On the ring $R=F[x_1,\cdots ,x_n]$ of polynomials in n variables over a field $F$ special isomorphisms $A$’s of $R$ into $R$ are defined which preserve the greatest common divisor of two polynomials. The ring $R$ is extended to the ring $S=F{\left[[x_1,\cdots ,x_n]\right]}^{+}$ and the ring $T=F\left[[x_1,\cdots ,x_n]\right]$ of generalized polynomials in such a way that the exponents of the variables are non-negative rational numbers and rational numbers, respectively. The isomorphisms $A$’s are extended to automorphisms $B$’s of the ring $S$. Using the property that the isomorphisms $A$’s preserve...

Recall that a space $X$ is a c-semistratifiable (CSS) space, if the compact sets of $X$ are $G_{\delta }$-sets in a uniform way. In this note, we introduce another class of spaces, denoting it by k-c-semistratifiable (k-CSS), which generalizes the concept of c-semistratifiable. We discuss some properties of k-c-semistratifiable spaces. We prove that a $T_2$-space $X$ is a k-c-semistratifiable space if and only if $X$ has a $g$ function which satisfies the following conditions: (1) For each $x\in X$, $\left\{x\right\}=\bigcap \left\{g\right(x,n):n\in \mathbb{N}\}$ and $g(x,n+1)\subseteq g(x,n)$ for each...

In this paper, we determine the forbidden set and give an explicit formula for the solutions of the difference equation $$x_{n+1}=\frac{a{x}_n{x}_{n-1}}{-b{x}_n+c{x}_{n-2}},n\in {\mathbb{N}}_0$$ where $a$, $b$, $c$ are positive real numbers and the initial conditions $x_{-2}$, $x_{-1}$, $x_0$ are real numbers. We show that every admissible solution of that equation converges to zero if either $a<c$ or $a>c$ with $(a-c)/b<1$. When $a>c$ with $(a-c)/b>1$, we prove that every admissible solution is unbounded. Finally, when $a=c$, we prove that every admissible solution converges to zero.

We study the solutions and attractivity of the difference equation $x_{n+1}=x_{n-3}/(-1+x_n{x}_{n-1}{x}_{n-2}{x}_{n-3})$ for $n=0,1,2,\cdots$ where $x_{-3},x_{-2},x_{-1}$ and $x_0$ are real numbers such that $x_0{x}_{-1}{x}_{-2}{x}_{-3}\ne 1.$

Let $a_{d-1},\cdots ,a_0\in \mathbb{Z}$, where $d\in \mathbb{N}$ and $a_0\ne 0$, and let $X={\left(x_n\right)}_{n=1}^{\infty }$ be a sequence of integers given by the linear recurrence $x_{n+d}=a_{d-1}{x}_{n+d-1}+\cdots +a_0{x}_n$ for $n=1,2,3,\cdots$. We show that there are a prime number $p$ and $d$ integers $x_1,\cdots ,x_d$ such that no element of the sequence $X={\left(x_n\right)}_{n=1}^{\infty }$ defined by the above linear recurrence is divisible by $p$. Furthermore, for any nonnegative integer $s$ there is a prime number $p\ge 3$ and $d$ integers $x_1,\cdots ,x_d$ such that every element of the sequence $X={\left(x_n\right)}_{n=1}^{\infty }$ defined as above modulo $p$ belongs to the set $\{s+1,s+2,\cdots ,p-s-1\}$.

The main result says in particular that if $t\left(\zeta \right):={\sum }_{\nu =-n}ⁿc_{\nu }{e}^{i\nu \zeta }$ is a trigonometric polynomial of degree n having all its zeros in the open upper half-plane such that |t(ξ)| ≥ μ on the real axis and cₙ ≠ 0, then |t’(ξ)| ≥ μn for all real ξ.