Interiors of completely positive cones

A symmetric matrix $A$ is completely positive (CP) if there exists an entrywise nonnegative matrix $B$ such that $A = BB^T$. We characterize the interior of the CP cone. A semidefinite algorithm is proposed for checking interiors of the CP cone, and its properties are studied. A CP-decomposition of a matrix in Dickinson's form can be obtained if it is an interior of the CP cone. Some computational experiments are also presented

On the Hecke Eigenvalues of Maass Forms

Let $\phi$ denote a primitive Hecke-Maass cusp form for $\Gamma_o(N)$ with the Laplacian eigenvalue $\lambda_\phi=1/4+t_{\phi}^2$. In this work we show that there exists a prime $p$ such that $p\nmid N$, $|\alpha_{p}|=|\beta_{p}| = 1$, and $p\ll(N(1+|t_{\phi}|))^c$, where $\alpha _{p},\;\beta _{p}$ are the Satake parameters of $\phi$ at $p$, and $c$ is an absolute constant with $0<c<1$. In fact, $c$ can be taken as $0.27332$. In addition, we prove that the natural density of such primes $p$ ($p\nmid N$ and $|\alpha_{p}|=|\beta_{p}| = 1$) is at least $34/35$.Comment: Version 2: typos corrected and a new section on natural density adde

An updated catalog of M31 globular-like clusters: UBVRI photometry, ages, and masses

We present an updated UBVRI photometric catalog containing 970 objects in the field of M31, selected from the Revised Bologna Catalog (RBC v.4.0), including 965, 967, 965, 953, and 827 sources in the individual UBVRI bands, respectively, of which 205, 123, 14, 126, and 109 objects do not have previously published photometry. Photometry is performed using archival images from the Local Group Galaxies Survey, which covers 2.2 deg^2 along the major axis of M31. We focus on 445 confirmed `globular-like' clusters and candidates, comprising typical globular and young massive clusters. The ages and masses of these objects are derived by comparison of their observed spectral-energy distributions with simple stellar population synthesis. Approximately half of the clusters are younger than 2 Gyr, suggesting that there has been significant recent active star formation in M31, which is consistent with previous results. We note that clusters in the halo (r_ projected>30kpc) are composed of two different components, older clusters with ages >10 Gyr and younger clusters with ages around 1 Gyr. The spatial distributions show that the young clusters (<2 Gyr) are spatially coincident with the galaxy's disk, including the `10 kpc ring,' the `outer ring,' and the halo of M31, while the old clusters (> 2 Gyr) are spatially correlated with the bulge and halo. We also estimate the masses of the 445 confirmed clusters and candidates in M31 and find that our estimates agree well with previously published values. We find that none of the young disk clusters can survive the inevitable encounters with giant molecular clouds in the galaxy's disk and that they will eventually disrupt on timescales of a few Gyr. Specifically, young disk clusters with a mass of 10^4 M_\odot are expected to dissolve within 3.0 Gyr and will, thus, not evolve to become globular clusters.Comment: 35 pages, 20 figures and 5 tables, accepted for publication in Ap