‎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...

let $f$ be a locally univalent function on the unit disk $u$. we consider the normalized extensions of $f$ to the euclidean unit ball $b^nsubseteqmathbb{c}^n$ given by$$phi_{n,gamma}(f)(z)=left(f(z_1),(f'(z_1))^gammahat{z}right),$$ where $gammain[0,1/2]$, $z=(z_1,hat{z})in b^n$ and$$psi_{n,beta}(f)(z)=left(f(z_1),(frac{f(z_1)}{z_1})^betahat{z}right),$$in which $betain[0,1]$, $f(z_1)neq 0$ and $...

We take into account the stochastic Boiti–Leon–Manna–Pempinelli equation (SBLMPE), which is perturbed by a multiplicative Brownian motion. By applying He’s semi-inverse method and Riccati mapping method, we can acquire solutions of SBLMPE. Since utilized to explain incompressible liquid in fluid mechanics, acquired may be applied lot fascinating physical phenomena. To address how motion effects...

The fractional-stochastic Drinfel’d–Sokolov–Wilson equations (FSDSWEs) perturbed by the multiplicative Wiener process are studied. mapping method is used to obtain rational, hyperbolic, and elliptic stochastic solutions for FSDSWEs. Due importance of FSDSWEs in describing propagation shallow water waves, derived significantly more useful effective understanding various important challenging phy...

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...

Let $\Re$ and $\Re'$ unital $2$,$3$-torsion free alternative rings $\varphi: \Re \rightarrow \Re'$ be a surjective Lie multiplicative map that preserves idempotents. Assume has nontrivial Under certain assumptions on $\Re$, we prove $\varphi$ is of the form $\psi + \tau$, where $\psi$ either an isomorphism or negative anti-isomorphism onto $\tau$ additive mapping into centre which maps commutat...

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...