Let Pn,r (x) be the generalized weighted means. Let F(x) be a C1 function, y = y(x) an implicit decreasing function defined by f(x,y) = 0 and 0 < m < M ≤ m′, n ≥ 2, xi ∈ [m,M], yi ∈ [m′,M′]. Then for −1 ≤ r ≤ 1, if f ′ x/f ′ y ≤ 1, |(F(Pn,1(y))− F(Pn,r (y)))/(F(Pn,1(x)) − F(Pn,r (x)))| < (maxm′≤ξ≤M′ |F ′(ξ)|)/(minm≤η≤M |F ′(η)|) · M/m′. A similar result exists for f ′ x/f ′ y ≥ 1. By specifying...

We will introduce new means of Cauchy’s type M r,l f, μ defined, for example, as M s r,l f, μ l l− s /r r − s M r f, μ −Mr s f, μ /M l f, μ −Ml s f, μ 1/ r−l , in the case when l / r / s, l, r / 0. We will show that this new Cauchy’s mean is monotonic, that is, the following result. Theorem. Let t, r, u, v ∈ R, such that t ≤ v, r ≤ u. Then for M r,l f, μ , one has M t,r ≤ M v,u. We will also gi...

let $m$ be an $r$-module and $0 neq fin m^*={rm hom}(m,r)$. we associate an undirected graph $gf$ to $m$ in which non-zero elements $x$ and $y$ of $m$ are adjacent provided that $xf(y)=0$ or $yf(x)=0$. weobserve that over a commutative ring $r$, $gf$ is connected anddiam$(gf)leq 3$. moreover, if $gamma (m)$ contains a cycle,then $mbox{gr}(gf)leq 4$. furthermore if $|gf|geq 1$, then$gf$ is finit...

we consider the class $mathfrak m$ of $bf r$--modules where $bf r$ is an associative ring. let $a$ be a module over a group ring $bf r$$g$, $g$ be a group and let $mathfrak l(g)$ be the set of all proper subgroups of $g$. we suppose that if $h in mathfrak l(g)$ then $a/c_{a}(h)$ belongs to $mathfrak m$. we investigate an $bf r$$g$--module $a$ such that $g not = g&apos;$, $c_{g}(a) = 1$. we stud...

Let $M$ be an $R$-module and $0 neq fin M^*={rm Hom}(M,R)$. We associate an undirected graph $gf$ to $M$ in which non-zero elements $x$ and $y$ of $M$ are adjacent provided that $xf(y)=0$ or $yf(x)=0$. Weobserve that over a commutative ring $R$, $gf$ is connected anddiam$(gf)leq 3$. Moreover, if $Gamma (M)$ contains a cycle,then $mbox{gr}(gf)leq 4$. Furthermore if $|gf|geq 1$, then$gf$ is finit...

The Five different cropping patterns such as Fallow-Fallow-Fallow (F-F-F), Sugarcane-Sugarcane-T Aman (S-S-R), Vegetables-Fallow-T (V-F-R), Fallow-Fallow-T (F-F-R) and Fallow-Mung bean- T (F-M-R) from Jessore district were investigated for carbon sequestration in soil aggreagates. Highest value of bulk density (1.53 g cm-3) was found S-S-R lowest (1.27 F-M-R. under V-F-R exhibited highest poros...