Untangling polygons and graphs

Untangling is a process in which some vertices of a planar graph are moved to obtain a straight-line plane drawing. The aim is to move as few vertices as possible. We present an algorithm that untangles the cycle graph C_n while keeping at least \Omega(n^{2/3}) vertices fixed. For any graph G, we also present an upper bound on the number of fixed vertices in the worst case. The bound is a function of the number of vertices, maximum degree and diameter of G. One of its consequences is the upper bound O((n log n)^{2/3}) for all 3-vertex-connected planar graphs.Comment: 11 pages, 3 figure

Study of the oscillation of a pendulum in a magnetic field

RVO: 6138899

Recognition and coacervation of G-quadruplexes by a multifunctional disordered region in RECQ4 helicase

Biomolecular polyelectrolyte complexes can be formed between oppositely charged intrinsically disordered regions (IDRs) of proteins or between IDRs and nucleic acids. Highly charged IDRs are abundant in the nucleus, yet few have been functionally characterized. Here, we show that a positively charged IDR within the human ATP-dependent DNA helicase Q4 (RECQ4) forms coacervates with G-quadruplexes (G4s). We describe a three-step model of charge-driven coacervation by integrating equilibrium and kinetic binding data in a global numerical model. The oppositely charged IDR and G4 molecules form a complex in the solution that follows a rapid nucleation-growth mechanism leading to a dynamic equilibrium between dilute and condensed phases. We also discover a physical interaction with Replication Protein A (RPA) and demonstrate that the IDR can switch between the two extremes of the structural continuum of complexes. The structural, kinetic, and thermodynamic profile of its interactions revealed a dynamic disordered complex with nucleic acids and a static ordered complex with RPA protein. The two mutually exclusive binding modes suggest a regulatory role for the IDR in RECQ4 function by enabling molecular handoffs. Our study extends the functional repertoire of IDRs and demonstrates a role of polyelectrolyte complexes involved in G4 binding

The Radius of Metric Subregularity

There is a basic paradigm, called here the radius of well-posedness, which quantifies the "distance" from a given well-posed problem to the set of ill-posed problems of the same kind. In variational analysis, well-posedness is often understood as a regularity property, which is usually employed to measure the effect of perturbations and approximations of a problem on its solutions. In this paper we focus on evaluating the radius of the property of metric subregularity which, in contrast to its siblings, metric regularity, strong regularity and strong subregularity, exhibits a more complicated behavior under various perturbations. We consider three kinds of perturbations: by Lipschitz continuous functions, by semismooth functions, and by smooth functions, obtaining different expressions/bounds for the radius of subregularity, which involve generalized derivatives of set-valued mappings. We also obtain different expressions when using either Frobenius or Euclidean norm to measure the radius. As an application, we evaluate the radius of subregularity of a general constraint system. Examples illustrate the theoretical findings.Comment: 20 page

Slogging and Stumbling Toward Social Justice in a Private Elementary School: The Complicated Case of St. Malachy

This case study examines St. Malachy, an urban Catholic elementary school primarily serving children traditionally marginalized by race, class, linguistic heritage, and disability. As a private school, St. Malachy serves the public good by recruiting and retaining such traditionally marginalized students. As empirical studies involving Catholic schools frequently juxtapose them with public schools, the author presents this examination from a different tack. Neither vilifying nor glorifying Catholic schooling, this study critically examines the pursuit of social justice in this school context. Data gathered through a 1-year study show that formal and informal leaders in St. Malachy adapted their governance, aggressively sought community resources, and focused their professional development to build the capacity to serve their increasingly pluralistic student population. The analysis confirms the deepening realization that striving toward social justice is a messy, contradictory, and complicated pursuit, and that schools in both public and private sectors are allies in this pursuit

Variational Analysis Down Under Open Problem Session

© 2018, Springer Science+Business Media, LLC, part of Springer Nature. We state the problems discussed in the open problem session at Variational Analysis Down Under conference held in honour of Prof. Asen Dontchev on 19–21 February 2018 at Federation University Australia