By a topological ring we mean a ring X with a Hausdorff topology such that (x, y) 7→ x + y, x 7→ −x, (x, y) 7→ x · y are continuous. A morphism of topological rings is a continuous homomorphism of rings. An inverse system of topological rings is a family of topological rings Xi and a family of morphisms πi,j : Xi → Xj for i, j ∈ I with i ≥ j, such that when i ≥ j ≥ k, πi,k = πj,k ◦ πi,j . If Y ...

Using the variational method and critical point theory, the authors study the existence of infinitely many homoclinic solutions to the difference equation −∆ ( a(k)φp(∆u(k − 1)) ) + b(k)φp(u(k)) = λf(k, u(k))), k ∈ Z, where p > 1 is a real number, φp(t) = |t|p−2t for t ∈ R, λ > 0 is a parameter, a, b : Z→ (0,∞), and f : Z × R → R is continuous in the second variable. Related results in the lite...

A Wilf–Zeilberger pair (F,G) in the discrete case satisfies the equation F(n+ 1,k)−F(n,k) = G(n,k+ 1)−G(n,k). We present a structural description of all possible rational Wilf–Zeilberger pairs and their continuous and mixed analogues.

background: cross-contamination between cells is a usual mistake at cell culture laboratories and cell banks worldwide. mrc5 diploid cell and rk-13, vero and hela continuous cell lines are used in different stages of human viral vaccines propagation at razi vaccine and serum research institute of iran. however diploid and continuous cells are propagated at separated cell culture laboratory and ...

We prove that any minimal valid function for the k-dimensional infinite group relaxation that is continuous piecewise linear with at most k+ 1 slopes and does not factor through a linear map with non-trivial kernel is extreme. This generalizes a theorem of Gomory and Johnson for k = 1, and Cornuéjols and Molinaro for k = 2.

This paper is a continuation of [5]. We apply the upper envelope representation method (or the quasilinearization method) in vector lattices developed in [4, 5] to the homogeneous functional calculus of linear and bilinear operators. Explicit formulae for computing φ̂(T1, . . . , TN ) for any finite sequence of regular linear or bilinear operators T1, . . . , TN are derived. For the theory of ve...

A general setting for constrained Z,1-approximation is presented. Let Un be a finite dimensional subspace of C[a, b] and L be a linear operator from Un to C(K) (r = 0, 1) where K is a finite union of disjoint, closed, bounded intervals. For v , u e C(K) with v < u, the approximating set is Univ, u) = {p e Un : v < Lp < u on K} and the norm is \\f\\w = Xf \f\wdx where w a positive continuous fun...

In this note, we give an elementary proof of the random walk hitting time theorem, which states that, for a left-continuous random walk on Z starting at a nonnegative integer k, the conditional probability that the walk hits the origin for the first time at time n, given that it does hit zero at time n, is equal to k/n. Here, a walk is called left-continuous when its steps are bounded from belo...

We show that if U is a bounded open set in a complete CAT(0) spaceX , and if f : U → X is nonexpansive, then f always has a fixed point if there exists p ∈U such that x / ∈ [p, f (x)) for all x ∈ ∂U . It is also shown that if K is a geodesically bounded closed convex subset of a complete R-tree with int(K) = ∅, and if f : K → X is a continuous mapping for which x / ∈ [p, f (x)) for some p ∈ int...