If $A$ is a subring of a commutative ring $B$ and if $n$ is a positive integer, a number of sufficient conditions are given for “$A[[X]]$ is $n$-root closed in $B[[X]]$” to be equivalent to “$A$ is $n$-root closed in $B$.” In addition, it is shown that if $S$ is a multiplicative submonoid of the positive integers $\mathbb{P}$ which is generated by primes, then there exists a one-dimensional quasilocal integral domain $A$ (resp., a von Neumann regular ring $A$) such that $S=\{n\in \mathbb{P}|A\,\,\text{is}\,n-\text{root}\,\text{closed}\}$ (resp., $S=\{n\in \mathbb{P}\,|\,\,A[[X]]\,\,\text{is}\,n-\text{root}\,\text{closed}\}$).

The two theorems proved yield simple yet reasonably general conditions for triangular matrices to be bounded operators on ${{l}_{p}}$. The theorems are applied to Nörlund and weighted mean matrices.

Garrison [3], Forman [2], and Abel and Siebert [1] showed that for all positive integers $k$ and $N$, there exists a positive integer $\lambda $ such that ${{n}^{k}}\,+\,\lambda $ is prime for at least $N$ positive integers $n$. In other words, there exists $\lambda $ such that ${{n}^{k}}\,+\,\lambda $, represents at least $N$ primes.

We give a quantitative version of this result.We show that there exists $\lambda \le {{x}^{k}}$ such that ${{n}^{k}}\,+\,\lambda $, 1 ≤ n ≤ x, represents at least $\left( \frac{1}{k}\,+\,o\left( 1 \right) \right)\,\pi \left( x \right)$ primes, as $x\to \infty $. We also give some related results.

Let $G$ be a finite group, $H$ a copy of its $p$-Sylow subgroup, and $K{{\left( n \right)}^{*}}\left( - \right)$ the $n$-th Morava $K$-theory at $p$. In this paper we prove that the existence of an isomorphism between $K{{(n)}^{*}}(BG)$ and $K{{(n)}^{*}}(BH)$ is a sufficient condition for $G$ to be $p$-nilpotent.

The Gilbert-Pearson characterization of the spectrum is established for a generalized Sturm-Liouville equation with two singular endpoints. It is also shown that strong absolute continuity for the one singular endpoint problem guarantees absolute continuity for the two singular endpoint problem. As a consequence, we obtain the result that strong nonsubordinacy, at one singular endpoint, of a particular solution guarantees the nonexistence of subordinate solutions at both singular endpoints.

We consider graded connected Gorenstein algebras with respect to the evaluation map $\text{e}{{\text{v}}_{G\,}}\,=\,\text{Ex}{{\text{t}}_{G}}(k,\varepsilon )\,::\,\text{Ex}{{\text{t}}_{G}}(k,G)\,\to \,\text{Ex}{{\text{t}}_{G}}(k,k)$. We prove that if $e{{v}_{G}}\,\ne \,0$, then the global dimension of $G$ is finite.

Our main result is showing the asymptotic existence of tight $\text{OMEPs}$. More precisely, for each fixed number $k$ of rows, and with the exception of $\text{OMEPs}$ of the form $2\times 2\times \cdot \cdot \cdot 2\times 2s\,//\,4s\,\,$ with $s$ odd and with more than three rows, there are only a finite number of tight $\text{OMEP}$ parameters for which the tight $\text{OMEP}$ does not exist.

Given an integral functional defined on ${{L}_{p}}$, $1\le p<\infty $, under a growth condition we give an upper bound of the Clarke directional derivative and we obtain a nice inclusion between the Clarke subdifferential of the integral functional and the set of selections of the subdifferential of the integrand.

We study the stability of linear filters associated with certain types of linear difference equations with variable coefficients. We show that stability is determined by the locations of the poles of a rational transfer function relative to the spectrum of an associated weighted shift operator. The known theory for filters associated with constant-coefficient difference equations is a special case.

In this paper we prove the following:

1. 1. Let $m\ge 2,\,n\ge 1$ be integers and let $G$ be a group such that ${{(XY)}^{n}}\,=\,{{(YX)}^{n}}$ for all subsets $X,Y$ of size $m$ in $G$. Then

   1. a) $G$ is abelian or a $\text{BFC}$-group of finite exponent bounded by a function of $m$ and $n$.

   2. b) If $m\ge n$ then $G$ is abelian or $|G|$ is bounded by a function of $m$ and $n$.

2. 2. The only non-abelian group $G$ such that ${{(XY)}^{2}}\,=\,{{(YX)}^{2}}$ for all subsets $X,Y$ of size 2 in $G$ is the quaternion group of order 8.

3. 3. Let $m$, $n$ be positive integers and $G$ a group such that

   $${{X}_{1}}\cdot \cdot \cdot \,{{X}_{n}}\,\subseteq \,\bigcup\limits_{\sigma \in {{S}_{n}}\,\backslash \,1}{{{X}_{\sigma (1)}}\cdot \cdot \cdot \,{{X}_{\sigma (n)}}}$$

for all subsets ${{X}_{i}}$ of size $m$ in $G$. Then $G$ is $n$-permutable or $|G|$ is bounded by a function of $m$ and $n$.

The splitting pattern of a quadratic form $q$ over a field $k$ consists of all distinct Witt indices that occur for $q$ over extension fields of $k$. In small dimensions, the complete list of splitting patterns of quadratic forms is known. We show that all splitting patterns of quadratic forms of dimension at most nine can be realized by trace forms.

We construct a ring $R$ which is a sum of two subrings $A$ and $B$ such that the Levitzki radical of $R$ does not contain any of the hyperannihilators of $A$ and $B$. This answers an open question asked by Kegel in 1964.

The main result shows that if $R$ is a semiprime ring satisfying a polynomial identity, and if $Z(R)$ is the center of $R$, then card $R\,\le \,{{2}^{\text{card}\,Z\text{(}R\text{)}}}$. Examples show that this bound can be achieved, and that the inequality fails to hold for rings which are not semiprime.

We discuss the $q$-exponential functions ${{e}_{q}},\,{{E}_{q}}$ for $q$ on the unit circle, especially their continuity in $q$, and analogues of the limit relation ${{\lim }_{q\to 1}}\,{{e}_{q}}\,((1-q)z)\,=\,{{e}^{z}}.$

We investigate the problem of determining when $\text{IA}({{F}_{n}}({{\mathbf{A}}_{m}}\mathbf{A}))$ is finitely generated for all $n$ and $m$, with $n\ge 2$ and $m\ne 1$. If $m$ is a nonsquare free integer then $\text{IA}({{F}_{n}}({{\mathbf{A}}_{m}}\mathbf{A}))$ is not finitely generated for all $n$ and if $m$ square free integer then $\text{IA}({{F}_{n}}({{\mathbf{A}}_{m}}\mathbf{A}))$ is finitely generated for all $n$, with $n\ne 3$, and $\text{IA}({{F}_{3}}({{\mathbf{A}}_{m}}\mathbf{A}))$ is not finitely generated. In case $m$ is square free, Bachmuth and Mochizuki claimed in ([7], Problem 4) that $\text{TR}({{\mathbf{A}}_{m}}\mathbf{A})$ is 1 or 4. We correct their assertion by proving that $\text{TR}({{\mathbf{A}}_{m}}\mathbf{A})=\infty$.

Quaternionic numerical range is not always a convex set. In this note, an explicit criterion is given for the convexity of quaternionic numerical range.

Let $G$ be any group, and $H$ be a normal subgroup of $G$. Then M. Hartl identified the subgroup $G\,\cap \,(1+\,{{\Delta }^{3}}\,(G)\,+\,\Delta (G)\Delta (H))$ of $G$. In this note we give an independent proof of the result of Hartl, and we identify two subgroups $G\,\cap \,(1\,+\,\Delta (H)\Delta (G)\Delta (H)\,+\,\Delta (\left[ H,\,G \right]\Delta (H)),\,G\,\cap \,(1\,+\,{{\Delta }^{2}}\,(G)\Delta (H)\,+\,\Delta (K)\Delta (H))$ of $G$ for some subgroup $K$ of $G$ containing $[H,G]$.

Let ${{M}_{n}}(F)$ be the algebra of $n\times n$ matrices over a field $F$ of characteristic $p>2$ and let $*$ be an involution on ${{M}_{n}}(F)$. If ${{s}_{1}},...,{{s}_{r}}$ are symmetric variables we determine the smallest $r$ such that the polynomial

$${{P}_{r}}({{S}_{1}},...,{{S}_{r}})\,=\,\sum\limits_{\sigma \in {{S}_{r}}}{{{S}_{\sigma (1)}}...{{S}_{\sigma (r)}}}$$

is a $*$-polynomial identity of ${{M}_{n}}(F)$ under either the symplectic or the transpose involution. We also prove an analogous result for the polynomial

$${{C}_{r}}\left( {{k}_{1}},...,{{k}_{r,}}{{{{k}'}}_{1}},...,{{{{k}'}}_{r}} \right)=\sum\limits_{\sigma ,\tau \in {{S}_{r}}}{{{k}_{\sigma \left( 1 \right)}}{{{{k}'}}_{\tau \left( 1 \right)}}\cdot \cdot \cdot {{k}_{\sigma \left( r \right)}}{{{{k}'}}_{\tau \left( r \right)}}}$$

where ${{k}_{1}},...,{{k}_{r}},k_{1}^{'},...,k_{r}^{'}$ are skew variables under the transpose involution.

If $f({{x}_{1}},...,{{x}_{k}})$ is a polynomial with complex coefficients, the Mahler measure of $f$ , $M(f)$ is defined to be the geometric mean of $|f|$ over the $k$-torus ${{\mathbb{T}}^{k}}$. We construct a sequence of approximations ${{M}_{n}}\,(f)$ which satisfy $-d{{2}^{-n}}\,\log \,2\,+\,\log \,{{M}_{n}}(f)\,\le \,\log \,M(f)\,\le \,\log \,{{M}_{n}}(f)$. We use these to prove that $M(f)$ is a continuous function of the coefficients of $f$ for polynomials of fixed total degree $d$. Since ${{M}_{n}}\,(f)$ can be computed in a finite number of arithmetic operations from the coefficients of $f$ this also demonstrates an effective (but impractical) method for computing $M(f)$ to arbitrary accuracy.