On the Abundance of Primordial Helium

We have used recent observations of helium-4, nitrogen and oxygen from some four dozen, low metallicity, extra-galactic HII regions to define mean $N$ versus $O$, $^4He$ versus $N$ and $^4He$ versus $O$ relations which are extrapolated to zero metallicity to determine the primordial $^4He$ mass fraction $Y_P$. The data and various subsets of the data, selected on the basis of nitrogen and oxygen, are all consistent with $Y_P = 0.232 \pm 0.003$. For the 2$\sigma$ (statistical) upper bound we find $Y_P^{2\sigma} \le 0.238$. Estimating a 2\% systematic uncertainty $(\sigma _{syst} = \pm 0.005)$ leads to a maximum upper bound to the primordial helium mass fraction: $Y_P^{MAX} = Y_P^{2\sigma} + \sigma_{syst} \le 0.243$. We compare these upper bounds to $Y_P$ with recent calculations of the predicted yield from big bang nucleosynthesis to derive upper bounds to the nucleon-to-photon ratio $\eta$ ($\eta_{10} \equiv 10^{10}\eta$) and the number of equivalent light (\lsim 10 MeV) neutrino species. For $Y_P \le 0.238$ ($0.243$), we find $\eta_{10} \le 2.5 (3.9)$ and $N_\nu \leq 2.7 (3.1)$. If indeed $Y_P \le 0.238$, then BBN predicts enhanced production of deuterium and helium-3 which may be in conflict with the primordial abundances inferred from model dependent (chemical evolution) extrapolations of solar system and interstellar observations. Better chemical evolution models and more data - especially $D$-absorption in the QSO Ly-$\alpha$ clouds - will be crucial to resolve this potential crisis for BBN. The larger upper bound, $Y_P \leq 0.243$ is completelyComment: 21 pages, LaTeX, 6 postscript figures available upon request, UMN-TH-123

Synchronisation Properties of Trees in the Kuramoto Model

We consider the Kuramoto model of coupled oscillators, specifically the case of tree networks, for which we prove a simple closed-form expression for the critical coupling. For several classes of tree, and for both uniform and Gaussian vertex frequency distributions, we provide tight closed form bounds and empirical expressions for the expected value of the critical coupling. We also provide several bounds on the expected value of the critical coupling for all trees. Finally, we show that for a given set of vertex frequencies, there is a rearrangement of oscillator frequencies for which the critical coupling is bounded by the spread of frequencies.Comment: 21 pages, 19 Figure

Rounding Algorithms for a Geometric Embedding of Minimum Multiway Cut

The multiway-cut problem is, given a weighted graph and k >= 2 terminal nodes, to find a minimum-weight set of edges whose removal separates all the terminals. The problem is NP-hard, and even NP-hard to approximate within 1+delta for some small delta > 0. Calinescu, Karloff, and Rabani (1998) gave an algorithm with performance guarantee 3/2-1/k, based on a geometric relaxation of the problem. In this paper, we give improved randomized rounding schemes for their relaxation, yielding a 12/11-approximation algorithm for k=3 and a 1.3438-approximation algorithm in general. Our approach hinges on the observation that the problem of designing a randomized rounding scheme for a geometric relaxation is itself a linear programming problem. The paper explores computational solutions to this problem, and gives a proof that for a general class of geometric relaxations, there are always randomized rounding schemes that match the integrality gap.Comment: Conference version in ACM Symposium on Theory of Computing (1999). To appear in Mathematics of Operations Researc

Ground state of the Bethe-lattice spin glass and running time of an exact optimization algorithm

We study the Ising spin glass on random graphs with fixed connectivity z and with a Gaussian distribution of the couplings, with mean \mu and unit variance. We compute exact ground states by using a sophisticated branch-and-cut method for z=4,6 and system sizes up to N=1280 for different values of \mu. We locate the spin-glass/ferromagnet phase transition at \mu = 0.77 +/- 0.02 (z=4) and \mu = 0.56 +/- 0.02 (z=6). We also compute the energy and magnetization in the Bethe-Peierls approximation with a stochastic method, and estimate the magnitude of replica symmetry breaking corrections. Near the phase transition, we observe a sharp change of the median running time of our implementation of the algorithm, consistent with a change from a polynomial dependence on the system size, deep in the ferromagnetic phase, to slower than polynomial in the spin-glass phase.Comment: 10 pages, RevTex, 10 eps figures. Some changes in the tex