Joint research work with W. Jäger, an employee of Heidelberg University. Results on the existence, uniqueness and periodicity of solutions in the case of the heat equation were obtained. It was proved that the existence of a periodic average solution implies the existence of a strong periodic solution with the same period. An example is given in which there is a single periodic average, and hence the only strong periodic solution. The results of the work are reflected in the article: P. Gurevich, W. Jäger, and A. Skubachevskii. On periodicity of solutions for thermocontrol problems with hysteresis-type switches. // SIAM J. Math. Anal. 41, No. 2, 2009, pp. 733-752.

Joint research work with H.-O.-Walther, an employee of Justus Liebig University. Sufficient terms of hyperbolicity and stability of periodic solutions of nonlinear functional-differential equations were obtained. The results of the work are reflected in the articles: - Walter H.-O., Skubachevskii A. L. On hyperbolicity of rapidly oscillating periodic solutions to functional differential equations. // Journal “Functional analysis and its applications”, Vol. 39, is. 1, M., 2005, p. 82-85. - Walter X-Skubachevskii A. L. On hyperbolicity of solutions with irrational periods of some functional differential equations. // Journal “Proceedings of the Russian Higher School Academy of Sciences”, Vol. 402. №2, M., 2005, p. 151-154.

We are working on the joint article “Spherical symmetric stationary solutions of the Vlasov – Poisson equation” with J. Batt, an employee of Ludwig Maximilian University of Munich.

Joint research work with D. Haroske, an employee of Friedrich Schiller University Jena. Evaluation of the uniform modulus of continuity for Bessel potentials, accurate evaluation of the majorant of modules of continuity and optimal embedding for generalized Bessel potentials were obtained, optimal Calderon space for Bessel potentials and optimal Calderon space for generalized Bessel potentials were built.

Goldman M.L., Haroske D. Estimates for continuity envelopes and approximation numbers of Bessel potentials // Journal of Approximation Theory. 2013. Vol. 172. P. 58–85.

Goldman M.L., Haroske D. Optimal Calderon spaces for the generalized Bessel potentials / // Doklady Mathematics. 2015. Vol. 92. № 1. P. 404–407.

Mathematicians of RUDN University (Professor B. Yu. Sternin, Professor A. Y. Savin) with mathematicians of the University of Hannover (Prof. Dr. Elmar Schrohe) analyze actual problems of elliptic theory and noncommutative geometry in joint scientific works. In particular, an explicit uniformization for elliptic operators associated with groups of shift operators was found out. The terms for the Fredholm property of the operators associated with groups of quantized canonical transformations were obtained.

A.Yu. Savin, B.Yu. Sternin, E. Schrohe, “Index problem for elliptic operators associated with a diffeomorphism of a manifold and uniformization”, Proceedings of the Russian Higher School Academy of Sciences, 441:5 (2011), 593–596; A.Yu. Savin, B.Yu. Sternin, E. Schrohe, “Index problem for elliptic operators associated with a diffeomorphism of a manifold and uniformization”, Dokl. Math., 84:3 (2011), 846-849

A.Yu. Savin, B.Yu. Sternin, E. Schrohe, “On the Index Formula for an Isometric Diffeomorphism”, Journal of Mathematical Sciences, 201:6 (2014), 818–829

A.Yu. Savin, E. Schrohe, B. Sternin, “Uniformization and Index of Elliptic Operators Associated with Diffeomorphisms of a Manifold”, Russian Journal of Mathematical Physics, 22:3 (2015), “410–420”

Accurate stability evaluation for the variation of eigenvalues of nonnegative selfadjoint elliptic operators of arbitrary even order when changing the open sets on which they are defined were obtain with P. D. Lamberti, M. Lanza de Cristoforis, Feleqi E. This evaluation is expressed in terms of Lebesgue measure of the symmetric difference of open sets. The boundary terms of Dirichlet and Neumann were analyzed.

A full list of joint works can be found here

The problem of the limitation of various operators (Riesz potential, the maximum operator, Hardy operator) in local spaces of the Morrie type was studied together with the staff of this Institute (V. S. Guliyev, H. V. Guliyev, R. Mustafayev). A full list of collaborations can be found here.

The terms that ensure the boundedness of Hausdorff operators in Morrie spaces were obtained together with E. Liflyand. The classes of Hausdorff operators for which the necessary and sufficient terms of boundedness coincide were described.

Necessary and sufficient terms for the boundedness of fractional maximal operators in general Morrie local spaces are for a wide class of admissible values of numerical parameters were obtained together with A. Gogatishvili. Sufficient and necessary terms for this kind of limitation for a certain range of parameters were obtained.

Joint research work with A. Bouchnita. Mathematical models and computer programs for describing the coagulation of blood in the stream considering biochemical reactions in the plasma and aggregation of platelets were developed. The terms of normal clot growth and thrombosis were obtained. Two articles are in print this year.

Joint research work with M. Koury. The mathematical model was developed, computer calculations and comparisons with experimental data of erythropoiesis functioning on the basis of erythroblastic islands were carried out. The development of multiple myeloma and its effect on erythropoiesis were investigated. Various treatment protocols of patients were modeled and compared with clinical data.

Joint research work with A. Meyerhans. The study of the spread of viral infection in body tissues on the basis of reaction-diffusion and hybrid discrete-continuous models of immune response.

Joint research work with C. Dumontet, P. Nony. Mathematical modeling of variousdiseases: leukaemia, lymphoma, lymphodema, mucoviscidosis, thrombosis. Simulation of clinical trials.

Parametric models of the geometry of the lymph node were developed. Models were developed and topological properties of a network of fibroblastic reticular cells were analyzed. The features of lymph flow in the network of lymph node conduits were studied.

The components of the system model describing the regulation of energy metabolism of cells in normoxia and hypoxia were developed.