David Natroshvili (Department of Mathematics, Georgian Technical University)

Application of the potential method to transmission problems for composite layered elastic structures containing interfacial cracks

Abstract: We investigate mixed dynamical boundary-transmission problems of the generalized thermo-electro-magneto elasticity theory for anisotropic elastic multi-layered structures containing interfacial cracks. We apply the potential method and the theory of pseudodifferential equations and analyze smoothness properties and asymptomatic behaviour of solutions near the edges of cracks and near the curves where different types of boundary conditions collide. We describe the stress singularity exponents explicitly and analyze their dependance on material parameters.
This is a joint work with O.Chkadua and T.Buchukuri.
Acknowledgments. This work was supported by Shota Rustaveli National Science Foundation of Georgia (SRNSF) (Grant number FR-23-267).

Radu Purice (Simion Stoilow Institute of Mathematics of the Romanian Academy)

An algebra of infinite matrices associated to the Weyl pseudo-differential calculus

Abstract: We consider the description of Hoermander type pseudo-differential operators in d dimensions with respect to a tight Gabor frame. We show that all these operators can be identified with some infinitely dimensional matrices whose elements are strongly localized near the diagonal. Using this matrix representation, we give short proofs to classical results like the Calderon-Vaillancourt theorem and Beals commutator criterion, and also some local trace-class criteria.

Goerge Jaiani (I.Vekua Institute of Applied Mathematics of I. Javakhishvili Tbilisi State University)

Even Order Singular Elliptic Equations

Abstract: The present talk is devoted to singular partial differential equations, i.e., to partial differential equations with the order degeneracy. The results stated in the talk are applied in investigations of cusped prismatic shells and bars and of motion of fluids in angular ducts.

Oana Lupascu-Stamate (Gheorghe Mihoc-Caius Iacob Institute of Mathematical Statistics and Applied Mathematics of the Romanian Academy, Bucharest, Romania)

Stochastic fragmentation processes for a system of particles with spatial position

Abstract: We investigate random multiple-fragmentation associated to a flow of particles on a surface. The mathematical problem models the time evolution of a system of particles which move on an Euclidean surface driven by a given force (e.g., gravitational, fluid interaction, repulsion/attraction), and split in fragments with smaller masses and velocities. We establish a stochastic multiple-fragmentation process and we solve the corresponding stochastic integro-differential equation. Finally, we present several numerical simulations of such processes.
These results are obtained jointly with Lucian Beznea (Bucharest) and Ioan R. Ionescu (Paris).

Giorgi Rukhaia (I.Vekua Institute of Applied Mathematics of I. Javakhishvili Tbilisi State University)

Monge-Ampere equations in freeform optics, and the corresponding optimal transport solvers

Abstract: This talk reviews the derivation of Monge-Ampere equation for the freeform optics point-source to far-field reflector problem, and discusses the optimal transport solvers used in that (and many other) context. We also discuss a way of using optimal transport solvers to solve a more general problem of extended-source far-field reflector problem.

Mariam Kokhreidze (I. Javakhishvili Tbilisi State University)

Construction of Hierarchical Mathematical Models for a Linearized Reiner Elastic Materials

Abstract: The present talk is devoted to construction, using I.Vekuas dimension reduction method, for the Reiner elastic bodies, in the some sense linearized case. The shell-like domain Ω ⊆ R3 with both Lipschitz and Non-Lipschitz boundaries is occupied by the Reiner elastic material. In the N-th approximation initial-boundary value problems are posed. The peculiarities of settling the boundary conditions for cusped prismatic shells is discussed in the projection ω ⊆ R2 of Ω.