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.

We investigate the structure of rank-to-rank elementary embeddings at successor rank, working in $\mathrm{ZF}$ set theory without Axiom Choice. Recall that set-theoretic universe is naturally stratified by cumulative hierarchy, whose levels $V\_\alpha$ are defined via iterated application power operation, starting from $V\_0=\emptyset$, setting $V\_{\alpha+1}=\mathcal{P}(V\_\alpha)$, and taking...

let $r$ be a 2-torsion free ring and $u$ be a square closed lie ideal of $r$. suppose that $alpha, beta$ are automorphisms of $r$. an additive mapping $delta: r longrightarrow r$ is said to be a jordan left $(alpha,beta)$-derivation of $r$ if $delta(x^2)=alpha(x)delta(x)+beta(x)delta(x)$ holds for all $xin r$. in this paper it is established that if $r$ admits an additive mapping $g : rlongrigh...