In this paper, we count the number of non-attacking bishop and king positions on the regular and cylindrical m×n (where m = 1, 2, 3) chessboards. This is accomplished through the use of scientific computing, recurrence relations, generating functions and closed-form formulas.

We find the solution of the partial difference equation u(x, t + 1) = l u(x + 1, t)+ r u(x − 1, t) for x ∈ [1,m] subject to the absorbing boundary conditions at x = 0 and x = m+1. Green’s function will be determined using random walk techniques applying the reflective and inclusion-exclusion principles. AMS subject classification: 39A12, 37H10.

The authors consider the diierence equation () m yn ? pny n?k ] + qny (n+m?1) = 0 f(n)g is a sequence of integers with (n) n and limn!1 (n) = 1. They obtain results on the classiication of the set of nonoscillatory solutions of () and use a xed point method to show the existence of solutions having certain types of asymptotic behavior. Examples illustrating the results are included.

We show that the difference equation xn = f3(xn−1) f2(xn−2) f1(xn−3), n ∈ N0, where fi ∈ C[(0,∞),(0,∞)], i ∈ {1,2,3}, is periodic with period 4 if and only if fi(x) = ci/x for some positive constants ci, i ∈ {1,2,3} or if fi(x) = ci/x when i = 2 and fi(x) = cix if i ∈ {1,3}, with c1c2c3 = 1. Also, we prove that the difference equation xn = f4(xn−1) f3(xn−2) f2(xn−3) f1(xn−4), n ∈ N0, where fi ∈...

A review is made of recent efforts to define linear connections and their corresponding curvature within the context of noncommutative geometry. As an application it is suggested that it is possible to identify the gravitational field as a phenomenological manifestation of space-time commutation relations and to thereby clarify its role as an ultraviolet regularizer.

We investigate the global dynamics of solutions of two distinct competitive rational systems of difference equations in the plane. We show that the basins of attraction of different locally asymptotically stable equilibrium points are separated by the global stable manifolds of either saddle points or of non-hyperbolic equilibrium points. Our results give complete answer to Open Problem 1 posed...

We study differential-difference equation of the form d dx t(n+ 1, x) = f(t(n, x), t(n + 1, x), d dx t(n, x)) with unknown t(n, x) depending on continuous and discrete variables x and n. Equation of such kind is called Darboux integrable, if there exist two functions F and I of a finite number of arguments x, {t(n ± k, x)}k=−∞, { d dxk t(n, x) }∞ k=1 , such that DxF = 0 and DI = I, where Dx is ...

3 Analytic classification of admissible q-difference modules 10 3.1 Generalities on q-difference modules . . . . . . . . . . . . . . . . . . . . . . . 11 3.2 Simple objects in the category of admissible q-difference modules . . . . . . . 11 3.3 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.4 Analytic vs formal classification . . . . . . . . . . . . ...

In this work we give sufficient conditions in order that the difference equation with advanced arguments x(n + 1) = A(n)x(n) + B(n)x(σ1(n)) + f(n, x(n), x(σ2(n))), σi(n) ≥ n + 1 has Φ−bounded solutions in the space `Φ of Φ−bounded sequences, are given.

Denote by x = (x0, x1, ..., xn) = (x0, x′) the coordinates in R × R. Sometimes we refer to the first coordinate x0 as “time” and to x′ as “space” coordinates. The Strichartz estimates for solutions to the homogeneous wave equation in R× R, 2u = 0, u(0) = u0, ut(0) = u1, have the form ‖|D|u‖Lp(Lq) . ‖∇u0‖L2 + ‖u1‖L2. (1.1) Such an estimate holds for all pairs (ρ, p, q) satisfying the relations 2...