Optical scalars in spherical spacetimes

Consider a spherically symmetric spacelike slice through a spherically symmetric spacetime. One can derive a universal bound for the optical scalars on any such slice. The only requirement is that the matter sources satisfy the dominant energy condition and that the slice be asymptotically flat and regular at the origin. This bound can be used to derive new conditions for the formation of apparent horizons. The bounds hold even when the matter has a distribution on a shell or blows up at the origin so as to give a conical singularity

Schwarzschild horizon and the gravitational redshift formula

The gravitational redshift formula is usually derived in the geometric optics approximation. In this note we consider an exact formulation of the problem in the Schwarzschild space-time, with the intention to clarify under what conditions this redshift law is valid. It is shown that in the case of shocks the radial component of the Poynting vector can scale according to the redshift formula, under a suitable condition. If that condition is not satisfied, then the effect of the backscattering can lead to significant modifications. The obtained results imply that the energy flux of the short wavelength radiation obeys the standard gravitational redshift formula while the energy flux of long waves can scale differently, with redshifts being dependent on the frequency.Comment: Revtex, 5 p. Rewritten Sec. II, minor changes in Secs III - VII. To appear in the Classical and Quantum Gravit

Saturation of Spin-Polarized Current in Nanometer Scale Aluminum Grains

We describe measurements of spin-polarized tunnelling via discrete energy levels of single Aluminum grains. In high resistance samples ($\sim G\Omega$), the spin-polarized tunnelling current rapidly saturates as a function of the bias voltage. This indicates that spin-polarized current is carried only via the ground state and the few lowest in energy excited states of the grain. At the saturation voltage, the spin-relaxation rate $T_1^{-1}$ of the highest excited states is comparable to the electron tunnelling rate: $T_1^{-1}\approx 1.5\cdot 10^6 s^{-1}$ and $10^7s^{-1}$ in two samples. The ratio of $T_1^{-1}$ to the electron-phonon relaxation rate is in agreement with the Elliot-Yafet scaling, an evidence that spin-relaxation in Al grains is governed by the spin-orbit interaction.Comment: 5 pages, 4 figure

Modelling Electron Spin Accumulation in a Metallic Nanoparticle

A model describing spin-polarized current via discrete energy levels of a metallic nanoparticle, which has strongly asymmetric tunnel contacts to two ferromagnetic leads, is presented. In absence of spin-relaxation, the model leads to a spin-accumulation in the nanoparticle, a difference ($\Delta\mu$) between the chemical potentials of spin-up and spin-down electrons, proportional to the current and the Julliere's tunnel magnetoresistance. Taking into account an energy dependent spin-relaxation rate $\Omega (\omega)$, $\Delta\mu$ as a function of bias voltage ($V$) exhibits a crossover from linear to a much weaker dependence, when $|e|\Omega (\Delta\mu)$ equals the spin-polarized current through the nanoparticle. Assuming that the spin-relaxation takes place via electron-phonon emission and Elliot-Yafet mechanism, the model leads to a crossover from linear to $V^{1/5}$ dependence. The crossover explains recent measurements of the saturation of the spin-polarized current with $V$ in Aluminum nanoparticles, and leads to the spin-relaxation rate of $\approx 1.6 MHz$ in an Aluminum nanoparticle of diameter $6nm$, for a transition with an energy difference of one level spacing.Comment: 37 pages, 7 figure

Transport in Graphene Tunnel Junctions

We present a technique to fabricate tunnel junctions between graphene and Al and Cu, with a Si back gate, as well as a simple theory of tunneling between a metal and graphene. We map the differential conductance of our junctions versus probe and back gate voltage, and observe fluctuations in the conductance that are directly related to the graphene density of states. The conventional strong-suppression of the conductance at the graphene Dirac point can not be clearly demonstrated, but a more robust signature of the Dirac point is found: the inflection in the conductance map caused by the electrostatic gating of graphene by the tunnel probe. We present numerical simulations of our conductance maps, confirming the measurement results. In addition, Al causes strong n-doping of graphene, Cu causes a moderate p-doping, and in high resistance junctions, phonon resonances are observed, as in STM studies.Comment: 22 pages, 5 figure

Electronic Properties of Clean Au-Graphene Contacts

The effects of Au grains on graphene conduction and doping are investigated in this report. To obtain a clean Au-graphene contact, Au grains are deposited over graphene at elevated temperature and in high vacuum, before any chemical processing. The bulk and the effective contact resistance versus gate voltage demonstrate that Au grains cause p-doping in graphene. The Fermi level shift is in agreement with first principles calculations, but the equilibrium separation we find between the graphene and the top-most Au layer is larger than predicted. Nonequilibrium electron transport displays giant-phonon thresholds observed in graphene tunnel junctions, demonstrating the tunneling nature of the contact, even though there are no dielectrics involved.Comment: 11 pages, 4 figure

Three-dimensional shapelets and an automated classification scheme for dark matter haloes

We extend the two-dimensional Cartesian shapelet formalism to d-dimensions. Concentrating on the three-dimensional case, we derive shapelet-based equations for the mass, centroid, root-mean-square radius, and components of the quadrupole moment and moment of inertia tensors. Using cosmological N-body simulations as an application domain, we show that three-dimensional shapelets can be used to replicate the complex sub-structure of dark matter halos and demonstrate the basis of an automated classification scheme for halo shapes. We investigate the shapelet decomposition process from an algorithmic viewpoint, and consider opportunities for accelerating the computation of shapelet-based representations using graphics processing units (GPUs).Comment: 19 pages, 11 figures, accepted for publication in MNRA

How to Influence People with Partial Incentives

We study the power of fractional allocations of resources to maximize influence in a network. This work extends in a natural way the well-studied model by Kempe, Kleinberg, and Tardos (2003), where a designer selects a (small) seed set of nodes in a social network to influence directly, this influence cascades when other nodes reach certain thresholds of neighbor influence, and the goal is to maximize the final number of influenced nodes. Despite extensive study from both practical and theoretical viewpoints, this model limits the designer to a binary choice for each node, with no way to apply intermediate levels of influence. This model captures some settings precisely, e.g. exposure to an idea or pathogen, but it fails to capture very relevant concerns in others, for example, a manufacturer promoting a new product by distributing five "20% off" coupons instead of giving away one free product. While fractional versions of problems tend to be easier to solve than integral versions, for influence maximization, we show that the two versions have essentially the same computational complexity. On the other hand, the two versions can have vastly different solutions: the added flexibility of fractional allocation can lead to significantly improved influence. Our main theoretical contribution is to show how to adapt the major positive results from the integral case to the fractional case. Specifically, Mossel and Roch (2006) used the submodularity of influence to obtain their integral results; we introduce a new notion of continuous submodularity, and use this to obtain matching fractional results. We conclude that we can achieve the same greedy $(1-1/e-\epsilon)$-approximation for the fractional case as the integral case. In practice, we find that the fractional model performs substantially better than the integral model, according to simulations on real-world social network data

Global solutions of a free boundary problem for selfgravitating scalar fields

The weak cosmic censorship hypothesis can be understood as a statement that there exists a global Cauchy evolution of a selfgravitating system outside an event horizon. The resulting Cauchy problem has a free null-like inner boundary. We study a selfgravitating spherically symmetric nonlinear scalar field. We show the global existence of a spacetime with a null inner boundary that initially is located outside the Schwarzschild radius or, more generally, outside an apparent horizon. The global existence of a patch of a spacetime that is exterior to an event horizon is obtained as a limiting case.Comment: 31 pages, revtex, to appear in the Classical and Quantum Gravit