Computationally Efficient Nonparametric Importance Sampling

The variance reduction established by importance sampling strongly depends on the choice of the importance sampling distribution. A good choice is often hard to achieve especially for high-dimensional integration problems. Nonparametric estimation of the optimal importance sampling distribution (known as nonparametric importance sampling) is a reasonable alternative to parametric approaches.In this article nonparametric variants of both the self-normalized and the unnormalized importance sampling estimator are proposed and investigated. A common critique on nonparametric importance sampling is the increased computational burden compared to parametric methods. We solve this problem to a large degree by utilizing the linear blend frequency polygon estimator instead of a kernel estimator. Mean square error convergence properties are investigated leading to recommendations for the efficient application of nonparametric importance sampling. Particularly, we show that nonparametric importance sampling asymptotically attains optimal importance sampling variance. The efficiency of nonparametric importance sampling algorithms heavily relies on the computational efficiency of the employed nonparametric estimator. The linear blend frequency polygon outperforms kernel estimators in terms of certain criteria such as efficient sampling and evaluation. Furthermore, it is compatible with the inversion method for sample generation. This allows to combine our algorithms with other variance reduction techniques such as stratified sampling. Empirical evidence for the usefulness of the suggested algorithms is obtained by means of three benchmark integration problems. As an application we estimate the distribution of the queue length of a spam filter queueing system based on real data.Comment: 29 pages, 7 figure

The Principal Axis of the Virgo Cluster

Using accurate distances to individual Virgo cluster galaxies obtained by the method of Surface Brightness Fluctuations, we show that Virgo's brightest ellipticals have a remarkably collinear arrangement in three dimensions. This axis, which is inclined by 10 to 15 degrees from the line of sight, can be traced to even larger scales where it appears to join a filamentary bridge of galaxies connecting Virgo to the rich cluster Abell 1367. The orientations of individual Virgo ellipticals also show some tendency to be aligned with the cluster axis, as does the jet of the supergiant elliptical M87. These results suggest that the formation of the Virgo cluster, and its brightest member galaxies, have been driven by infall of material along the Virgo-A1367 filament.Comment: 8 pages, 4 figures, accepted for publication in ApJ Letter

E_11 and M Theory

We argue that eleven dimensional supergravity can be described by a non-linear realisation based on the group E_{11}. This requires a formulation of eleven dimensional supergravity in which the gravitational degrees of freedom are described by two fields which are related by duality. We show the existence of such a description of gravity.Comment: 21 pages, some typos corrected and two references adde

E11, generalised space-time and equations of motion in four dimensions

We construct the non-linear realisation of the semi-direct product of E11 and its first fundamental representation at low levels in four dimensions. We include the fields for gravity, the scalars and the gauge fields as well as the duals of these fields. The generalised space-time, upon which the fields depend, consists of the usual coordinates of four dimensional space-time and Lorentz scalar coordinates which belong to the 56-dimensional representation of E7. We demand that the equations of motion are first order in derivatives of the generalised space-time and then show that they are essentially uniquely determined by the properties of the E11 Kac-Moody algebra and its first fundamental representation. The two lowest equations correctly describe the equations of motion of the scalars and the gauge fields once one takes the fields to depend only on the usual four dimensional space-time

The Low-level Spectrum of the W3W_3 String

We investigate the spectrum of physical states in the $W_3$ string theory, up to level 2 for a multi-scalar string, and up to level 4 for the two-scalar string. The (open) $W_3$ string has a photon as its only massless state. By using screening charges to study the null physical states in the two-scalar $W_3$ string, we are able to learn about the gauge symmetries of the states in the multi-scalar $W_3$ string.Comment: 31 pages, Plain Tex, CTP TAMU-70/92, Goteborg ITP 92-43, Imperial/TP/91-92/22, KCL-TH-92-

Very Extended E8E_8 and A8A_8 at low levels, Gravity and Supergravity

We define a level for a large class of Lorentzian Kac-Moody algebras. Using this we find the representation content of very extended $A_{D-3}$ and $E_8$ (i.e. $E_{11}$) at low levels in terms of $A_{D-1}$ and $A_{10}$ representations respectively. The results are consistent with the conjectured very extended $A_8$ and $E_{11}$ symmetries of gravity and maximal supergravity theories given respectively in hep-th/0104081 and hep-th/0107209. We explain how these results provided further evidence for these conjectures.Comment: 16 pages, plain tex (equation 3.3 modified and one reference expanded

Current-induced nuclear-spin activation in a two-dimensional electron gas

Electrically detected nuclear magnetic resonance was studied in detail in a two-dimensional electron gas as a function of current bias and temperature. We show that applying a relatively modest dc-current bias, I_dc ~ 0.5 microAmps, can induce a re-entrant and even enhanced nuclear spin signal compared with the signal obtained under similar thermal equilibrium conditions at zero current bias. Our observations suggest that dynamic nuclear spin polarization by small current flow is possible in a two-dimensional electron gas, allowing for easy manipulation of the nuclear spin by simple switching of a dc current.Comment: 5 pages, 3 fig