21 research outputs found
On Some Open Problems in Many-Electron Theory
Mel Levy and Elliott Lieb are two of the most prominent researchers who have
dedicated their efforts to the investigation of fundamental questions in
many-electron theory. Their results have not only revolutionized the
theoretical approach of the field, but, directly or indirectly, allowed for a
quantum jump in the computational treatment of realistic systems as well. For
this reason, at the conclusion of our book where the subject is treated across
different disciplines, we have asked Mel Levy and Elliott Lieb to provide us
with some open problems, which they believe will be a worth challenge for the
future also in the perspective of a synergy among the various disciplines.Comment: "Epilogue" chapter in "Many-Electron Approaches in Physics, Chemistry
and Mathematics: A Multidisciplinary View", Volker Bach and Luigi Delle Site
Eds. pages 411-416; Book Series: Mathematical Physics Studies, Springer
International Publishing Switzerland, 2014. The original title has been
modified in order to clarify the subject of the chapter out of the context of
the boo
The Evolution of Travelling Waves in a KPP Reaction-Diffusion Model with Cut-off Reaction Rate. I. Permanent Form Travelling Waves
We consider Kolmogorov--Petrovskii--Piscounov (KPP) type models in the
presence of a discontinuous cut-off in reaction rate at concentration .
In Part I we examine permanent form travelling wave solutions (a companion
paper, Part II, is devoted to their evolution in the large time limit). For
each fixed cut-off value , we prove the existence of a unique
permanent form travelling wave with a continuous and monotone decreasing
propagation speed . We extend previous asymptotic results in the
limit of small and present new asymptotic results in the limit of large
which are respectively obtained via the systematic use of matched and
regular asymptotic expansions. The asymptotic results are confirmed against
numerical results obtained for the particular case of a cut-off Fisher reaction
function
Isoperimetric inequalities for some integral operators arising in potential theory
In this paper we review our previous isoperimetric results for the
logarithmic potential and Newton potential operators. The main reason why the
results are useful, beyond the intrinsic interest of geometric extremum
problems, is that they produce a priori bounds for spectral invariants of
operators on arbitrary domains. We demonstrate these in explicit examples.Comment: This conference paper gives a review of our previous results in the
subjec
Attractor bifurcation for the extended Fisher-Kolmogorov equation with periodic boundary condition
Variational Characterization of the Speed of Reaction Diffusion Fronts for Gradient Dependent Diffusion
On first and second eigenvalues of Riesz transforms in spherical and hyperbolic geometries
This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were madeTo appear in Bull. Math. SciThe authors were supported in parts by the EPSRC Grant EP/K039407/1 and by the Leverhulme Grant RPG-2014-02, as well as by the MESRK Grant 5127/GF4
Sharp Fractional Hardy Inequalities in Half-Spaces
We determine the sharp constant in the Hardy inequality for fractional
Sobolev spaces on half-spaces. Our proof relies on a non-linear and non-local
version of the ground state representation.Comment: 6 pages; dedicated to V. G. Maz'y