A Banach algebra is Arens-regular when all its continuous functionals are weakly almost periodic, in symbols A⁎=WAP(A). To identify the opposite behaviour, Granirer called a extremely non-Arens regular (enAr, for short) quotient A⁎/WAP(A) contains closed subspace that has A⁎ as quotient. In this paper we propose simplification and quantification of concept. We say r-enAr, with r≥1, there an iso...

‎Let G=(V,E) be a simple connected graph with vertex set V and edge set E. The first, second and third Zagreb indices of G are respectivly defined by: $M_1(G)=sum_{uin V} d(u)^2, hspace {.1 cm} M_2(G)=sum_{uvin E} d(u).d(v)$ and $ M_3(G)=sum_{uvin E}| d(u)-d(v)| $ , where d(u) is the degree of vertex u in G and uv is an edge of G connecting the vertices u and v. Recently, the first and second m...

We take into account the (2 + 1)-dimensional stochastic Kadomtsev–Petviashvili equation with beta-derivative (SKPE-BD) in this paper. To develop new hyperbolic, trigonometric, elliptic, and rational solutions, Riccati Jacobi elliptic function methods are employed. Because KP is required for explaining development of quasi-one-dimensional shallow-water waves, solutions obtained can be used to in...

We study the stochastic heat equation (SHE) $$\partial _t u = \frac{1}{2} \Delta + \beta \xi $$ driven by a multiplicative Lévy noise $$\xi with positive jumps and coupling constant $$\beta >0$$ , in arbitrary dimension $$d\ge 1$$ . prove existence of solutions under an optimal condition if $$d=1,2$$ close-to-optimal 3$$ Under assumption that is general enough to include stable noises, we furth...

We consider a Callan–Symanzik and a Wilson renormalization group (RG) approach to the infrared problem for interacting fermions in one dimension with backscattering. We compute the third order (two–loop) approximation of the beta function using both methods and compare it with the well known multiplicative Gell–Mann Low approach. We point out a previously unnoticed strong instability of the thi...

We consider a Callan–Symanzik and a Wilson Renormalization Group approach to the infrared problem for interacting fermions in one dimension with backscattering. We compute the third order (two–loop) approximation of the beta function using both methods and compare it with the well known multiplicative Gell–Mann Low approach. We point out an unnoticed qualitative dependence of the third order fi...

Todeschini et al. have recently suggested to consider multiplicative variants of additive graph invariants, which applied to the Zagreb indices would lead to the multiplicative Zagreb indices of a graph G, denoted by ( ) 1  G and ( ) 2  G , under the name first and second multiplicative Zagreb index, respectively. These are define as     ( ) 2 1 ( ) ( ) v V G G G d v and ( ) ( ) ( ) ( ) 2...