VictorJuiz/thesis huggingface dataset:
Lower Semicontinuity and Young Measures for Integral Functionals with Linear Growth
Keywords: Alberti?s Rank One Theorem
Bounded variation
bounded deformation
BV
lower semicontinuity
BD
differential inclusion
Young measure
rigidity

The first contribution of this thesis is a new proof of sequential weak* lower semicontinuity in BV(Ω;Rm)
for integral functionals of the form
F(u) :=∫Ωf(x,∇u)dx+∫Ωf∞(x,
dDsu
d|Dsu|
)d|Dsu|
+∫∂Ωf∞(x,u|∂Ω⊗nΩ)dHd−1,u∈BV(Ω;Rm),
where f:
¯
Ω
×Rm×d→R
is a quasiconvex Carath\'{e}odory integrand with linear growth at infinity, i.e.\ |f(x,A)|≤M(1+|A|)
for some M>0
, and f
is such that the recession function f∞(x,A):=limx′→x,t→∞t−1f(x′,tA)
exists and is (jointly) continuous. In contrast to the classical proofs by Ambrosio \& Dal Maso [\textit{J.\ Funct.\ Anal.} 109 (1992), 76--97] and Fonseca \& M\"{u}ller [\textit{Arch.\ Ration.\ Mech.\ Anal.} 123 (1993), 1--49], the one presented here does not use Alberti's Rank One Theorem [\textit{Proc.\ Roy.\ Soc.\ Edinburgh Sect.\ A} 123 (1993), 239--274], but combines the usual blow-up method with a rigidity result for gradients.
The second main result is the first general weak* lower semicontinuity theorem in the space BD(Ω)
of functions of bounded deformation for functionals of the form
F(u) :=∫Ωf(x,Eu)dx+∫Ωf∞(x,
dEsu
d|Esu|
)d|Esu|
+∫∂Ωf∞(x,u|∂Ω⊙nΩ)dHd−1,u∈BD(Ω),
where f
satisfies similar assumptions as before, in particular it is assumed symmetric-quasiconvex. The main novelty is that we allow for non-vanishing Cantor-parts in the symmetrized derivative Eu
. The proof hinges on a construction of good blow-ups, which in turn is based on local rigidity arguments for some differential inclusions involving symmetrized gradients and an iteration of the blow-up construction. This strategy allows us to establish the lower semicontinuity result without an Alberti-type theorem in BD(Ω)
, which is not available at present.
This dissertation also presents an improved functional-analytic framework for generalized Young measures (with linear growth) as introduced by DiPerna \& Majda [\textit{Comm. Math. Phys.} 108 (1987), 667--689]. Localization principles and Jensen-type inequalities for such Young measures form the technical backbone of the lower semicontinuity proofs.
Direct wavefront sensing in adaptive microscopy
Keywords: Shack-Hartmann
specimen-induced aberration
backscattered light
Direct wavefront sensing
microscopy

Specimen-induced aberrations are frequently encountered in high resolution microscopy, particularly when high numerical aperture lenses are used to image deep into biological specimens. These aberrations distort the focal spot causing reduction in resolution, signal level and contrast. Aberrations arise due to refractive index variations in the specimen. Adaptive optics has been used in microscopes to correct for these effects and restore optimum imaging performance. An adaptive element, such as a deformable mirror, is used to compensate for aberrations by introducing equal but opposite phase distortions to that introduced by the specimen. Most adaptive optics systems in microscopy have employed indirect optimisation schemes that require several exposures of the specimen before the correction phase can be determined. This could lead to photo-toxicity and is sometimes unsuitable for real-time imaging. Direct wavefront sensing can significantly decrease the time required for aberration measurements by using dedicated sensors, but has limitations, particularly when backscattered light is used from thick specimens. While similar research has been undertaken in ophthalmology where the imaged retina is approximately two-dimensional, this report extends that work to three-dimensional specimens in microscopes. Aberration correction has already been demonstrated in single-pass systems for scattering specimens, but have employed either indirect wavefront sensing techniques or fluorescent beads as guide stars. The use of a Shack-Hartmann wavefront sensor for direct wavefront sensing using backscattered light in a dual-pass system is investigated here. Out-of-focus rejection using an axial selectivity pinhole is studied, along with the phase smoothing effect that is dependant on the pinhole size. The effectiveness of using asymmetric illumination and detection paths, and polarisation filtering in the retrieval of phase information are also demonstrated. It is shown that whilst direct quantitative measurements are not possible for scattering specimens, the measurements could be used in closed-loop adaptive optics systems to correct aberrations.
Particle systems and stochastic PDEs on the half-line
Keywords: Particles system ; stochastic PDEs ; Skorokhod topologies

The purpose of this thesis is to develop techniques for analysing interacting particle systems on the half-line. When the number of particles becomes large, stochastic partial differential equations (SPDEs) with Dirichlet boundary conditions will be the natural objects for describing the dynamics of the population's empirical measure.
As a source of motivation, we consider systems that arise naturally as models for the pricing of portfolio credit derivatives, although similar applications are found in mathematical neuroscience, stochastic filtering and mean-field games.
We will focus on a stochastic McKean--Vlasov system in which a collection of Brownian motions interact through a correlation which is a function of the proportion of particles that have been absorbed at level zero. We prove a law of large numbers where the limiting object is the unique solution to (the weak formulation of) the loss-dependent SPDE:
dVt(x) = 1/2 ∂xxVt(x)dt - p(Lt)∂xVt(x)dWt, Vt(0)=0,
where Lt = 1-&lmoust;∞t
Vt(x)dx, V is a density process on the half-line and W is a Brownian motion. The correlation function is assumed to be piecewise Lipschitz, which encompasses a natural class of credit models.
The first of our theoretical developments is to introduce the kernel smoothing method in the dual of the first Sobolev space, H-1, with the aim of proving uniqueness results for SPDEs. A benefit of this approach is that only first order moment estimates of solutions are required, and in the particle setting this translates into studying the particles at an individual level rather than as a correlated collection. The second idea is to extend Skorokhod's M1 topology to the space of processes that take values in the tempered distributions. The benefit we gain is that monotone functions have zero modulus of continuity under this topology, so the loss process, L, is easy to control.
As a final example, we consider the fluctuations in the convergence of a basic particle system with constant correlation. This gives rise to a central limit theorem, for which the limiting object is a solution to an SPDE with random transport and an additive idiosyncratic driver acting on the first derivative terms. Conditional on the systemic random variables, this driver is a space-time white noise with intensity controlled by the empirical measure of the underlying system. The SPDE has insufficient regularity for us to work in any Sobolev space higher than H-1, hence we have an example of where our extension to the kernel smoothing method is necessary.
Uncertainty and firm investment
Keywords: firm uncertainty
firm investment

This thesis explores effects of uncertainty on firm investment that are described in estimates of firm level investment specifications which include proxies for uncertainty over expected future firm profitability. A panel data set of UK firms covering the period 1987-2000 is used to estimate firm level investment specifications. Within year volatility in stock returns - a common proxy for firm specific uncertainty in previous literature - is compared with covariance measures between stock returns and market returns representing un-diversifiable risk from the CAPM; and with alternative uncertainty proxies based on volatility in I/B/E/S securities analysts' forecasts of earnings per share. Within estimates of firm level investment specifications, the thesis investigates the sensitivity of coefficients on uncertainty terms to the choice of underlying investment specification: error correction model between the natural logarithms of capital and sales; or the Hayashi (1982) Q model of investment. Coefficients on stock return volatility measures of uncertainty terms are found to vary significantly between estimates of error correction and average q specifications. Differences between coefficients estimated on uncertainty terms across estimates of these two investment specifications are supported with simulated data. Uncertainty measures based on volatility in I/B/E/S securities analysts' forecasts of earnings per share are found to be much more informative of investment behaviour than within year stock return volatility in estimates of both error correction and average q specifications. Coefficients on I/B/E/S uncertainty proxies imply more consistent investment-uncertainty relationships across estimates of error correction and average q specifications for the UK panel data set.
Teresa de Cartagena: a late medieval woman's theological approach to disability
Keywords: converso
deafness
history of Spanish Jews
authorship
gender
disability
medieval female spirituality

This thesis studies, through a literary and theological analysis of her writings and an examination of her background, how a fifteenth century Spanish nun called Teresa de Cartagena dealt spiritually with disability. She was physically disabled, having become deaf as an adult but also having endured many illnesses. Her first book, Arboleda de los enfermos, was written to pass on to other sufferers the spiritual lessons she had learned from her own suffering; that suffering was good because it had saved her from sin and had brought her to God. Her second work, Admiraçión operum Dey, was written to answer those who had criticised her for the act of writing because of her gender, at that time a disability for any woman wishing to write or teach. She justified her writing as a special work of God, but did not claim mystical direct divine inspiration.
Teresa was a member of a prominent family of Jewish Christians (conversos). At the time she was writing, the second half of the fifteenth century, anti-converso prejudice and violence were growing in Spain. This culminated in the introduction of the Inquisition in order to deal with the so-called 'judaising conversos'. In these circumstances her conversa status was a distinct social disability, but there is no express mention of this in her writings. However, there are traces in her writings of converso concerns, and of a specifically converso theology.
Although there have been many studies of Teresa de Cartagena from the viewpoints of medieval Spanish literature, disability studies, feminist history and her use of rhetorical techniques, there has been no in depth study of her theology and spirituality. This thesis demonstrates that, although in general these were orthodox and unoriginal, they were unusual for a woman of her time and background.
Towards a computational model of the colonic crypt with a realistic, deformable geometry
Keywords: mathematical modelling
Colorectal cancer

Colorectal cancer (CRC) is one of the most prevalent and deadly forms of cancer. Its high mortality rate is associated with difficulties in early detection, which is crucial to survival. The onset of CRC is marked by macroscopic changes in intestinal tissue, originating from a deviation in the healthy cell dynamics of glands known as the crypts of Lieberkuhn. It is believed that accumulated genetic alterations confer on mutated cells the ability to persist in the crypts, which can lead to the formation of a benign tumour through localised proliferation. Stress on the crypt walls can lead to buckling, or crypt fission, and the further spread of mutant cells. Elucidating the initial perturbations in crypt dynamics is not possible experimentally, but such investigations could be made using a predictive, computational model. This thesis proposes a new discrete crypt model, which focuses on the interaction between cell- and tissue-level behaviour, while incorporating key subcellular components. The model contains a novel description of the role of the surrounding tissue and musculature, which allows the shape of the crypt to evolve and deform. A two-dimensional (2D) cross-sectional geometry is considered. Simulation results reveal how the shape of the crypt base may contribute mechanically to the asymmetric division events typically associated with the stem cells in this region. The model predicts that epithelial cell migration may arise due to feedback between cell loss at the crypt collar and density-dependent cell division, an hypothesis which can be investigated in a wet lab. Further, in silico experiments illustrate how this framework can be used to investigate the spread of mutations, and conclude that a reduction in cell migration is key to confer persistence on mutant cell populations. A three-dimensional (3D) model is proposed to remove the spatial restrictions imposed on cell migration in 2D, and preliminary simulation results agree with the hypotheses generated in 2D. Computational limitations that currently restrict extension to a realistic 3D geometry are discussed. These models enable investigation of the role that mechanical forces play in regulating tissue homeostasis, and make a significant contribution to the theoretical study of the onset of crypt deformation under pre-cancerous conditions.
Magneto-optics of complex oxides at terahertz frequencies
Keywords: multiferroic
terahertz spectroscopy
magnon
electromagnon

This thesis presents experimental results on two complex oxide systems: Cu1-xZnxO and La0.7Sr0.3MnO3:ZnO. The dynamic magnetoelectric response of these materials is obtained using terahertz time-domain spectroscopy, supported by Fourier-transform infrared spectroscopy, Raman spectroscopy and X-ray diffraction.
Evidence for an electromagnon in the multiferroic phase of CuO is presented for the first time. This high temperature (213-230K) electromagnon is linked to intersublattice exchange between two Cu sublattices. The temperature dependence of a magnon in the collinear antiferromagnetic phase is indicative of biquadratic exchange.
Broadening of the multiferroic phase on substitution of copper with zinc is reported along with a 25% depression of the Néel temperature due to spin dilution. Phonons and magnons broaden and shift in energy on alloying. However, the electromagnon is relatively insensitive, increasing in energy without widening. This indicates that electromagnons and dynamic magnetoelectric coupling can be mantained even in disordered spin systems. Strong spin-phonon coupling is present in both magnetically ordered phases as shown by the anomalous behavior of the A3u phonon at TN1 and a Raman-active mode associated with a magnetic modulation of a zone-folded acoustic phonon.
Dynamic 1THz magnetoresistance is found to be significantly larger than static magnetoresistance in La0.7Sr0.3MnO3:ZnO vertically-aligned nanocolumns on LaAlO3 substrates. The metal-insulator transition temperature is determined to be 300 K. Temperature dependent static and dynamic resistivity and magnetoresistance are discussed in terms of strain and grain boundary effects. Negative photoconductivity is observed and the dynamic response analysed.
Playing with fire: an MNC’s inability to translate its market logic in a culturally complex exchange setting in rural India
Keywords: glocalization
base of the pyramid
market-based poverty alleviation
social structures
neo-institutional theory
last mile distribution
private sector
clean cookstoves
rural India
corporate partnerships
women's empowerment
behaviour change
social entrepreneurship
economic development

This dissertation describes the manner by which a multinational corporation (MNC) enacts a market-based logic with a locally embedded partner in a complex and unfamiliar operating setting to fulfil both business and social objectives. It examines a hybrid partnership between BP, an MNC, and SSP, a rural Indian non-governmental organisation (NGO). Together, the organisations trained rural women, who were affiliated with SSP, as agents to distribute and sell BP’s ‘smokeless’ cookstoves and fuel pellets to households who cook on smoky firewood stoves. The research draws on two theories—neo-institutional organizational theory and real markets theory—to examine the process by which logics are aligned across partners and projected and translated into the rural Indian exchange setting. It constructs a four-actor model (MNC, NGO, agent, customer) to explore the exchange relationships between the actors at the meso- and micro-levels. At the meso-level, it explains how the MNC and NGO’s non-aligned logics, asymmetric power dynamics, and lack of mutual trust contribute to the venture’s failure. In addition, the NGO was so determined to succeed as a professional, market-driven, channel partner that it shed part of its identity as a civil advocacy organisation and adopted mainstream commercial practices that were not sensitive to the needs of its local stakeholders. At the micro-level, the partners did not come to a common understanding with the agents regarding the cultural challenges they faced marketing the stove. Moreover, the marketing strategy glossed over the multi-layered social relationships and culinary, behavioural, and religious practices that needed to be translated for the technology to meet the needs of consumers. Using gritty ethnographic data, the dissertation highlights a challenge that large, foreign companies face when entering ‘Base of the Pyramid’ markets, namely the inconsistency between the MNC’s market logic and the wider associational logics that motivate village agents and customers.
The duality between two-index potentials and the non-linear sigma model in field theory
Keywords: K-Theory
Superstring theory
M-Theory
Index Theory
Foliations
Global Analysis

We interpret the generalised gauge symmetry introduced in string theory and M-Theory as a special case of Grothendieck's stability equivalence relation in the definition of the 0th K-group and we calculate the Euler number of the elliptic de Rham complex twisted by a flat connection. Then using Polyakov's classical equivalence of flat bundles with non-linear sigma models we define a new topological invariant for foliations using techniques from noncommutative geometry, in particular the Connes' pairing between K-Theory and cyclic cohomology. This new invariant classifies foliations up to Morita equivalence.
Spin(7)-manifolds and calibrated geometry
Keywords: special holonomy
spin(7)-manifolds
Cayley submanifolds

In this thesis we study Spin(7)-manifolds, that is Riemannian 8-manifolds with torsion-free Spin(7)-structures, and Cayley submanifolds of such manifolds. We use a construction of compact Spin(7)-manifolds from Calabi–Yau 4-orbifolds with antiholomorphic involutions, due to Joyce, to find new examples of compact Spin(7)-manifolds. We search the class of well-formed quasismooth hypersurfaces in weighted projective spaces for suitable Calabi–Yau 4-orbifolds. We consider antiholomorphic involutions induced by the restriction of an involution of the ambient weighted projective space and we classify anti-holomorphic involutions of weighted projective spaces.
We consider the moduli problem for Cayley submanifolds of Spin(7)-manifolds and show that there is a fine moduli space of unobstructed Cayley submanifolds. This result improves on the work of McLean in that we consider the global issues of how to patch together the local result of McLean. We also use the work of Kriegl and Michor on ‘convenient manifolds’ to show that this moduli space carries a universal family of Cayley submanifolds.
Using the analysis necessary for the study of the moduli problem of Cayleys we find examples of compact Cayley submanifolds in any compact Spin(7)-manifold arising, using Joyce’s construction, from a suitable Calabi–Yau 4-orbifold with antiholomorphic involution. For the analysis to work, we need to show that a given Cayley submanifold is unobstructed. To show that particular examples of Cayley submanifolds are unobstructed, we relate the obstructions of complex surfaces in Calabi–Yau 4-folds as complex submanifolds to the obstructions as Cayley submanifolds.
Making home safe? The role of criminal law and punishment in British immigration controls
Keywords: regulation
immigration
criminalisation
control of immigration

This thesis is an enquiry into the regulation of immigration through criminal law and its institutions. It looks at the range of immigration offences in British legislation, and whether and how they are being used in practice. The criminalisation of immigration status has historically served functions of exclusion and control against those who defy the state’s powers over its territory and population. In the last two decades, the prerogatives to exclude and punish have been enhanced by the expansion of the catalogue of immigration offences and the more systematic enforcement of these powers. The great reliance on the criminal law to regulate immigration is distinctive of a period in which crime and immigration have been increasingly politicised. As a consequence, more offences have been created and more individuals have been subject to the hybrid immigration and criminal justice system. While immigration offences largely remain under-enforced, some of them –particularly those penalising document fraud and identity stripping- are used against foreign nationals who cannot be removed from the country.
In this thesis I explain what I consider to be the most pernicious consequences of this expansion of formal and substantive criminalisation of immigration breaches. The existence of a parallel system of sanctions allows enforcement agencies wide margins of discretion. Therefore, similar cases may be dealt with in very different ways. When the criminal route is chosen, the use of criminal law in the vast majority of cases reaching the criminal courts is unnecessary, disproportionate and extremely harmful. Both the decision to prosecute and the sanction eventually imposed are justified by preventive and regulatory purposes. The actual practice of criminalisation reveals that the criminal procedural safeguards are weakened and those accused of immigration crimes are likely to be convicted and imprisoned for these offences.
I conclude that the formal and substantive criminalisation of immigration represents a departure from liberal criminal law principles and the purposes of criminal punishment. These conclusions cast doubts about the pragmatic, non-principled use of criminal law to regulate immigration flows, and call for the need to look at other, more humane alternatives in the treatment of ‘unwelcome’ migrants.