Pseudostarlike and pseudoconvex solutions of a differential equation with exponential coefficients

Dirichlet series $F(s)=e^{s}+\sum_{k=1}^{\infty}f_ke^{s\lambda_k}$ with the exponents $1<\lambda_k\uparrow+\infty$ and the abscissa of absolute convergence $\sigma_a[F]\ge 0$ is said to be pseudostarlike of order $\alpha\in [0,\,1)$ and type $\beta \in (0,\,1]$ if$\left|\dfrac{F'(s)}{F(s)}-1\right|<\beta\left|\dfrac{F'(s)}{F(s)}-(2\alpha-1)\right|$\ for all\ $s\in \Pi_0=\{s\colon \,\text{Re}\,s<0\}$. Similarly, the function $F$ is said to be pseudoconvex of order $\alpha\in [0,\,1)$ and type $\beta \in (0,\,1]$ if$\left|\dfrac{F''(s)}{F'(s)}-1\right|<\beta\left|\dfrac{F''(s)}{F'(s)}-(2\alpha-1)\right|$\ for all\ $s\in \Pi_0$. Some conditions are found on the parameters $b_0,\,b_1,\,c_0,\,c_1,\,\,c_2$ and the coefficients $a_n$, under which the differential equation $\dfrac{d^2w}{ds^2}+(b_0e^{s}+b_1)\dfrac{dw}{ds}+(c_0e^{2s}+c_1e^{s}+c_2)w=\sum\limits_{n=1}^{\infty}a_ne^{ns}$has an entire solution which is pseudostarlike or pseudoconvex of order $\alpha\in [0,\,1)$ and type $\beta \in (0,\,1]$. It is proved that by some conditions for such solution the asymptotic equality holds $\ln\,\max\{|F(\sigma+it)|\colon t\in {\mathbb R}\}=\dfrac{1+o(1)}{2}\left(|b_0|+\sqrt{|b_0|^2+4|c_0|}\right)$ as $\sigma \to+\infty$.

Method for transforming symbolic radar marks of low-noticeable moving objects based on the Talbot effect

In this paper a method to transform radar images of moving aerial objects with scintillating inter-period fluctuations, sometimes resulting to complete signal fading, using the Talbot effect is considered. These transformations are reduced to the establishment of a certain correspondence of the asymptotic equality of perception of visual images, arbitrarily changing in time and space, in the statement about the conditions of simple equality of perception of images of radar marks that have different frequencies of fluctuations. It is shown how this approach can be used to analyze radar data by transforming and smoothing scintillating signal fluctuations, invisible in the presence of interference, into visible symbolic images. First, to detect and recognize the aerial objects from the analysis of relations and functional (semantic) dependencies between attributes, second, to make a decision based on semantic components of symbolic radar images. The possibility of using such transformation to generate pulse-frequency code of fluctuations of the symbolic radar angel-echo images as an important characteristic for their recognition has been experimentally verified. Algorithms for generating symbolic images in asynchronous and synchronous pulse-frequency code are formulated. The symbolic image represented by such a code is considered as an additional feature for recognizing and filtering out natural interferences such as angel-echoes.

Approximation of the function |sin x| s by the partial sums of the trigonmometric rational fourier series

In the present article, the approximation of the function |sin x| s by the partial sums of the rational trigonometric Fourier series is considered. An integral representation, uniform and point estimates for the above-mentioned approximation were obtained. Based on them, several special cases of the selection of poles were studied. In the case of the approximation by the partial sums of the polynomial trigonometric Fourier series, an asymptotic equality was found. A detailed study is made of a fixed number of geometrically different poles.

Rational interpolation of the function |x|α by an extended system of Chebyshev – Markov nodes

In this paper, we study the approximations of a function |x|α, α > 0 by interpolation rational Lagrange functions on a segment [–1,1]. The zeros of the even Chebyshev – Markov rational functions and a point x = 0 are chosen as the interpolation nodes. An integral representation of an interpolation remainder and an upper bound for the considered uniform approximations are obtained. Based on them, a detailed study is made:a) the polynomial case. Here, the authors come to the famous asymptotic equality of M. N. Hanzburg;b) at a fixed number of geometrically different poles, the upper estimate is obtained for the corresponding uniform approximations, which improves the well-known result of K. N. Lungu;c) when approximating by general Lagrange rational interpolation functions, the estimate of uniform approximations is found and it is shown that at the ends of the segment [–1,1] it can be improved.The results can be applied in theoretical research and numerical methods.

THERE ARE ASYMPTOTICALLY THE SAME NUMBER OF LATIN SQUARES OF EACH PARITY

A Latin square is reduced if its first row and first column are in natural order. For Latin squares of a particular order$n$, there are four possible different parities. We confirm a conjecture of Stones and Wanless by showing asymptotic equality between the numbers of reduced Latin squares of each possible parity as the order$n\rightarrow \infty$.

HEIGHTS AND LOGARITHMIC gcd ON ALGEBRAIC CURVES

Let F(x,y) be an irreducible polynomial over ℚ, satisfying F(0,0) = 0. Skolem proved that the integral solutions of F(x,y) = 0 with fixed gcd are bounded [13] and Walsh gave an explicit bound in terms of d = gcd (x,y) and F [16]. Assuming that (0,0) is a non-singular point of the plane curve F(x,y) = 0, we extend this result to algebraic solution, and obtain an asymptotic equality instead of inequality. We show that for any algebraic solution (α,β), the quotient h(α)/ log d is approximatively equal to degyF and the quotient h(β)/ log d to deg x F; here h(·) is the absolute logarithmic height and d is the (properly defined) "greatest common divisor" of α and β.