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 $\alpha\le \frac 14$. 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 $\alpha\in(0,1)$). As a corollary, we obtain an improved global regularity criterion for $H^3$ patch solutions when $\alpha\le\frac 14$, namely that finite time singularities cannot occur while the $H^3$ norms of patch boundaries remain bounded