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We propose an iterative algorithm for the numerical computation of sums of squares of polynomials approximating given data at prescribed interpolation points. The method is based on the definition of a convex functional $G$ arising from the dualization of a quadratic regression over the Cholesky factors of the sum of squares decomposition. In order to justify the construction, the domain of $G$, the boundary of the domain and the behavior at infinity are analyzed in details. When the data interpolate a positive univariate polynomial, we show that in the context of the Lukacs sum of squares representation, $G$ is coercive and strictly convex which yields a unique critical point and a corresponding decomposition in sum of squares. For multivariate polynomials which admit a decomposition in sum of squares and up to a small perturbation of size $\varepsilon$, $G^\varepsilon$ is always coercive and so it minimum yields an approximate decomposition in sum of squares. Various unconstrained descent algorithms are proposed to minimize $G$. Numerical examples are provided, for univariate and bivariate polynomials.

We study the spreading of information in a wide class of quantum systems, with variable-range interactions. We show that, after a quench, it generally features a double structure, whose scaling laws are related to a set of universal microscopic exponents that we determine. When the system supports excitations with a finite maximum velocity, the spreading shows a twofold ballistic behavior. While the correlation edge spreads with a velocity equal to twice the maximum group velocity, the dominant correlation maxima propagate with a different velocity that we derive. When the maximum group velocity diverges, as realizable with long-range interactions, the correlation edge features a slower-than-ballistic motion. The motion of the maxima is, instead, either faster-than-ballistic, for gapless systems, or ballistic, for gapped systems. The phenomenology that we unveil here provides a unified framework, which encompasses existing experimental observations with ultracold atoms and ions. It also paves the way to simple extensions of those experiments to observe the structures we describe in their full generality.

We study the Majorana bound states arising in quasi-one-dimensional systems with Rashba spin-orbit coupling in the presence of an in-plane Zeeman magnetic field. Using two different methods, first, the numerical diagonalization of the tight-binding Hamiltonian, and second, finding the singular points of the Hamiltonian (see Refs. [1-4]), we obtain the topological phase diagram for these systems as a function of the chemical potential and the magnetic field, and we demonstrate the consistency of these two methods. By introducing disorder into these systems we confirm that the states with even number of Majorana pairs are not topologically protected. Finally, we show that a formal calculation of the $\mathbb{Z}_2$ topological invariants recovers correctly the parity of the number of Majorana bound states pairs, and it is thus fully consistent with the phase diagrams of the disordered systems., Comment: 8,5 pages, 8 figures (+2,5 pages, 1 figure in Appendixes)

We review sharpening methods for finite volume schemes, with an emphasis on the basic structure of sharpening methods. It covers high order methods and non linear techniques for linear advection, Glimm's method, anti-diffusion techniques, the interaction of these techniques with the PDE structures. Additional approaches like level sets, interface reconstruction and Vofire are also discussed. We also present the algorithmic structure of the downwind method for a simple two components problem.

Information Retrieval (IR) models need to deal with two difficult issues, vocabulary mismatch and term dependencies. Vocabulary mismatch corresponds to the difficulty of retrieving relevant documents that do not contain exact query terms but semantically related terms. Term dependencies refers to the need of considering the relationship between the words of the query when estimating the relevance of a document. A multitude of solutions has been proposed to solve each of these two problems, but no principled model solve both. In parallel, in the last few years, language models based on neural networks have been used to cope with complex natural language processing tasks like emotion and paraphrase detection. Although they present good abilities to cope with both term dependencies and vocabulary mismatch problems, thanks to the distributed representation of words they are based upon, such models could not be used readily in IR, where the estimation of one language model per document (or query) is required. This is both computationally unfeasible and prone to over-fitting. Based on a recent work that proposed to learn a generic language model that can be modified through a set of document-specific parameters, we explore use of new neural network models that are adapted to ad-hoc IR tasks. Within the language model IR framework, we propose and study the use of a generic language model as well as a document-specific language model. Both can be used as a smoothing component, but the latter is more adapted to the document at hand and has the potential of being used as a full document language model. We experiment with such models and analyze their results on TREC-1 to 8 datasets.

We study the long-time asymptotic of the solutions to Maxwell's equation in the case of a upper-hybrid resonance in the cold plasma model. We base our analysis in the transfer to the time domain of the recent results of B. Despr\'es, L.M. Imbert-G\'erard and R. Weder, J. Math. Pures Appl. {\bf 101} ( 2014) 623-659, where the singular solutions to Maxwell's equations in the frequency domain were constructed by means of a limiting absorption principle and a formula for the heating of the plasma in the limit of vanishing collision frequency was obtained. Currently there is considerable interest in these problems, in particular, because upper-hybrid resonances are a possible scenario for the heating of plasmas, and since they can be a model for the diagnostics involving wave scattering in plasmas., Comment: This published version has been edited to improve the presentation of the results

The Vapnik-Chervonenkis (VC) dimension of a collection of subsets of a set is an important combinatorial concept in settings such as discrete geometry and machine learning. In this paper we prove that the VC dimension of the family of $d$-dimensional cubes in $\mathbb R^d$ is $\lfloor(3d+1)/2\rfloor$., Comment: Derived from Fall 2014 Honours research project done under the supervision of Dr. Vladimir Pestov at the University of Ottawa; 4 pages; significantly simplified the constructions, removed the final two sections (which did not fit well with the rest)

In this paper we propose an asymptotic preserving scheme for a family of Friedrichs systems on unstructured meshes based on a decomposition between the hyperbolic heat equation and a linear hyperbolic which not involved in the di usive regime. For the hyperbolic heat equation we use asymptotic preserving schemes recently designed previously. To discretize the second part we use classical Rusanov or upwind schemes. To nish we apply this method for the discretization of the PN and SN models which are widely used in transport codes.

Hybrid resonance is a physical mechanism for the heating of a magnetic plasma. In our context hybrid resonance is a solution of the time harmonic Maxwell's equations with smooth coefficients, where the dielectric tensor is a non diagonal hermitian matrix. The main part of this work is dedicated to the construction and analysis of a mathematical solution of the hybrid resonance with the limit absorption principle. We prove that the limit solution is singular: it is constituted of a Dirac mass at the origin plus a principle value and a smooth square integrable function. The formula obtained for the plasma heating is directly related to the singularity., Comment: This published version has been edited to improve the presentation of the results

The mathematical description of laboratory fusion plasmas produced in Tokamaks is still challenging. Complete models for electrons and ions, as Vlasov-Maxwell systems, are computationally too expensive because they take into account all details and scales of magneto-hydrodynamics. In particular, for most of the relevant studies, the mass electron is negligible and the velocity of material waves is much smaller than the speed of light. Therefore it is useful to understand simplified models. Here we propose and study one of those which keeps both the complexity of the Vlasov equation for ions and the Hall effect in Maxwell's equation. Based on energy dissipation, a fundamental physical property, we show that the model is nonlinear stable and consequently prove existence.

One important class of online videos is that of news broadcasts. Most news organisations provide near-immediate access to topical news broadcasts over the Internet, through RSS streams or podcasts. Until lately, technology has not made it possible for a user to automatically go to the smaller parts, within a longer broadcast, that might interest them. Recent advances in both speech recognition systems and natural language processing have led to a number of robust tools that allow us to provide users with quicker, more focussed access to relevant segments of one or more news broadcast videos. Here we present our new interface for browsing or searching news broadcasts (video/audio) that exploits these new language processing tools to (i) provide immediate access to topical passages within news broadcasts, (ii) browse news broadcasts by events as well as by people, places and organisations, (iii) perform cross lingual search of news broadcasts, (iv) search for news through a map interface, (v) browse news by trending topics, and (vi) see automatically-generated textual clues for news segments, before listening. Our publicly searchable demonstrator currently indexes daily broadcast news content from 50 sources in English, French, Chinese, Arabic, Spanish, Dutch and Russian., Comment: NEM Summit, Torino : Italy (2011)

The future of high-performance computing is aligning itself towards the efficient use of highly parallel computing environments. One application where the use of massive parallelism comes instinctively is Monte Carlo simulations, where a large number of independent events have to be simulated. At the core of the Monte Carlo simulation lies the Random Number Generator (RNG). In this paper, the massively parallel implementation of a collection of pseudo-random number generators on a graphics processing unit (GPU) is presented. The results of the GPU implementation, in terms of samples/s, effective bandwidth and operations per second, are presented. A comparison with other implementations on different hardware platforms, in terms of samples/s, power efficiency and cost-benefit, is also presented. Random numbers generation throughput of up to ~18MSamples/s have been achieved on the graphics hardware used. Efficient hardware utilization, in terms of operations per second, has reached ~98% of the possible integer operation throughput.

Purpose: Monte Carlo methods are considered the gold standard for dosimetric computations in radiotherapy. Their execution time is however still an obstacle to the routine use of Monte Carlo packages in a clinical setting. To address this problem, a completely new, and designed from the ground up for the GPU, Monte Carlo dose calculation package for voxelized geometries is proposed: GPUMCD. Method : GPUMCD implements a coupled photon-electron Monte Carlo simulation for energies in the range 0.01 MeV to 20 MeV. An analogue simulation of photon interactions is used and a Class II condensed history method has been implemented for the simulation of electrons. A new GPU random number generator, some divergence reduction methods as well as other optimization strategies are also described. GPUMCD was run on a NVIDIA GTX480 while single threaded implementations of EGSnrc and DPM were run on an Intel Core i7 860. Results : Dosimetric results obtained with GPUMCD were compared to EGSnrc. In all but one test case, 98% or more of all significant voxels passed a gamma criteria of 2%-2mm. In terms of execution speed and efficiency, GPUMCD is more than 900 times faster than EGSnrc and more than 200 times faster than DPM, a Monte Carlo package aiming fast executions. Absolute execution times of less than 0.3 s are found for the simulation of 1M electrons and 4M photons in water for monoenergetic beams of 15 MeV, including GPU-CPU memory transfers. Conclusion : GPUMCD, a new GPU-oriented Monte Carlo dose calculation platform, has been compared to EGSnrc and DPM in terms of dosimetric results and execution speed. Its accuracy and speed make it an interesting solution for full Monte Carlo dose calculation in radiation oncology., Comment: Accepted for publication in Medical Physics

We study two statistical models for short-length categorical (or ordinal) time series. The first one is a regression model based on generalized linear model. The second one is a parametrized Markovian model, particularizing the discrete autoregressive model to the case of categorical data. These models are used to analyze two data-sets: annual larch cone production and weekly planktonic abundance., Comment: 18 pages