Sensitivity of helioseismic travel-times to the imposition of a Lorentz force limiter in computational helioseismology

The rapid exponential increase in the Alfv\'en wave speed with height above the solar surface presents a serious challenge to physical modelling of the effects of magnetic fields on solar oscillations, as it introduces a significant CFL time-step constraint for explicit numerical codes. A common approach adopted in computational helioseismology, where long simulations in excess of 10 hours (hundreds of wave periods) are often required, is to cap the Alfv\'en wave speed by artificially modifying the momentum equation when the ratio between Lorentz and hydrodynamic forces becomes too large. However, recent studies have demonstrated that the Alfv\'en wave speed plays a critical role in the MHD mode conversion process, particularly in determining the reflection height of the upward propagating helioseismic fast wave. Using numerical simulations of helioseismic wave propagation in constant inclined (relative to the vertical) magnetic fields we demonstrate that the imposition of such artificial limiters significantly affects time-distance travel times unless the Alfv\'en wave-speed cap is chosen comfortably in excess of the horizontal phase speeds under investigation.Comment: 8 pages, 5 figures, accepted by ApJ

Interacting Dark Energy and the Cosmic Coincidence Problem

The introduction of an interaction for dark energy to the standard cosmology offers a potential solution to the cosmic coincidence problem. We examine the conditions on the dark energy density that must be satisfied for this scenario to be realized. Under some general conditions we find a stable attractor for the evolution of the Universe in the future. Holographic conjectures for the dark energy offer some specific examples of models with the desired properties.Comment: 8 pages, 3 figures, Phys. Rev. D versio

Polar Codes: Robustness of the Successive Cancellation Decoder with Respect to Quantization

Polar codes provably achieve the capacity of a wide array of channels under successive decoding. This assumes infinite precision arithmetic. Given the successive nature of the decoding algorithm, one might worry about the sensitivity of the performance to the precision of the computation. We show that even very coarsely quantized decoding algorithms lead to excellent performance. More concretely, we show that under successive decoding with an alphabet of cardinality only three, the decoder still has a threshold and this threshold is a sizable fraction of capacity. More generally, we show that if we are willing to transmit at a rate $\delta$ below capacity, then we need only $c \log(1/\delta)$ bits of precision, where $c$ is a universal constant.Comment: In ISIT 201

The Space of Solutions of Coupled XORSAT Formulae

The XOR-satisfiability (XORSAT) problem deals with a system of $n$ Boolean variables and $m$ clauses. Each clause is a linear Boolean equation (XOR) of a subset of the variables. A $K$-clause is a clause involving $K$ distinct variables. In the random $K$-XORSAT problem a formula is created by choosing $m$ $K$-clauses uniformly at random from the set of all possible clauses on $n$ variables. The set of solutions of a random formula exhibits various geometrical transitions as the ratio $\frac{m}{n}$ varies. We consider a {\em coupled} $K$-XORSAT ensemble, consisting of a chain of random XORSAT models that are spatially coupled across a finite window along the chain direction. We observe that the threshold saturation phenomenon takes place for this ensemble and we characterize various properties of the space of solutions of such coupled formulae.Comment: Submitted to ISIT 201

How to Achieve the Capacity of Asymmetric Channels

We survey coding techniques that enable reliable transmission at rates that approach the capacity of an arbitrary discrete memoryless channel. In particular, we take the point of view of modern coding theory and discuss how recent advances in coding for symmetric channels help provide more efficient solutions for the asymmetric case. We consider, in more detail, three basic coding paradigms. The first one is Gallager's scheme that consists of concatenating a linear code with a non-linear mapping so that the input distribution can be appropriately shaped. We explicitly show that both polar codes and spatially coupled codes can be employed in this scenario. Furthermore, we derive a scaling law between the gap to capacity, the cardinality of the input and output alphabets, and the required size of the mapper. The second one is an integrated scheme in which the code is used both for source coding, in order to create codewords distributed according to the capacity-achieving input distribution, and for channel coding, in order to provide error protection. Such a technique has been recently introduced by Honda and Yamamoto in the context of polar codes, and we show how to apply it also to the design of sparse graph codes. The third paradigm is based on an idea of B\"ocherer and Mathar, and separates the two tasks of source coding and channel coding by a chaining construction that binds together several codewords. We present conditions for the source code and the channel code, and we describe how to combine any source code with any channel code that fulfill those conditions, in order to provide capacity-achieving schemes for asymmetric channels. In particular, we show that polar codes, spatially coupled codes, and homophonic codes are suitable as basic building blocks of the proposed coding strategy.Comment: 32 pages, 4 figures, presented in part at Allerton'14 and published in IEEE Trans. Inform. Theor