Certifying reality of projections

Computational tools in numerical algebraic geometry can be used to numerically approximate solutions to a system of polynomial equations. If the system is well-constrained (i.e., square), Newton's method is locally quadratically convergent near each nonsingular solution. In such cases, Smale's alpha theory can be used to certify that a given point is in the quadratic convergence basin of some solution. This was extended to certifiably determine the reality of the corresponding solution when the polynomial system is real. Using the theory of Newton-invariant sets, we certifiably decide the reality of projections of solutions. We apply this method to certifiably count the number of real and totally real tritangent planes for instances of curves of genus 4.Comment: 9 page

Renal function impairment predicts mortality in patients with chronic heart failure treated with resynchronization therapy

Background: The use of cardiac resynchronization therapy (CRT) and implantable cardioverter-defibrillator (ICD) for advanced heart failure (HF) is increasing. Renal dysfunction is a common condition in HF which is associated with a worse survival. The study aims at identifying in patients with advanced HF treated with CRT the effect of baseline glomerular filtration rate (GFR), GFR improvement and left ventricular ejection fraction (LVEF) change, after 6-months of CRT implant, on survival. Methods: The study population consisted of 375 advanced HF patients who received a CRT between 1999 and 2009, of these 277 received also an ICD implant. Clinical characteristics (New York Heart Association [NYHA] functional class, ischemic vs. non-ischemic etiology, atrial fibrillation, diabetes, hypertension, LVEF, QRS duration and GFR were recorded. The use of common used drugs was evaluated. Cox proportional hazards analysis was calculated in order to evaluate variables associated to mortality. Results: During a median follow-up of 43.0 months, 93 (24.8%) patients died. Patients deceased during the study had at baseline higher NYHA class and lower LVEF and GFR. In Cox regression analysis, GFR predicts long-term mortality (hazard ratio [HR] 0.983; 95% confidence interval [CI] 0.969\u20130.998; p = 0.023) independently from the effect of others covariates. In addition, a positive GFR improvement 6 months after CRT implant is significantly associated with a lower hazard of mortality (for each 10 mL/min of GFR improvement HR 0.86; 95% CI 0.75\u20130.99; p = 0.038). Conclusions: GFR is a significant predictor of mortality in advanced HF patients who received CRT. A GFR improvement 6 months after CRT implant is significantly associated with a lower hazard of mortality

Polynomial-Time Amoeba Neighborhood Membership and Faster Localized Solving

We derive efficient algorithms for coarse approximation of algebraic hypersurfaces, useful for estimating the distance between an input polynomial zero set and a given query point. Our methods work best on sparse polynomials of high degree (in any number of variables) but are nevertheless completely general. The underlying ideas, which we take the time to describe in an elementary way, come from tropical geometry. We thus reduce a hard algebraic problem to high-precision linear optimization, proving new upper and lower complexity estimates along the way.Comment: 15 pages, 9 figures. Submitted to a conference proceeding

Numerical Algebraic Geometry: A New Perspective on String and Gauge Theories

The interplay rich between algebraic geometry and string and gauge theories has recently been immensely aided by advances in computational algebra. However, these symbolic (Gr\"{o}bner) methods are severely limited by algorithmic issues such as exponential space complexity and being highly sequential. In this paper, we introduce a novel paradigm of numerical algebraic geometry which in a plethora of situations overcomes these short-comings. Its so-called 'embarrassing parallelizability' allows us to solve many problems and extract physical information which elude the symbolic methods. We describe the method and then use it to solve various problems arising from physics which could not be otherwise solved.Comment: 36 page

Four lectures on secant varieties

This paper is based on the first author's lectures at the 2012 University of Regina Workshop "Connections Between Algebra and Geometry". Its aim is to provide an introduction to the theory of higher secant varieties and their applications. Several references and solved exercises are also included.Comment: Lectures notes to appear in PROMS (Springer Proceedings in Mathematics & Statistics), Springer/Birkhause

Statistical-Thermodynamic Model for Light Scattering from Eye Lens Protein Mixtures

We model light-scattering cross sections of concentrated aqueous mixtures of the bovine eye lens proteins γB- and α-crystallin by adapting a statistical-thermodynamic model of mixtures of spheres with short-range attractions. The model reproduces measured static light scattering cross sections, or Rayleigh ratios, of γB-α mixtures from dilute concentrations where light scattering intensity depends on molecular weights and virial coefficients, to realistically high concentration protein mixtures like those of the lens. The model relates γB-γB and γB-α attraction strengths and the γB-α size ratio to the free energy curvatures that set light scattering efficiency in tandem with protein refractive index increments. The model includes (i) hard-sphere α-α interactions, which create short-range order and transparency at high protein concentrations, (ii) short-range attractive plus hard-core γ-γ interactions, which produce intense light scattering and liquid-liquid phase separation in aqueous γ-crystallin solutions, and (iii) short-range attractive plus hard-core γ-α interactions, which strongly influence highly non-additive light scattering and phase separation in concentrated γ-α mixtures. The model reveals a new lens transparency mechanism, that prominent equilibrium composition fluctuations can be perpendicular to the refractive index gradient. The model reproduces the concave-up dependence of the Rayleigh ratio on α/γ composition at high concentrations, its concave-down nature at intermediate concentrations, non-monotonic dependence of light scattering on γ-α attraction strength, and more intricate, temperature-dependent features. We analytically compute the mixed virial series for light scattering efficiency through third order for the sticky-sphere mixture, and find that the full model represents the available light scattering data at concentrations several times those where the second and third mixed virial contributions fail. The model indicates that increased γ-γ attraction can raise γ-α mixture light scattering far more than it does for solutions of γ-crystallin alone, and can produce marked turbidity tens of degrees celsius above liquid-liquid separation

Semidefinite Characterization and Computation of Real Radical Ideals

For an ideal $I\subseteq\mathbb{R}[x]$ given by a set of generators, a new semidefinite characterization of its real radical $I(V_\mathbb{R}(I))$ is presented, provided it is zero-dimensional (even if $I$ is not). Moreover we propose an algorithm using numerical linear algebra and semidefinite optimization techniques, to compute all (finitely many) points of the real variety $V_\mathbb{R}(I)$ as well as a set of generators of the real radical ideal. The latter is obtained in the form of a border or Gr\"obner basis. The algorithm is based on moment relaxations and, in contrast to other existing methods, it exploits the real algebraic nature of the problem right from the beginning and avoids the computation of complex components.Comment: 41 page