On the stochasticity parameter of quadratic residues

Following V. I. Arnold, we define the stochasticity parameter $S(U)$ of a set $U\subseteq \mathbb{Z}_M$ to be the sum of squares of the consecutive distances between elements of $U$. We study the stochasticity parameter of the set $R_M$ of quadratic residues modulo $M$. Denote by $s(k)=s(k,\mathbb{Z}_M)$ the average value of $S(U)$ over all subsets $U\subseteq \mathbb{Z}_M$ of size $k$, which can be thought of as the stochasticity parameter of a random set of size $k$. We prove that a) $\varliminf_{M\to\infty}\frac{S(R_M)}{s(|R_M|)}<1<\varlimsup_{M\to\infty}\frac{S(R_M)}{s(|R_M|)}$; b) the set $\{ M\in \mathbb{N}: S(R_M)<s(|R_M|) \}$ has positive lower density

Prime avoiding numbers is a basis of order 22

For a positive integer $n$, we denote by $F(n)$ the distance from $n$ to the nearest prime number. We prove that every sufficiently large positive integer $N$ can be represented as the sum $N=n_1+n_2$, where $$F(n_i) \geqslant (\log N)(\log\log N)^{1/325565},$$ for $i=1,2$. This improves the corresponding "trivial" statement where only $F(n_i)\gg \log N$ is required

Sets whose differences avoid squares modulo m

We prove that if $\varepsilon(m)\to 0$ arbitrarily slowly, then for almost all $m$ and any $A\subset\mathbb{Z}_m$ such that $A-A$ does not contain non-zero quadratic residues we have $|A|\leq m^{1/2-\varepsilon(m)}.$Comment: 14 page

Long strings of consecutive composite values of polynomials

We show that for any polynomial $f$ from the integers to the integers, with positive leading coefficient and irreducible over the rationals, if $x$ is large enough then there is a string of $(\log x)(\log\log x)^{1/835}$ consecutive integers $n \in [1,x]$ for which $f(n)$ is composite. This improves a result of the first author, Konyagin, Maynard, Pomerance and Tao, which states that there are such strings of length $(\log x)(\log\log x)^{c_f}$, where $c_f$ depends on $f$ and $c_f$ is exponentially small in the degree of $f$ for some polynomials.Comment: 22 page

Mechanisms of Integration of the Republic of Kazakhstan in the Processes of Global Competition

This article considers issues related to integration of national economy of Kazakhstan in global microeconomic processes and also issues of formation of highly industrial economy based on innovational modernization of priority industries and sectors. Besides this assignment the article comprise contains comprehensive practical recommendations for optimization of Kazakhstan economy structure during post-crisis development through activation of institutes of development.У статті розглядаються питання інтеграції національної економіки Казахстану в глобальні світогосподарські процеси, а також формування високоіндустріальної економіки на основі інноваційної модернізації пріоритетних галузей і сфер. Крім того робота містить комплексні практичні рекомендації щодо оптимізації структури економіки Казахстану в посткризовий період розвитку шляхом активізації інститутів розвитку.В статье рассматриваются вопросы интеграции национальной экономики Казахстана в глобальные мирохозяйственные процессы, а также формирования высокоиндустриальной экономики на основе инновационной модернизации приоритетных отраслей и сфер. Кроме того, работа содержит комплексные практические рекомендации по оптимизации структуры экономики Казахстана в посткризисный период развития посредствам активизации институтов развития

Karatsuba's divisor problem and related questions

We prove that $$\sum_{p \leq x} \frac{1}{\tau(p-1)} \asymp \frac{x}{(\log x)^{3/2}}, \quad \quad \sum_{n \leq x} \frac{1}{\tau(n^2+1)} \asymp \frac{x}{(\log x)^{1/2}},$$ where $\tau(n)=\sum_{d|n}1$ is the number of divisors of $n$, and the summation in the first sum is over primes

Numbers of the form kf(k)kf(k)

For a function $f\colon \mathbb{N}\to\mathbb{N}$, define N^{\times}_{f}(x)=\#\{n\leq x: n=kf(k) \mbox{ for some k} \}. Let $\tau(n)=\sum_{d|n}1$ be the divisor function, $\omega(n)=\sum_{p|n}1$ be the prime divisor function, and $\varphi(n)=\#\{1\leq k\leq n: (k,n)=1 \}$ be Euler's totient function. We prove that \begin{gather*} \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! 1) \quad N^{\times}_{\tau}(x) \asymp \frac{x}{(\log x)^{1/2}}; \\ 2) \quad N^{\times}_{\omega}(x) = (1+o(1))\frac{x}{\log\log x}; \\ \!\!\!\!\!\!\!\!\! 3) \quad N^{\times}_{\varphi}(x) = (c_0+o(1))x^{1/2}, \end{gather*} where $c_0=1.365...$\,.Comment: The error term in Theorem 1.2 is improved in this version of the pape

Problems and urban infrastructure development in Russia (for example, the city of naberezhnyechelny)

Over the last few years the number of cars owned by Russian citizens, has dramatically increased, but to levels of Europe and the United States still need to "grow". Some regions in motorization have already caught up with the European countries, and the capital of Russia is not on the leading places for this indicator. At the moment the country is at the level of European countries in the 70-ies of the last century. But there is another important problem of insufficient roadareas and parking lots in the infrastructure of cities of Russia. In NaberezhnyeChelny.The city's population is 522 thousand people, city area is 17103 ha. Ratio of "metropolis" in NaberezhnyeChelny is 3.9. For comparison, this ratio in Moscow (within MKAD) - 35, Paris - 28, Sydney - 4, Kazan - 5,9. The coefficient of NaberezhnyeChelny in the density of the metropolis is about on par with Sydney, it is better to Kazan. This speaks to the amazing possibilities of NaberezhnyeChelny for the harmonious development (extension) adjoining roads and construction (extension) of the commercial parking lots within walking distance, parks and green spaces. Taking into account the ratio of the reserve for the development of road infrastructure exists. The observed infrastructural problems of the city of NaberezhnyeChelny: the lack of local parking lots; a huge amount of garbage in the form of leaves in the spring and autumn, mainly from not presentable, dangerous and large trees (poplar, birch); parking on lawns, the result of which is pollution of roads. We cannot quickly catch up with Europe but we can develop the infrastructure in this direction

Experimental study of energy distribution in ion-beam lithography

The paper reports two important results. Conducted a rigorous comparison of the sensitivity of the resist is polymethylmethacrylate (PMMA) to the irradiation of electron and ion beams. It is shown that, as in the case of electron irradiation, the resist shows both positive (at low doses) and negative (at higher doses) behavior of sensitivity. But compared with the electronic exposure, sensitivity of the resist is approximately a thousand times higher to the ion exposure, both the positive and negative areas....

3D modelling and simulation of a crawler robot in ROS/Gazebo

1. Modelling and animation of crawler UGV's caterpillars is a complicated task, which has not been completely resolved in ROS/Gazebo simulators. In this paper, we proposed an approximation of track-terrain interaction of a crawler UGV, perform modelling and simulation of Russian crawler robot "Engineer" within ROS/Gazebo and visualize its motion in ROS/RViz software. Finally, we test the proposed model in heterogeneous robot group navigation scenario within uncertain Gazebo environment. Copyright is held by the owner/author(s). Publication rights licensed to ACM