for a graph $g$ with edge set $e(g)$, the multiplicative sum zagreb index of $g$ is defined as$pi^*(g)=pi_{uvin e(g)}[d_g(u)+d_g(v)]$, where $d_g(v)$ is the degree of vertex $v$ in $g$.in this paper, we first introduce some graph transformations that decreasethis index. in application, we identify the fourteen class of trees, with the first through fourteenth smallest multiplicative sum zagreb ...

based on the {s+, i-} pathway, the concentration of surfactant and surface charge density of silanolate groups control the phase transition from lamellar, to hexagonal through cubic form. the high surface charge density of silanolate groups was observed for the lamellar phase. with decrease of molar concentration of surfactant on the gel, the yield of reaction decreases and the stability of mes...

It has long been known that the left-most or right-most particle in a one dimensional dyadic branching Brownian motion with constant branching rate β > 0 has almost sure asymptotic speed √ 2β, (cf. [18]). Recently similar results for higher dimensional branching Brownian motion and super-Brownian motion have also been established but in the weaker sense of convergence in probability; see [20] a...

Evans [7] described the semi-group of a superprocess with quadratic branching mechanism under a martingale change of measure in terms of the semi-group of an immortal particle and the semigroup of the superprocess prior to the change of measure. This result, commonly referred to as the spine decomposition, alludes to a pathwise decomposition in which independent copies of the original process ‘...

Renal malformations are a major cause of childhood renal failure. During the development of the kidney, ureteric bud (UB) branching morphogenesis is critical for normal nephrogenesis. These studies investigated whether renal UB branching morphogenesis is altered by a high ambient glucose environment and studied underlying mechanism(s). Kidney explants that were isolated from different periods o...

Vascular endothelial growth factor (VEGF) is transcribed in the VEGF(120), VEGF(164), or VEGF(188) isoforms, which differ in receptor binding, matrix association, and angiogenic activity. This vascular growth factor has been implicated in the development of the renal vasculature, but the role of the distinct VEGF isoforms remains unknown. In the present report, renal angiogenesis and arteriogen...