Numerical algebraic geometry for model selection and its application to the life sciences

Researchers working with mathematical models are often confronted by the related problems of parameter estimation, model validation, and model selection. These are all optimization problems, well-known to be challenging due to non-linearity, non-convexity and multiple local optima. Furthermore, the challenges are compounded when only partial data is available. Here, we consider polynomial models (e.g., mass-action chemical reaction networks at steady state) and describe a framework for their analysis based on optimization using numerical algebraic geometry. Specifically, we use probability-one polynomial homotopy continuation methods to compute all critical points of the objective function, then filter to recover the global optima. Our approach exploits the geometric structures relating models and data, and we demonstrate its utility on examples from cell signaling, synthetic biology, and epidemiology.Comment: References added, additional clarification

Federal Pell Grant Eligibility and Receipt: Explaining Nonreceipt and Changes to EFC Using National and Institutional Data

In examining national data on Federal Pell Grant eligibility in the National Postsecondary Student Aid Study (NPSAS), we were puzzled to discover that many students who appear to have eligible Expected Family Contributions (EFCs) do not receive the award. We use institutional data from a large public university to understand and enumerate changes from initial Free Application for Student Financial Aid (FAFSA) EFC to final Pell Grant EFC and explore why EFC changes occur. We determine that the nonreceipt of Pell Grant observed in NPSAS is likely due to NPSAS not reporting final Pell Grant EFCs. We examine how the verification process results in changes to EFC and describe how nearly half of students who experienced a change in EFC during the award year were not asked to verify. We also observe that selection for Quality Assurance verification and EFC changes varied based on students’ demographics characteristics. The paper concludes with discussion of improving the verification process

Multi-site mean-field theory for cold bosonic atoms in optical lattices

We present a detailed derivation of a multi-site mean-field theory (MSMFT) used to describe the Mott-insulator to superfluid transition of bosonic atoms in optical lattices. The approach is based on partitioning the lattice into small clusters which are decoupled by means of a mean field approximation. This approximation invokes local superfluid order parameters defined for each of the boundary sites of the cluster. The resulting MSMFT grand potential has a non-trivial topology as a function of the various order parameters. An understanding of this topology provides two different criteria for the determination of the Mott insulator superfluid phase boundaries. We apply this formalism to $d$-dimensional hypercubic lattices in one, two and three dimensions, and demonstrate the improvement in the estimation of the phase boundaries when MSMFT is utilized for increasingly larger clusters, with the best quantitative agreement found for $d=3$. The MSMFT is then used to examine a linear dimer chain in which the on-site energies within the dimer have an energy separation of $\Delta$. This system has a complicated phase diagram within the parameter space of the model, with many distinct Mott phases separated by superfluid regions.Comment: 30 pages, 23 figures, accepted for publication in Phys. Rev.

Light-cone coordinates based at a geodesic world line

Continuing work initiated in an earlier publication [Phys. Rev. D 69, 084007 (2004)], we construct a system of light-cone coordinates based at a geodesic world line of an arbitrary curved spacetime. The construction involves (i) an advanced-time or a retarded-time coordinate that labels past or future light cones centered on the world line, (ii) a radial coordinate that is an affine parameter on the null generators of these light cones, and (iii) angular coordinates that are constant on each generator. The spacetime metric is calculated in the light-cone coordinates, and it is expressed as an expansion in powers of the radial coordinate in terms of the irreducible components of the Riemann tensor evaluated on the world line. The formalism is illustrated in two simple applications, the first involving a comoving world line of a spatially-flat cosmology, the other featuring an observer placed on the axis of symmetry of Melvin's magnetic universe.Comment: 11 pages, 1 figur

Principles, limitations, and performance of multiconjugate adaptive optics

Multi-Conjugate Adaptive Optics (MCAO) holds the promise of moderate to large adaptively compensated field of view with uniform image quality. This paper is a first effort to analyze the fundamental limitations of such systems, and that are mainly related to the finite number of deformable mirrors and guide stars. We demonstrate that the ultimate limitation is due to the vertical discretization of the correction. This effect becomes more severe quite rapidly with increasing compensated field of view or decreasing wavelength, but does not depend at first order on the telescope aperture. We also discuss limitations associated with the use of laser guide stars and ELT related issues

The post-common-envelope, binary central star of the planetary nebula Hen 2-11

We present a detailed photometric study of the central star system of the planetary nebula Hen 2-11, selected for study because of its low-ionisation filaments and bipolar morphology - traits which have been strongly linked with central star binarity. Photometric monitoring with NTT-EFOSC2 reveals a highly irradiated, double-eclipsing, post-common-envelope system with a period of 0.609 d. Modelling of the lightcurve indicates that the nebular progenitor is extremely hot, while the secondary in the system is probably a K-type main sequence star. The chemical composition of the nebula is analysed, showing Hen 2-11 to be a medium-excitation non-Type I nebula. A simple photoionisation model is constructed determining abundance ratios of C/O and N/O which would be consistent with the common-envelope cutting short the AGB evolution of the nebular progenitor. The detection of a post-common-envelope binary system at the heart of Hen 2-11 further strengthens the link between binary progeny and the formation of axisymmetric planetary nebulae with patterns of low-ionisation filaments, clearly demonstrating their use as morphological indicators of central star binarity.Comment: Accepted for publication in A&A. 9 pages, 5 figures, 4 table

Success Despite Socioeconomics: A Case Study of a High-Achieving, High-Poverty School

Effective school leadership is becoming more difficult than ever with the challenges of increased accountability and high stakes testing that are components of federal No Child Left Behind (NCLB) legislation. These challenges are more pronounced in schools with high rates of poverty. This was a case study of a high performing, high poverty school that has consistently been one of the highest performing elementary schools in the state. The purpose of the study was to describe the leadership that exists at the school, the culture of the school, and programs that contribute to the school’s success. The researcher conducted observations at the school site and interviews with school personnel. School personnel also completed the School Culture Survey and the Vanderbilt Assessment of Leadership in Education (VAL-ED). The study revealed school leadership that had high expectations for staff members and emphasized small group instruction, collaboration, and continuous improvement in instructional practices. The culture of the school was that of excellence, continuous improvement, school pride, and collaboration