Let $F$ and $G$ be distributions in ${𝒟}^{\text{'}}$ and let $f$ be an infinitely differentiable function with $f^{\text{'}}\left(x\right)>0$, (or $<0$). It is proved that if the neutrix product $F\circ G$ exists and equals $H$, then the neutrix product $F\left(f\right)\circ G\left(f\right)$ exists and equals $H\left(f\right)$.

Let $p\ge 3$ be a prime. Let $n\in \mathbb{N}$ such that $n\ge 1$, let ${\chi }_1,...,{\chi }_n$ be characters of conductor $d$ not divided by $p$ and let $\omega$ be the Teichmüller character. For all $i$ between $1$ and $n$, for all $j$ between $0$ and $(p-3)/2$, set $${\theta }_{i,j}=\left\{\begin{array}{cc}{\chi }_i{\omega }^{2j+1}\hfill & \text{if}{\chi }_i\mathrm{is}\mathrm{odd};\hfill \\ {\chi }_i{\omega }^{2j}\hfill & \text{if}{\chi }_i\mathrm{is}\mathrm{even}.\hfill \end{array}\right.$$ Let $K={\mathbb{Q}}_p({\chi }_1,...,{\chi }_n)$ and let $\pi$ be a prime of the valuation ring ${𝒪}_K$ of $K$. For all $i,j$ let $f(T,{\theta }_{i,j})$ be the Iwasawa series associated to ${\theta }_{i,j}$ and $\overline{f(T,{\theta }_{i,j})}$ its reduction modulo $\left(\pi \right)$. Finally let $\overline{{𝔽}_p}$ be an algebraic closure of ${𝔽}_p$. Our main result is that if the characters ${\chi }_i$ are all distinct modulo $\left(\pi \right)$, then $1$ and the series...

We consider the Robin eigenvalue problem $\Delta u+\lambda u=0$ in $\Omega$, $\partial u/\partial \nu +\alpha u=0$ on $\partial \Omega$ where $\Omega \subset {ℝ}^n$, $n\ge 2$ is a bounded domain and $\alpha$ is a real parameter. We investigate the behavior of the eigenvalues ${\lambda }_k\left(\alpha \right)$ of this problem as functions of the parameter $\alpha$. We analyze the monotonicity and convexity properties of the eigenvalues and give a variational proof of the formula for the derivative ${\lambda }_1^{\text{'}}\left(\alpha \right)$. Assuming that the boundary $\partial \Omega$ is of class $C^2$ we obtain estimates to the difference ${\lambda }_k^D-{\lambda }_k\left(\alpha \right)$ between the $k$-th eigenvalue of the Laplace operator with...