Concentration inequalities for non-Lipschitz functions with bounded derivatives of higher order

Building on the inequalities for homogeneous tetrahedral polynomials in independent Gaussian variables due to R. Lata{\l}a we provide a concentration inequality for non-necessarily Lipschitz functions $f\colon \R^n \to \R$ with bounded derivatives of higher orders, which hold when the underlying measure satisfies a family of Sobolev type inequalities \|g- \E g\|_p \le C(p)\|\nabla g\|_p. Such Sobolev type inequalities hold, e.g., if the underlying measure satisfies the log-Sobolev inequality (in which case $C(p) \le C\sqrt{p}$) or the Poincar\'e inequality (then $C(p) \le Cp$). Our concentration estimates are expressed in terms of tensor-product norms of the derivatives of $f$. When the underlying measure is Gaussian and $f$ is a polynomial (non-necessarily tetrahedral or homogeneous), our estimates can be reversed (up to a constant depending only on the degree of the polynomial). We also show that for polynomial functions, analogous estimates hold for arbitrary random vectors with independent sub-Gaussian coordinates. We apply our inequalities to general additive functionals of random vectors (in particular linear eigenvalue statistics of random matrices) and the problem of counting cycles of fixed length in Erd\H{o}s-R{\'e}nyi random graphs, obtaining new estimates, optimal in a certain range of parameters

Coloring and Recognizing Directed Interval Graphs

A \emph{mixed interval graph} is an interval graph that has, for every pair of intersecting intervals, either an arc (directed arbitrarily) or an (undirected) edge. We are particularly interested in scenarios where edges and arcs are defined by the geometry of intervals. In a proper coloring of a mixed interval graph $G$, an interval $u$ receives a lower (different) color than an interval $v$ if $G$ contains arc $(u,v)$ (edge $\{u,v\}$). Coloring of mixed graphs has applications, for example, in scheduling with precedence constraints; see a survey by Sotskov [Mathematics, 2020]. For coloring general mixed interval graphs, we present a $\min \{\omega(G), \lambda(G)+1 \}$-approximation algorithm, where $\omega(G)$ is the size of a largest clique and $\lambda(G)$ is the length of a longest directed path in $G$. For the subclass of \emph{bidirectional interval graphs} (introduced recently for an application in graph drawing), we show that optimal coloring is NP-hard. This was known for general mixed interval graphs. We introduce a new natural class of mixed interval graphs, which we call \emph{containment interval graphs}. In such a graph, there is an arc $(u,v)$ if interval $u$ contains interval $v$, and there is an edge $\{u,v\}$ if $u$ and $v$ overlap. We show that these graphs can be recognized in polynomial time, that coloring them with the minimum number of colors is NP-hard, and that there is a 2-approximation algorithm for coloring.Comment: To appear in Proc. ISAAC 202

Hubbard Model with L\"uscher fermions

We study the basic features of the two-dimensional quantum Hubbard Model at half-filling by means of the L\"uscher algorithm and the algorithm based on direct update of the determinant of the fermionic matrix. We implement the L\"uscher idea employing the transfer matrix formalism which allows to formulate the problem on the lattice in $(2+1)$ dimensions. We discuss the numerical complexity of the L\"uscher technique, systematic errors introduced by polynomial approximation and introduce some improvements which reduce long autocorrelations. In particular we show that preconditioning of the fermionic matrix speeds up the algorithm and extends the available range of parameters. We investigate the magnetic and the one-particle properties of the Hubbard Model at half-filling and show that they are in qualitative agreement with the existing Monte Carlo data and the mean-field predictions.Comment: 49 pages, Latex, 11 Postscript figures, uses psfrag packag

Pathogenetics of alveolar capillary dysplasia with misalignment of pulmonary veins.

Alveolar capillary dysplasia with misalignment of pulmonary veins (ACDMPV) is a lethal lung developmental disorder caused by heterozygous point mutations or genomic deletion copy-number variants (CNVs) of FOXF1 or its upstream enhancer involving fetal lung-expressed long noncoding RNA genes LINC01081 and LINC01082. Using custom-designed array comparative genomic hybridization, Sanger sequencing, whole exome sequencing (WES), and bioinformatic analyses, we studied 22 new unrelated families (20 postnatal and two prenatal) with clinically diagnosed ACDMPV. We describe novel deletion CNVs at the FOXF1 locus in 13 unrelated ACDMPV patients. Together with the previously reported cases, all 31 genomic deletions in 16q24.1, pathogenic for ACDMPV, for which parental origin was determined, arose de novo with 30 of them occurring on the maternally inherited chromosome 16, strongly implicating genomic imprinting of the FOXF1 locus in human lungs. Surprisingly, we have also identified four ACDMPV families with the pathogenic variants in the FOXF1 locus that arose on paternal chromosome 16. Interestingly, a combination of the severe cardiac defects, including hypoplastic left heart, and single umbilical artery were observed only in children with deletion CNVs involving FOXF1 and its upstream enhancer. Our data demonstrate that genomic imprinting at 16q24.1 plays an important role in variable ACDMPV manifestation likely through long-range regulation of FOXF1 expression, and may be also responsible for key phenotypic features of maternal uniparental disomy 16. Moreover, in one family, WES revealed a de novo missense variant in ESRP1, potentially implicating FGF signaling in the etiology of ACDMPV