‎The first multiplicative Zagreb index $Pi_1(G)$ is equal to the‎ ‎product of squares of the degree of the vertices and the second‎ ‎multiplicative Zagreb index $Pi_2(G)$ is equal to the product of‎ ‎the products of the degree of pairs of adjacent vertices of the‎ ‎underlying molecular graphs $G$‎. ‎Also‎, ‎the multiplicative sum Zagreb index $Pi_3(G)$ is equal to the product of‎ ‎the sum...

An effective field theory for the three-body system with large two-body scattering length a is applied to three-body recombination into deep bound states in a Bose gas. The recombination constant alpha is calculated to first order in the short-distance interactions that allow the recombination. For a < 0, the dimensionless combination m alpha/(Planck's constant a(4)) is a periodic function of l...

background: for psychiatric disorders, comorbidity is a rule rather than exception. thus it is particularly important to study additive and multiplicative effects of multiple mental disorders on suicidal behaviors. objectives: the aim of this study was to investigate the ethnic differences in multiplicative effects of mental disorders on suicidal ideation among black adults in the united states...

Abstract For positive integers $K$ and $L$, we introduce study the notion of $K$-multiplicative dependence over algebraic closure ${\overline{{\mathbb{F}}}}_p$ a finite prime field ${\mathbb{F}}_p$, as well $L$-linear points on elliptic curves in reduction modulo primes. One our main results shows that, given non-zero rational functions $\varphi _1,\ldots ,\varphi _m, \varrho ,\varrho _n\in{\ma...

Let $${\mathbb {F}}_{q^n}$$ be a finite field with $$q^n$$ elements and r positive divisor of $$q^n-1$$ . An element $$\alpha \in {\mathbb {F}}_{q^n}^*$$ is called r-primitive if its multiplicative order $$(q^n-1)/r$$ Also, k-normal over {F}}_q$$ the greatest common polynomials $$g_{\alpha }(x) = \alpha x^{n-1}+ ^q x^{n-2} + \ldots ^{q^{n-2}}x ^{q^{n-1}}$$ $$x^n-1$$ in {F}}_{q^n}[x]$$ has degre...

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 $(q_n)$,...

Abstract We extend the Matomäki–Radziwiłł theorem to a large collection of unbounded multiplicative functions that are uniformly bounded, but not necessarily bounded by 1, on primes. Our result allows us estimate averages such function f in typical intervals length $$h(\log X)^c$$ h (</mml:m...

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 earlier work we showed that in the bulk, correlation of gaps dimer systems on hexagonal lattice is governed, fine mesh limit, by Coulomb’s law for 2D electrostatics. We also proved scaling limit discrete field $$\mathbf{F}$$ average tile orientations is, up to a multiplicative constant, electric produced system charges corresponding gaps. this paper show relative change $$T_{\alpha ,\beta }$...

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...