On the extreme eigenvalues of regular graphs

In this paper, we present an elementary proof of a theorem of Serre concerning the greatest eigenvalues of $k$-regular graphs. We also prove an analogue of Serre's theorem regarding the least eigenvalues of $k$-regular graphs: given $\epsilon>0$, there exist a positive constant $c=c(\epsilon,k)$ and a nonnegative integer $g=g(\epsilon,k)$ such that for any $k$-regular graph $X$ with no odd cycles of length less than $g$, the number of eigenvalues $\mu$ of $X$ such that $\mu \leq -(2-\epsilon)\sqrt{k-1}$ is at least $c|X|$. This implies a result of Winnie Li.Comment: accepted to J.Combin.Theory, Series B. added 5 new references, some comments on the constant c in Section

How would GW150914 look with future GW detector networks?

The first detected gravitational wave signal, GW150914, was produced by the coalescence of a stellar-mass binary black hole. Along with the subsequent detection of GW151226, GW170104 and the candidate event LVT151012, this gives us evidence for a population of black hole binaries with component masses in the tens of solar masses. As detector sensitivity improves, this type of source is expected to make a large contribution to the overall number of detections, but has received little attention compared to binary neutron star systems in studies of projected network performance. We simulate the observation of a system like GW150914 with different proposed network configurations, and study the precision of parameter estimates, particularly source location, orientation and masses. We find that the improvements to low frequency sensitivity that are expected with continued commissioning will improve the precision of chirp mass estimates by an order of magnitude, whereas the improvements in sky location and orientation are driven by the expanded network configuration. This demonstrates that both sensitivity and number of detectors will be important factors in the scientific potential of second generation detector networks.Comment: 18 pages, 5 figures, 2 table

SS433's circumbinary ring and accretion disc viewed through its attenuating disc wind

We present optical spectroscopy of the microquasar SS433 covering a significant fraction of a precessional cycle of its jet axis. The components of the prominent stationary H-alpha and H-beta lines are mainly identified as arising from three emitting regions: (i) a super-Eddington accretion disc wind, in the form of a broad component accounting for most of the mass loss from the system, (ii) a circumbinary disc of material that we presume is being excreted through the binary's L2 point, and (iii) the accretion disc itself as two remarkably persistent components. The accretion disc components move with a Keplerian velocity of ~600 km/s in the outer region of the disc. A direct result of this decomposition is the determination of the accretion disc size, whose outer radius attains ~8 R_sun in the case of Keplerian orbits around a black hole mass of 10 M_sun. We determine an upper limit for the accretion disc inner to outer radius ratio in SS433, R_in/R_out ~ 0.2, independent of the mass of the compact object. The Balmer decrements, H-alpha/H-beta, are extracted from the appropriate stationary emission lines for each component of the system. The physical parameters of the gaseous components are derived. The circumbinary ring decrement seems to be quite constant throughout precessional phase, implying a constant electron density of log N_e(cm^-3) ~ 11.5 for the circumbinary disc. The accretion disc wind shows a larger change in its decrements exhibiting a clear dependence on precessional phase, implying a sinusoid variation in its electron density log N_e(cm^-3) along our line-of-sight between 10 and 13. This dependence of density on direction suggests that the accretion disc wind is polloidal in nature.Comment: 7 pages, 5 figures. Accepted for publication in MNRAS Main Journal

Using GWAS Data to Identify Copy Number Variants Contributing to Common Complex Diseases

Copy number variants (CNVs) account for more polymorphic base pairs in the human genome than do single nucleotide polymorphisms (SNPs). CNVs encompass genes as well as noncoding DNA, making these polymorphisms good candidates for functional variation. Consequently, most modern genome-wide association studies test CNVs along with SNPs, after inferring copy number status from the data generated by high-throughput genotyping platforms. Here we give an overview of CNV genomics in humans, highlighting patterns that inform methods for identifying CNVs. We describe how genotyping signals are used to identify CNVs and provide an overview of existing statistical models and methods used to infer location and carrier status from such data, especially the most commonly used methods exploring hybridization intensity. We compare the power of such methods with the alternative method of using tag SNPs to identify CNV carriers. As such methods are only powerful when applied to common CNVs, we describe two alternative approaches that can be informative for identifying rare CNVs contributing to disease risk. We focus particularly on methods identifying de novo CNVs and show that such methods can be more powerful than case-control designs. Finally we present some recommendations for identifying CNVs contributing to common complex disorders.Comment: Published in at http://dx.doi.org/10.1214/09-STS304 the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org

Optimal redundancy against disjoint vulnerabilities in networks

Redundancy is commonly used to guarantee continued functionality in networked systems. However, often many nodes are vulnerable to the same failure or adversary. A "backup" path is not sufficient if both paths depend on nodes which share a vulnerability.For example, if two nodes of the Internet cannot be connected without using routers belonging to a given untrusted entity, then all of their communication-regardless of the specific paths utilized-will be intercepted by the controlling entity.In this and many other cases, the vulnerabilities affecting the network are disjoint: each node has exactly one vulnerability but the same vulnerability can affect many nodes. To discover optimal redundancy in this scenario, we describe each vulnerability as a color and develop a "color-avoiding percolation" which uncovers a hidden color-avoiding connectivity. We present algorithms for color-avoiding percolation of general networks and an analytic theory for random graphs with uniformly distributed colors including critical phenomena. We demonstrate our theory by uncovering the hidden color-avoiding connectivity of the Internet. We find that less well-connected countries are more likely able to communicate securely through optimally redundant paths than highly connected countries like the US. Our results reveal a new layer of hidden structure in complex systems and can enhance security and robustness through optimal redundancy in a wide range of systems including biological, economic and communications networks.Comment: 15 page