The lonely runner problem for many runners

The lonely runner conjecture asserts that for any positive integer n and any positive numbers v1 < ... < vn there exists a positive number t such that ||vi t|| ≥ 1/(n+1) for every i=1, ...,n. We verify this conjecture for n ≥ 16342 under assumption that the speeds of the runners satisfy vj+1/vj ≥ 1+33 log n/n for j=1, ...,n-1

On the smallest integer vector at which a multivariable polynomial does not vanish

We prove that for any polynomial $P$ of degree $d$ in $C[x_1,dots,x_n]$ there exists a vector $(u_1,dots,u_n) in Z^n$ such that $P(u_1,dots,u_n) ne 0$ and $sum_{i=1}^n |u_i| leq min{d, lfloor (d+n)/2 rfloor}$. We also show that this bound is best possible. Similarly, for any $P in C[x_1,dots,x_n]$ of degree $d$ and any real number $p geq log 3/log 2$ there is a vector $(u_1,dots,u_n) in Z^n$ such that $P(u_1,dots,u_n) ne 0$ and $sum_{i=1}^n |u_i|^p leq max{1+lfloor d/2 rfloor^p, lfloor (d+1)/2 rfloor^p}$. The latter bound is also best possible for every $n geq 2$

On the minimum of certain functional related to the Schrödinger equation

We consider the infimum $\inf_f \max \limits_{j=1,2,3} \|f^{(j)}\|_{L^{\infty}(0,T_0)}$, where the infimum is taken over every function $f$ which runs through the set $KC^3(0,T_0)$ consisting of all functions $f : [0,T_0] \to \mathbb{R}$ satisfying the boundary conditions $f^{(j)}(0)=a_j$, $f^{(j)}(T_0)=0$ for $j=0,1,2$, whose derivatives $f^{(j)}$ are continuous for $j=0,1,2$ and the third derivative $f^{(3)}$ may have a finite number of discontinuities in the interval $(0,T_0)$, and find this infimum explicitly for certain choice of boundary conditions. This problem is motivated by some conditions under which the solution of the nonlinear Schrödinger equation with periodic boundary condition blows up in a finite time

Explicit form of Cassels' pp-adic embedding theorem for number fields

In this paper, we mainly give a general explicit form of Cassels' $p$-adic embedding theorem for number fields. We also give its refined form in the case of cyclotomic fields. As a byproduct, given an irreducible polynomial $f$ over $Z$, we give a general unconditional upper bound for the smallest prime number $p$ such that $f$ has a simple root modulo $p$

О многочленах Нюмена без корней на единичном круге

In this note we give a necessary and sufficient condition on the triplet of nonnegative integers a &lt; b &lt; c for which the Newman polynomial $$\sum_{j=0}^a x^j + \sum_{j=b}^c x^j$$ has a root on the unit circle. From this condition we derive that for each $$d \geq 3$$ there is a positive integer n>d such that the Newman polynomial $$1+x+\dots+x^{d-2}+x^n$$ of length d has no roots on the unit circle.В настоящей заметке мы получим необходимое и достаточное условие на тройку неотрицательных целых чисел a &lt; b &lt; c при выполнении которого многочлен Нюмена $$\sum_{j=0}^a x^j + \sum_{j=b}^c x^j$$ имеет корень на единичном круге. Изпользуя это условие мы докажем, что для каждого $$d \geq 3$$ существует такое целое положительное число n &gt; d, что многочлен Нюмена $$1+x+\dots+x^{d-2}+x^n$$ длины d не имеет корней на единичном круге

Некоторые моменты из жизни Антанаса Лауринчикаса: в поисках Универсальности

This article is dedicated to Lithuanian number theorist Professor Antanas Laurinˇcikas onthe occasion of his 70th birthday. We sketch the main stages in the development of his scientificcareer. Although A. Laurinˇcikas started with probabilistic number theory, later on he becameone of the leading world scientists in the theory of zeta-functions, especially concerning theiruniversality. In the review we give a brief account of his pre-university life and the developmentof his career as a mathematician from the time he entered Vilnius University. We review someresults of Antanas starting with early ones and then higlight the main results. At the end a listof scientific publications of A. Laurinˇcikas is presented.Эта статья посвящена литовскому теоретико-числовику профессору Антанасу Лаурин-чикасу по случаю его 70-летия. Очерчиваются основные этапы развития его научной ка-рьеры. Хотя А. Лауринчикас начал с вероятностной теории чисел, впоследствии он сталодним из ведущих мировых ученых в области теории дзета-функций, особенно в отноше-нии их универсальности. Приводится краткий обзор его довузовской жизни и описываетсяразвитие его карьеры математика с момента поступления в Вильнюсский университет.Мы рассмотрим некоторые результаты Антанаса, начиная с ранних, а затем осветимосновные результаты.В конце представлен список научных публикаций А. Лауринчикаса