Let $F$ be a finite field of odd cardinality $q$, $A=F[T]$ the polynomial ring over $F$, $k=F(T)$ the rational function field over $F$ and $\mathcal{H}$ the set of square-free monic polynomials in $A$ of degree odd. If $D\in\mathcal{H}$, we denote by $\mathcal{O}_{D}$ the integral closure of $A$ in $k(\sqrt{D})$. In this note we give a simple proof for the average value of the size of the groups $K_{2}(\mathcal{O}_{D})$ as $D$ varies over the ensemble $\mathcal{H}$ and $q$ is kept fixed. The proof is based on character sums estimates and in the use of the Riemann hypothesis for curves over finite fields.
Composition polynomials of RNA matrix and $B$-composition polynomials of Riordan pseudo-involution
令$\left( g\left( x \right),xg\left( x \right) \right)$是Bell子群中的Riordan矩阵。我们表示为${{\left( g\left( x \right),xg\left( x \right) \right)}^{φ}}=\left( {{g}^{\left( φ\right)}}\left( x \right),x{{g}^{\left( φ\right)}}\left( x \right) \right)$，其中矩阵幂以标准方式定义。多项式${{c}_{n}}\left( x \right)$使得${{g}^{\left( φ\right)}}\left( x \right)=\sum\nolimits_{n=0}^{\infty }{{{c}_{n}}}\left( φ\right){{x}^{n}}$将被称为复合多项式。我们考虑RNA矩阵的复合多项式。与这些多项式相关联的构造允许以下概括。如果矩阵$\left( g\left( x \right),xg\left( x \right) \right)$是一个伪对合，则存在一个具有生成函数$B\left( x \right)$的数值序列（$B$ -序列），使得$g\left( x \right)=1+xg\left( x \right)B\left( {{x}^{2}}g…
$α$, $β$-expansions of the Riordan matrices of the associated subgroup
我们考虑矩阵$\left( 1,g\left( x \right) \right)$的群同构于复合下的形式幂级数$g\left( x \right)=x+{{g}_{2}}{{x}^{2}}+...$的群：$\left( 1,{{g}_{2}}\left( x \right) \right)\left( 1,{{g}_{1}}\left( x \right) \right)=\left( 1,{{g}_{1}}\left( {{g}_{2}}\left( x \right) \right) \right)$。表示为$P_{k}^{α}=\left( 1,x{{\left( 1-kα{{x}^{k}} \right)}^{{-1}/{k}\;}} \right)$。矩阵$\left( 1,g\left( x \right) \right)$以两种方式被分解成具有适当指数的矩阵$P_{k}^{α}$的无穷乘积：关于矩阵$P_{1}^{{{α}_{1}}={{β}_{1}}}$的左手和右手乘积：$\left( 1,g\left( x \right) \right)=...P_{k}^{{{α}_{k}}}...P_{2}^{{{α}_{2}}}P_{1}^{{{α}_{1}}}=P_{1}^{{{β}_{1}}}P_{2}^{{{β}_{2}}}...P_{k}^{{{β}_{k}}}.…
$B$-expansion of pseudo-involution in the Riordan group
具有生成函数$B\left( x \right)$的每个数字序列$\left( {{b}_{0}},{{b}_{1}},{{b}_{2}},... \right)$定义Riordan群$\left( 1,xg\left( x \right) \right)$中的伪对合，使得$g\left( x \right)=1+xg\left( x \right)B\left( {{x}^{2}}g\left( x \right) \right)$。在本文中，我们实现了一个简单的想法：表示系列${{g}^{m}}\left( x \right)$的系数的系列$B\left( x \right)$。所得到的展开式具有鲜明的组合特征，揭示了Riordan群中的伪对合与广义二项式级数的联系，也有助于由给定级数$B\left( x \right)$求级数$g\left( x \right)$.我们比较这个扩展与序列$\left( 1,{{a}_{1}},{{a}_{2}},... \right)$的生成函数$A\left( x \right)$的类似扩展，使得$g\left( x \right)=A\left( xg\left( x \right) \right)$。
Riordan矩阵是形式幂级数空间中某些算子所对应的无限下三角形矩阵。广义欧拉多项式${{g}_{n}}\left( x \right)={{\left( 1-x \right)}^{n+1}}\sum\nolimits_{m=0}^{\infty }{{{p}_{n}}}\left( m \right){{x}^{m}}$是普通Riordan矩阵对角线母函数的分子多项式，其中${{p}_{n}}\left( m \right)$是$\le n$次多项式.广义Narayana多项式${{h}_{n}}\left( x \right)={{\left( 1-x \right)}^{2n+1}}\sum\nolimits_{m=0}^{\infty }{\left( m+1 \right)...\left( m+n \right){{p}_{n}}}\left( m \right){{x}^{m}}$是指数Riordan矩阵的对角生成函数的分子多项式。本文讨论了这两类分子多项式的性质以及它们之间的构造关系。单独注意与系列$_{\left( β\right)}a\left( x \right)=a\left( x{}_{\left( β\right)}{{a}^{β}}\left( x \right) \right)$的家庭的Riordan矩阵的分子多项式。
Exponential Riordan arrays and generalized Narayana polynomials
广义欧拉多项式${{α}_{n}}\left( x \right)={{\left( 1-x \right)}^{n+1}}\sum\nolimits_{m=0}^{\infty }{{{p}_{n}}}\left( m \right){{x}^{m}}$是普通Riordan阵列对角线母函数的分子多项式，其中${{p}_{n}}\left( x \right)$是$n$次多项式.广义Narayana多项式${{φ}_{n}}\left( x \right)={{\left( 1-x \right)}^{2n+1}}\sum\nolimits_{m=0}^{\infty }{\left( m+1 \right)...\left( m+n \right){{p}_{n}}}\left( m \right){{x}^{m}}$是指数Riordan阵列对角线的生成函数的分子多项式。本文考虑这两类分子多项式之间的构造关系。
设$S=\left\langle s_1,\ldots,s_n\right\rangle$是由互质正整数$s_1,\ldots,s_n$生成的数值半群.令$k\geqslant 2$为整数。本文考虑$S$的Frobenius数的下列$k$ -幂变式定义为$${}^{k\!}r\!\left(S\right):= \text{ the largest } k \text{-power integer not belonging to } S.$$。给出了由{ \em算术级数}生成的无限半群族$S_A$的${}^{2\!}r\!\left(S_A\right)$的一个上界.在某些条件下，后者是${}^{2\!}r\!\left(\left\langle s_1,s_2\right\rangle\right)$的精确值。当$d=3,4$和$5$时，给出了${}^{2\!}r\!\left(\left\langle s_1,s_1+d \right\rangle\right)$的精确公式，研究了${}^{2\!}r\!\left(\left\langle s_1,s_1+1 \right\rangle\right)$和${}^{2\!}r\!\left(\left\langle s_1,s_1+2 \right\rangle\right)$，并给出了两个相应的公式。最后讨论了相关问题。
On the Gross-Prasad conjecture with its refinement for $\left(\mathrm{SO}\left(5\right),\mathrm{SO}\left(2\right)\right)$ and the generalized Böcherer conjecture
Metric results on the discrepancy of sequences $\left(a_{n} α\right)_{n \geq 1}$ modulo one for integer sequences $\left(a_{n}\right)_{n \geq 1}$ of polynomial growth
