Existing general definitions of beta diversity often produce a beta with a hidden dependence on alpha. Such a beta cannot be used to compare regions that differ in alpha diversity. To avoid misinterpretation, existing definitions of alpha and beta must be replaced by a definition that partitions diversity into independent alpha and beta components. Such a unique definition is derived here. When...

‎Bucket recursive trees are an interesting and natural‎ ‎generalization of ordinary recursive trees and have a connection‎ to mathematical chemistry‎. ‎In this paper‎, ‎we give the lower and upper bounds for the moment generating‎ ‎function and moments of the multiplicative Zagreb indices in a‎ ‎randomly chosen bucket recursive tree of size $n$ with maximal bucket size $bgeq1$‎. Also, ‎we consi...

for fr$acute{mathbf{text{e}}}$chet algebras $(a, (p_n))$ and $(b, (q_n))$, a linear map $t:arightarrow b$ is textit{almost multiplicative} with respect to $(p_n)$ and $(q_n)$, if there exists $varepsilongeq 0$ such that $q_n(tab - ta tb)leq varepsilon p_n(a) p_n(b),$ for all $n in mathbb{n}$, $a, b in a$, and it is called textit{weakly almost multiplicative} with respect to $(p_n)$ and $(q_n)$,...

We study quasilinear degenerate parabolic-hyperbolic stochastic partial differential equations with general multiplicative noise within the framework of kinetic solutions. Our results are twofold: First, we establish new regularity results based on averaging techniques. Second, we prove the existence and uniqueness of solutions in a full L1 setting requiring no growth assumptions on the nonline...

We present measurements of broad emission lines and virial estimates supermassive black hole masses ($M_{BH}$) for a large sample ultra-hard X-ray selected active galactic nuclei (AGNs) as part the second data release BAT AGN Spectroscopic Survey (BASS/DR2). Our catalog includes $M_{BH}$ total 689 AGNs, determined from H$\alpha$, H$\beta$, $MgII\lambda2798$, and/or $CIV\lambda1549$ lines. The c...

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 ...

‎for a graph $g$ with edge set $e(g)$‎, ‎the multiplicative second zagreb index of $g$ is defined as‎ ‎$pi_2(g)=pi_{uvin e(g)}[d_g(u)d_g(v)]$‎, ‎where $d_g(v)$ is the degree of vertex $v$ in $g$‎. ‎in this paper‎, ‎we identify the eighth class of trees‎, ‎with the first through eighth smallest multiplicative second zagreb indeces among all trees of order $ngeq 14$‎.