Given a large prime number $p$ and a rational function $r\left(X\right)$ defined over ${𝔽}_p=\mathbb{Z}/p\mathbb{Z}$, we investigate the size of the set $\left\{x\in {𝔽}_p:\tilde{r}\left(x\right)\>\tilde{r}(x+1)\right.$, where $\tilde{r}\left(x\right)$ and $\tilde{r}(x+1)$ denote the least positive representatives of $r\left(x\right)$ and $r(x+1)$ in $\mathbb{Z}$ modulo $p\mathbb{Z}$.

For an arbitrary (not totally real) number field $K$ of degree $\ge 3$, we ask how many perfect powers ${\gamma }^p$ of algebraic integers $\gamma$ in $K$ exist, such that $\mu \left(\tau \left({\gamma }^p\right)\right)\le X$ for each embedding $\tau$ of $K$ into the complex field. ($X$ a large real parameter, $p\ge 2$ a fixed integer, and $\mu \left(z\right)=max\left(\right|\mathrm{Re}\left(z\right)|,|\mathrm{Im}\left(z\right)\left|\right)$ for any complex $z$.) This quantity is evaluated asymptotically in the form $c_{p,K}{X}^{n/p}+R_{p,K}\left(X\right)$, with sharp estimates for the remainder $R_{p,K}\left(X\right)$. The argument uses techniques from lattice point theory along with W. Schmidt’s multivariate extension of K.F. Roth’s result...

Let D be the unit disc in the complex plane ℂ. Then for every complex linear subspace H in ${ℂ}^n$ of codimension 1, $vo{l}_{2n-2}\left(D^{n-1}\right)\le vo{l}_{2n-2}\left(H\cap D^n\right)\le 2vo{l}_{2n-2}\left(D^{n-1}\right)$. The lower bound is attained if and only if H is orthogonal to the versor $e_j$ of the jth coordinate axis for some j = 1,...,n; the upper bound is attained if and only if H is orthogonal to a vector $e_j+\sigma e_k$ for some 1 ≤ j < k ≤ n and some σ ∈ ℂ with |σ| = 1. We identify ${ℂ}^n$ with ${ℝ}^{2n}$; by $vo{l}_k\left(·\right)$ we denote the usual k-dimensional volume in ${ℝ}^{2n}$. The result is a complex counterpart of Ball’s [B1]...

We study the ramification properties of the extensions $\mathbb{Q}({\zeta }_m,\sqrt[m]{a})/\mathbb{Q}$ under the hypothesis that $m$ is odd and if $p\mid m$ than either $p\nmid v_p\left(a\right)$ or $p^{v_p\left(m\right)}\mid v_p\left(a\right)$ ($v_p\left(a\right)$ and $v_p\left(m\right)$ are the exponents with which $p$ divides $a$ and $m$). In particular we determine the higher ramification groups of the completed extensions and the Artin conductors of the characters of their Galois group. As an application, we give formulas for the $p$-adique valuation of the discriminant of the studied global extensions with $m=p^r$.

Let G be a graph with |V(G)| ≥ 10. We prove that if both G and G̅ are claw-free, then minΔ(G), Δ(G̅) ≤ 2. As a generalization of this result in the case where |V(G)| is sufficiently large, we also prove that if both G and G̅ are $K_{1,t}$-free, then minΔ(G),Δ(G̅) ≤ r(t- 1,t)-1 where r(t-1,t) is the Ramsey number.

In this paper we consider the asymptotic formula for the number of the solutions of the equation $p_1+p_2+p_3=N$ where $N$ is an odd integer and the unknowns $p_i$ are prime numbers of the form $p_i=\left[n^{1/{\gamma }_i}\right]$. We use the two-dimensional van der Corput’s method to prove it under less restrictive conditions than before. In the most interesting case ${\gamma }_1={\gamma }_2={\gamma }_3=\gamma$ our theorem implies that every sufficiently large odd integer $N$ may be written as the sum of three Piatetski-Shapiro primes of type $\gamma$ for $50/53$ &lt; $\gamma$ &lt; $1$. ...

We compute the moments and free cumulants of the measure ${\rho }_t:={\pi }_t⊠{\pi }_t$, where ${\pi }_t$ denotes the free Poisson law with parameter t > 0. We also compute free cumulants of the symmetrization of ${\rho }_t$. Finally, we introduce the free symmetrization of a probability measure on ℝ and provide some examples.

The Lukacs property of the free Poisson distribution is studied. We prove that if free and are free Poisson distributed with suitable parameters, then + and ${(+)}^{-1/2}{(+)}^{-1/2}$ are free. As an auxiliary result we compute the joint cumulants of and ${}^{-1}$ for free Poisson distributed . We also study the Lukacs property of the free Gamma distribution.

Let $\Gamma$ be a non-pluripolar set in ${ℂ}^N$. Let $f$ be a function holomorphic in a connected open neighborhood $G$ of $\Gamma$. Let $\left\{P_n\right\}$ be a sequence of polynomials with $deg{P}_n\le d_n(d_n\&lt;d_{n+1})$ such that $$\underset{n\to \infty }{lim\; sup}{|f\left(z\right)-P_n\left(z\right)|}^{1/d_n}\&lt;1,z\in \Gamma .$$ We show that if $$\underset{n\to \infty }{lim\; sup}{\left|P_n\left(z\right)\right|}^{1/d_n}\le 1,z\in E,$$ where $E$ is a set in ${ℂ}^N$ such that the global extremal function $V_E\equiv 0$ in ${ℂ}^N$, then the maximal domain of existence $G_f$ of $f$ is one-sheeted, and $$\underset{n\to \infty }{lim\; sup}{\parallel f-P_n\parallel }_K^{\frac{1}{d_n}}\&lt;1$$ for every compact set $K\subset G_f$. If, moreover, the sequence $\{d_{n+1}/d_n\}$ is bounded then $G_f={ℂ}^N$. If $E$ is a closed...