We study a discrete model of the $SU\left(2\right)$ Yang-Mills equations on a combinatorial analog of ${ℝ}^4$. Self-dual and anti-self-dual solutions of discrete Yang-Mills equations are constructed. To obtain these solutions we use both the techniques of a double complex and the quaternionic approach.

In this paper, by using an iterative scheme, we advance the main oscillation result of Zhang and Liu (1997). We not only extend this important result but also drop a superfluous condition even in the noniterated case. Moreover, we present some illustrative examples for which the previous results cannot deliver answers for the oscillation of solutions but with our new efficient test, we can give affirmative answers for the oscillatory behaviour of solutions. For a visual explanation of the examples,...

This paper is concerned with the nonlinear advanced difference equation with constant coefficients $$x_{n+1}-x_n+\sum _{i=1}^m{p}_i{f}_i\left(x_{n-k_i}\right)=0,n=0,1,\cdots$$ where $p_i\in (-\infty ,0)$ and $k_i\in \{\cdots ,-2,-1\}$ for $i=1,2,\cdots ,m$. We obtain sufficient conditions and also necessary and sufficient conditions for the oscillation of all solutions of the difference equation above by comparing with the associated linearized difference equation. Furthermore, oscillation criteria are established for the nonlinear advanced difference equation with variable coefficients $$x_{n+1}-x_n+\sum _{i=1}^m{p}_{in}{f}_i\left(x_{n-k_i}\right)=0,n=0,1,\cdots$$ where $p_{in}\le 0$ and $k_i\in \{\cdots ,-2,-1\}$ for $i=1,2,\cdots ,m$.

We consider the maximal regularity problem for the discrete time evolution equation $u_{n+1}-Tuₙ=fₙ$ for all n ∈ ℕ₀, u₀ = 0, where T is a bounded operator on a UMD space X. We characterize the discrete maximal regularity of T by two types of conditions: firstly by R-boundedness properties of the discrete time semigroup ${\left(Tⁿ\right)}_{n\in \mathbb{N}₀}$ and of the resolvent R(λ,T), secondly by the maximal regularity of the continuous time evolution equation u’(t) - Au(t) = f(t) for all t > 0, u(0) = 0, where A:= T - I. By recent results of...

We consider a motion of spiral-shaped piecewise linear curves governed by a crystalline curvature flow with a driving force and a tip motion which is a simple model of a step motion of a crystal surface. We extend our previous result on global existence of a spiral-shaped solution to a linear crystalline motion for a power type nonlinear crystalline motion with a given rotating tip motion. We show that self-intersection of the solution curves never occurs and also show that facet extinction never...