Page 472 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 472
CA18232: VARIATIONAL METHODS AND EQUATIONS ON GRAPHS (MS-40)
further assumptions. Our proofs are based on careful variational arguments and rearrangement
techniques, and hold both for the subcritical regime p ∈ (2, 6) and in the critical case p = 6.
This is a joint work with Enrico Serra and Paolo Tilli.
Convergence to equilibrium of stochastic semigroups and an application
to buffered networks flows
Jochen Glück, jochen.glueck@uni-passau.de
University of Passau, Germany
Coauthor: Florian Martin
The long-time behaviour of flows on finite metric graphs is known to depend heavily on the
network topology. Depending on this topology, the lengths of the edges and the flow velocities,
the flow might converge or behave asymptotically periodic as t → ∞.
In this talk we show that the situation changes if we introduce a mass buffer in at least one
of the vertices. Such a buffer has a smoothing effect on the flow and thus enforces convergence
as t → ∞. As we consider finite graphs only, one would even expect that the convergence is
uniform (i.e., with respect to the operator norm over the L1-space over the graph). In order to
prove that this is indeed true we employ a novel characterisation of operator norm convergence
to equilibrium for stochastic C0-semigroups.
Trace formulas for general Hermitian matrices: a scattering approach on
their associated graphs
Sven Gnutzmann, sven.gnutzmann@nottingham.ac.uk
School of Mathematical Sciences, University of Nottingham, United Kingdom
Coauthor: Uzy Smilansky
Two trace formulas for the spectra of arbitrary Hermitian matrices are presented. In either
case the one associates a unitary scattering matrix to the given Hermitian matrix H such that
the unitary matrix depends on the spectral parameter. In the first type the unitary matrix is
obtained by exponentiation. The new feature in this case is that the spectral parameter appears
in the final form as an argument of Eulerian polynomials—thus connecting the periodic orbits to
combinatorial objects in a novel way. To obtain the second type, one expresses the input in terms
of a unitary scattering matrix in a larger Hilbert space. One of the surprising features here is
that the locations and radii of the spectral discs of Gershgorin’s theorem appear naturally as the
pole parameters of the scattering matrix. Both formulas are discussed and possible applications
are outlined.
470
further assumptions. Our proofs are based on careful variational arguments and rearrangement
techniques, and hold both for the subcritical regime p ∈ (2, 6) and in the critical case p = 6.
This is a joint work with Enrico Serra and Paolo Tilli.
Convergence to equilibrium of stochastic semigroups and an application
to buffered networks flows
Jochen Glück, jochen.glueck@uni-passau.de
University of Passau, Germany
Coauthor: Florian Martin
The long-time behaviour of flows on finite metric graphs is known to depend heavily on the
network topology. Depending on this topology, the lengths of the edges and the flow velocities,
the flow might converge or behave asymptotically periodic as t → ∞.
In this talk we show that the situation changes if we introduce a mass buffer in at least one
of the vertices. Such a buffer has a smoothing effect on the flow and thus enforces convergence
as t → ∞. As we consider finite graphs only, one would even expect that the convergence is
uniform (i.e., with respect to the operator norm over the L1-space over the graph). In order to
prove that this is indeed true we employ a novel characterisation of operator norm convergence
to equilibrium for stochastic C0-semigroups.
Trace formulas for general Hermitian matrices: a scattering approach on
their associated graphs
Sven Gnutzmann, sven.gnutzmann@nottingham.ac.uk
School of Mathematical Sciences, University of Nottingham, United Kingdom
Coauthor: Uzy Smilansky
Two trace formulas for the spectra of arbitrary Hermitian matrices are presented. In either
case the one associates a unitary scattering matrix to the given Hermitian matrix H such that
the unitary matrix depends on the spectral parameter. In the first type the unitary matrix is
obtained by exponentiation. The new feature in this case is that the spectral parameter appears
in the final form as an argument of Eulerian polynomials—thus connecting the periodic orbits to
combinatorial objects in a novel way. To obtain the second type, one expresses the input in terms
of a unitary scattering matrix in a larger Hilbert space. One of the surprising features here is
that the locations and radii of the spectral discs of Gershgorin’s theorem appear naturally as the
pole parameters of the scattering matrix. Both formulas are discussed and possible applications
are outlined.
470
