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Explicitly solvable models and their applications to current challenges in mechanics and wave propagation

Venue: CIMAT, Unidad Mérida. 

The meeting will provide an overview and a discussion forum for modelling the behaviour of highly heterogeneous media using "explicitly solvable" formulations, e.g. by replacing an operator in an infinite-dimensional space by a finite-size matrix. The analysis of the related approximation error and the implications of such effective formulations on spectral properties and wave propagation will span the range of the discussion topics.


Talk schedule

Wednesday 4 January

10:00--10:30: Arrival and coffee

10:30--11:30: Kirill Cherednichenko "Operator-norm resolvent estimates for thin elastic periodically heterogeneous rods in moderate contrast"

Abstract: I shall discuss the construction of an operator-norm resolvent asymptotics for the system of linearised elasticity for a thin elastic rod with highly oscillating properties, where the thickness of the rod and the period of material oscillations are taken to be of the same order. The related analysis carried out separately on two invariant subspaces pertaining to the out-of-line and in-line displacements. This is joint work with I. Velčić and J. Josip Žubrinić.

11:30--12:00: Coffee

12:00--13:00: Mikhail Cherdantsev "High-contrast random composites: homogenisation framework and new spectral phenomena"


13:00-14:00: Lunch at CIMAT

14:00--15:00: Luis Octavio Silva "Point mass perturbations of spectral measures" 

Abstract: This talk is motivated by the intriguing fact that very small perturbations of a measure may destroy the density of polynomials in the space of square integrable functions with respect to that measure. This instability phenomenon appears when the measure is a solution to the classical Hamburger moment problem, but it is also present in a much more general setting. Departing from the fact that a solution to the moment problem is the spectral measure of a Jacobi operator, we study the spectral measures of the class of closed, symmetric, regular operators with deficiency indices (1,1) of which the Jacobi operators constitute a subclass. On the basis of Krein's theory of representation for symmetric operators, Naimark's extension theory, and Livshic's proposal for a generalization of the moment problem, we construct the theoretical scaffold for proving various results on the point mass perturbation of the spectral measures of closed, symmetric, regular operators with deficiency indices (1,1). Regular and singular Schrödinger operators are within this class of operators and results on point mass perturbations of their spectral measures are provided.

15:00--16:00: Coffee and open research discussion

Thursday 5 January

10:00--11:00: Danila Prikazchikov "Nonlocally elastic Rayleigh-type waves"

Abstract: Nonlocally elastic surface waves are studied within both integral and differential formulations, and the issue of their non-equivalence is discussed. Consistent differential constitutive relations assuming an extra boundary condition along the surface are proposed. Both antiplane shear and Rayleigh waves are considered. The asymptotic procedure is developed, incorporating the contribution of nonlocal boundary layers. It is shown that the associated nonlocal correction to the classical Rayleigh wave speed is of order of microscale parameter, which is by order of magnitude higher than that originating from the equations of motion obtained by Eringen. In addition, it appears that the nonlocal boundary layers also support nonlocal shear surface waves which are not the feature of classical elasticity. An explicit formulation for the nonlocally elastic Rayleigh wave excited by a prescribed surface loading is also derived.

11:00--11:30: Coffee

11:30--12:30: Alexander Kiselev "An example of a phase transition in a periodic tubular structure"


12:30-13:30: Lunch at CIMAT

13:30--14:30: Yi Sheng Lim "Model spaces and the compressed shift operator"


14:30--15:00: Coffee

15:00--16:00: Julius Kaplunov "Asymptotic analysis of thin walled transversely inhomogenenous structures"


Monday 9 January

10:00--10:30: Coffee

10:30--11:30: Gerardo Martin Franco "Levinson theorem for matrix-valued Schrödinger operators on the discrete line"

Abstract: We consider matrix-valued Schrödinger operators on the discrete line with a non-compactly supported perturbation whose first moment is assumed to exist. We derive explicit formulas for the scattering matrix and extend them to the band edges. We then prove a Levinson theorem, which establishes a relation between scattering data and spectral properties (bound and half-bound states) of the corresponding Hamiltonians.

11:30--12:30: Miguel Ballesteros "Scattering theory and resonances in quantum field theory"

Abstract: We touch upon the particularities of scattering theory in the context of Quantum Field Theory (QFT). Within this theoretical framework, we study scattering and resonances for some models of QFT.

12:30-13:30: Lunch at CIMAT

13:30--14:30: Yulia Ershova "Derivation of instantaneous frequencies of tectonic plates from gravimetric data"

Abstract: We present a mathematically rigorous procedure to obtain spectrograms of tectonic plate eigenmodes in an extremely low-frequency band, corresponding to oscillation periods of 15 min to 8 hours. The data is sourced from superconducting gravimeters (IGETS network). The motivation for this research stems from the proposal by Pavlov et al to use this spectral data in monitoring of the state of active zones located along plate boundaries.

14:30--15:30: Coffee and open research discussion



Kirill Cherednichenko
Alexander Kiselev
Yi Sheng Lim
Matthew Glenn Dawson
Rafael Herrera Guzmán
Adolfo Sánchez Valenzuela
Miguel Ballesteros
Gerardo Martin Franco
Luis Octavio Silva
Mikhail Cherdantsev (Cardiff)
Yulia Ershova (Texas A&M)
Julius Kaplunov (Keele)
Danila Prikazchikov (Keele)