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We present an exact formula for the dynamics of $N$ hard spheres of radius $r>0$ on an infinite line which evolve under the assumption that total linear momentum and kinetic energy of the system is conserved for all times. This model is commonly known as the one-dimensional Tonks gas or the hard rod gas model. Our exact formula is expressed as a sum over the Weyl group associated to the root system $A_{N-1}$ and is valid for all initial data in a full-measure subset of the tangent bundle of the hard sphere table. As an application of our explicit formula, we produce a simple proof that the associated billiard flow admits the Liouville measure on the tangent bundle of the hard sphere table as an invariant measure., Comment: 1 figure

For any $N\geq 3$, we study invariant measures of the dynamics of $N$ hard spheres whose centres are constrained to lie on a line. In particular, we study the invariant submanifold $\mathcal{M}$ of the tangent bundle of the hard sphere billiard table comprising initial data that lead to the simultaneous collision of all $N$ hard spheres. Firstly, we obtain a characterisation of those continuously-differentiable $N$-body scattering maps which generate a billiard dynamics on $\mathcal{M}$ admitting a canonical weighted Hausdorff measure on $\mathcal{M}$ (that we term the Liouville measure on $\mathcal{M}$) as an invariant measure. We do this by deriving a second boundary-value problem for a fully nonlinear PDE that all such scattering maps satisfy by necessity. Secondly, by solving a family of functional equations, we find sufficient conditions on measures which are absolutely continuous with respect to the Hausdorff measure in order that they be invariant for billiard flows that conserve momentum and energy. Finally, we show that the unique momentum- and energy-conserving linear $N$-body scattering map yields a billiard dynamics which admits the Liouville measure on $\mathcal{M}$ as an invariant measure., Comment: 44 pages, 1 figure

We describe a London Quantum-Secured Metro Network using Quantum Key Distribution between three London nodes together with customer access tails. The commercially- eady solution is fully integrated into the BT network and on-boarded its first customer., Comment: https://opg.optica.org/abstract.cfm?uri=OFC-2023-W4K.4

We present a new implementation of the geometric method of Cullen & Purser (1984) for solving the semi-geostrophic Eady slice equations which model large scale atmospheric flows and frontogenesis. The geometric method is a Lagrangian discretisation, where the PDE is approximated by a particle system. An important property of the discretisation is that it is energy conserving. We restate the geometric method in the language of semi-discrete optimal transport theory and exploit this to develop a fast implementation that combines the latest results from numerical optimal transport theory with a novel adaptive time-stepping scheme. Our results enable a controlled comparison between the Eady-Boussinesq vertical slice equations and their semi-geostrophic approximation. We provide further evidence that weak solutions of the Eady-Boussinesq vertical slice equations converge to weak solutions of the semi-geostrophic Eady slice equations as the Rossby number tends to zero., Comment: 43 pages, 9 figures

By studying scattering Lie groups and their associated Lie algebras, we introduce a new method for the characterisation of collision invariants for physical scattering families associated to smooth, convex hard particles in the particular case that the collision invariant is of class $\mathscr{C}^{1}$. This work extends that of Saint-Raymond and Wilkinson (Communications on Pure and Applied Mathematics (2018), 71(8), pp. 1494-1534), in which the authors characterise collision invariants only in the case of the so-called canonical physical scattering family. Indeed, our method extends to the case of non-canonical physical scattering, whose existence was reported in Wilkinson (Archive for Rational Mechanics and Analysis (2020), 235(3), pp. 2055-2083). Moreover, our new method improves upon the work in Saint-Raymond and Wilkinson as we place no symmetry hypotheses on the underlying non-spherical particles which make up the gas under consideration. The techniques established in this paper also yield a new proof of the result of Boltzmann for collision invariants of class $\mathscr{C}^{1}$ in the classical case of hard spheres., Comment: 30 pages

Recently, an increasing number of wide binaries has been discovered. Their chemical and dynamical properties are studied through extensive surveys and pointed observations. However, the formation of these wide binaries is far from clear, although several scenarios have been suggested. In order to investigate the chemical compositions of these systems, we analysed high-resolution spectroscopy of three wide binary pairs belonging to the Galactic halo. In total, another three candidates from our original sample of 11 candidates observed at various resolutions with various instruments were refuted as co-moving pairs because their radial velocities are significantly different. Within our sample of wide binaries, we found homogeneity amongst the pair components in dynamical properties (proper motion and line-of-sight velocities) and also in chemical composition. Their metallicities are -1.16, -1.42, and -0.79 dex in [Fe/H] for each wide binary pair, which places these stars on the metal-poor side of wide binaries reported in the literature. In particular, the most metal-poor pair in our sample (WB2 = HD134439/HD134440) shows a lower [$\alpha$/Fe] abundance ratio than Milky Way field stars, which is a clear signature of an accreted object. We also confirmed that this wide binary shares remarkably similar orbital properties with stars and globular clusters associated with the Sequoia event. Thus, it appears that the WB2 pair was formed in a dwarf galaxy environment and subsequently dissolved into the Milky Way halo. Although the other two wide binaries appear to arise from a different formation mechanism, our results provide a novel opportunity for understanding the formation of wide binaries and the assembly process of the Milky Way., Comment: 13 pages, 7 figures, accepted for publication in A&A

We use a symmetry-motivated approach to analyse neutron pair distribution function data to investigate the mechanism of negative thermal expansion (NTE) in ReO$_3$. This analysis shows that the local structure of ReO$_3$ is dominated by an in-phase octahedral tilting mode and that the octahedral units are far less flexible to scissoring type deformations than the octahedra in the related compound ScF$_3$. These results support the idea that structural flexibility is an important factor in NTE materials, allowing the phonon modes that drive a volume contraction of the lattice to occupy a greater volume in reciprocal space. The lack of flexibility in ReO$_3$ restricts the NTE-driving phonons to a smaller region of reciprocal space, limiting the magnitude and temperature range of NTE. In addition, we investigate the thermal expansion properties of the material at high temperature and do not find the reported second NTE region. Finally, we show that the local fluctuations, even at elevated temperatures, respect the symmetry and order parameter direction of the observed $P4/mbm$ high pressure phase of ReO$_3$. The result indicates that the motions associated with rigid unit modes are highly anisotropic in these systems., Comment: 9 pages, 6 figures

This document captures the discussion and deliberation of the FAIR for Research Software (FAIR4RS) subgroup that took a fresh look at the applicability of the FAIR Guiding Principles for scientific data management and stewardship for research software. We discuss the vision of research software as ideally reproducible, open, usable, recognized, sustained and robust, and then review both the characteristic and practiced differences of research software and data. This vision and understanding of initial conditions serves as a backdrop for an attempt at translating and interpreting the guiding principles to more fully align with research software. We have found that many of the principles remained relatively intact as written, as long as considerable interpretation was provided. This was particularly the case for the "Findable" and "Accessible" foundational principles. We found that "Interoperability" and "Reusability" are particularly prone to a broad and sometimes opposing set of interpretations as written. We propose two new principles modeled on existing ones, and provide modified guiding text for these principles to help clarify our final interpretation. A series of gaps in translation were captured during this process, and these remain to be addressed. We finish with a consideration of where these translated principles fall short of the vision laid out in the opening.

We consider a class of steady solutions of the semi-geostrophic equations on $\mathbb{R}^3$ and derive the linearised dynamics around those solutions. The linear PDE which governs perturbations around those steady states is a transport equation featuring a pseudo-differential operator of order 0. We study well-posedness of this equation in $L^2(\mathbb{R}^3;\mathbb{R}^3)$ introducing a representation formula for the solutions, and extend the result to the space of tempered distributions on $\mathbb{R}^{3}$. We investigate stability of the steady solutions by looking at plane wave solutions of the linearised problem, and discuss differences in the case of the quasi-geostrophic equations., Comment: 20 pages

We give a new and constructive proof of the existence of global-in-time weak solutions of the 3-dimensional incompressible semi-geostrophic equations (SG) in geostrophic coordinates, for arbitrary initial measures with compact support. This new proof, based on semi-discrete optimal transport techniques, works by characterising discrete solutions of SG in geostrophic coordinates in terms of trajectories satisfying an ordinary differential equation. It is advantageous in its simplicity and its explicit relation to Eulerian coordinates through the use of Laguerre tessellations. Using our method, we obtain improved time-regularity for a large class of discrete initial measures, and we compute explicitly two discrete solutions. The method naturally gives rise to an efficient numerical method, which we illustrate by presenting simulations of a 2-dimensional semi-geostrophic flow in geostrophic coordinates generated using a numerical solver for the semi-discrete optimal transport problem coupled with an ordinary differential equation solver., Comment: 35 pages, 2 figures

The semi-geostrophic equations have attracted the attention of the physical and mathematical communities since the work of Hoskins in the 1970s owing to their ability to model the formation of fronts in rotation-dominated flows, and also to their connection with optimal transport theory. In this paper, we study an active scalar equation, whose activity is determined by way of a Neumann-to-Dirichlet map associated to a fully nonlinear second-order Neumann boundary value problem on the infinite strip $\mathbb{R}^{2}\times(0,1)$, that models a semi-geostrophic flow in regime of constant potential vorticity. This system is an expression of an Eulerian semi-geostrophic flow in a co-ordinate system originally due to Hoskins, to which we shall refer as {\em Hoskins' coordinates}. We obtain results on the local-in-time existence and uniqueness of classical solutions of this active scalar equation in H\"older spaces., Comment: 27 pages. Typos corrected. To appear in Calc of Var & PDEs

The semi-geostrophic equations are used widely in the modelling of large-scale atmospheric flows. In this note, we prove the global existence of weak solutions of the incompressible semi-geostrophic equations, in geostrophic coordinates, in a three-dimensional domain with a free upper boundary. The proof, based on an energy minimisation argument originally inspired by Cullen's Stability Principle, uses optimal transport results as well as the analysis of Hamiltonian ODEs in spaces of probability measures as studied by Ambrosio and Gangbo. We also give a general formulation of Cullen's Stability Principle in a rigorous mathematical framework.

We present an innovative approach to the methodology of dynamical modelling, allowing practical reconstruction of the underlying dark matter mass without assuming both the density and anisotropy functions. With this, the mass-anisotropy degeneracy is reduced to simple model inference, incorporating the uncertainties inherent with observational data, statistically circumventing the mass-anisotropy degeneracy in spherical collisionless systems. We also tackle the inadequacy that the Jeans method of moments has on small datasets, with the aid of Generative Adversarial Networks: we leverage the power of artificial intelligence to reconstruct non-parametrically the projected line-of-sight velocity distribution. We show with realistic numerical simulations of dwarf spheroidal galaxies that we can distinguish between competing dark matter distributions and recover the anisotropy and mass profile of the system., Comment: 19 pages, 13 figures, accepted for publication in MNRAS

We outline a new method of construction of global-in-time weak solutions of the Liouville equation - and also of the associated BBGKY hierarchy - corresponding to the hard sphere singular Hamiltonian. Our method makes use only of geometric reflection arguments on phase space. As a consequence of our method, in the case of $N=2$ hard spheres, we show for any chaotic initial data, the unique global-in-time weak solution $F$ of the Liouville equation is realised as $F=\mathsf{R}(f\otimes f)$ in the sense of tempered distributions on $T\mathbb{R}^{6}\times (-\infty, \infty)$, where $\mathsf{R}$ is a 'reflection-type' operator on Schwartz space, and $f$ is a global-in-time classical solution of the 1-particle free transport Liouville equation on $T\mathbb{R}^{3}$., Comment: 31 pages

We prove the existence of local-in-time smooth solutions of the incompressible semi-geostrophic equations expressed in Eulerian co-ordinates in 3-dimensional smooth bounded simply-connected domains. Our solutions adhere to Cullen's Stability Principle in that the geopotential is guaranteed to be a convex map for all times of its existence. We achieve our results by appealing to the theory of so-called div-curl systems (or Hodge systems), making use of recent results of Wang, which yield useful estimates on the ageostrophic velocity field. To our knowledge, this work constitutes the first time that any notion of bounded solution of the semi-geostrophic equations in Eulerian co-ordinates has been constructed on a bounded domain. Indeed, our work solves an open problem as highlighted by, among others, A. Figalli in his CIME lectures on the semi-geostrophic equations. Our methods are largely elementary. We discuss the application of the novel ideas in this work to the case of variable Coriolis force in the final section of the article., Comment: 17 pages; some typos corrected; statement of main result modified; more details on proof of main result provided

We present a novel and efficient method for fitting dynamical models of stellar kinematic data in dwarf spheroidal galaxies (dSph). Our approach is based on Gaussian-process emulation (GPE), which is a sophisticated form of curve fitting that requires fewer training data than alternative methods. We use a set of validation tests and diagnostic criteria to assess the performance of the emulation procedure. We have implemented an algorithm in which both the GPE procedure and its validation are fully automated. Applying this method to synthetic data, with fewer than 100 model evaluations we are able to recover a robust confidence region for the three-dimensional parameter vector of a toy model of the phase-space distribution function of a dSph. Although the dynamical model presented in this paper is low-dimensional and static, we emphasize that the algorithm is applicable to any scheme that involves the evaluation of computationally expensive models. It therefore has the potential to render tractable previously intractable problems, for example, the modelling of individual dSphs using high-dimensional, time-dependent N-body simulations., Comment: 14 pages 11 figures

We prove that to each initial datum in a set of positive measure in phase space, there exist uncountably-many associated weak solutions of Newton's equations of motion which govern the dynamics of two non-spherical sets with real-analytic boundaries subject to the conservation of linear momentum, angular momentum and kinetic energy. We prove this result by first exhibiting non-uniqueness of classical solution to a constrained Monge-Amp\`ere equation posed on Euclidean space, and then applying the deep existence theory of Ballard for hard particle dynamics. In the final section of the article, we discuss the relevance of this observation to the kinetic theory of hard particle systems.

In the first of two papers, we study the initial boundary-value problem that underlies the theory of the Boltzmann equation for general non-spherical hard particles. In this work, for two congruent ellipses and for a large class of associated boundary conditions, we identify initial conditions for which there do not exist local-in-time weak solutions of Newton's equations of motion. To our knowledge, this is the first time the necessity of rolling in the energy-conserving dynamics of strictly-convex rigid bodies has been demonstrated. This study was, in part, motivated by a recent observation of Palffy-Muhoray, Virga, Wilkinson and Zheng on the interpenetration of strictly-convex rigid bodies.