The Pythagoras number and the uu-invariant of Laurent series fields in several variables

We show that every sum of squares in the three-variable Laurent series field $\mathbb{R}((x,y,z))$ is a sum of 4 squares, as was conjectured in a paper of Choi, Dai, Lam and Reznick in the 1980's. We obtain this result by proving that every sum of squares in a finite extension of $\mathbb{R}((x,y))$ is a sum of $3$ squares. It was already shown in Choi, Dai, Lam and Reznick's paper that every sum of squares in $\mathbb{R}((x,y))$ itself is a sum of two squares. We give a generalization of this result where $\mathbb{R}$ is replaced by an arbitrary real field. Our methods yield similar results about the $u$-invariant of fields of the same type.Comment: final version, major revisions in the style of writing (abstract and introduction rewritten) compared to v.

Additivity of Entangled Channel Capacity for Quantum Input States

An elementary introduction into algebraic approach to unified quantum information theory and operational approach to quantum entanglement as generalized encoding is given. After introducing compound quantum state and two types of informational divergences, namely, Araki-Umegaki (a-type) and of Belavkin-Staszewski (b-type) quantum relative entropic information, this paper treats two types of quantum mutual information via entanglement and defines two types of corresponding quantum channel capacities as the supremum via the generalized encodings. It proves the additivity property of quantum channel capacities via entanglement, which extends the earlier results of V. P. Belavkin to products of arbitrary quantum channels for quantum relative entropy of any type.Comment: 17 pages. See the related papers at http://www.maths.nott.ac.uk/personal/vpb/research/ent_com.htm

A remark on the Restricted Isometry Property in Orthogonal Matching Pursuit

This paper demonstrates that if the restricted isometry constant $\delta_{K+1}$ of the measurement matrix $A$ satisfies $$\delta_{K+1} < \frac{1}{\sqrt{K}+1},$$ then a greedy algorithm called Orthogonal Matching Pursuit (OMP) can recover every $K$--sparse signal $\mathbf{x}$ in $K$ iterations from A\x. By contrast, a matrix is also constructed with the restricted isometry constant $$\delta_{K+1} = \frac{1}{\sqrt{K}}$$ such that OMP can not recover some $K$-sparse signal $\mathbf{x}$ in $K$ iterations. This result positively verifies the conjecture given by Dai and Milenkovic in 2009

GRB 030226 in a Density-Jump Medium

We present an explanation for the unusual temporal feature of the GRB 030226 afterglow. The R-band afterglow of this burst faded as ~ t^{-1.2} in ~ 0.2 days after the burst, rebrightened during the period of ~ 0.2 - 0.5 days, and then declined with ~ t^{-2.0}. To fit such a light curve, we consider an ultrarelativistic jetted blast wave expanding in a density-jump medium. The interaction of the blast wave with a large density jump produces relativistic reverse and forward shocks. In this model, the observed rebrightening is due to emissions from these newly forming shocks, and the late-time afterglow is caused by sideways expansion of the jet. Our fitting implies that the progenitor star of GRB 030226 could have produced a stellar wind with a large density jump prior to the GRB onset.Comment: 9 pages, 1 figure, accepted for publication in ApJ Letter

Harmonic forms on manifolds with edges

Let $(X,g)$ be a compact Riemannian stratified space with simple edge singularity. Thus a neighbourhood of the singular stratum is a bundle of truncated cones over a lower dimensional compact smooth manifold. We calculate the various polynomially weighted de Rham cohomology spaces of $X$, as well as the associated spaces of harmonic forms. In the unweighted case, this is closely related to recent work of Cheeger and Dai \cite{CD}. Because the metric $g$ is incomplete, this requires a consideration of the various choices of ideal boundary conditions at the singular set. We also calculate the space of $L^2$ harmonic forms for any complete edge metric on the regular part of $X$

Large deviations analysis for the M/H2/n+MM/H_2/n + M queue in the Halfin-Whitt regime

We consider the FCFS $M/H_2/n + M$ queue in the Halfin-Whitt heavy traffic regime. It is known that the normalized sequence of steady-state queue length distributions is tight and converges weakly to a limiting random variable W. However, those works only describe W implicitly as the invariant measure of a complicated diffusion. Although it was proven by Gamarnik and Stolyar that the tail of W is sub-Gaussian, the actual value of $\lim_{x \rightarrow \infty}x^{-2}\log(P(W >x))$ was left open. In subsequent work, Dai and He conjectured an explicit form for this exponent, which was insensitive to the higher moments of the service distribution. We explicitly compute the true large deviations exponent for W when the abandonment rate is less than the minimum service rate, the first such result for non-Markovian queues with abandonments. Interestingly, our results resolve the conjecture of Dai and He in the negative. Our main approach is to extend the stochastic comparison framework of Gamarnik and Goldberg to the setting of abandonments, requiring several novel and non-trivial contributions. Our approach sheds light on several novel ways to think about multi-server queues with abandonments in the Halfin-Whitt regime, which should hold in considerable generality and provide new tools for analyzing these systems