An Erratum to this paper has been published: https://doi.org/10.1134/S1547477125010017

In this paper, we obtain necessary and sufficient conditions for the existence of variational principles for a given first-order differential-difference operator equation with a special form of the linear operator Pλ(t) depending on t and the nonlinear operator Q. Under the corresponding conditions the functional is constructed. These conditions are obtained thanks to the well-known criterion of potentiality. Examples show how the inverse problem of the calculus of variations is constructed for given differential-difference operators.

We designed a study with the objective to determine the long-term radiation effects of gamma rays, originating from a single shot of Co60 at a dose of 2 Gy on the 7-month-old male mice of the ICR line in 30 days after the irradiation. The aim of this study was to characterize the behavior of animals using the Open Field test, immuno-hematological status, and morpho-functional changes in the central nervous system of mice. Irradiated animals displayed significantly different behavior in the OF in comparison with the control group. The radiation damage was confirmed by assessing the ratio of leukocytes in the peripheral blood of mice at a later date after exposure to Co60. After irradiation, a decrease in the glioneuronal complex was observed in the irritated group as well as histological changes of brain cells. To sum up, not only was the hematological status of mice altered upon the total gamma irradiation, but also their behavior, which was most probably due to significant alterations in the CNS. Study of influence of ionizing radiation on female mice, comparison between different age groups. Open Field test on the 30 days after 2 Gy of γ-rays and histological analysis indicated changes in behavioral patterns, leucocytes, and brain tissue.

The aim of this research was to study behavioral reactions and morphological changes in the brain of adult female Sprague Dawley rats after exposure to 170 MeV and 70 MeV protons and gamma radiation (60Co) at a dose of 1 Gy. The analysis of the behavioral reactions in the T-maze showed that exposure to ionizing radiation with different LETs led to an increase in number of repeated entries into the arms of the maze in the spontaneous alternation test. In the Open Field test a decrease in overall motor activity in the group of irradiated animals (70 MeV protons at the Bragg peak) was observed. A decrease in the number of standing positions was seen in all groups of irradiated animals. Morphological analysis showed the development of early amyloidosis, autolysis of the ependymal layer, an increase in the number of neurodegenerative changes in various structures of the brain, and the development of neuronal hypertrophy on the 30th day after irradiation in the cerebellum and hippocampal hilus. Exposure to protons at a dose of 1 Gy leads to the development of structural and functional disorders of the central nervous system of animals on the 30th day after irradiation. These data indicate a damage of short-term memory, a decrease in motor activity and exploratory behavior of animals. With an increase in LET, there is an increase in the number of amyloid plaques in the forebrain of rats, autolysis of the ependymal layer of the ventricles, and the development of dystrophic changes. Investigations of behavioral reactions and morphological changes in various parts of the brain of adult rats on the 30th day after influence of ionizing radiation with different physical characteristics at a dose of 1 Gy. Various negative patho-morphological and cognitive-behavioral changes observed.

Fluorescence and chromatographic analysis of bisretinoids from the retina and retinal pigment epithelium of mouse eyes was carried out before and after exposure to accelerated protons in the Bragg peak. It has been shown that ionizing radiation at doses of 1–4 Gy leads to a shift in the short-wave region of the maximum of the fluorescence spectrum in the chloroform extract obtained from both the retinal pigment epithelium and the retina. Chromatographic analysis of these extracts has shown a change in the relative content of individual bisretinoids. The obtained spectral and chromatographic data indicate that the exposure of mice to accelerated protons in the Bragg peak at doses of 1–4 Gy leads to radiation oxidation of bisretinoids in eye tissues.

We obtain necessary and sufficient conditions for a system of partial differential-difference equations to admit a group of infinitesimal point transformations. We use examples to illustrate how one can find such groups and generators of symmetries as well as first integrals and functionals for potential operators.

In the present paper, we obtain necessary and sufficient conditions for the existence of symmetry groups for systems of differential-difference equations with partial derivatives. Constructions of symmetry groups and recursion operators for various differential-difference equations are described. The corresponding symmetry generators that transform solutions of given equations into other solutions are obtained.

For a differential-difference evolution operator, we obtain necessary and sufficient conditions for the existence of a variational principle. We describe the structure of the operators P λ and Q for which the corresponding differential-difference equation admits a direct variational statement.

Necessary and sufficient conditions for the existence of variational principles for a given wide class of evolutionary differential-difference operator equation are obtained. The theoretic results are illustrated by two examples.