Let $$\mathbb {F}_{q^n}$$ be a finite field with $$q^n$$ elements, and let $$m_1$$ $$m_2$$ positive integers. Given polynomials $$f_1(x), f_2(x) \in \mathbb {F}_{q^n}[x]$$ $$\deg (f_i(x)) \le m_i$$ , for $$i = 1, 2$$ such that the rational function $$f_1(x)/f_2(x)$$ satisfies certain conditions which we define, present sufficient condition existence of primitive element $$\alpha normal over {F}...

In this paper we establish eigenvector delocalization and bulk universality for Levy matrices, which are real, symmetric, $N \times N$ random matrices $\textbf{H}$ whose upper triangular entries independent, identically distributed $\alpha$-stable laws. First, if $\alpha \in (1, 2)$ $E \mathbb{R}$ is any energy bounded away from $0$, show that every of corresponding to an eigenvalue near $E$ co...

Fix a finite ordinal \(n\geq 3\) and let \(\alpha\) be an arbitrary ordinal. Let \(\mathsf{CA}_n\) denote the class of cylindric algebras dimension \(n\) \(\sf RA\) relation algebras. \(\mathbf{PA}_{\alpha}(\mathsf{PEA}_{\alpha})\) stand for polyadic (equality) \(\alpha\). We reprove that \(\mathsf{CRCA}_n\) completely representable \(\mathsf{CA}_n\)s, CRRA\) \(\mathsf{RA}\)s are not elementary...

We study the kinetics after a low temperature quench of one-dimensional Ising model with long range interactions between spins at distance r r decaying as r^{-\alpha} display="inline">r−α . F...

Let G be a random d-regular graph on n vertices. We prove that for every constant $$\alpha > 0$$ , with high probability eigenvector of the adjacency matrix eigenvalue less than $$-2\sqrt{d-2}-\alpha $$ has $$\Omega (n/\textrm{polylog}(n))$$ nodal domains.

in this paper‎, ‎following a very recent‎ ‎and new approach‎, ‎we further generalize recently introduced‎ ‎summability methods‎, ‎namely‎, ‎$i$-statistical convergence and‎ ‎$i$-lacunary statistical convergence (which extend the important‎ ‎summability methods‎, ‎statistical convergence and lacunary‎ ‎statistical convergence using ideals of $mathbb{n}$) and‎ ‎introduce the notions of $i$-statis...

In this paper, we study the so-called intermediate disorder regime for a directed polymer in random environment with heavy-tail. Consider simple symmetric walk $$(S_n)_{n\ge 0}$$ on $$\mathbb {Z}^d$$ , $$d\ge 1$$ and modify its law using Gibbs weights product form $$\prod _{n=1}^{N} (1+\beta \eta _{n,S_n})$$ where $$(\eta _{n,x})_{n\ge 0, x\in {\mathbb {Z}}^d}$$ is field of i.i.d. variables who...

Let ( a n ) = 1 ? $(a_{n})_{n=1}^{\infty }$ be lacunary sequence of positive real numbers. Rudnick and Technau showed that for almost all ? ? R $\alpha \in \mathbb {R}$ , the pair correlation $(\alpha a_{n})_{n=1}^{\infty mod is Poissonian. We show higher correlations hence nearest-neighbour spacing distribution are Poissonian as well, thereby extending result Zaharescu to real-valued sequences.