Let $f\left(X\right)\in \mathbb{Q}\left[X\right]$ be a monic polynomial which is a power of a polynomial $g\left(X\right)\in \mathbb{Q}\left[X\right]$ of degree $\mu \ge 2$ and having simple real roots. For given positive integers $d_1,d_2,\ell ,m$ with $\ell \<m$ and gcd$(\ell ,m)=1$ with $\mu \le m+1$ whenever $m\<2$, we show that the equation $$f\left(x\right)f(x+d_1)\cdots f(x+(\ell k-1)d_1)=f\left(y\right)f(y+d_2)\cdots f(y+(mk-1)d_2)$$ with $f(x+j{d}_1)\ne 0$ for $0\le j\<\ell k$ has only finitely many solutions in integers $x,y$ and $k\ge 1$ except in the case $$m=\mu =2,\ell =k=d_2=1,f\left(X\right)=g\left(X\right),x=f\left(y\right)+y.$$

Let p be a prime number, and let [...] Q¯ p${\tilde{\overline{\mathbf{\text{Q}}}}}_{𝐩}$ be the completion of Q with respect to the pseudovaluation w which extends the p-adic valuation vp. In this paper our goal is to give a characterization of closed subfields of [...] Q¯ p ${\tilde{\overline{\mathbf{\text{Q}}}}}_{𝐩}$, the completion of Q with respect w, i.e. the spectral extension of the p-adic valuation vp on Q.

We prove necessary and sufficient conditions for the validity of the classical chain rule in the Sobolev space $W_{\text{loc}}^{1,1}({ℝ}^N;{ℝ}^d)$ and in the space $B{V}_{\text{loc}}({ℝ}^N;{ℝ}^d)$ of functions of bounded variation.

Let $K$ be a $p$-adic local field with residue field $k$ such that $[k:k^p]=p^e\<+\infty$ and $V$ be a $p$-adic representation of $\text{Gal}(\overline{K}/K)$. Then, by using the theory of $p$-adic differential modules, we show that $V$ is a Hodge-Tate (resp. de Rham) representation of $\text{Gal}(\overline{K}/K)$ if and only if $V$ is a Hodge-Tate (resp. de Rham) representation of $\text{Gal}(\overline{K^{\text{pf}}}/K^{\text{pf}})$ where $K^{\text{pf}}/K$ is a certain $p$-adic local field with residue field the smallest perfect field $k^{\text{pf}}$ containing $k$.

Let $X\subset {{ℂ}^n}^n$ be a closed real-analytic subset and put $$𝒜:=\{z\in X\mid \exists A\subset X,\text{germ}\text{of}\text{a}\text{complex-analytic}\text{set,}z\in A,{dim}_{z}A\>0\}$$ This article deals with the question of the structure of $𝒜$. In the main result a natural proof is given for the fact, that $𝒜$ always is closed. As a main tool an interesting relation between complex analytic subsets of $X$ of positive dimension and the Segre varieties of $X$ is proved and exploited.

Let $R$ be a ring. In two previous articles [12, 14] we studied the homotopy category $𝐊\left(R\text{-}\mathrm{Proj}\right)$ of projective $R$-modules. We produced a set of generators for this category, proved that the category is ${\aleph }_1$-compactly generated for any ring $R$, and showed that it need not always be compactly generated, but is for sufficiently nice $R$. We furthermore analyzed the inclusion $j_{!}^{}:𝐊\left(R\text{-}\mathrm{Proj}\right)\to 𝐊\left(R\text{-}\mathrm{Flat}\right)$ and the orthogonal subcategory $𝒮={𝐊\left(R\text{-}\mathrm{Proj}\right)}^{\perp }$. And we even showed that the inclusion $𝒮\to 𝐊\left(R\text{-}\mathrm{Flat}\right)$ has a right adjoint; this forces some natural map to be...

Let $\Delta$ be a numerical semigroup. In this work we show that $𝒥\left(\Delta \right)=\{I\cup \{0\}:I\text{is}\text{an}\text{ideal}\text{of}\Delta \}$ is a distributive lattice, which in addition is a Frobenius restricted variety. We give an algorithm which allows us to compute the set ${𝒥}_a\left(\Delta \right)=\{S\in 𝒥\left(\Delta \right):max(\Delta \setminus S)=a\}$ for a given $a\in \Delta .$ As a consequence, we obtain another algorithm that computes all the elements of $𝒥\left(\Delta \right)$ with a fixed genus.

The paper deals with very weak solutions $u\in \theta +W_0^{1,r}\left(\Omega \right)$, $max\{1,p-1\}<r<p<n$, to boundary value problems of the $p$-harmonic equation $$\left\{\begin{array}{cc}-\text{div}\left(\right|\nabla u\left(x\right){|}^{p-2}\nabla u\left(x\right))=0,\hfill & x\in \Omega ,\hfill \\ u\left(x\right)=\theta \left(x\right),\hfill & x\in \partial \Omega .\hfill \end{array}\right.(*)$$ We show that, under the assumption $\theta \in W^{1,q}\left(\Omega \right)$, $q>r$, any very weak solution $u$ to the boundary value problem ($*$) is integrable with $$u\in \left\{\begin{array}{cc}\theta +L_{\mathrm{weak}}^{q^{*}}\left(\Omega \right)\hfill & \text{for}q<n,\hfill \\ \theta +L_{\mathrm{weak}}^{\tau }\left(\Omega \right)\hfill & \text{for}q=n\text{and}\text{any}\tau <\infty ,\hfill \\ \theta +L^{\infty }\left(\Omega \right)\hfill & \text{for}q>n,\hfill \end{array}\right.$$ provided that $r$ is sufficiently close to $p$.

Let $p$ be an odd prime, and let $a$ be an integer not divisible by $p$. When $m$ is a positive integer with $p\equiv 1(mod2m)$ and $2$ is an $m$th power residue modulo $p$, we determine the value of the product ${\prod }_{k\in R_m\left(p\right)}(1+tan(\pi ak/p))$, where $$R_m\left(p\right)=\{0<k<p:k\in \mathbb{Z}\text{is}\text{an}m\text{th}\text{power}\text{residue}\text{modulo}p\}.$$ In particular, if $p=x^2+64y^2$ with $x,y\in \mathbb{Z}$, then $$\prod _{k\in R_4\left(p\right)}\left(1+tan\pi \frac{ak}{p}\right)={(-1)}^y{(-2)}^{(p-1)/8}.$$