224 research outputs found
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
that is the union of elementary regions in .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
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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
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