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.

Let $E$ be the self-similar set generated by {\it iterated function system} {\[ f_0(x)=\frac{x}{\beta},\quad f_1(x)=\frac{x+1}{\beta}, \quad f_{\beta+1}=\frac{x+\beta+1}{\beta} \]}with $\beta\ge 3$. {Then} is a with complete {overlaps}, i.e., $f_{0}\circ f_{\beta+1}=f_{1}\circ f_1$, but not totally self-similar. We investigate all its generating systems, give spectrum of $E$, and determine Haus...

The Freud ensemble of random matrices is the unitary invariant corresponding to weight $\exp(-n |x|^{\beta})$, $\beta>0$, on real line. We consider local behaviour eigenvalues near zero, which exhibits a transition in $\beta$. If $\beta\ge 1$, it described by standard sine process. Below critical value $\beta=1$, process depending $\beta$, and we determine first two terms large gap probability ...

The maximum sustainable amplitude, so-called wave breaking limit, of a nonlinear plasma in arbitrary mass ratio warm plasmas is obtained the non-relativistic regime. Using method Sagdeev potential, general formula derived by taking into account dynamics both species having finite temperature. It found that amplitude decreases monotonically with increase temperature β − $$ {\beta}_{-} negative (...