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

On Determining the Volume of Repeated and Non-Repeated Sampling in the Preparation of Clinical Studies

In this article, non-repeatable and repeatable sampling methods were analyzed. Main objectives connected with the use of sampling research method were determined. The problem of non-repeatable sampling amount defining at a given rate of bronchial asthma prevalence was analyzed. Key preconditions and assumptions used when constructing the expression to determine the non-repeatable and repeatable sample amount were depicted

Use of the complex of models of regression for analysis of the factors that determine the severity of bronchial asthma

Background: According to an International Study of Asthma and Allergies in Childhood (ISAAC), the prevalence of asthma in children of 6-7 years old has increased by 10%, and at the age of 13-14 years by 16% over the last decade. Determining the factors that are keys to the occurrence of the disease and its severity is important in explaining the pathogenesis of bronchial asthma. Methods: Analyzed 142 indicators of clinical and paraclinical examination of 70 children with asthma. To select factors that could be significant in the formation of severe asthma, applied the method of logistic regression with step-by-step inclusion of predictors. Both quantitative and qualitative characteristics were selected. Each qualitative attribute was coded “1” if the child had this characteristic, or “0” if this characteristic had not been established. The formation of a severe asthma course was accepted according to (1) and the absence of a severe asthma flow formation as (0). Results: Analyzed the model of paired regression, the boundary value of thymic stromal lymphopoietin was established, exceeding which indicates the high probability of the presence of severe bronchial asthma. Increasing the value of thymic stromal lymphopoietin by 10 pg/mL suggests an increase in the likelihood of severe asthma by 10%. Conclusions: A complex of steam regression models has been developed to determine the factors characterizing the severity of bronchial asthma. The risk of developing severe bronchial asthma in children has been determined and 15 factors have been identified that affect severe asthma