Cores with distinct parts and bigraded Fibonacci numbers

The notion of $(a,b)$-cores is closely related to rational $(a,b)$ Dyck paths due to Anderson's bijection, and thus the number of $(a,a+1)$-cores is given by the Catalan number $C_a$. Recent research shows that $(a,a+1)$ cores with distinct parts are enumerated by another important sequence- Fibonacci numbers $F_a$. In this paper, we consider the abacus description of $(a,b)$-cores to introduce the natural grading and generalize this result to $(a,as+1)$-cores. We also use the bijection with Dyck paths to count the number of $(2k-1,2k+1)$-cores with distinct parts. We give a second grading to Fibonacci numbers, induced by bigraded Catalan sequence $C_{a,b} (q,t)$

Crystal analysis of type CC Stanley symmetric functions

Combining results of T.K. Lam and J. Stembridge, the type $C$ Stanley symmetric function $F_w^C(\mathbf{x})$, indexed by an element $w$ in the type $C$ Coxeter group, has a nonnegative integer expansion in terms of Schur functions. We provide a crystal theoretic explanation of this fact and give an explicit combinatorial description of the coefficients in the Schur expansion in terms of highest weight crystal elements.Comment: 39 page

Estimating Graphlet Statistics via Lifting

Exploratory analysis over network data is often limited by the ability to efficiently calculate graph statistics, which can provide a model-free understanding of the macroscopic properties of a network. We introduce a framework for estimating the graphlet count---the number of occurrences of a small subgraph motif (e.g. a wedge or a triangle) in the network. For massive graphs, where accessing the whole graph is not possible, the only viable algorithms are those that make a limited number of vertex neighborhood queries. We introduce a Monte Carlo sampling technique for graphlet counts, called {\em Lifting}, which can simultaneously sample all graphlets of size up to $k$ vertices for arbitrary $k$. This is the first graphlet sampling method that can provably sample every graphlet with positive probability and can sample graphlets of arbitrary size $k$. We outline variants of lifted graphlet counts, including the ordered, unordered, and shotgun estimators, random walk starts, and parallel vertex starts. We prove that our graphlet count updates are unbiased for the true graphlet count and have a controlled variance for all graphlets. We compare the experimental performance of lifted graphlet counts to the state-of-the art graphlet sampling procedures: Waddling and the pairwise subgraph random walk

Thz range low-noise sis receivers for space and ground-based radio astronomy

We report on research in the field of low-noise receiving systems in the sub-terahertz (THz) range, carried out in recent years, aimed at developing receivers with quantum sensitivity for implementation in space and ground-based radio telescopes. Superconductor-Insulator-Superconductor (SIS) mixers based on high-quality tunnel junctions are the key elements of the most sensitive sub-THz heterodyne receivers. Motivations and physical background for technology improvement and optimization, as well as fabrication details, are described. This article presents the results of the SIS receiver developments for the 211–275 GHz and 790–950 GHz frequency ranges with a noise temperature in the double sideband (DSB) mode of approximTELY 20 K and 200 K, respectively. These designs and achievements are implemented in the development of the receiving systems for the Russian Space Agency mission “Millimetron”, and for the ground-based APEX (Atacama Pathfinder EXperiment) telescope

Search for dark matter produced in association with bottom or top quarks in √s = 13 TeV pp collisions with the ATLAS detector

A search for weakly interacting massive particle dark matter produced in association with bottom or top quarks is presented. Final states containing third-generation quarks and miss- ing transverse momentum are considered. The analysis uses 36.1 fb−1 of proton–proton collision data recorded by the ATLAS experiment at √s = 13 TeV in 2015 and 2016. No significant excess of events above the estimated backgrounds is observed. The results are in- terpreted in the framework of simplified models of spin-0 dark-matter mediators. For colour- neutral spin-0 mediators produced in association with top quarks and decaying into a pair of dark-matter particles, mediator masses below 50 GeV are excluded assuming a dark-matter candidate mass of 1 GeV and unitary couplings. For scalar and pseudoscalar mediators produced in association with bottom quarks, the search sets limits on the production cross- section of 300 times the predicted rate for mediators with masses between 10 and 50 GeV and assuming a dark-matter mass of 1 GeV and unitary coupling. Constraints on colour- charged scalar simplified models are also presented. Assuming a dark-matter particle mass of 35 GeV, mediator particles with mass below 1.1 TeV are excluded for couplings yielding a dark-matter relic density consistent with measurements