Conditional Probabilities of Multivariate Poisson Distributions

Multivariate Poisson distributions have numerous applications. Fast computation of these distributions, holding constant a fixed set of linear combinations of these variables, has been explored by Sontag and Zeilberger. This elaborates on their work

Actions of nilpotent groups on nilpotent groups

For finite nilpotent groups J and N, suppose J acts on N via automorphisms. We exhibit a decomposition of the first cohomology set in terms of the first cohomologies of the Sylow p-subgroups of J that mirrors the primary decomposition of H1(J, N) for abelian N. We then show that if N ⋊ J acts on some non-empty set Ω, where the action of N is transitive and for each prime p a Sylowp-subgroup of J fixes an element of Ω, then J fixes an element of Ω

Local conjugacy and primary-type decompositions in nonabelian cohomology

In 1979, Losey and Stonehewer (Warwick) introduced local conjugacy and provided some conditions under which locally conjugate complements J & J' to a normal nilpotent subgroup N must be conjugate. We will interpret their results in the language of nonabelian group cohomology and relate them to the existence of a primary-type decomposition of the first cohomology set H^1(J,N). We will cover the cases: N is abelian, J is nilpotent, and N⋊J is supersoluble. This decomposition will also provide us with an inclusion-based version of their result. We will motivate our discussion in terms of fixed point results in the style of Glauberman (Chicago)

Discriminative Bayesian filtering lends momentum to the stochastic Newton method for minimizing log-convex functions

To minimize the average of a set of log-convex functions, the stochastic Newton method iteratively updates its estimate using subsampled versions of the full objective's gradient and Hessian. We contextualize this optimization problem as sequential Bayesian inference on a latent state-space model with a discriminatively-specified observation process. Applying Bayesian filtering then yields a novel optimization algorithm that considers the entire history of gradients and Hessians when forming an update. We establish matrix-based conditions under which the effect of older observations diminishes over time, in a manner analogous to Polyak's heavy ball momentum. We illustrate various aspects of our approach with an example and review other relevant innovations for the stochastic Newton method

The Short Rotation Period of Hi'iaka, Haumea's Largest Satellite

Hi'iaka is the larger outer satellite of the dwarf planet Haumea. Using relative photometry from the Hubble Space Telescope and Magellan and a phase dispersion minimization analysis, we have identified the rotation period of Hi'iaka to be ~9.8 hrs (double-peaked). This is ~120 times faster than its orbital period, creating new questions about the formation of this system and possible tidal evolution. The rapid rotation suggests that Hi'iaka could have a significant obliquity and spin precession that could be visible in light curves within a few years. We then turn to an investigation of what we learn about the (presently unclear) formation of the Haumea system and family based on this unexpectedly rapid rotation rate. We explore the importance of the initial semi-major axis and rotation period in tidal evolution theory and find they strongly influence the time required to despin to synchronous rotation, relevant to understanding a wide variety of satellite and binary systems. We find that despinning tides do not necessarily lead to synchronous spin periods for Hi'iaka, even if it formed near the Roche limit. Therefore the short rotation period of Hi'iaka does not rule out significant tidal evolution. Hi'iaka's spin period is also consistent with formation near its current location and spin up due to Haumea-centric impactors.Comment: 21 pages with 6 figures, to be published in The Astronomical Journa

The orbits of the quadruple star system 88 Tau A from PHASES differential astrometry and radial velocity

We have used high precision differential astrometry from the Palomar High-precision Astrometric Search for Exoplanet Systems (PHASES) project and radial velocity measurements covering a time-span of 20 years to determine the orbital parameters of the 88 Tau A system. 88 Tau is a complex hierarchical multiple system comprising a total of six stars; we have studied the brightest 4, consisting of two short-period pairs orbiting each other with an 18-year period. We present the first orbital solution for one of the short-period pairs, and determine the masses of the components and distance to the system to the level of a few percent. In addition, our astrometric measurements allow us to make the first determination of the mutual inclinations of the orbits. We find that the sub-systems are not coplanar.Comment: Corrected Author Ordering; 12 Pages, Accepted for publication in Ap

Unsupervised multimodal modeling of cognitive and brain health trajectories for early dementia prediction

Predicting the course of neurodegenerative disorders early has potential to greatly improve clinical management and patient outcomes. A key challenge for early prediction in real-world clinical settings is the lack of labeled data (i.e., clinical diagnosis). In contrast to supervised classification approaches that require labeled data, we propose an unsupervised multimodal trajectory modeling (MTM) approach based on a mixture of state space models that captures changes in longitudinal data (i.e., trajectories) and stratifies individuals without using clinical diagnosis for model training. MTM learns the relationship between states comprising expensive, invasive biomarkers (β-amyloid, grey matter density) and readily obtainable cognitive observations. MTM training on trajectories stratifies individuals into clinically meaningful clusters more reliably than MTM training on baseline data alone and is robust to missing data (i.e., cognitive data alone or single assessments). Extracting an individualized cognitive health index (i.e., MTM-derived cluster membership index) allows us to predict progression to AD more precisely than standard clinical assessments (i.e., cognitive tests or MRI scans alone). Importantly, MTM generalizes successfully from research cohort to real-world clinical data from memory clinic patients with missing data, enhancing the clinical utility of our approach. Thus, our multimodal trajectory modeling approach provides a cost-effective and non-invasive tool for early dementia prediction without labeled data (i.e., clinical diagnosis) with strong potential for translation to clinical practice

The Two States of Star Forming Clouds

We examine the effects of self-gravity and magnetic fields on supersonic turbulence in isothermal molecular clouds with high resolution simulations and adaptive mesh refinement. These simulations use large root grids (512^3) to capture turbulence and four levels of refinement to capture high density, for an effective resolution of 8,196^3. Three Mach 9 simulations are performed, two super-Alfv\'enic and one trans-Alfv\'enic. We find that gravity splits the clouds into two populations, one low density turbulent state and one high density collapsing state. The low density state exhibits properties similar to non-self-gravitating in this regime, and we examine the effects of varied magnetic field strength on statistical properties: the density probability distribution function is approximately lognormal; velocity power spectral slopes decrease with field strength; alignment between velocity and magnetic field increases with field; the magnetic field probability distribution can be fit to a stretched exponential. The high density state is characterized by self-similar spheres; the density PDF is a power-law; collapse rate decreases with increasing mean field; density power spectra have positive slopes, P({\rho},k) \propto k; thermal-to-magnetic pressure ratios are unity for all simulations; dynamic-to-magnetic pressure ratios are larger than unity for all simulations; magnetic field distribution is a power-law. The high Alfv\'en Mach numbers in collapsing regions explain recent observations of magnetic influence decreasing with density. We also find that the high density state is found in filaments formed by converging flows, consistent with recent Herschel observations. Possible modifications to existing star formation theories are explored.Comment: 19 pages, 20 figure

Robust and interpretable AI-guided marker for early dementia prediction in real-world clinical settings

Background: Predicting dementia early has major implications for clinical management and patient outcomes. Yet, we still lack sensitive tools for stratifying patients early, resulting in patients being undiagnosed or wrongly diagnosed. Despite rapid expansion in machine learning models for dementia prediction, limited model interpretability and generalizability impede translation to the clinic.Methods: We build a robust and interpretable predictive prognostic model (PPM) and validate its clinical utility using real-world, routinely-collected, non-invasive, and low-cost (cognitive tests, structural MRI) patient data. To enhance scalability and generalizability to the clinic, we: 1) train the PPM with clinically-relevant predictors (cognitive tests, grey matter atrophy) that are common across research and clinical cohorts, 2) test PPM predictions with independent multicenter real-world data from memory clinics across countries (UK, Singapore).Findings: PPM robustly predicts (accuracy: 81.66%, AUC: 0.84, sensitivity: 82.38%, specificity: 80.94%) whether patients at early disease stages (MCI) will remain stable or progress to Alzheimer's Disease (AD). PPM generalizes from research to real-world patient data across memory clinics and its predictions are validated against longitudinal clinical outcomes. PPM allows us to derive an individualized AI-guided multimodal marker (i.e. predictive prognostic index) that predicts progression to AD more precisely than standard clinical markers (grey matter atrophy, cognitive scores; PPM-derived marker: hazard ratio = 3.42, p = 0.01) or clinical diagnosis (PPM-derived marker: hazard ratio = 2.84, p &lt; 0.01), reducing misdiagnosis.Interpretation: Our results provide evidence for a robust and explainable clinical AI-guided marker for early dementia prediction that is validated against longitudinal, multicenter patient data across countries, and has strong potential for adoption in clinical practice.Funding: Wellcome Trust, Royal Society, Alzheimer’s Research UK, Alzheimer’s Drug Discovery Foundation Diagnostics Accelerator, Alan Turing Institute.</div