Validation of radiative transfer computation with Monte Carlo method for ultra-relativistic background flow

We developed a three-dimensional radiative transfer code for an ultra-relativistic background flow-field by using the Monte Carlo (MC) method in the context of gamma-ray burst (GRB) emission. For obtaining reliable simulation results in the coupled computation of MC radiation transport with relativistic hydrodynamics which can reproduce GRB emission, we validated radiative transfer computation in the ultra-relativistic regime and assessed the appropriate simulation conditions. The radiative transfer code was validated through two test calculations: (1) computing in different inertial frames and (2) computing in flow-fields with discontinuous and smeared shock fronts. The simulation results of the angular distribution and spectrum were compared among three different inertial frames and in good agreement with each other. If the time duration for updating the flow-field was sufficiently small to resolve a mean free path of a photon into ten steps, the results were thoroughly converged. The spectrum computed in the flow-field with a discontinuous shock front obeyed a power-law in frequency whose index was positive in the range from 1 to 10 MeV. The number of photons in the high-energy side decreased with the smeared shock front because the photons were less scattered immediately behind the shock wave due to the small electron number density. The large optical depth near the shock front was needed for obtaining high-energy photons through bulk Compton scattering. Even one-dimensional structure of the shock wave could affect the results of radiation transport computation. Although we examined the effect of the shock structure on the emitted spectrum with a large number of cells, it is hard to employ so many computational cells per dimension in multi-dimensional simulations. Therefore, a further investigation with a smaller number of cells is required for obtaining realistic high-energy photons with multi-dimensional computations

Toward Better Formula Lower Bounds: An Information Complexity Approach to the KRW Composition Conjecture

One of the major open problems in complexity theory is proving super-polynomial lower bounds for circuits with logarithmic depth (i.e., P ̸ ⊆ NC1). This problem is interesting for two reasons: first, it is tightly related to understanding the power of parallel computation and of small-space computation; second, it is one of the first milestones toward proving superpolynomial circuit lower bounds. Karchmer, Raz, and Wigderson [KRW95] suggested to approach this problem by proving the following conjecture: given two boolean functions f and g, the depth complexity of the composed function g ◦ f is roughly the sum of the depth complexities of f and g. They showed that the validity of this conjecture would imply that P ̸ ⊆ NC1. As a starting point for studying the composition of functions, they introduced a relation called “the universal relation”, and suggested to study the composition of universal relations. This suggestion proved fruitful, and an analogue of the KRW conjecture for the universal relation was proved by Edmonds et. al. [EIRS01]. An alternative proof was given later by H˚astad and Wigderson [HW93]. However, studying the composition of functions seems more difficult, and the KRW conjecture is still wide open. In this work, we make a natural step in this direction, which lies between what is known and the original conjecture: we show that an analogue of the conjecture holds for the composition of a function with a universal relation. We also suggest a candidate for the next step and provide initial results toward it. Our main technical contribution is developing an approach based on the notion of information complexity for analyzing KW relations – communication problems that are closely related to questions on circuit depth and formula complexity. Recently, information complexity has proved to be a powerful tool, and underlined some major progress on several long-standing open problems in communication complexity. In this work, we develop general tools for analyzing the information complexity of KW relations, which may be of independent interest.

A new method of correcting radial velocity time series for inhomogeneous convection

Magnetic activity strongly impacts stellar RVs and the search for small planets. We showed previously that in the solar case it induces RV variations with an amplitude over the cycle on the order of 8 m/s, with signals on short and long timescales. The major component is the inhibition of the convective blueshift due to plages. We explore a new approach to correct for this major component of stellar radial velocities in the case of solar-type stars. The convective blueshift depends on line depths; we use this property to develop a method that will characterize the amplitude of this effect and to correct for this RV component. We build realistic RV time series corresponding to RVs computed using different sets of lines, including lines in different depth ranges. We characterize the performance of the method used to reconstruct the signal without the convective component and the detection limits derived from the residuals. We identified a set of lines which, combined with a global set of lines, allows us to reconstruct the convective component with a good precision and to correct for it. For the full temporal sampling, the power in the range 100-500~d significantly decreased, by a factor of 100 for a RV noise below 30 cm/s. We also studied the impact of noise contributions other than the photon noise, which lead to uncertainties on the RV computation, as well as the impact of the temporal sampling. We found that these other sources of noise do not greatly alter the quality of the correction, although they need a better noise level to reach a similar performance level. A very good correction of the convective component can be achieved providing very good RV noise levels combined with a very good instrumental stability and realistic granulation noise. Under the conditions considered in this paper, detection limits at 480~d lower than 1 MEarth could be achieved for RV noise below 15 cm/s.Comment: Accepted in A&A 18 July 201

Small Depth Quantum Circuits

Small depth quantum circuits have proved to be unexpectedly powerful in comparison to their classical counterparts. We survey some of the recent work on this and present some open problems.National Security Agency; Advanced Research and Development Agency under Army Research Office (DAAD 19-02-1-0058