The averaged characteristic polynomial for the Gaussian and chiral Gaussian ensembles with a source

In classical random matrix theory the Gaussian and chiral Gaussian random matrix models with a source are realized as shifted mean Gaussian, and chiral Gaussian, random matrices with real $(\beta = 1)$, complex ($\beta = 2)$ and real quaternion $(\beta = 4$) elements. We use the Dyson Brownian motion model to give a meaning for general $\beta > 0$. In the Gaussian case a further construction valid for $\beta > 0$ is given, as the eigenvalue PDF of a recursively defined random matrix ensemble. In the case of real or complex elements, a combinatorial argument is used to compute the averaged characteristic polynomial. The resulting functional forms are shown to be a special cases of duality formulas due to Desrosiers. New derivations of the general case of Desrosiers' dualities are given. A soft edge scaling limit of the averaged characteristic polynomial is identified, and an explicit evaluation in terms of so-called incomplete Airy functions is obtained.Comment: 21 page

Explicit formulas for the generalized Hermite polynomials in superspace

We provide explicit formulas for the orthogonal eigenfunctions of the supersymmetric extension of the rational Calogero-Moser-Sutherland model with harmonic confinement, i.e., the generalized Hermite (or Hi-Jack) polynomials in superspace. The construction relies on the triangular action of the Hamiltonian on the supermonomial basis. This translates into determinantal expressions for the Hamiltonian's eigenfunctions.Comment: 19 pages. This is a recasting of the second part of the first version of hep-th/0305038 which has been splitted in two articles. In this revised version, the introduction has been rewritten and a new appendix has been added. To appear in JP

Macdonald polynomials in superspace: conjectural definition and positivity conjectures

We introduce a conjectural construction for an extension to superspace of the Macdonald polynomials. The construction, which depends on certain orthogonality and triangularity relations, is tested for high degrees. We conjecture a simple form for the norm of the Macdonald polynomials in superspace, and a rather non-trivial expression for their evaluation. We study the limiting cases q=0 and q=\infty, which lead to two families of Hall-Littlewood polynomials in superspace. We also find that the Macdonald polynomials in superspace evaluated at q=t=0 or q=t=\infty seem to generalize naturally the Schur functions. In particular, their expansion coefficients in the corresponding Hall-Littlewood bases appear to be polynomials in t with nonnegative integer coefficients. More strikingly, we formulate a generalization of the Macdonald positivity conjecture to superspace: the expansion coefficients of the Macdonald superpolynomials expanded into a modified version of the Schur superpolynomial basis (the q=t=0 family) are polynomials in q and t with nonnegative integer coefficients.Comment: 18 page

Spectral dimension reduction of complex dynamical networks

Dynamical networks are powerful tools for modeling a broad range of complex systems, including financial markets, brains, and ecosystems. They encode how the basic elements (nodes) of these systems interact altogether (via links) and evolve (nodes' dynamics). Despite substantial progress, little is known about why some subtle changes in the network structure, at the so-called critical points, can provoke drastic shifts in its dynamics. We tackle this challenging problem by introducing a method that reduces any network to a simplified low-dimensional version. It can then be used to describe the collective dynamics of the original system. This dimension reduction method relies on spectral graph theory and, more specifically, on the dominant eigenvalues and eigenvectors of the network adjacency matrix. Contrary to previous approaches, our method is able to predict the multiple activation of modular networks as well as the critical points of random networks with arbitrary degree distributions. Our results are of both fundamental and practical interest, as they offer a novel framework to relate the structure of networks to their dynamics and to study the resilience of complex systems.Comment: 16 pages, 8 figure

Airborne radar quality control and analysis of the rapid intensification of Hurricane Michael (2018)

2020 Fall.Includes bibliographical references.Improvements made by the National Hurricane Center (NHC) in track forecasts have outpaced advances in intensity forecasting. Rapid intensification (RI), an increase of at least 30 knots in the maximum sustained winds of a tropical cyclone (TC) in a 24 hour period, is poorly understood and provides a considerable hurdle to intensity forecasting. RI depends on internal processes which require detailed inner core information to better understand. Close range measurements of TCs from aircraft reconnaissance with tail Doppler radar (TDR) allow for the retrieval of the kinematic state of the inner core. Fourteen consecutive passes were flown through Hurricane Michael (2018) as it underwent RI on its way to landfall at category 5 intensity. The TDR data collected offered an exceptional opportunity to diagnose mechanisms that contributed to RI. Quality Control (QC) is required to remove radar gates originating from non meteorological sources which can impair dual-Doppler wind synthesis techniques. Automation of the time-consuming manual QC process was needed to utilize all TDR data collected in Hurricane Michael in a timely manner. The machine learning (ML) random forest technique was employed to create a generalized QC method for TDR data collected in convective environments. The complex decision making ability of ML offered an advantage over past approaches. A dataset of radar scans from a tornadic supercell, bow echo, and mature and developing TCs collected by the Electra Doppler Radar (ELDORA) containing approximately 87.9 million radar gates was mined for predictors. Previous manual QC performed on the data was used to classify each data point as weather or non-weather. This varied dataset was used to train a model which classified over 99% of the radar gates in the withheld testing data succesfully. Creation of a dual-Doppler analysis from a tropical depression using ML efforts that was comparable to manual QC confirmed the utility of this new method. The framework developed was capable of performing QC on the majority of the TDR data from Hurricane Michael. Analyses of the inner core of Hurricane Michael were used to document inner core changes throughout RI. Angular momentum surfaces moved radially inward and became more vertically aligned over time. The hurricane force wind field expanded radially outward and increased in depth. Intensification of the storm became predominantly axisymmetric as RI progressed. TDR-derived winds are used to infer upper-level processes that influenced RI at the surface. Tilting of ambient horizontal vorticity, created by the decay of tangential winds aloft, by the axisymmetric updraft created a positive vorticity tendency atop the existing vorticity tower. A vorticity budget helped demonstrate how the axisymmetric vorticity tower built both upward and outward in the sloped eyewall. A retrieval of the radial gradient of density temperature provided evidence for an increasing warm core temperature perturbation in the eye. Growth of the warm core temperature perturbation in upper levels aided by subsidence helped lower the minimum sea level pressure which correlated with intensification of the near-surface wind field

Jack superpolynomials with negative fractional parameter: clustering properties and super-Virasoro ideals

The Jack polynomials P_\lambda^{(\alpha)} at \alpha=-(k+1)/(r-1) indexed by certain (k,r,N)-admissible partitions are known to span an ideal I^{(k,r)}_N of the space of symmetric functions in N variables. The ideal I^{(k,r)}_N is invariant under the action of certain differential operators which include half the Virasoro algebra. Moreover, the Jack polynomials in I^{(k,r)}_N admit clusters of size at most k: they vanish when k+1 of their variables are identified, and they do not vanish when only k of them are identified. We generalize most of these properties to superspace using orthogonal eigenfunctions of the supersymmetric extension of the trigonometric Calogero-Moser-Sutherland model known as Jack superpolynomials. In particular, we show that the Jack superpolynomials P_{\Lambda}^{(\alpha)} at \alpha=-(k+1)/(r-1) indexed by certain (k,r,N)-admissible superpartitions span an ideal {\mathcal I}^{(k,r)}_N of the space of symmetric polynomials in N commuting variables and N anticommuting variables. We prove that the ideal {\mathcal I}^{(k,r)}_N is stable with respect to the action of the negative-half of the super-Virasoro algebra. In addition, we show that the Jack superpolynomials in {\mathcal I}^{(k,r)}_N vanish when k+1 of their commuting variables are equal, and conjecture that they do not vanish when only k of them are identified. This allows us to conclude that the standard Jack polynomials with prescribed symmetry should satisfy similar clustering properties. Finally, we conjecture that the elements of {\mathcal I}^{(k,2)}_N provide a basis for the subspace of symmetric superpolynomials in N variables that vanish when k+1 commuting variables are set equal to each other.Comment: 36 pages; the main changes in v2 are : 1) in the introduction, we present exceptions to an often made statement concerning the clustering property of the ordinary Jack polynomials for (k,r,N)-admissible partitions (see Footnote 2); 2) Conjecture 14 is substantiated with the extensive computational evidence presented in the new appendix C; 3) the various tests supporting Conjecture 16 are reporte

Curricular initiatives that enhance student knowledge and perceptions of sexual and gender minority groups: a critical interpretive synthesis

Background: There is no accepted best practice for optimizing tertiary student knowledge, perceptions, and skills to care for sexual and gender diverse groups. The objective of this research was to synthesize the relevant literature regarding effective curricular initiatives designed to enhance tertiary level student knowledge, perceptions, and skills to care for sexual and gender diverse populations.Methods: A modified Critical Interpretive Synthesis using a systematic search strategy was conducted in 2015. This method was chosen to synthesize the relevant qualitative and quantitative literature as it allows for the depth and breadth of information to be captured and new constructs to be illuminated. Databases searched include AMED, CINAHL EBM Reviews, ERIC, Ovid MEDLINE, Ovid Nursing Database, PsychInfo, and Google Scholar. Results: Thirty-one articles were included in this review. Curricular initiatives ranging from discrete to multimodal approaches have been implemented. Successful initiatives included discrete sessions with time for processing, and multi-modal strategies. Multi-modal approaches that encouraged awareness of one’s lens and privilege in conjunction with facilitated communication seemed the most effective.Conclusions: The literature is limited to the evaluation of explicit curricula. The wider cultural competence literature offers further insight by highlighting the importance of broad and embedded forces including social influences, the institutional climate, and the implicit, or hidden, curriculum. A combined interpretation of the complementary cultural competence and sexual and gender diversity literature provides a novel understanding of the optimal content and context for the delivery of a successful curricular initiative

Time Series Forecasting by Recommendation: An Empirical Analysis on Amazon Marketplace

This study proposes a forecasting methodology for univari ate time series (TS) using a Recommender System (RS). The RS is built from a given TS as only input data and following an item-based Collabo rative Filtering approach. A set of top-N values is recommended for this TS which represent the forecasts. The idea is to emulate RS elements (the users, items and ratings triple) from the TS. Two TS obtained from Italy’s Amazon webpage were used to evaluate this methodology and very promising performance results were obtained, even the difficult environ ment chosen to conduct forecasting (short length and unevenly spaced TS). This performance is dependent on the similarity measure used and suffers from the same problems that other RSs (e.g., cold-start). However, this approach does not require high computational power to perform and its intuitive conception allows for being deployed with any programming language

Universality of the stochastic block model

Mesoscopic pattern extraction (MPE) is the problem of finding a partition of the nodes of a complex network that maximizes some objective function. Many well-known network inference problems fall in this category, including, for instance, community detection, core-periphery identification, and imperfect graph coloring. In this paper, we show that the most popular algorithms designed to solve MPE problems can in fact be understood as special cases of the maximum likelihood formulation of the stochastic block model (SBM), or one of its direct generalizations. These equivalence relations show that the SBM is nearly universal with respect to MPE problems.Comment: 13 pages, 4 figure