On LL-close Sperner systems

For a set $L$ of positive integers, a set system $\mathcal{F} \subseteq 2^{[n]}$ is said to be $L$-close Sperner, if for any pair $F,G$ of distinct sets in $\mathcal{F}$ the skew distance $sd(F,G)=\min\{|F\setminus G|,|G\setminus F|\}$ belongs to $L$. We reprove an extremal result of Boros, Gurvich, and Milani\v c on the maximum size of $L$-close Sperner set systems for $L=\{1\}$ and generalize to $|L|=1$ and obtain slightly weaker bounds for arbitrary $L$. We also consider the problem when $L$ might include 0 and reprove a theorem of Frankl, F\"uredi, and Pach on the size of largest set systems with all skew distances belonging to $L=\{0,1\}$

Ubiquitination of tombusvirus p33 replication protein plays a role in virus replication and binding to the host Vps23p ESCRT protein

AbstractPost-translational modifications of viral replication proteins could be widespread phenomena during the replication of plus-stranded RNA viruses. In this article, we identify two lysines in the tombusvirus p33 replication co-factor involved in ubiquitination and show that the same lysines are also important for the p33 to interact with the host Vps23p ESCRT-I factor. We find that the interaction of p33 with Vps23p is also affected by a “late-domain”-like sequence in p33. The combined mutations of the two lysines and the late-domain-like sequences in p33 reduced replication of a replicon RNA of Tomato bushy stunt virus in yeast model host, in plant protoplasts, and plant leaves, suggesting that p33-Vps23p ESCRT protein interaction affects tombusvirus replication. Using ubiquitin-mimicking p33 chimeras, we demonstrate that high level of p33 ubiquitination is inhibitory for TBSV replication. These findings argue that optimal level of p33 ubiquitination plays a regulatory role during tombusvirus infections

The asymptotic geometry of G2\rm G_2-monopoles

This article investigates the asymptotics of $\rm G_2$-monopoles. First, we prove that when the underlying $\rm G_2$-manifold has polynomial volume growth strictly faster than $r^{7/2}$, finite intermediate energy monopoles with bounded curvature have finite mass. The second main result restricts to the case when the underlying $\rm G_2$-manifold is asymptotically conical. In this situation, we deduce sharp decay estimates and that the connection converges, along the end, to a pseudo-Hermitian--Yang--Mills connection over the asymptotic cone. Finally, our last result exhibits a Fredholm setup describing the moduli space of finite intermediate energy monopoles on an asymptotically conical $\rm G_2$-manifold.Comment: 80 pages. v2: added 4 new sections, new results, including a third main result; previous sections fully revised, exposition improved, corrected typos and reworked some proof