In this paper, we investigate the sums of multiple zeta(-star) values height one: $$Z_{\pm }(n)=\sum _{a+b=n} (\pm 1)^b\zeta (\{1\}^a,b+2)$$ , }^{\star ^{\star }(\{1\}^a,b+2)$$ . particular, prove that weighted sum $$\begin{aligned}\sum _{\begin{array}{c} 0\le m\le p\\ m: \mathrm{even} \end{array}} \sum _{\mid \varvec{\alpha }\mid =p+3} 2^{\alpha _{m+1}+1}\zeta (\alpha _0,\alpha _1,\ldots ,\alp...

This paper considers solutions $u_\alpha$ of the three-dimensional Navier--Stokes equations on periodic domains $Q_\alpha:=(-\alpha,\alpha)^3$ as domain size $\alpha\to\infty$, and compares them to same whole space. For compactly-supported initial data $u_\alpha^0\in H^1(Q_\alpha)$, an appropriate extension converges a solution $u$ ${\mathbb R}^3$, strongly in $L^r(0,T;H^1({\mathbb R}^3))$, $r\...

We consider the punctured plane with volume density $|x|^\alpha$ and perimeter $|x|^\beta$. show that centred balls are uniquely isoperimetric for indices $(\alpha,\beta)$ which satisfy conditions $\alpha-\beta+1>0$, $\alpha\leq 2\beta$ $\alpha(\beta+1)\leq\beta^2$ except in case $\alpha=\beta=0$ corresponds to classical inequality.