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Let $p$, $q$ and $r$ be fixed non-negative integers. In this note, it is shown that if $R$ is left (right) $s$-unital ring satisfying $[f\left(x^p{y}^q\right)-x^{r}y,x]=0$ ($[f\left(x^p{y}^q\right)-y{x}^r,x]=0$, respectively) where $f\left(\lambda \right)\in {\lambda }^2\mathbb{Z}\left[\lambda \right]$, then $R$ is commutative. Moreover, commutativity of $R$ is also obtained under different sets of constraints on integral exponents. Also, we provide some counterexamples which show that the hypotheses are not altogether superfluous. Thus, many well-known commutativity theorems become corollaries of our results.

Suppose that $R$ is an associative ring with identity $1$, $J\left(R\right)$ the Jacobson radical of $R$, and $N\left(R\right)$ the set of nilpotent elements of $R$. Let $m\ge 1$ be a fixed positive integer and $R$ an $m$-torsion-free ring with identity $1$. The main result of the present paper asserts that $R$ is commutative if $R$ satisfies both the conditions (i) $[x^m,y^m]=0$ for all $x,y\in R\setminus J\left(R\right)$ and (ii) $[{\left(xy\right)}^m+y^m{x}^m,x]=0=[{\left(yx\right)}^m+x^m{y}^m,x]$, for all $x,y\in R\setminus J\left(R\right)$. This result is also valid if (i) and (ii) are replaced by (i)${}^{\text{'}}$ $[x^m,y^m]=0$ for all $x,y\in R\setminus N\left(R\right)$ and (ii)${}^{\text{'}}$ $[{\left(xy\right)}^m+y^m{x}^m,x]=0=[{\left(yx\right)}^m+x^m{y}^m,x]$ for all $x,y\in R\setminus N\left(R\right)$. Other similar commutativity...

In this paper we investigate commutativity of rings with unity satisfying any one of the properties: $$\begin{array}{cc}& \{1-g\left(y{x}^m\right)\}[y{x}^m-x^{r}f\left(y{x}^m\right)x^s,x]\{1-h\left(y{x}^m\right)\}=0,\hfill \\ & \{1-g\left(y{x}^m\right)\}[x^{m}y-x^{r}f\left(y{x}^m\right)x^s,x]\{1-h\left(y{x}^m\right)\}=0,\hfill \\ & y^t[x,y^n]=g\left(x\right)[f\left(x\right),y]h\left(x\right)\mathrm{a}nd[x,y^n]y^t=g\left(x\right)[f\left(x\right),y]h\left(x\right)\hfill \end{array}$$ for some $f\left(X\right)$ in $X^2\mathbb{Z}\left[X\right]$ and $g\left(X\right)$, $h\left(X\right)$ in $\mathbb{Z}\left[X\right]$, where $m\ge 0$, $r\ge 0$, $s\ge 0$, $n>0$, $t>0$ are non-negative integers. We also extend these results to the case when integral exponents in the underlying conditions are no longer fixed, rather they depend on the pair of ring elements $x$ and $y$ for their values. Further, under different appropriate constraints on commutators, commutativity of rings has been studied. These results generalize...

Let $R$ be an associative ring with identity $1$ and $J\left(R\right)$ the Jacobson radical of $R$. Suppose that $m\ge 1$ is a fixed positive integer and $R$ an $m$-torsion-free ring with $1$. In the present paper, it is shown that $R$ is commutative if $R$ satisfies both the conditions (i) $[x^m,y^m]=0$ for all $x,y\in R\setminus J\left(R\right)$ and (ii) $[x,[x,y^m]]=0$, for all $x,y\in R\setminus J\left(R\right)$. This result is also valid if (ii) is replaced by (ii)’ $[{\left(yx\right)}^m{x}^m-x^m{\left(xy\right)}^m,x]=0$, for all $x,y\in R\setminus N\left(R\right)$. Our results generalize many well-known commutativity theorems (cf. [1], [2], [3], [4], [5], [6], [9], [10], [11] and [14]).

Gli Autori studiano spazi di Finsler ricorrenti di secondo ordine: cioè tali che le componenti delle derivate seconde covarianti del tensore di Berwald si esprimano come prodotti delle componenti stesse per quelle di un campo tensoriale doppio.

Estensione di alcuni risultati di Takano [3] al caso di spazi di Finsler ricorrenti in relazione ai tensori di curvatura secondo Berwald.

In questa Nota vengono definiti e studiati gli spazi ricorrenti di Finsler in rapporto al campo di tensori di curvatura secondo Cartan.

In this paper, we develop a generalized quasilinearization technique for a nonlinear second order periodic boundary value problem and obtain a sequence of approximate solutions converging uniformly and quadratically to a solution of the problem. Then we improve the convergence of the sequence of approximate solutions by establishing the convergence of order $k$ $(k\ge 2)$.

2000 Mathematics Subject Classification: 53C40, 53B25. In the present note we study totally umbilical pseudo-slant submanifolds of a nearly cosymplectic manifold. We have obtained a classification theorem for totally umbilical pseudo-slant submanifolds of a nearly cosymplectic manifold.

We study a mathematical model which was originally suggested by Greenhalgh and Das and takes into account the delay in the recruitment of infected persons. The stability of the equilibria are also discussed. In addition, we show that the introduction of a time delay in the transmission term can destabilize the system and periodic solutions can arise by Hopf bifurcation.