Let G = (V,E) be a graph and f be a function f : V → {0, 1, 2}. A vertex u with f(u) = 0 is said to be undefended with respect to f , if it is not adjacent to a vertex with positive weight. The function f is a weak Roman dominating function (WRDF) if each vertex u with f(u) = 0 is adjacent to a vertex v with f(v) > 0 such that the function f ′ : V → {0, 1, 2} defined by f ′ (u) = 1, f ′ (v) = f...

Let f be a function in an Orlicz space L and μ(f,L) be the set of all the best Φ-approximants to f, given a σ−lattice L. Weak type inequalities are proved for the maximal operator f∗ = supn |fn|, where fn is any selection of functions in μ(f,Ln), and Ln is an increasing sequence of σ-lattices. Strong inequalities are proved in an abstract set up which can be used for an operator as f∗.

In the crystal structure of the title salt, C11H18N(+)·PF6(-), the cation and anion are connected via an N-H⋯F hydrogen bond; weak C-H⋯F hydrogen bonding also occurs between the cations and anions.

We propose a finite volume scheme to discretize the one-dimensional Vlasov–Poisson system. We prove that, if the initial data are positive, bounded, continuous, and have their first moment bounded, then the numerical approximation converges to the weak solution of the system for the weak topology of L. Moreover, if the initial data belong to BV , the convergence is strong in C(0, T ;L loc ). To...

In this paper, we consider a variant of Pillai's problem over function fields F in one variable C. For given simple linear recurrence sequences Gn and Hm, defined satisfying some weak conditions, will prove that the equation Gn−Hm=f has only finitely many solutions (n,m)∈N2 for any non-zero f∈F, which can be effectively bounded. Furthermore, under suitable assumptions there are computable f wit...

In this article, we describe spaces P such that : if u is a weak (in the sense of Leray [26]) solution of the Navier-Stokes system for some initial data u0, and if u belongs to P, then u is unique in the class of weak solutions. We say then that weak-strong uniqueness holds. It turns out that the proof of such results relies on the boundedness of a trilinear functional F : L2/αḢα × L2/βḢβ × P →...

Bumps, the elem entary excitatory events o f the Limulus ventral nerve photorecep tor follow ­ ing a weak flash o f light were recorded under voltage clam p conditions. The statistical d is tri­ bution o f various bum p param eters and their changes caused by weak conditioning pre-illumination are described, and the influence o f lowered external C a2+-concentration together with norm al or rai...

The weak intramolecular F∙∙∙N and S∙∙∙N interactions in the novel pyridine-containing perfluorodiarylethenes achieved high cyclization quantum yields in solutions. While the F∙∙∙N interactions stabilize the photoreactive conformation, the S∙∙∙N contacts contribute from the energetic aspect.

Certain corrections are needed in this paper concerning the definition and properties of the weak topology on 12. Instead of characterising it in terms of converging sequences as on p. 7, one should define the weak topology as the coarsest topology for which the evaluations f --+ f o h(t)dr(t), f E ]2, are continuous for all continuous h(t) of compact support. With this topology, 12 is a Tihono...

and Applied Analysis 3 R1 F T / ∅; R2 φ p, Tx ≤ φ p, x , for all x ∈ C, p ∈ F T ; R3 F T F̂ T . Definition 1.4. A point p ∈ C is said to be an strong asymptotic fixed point of T if C contains a sequence {xn}n 0 which converges strongly to p and limn→∞‖xn − Txn‖ 0. The set of strong asymptotic fixed points of T is denoted by F̃ T . We say that a mapping T is weak relatively nonexpansive see, e.g.,...