Computing Linear Matrix Representations of Helton-Vinnikov Curves

Helton and Vinnikov showed that every rigidly convex curve in the real plane bounds a spectrahedron. This leads to the computational problem of explicitly producing a symmetric (positive definite) linear determinantal representation for a given curve. We study three approaches to this problem: an algebraic approach via solving polynomial equations, a geometric approach via contact curves, and an analytic approach via theta functions. These are explained, compared, and tested experimentally for low degree instances.Comment: 19 pages, 3 figures, minor revisions; Mathematical Methods in Systems, Optimization and Control, Birkhauser, Base

Instabilities of one-dimensional stationary solutions of the cubic nonlinear Schrodinger equation

The two-dimensional cubic nonlinear Schrodinger equation admits a large family of one-dimensional bounded traveling-wave solutions. All such solutions may be written in terms of an amplitude and a phase. Solutions with piecewise constant phase have been well studied previously. Some of these solutions were found to be stable with respect to one-dimensional perturbations. No such solutions are stable with respect to two-dimensional perturbations. Here we consider stability of the larger class of solutions whose phase is dependent on the spatial dimension of the one-dimensional wave form. We study the spectral stability of such nontrivial-phase solutions numerically, using Hill's method. We present evidence which suggests that all such nontrivial-phase solutions are unstable with respect to both one- and two-dimensional perturbations. Instability occurs in all cases: for both the elliptic and hyperbolic nonlinear Schrodinger equations, and in the focusing and defocusing case.Comment: Submitted: 13 pages, 3 figure

Linearly Coupled Bose-Einstein Condensates: From Rabi Oscillations and Quasi-Periodic Solutions to Oscillating Domain Walls and Spiral Waves

In this paper, an exact unitary transformation is examined that allows for the construction of solutions of coupled nonlinear Schr{\"o}dinger equations with additional linear field coupling, from solutions of the problem where this linear coupling is absent. The most general case where the transformation is applicable is identified. We then focus on the most important special case, namely the well-known Manakov system, which is known to be relevant for applications in Bose-Einstein condensates consisting of different hyperfine states of $^{87}$Rb. In essence, the transformation constitutes a distributed, nonlinear as well as multi-component generalization of the Rabi oscillations between two-level atomic systems. It is used here to derive a host of periodic and quasi-periodic solutions including temporally oscillating domain walls and spiral waves.Comment: 6 pages, 4 figures, Phys. Rev. A (in press

Predictive control for multi-market trade of aggregated demand response using a black box approach

Aggregated demand response for smart grid services is a growing field of interest especially for market participation. To minimize economic and network instability risks, flexibility characteristics such as shiftable capacity must be known. This is traditionally done using lower level, end user, device specifications. However, with these large numbers, having lower level information, has both privacy and computational limitations. Previous studies have shown that black box forecasting of shiftable capacity, using machine learning techniques, can be done accurately for a homogeneous cluster of heating devices. This paper validates the machine learning model for a heterogeneous virtual power plant. Further it applies this model to a control strategy to offer flexibility on an imbalance market while maintaining day ahead market obligations profitably. It is shown that using a black box approach 89% optimal economic performance is met. Further, by combining profits made on imbalance market and the day ahead costs, the overall monthly electricity costs are reduced 20%

Cerenkov-like radiation in a binary Schr{\"o}dinger flow past an obstacle

We consider the dynamics of two coupled miscible Bose-Einstein condensates, when an obstacle is dragged through them. The existence of two different speeds of sound provides the possibility for three dynamical regimes: when both components are subcritical, we do not observe nucleation of coherent structures; when both components are supercritical they both form dark solitons in one dimension (1D) and vortices or rotating vortex dipoles in two dimensions (2D); in the intermediate regime, we observe the nucleation of a structure in the form of a dark-antidark soliton in 1D; subcritical component; the 2D analog of such a structure, a vortex-lump, is also observed.Comment: 4 pages, 4 figures, submitted to Phys Rev

Nonsense mutations in alpha-II spectrin in three families with juvenile onset hereditary motor neuropathy

Distal hereditary motor neuropathies are a rare subgroup of inherited peripheral neuropathies hallmarked by a length-dependent axonal degeneration of lower motor neurons without significant involvement of sensory neurons. We identified patients with heterozygous nonsense mutations in the alpha II-spectrin gene, SPTAN1, in three separate dominant hereditary motor neuropathy families via next-generation sequencing. Variable penetrance was noted for these mutations in two of three families, and phenotype severity differs greatly between patients. The mutant mRNA containing nonsense mutations is broken down by nonsense-mediated decay and leads to reduced protein levels in patient cells. Previously, dominant-negative alpha II-spectrin gene mutations were described as causal in a spectrum of epilepsy phenotypes

On the numerical evaluation of algebro-geometric solutions to integrable equations

Physically meaningful periodic solutions to certain integrable partial differential equations are given in terms of multi-dimensional theta functions associated to real Riemann surfaces. Typical analytical problems in the numerical evaluation of these solutions are studied. In the case of hyperelliptic surfaces efficient algorithms exist even for almost degenerate surfaces. This allows the numerical study of solitonic limits. For general real Riemann surfaces, the choice of a homology basis adapted to the anti-holomorphic involution is important for a convenient formulation of the solutions and smoothness conditions. Since existing algorithms for algebraic curves produce a homology basis not related to automorphisms of the curve, we study symplectic transformations to an adapted basis and give explicit formulae for M-curves. As examples we discuss solutions of the Davey-Stewartson and the multi-component nonlinear Schr\"odinger equations.Comment: 29 pages, 20 figure

Pole dynamics for the Flierl-Petviashvili equation and zonal flow

We use a systematic method which allows us to identify a class of exact solutions of the Flierl-Petvishvili equation. The solutions are periodic and have one dimensional geometry. We examine the physical properties and find that these structures can have a significant effect on the zonal flow generation.Comment: Latex 40 pages, seven figures eps included. Effect of variation of g_3 is studied. New references adde