It is shown that if two cosine families with values in a normed algebra with unity, both indexed by t running over all real numbers, of which one consists of the multiples of the unity of the algebra by numbers of the form cos at for some real a, differ in norm by less than 8/(3 √ 3) uniformly in t, then these families coincide. For a = 0, the constant 8/(3 √ 3) is optimal and cannot be replaced by any larger number.
We use Kelvin's method of images (Bobrowski in J. Evol. Equ. 10(3): 663-675, 2010; Semigroup Forum 81(3):435-445, 2010) to show that given two nonnegative integers i = j there exists a unique cosine family generated by a restriction of the Laplace operator in C [0,1], that preserves the moments of order i and j about 0, if and only if precisely one of these integers is zero.
We study a diffusion process on a finite graph with semipermeable membranes on vertices. We prove, in $$L^1$$
L
1
and $$L^2$$
L
2
-type spaces that for a large class of boundary conditions, describing communication between the edges of the graph, the process is governed by a strongly continuous semigroup of operators, and we describe asymptotic behaviour of the diffusion semigroup as the diffusions’ speed increases at the same rate as the membranes’ permeability decreases. Such a process, in which communication is based on the Fick law, was studied by Bobrowski (Ann. Henri Poincaré 13(6):1501–1510, 2012) in the space of continuous functions on the graph. His results were generalized by Banasiak et al. (Semigroup Forum 93(3):427–443, 2016). We improve, in a way that cannot be obtained using a very general tool developed recently by Engel and Kramar Fijavž (Evolut. Equ. Control Theory 8(3)3:633–661, 2019), the results of J. Banasiak et al.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.