A graph theoretic characterization of the classical generalized hexagon on 364364 vertices

A tetravalent $2$-arc-transitive graph of order $728$ is either the known $7$-arc-transitive incidence graph of the classical generalized hexagon $GH(3,3)$ or a normal cover of a $2$-transitive graph of order $182$ denoted $A[182,1]$ or $A[182,2]$ in the $2009$ list of Poto\v{c}nik.Comment: 11 page

Set-based differential covariance testing for high-throughput data

The problem of detecting changes in covariance for a single pair of features has been studied in some detail, but may be limited in importance or general applicability. In contrast, testing equality of covariance matrices of a {\it set} of features may offer increased power and interpretability. Such approaches have received increasing attention in recent years, especially in the context of high-dimensional testing. These approaches have been limited to the two-sample problem and involve varying assumptions on the number of features $p$ vs. the sample size $n$. In addition, there has been little discussion of the motivating principles underlying various choices of statistic, and no general approaches to test association of covariances with a continuous outcome. We propose a uniform framework to test association of covariance matrices with an experimental variable, whether discrete or continuous. We describe four different summary statistics, to ensure power and flexibility under various settings, including a new "connectivity" statistic that is sensitive to changes in overall covariance magnitude. The approach is not limited by the data dimensions, and is applicable to situations where $p >> n$. For several statistics we obtain asymptotic $p$-values under relatively mild conditions. For the two-sample special case, we show that the proposed statistics are permutationally equivalent or similar to existing proposed statistics. We demonstrate the power and utility of our approaches via simulation and analysis of real data.Comment: arXiv admin note: substantial text overlap with arXiv:1609.0073

Estimation And Selection Via Absolute Penalized Convex Minimization And Its Multistage Adaptive Applications

The $\ell_1$-penalized method, or the Lasso, has emerged as an important tool for the analysis of large data sets. Many important results have been obtained for the Lasso in linear regression which have led to a deeper understanding of high-dimensional statistical problems. In this article, we consider a class of weighted $\ell_1$-penalized estimators for convex loss functions of a general form, including the generalized linear models. We study the estimation, prediction, selection and sparsity properties of the weighted $\ell_1$-penalized estimator in sparse, high-dimensional settings where the number of predictors $p$ can be much larger than the sample size $n$. Adaptive Lasso is considered as a special case. A multistage method is developed to apply an adaptive Lasso recursively. We provide $\ell_q$ oracle inequalities, a general selection consistency theorem, and an upper bound on the dimension of the Lasso estimator. Important models including the linear regression, logistic regression and log-linear models are used throughout to illustrate the applications of the general results

Communities of solutions in single solution clusters of a random K-Satisfiability formula

The solution space of a K-satisfiability (K-SAT) formula is a collection of solution clusters, each of which contains all the solutions that are mutually reachable through a sequence of single-spin flips. Knowledge of the statistical property of solution clusters is valuable for a complete understanding of the solution space structure and the computational complexity of the random K-SAT problem. This paper explores single solution clusters of random 3- and 4-SAT formulas through unbiased and biased random walk processes and the replica-symmetric cavity method of statistical physics. We find that the giant connected component of the solution space has already formed many different communities when the constraint density of the formula is still lower than the solution space clustering transition point. Solutions of the same community are more similar with each other and more densely connected with each other than with the other solutions. The entropy density of a solution community is calculated using belief propagation and is found to be different for different communities of the same cluster. When the constraint density is beyond the clustering transition point, the same behavior is observed for the solution clusters reached by several stochastic search algorithms. Taking together, the results of this work suggests a refined picture on the evolution of the solution space structure of the random K-SAT problem; they may also be helpful for designing new heuristic algorithms.Comment: Extensively revised and expanded into 15 pages with 10 figures. New mean-field calculations and simulation results adde

Where to Find Habitable "Earths" in Circumbinary Systems

Hitherto, six P-type planets are found around five binary systems, i.e. Kepler-16 b, 34 b, 35 b, 38 b, 47 b, c, which are all Neptune or Jupiter-like planets. The stability of planets and the habitable zones are influenced by the gravitational and radiative perturbations of binary companions. In this Letter, we check the stability of an additional habitable Earth-mass planet in each system. Based on our simulations in 10 Myr, a habitable "Earth" is hardly stable in Kepler-16 while a stable "Earth" in Kepler-47 close to the boundaries of the habitable zone is possible. In contrast, Kepler-34, 35 and 38 seem to have high probabilities of being able to tolerate a stable "Earth" in their habitable zones. The affects of transit time variations are quite small due to the small mass of an undetected "Earth", except that of Kepler-16 b. With a time precision of 0.001 day(~ 88 s), an "Earth" in the corotational resonance with Kepler-16 b can be detected in 3 years, while habitable "Earths" in Kepler-34 and 38 systems can be detected in 10 years. Habitable "Earths" in Kepler-35 and 47 are not likely to be detected in 10 years under this precision.Comment: 7 pages, 3 figures, 2 table

An Explicit Cross Entropy Scheme for Mixtures

The key issue in importance sampling is the choice of the alternative sampling distribution, which is often chosen from the exponential tilt family of the underlying distribution. However, when the problem exhibits certain kind of nonconvexity, it is very likely that a single exponential change of measure will never attain asymptotic optimality and can lead to erroneous estimates. In this paper we introduce an explicit iterative scheme which combines the traditional cross-entropy method and the EM algorithm to find an efficient alternative sampling distribution in the form of mixtures. We also study the applications of this scheme to the estimation of rainbow option prices

Topological Superconductivity Intertwined with Broken Symmetries

Recently the superconductor and topological semimetal PbTaSe$_2$ was experimentally found to exhibit surface-only lattice rotational symmetry breaking below $T_c$. We exploit the Ginzburg-Landau free energy and propose a microscopic two-channel model to study possible superconducting states on the surface of PbTaSe$_2$. We identify two types of topological superconducting states. One is time-reversal invariant and preserves the lattice hexagonal symmetry while the other breaks both symmetries. We find that such time-reversal symmetry breaking is unavoidable for a superconducting state in a two dimensional irreducible representation of crystal point group in a system where the spatial inversion symmetry is broken and the strong spin-orbit coupling is present. Our findings will guide the search for topological chiral superconductors.Comment: 4+5 pages, 5 figure

On the orbital evolution of a giant planet pair embedded in a gaseous disk. I: Jupiter-Saturn configuration

We carry out a series of high resolution ($1024\times 1024$) hydrodynamical simulations to investigate the orbital evolution of Jupiter and Saturn embedded in a gaseous protostellar disk. Our work extends the results in the classical papers of Masset & Snellgrove (2001) and Morbidelli & Crida (2007) by exploring various surface density profiles ($\sigma$), where $\sigma \propto r^{-\alpha}$. The stability of the mean motion resonances(MMRs) caused by the convergent migration of the two planets is studied as well. Our results show that:(1) The gap formation process of Saturn is greatly delayed by the tidal perturbation of Jupiter. These perturbations cause inward or outward runaway migration of Saturn, depending on the density profiles on the disk. (2) The convergent migration rate increases as $\alpha$ increases and the type of MMRs depends on $\alpha$ as well. When $0<\alpha<1$, the convergent migration speed of Jupiter and Saturn is relatively slow, thus they are trapped into 2:1 MMR. When $\alpha>4/3$, Saturn passes through the $2:1$ MMR with Jupiter and is captured into the $3:2$ MMR. (3) The $3:2$ MMR turns out to be unstable when the eccentricity of Saturn ($e_s$) increases too high. The critical value above which instability will set in is $e_s \sim 0.15$. We also observe that the two planets are trapped into $2:1$ MMR after the break of $3:2$ MMR. This process may provide useful information for the formation of orbital configuration between Jupiter and Saturn in the Solar System.Comment: 21 pages, 23 figure

Hypothesis testing at the extremes: fast and robust association for high-throughput data

A number of biomedical problems require performing many hypothesis tests, with an attendant need to apply stringent thresholds. Often the data take the form of a series of predictor vectors, each of which must be compared with a single response vector, perhaps with nuisance covariates. Parametric tests of association are often used, but can result in inaccurate type I error at the extreme thresholds, even for large sample sizes. Furthermore, standard two-sided testing can reduce power compared to the doubled $p$-value, due to asymmetry in the null distribution. Exact (permutation) testing is attractive, but can be computationally intensive and cumbersome. We present an approximation to exact association tests of trend that is accurate and fast enough for standard use in high-throughput settings, and can easily provide standard two-sided or doubled $p$-values. The approach is shown to be equivalent under permutation to likelihood ratio tests for the most commonly used generalized linear models. For linear regression, covariates are handled by working with covariate-residualized responses and predictors. For generalized linear models, stratified covariates can be handled in a manner similar to exact conditional testing. Simulations and examples illustrate the wide applicability of the approach

HAWC Observation of Supernova Remnants and Pulsar Wind Nebulae

The majority of Galactic TeV gamma-ray sources are pulsar wind nebulae (PWNe) and supernova remnants (SNRs), and the most common association for unidentified sources is PWN. Many of these sources were discovered in TeV by imaging air Cherenkov telescopes using overlapping pointed observations over sections of the Galactic plane. The HAWC observatory is a survey type instrument in the Northern hemisphere with an energy range of 100 GeV to 100 TeV. Preliminary analysis of data recorded with the partially completed HAWC array taken since 2013 shows extended detections that are coincident with known TeV SNRs and PWNe. The full array became operational in early 2015 and has been steadily surveying the Northern sky since. I will discuss detections in HAWC data taken since 2013 associated with PWNe and SNRs.Comment: Presented at the 34th International Cosmic Ray Conference (ICRC2015), The Hague, The Netherlands. See arXiv:1508.03327 for all HAWC contribution