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 $X\subset {{ℂ}^n}^n$ 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.

In the two dimensional real vector space ${ℝ}^2$ one can define analogs of the well-known $q$-adic number systems. In these number systems a matrix $M$ plays the role of the base number $q$. In the present paper we study the so-called fundamental domain $ℱ$ of such number systems. This is the set of all elements of ${ℝ}^2$ having zero integer part in their “$M$-adic” representation. It was proved by Kátai and Környei, that $ℱ$ is a compact set and certain translates of it form a tiling of the ${ℝ}^2$. We construct...

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...

It is shown, that the function $$\begin{array}{cc}\hfill H\left(x\right)& =\sum _{k=-\infty }^{\infty }{e}^{-k^{2}x}\hfill \\ \multicolumn{2}{c}{\text{satisfies}\text{the}\text{relation}}\\ \hfill H\left(x\right)& =\sum _{n=0}^{\infty }\frac{{\left(2\pi \right)}^{2n}}{\left(2n\right)!}{H}^{\left(n\right)}\left(x\right).\hfill \end{array}$$

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 $p$.

We consider the problem $$\left\{\begin{array}{cc}-\Delta u=u^{\frac{N+2}{N-2}+\lambda }\hfill & \text{in}\Omega \setminus \epsilon \omega ,\hfill \\ u\&gt;0\hfill & \text{in}\Omega \setminus \epsilon \omega ,\hfill \\ u=0\hfill & \text{on}\partial \left(\Omega \setminus \epsilon \omega \right),\hfill \end{array}\right.$$ where $\Omega$ and $\omega$ are smooth bounded domains in ${ℝ}^N$, $N\ge 3$, $\epsilon \&gt;0$ and $\lambda \in ℝ.$ We prove that if the size of the hole $\epsilon$ goes to zero and if, simultaneously, the parameter $\lambda$ goes to zero at the appropriate rate, then the problem has a solution which blows up at the origin.

There are nontrivial dualities and parallels between polynomial algebras and the Grassmann algebras (e.g., the Grassmann algebras are dual of polynomial algebras as quadratic algebras). This paper is an attempt to look at the Grassmann algebras at the angle of the Jacobian conjecture for polynomial algebras (which is the question/conjecture about the $$ Jacobian set– the set of all algebra endomorphisms of a polynomial algebra with the Jacobian...

Let $K$ be a $p$-adic local field with residue field $k$ such that $[k:k^p]=p^e\&lt;+\infty$ 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 $k$.