is to study the boundedness character, the periodicity character, the convergence and the global stability of positive solutions of the difference equation $$x_{n+1}=\frac{{\alpha }_0{x}_n+{\alpha }_1{x}_{n-l}+{\alpha }_2{x}_{n-k}}{{\beta }_0{x}_n+{\beta }_1{x}_{n-l}+{\beta }_2{x}_{n-k}},n=0,1,2,\cdots$$ where the coefficients ${\alpha }_i,{\beta }_i\in (0,\infty )$ for $i=0,1,2,$ and $l$, $k$ are positive integers. The initial conditions $x_{-k},\cdots ,x_{-l},\cdots ,x_{-1},x_0$ are arbitrary positive real numbers such that $l<k$. Some numerical experiments are presented.