Uniform estimates for polyharmonic Green functions in domains with small holes

We prove a pointwise control for the Green's function of polyharmonic operators with holes: this control is uniform while holes shrink. For the usual Laplacian, such a control is given by the maximum principle; the techniques developed here applies to general polyharmonic operators for which there is no comparison principle

Optimal estimates from below for biharmonic Green functions

Optimal pointwise estimates are derived for the biharmonic Green function under Dirichlet boundary conditions in arbitrary $C^{4,\gamma}$-smooth domains. Maximum principles do not exist for fourth order elliptic equations and the Green function may change sign. It prevents using a Harnack inequality as for second order problems and hence complicates the derivation of optimal estimates. The present estimate is obtained by an asymptotic analysis. The estimate shows that this Green function is positive near the singularity and that a possible negative part is small in the sense that it is bounded by the product of the squared distances to the boundary.Comment: 11 pages. To appear in "Proceedings of the AMS

Adapting kk-means algorithms for outliers

This paper shows how to adapt several simple and classical sampling-based algorithms for the $k$-means problem to the setting with outliers. Recently, Bhaskara et al. (NeurIPS 2019) showed how to adapt the classical $k$-means++ algorithm to the setting with outliers. However, their algorithm needs to output $O(\log (k) \cdot z)$ outliers, where $z$ is the number of true outliers, to match the $O(\log k)$-approximation guarantee of $k$-means++. In this paper, we build on their ideas and show how to adapt several sequential and distributed $k$-means algorithms to the setting with outliers, but with substantially stronger theoretical guarantees: our algorithms output $(1+\varepsilon)z$ outliers while achieving an $O(1 / \varepsilon)$-approximation to the objective function. In the sequential world, we achieve this by adapting a recent algorithm of Lattanzi and Sohler (ICML 2019). In the distributed setting, we adapt a simple algorithm of Guha et al. (IEEE Trans. Know. and Data Engineering 2003) and the popular $k$-means$\|$ of Bahmani et al. (PVLDB 2012). A theoretical application of our techniques is an algorithm with running time $\tilde{O}(nk^2/z)$ that achieves an $O(1)$-approximation to the objective function while outputting $O(z)$ outliers, assuming $k \ll z \ll n$. This is complemented with a matching lower bound of $\Omega(nk^2/z)$ for this problem in the oracle model

eL-DASionator: an LDAS upload file generator

BACKGROUND: The Distributed Annotation System (DAS) allows merging of DNA sequence annotations from multiple sources and provides a single annotation view. A straightforward way to establish a DAS annotation server is to use the "Lightweight DAS" server (LDAS). Onto this type of server, annotations can be uploaded as flat text files in a defined format. The popular Ensembl ContigView uses the same format for the transient upload and display of user data. RESULTS: In order to easily generate LDAS upload files we developed a software tool that is accessible via a web-interface . Users can submit their DNA sequences of interest. Our program (i) aligns these sequences to the reference sequences of Ensembl, (ii) determines start and end positions of each sequence on the reference sequence, and (iii) generates a formatted annotation file. This file can be used to load any LDAS annotation server or it can be uploaded to the Ensembl ContigView. CONCLUSION: The eL-DASionator is an on-line tool that is intended for life-science researchers with little bioinformatics background. It conveniently generates LDAS upload files, and makes it possible to generate annotations in a standard format that permits comfortable sharing of this data

Faster Deterministic Distributed MIS and Approximate Matching

$\renewcommand{\tilde}{\widetilde}$We present an $\tilde{O}(\log^2 n)$ round deterministic distributed algorithm for the maximal independent set problem. By known reductions, this round complexity extends also to maximal matching, $\Delta+1$ vertex coloring, and $2\Delta-1$ edge coloring. These four problems are among the most central problems in distributed graph algorithms and have been studied extensively for the past four decades. This improved round complexity comes closer to the $\tilde{\Omega}(\log n)$ lower bound of maximal independent set and maximal matching [Balliu et al. FOCS '19]. The previous best known deterministic complexity for all of these problems was $\Theta(\log^3 n)$. Via the shattering technique, the improvement permeates also to the corresponding randomized complexities, e.g., the new randomized complexity of $\Delta+1$ vertex coloring is now $\tilde{O}(\log^2\log n)$ rounds. Our approach is a novel combination of the previously known two methods for developing deterministic algorithms for these problems, namely global derandomization via network decomposition (see e.g., [Rozhon, Ghaffari STOC'20; Ghaffari, Grunau, Rozhon SODA'21; Ghaffari et al. SODA'23]) and local rounding of fractional solutions (see e.g., [Fischer DISC'17; Harris FOCS'19; Fischer, Ghaffari, Kuhn FOCS'17; Ghaffari, Kuhn FOCS'21; Faour et al. SODA'23]). We consider a relaxation of the classic network decomposition concept, where instead of requiring the clusters in the same block to be non-adjacent, we allow each node to have a small number of neighboring clusters. We also show a deterministic algorithm that computes this relaxed decomposition faster than standard decompositions. We then use this relaxed decomposition to significantly improve the integrality of certain fractional solutions, before handing them to the local rounding procedure that now has to do fewer rounding steps

The Dirichlet problem for supercritical biharmonic equations with power-type nonlinearity

AbstractFor a semilinear biharmonic Dirichlet problem in the ball with supercritical power-type nonlinearity, we study existence/nonexistence, regularity and stability of radial positive minimal solutions. Moreover, qualitative properties, and in particular the precise asymptotic behaviour near x=0 for (possibly existing) singular radial solutions, are deduced. Dynamical systems arguments and a suitable Lyapunov (energy) function are employed