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

boostorg/charconv: Boost 1.85.0

<h2>What's Changed</h2> <ul> <li>fix typo in fast_float.hpp by @akrzemi1 in https://github.com/boostorg/charconv/pull/135</li> <li>Add fuzzing to CI by @mborland in https://github.com/boostorg/charconv/pull/134</li> <li>Conditionally link quadmath by @mborland in https://github.com/boostorg/charconv/pull/132</li> <li>Add testing and fix handling of leading spaces by @mborland in https://github.com/boostorg/charconv/pull/144</li> <li>Reverse behavior of from_chars and from_chars_strict by @mborland in https://github.com/boostorg/charconv/pull/141</li> <li>Collected documentation updates from review comments by @mborland in https://github.com/boostorg/charconv/pull/145</li> <li>Post-Review Refactoring by @mborland in https://github.com/boostorg/charconv/pull/146</li> <li>Add string_view interface to from_chars by @mborland in https://github.com/boostorg/charconv/pull/147</li> <li>Fix long double and Float128 negative sign being overwritten by @mborland in https://github.com/boostorg/charconv/pull/150</li> <li>Fix to_chars_hex exp for Float128 types by @mborland in https://github.com/boostorg/charconv/pull/151</li> <li>Add dragonbox, floff and Junekey Jeon to sources and acknowledgements by @mborland in https://github.com/boostorg/charconv/pull/155</li> <li>Add cap to output lenght of <code>to_chars</code> for <code>std::bfloat16_t</code> with interchange format by @mborland in https://github.com/boostorg/charconv/pull/157</li> <li>Add additional bounds checking in <code>to_chars</code> for floats by @mborland in https://github.com/boostorg/charconv/pull/153</li> <li>Fix 80-bit long double hex formatting by @mborland in https://github.com/boostorg/charconv/pull/159</li> <li>Add fixed format printing to floff by @mborland in https://github.com/boostorg/charconv/pull/161</li> <li>Use floff to consistently generate correctly rounded results by @mborland in https://github.com/boostorg/charconv/pull/164</li> <li>Fix to_chars errc value for overflow by @mborland in https://github.com/boostorg/charconv/pull/167</li> <li>Try to fix #158 by @jk-jeon in https://github.com/boostorg/charconv/pull/169</li> <li>Fix for issue #166 negative precision by @mborland in https://github.com/boostorg/charconv/pull/168</li> <li>Issue 158 for 80 and 128 bit types by @mborland in https://github.com/boostorg/charconv/pull/171</li> </ul> <h2>New Contributors</h2> <ul> <li>@akrzemi1 made their first contribution in https://github.com/boostorg/charconv/pull/135</li> <li>@jk-jeon made their first contribution in https://github.com/boostorg/charconv/pull/169</li> </ul> <p><strong>Full Changelog</strong>: https://github.com/boostorg/charconv/compare/v1.2.0...boost-1.85.0</p&gt