We study the microlocal analyticity of solutions $u$ of the nonlinear equation $$u_t=f(x,t,u,u_x)$$ where $f(x,t,{\zeta }_0,\zeta )$ is complex-valued, real analytic in all its arguments and holomorphic in $({\zeta }_0,\zeta )$. We show that if the function $u$ is a $C^2$ solution, $\sigma \in \text{Char}{L}^u$ and $\frac{1}{i}\sigma \left([L^u,\overline{L^u}]\right)\<0$ or if $u$ is a $C^3$ solution, $\sigma \in \text{Char}{L}^u$, $\sigma \left([L^u,\overline{L^u}]\right)=0$, and $\sigma \left([L^u,[L^u,\overline{L^u}]]\right)\ne 0$, then $\sigma \notin W{F}_{a}u$. Here $W{F}_{a}u$ denotes the analytic wave-front set of $u$ and Char$L^u$ is the characteristic set of the linearized operator. When $m=1$, we prove a more general result involving the repeated brackets of $L^u$ and $\overline{L^u}$ of any...

We prove that, for a sequence of positive numbers δ(n), if $n^{1/2}\delta \left(n\right)\neg \to \infty$ as $n\to \infty$, to guarantee that the modified Szász-Mirakjan operators $S_{n,\delta }(f,x)$ converge to f(x) at every point, f must be identically zero.

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...

The present paper deals with the summatory function of functions acting on the digits of an $q$-ary expansion. In particular let $n$ be a positive integer, then we call $$\begin{array}{c}n=\sum _{r=0}^{\ell }{d}_r\left(n\right)q^r\text{with}{d}_r\left(n\right)\in \{0,...,q-1\}\end{array}$$ its $q$-ary expansion. We call a function $f$ strictly $q$-additive, if for a given value, it acts only on the digits of its representation, i.e., $$f\left(n\right)=\sum _{r=0}^{\ell }f\left(d_r\left(n\right)\right).$$ Let $p\left(x\right)={\alpha }_0{x}^{{\beta }_0}+\cdots +{\alpha }_d{x}^{{\beta }_d}$ with ${\alpha }_0,{\alpha }_1,...,{\alpha }_d,\in ℝ$, ${\alpha }_0\>0$, ${\beta }_0\>\cdots \>{\beta }_d\ge 1$ and at least one ${\beta }_i\notin \mathbb{Z}$. Then we call $p$ a pseudo-polynomial. ...

For a prime $p$ and positive integers $\ell \<k\<h\<p$ with $d=(h,k,\ell ,p-1)$, we show that $M$, the number of simultaneous solutions $x,y,z,w$ in ${\mathbb{Z}}_p^{*}$ to $x^h+y^h=z^h+w^h$, $x^k+y^k=z^k+w^k$, $x^{\ell }+y^{\ell }=z^{\ell }+w^{\ell }$, satisfies $${\displaystyle M\le 3d^2{(p-1)}^2+25hk\ell (p-1).}$$ When $hk\ell =o\left(p{d}^2\right)$ we obtain a precise asymptotic count on $M$. This leads to the new twisted exponential sum bound $${\displaystyle \left|\sum _{x=1}^{p-1}\chi \left(x\right)e^{2\pi if\left(x\right)/p}\right|\le 3^{\frac{1}{4}}{d}^{\frac{1}{2}}{p}^{\frac{7}{8}}+\sqrt{5}{\left(hk\ell \right)}^{\frac{1}{4}}{p}^{\frac{5}{8}},}$$ for trinomials $f=a{x}^h+b{x}^k+c{x}^{\ell }$, and to results on the average size of such sums.