Fredholm Property of Nonlocal Problems for Integro-Differential Hyperbolic Systems

The paper concerns nonlocal time-periodic boundary value problems for first-order Volterra integro-differential hyperbolic systems with boundary inputs. The systems are subjected to integral boundary conditions. Under natural regularity assumptions on the data it is shown that the problems display completely non-resonant behaviour and satisfy the Fredholm alternative in the spaces of continuous and time-periodic functions.Comment: Final versio

Fredholm Solvability of a Periodic Neumann Problem for a Linear Telegraph Equation

We investigate a periodic problem for the linear telegraph equation utt − uxx + 2µut = f(x, t) with Neumann boundary conditions. We prove that the operator of the problem is modeled by a Fredholm operator of index zero in the scale of Sobolev spaces of periodic functions. This result is stable under small perturbations of the equation where µ becomes variable and discontinuous or an additional zero-order term appears. We also show that the solutions of this problem possess smoothing properties.Дослiджується перiодична задача для лiнiйного телеграфного рiвняння utt − uxx + 2µut = f(x, t) з крайовими умовами Неймана. Доведено, що оператор задачi моделюється фредгольмовим оператором нульового iндексу у шкалi просторiв Соболєва перiодичних функцiй. Цей результат є стiйким щодо малих збурень рiвняння, де µ стає змiнною i розривною або з’являється додатковий член нульового порядку. Також показано, що розв’язки задачi мають властивiсть пiдвищення гладкостi

Heart rate variability and cortisol levels in school-age children with different cognitive tests

Background. An urgent task of age-related physiology is to study the functional state of the body of school-age children in cognitive activity due to the large academic load and the use of information and computer technologies in the educational process to identify the characteristics of the reactivity of the body of students when performing cognitive load of various types, including on electronic devices, is necessary for the organization of the school educational environment.The aim. To assess the nature of vegetative, cardiovascular and hormonal reactivity in cognitive load of various types in school-age children.Materials and methods. By methods of heart rate variability analysis, electrocardiography, tonometry and enzyme immunoassay of cortisol in saliva, 117 school-age children were examined while performing cognitive load of various types.Results. There is a change in heart rate variability indicators while performing cognitive load. Oral counting causes an increase in sympathetic influences on the heart rate with a decrease in parasympathetic activity, as well as a shift in the vagosympathetic balance. Operation on the electronic devices causes a decrease in the total power density of the spectrum and an increase in the index of low-frequency and highfrequency vibrations ratio due to a decrease in parasympathetic activity. Two types of reaction were revealed: type I – an increase in the concentration of cortisol in saliva, an increase in sympathetic effects on Heart rate with a simultaneous decrease in parasympathetic activity (counting), a decrease in the total power of the spectrum density (laptop), a decrease in parasympathetic activity (tablet). Type II – a decrease in hormone levels and a decrease in very low-frequency vibrations and parasympathetic activity, regardless of the type of load presentedConclusion. The results obtained indicate that the nature of the reactivity of heart rate indicators and the stress hormone cortisol in students depends not so much on which electronic device it is performed on, but on the type of cognitive load

Initial-boundary problems for semilinear hiperbolic systems with singular coefficients

In the paper we use the framework of Colombeau algebras of generalized functions to study existence and uniqueness of global generalized solutions to mixed non-local problems for a semilinear hyperbolic system. Coefficients of the system as well as initial and boundary data are allowed to be strongly singular, as the Dirac delta function and derivatives thereof. To obtain the existence-uniqueness result we prove a criterion of invertibilitv in the full version of the Colombeau algebras