arXiv subjects

David M. Meier

Publications and source records attributed to David M. Meier.

10 recordsLinked to original sources

arXiv2021

Mathematical models for fake news

Over the past decade it has become evident that intentional disinformation in the political context -- so-called fake news -- is a danger to democracy. However, until now there has been no clear understanding of how to define fake news, much less how to model it. This paper addresses both of these issues. A definition of fake news is given, and two approaches for the modelling of fake news and its impact in elections and referendums are introduced. The first approach, based on the idea of a representative voter, is shown to be suitable for obtaining a qualitative understanding of phenomena associated with fake news at a macroscopic level. The second approach, based on the idea of an election microstructure, describes the collective behaviour of the electorate by modelling the preferences of individual voters. It is shown through a simulation study that the mere knowledge that fake news may be in circulation goes a long way towards mitigating the impact of fake news.

stat.AP↗

arXiv2018

Laser spectroscopic characterization of the nuclear clock isomer $^{229m}$Th

The isotope $^{229}$Th is the only nucleus known to possess an excited state $^{229m}$Th in the energy range of a few electron volts, a transition energy typical for electrons in the valence shell of atoms, but about four orders of magnitude lower than common nuclear excitation energies. A number of applications of this unique nuclear system, which is accessible by optical methods, have been proposed. Most promising among them appears a highly precise nuclear clock that outperforms existing atomic timekeepers. Here we present the laser spectroscopic investigation of the hyperfine structure of $^{229m}$Th$^{2+}$, yielding values of fundamental nuclear properties, namely the magnetic dipole and electric quadrupole moments as well as the nuclear charge radius. After the recent direct detection of this long-searched-for isomer, our results now provide detailed insight into its nuclear structure and present a method for its non-destructive optical detection.

nucl-ex↗

arXiv2016

Lévy-Vasicek Models and the Long-Bond Return Process

The classical derivation of the well-known Vasicek model for interest rates is reformulated in terms of the associated pricing kernel. An advantage of the pricing kernel method is that it allows one to generalize the construction to the Lévy-Vasicek case, avoiding issues of market incompleteness. In the Lévy-Vasicek model the short rate is taken in the real-world measure to be a mean-reverting process with a general one-dimensional Lévy driver admitting exponential moments. Expressions are obtained for the Lévy-Vasicek bond prices and interest rates, along with a formula for the return on a unit investment in the long bond, defined by $L_t = \lim_{T \rightarrow \infty} P_{tT} / P_{0T}$, where $P_{tT}$ is the price at time $t$ of a $T$-maturity discount bond. We show that the pricing kernel of a Lévy-Vasicek model is uniformly integrable if and only if the long rate of interest is strictly positive.

q-fin.MF↗

arXiv2016

A Riemannian approach to Randers geodesics

In certain circumstances tools of Riemannian geometry are sufficient to address questions arising in the more general Finslerian context. We show that one such instance presents itself in the characterisation of geodesics in Randers spaces of constant flag curvature. To achieve a simple, Riemannian derivation of this special family of curves, we exploit the connection between Randers spaces and the Zermelo problem of time-optimal navigation in the presence of background fields. The characterisation of geodesics is then proven by generalising an intuitive argument developed recently for the solution of the quantum Zermelo problem.

math-ph↗

arXiv2015

Fragile entanglement statistics

If X and Y are independent, Y and Z are independent, and so are X and Z, one might be tempted to conclude that X, Y, and Z are independent. But it has long been known in classical probability theory that, intuitive as it may seem, this is not true in general. In quantum mechanics one can ask whether analogous statistics can emerge for configurations of particles in certain types of entangled states. The explicit construction of such states, along with the specification of suitable sets of observables that have the purported statistical properties, is not entirely straightforward. We show that an example of such a configuration arises in the case of an N-particle GHZ state, and we are able to identify a family of observables with the property that the associated measurement outcomes are independent for any choice of 2, 3, ..., N-1 of the particles, even though the measurement outcomes for all N particles are not independent. Although such states are highly entangled, the entanglement turns out to be 'fragile', i.e. the associated density matrix has the property that if one traces out the freedom associated with even a single particle, the resulting reduced density matrix is separable.

quant-ph↗

arXiv2015

Time-optimal navigation through quantum wind

The quantum navigation problem of finding the time-optimal control Hamiltonian that transports a given initial state to a target state through quantum wind, that is, under the influence of external fields or potentials, is analysed. By lifting the problem from the state space to the space of unitary gates realising the required task, we are able to deduce the form of the solution to the problem by deriving a universal quantum speed limit. The expression thus obtained indicates that further simplifications of this apparently difficult problem are possible if we switch to the interaction picture of quantum mechanics. A complete solution to the navigation problem for an arbitrary quantum system is then obtained, and the behaviour of the solution is illustrated in the case of a two-level system.

quant-ph↗

arXiv2014

Elementary solution to the time-independent quantum navigation problem

A quantum navigation problem concerns the identification of a time-optimal Hamiltonian that realises a required quantum process or task, under the influence of a prevailing `background' Hamiltonian that cannot be manipulated. When the task is to transform one quantum state into another, finding the solution in closed form to the problem is nontrivial even in the case of time-independent Hamiltonians. An elementary solution, based on trigonometric analysis, is found here when the Hilbert space dimension is two. Difficulties arising from generalisations to higher-dimensional systems are discussed.

quant-ph↗

arXiv2012

Quantum splines

A quantum spline is a smooth curve parameterised by time in the space of unitary transformations, whose associated orbit on the space of pure states traverses a designated set of quantum states at designated times, such that the trace norm of the time rate of change of the associated Hamiltonian is minimised. The solution to the quantum spline problem is obtained, and is applied in an example that illustrates quantum control of coherent states. An efficient numerical scheme for computing quantum splines is discussed and implemented in the examples.

quant-ph↗

arXiv2011

Invariant higher-order variational problems II

Motivated by applications in computational anatomy, we consider a second-order problem in the calculus of variations on object manifolds that are acted upon by Lie groups of smooth invertible transformations. This problem leads to solution curves known as Riemannian cubics on object manifolds that are endowed with normal metrics. The prime examples of such object manifolds are the symmetric spaces. We characterize the class of cubics on object manifolds that can be lifted horizontally to cubics on the group of transformations. Conversely, we show that certain types of non-horizontal geodesics on the group of transformations project to cubics. Finally, we apply second-order Lagrange--Poincaré reduction to the problem of Riemannian cubics on the group of transformations. This leads to a reduced form of the equations that reveals the obstruction for the projection of a cubic on a transformation group to again be a cubic on its object manifold.

math.OC↗

arXiv2011

Geometric integrators for higher-order mechanics on Lie groups

This paper develops a structure-preserving numerical integration scheme for a class of higher-order mechanical systems. The dynamics of these systems are governed by invariant variational principles defined on higher-order tangent bundles of Lie groups. The variational principles admit Lagrangians that depend on acceleration, for example. The symmetry reduction method used in the Hamilton--Pontryagin approach for developing variational integrators of first-order mechanics is extended here to higher order. The paper discusses the general approach and then focuses on the primary example of Riemannian cubics. Higher-order variational integrators are developed both for the discrete-time integration of the initial value problem and for a particular type of trajectory-planning problem. The solution of the discrete trajectory-planning problem for higher-order interpolation among points on the sphere illustrates the approach.

math.DS↗