3 research outputs found
A Generalized Typicality for Abstract Alphabets
A new notion of typicality for arbitrary probability measures on standard
Borel spaces is proposed, which encompasses the classical notions of weak and
strong typicality as special cases. Useful lemmas about strong typical sets,
including conditional typicality lemma, joint typicality lemma, and packing and
covering lemmas, which are fundamental tools for deriving many inner bounds of
various multi-terminal coding problems, are obtained in terms of the proposed
notion. This enables us to directly generalize lots of results on finite
alphabet problems to general problems involving abstract alphabets, without any
complicated additional arguments. For instance, quantization procedure is no
longer necessary to achieve such generalizations. Another fundamental lemma,
Markov lemma, is also obtained but its scope of application is quite limited
compared to others. Yet, an alternative theory of typical sets for Gaussian
measures, free from this limitation, is also developed. Some remarks on a
possibility to generalize the proposed notion for sources with memory are also
given.Comment: 44 pages; submitted to IEEE Transactions on Information Theor
On evolution of corner-like gSQG patches
We study the evolution of corner-like patch solutions to the generalized SQG
equations. Depending on the angle size and order of the velocity kernel, the
corner instantaneously bents either downward or upward. In particular, we
obtain the existence of strictly convex and smooth patch solutions which become
immediately non-convex.Comment: 10 pages, 3 figure
An Improved Regularity Criterion and Absence of Splash-like Singularities for g-SQG Patches
We prove that splash-like singularities cannot occur for sufficiently regular
patch solutions to the generalized surface quasi-geostrophic equation on the
plane or half-plane with parameter . This includes
potential touches of more than two patch boundary segments in the same
location, an eventuality that has not been excluded previously and presents
nontrivial complications (in fact, if we do a priori exclude it, then our
results extend to all ). As a corollary, we obtain an improved
global regularity criterion for patch solutions when ,
namely that finite time singularities cannot occur while the norms of
patch boundaries remain bounded