Counting Hamilton cycles in sparse random directed graphs

Let D(n,p) be the random directed graph on n vertices where each of the n(n-1) possible arcs is present independently with probability p. A celebrated result of Frieze shows that if $p\ge(\log n+\omega(1))/n$ then D(n,p) typically has a directed Hamilton cycle, and this is best possible. In this paper, we obtain a strengthening of this result, showing that under the same condition, the number of directed Hamilton cycles in D(n,p) is typically $n!(p(1+o(1)))^{n}$. We also prove a hitting-time version of this statement, showing that in the random directed graph process, as soon as every vertex has in-/out-degrees at least 1, there are typically $n!(\log n/n(1+o(1)))^{n}$ directed Hamilton cycles

Astrometric Effects of a Stochastic Gravitational Wave Background

A stochastic gravitational wave background causes the apparent positions of distant sources to fluctuate, with angular deflections of order the characteristic strain amplitude of the gravitational waves. These fluctuations may be detectable with high precision astrometry, as first suggested by Braginsky et al. in 1990. Several researchers have made order of magnitude estimates of the upper limits obtainable on the gravitational wave spectrum \Omega_gw(f), at frequencies of order f ~ 1 yr^-1, both for the future space-based optical interferometry missions GAIA and SIM, and for VLBI interferometry in radio wavelengths with the SKA. For GAIA, tracking N ~ 10^6 quasars over a time of T ~ 1 yr with an angular accuracy of \Delta \theta ~ 10 \mu as would yield a sensitivity level of \Omega_gw ~ (\Delta \theta)^2/(N T^2 H_0^2) ~ 10^-6, which would be comparable with pulsar timing. In this paper we take a first step toward firming up these estimates by computing in detail the statistical properties of the angular deflections caused by a stochastic background. We compute analytically the two point correlation function of the deflections on the sphere, and the spectrum as a function of frequency and angular scale. The fluctuations are concentrated at low frequencies (for a scale invariant stochastic background), and at large angular scales, starting with the quadrupole. The magnetic-type and electric-type pieces of the fluctuations have equal amounts of power.Comment: 23 pages, 2 figures, references added and minor text correction

Program

Program for the 44th Annual Fay B. Kaigler Children\u27s Book Festiva

Program

Program for the 41st annual Fay B. Kaigler Children\u27s Book Festiva

Program

Program for the 40th annual Fay B. Kaigler Children\u27s Book Festiva

Program

Program for the 40th annual Fay B. Kaigler Children\u27s Book Festiva

Program

Program for the 43rd annual Fay B. Kaigler Children\u27s Book Festiva

Long-term Evolution of Protostellar and Protoplanetary Disks. I. Outbursts

As an initial investigation into the long-term evolution of protostellar disks, we explore the conditions required to explain the large outbursts of disk accretion seen in some young stellar objects. We use one-dimensional time-dependent disk models with a phenomenological treatment of the magnetorotational instability (MRI) and gravitational torques to follow disk evolution over long timescales. Comparison with our previous two-dimensional disk model calculations (Zhu et al. 2009b, Z2009b) indicates that the neglect of radial effects and two-dimensional disk structure in the one-dimensional case makes only modest differences in the results; this allows us to use the simpler models to explore parameter space efficiently. We find that the mass infall rates typically estimated for low-mass protostars generally result in AU-scale disk accretion outbursts, as predicted by our previous analysis (Zhu et al. 2009a,Z2009a). We also confirm quasi-steady accretion behavior for high mass infall rates if the values of $\alpha$-parameter for the magnetorotational instability is small, while at this high accretion rate convection from the thermal instability may lead to some variations. We further constrain the combinations of the $\alpha$-parameter and the MRI critical temperature, which can reproduce observed outburst behavior. Our results suggest that dust sublimation may be connected with full activation of the MRI. This is consistent with the idea that small dust captures ions and electrons to suppress the MRI. In a later paper we will explore both long-term outburst and disk evolution with this model, allowing for infall from protostellar envelopes with differing angular momenta.Comment: Accepted to publish in Ap

Experimental investigation of the fundamental modes of a collisionless plasma Final report, 10 Mar. 1964 - 31 Oct. 1967

Propagation of electron cyclotron waves and effects of low frequency noise in collisionless plasm