Doctoral Theses (Mathematical & Computer Sciences)http://hdl.handle.net/10399/372022-12-02T04:28:04Z2022-12-02T04:28:04ZAge heaping in population data of emerging countriesBarajas Paz, Andreshttp://hdl.handle.net/10399/45732022-12-01T22:04:04Z2022-02-01T00:00:00ZAge heaping in population data of emerging countries
Barajas Paz, Andres
Mortality analyses have commonly focused on countries represented in the Human
Mortality Database that have good quality mortality data. In this thesis, we address
the challenge that, in many countries, population and deaths data can be somewhat
unreliable. In many countries, for example, there is significant misreporting of age
in both census and deaths data: referred to as “age heaping”. The purpose of our
research is to develop Bayesian computational methods for fitting a new model for
misreporting of age for countries where their population data and death counts have
been affected by age heaping. The innovation of our model is that it allows us
to detect misreporting, identify age preferences and estimate the true underlying
distribution of ages.
2022-02-01T00:00:00ZNon-linear partial differential equations of kinetic typeLazaridou, Christinahttp://hdl.handle.net/10399/45622022-11-26T22:04:19Z2022-02-01T00:00:00ZNon-linear partial differential equations of kinetic type
Lazaridou, Christina
This thesis is concerned with the analytical study of non-linear partial differential equations (PDEs) of kinetic type which admit multiple stationary solutions. We consider
a kinetic model which is given by a non-linear PDE in the sense of McKean and describes the time-evolution of the density ft = ft(x, v), (x, v) 2 T ⇥ R, of a collection
of interacting particles moving in the one dimensional torus. We focus on tackling the
main diculties which arise from the fact that the non-linear PDE has unbounded coecients, is non-elliptic and not in gradient form. In particular, we employ techniques
to show well-posedness of the solution ft in a weighted Lp space. When the density ft
does not depend on the spatial variable, we study the long time behavior of the spacehomogeneous PDE by using and comparing two different approaches: hypocoercivity
theory and gradient flow theory.
2022-02-01T00:00:00ZLinear and nonlinear wave equation models with power law attenuationBaker, Katherinehttp://hdl.handle.net/10399/45602022-11-26T22:04:15Z2022-02-01T00:00:00ZLinear and nonlinear wave equation models with power law attenuation
Baker, Katherine
Motivated by the need to model high intensity focused ultrasound in lossy media
we study linear and nonlinear wave equations that contain non local time fractional
derivatives, whose inclusion in our models incorporates the effects of acoustic attenuation. This is characterized by a frequency dependent power law parametrized
by a non integer γ ∈ (0, 2), leading to the need for fractional derivative damping.
Issues with such integro-differential equations arise in the continuous and discrete
analysis due to singularities that occur at t = 0 and as a result of their non local
nature. To address these issues we present results that carefully show how to treat
such equations for smooth and non smooth solutions, and we derive fast and efficient
numerical schemes that pay particular attention to the handling of these non local
operators.
As well as acoustic attenuation the models we consider will need to account for
nonlinear propagation effects that result from the focusing of the ultrasound waves
and be modelled on an unbounded domain. To combat this issue, we use a perfectly
matched layer. That is, we truncate the unbounded domain to a finite computational
domain and impose an absorbing, non reflecting boundary layer around it.
2022-02-01T00:00:00ZA pseudodifferential approach to the fractional Laplacian and magnitudeLouca, Nikolettahttp://hdl.handle.net/10399/45572022-11-26T22:04:10Z2022-01-01T00:00:00ZA pseudodifferential approach to the fractional Laplacian and magnitude
Louca, Nikoletta
In this thesis we consider two problems related to the fractional Laplacian. One of
these is the study of the singular behaviour of solutions of boundary value problems
for the fractional Laplacian in smooth and polygonal domains. We study these
problems in model geometries using a harmonic extension of the solution to the
upper half-space. We then discuss how these problems fit in the theory of edge
pseudodifferential calculus and outline the additional steps necessary to address the
general case.
On the other hand, we study magnitude, a metric invariant of compact metric
spaces, for Euclidean domains or compact manifolds with boundary. We provide a
framework for its analysis, relating it to a boundary problem for a non-local integral
operator, which upon localisation near the diagonal is a negative order parameter-elliptic pseudodifferential operator coinciding with a negative order fractional Laplacian up to lower order terms. The inverse of this operator, when acting between
suitable Sobolev spaces, is asymptotically computed using Wiener-Hopf factorizations and methods from semiclassical analysis. As a main result, we obtain an
asymptotic variant of the convex magnitude conjecture by Leinster and Willerton
by explicit computation.
2022-01-01T00:00:00Z