Two Approaches to Sidorenko's Conjecture

Sidorenko's conjecture states that for every bipartite graph $H$ on $\{1,\cdots,k\}$, $\int \prod_{(i,j)\in E(H)} h(x_i, y_j) d\mu^{|V(H)|} \ge \left( \int h(x,y) \,d\mu^2 \right)^{|E(H)|}$ holds, where $\mu$ is the Lebesgue measure on $[0,1]$ and $h$ is a bounded, non-negative, symmetric, measurable function on $[0,1]^2$. An equivalent discrete form of the conjecture is that the number of homomorphisms from a bipartite graph $H$ to a graph $G$ is asymptotically at least the expected number of homomorphisms from $H$ to the Erd\H{o}s-R\'{e}nyi random graph with the same expected edge density as $G$. In this paper, we present two approaches to the conjecture. First, we introduce the notion of tree-arrangeability, where a bipartite graph $H$ with bipartition $A \cup B$ is tree-arrangeable if neighborhoods of vertices in $A$ have a certain tree-like structure. We show that Sidorenko's conjecture holds for all tree-arrangeable bipartite graphs. In particular, this implies that Sidorenko's conjecture holds if there are two vertices $a_1, a_2$ in $A$ such that each vertex $a \in A$ satisfies $N(a) \subseteq N(a_1)$ or $N(a) \subseteq N(a_2)$, and also implies a recent result of Conlon, Fox, and Sudakov \cite{CoFoSu}. Second, if $T$ is a tree and $H$ is a bipartite graph satisfying Sidorenko's conjecture, then it is shown that the Cartesian product $T \Box H$ of $T$ and $H$ also satisfies Sidorenko's conjecture. This result implies that, for all $d \ge 2$, the $d$-dimensional grid with arbitrary side lengths satisfies Sidorenko's conjecture.Comment: 20 pages, 2 figure

Sidorenko's conjecture for higher tree decompositions

This is a companion note to our paper 'Some advances on Sidorenko's conjecture', elaborating on a remark in that paper that the approach which proves Sidorenko's conjecture for strongly tree-decomposable graphs may be extended to a broader class, comparable to that given in work of Szegedy, through further iteration.Comment: 7 pages, unpublished not

Probing TeV scale Top-Philic Resonances with Boosted Top-Tagging at the High Luminosity LHC

We investigate the discovery potential of singly produced top-philic resonances at the high luminosity (HL) LHC in the four-top final state. Our analysis spans over the fully-hadronic, semi-leptonic, and same-sign dilepton channels where we present concrete search strategies adequate to a boosted kinematic regime and high jet-multiplicity environments. We utilize the Template Overlap Method (TOM) with newly developed template observables for tagging boosted top quarks, a large-radius jet variable $M_J$ and customized b-tagging tactics for background discrimination. Our results show that the same-sign dilepton channel gives the best sensitivity among the considered channels, with an improvement of significance up to 10%-20% when combined with boosted-top tagging. Both the fully-hadronic and semi-leptonic channels yield comparable discovery potential and contribute to further enhancements in the sensitivity by combining all channels. Finally, we show the sensitivity of a top-philic resonance at the LHC and HL-LHC by showing the $2\sigma$ exclusion limit and $5\sigma$ discovery reach, including a combination of all three channels.Comment: 19 pages, 14 figure

Boosted Event Topologies from TeV Scale Light Quark Composite Partners

We propose a new search strategy for quark partners which decay into a boosted Higgs and a light quark. As an example, we consider phenomenologically viable right handed up-type quark partners of mass $\sim 1$ TeV in composite pseudo-Nambu-Goldstone-boson Higgs models within the context of flavorful naturalness. Our results show that $S/B > 1$ and signal significance of $\sim 7\sigma$ is achievable at $\sqrt{s} = 14$ TeV LHC with 35 $fb^{-1}$ of integrated luminosity, sufficient to claim discovery of a new particle. A combination of a multi-dimensional boosted Higgs tagging technique, kinematics of pair produced heavy objects and $b$-tagging serves to efficiently diminish the large QCD backgrounds while maintaining adequate levels of signal efficiency. We present the analysis in the context of effective field theory, such that our results can be applied to any future search for pair produced vector-like quarks with decay modes to Higgs and a light jet.Comment: 18 pages, 7 figures, 5 tables, v2: short discussion added in Sec. 2, references added, corresponds to version published in JHE

Processing SPARQL queries with regular expressions in RDF databases

Background: As the Resource Description Framework (RDF) data model is widely used for modeling and sharing a lot of online bioinformatics resources such as Uniprot (dev.isb-sib.ch/projects/uniprot-rdf) or Bio2RDF (bio2rdf.org), SPARQL - a W3C recommendation query for RDF databases - has become an important query language for querying the bioinformatics knowledge bases. Moreover, due to the diversity of users' requests for extracting information from the RDF data as well as the lack of users' knowledge about the exact value of each fact in the RDF databases, it is desirable to use the SPARQL query with regular expression patterns for querying the RDF data. To the best of our knowledge, there is currently no work that efficiently supports regular expression processing in SPARQL over RDF databases. Most of the existing techniques for processing regular expressions are designed for querying a text corpus, or only for supporting the matching over the paths in an RDF graph. Results: In this paper, we propose a novel framework for supporting regular expression processing in SPARQL query. Our contributions can be summarized as follows. 1) We propose an efficient framework for processing SPARQL queries with regular expression patterns in RDF databases. 2) We propose a cost model in order to adapt the proposed framework in the existing query optimizers. 3) We build a prototype for the proposed framework in C++ and conduct extensive experiments demonstrating the efficiency and effectiveness of our technique. Conclusions: Experiments with a full-blown RDF engine show that our framework outperforms the existing ones by up to two orders of magnitude in processing SPARQL queries with regular expression patterns.X113sciescopu