A Note on the Recursive Sequence x n + 1 = p k x n + p k − 1 x n − 1 +...+ p 1 x n − k + 1

We present some comments on the behavior of solutions of the difference equation where p _i 0, i = 1,..., k, k N, and x _–k ,..., x _–1 R.     

Minimal Covers of Q +(2n + 1, q) by (n − 1)-Dimensional Subspaces

A t-cover of a quadric is a set C of t-dimensional subspaces contained in such that every point of is contained in at least one element of C.We consider (n – 1)-covers of the hyperbolic quadric Q ⁺(2n + 1, q). We show that such a cover must have at least q ^{n + 1} + 2q + 1 elements, give an example of this size for even q and describe what covers of this size should look like.     

Linking (n − 2)-dimensional panels inn-space I: (k − 1,k)-graphs and (k − 1,k)-frames

A (k – 1,k)-graph is a multi-graph satisfyinge (k – 1)v – k for every non-empty subset ofe edges onv vertices, with equality whene = |E(G)|. A (k – 1,k)-frame is a structure generalizing an (n – 2, 2)-framework inn-space, a structure consisting of a set of (n – 2)-dimensional bodies inn-space and a set of rigid bars each joining a pair of bodies using ball joints. We prove that a graph is the graph of a minimally rigid (with respect to edges) (k – 1,k)-frame if and only if it is a (k – 1,k)-graph. Rigidity here means infinitesimal rigidity or equivalently statical rigidity.     

Lokal-reguläre Bogen ohne (n−2,k)-Sekanten im projektivenP n (n≦k)

Ohne Zusammenfassung     

On the Behavior of Solutions of x n+1 = p+(x n−1/xn )

Our goal in this article is to complete the study of the behavior of solutions of the equation in the title when the parameter p is positive and the initial conditions are arbitrary positive numbers. Our main focus is the case 0 < p < 1. We will show that in this case, all solutions which do not monotonically converge to the equilibrium have a subsequence which converges to p and a subsequence which diverges to infinity. For the sake of completeness, we will also present the results (which were previously known) with alternative proofs for the case p = 1 and the case p > 1.     

Über einige Grundeigenschaften der Bogen ohne (n−2,k)-Sekanten im projektivenP n (n≦k)

Ohne Zusammenfassung     

More on the Difference Equation y n + 1 = ( p + y n −1 )/( qy n + y n −1 )

From previous studies of the equation in the title with positive parameters p and q and positive initial conditions we know that if q h 4 p + 1 then the equilibrium is a global attractor. We also know that if q > 4 p + 1 then every solution eventually enters and remains in the interval [ p / q , 1]. In this strip there exists a "unique" prime period two solution that is locally asymptotically stable. In this paper, we provide more insight as to the behavior of solutions of the equation in the title in the strip [ p / q , 1], where a one-dimensional stable manifold lives.     

Boundedness properties of the difference equation x n+1=(α+βx n +δx n−2)/(x n−1)

Consider the third-order difference equation x _n+1 = (α+βx _n +δx _n ? 2)/(x _n ? 1) with α ∈ [0,∞) and β,δ ∈ (0,∞). It is shown that this difference equation has unbounded solutions if and only if δ>β.     

Singular (k,n − k) boundary value problems between conjugate and right focal

For 1 ⩽k⩽n − 1 and 0 ⩽q⩽k − 1, solutions are obtained for the boundary value problem, (−1)^n−k = f(x,y), y⁽ⁱ⁾=0, 0⩽i⩽k − 1, and y⁽ⁱ⁾ = 0, q⩽j⩽n − k + q − 1, where f(x,y) is singular at y = 0. An application is made of a fixed point theorem for operators that are decreasing with respect to a cone.     

On the chromatic uniqueness of the graphW(n,n − 2, k)

This paper shows that the graphW(n, n – 2, k) is chromatically unique for any even integern 6 and any integerk 1.