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

A Special Homotopy Continuation Method For A Class of Polynomial Systems

A special homotopy continuation method, as a combination of the polyhedral homotopy and the linear product homotopy, is proposed for computing all the isolated solutions to a special class of polynomial systems. The root number bound of this method is between the total degree bound and the mixed volume bound and can be easily computed. The new algorithm has been implemented as a program called LPH using C++. Our experiments show its efficiency compared to the polyhedral or other homotopies on such systems. As an application, the algorithm can be used to find witness points on each connected component of a real variety

Numerical elimination and moduli space of vacua

We propose a new computational method to understand the vacuum moduli space of (supersymmetric) field theories. By combining numerical algebraic geometry (NAG) and elimination theory, we develop a powerful, efficient, and parallelizable algorithm toextract important information such as the dimension, branch structure, Hilbert series and subsequent operator counting, as well as variation according to coupling constants and mass parameters. We illustrate this method on a host of examples from gauge theory, string theory, and algebraic geometry

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

Efficacy and cost-effectiveness of two online interventions for children and adolescents at risk for depression (E.motion trial): study protocol for a randomized controlled trial within the ProHEAD consortium

Background: Depression is a serious mental health problem and is common in children and adolescents. Online interventions are promising in overcoming the widespread undertreatment of depression and in improving the help-seeking behavior in children and adolescents. Methods: The multicentre, randomized controlled E.motion trial is part of the German ProHEAD consortium (Promoting Help-seeking using E-technology for ADolescents). The objective of the trial is to investigate the efficacy and cost-effectiveness of two online interventions to reduce depressive symptomatology in high-risk children and adolescents with subsyndromal symptoms of depression in comparison to an active control group. Participants will be randomized to one of three conditions: (1) Intervention 1, a clinician-guided self-management program (iFightDepression®); (2) Intervention 2, a clinician-guided group chat intervention; and (3) Control intervention, a psycho-educational website on depressive symptoms. Interventions last six weeks. In total, N = 363 children and adolescents aged ≥ 12 years with Patient Health Questionnaire-9 modified for Adolescents (PHQ-A) scores in the range of 5–9 will be recruited at five study sites across Germany. Online questionnaires will be administered before onset of the intervention, at the end of the intervention, and at the six-month follow-up. Further, children and adolescents will participate in the baseline screening and the one- and two-year school-based follow-up assessments integrated in the ProHEAD consortium. The primary endpoint is depression symptomatology at the end of intervention as measured by the PHQ-A score. Secondary outcomes include depression symptomatology at all follow-ups, help-seeking attitudes, and actual face-to-face help-seeking, adherence to and satisfaction with the interventions, depression stigma, and utilization and cost of interventions. Discussion: This study represents the first randomized controlled trial (RCT) investigating efficacy and cost-effectiveness of two online interventions in children and adolescents aged ≥ 12 years at risk for depression. It aims to provide a better understanding of the help-seeking behavior of children and adolescents, potential benefits of E-mental health interventions for this age group, and new insights into so far understudied aspects of E-mental health programs, such as potential negative effects of online interventions. This knowledge will be used to tailor and improve future help offers and programs for children and adolescents and ways of treatment allocation. Trial registration: German Register for Clinical Trials (DRKS), DRKS00014668. Registered on 4 May 2018. International trial registration took place through the “international clinical trials registry platform” with the secondary ID S-086/2018

Energy-landscape analysis of the two-dimensional nearest-neighbor (phi)<SUP>4</SUP> model

NatuurwetenskappeInstituut Vir Teoretiese FisikaPlease help us populate SUNScholar with the post print version of this article. It can be e-mailed to: [email protected]

Energy-landscape analysis of the two-dimensional nearest-neighbor (phi)<SUP>4</SUP> model

NatuurwetenskappeInstituut Vir Teoretiese FisikaPlease help us populate SUNScholar with the post print version of this article. It can be e-mailed to: [email protected]