a ring $r$ is a strongly clean ring if every element in $r$ is the sum of an idempotent and a unit that commutate. we construct some classes of strongly clean rings which have stable range one. it is shown that such cleanness of $2 imes 2$ matrices over commutative local rings is completely determined in terms of solvability of quadratic equations.

The real Jacobi group $G^J_n(\mathbb{R})$, defined as the semidirect product of Heisenberg ${\rm H}_n(\R)$ with symplectic ${\mr {Sp}}(n,\mathbb{R})$, admits a matrix embedding in $\text{Sp}(n+1,\mathbb{R})$. modified pre-Iwasawa decomposition $\rm{Sp}(n,\mathbb{R})$ allows us to introduce convenient coordinatization $S_n$ which for $G^J_1(\mathbb{R})$ coincides $S$-coordinates. Invariant one-f...

Motivated by the Rudnick-Sarnak theorem we study limiting distribution of smoothed local correlations form $$ \sum_{j_1, j_2, \ldots, j_n} f(N\*(\theta_{j_2}-\theta_{j_1}), N\*(\theta_{j_3}-\theta_{j_1}), N\*(\theta_{j_n}-\theta_{j_1}))$$ for Circular United Ensemble random matrices sufficiently smooth test functions.

Suppose $j_N(\tau)$ and $j_N^{*}(\tau)$ are the Hauptmoduln of congruence subgroup $\Gamma_0(N)$ Fricke group $\Gamma^{*}_0(N)$, respectively. In [7], authors predicted that, like Klein's $j$-function, Fourier coefficients $j_{N}^{*}(\tau)$ in some arithmetic progression both even odd with density $\frac{1}{2}$. this article, we can find $n$ where $j_6(\tau)$ (resp. $j_6^{*}(\tau)$ $j_{10}(\tau...

We study some properties of the Rovelli-Smolin-DePietri volume operator in loop quantum gravity, which significantly simplify diagonalization problem and shed light on pattern degeneracy eigenstates. The is defined by its action spaces tensor products $\mathcal{H}_{j_1}\otimes \ldots \otimes \mathcal{H}_{j_N}$ irreducible SU(2) representation $\mathcal{H}_{j_i}, i=1,\ldots,N$, labelled with spi...

A ring $R$ is a strongly clean ring if every element in $R$ is the sum of an idempotent and a unit that commutate. We construct some classes of strongly clean rings which have stable range one. It is shown that such cleanness of $2 imes 2$ matrices over commutative local rings is completely determined in terms of solvability of quadratic equations.

the submodules with the property of the title ( a submodule $n$ of an $r$-module $m$ is called strongly dense in $m$, denoted by $nleq_{sd}m$, if for any index set $i$, $prod _{i}nleq_{d}prod _{i}m$) are introduced and fully investigated. it is shown that for each submodule $n$ of $m$ there exists the smallest subset $d'subseteq m$ such that $n+d'$ is a strongly dense submodule of $m$...