1,115 research outputs found
Global well-posedness for a slightly supercritical surface quasi-geostrophic equation
We use a nonlocal maximum principle to prove the global existence of smooth
solutions for a slightly supercritical surface quasi-geostrophic equation. By
this we mean that the velocity field is obtained from the active scalar
by a Fourier multiplier with symbol , where
is a smooth increasing function that grows slower than as
.Comment: 11 pages, second version with slightly stronger resul
Regularity and blow up for active scalars
We review some recent results for a class of fluid mechanics equations called
active scalars, with fractional dissipation. Our main examples are the surface
quasi-geostrophic equation, the Burgers equation, and the
Cordoba-Cordoba-Fontelos model. We discuss nonlocal maximum principle methods
which allow to prove existence of global regular solutions for the critical
dissipation. We also recall what is known about the possibility of finite time
blow up in the supercritical regime.Comment: 33 page
Shape-enforcing operators for point and interval estimators
https://arxiv.org/abs/1809.01038https://arxiv.org/abs/1809.01038First author draf
Pairwise-Stability and Nash Equilibria in Network Formation
Suppose that individual payoffs depend on the network connecting them. Consider the following simultaneous move game of network formation: players announce independently the links they wish to form, and links are formed only under mutual consent. We provide necessary and sufficient conditions on the network link marginal payoffs such that the set of pairwise stable, pairwise-Nash and proper equilibrium networks coincide, where pairwise stable networks are robust to one-link deviations, while pairwise-Nash networks are robust to one-link creation but multi-link severance. Under these conditions, proper equilibria in pure strategies are fully characterized by one-link deviation checks.Network formation, Pairwise-stability, Proper equilibrium
Different quantum f-divergences and the reversibility of quantum operations
The concept of classical -divergences gives a unified framework to
construct and study measures of dissimilarity of probability distributions;
special cases include the relative entropy and the R\'enyi divergences. Various
quantum versions of this concept, and more narrowly, the concept of R\'enyi
divergences, have been introduced in the literature with applications in
quantum information theory; most notably Petz' quasi-entropies (standard
-divergences), Matsumoto's maximal -divergences, measured
-divergences, and sandwiched and --R\'enyi divergences.
In this paper we give a systematic overview of the various concepts of
quantum -divergences with a main focus on their monotonicity under quantum
operations, and the implications of the preservation of a quantum
-divergence by a quantum operation. In particular, we compare the standard
and the maximal -divergences regarding their ability to detect the
reversibility of quantum operations. We also show that these two quantum
-divergences are strictly different for non-commuting operators unless
is a polynomial, and obtain some analogous partial results for the relation
between the measured and the standard -divergences.
We also study the monotonicity of the --R\'enyi divergences under
the special class of bistochastic maps that leave one of the arguments of the
R\'enyi divergence invariant, and determine domains of the parameters
where monotonicity holds, and where the preservation of the
--R\'enyi divergence implies the reversibility of the quantum
operation.Comment: 70 pages. v4: New Proposition 3.8 and Appendix D on the continuity
properties of the standard f-divergences. The 2-positivity assumption removed
from Theorem 3.34. The achievability of the measured f-divergence is shown in
Proposition 4.17, and Theorem 4.18 is updated accordingl
Uniqueness of Coalitional Equilibria
We provide an existence and a uniqueness result for coalitional equilibria of a game in strategic form. Both results are illustrated for a public good game and a homogeneous Cournot-oligopoly game.Existence and uniqueness of coalitional equilibrium, Game in strategic form
- âŠ