A digraph D=(V(D),A(D)) of order n≥3 is pancyclic, whenever D contains a directed cycle length k for each k∈{3,…,n}; and vertex-pancyclic iff, vertex v∈V(D) k∈{3,…,n}, passing through v. Let D1, D2, …, Dk be collection pairwise disjoint digraphs. The generalized sum (g.s.) Dk, denoted by ⊕i=1kDi or D1⊕D2⊕⋯⊕Dk, the set all digraphs satisfying: (i) V(D)=⋃i=1kV(Di), (ii) D〈V(Di)〉≅Di i=1,2,…,k, (ii...

PURPOSE To examine cell proliferation of the normal corneal epithelium and during extended rigid gas-permeable (RGP) lens wear. METHODS Twenty-three New Zealand White rabbits were fitted unilaterally with either a low oxygen transmissible (Dk/t) or hyper-Dk/t RGP lens, with the other eye serving as a control. The rabbits were injected with 5-bromo-2-deoxyuridine (BrdU) 24-hours later and kill...

The Marcinkiewicz Strong Law, lim n→∞ 1 n 1 p n ∑ k=1 (Dk −D) = 0 a.s. with p ∈ (1, 2), is studied for outer products Dk = XkX T k , where {Xk}, {Xk} are both twosided (multivariate) linear processes (with coefficient matrices (Cl), (Cl) and i.i.d. zero-mean innovations {Ξ}, {Ξ}). Matrix sequences Cl and Cl can decay slowly enough (as |l| → ∞) that {Xk, Xk} have long-range dependence while {Dk}...

[1] Douglass and Knox [2005, hereinafter referred to as DK], present an analysis of the observed tropospheric cooling following the 1991 Mt. Pinatubo eruption, and claim that these data imply a very low value for the climate sensitivity (equivalent to 0.6 C equilibrium warming for a CO2 doubling). We show here that their analysis is flawed and their results are incorrect. [2] We begin with a ve...

The hitting time expression Equation 14 involves three different types of contributions: a one-body term that depends on the destination node, C 1 jj Pn k1⁄41 dk; a two-body term that depends on the initial and final nodes, C 1 ji Pn k1⁄41 dk; and a series of three-body terms that depend on intermediate nodes, in addition to the two end points, Pn k1⁄41 C 1 ki C 1 kj dk. Derivation of Equatio...

Let dk,n and #k,n denote the dimension and the degree of the Grassmannian Gk,n, respectively. For each 1 ≤ k ≤ n−2 there are 2k,n ·#k,n (a priori complex) k-planes in Pn tangent to dk,n general quadratic hypersurfaces in Pn. We show that this class of enumerative problems is fully real, i.e., for 1 ≤ k ≤ n− 2 there exists a configuration of dk,n real quadrics in (affine) real space Rn so that a...

The one-dimensional Dunkl operator Dk with a non-negative parameter k, is considered under an arbitrary nonlocal boundary value condition. The right inverse operator of Dk, satisfying this condition is studied. An operational calculus of Mikusiński type is developed. In the frames of this operational calculi an extension of the Heaviside algorithm for solution of nonlocal Cauchy boundary value ...

Let [P,(")] be the class of all subsets P, ( ') of the k-dimensional Euclidean space consisting of n distinct points and having diameter 1 . Denote by dk (n, r) the maximum number of times a given distance r can occur among points of a set Pn ( k) . l'ut Dk (n) = max, d, (n, r) . In other words Dk (n) denotes the maximum number of times the same distance can occur between n suitably chosen poin...