A remark about positive polynomials

The following theorem is proved. {\bf Theorem.} {\it Let $P(x) = \sum_{k=0}^{2n} a_k x^k$ be a polynomial with positive coefficients. If the inequalities $\frac{a_{2k+1}^2}{a_{2k}a_{2k+ 2}} < \frac{1}{cos^2(\frac{\pi}{n+2})}$ hold for all $k=0, 1, ..., n-1,$ then $P(x)>0$ for every $x\in\mathbb{R}$ .} We show that the constant $\frac{1}{cos^2(\frac{\pi}{n+2})}$ in this theorem could not be increased. We also present some corollaries of this theorem.Comment: Submitted to the journal "Mathematical Inequalities and Applications" on September 29, 200

On sufficient conditions for the total positivity and for the multiple positivity of matrices

The following theorem is proved: Suppose $M = (a_{i,j})$ be a $k \times k$ matrix with positive entries and $a_{i,j}a_{i+1,j+1} > 4\cos ^2 \frac{\pi}{k+1} a_{i,j+1}a_{i+1,j} \quad (1 \leq i \leq k-1, 1 \leq j \leq k-1).$ Then $\det M > 0 .$ The constant $4\cos ^2 \frac{\pi}{k+1}$ in this Theorem is sharp. A few other results concerning totally positive and multiply positive matrices are obtained. Keywords: Multiply positive matrix; Totally positive matrix; Strictly totally positive matrix; Toeplitz matrix; Hankel matrix; P\'olya frequency sequence.Comment: 15 page

A note on zero sets of absolutely monotonic functions

A new necessary condition for a set to be the zero set of an absolutely monotonic function is given

Zero sets of entire absolutely monotonic functions

The paper is devoted to the problem of characterization of the class of subsets of $\mathbb {C}$ which can serve as zero sets of entire absolutely monotonic functions

A note on zero sets of absolutely monotonic functions

A new necessary condition for a set to be the zero set of an absolutely monotonic function is given

The first light of Mini-MegaTORTORA wide-field monitoring system

Here we describe the first light of the novel 9-channel wide-field optical monitoring system with sub-second temporal resolution, Mini-MegaTORTORA, which is being tested now at Special Astrophysical Observatory on Russian Caucasus. The system is able to observe the sky simultaneously in either wide (~900 square degrees) or narrow (~100 square degrees) fields of view, either in clear light or with any combination of color (Johnson B, V or R) polarimetric filters installed, with exposure times ranging from 100 ms to 100 s. The primary goal of the system is the detection of rapid -- with sub-second characteristic time-scales -- optical transients, but it may be also used for studying the variability of the sky objects on longer time scales.Comment: 9 pages, 9 figures, based on the talk presented on "Modern stellar astronomy-2014" (Rostov-on-Don