If $\Omega \subset {ℝ}^n$ is a bounded domain with Lipschitz boundary $\partial \Omega$ and $\Gamma$ is an open subset of $\partial \Omega$, we prove that the following inequality $${\left({\int }_{\Omega }{\left|A\left(x\right)\nabla u\left(x\right)\right|}^{p}dx\right)}^{1/p}+{\left({\int }_{\Gamma }{\left|u\left(x\right)\right|}^{p}d{\mathscr{H}}^{n-1}\left(x\right)\right)}^{1/p}\ge c{\parallel u\parallel }_{W^{1,p}\left(\Omega \right)}$$ holds for all $u\in W^{1,p}(\Omega ;{ℝ}^m)$ and $1<p<\infty$, where $${\left(A\left(x\right)\nabla u\left(x\right)\right)}_k=\sum _{i=1}^m\sum _{j=1}^n{a}_k^{ij}\left(x\right)\frac{\partial u_i}{\partial x_j}\left(x\right)(k=1,2,...,r;r\ge m)$$ defines an elliptic differential operator of first order with continuous coefficients on $\overline{\Omega }$. As a special case we obtain $${\int }_{\Omega }{\left|\nabla u\left(x\right)F\left(x\right)+{\left(\nabla u\left(x\right)F\left(x\right)\right)}^T\right|}^{p}dx\ge c{\int }_{\Omega }{\left|\nabla u\left(x\right)\right|}^{p}dx,(*)$$ for all $u\in W^{1,p}(\Omega ;{ℝ}^n)$ vanishing on $\Gamma$, where $F:\overline{\Omega }\to M^{n\times n}\left(ℝ\right)$ is a continuous mapping with $detF\left(x\right)\ge \mu >0$. Next we show that $(*)$ is not valid if $n\ge 3$, $F\in L^{\infty }\left(\Omega \right)$ and $detF\left(x\right)=1$, but does hold if $p=2$, $\Gamma =\partial \Omega$ and $F\left(x\right)$ is symmetric and positive definite in $\Omega$.

We study the integrability of Banach valued strongly measurable functions defined on $[0,1]$. In case of functions $f$ given by ${\sum }_{n=1}^{\infty }{x}_n{\chi }_{E_n}$, where $x_n$ belong to a Banach space and the sets $E_n$ are Lebesgue measurable and pairwise disjoint subsets of $[0,1]$, there are well known characterizations for the Bochner and for the Pettis integrability of $f$ (cf Musial (1991)). In this paper we give some conditions for the Kurzweil-Henstock and the Kurzweil-Henstock-Pettis integrability of such functions.

A Kurzweil-Henstock type integral on a zero-dimensional abelian group is used to recover by generalized Fourier formulas the coefficients of the series with respect to the characters of such groups, in the compact case, and to obtain an inversion formula for multiplicative integral transforms, in the locally compact case.

For a merely continuous partition of unity the PU integral is the Lebesgue integral.

This is an introduction to Witten’s analytic proof of the Morse inequalities. The text is directed primarily to readers whose main interest is in complex analysis, and the similarities to Hörmander’s $L^2$-estimates for the $\overline{\partial }$-equation is used as motivation. We also use the method to prove $L^2$-estimates for the $d$-equation with a weight $e^{-t\phi }$ where $\phi$ is a nondegenerate Morse function.

Let $P\left(z\right)={\sum }_{\nu =0}^n{a}_{\nu }{z}^{\nu }$ be a polynomial of degree at most $n$ which does not vanish in the disk $\left|z\right|<1$, then for $1\le p<\infty$ and $R>1$, Boas and Rahman proved $${∥P\left(Rz\right)∥}_p\le ({∥R^n+z∥}_p/{∥1+z∥}_p){∥P∥}_p.$$ In this paper, we improve the above inequality for $0\le p<\infty$ by involving some of the coefficients of the polynomial $P\left(z\right)$. Analogous result for the class of polynomials $P\left(z\right)$ having no zero in $\left|z\right|>1$ is also given.

En admettant la conjecture de Dickson, nous démontrons que, pour chaque couple d’entiers $q\&gt;0$ et $k\&gt;0$, il existe une partie infinie $L_{q,k}\subset \mathbb{N}$ telle que, pour chacun des entiers $n\in L_{q,k}$ et tout entier $s$ tel que $0\&lt;\left|s\right|\le q$, on ait $n+s=\left|s\right|t_1...t_k$ où $t_1\&lt;...\&lt;t_k$ sont des nombres premiers. De même, pour chaque couple d’entiers $q\&gt;0$ et $k\&gt;0$, il existe une partie infinie $M_{q,k}\subset \mathbb{N}$ telle que, pour chacun des entiers $n\in M_{q,k}$ et tout entier $s$ (nul ou non ) de l’intervalle $\left[-q,q\right]$, on ait $n+s=l{t}_1...t_k$ où $t_1\&lt;...\&lt;t_k$ sont des nombres premiers et l’entier $l$ appartient à l’intervalle $\left[1,2q+1\right]$. La lecture non standard...