The paper is concerned with estimating the number of integers smaller than x whose largest prime divisor is smaller than y, denoted ψ(x, y). Much of the related literature is concerned with approximating ψ(x, y) by Dickman’s function ρ(u), where u = lnx/ ln y. A typical such result is that ψ(x, y) = xρ(u)(1 + o(1)) (1) in a certain domain of the parameters x and y. In this paper a different typ...

We study the question of the existence of non-mitotic sets in NP. We show under various hypotheses that • 1-tt-mitoticity and m-mitoticity differ on NP. • 1-tt-reducibility and m-reducibility differ on NP. • There exist non-T-autoreducible sets in NP (by a result from Ambos-Spies, these sets are neither T-mitotic nor m-mitotic). • T-autoreducibility and T-mitoticity differ on NP (this contrasts...

Bonnet, Kim, Thomass\'{e}, and Watrigant (2020) introduced the twin-width of a graph. We show that an $n$-vertex graph is less than $(n+\sqrt{n\ln n}+\sqrt{n}+2\ln n)/2$, $m$-edge for positive $m$ $\sqrt{3m}+ m^{1/4} \sqrt{\ln m} / (4\cdot 3^{1/4}) + 3m^{1/4} 2$. Conference graphs order $n$ (when such exist) have at least $(n-1)/2$, we Paley achieve this lower bound. also Erd\H{o}s-R\'{e}nyi ra...

In the paper we consider a random analytic function of form$$f(z,\omega )=\sum\limits_{n=0}^{+\infty}\varepsilon_n(\omega_1)\xi_n(\omega_2)a_nz^n.$$Here $(\varepsilon_n)$ is sequence inde\-pendent Steinhausrandom variables, $(\xi_n)$ indepen\-dent standard complex Gaussianrandom and numbers $a_n\in\mathbb{C}$such that$a_0\neq0,\ \varlimsup\limits_{n\to+\infty}\sqrt[n]{|a_n|}=1,\ \sup\{|a_n|\col...

  Objective(s): Lupus nephritis (LN) is the main cause of mortality and disability in systemic lupus erythematosus (SLE) patients. Therefore, utilizing a reliable and non-invasive method for serial measurements of renal function seems to be necessary. The aim of this study was to evaluate the role of urinary lipocalin-2 as a biomarker of renal involvement in SLE patients.   Materi...

The present study prepares alkali lignin (AL) via acidification of black liquor, obtained from a pulp and paper factory. The average molecular weight of AL (equal to 2,530 g/mol) has been determined with gel permeation chromatography. AL has been modified by ethylene glycol, while lignin nanoparticles (LN) has been prepared through acid precipitation technology, their size being assessed by mea...