The debate on the properties and use of additive and multiplicative partitioning of species diversity exemplified in this forum reflects the broader challenge of quantifying and interpreting alpha, beta, and gamma diversity at multiple scales of sampling. As noted by Wilsey and Ricotta in their contributions, ecologists use a wide range of measures of beta diversity, partly because of differenc...

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 $(q_n)$,...

In the current study, we investigate stochastic Benjamin–Bona–Mahony equation with beta derivative (SBBME-BD). The considered term is multiplicative noise in Itô sense. By combining F-expansion approach two separate equations, such as Riccati and elliptic new hyperbolic, trigonometric, rational, Jacobi solutions for SBBME-BD can be generated. to are useful understanding various scientific pheno...

We consider presentations that were derived in [3] for the interval groups associated with proper quasi-Coxeter elements of Coxeter group W(Dn). use combinatorial methods to derive alternative groups, and these new show a element W(Dn) cannot be isomorphic Artin type Dn. While specific problems we solve arise from study their solution provides an illustration how techniques indicated by computa...

We prove that a class of stochastic differential equations with multiplicative noise has a unique solution in a noncommutative L2 space associated with a von Neumann algebra. As examples we consider usual L2 on a measure space, Hilbert-Schmidt operators and a hyperfinite II1-factor. A problem of finding an inverse of the solution is then discussed. Finally, we explain how a stochastic different...

In this paper, at first the elemantary and basic concepts of multiplicative discrete and continous differentian and integration introduced. Then for these kinds of differentiation invariant functions the general solution of discrete and continous multiplicative differential equations will be given. Finaly a vast class of difference equations with variable coefficients and nonlinear difference e...