Stability of Relative Equilibria in the Planar N-Vortex Problem

We study the linear and nonlinear stability of relative equilibria in the planar N-vortex problem, adapting the approach of Moeckel from the corresponding problem in celestial mechanics. After establishing some general theory, a topological approach is taken to show that for the case of positive circulations, a relative equilibrium is linearly stable if and only if it is a nondegenerate minimum of the Hamiltonian restricted to a level surface of the angular impulse (moment of inertia). Using a criterion of Dirichlet's, this implies that any linearly stable relative equilibrium with positive vorticities is also nonlinearly stable. Two symmetric families, the rhombus and the isosceles trapezoid, are analyzed in detail, with stable solutions found in each case.Comment: 23 pages, 3 figure

Systematic Physics Constrained Parameter Estimation of Stochastic Differential Equations

A systematic Bayesian framework is developed for physics constrained parameter inference ofstochastic differential equations (SDE) from partial observations. The physical constraints arederived for stochastic climate models but are applicable for many fluid systems. A condition isderived for global stability of stochastic climate models based on energy conservation. Stochasticclimate models are globally stable when a quadratic form, which is related to the cubic nonlinearoperator, is negative definite. A new algorithm for the efficient sampling of such negative definite matrices is developed and also for imputing unobserved data which improve the accuracy of theparameter estimates. The performance of this framework is evaluated on two conceptual climatemodels

Classifying Four-Body Convex Central Configurations

We classify the full set of convex central configurations in the Newtonian four-body problem. Particular attention is given to configurations possessing some type of symmetry or defining geometric property. Special cases considered include kite, trapezoidal, co-circular, equidiagonal, orthodiagonal, and bisecting-diagonal configurations. Good coordinates for describing the set are established. We use them to prove that the set of four-body convex central configurations with positive masses is three-dimensional, a graph over a domain $D$ that is the union of elementary regions in $\mathbb{R}^{+^3}$.Comment: 28 pages, 14 figure

A four-body convex central configuration with perpendicular diagonals is necessarily a kite

This is a post-peer-review, pre-copyedit version of an article published in Qualitative Theory of Dynamical Systems. The final authenticated version is available online at: http://dx.doi.org/10.1007/s12346-017-0238-z.We prove that any four-body convex central configuration with perpendicular diagonals must be a kite configuration. The result extends to general power-law potential functions, including the planar four-vortex problem.Peer ReviewedPostprint (author's final draft

Automatic Zig-Zag sampling in practice

Novel Monte Carlo methods to generate samples from a target distribution, such as a posterior from a Bayesian analysis, have rapidly expanded in the past decade. Algorithms based on Piecewise Deterministic Markov Processes (PDMPs), non-reversible continuous-time processes, are developing into their own research branch, thanks their important properties (e.g., correct invariant distribution, ergodicity, and super-efficiency). Nevertheless, practice has not caught up with the theory in this field, and the use of PDMPs to solve applied problems is not widespread. This might be due, firstly, to several implementational challenges that PDMP-based samplers present with and, secondly, to the lack of papers that showcase the methods and implementations in applied settings. Here, we address both these issues using one of the most promising PDMPs, the Zig-Zag sampler, as an archetypal example. After an explanation of the key elements of the Zig-Zag sampler, its implementation challenges are exposed and addressed. Specifically, the formulation of an algorithm that draws samples from a target distribution of interest is provided. Notably, the only requirement of the algorithm is a closed-form function to evaluate the target density of interest, and, unlike previous implementations, no further information on the target is needed. The performance of the algorithm is evaluated against another gradient-based sampler, and it is proven to be competitive, in simulation and real-data settings. Lastly, we demonstrate that the super-efficiency property, i.e. the ability to draw one independent sample at a lesser cost than evaluating the likelihood of all the data, can be obtained in practice.Comment: Small edits from previous version following some minor revisions requeste

Simulation from quasi-stationary distributions on reducible state spaces

Quasi-stationary distributions (QSDs) arise from stochastic processes that exhibit transient equilibrium behaviour on the way to absorption. QSDs are often mathematically intractable and even drawing samples from them is not straightforward. In this paper the framework of Sequential Monte Carlo samplers is utilized to simulate QSDs and several novel resampling techniques are proposed to accommodate models with reducible state spaces, with particular focus on preserving particle diversity on discrete spaces. Finally an approach is considered to estimate eigenvalues associated with QSDs, such as the decay parameter

Neptunium Reactivity During Co-Precipitation and Oxidation of Fe(II)/Fe(III) (Oxyhydr)oxides

Fe(II) bearing iron (oxyhydr)oxides were directly co-precipitated with Np(V)O2+ under anaerobic conditions to form Np doped magnetite and green rust. These environmentally relevant mineral phases were then characterised using geochemical and spectroscopic analyses. The Np doped mineral phases were then oxidised in air over 224 days with solution chemistry and end-point oxidation solid samples collected for further characterisation. Analysis using chemical extractions and X-ray absorption spectroscopy (XAS) techniques confirmed that Np(V) was initially reduced to Np(IV) during co-precipitation of both magnetite and green rust. Extended X-Ray Absorption Fine Structure (EXAFS) modelling suggested the Np(IV) formed a bidentate binuclear sorption complex to both minerals. Furthermore, following oxidation in air over several months, the sorbed Np(IV) was partially oxidised to Np(V), but very little remobilisation to solution occurred during oxidation. Here, linear combination fitting of the X-Ray Absorption Near Edge Structure (XANES) for the end-point oxidation samples for both mineral phases suggested approximately 50% oxidation to Np(V) had occurred over 7 months of oxidation in air. Both the reduction of Np(V) to Np(IV) and inner sphere sorption in association with iron (oxyhydr)oxides, and the strong retention of Np(IV) and Np(V) species with these phases under robust oxidation conditions, have important implications in understanding the mobility of neptunium in a range of engineered and natural environments

Neptunium Reactivity During Co-Precipitation and Oxidation of Fe(II)/Fe(III) (Oxyhydr)oxides

Fe(II) bearing iron (oxyhydr)oxides were directly co-precipitated with Np(V)O2+ under anaerobic conditions to form Np doped magnetite and green rust. These environmentally relevant mineral phases were then characterised using geochemical and spectroscopic analyses. The Np doped mineral phases were then oxidised in air over 224 days with solution chemistry and end-point oxidation solid samples collected for further characterisation. Analysis using chemical extractions and X-ray absorption spectroscopy (XAS) techniques confirmed that Np(V) was initially reduced to Np(IV) during co-precipitation of both magnetite and green rust. Extended X-Ray Absorption Fine Structure (EXAFS) modelling suggested the Np(IV) formed a bidentate binuclear sorption complex to both minerals. Furthermore, following oxidation in air over several months, the sorbed Np(IV) was partially oxidised to Np(V), but very little remobilisation to solution occurred during oxidation. Here, linear combination fitting of the X-Ray Absorption Near Edge Structure (XANES) for the end-point oxidation samples for both mineral phases suggested approximately 50% oxidation to Np(V) had occurred over 7 months of oxidation in air. Both the reduction of Np(V) to Np(IV) and inner sphere sorption in association with iron (oxyhydr)oxides, and the strong retention of Np(IV) and Np(V) species with these phases under robust oxidation conditions, have important implications in understanding the mobility of neptunium in a range of engineered and natural environments