Дослiдження розв’язкiв дисперсiйних рiвнянь старшого порядку з φ-субгауссовими початковими умовами

In this paper, there are studied sample paths properties of stochastic processes representing solutions of higher-order dispersive equations with random initial conditions given by φ-sub-Gaussian harmonizable processes. The main results are the bounds for the rate of growth of such stochastic processes considered over unbounded domains. The class of φ-sub-Gaussian processes with φ(x) = |x|^α/α, 1 < α <= 2, is a natural generalization of Gaussian processes. For such initial conditions the bounds for the distribution of supremum of solutions can be calculated in rather simple form. The bounds for the rate of growth of solution to higher-order partial differential equations with random initial conditions in the case of general φ were obtained in [9], the derivation was based on the sults stated in [1]. Here we use another approach, which allows us, for the particular case φ(x) = |x|^α/α, α є (1, 2], to present the expressions for the bounds in the closed form. Pages of the article in the issue: 78 - 84 Language of the article: EnglishIn this paper, there are studied sample paths properties of stochastic processes representing solutions of higher-order dispersive equations with random initial conditions given by φ-sub-Gaussian harmonizable processes. The main results are the bounds for the rate of growth of such stochastic processes considered over unbounded domains. The class of φ-sub-Gaussian processes with φ(x) = |x|^α/α, 1 < α <= 2, is a natural generalization of Gaussian processes. For such initial conditions the bounds for the distribution of supremum of solutions can be calculated in rather simple form. The bounds for the rate of growth of solution to higher-order partial differential equations with random initial conditions in the case of general φ were obtained in [9], the derivation was based on the sults stated in [1]. Here we use another approach, which allows us, for the particular case φ(x) = |x|^α/α, α є (1, 2], to present the expressions for the bounds in the closed form

Про один з методiв побудови моделi строго φ-субгауссового узагальненого дробового броунiвського руху

In the paper, we consider the problem of simulation of a strictly φ-sub-Gaussian generalized fracti-onal Brownian motion. Simulation of random processes and fields is used in many areas of natural and social sciences. A special place is occupied by methods of simulation of the Wiener process and fractional Brownian motion, as these processes are widely used in financial and actuarial mathematics, queueing theory etc. We study some specific class of processes of generalized fractional Brownian motion and derive conditions, under which the model based on a series representation approximates a strictly φ-sub-Gaussian generalized fractional Brownian motion with given reliability and accuracy in the space C([0; 1]) in the case, when φ(x) = exp{|x|} − |x| − 1, x ∈ R. In order to obtain these results, we use some results from the theory of φ-sub-Gaussian random processes. Necessary simulation parameters are calculated and models of sample pathes of corresponding processes are constructed for various values of the Hurst parameter H and for given reliability and accuracy using the R programming environment. Pages of the article in the issue: 18 - 25 Language of the article: UkrainianIn the paper, we consider the problem of simulation of a strictly φ-sub-Gaussian generalized fracti-onal Brownian motion. Simulation of random processes and fields is used in many areas of natural and social sciences. A special place is occupied by methods of simulation of the Wiener process and fractional Brownian motion, as these processes are widely used in financial and actuarial mathematics, queueing theory etc. We study some specific class of processes of generalized fractional Brownian motion and derive conditions, under which the model based on a series representation approximates a strictly φ-sub-Gaussian generalized fractional Brownian motion with given reliability and accuracy in the space C([0; 1]) in the case, when φ(x) = exp{|x|} − |x| − 1, x ∈ R. In order to obtain these results, we use some results from the theory of φ-sub-Gaussian random processes. Necessary simulation parameters are calculated and models of sample pathes of corresponding processes are constructed for various values of the Hurst parameter H and for given reliability and accuracy using the R programming environment.  Pages of the article in the issue: 18 - 25 Language of the article: Ukrainia

Властивостi розв’язкiв лiнiйного рiвняння KdV iз φ-субгауссовими початковими умовами

In this paper, there are studied sample paths properties of stochastic processes representing solutions (in L_2(Ω) sense) to the linear Korteweg–de Vries equation (called also the Airy equation) with random initial conditions given by φ-sub-Gaussian stationary processes. The main results are the bounds for the distributions of the suprema for such stochastic processes considered over bounded domains. Also, there are presented some examples to illustrate the results of the study. Pages of the article in the issue: 11 - 19 Language of the article: EnglishВажливий практичний аспект оцiнювання статистичних властивостей фiзичних систем спирається на ефективне представлення зв’язку мiж розв’язками рiвнянь з частинними похiдними та випадковими початковими умовами. У цiй роботi дослiджуються властивостi траєкторiй випадкових процесiв, що задають розв’язки (в L_2(Ω)) для рiвняння Айрi з φ-субгауссовими стацiонарними випадковими початковими умовами. Властивостi субгауссовостi та φ-субгауссовостi є важливими характеристиками випадкових процесiв, оскiльки вони дають можливiсть оцiнити рiзнi функцiонали вiд цих процесiв, i, зокрема, дослiдити поведiнку їх супремумiв. Основнi результати роботи – це оцiнки для розподiлiв супремумiв випадкових процесiв, що задають розв’язки для рiвняння Айрi, на обмежених множинах. Застосування отриманих результатiв проiлюстровано на прикладах у випадках гауссових початкових умов з рiзними допустимими функцiями та φ-субгауссових початкових умов з певними функцiями φ, зокрема φ(x) = exp{|x|} − |x| − 1, x \in R

Intermittent fasting causes metabolic stress and leucopenia in young mice

Overweight and obesity became the worldwide epidemic resulting from overeating especially when a so-called Western diet rich in carbohydrates and fats is used. It is widely accepted that limitation of food consumption could help to withstand such state of adult organism, but information about younger groups is contradictory. The present study was undertaken to characterize the effects of intermittent fasting, using an every other day (EOD) fasting/feeding protocol, on hematological parameters and biochemical blood plasma indices in young mice from one to two months old. It was shown that intermittently fasted mice were characterized by a reduced body weight, reduced total number of blood leucocytes, lower glucose and lactate levels and higher activity of alanine aminotransferase and aspartate aminotransferase in blood plasma as compared with the age-matched control mice. To gain the same mass EOD animals needed to eat more food than ad libitum fed animals. These differences may probably be explained by a need to expend certain resources to combat stress induced by intermittent fasting. Our data showed that EOD feeding at a young age may negatively influence young mammals

Estimation of distribution of suprema of a strictly ϕ-sub-Gaussian quasi shot noise process

In this paper, there are studied properties of a strictly ϕ-sub-Gaussian quasi shot noise process X(t) = integral_{-∞}^{+∞} g(t, u) dξ(u), t ∈ R, generated by the process ξ and the response function g. New estimates for distributions of suprema of such processes are derived. An example of application of the obtained results is given.Key words: shot noise processes, distrubution of suprema of a process, ϕ-sub-Gaussian processes.Pages of the article in the issue: 8 - 17Language of the article: Ukrainia

Simulation of a strictly ?-sub-Gaussian generalized fractional Brownian motion

In the paper, we consider the problem of simulation of a strictly ?-sub-Gaussian generalized fractional Brownian motion. Simulation of random processes and fields is used in many areas of natural and social sciences. A special place is occupied by methods of simulation of the Wiener process and fractional Brownian motion, as these processes are widely used in financial and actuarial mathematics, queueing theory etc. We study some specific class of processes of generalized fractional Brownian motion and derive conditions, under which the model based on a series representation approximates a strictly ?-sub-Gaussian generalized fractional Brownian motion with given reliability and accuracy in the space C([0; 1]) in the case, when ?(x) = (|x|^p)/p, |x| ? 1, p > 1. In order to obtain these results, we use some results from the theory of ?-sub-Gaussian random processes. Necessary simulation parameters are calculated and models of sample pathes of corresponding processes are constructed for various values of the Hurst parameter H and for given reliability and accuracy using the R programming environment. Pages of the article in the issue: 11 - 19 Language of the article: UkrainianIn the paper, we consider the problem of simulation of a strictly ?-sub-Gaussian generalized fractional Brownian motion. Simulation of random processes and fields is used in many areas of natural and social sciences. A special place is occupied by methods of simulation of the Wiener process and fractional Brownian motion, as these processes are widely used in financial and actuarial mathematics, queueing theory etc. We study some specific class of processes of generalized fractional Brownian motion and derive conditions, under which the model based on a series representation approximates a strictly ?-sub-Gaussian generalized fractional Brownian motion with given reliability and accuracy in the space C([0; 1]) in the case, when ?(x) = (|x|^p)/p, |x| ? 1, p > 1. In order to obtain these results, we use some results from the theory of ?-sub-Gaussian random processes. Necessary simulation parameters are calculated and models of sample pathes of corresponding processes are constructed for various values of the Hurst parameter H and for given reliability and accuracy using the R programming environment. Pages of the article in the issue: 11 - 19 Language of the article: Ukrainia

Оцінювання ймовірності виходу траєкторії строго φ\varphi-субгауссового процесу квазідробового ефекту за криву

In this paper, we continue to study the properties of a separable strictly φ-sub-Gaussian quasi shot noise process $X(t) = \int_{-\infty}^{+\infty} g(t,u) d\xi(u),  t\in\R$, generated by the response function g and the strictly φ-sub-Gaussian process ξ = (ξ(t), t ∈ R) with uncorrelated increments, such that E(ξ(t)−ξ(s))^2 = t−s, t>s ∈ R. We consider the problem of estimating the probability of exceeding some level by such a process on the interval [a;b], a,b ∈ R. The level is given by a continuous function f = {f(t), t ∈ [a;b]}, which satisfies some given conditions. In order to solve this problem, we apply the theorems obtained for random processes from a class V (φ, ψ), which generalizes the class of φ-sub-Gaussian processes. As a result, several estimates for probability of exceeding the curve f by sample pathes of a separable strictly φ-sub-Gaussian quasi shot noise process are obtained. Such estimates can be used in the study of shot noise processes that arise in the problems of financial mathematics, telecommunication networks theory, and other applications.Key words: shot noise processes, φ-sub-Gaussian processes.Pages of the article in the issue: 49 - 56Language of the article: UkrainianУ роботі  досліджуються властивості строго $\varphi$-субгауссового процесу квазідробового ефекту $X(t)=\int_{-\infty}^{+\infty}g(t,u)d\xi(u),$ $t\in\R$, породженого випадковим процесом $\xi$ та функцією відгуку~$g$. Отримано оцінки для ймовірності виходу траєкторії строго $\varphi$-субгауссового процесу квазідробового ефекту за криву

Properties of φ\varphi-sub-Gaussian stochastic processes related to the heat equation with random initial conditions

In this paper, there are studied sample paths properties of stochastic processes representing solutions (in $L_2(\Omega)$ sense) of the heat equation with random initial conditions given by $\varphi$-sub-Gaussian stationary processes. The main results are the bounds for the distributions of the suprema for such stochastic processes considered over bounded and unbounded domains.Key words: φ-sub-Gaussian processes, heat equation, random initial condition, distribution of supremum.Pages of the article in the issue: 17 - 24Language of the article: Englis