let $r=k[x_1,x_2,cdots, x_n]$ be a polynomial ring over a field $k$. we prove that for any positive integers $m, n$, $text{reg}(i^mj^nk)leq mtext{reg}(i)+ntext{reg}(j)+text{reg}(k)$ if $i, j, ksubseteq r$ are three monomial complete intersections ($i$, $j$, $k$ are not necessarily proper ideals of the polynomial ring $r$), and $i, j$ are of the form $(x_{i_1}^{a_1}, x_{i_2}^{a_2}, cdots, x_{i_l...

The paper considers a scalar differential equation of an advance-delay type \begin{equation*} \dot{y}(t)= -\left(a_0+\frac{a_1}{t}\right)y(t-\tau )+\left(b_0+\frac{b_1}{t}\right)y(t+\sigma )\,, \end{equation*} where constants $a_0$, $b_0$, $\tau $ and $\sigma are positive, $a_1$ $b_1$ arbitrary. behavior its solutions for $t\rightarrow \infty is analyzed provided that the transcendental \lambda...

Let $a_1,\ldots,a_n$ and $b_1,\ldots,b_m$ be fixed positive integers, let ${\mathcal S}$ denote the set of all nonnegative integer solutions equation $x_1a_1+\cdots +x_na_n=y_1b_1+\cdots...

We prove a Cohen-Macaulay version of result by Avramov-Golod and Frankild-J{\o}rgensen about Gorenstein rings, showing that if noetherian ring $A$ is Cohen-Macaulay, $a_1,\dots,a_n$ any sequence elements in $A$, then the Koszul complex $K(A;a_1,\dots,a_n)$ DG-ring. further generalize this result, it also holds for commutative DG-rings. In process proving this, we develop new technique to study ...

Let $R=k[x_1,x_2,cdots, x_N]$ be a polynomial ring over a field $k$. We prove that for any positive integers $m, n$, $text{reg}(I^mJ^nK)leq mtext{reg}(I)+ntext{reg}(J)+text{reg}(K)$ if $I, J, Ksubseteq R$ are three monomial complete intersections ($I$, $J$, $K$ are not necessarily proper ideals of the polynomial ring $R$), and $I, J$ are of the form $(x_{i_1}^{a_1}, x_{i_2}^{a_2}, cdots, x_{i_l...