Inverse optical tomography through PDE constrained optimisation in L∞

Fluorescent Optical Tomography (FOT) is a new bio-medical imaging method with wider industrial applications. It is currently intensely researched since it is very precise and with no side effects for humans, as it uses non-ionising red and infrared light. Mathematically, FOT can be modelled as an inverse parameter identification problem, associated with a coupled elliptic system with Robin boundary conditions. Herein we utilise novel methods of Calculus of Variations in L∞ to lay the mathematical foundations of FOT which we pose as a PDE-constrained minimisation problem in Lp and L∞

Existence and uniqueness of global solutions to fully nonlinear second order elliptic systems

We consider the problem of existence and uniqueness of strong a.e. solutions u:Rn⟶RNu:Rn⟶RN to the fully nonlinear PDE system F(⋅,D2u)=f, a.e. on Rn,(1) F(⋅,D2u)=f, a.e. on Rn,(1) when f∈L2(Rn)Nf∈L2(Rn)N and F is a Carathéodory map. (1) has not been considered before. The case of bounded domains has been studied by several authors, firstly by Campanato and under Campanato’s ellipticity condition on F. By introducing a new much weaker notion of ellipticity, we prove solvability of (1) in a tailored Sobolev “energy” space and a uniqueness estimate. The proof is based on the solvability of the linearised problem by Fourier transform methods, together with a “perturbation device” which allows to use Campanato’s near operators. We also discuss our hypothesis via counterexamples and give a stability theorem of strong global solutions for systems of the form (1)

HIV-infected sex workers with beneficial HLA-variants are potential hubs for selection of HIV-1 recombinants that may affect disease progression

Cytotoxic T lymphocyte (CTL) responses against the HIV Gag protein are associated with lowering viremia; however, immune control is undermined by viral escape mutations. The rapid viral mutation rate is a key factor, but recombination may also contribute. We hypothesized that CTL responses drive the outgrowth of unique intra-patient HIV-recombinants (URFs) and examined gag sequences from a Kenyan sex worker cohort. We determined whether patients with HLA variants associated with effective CTL responses (beneficial HLA variants) were more likely to carry URFs and, if so, examined whether they progressed more rapidly than patients with beneficial HLA-variants who did not carry URFs. Women with beneficial HLA-variants (12/52) were more likely to carry URFs than those without beneficial HLA variants (3/61) (p < 0.0055; odds ratio = 5.7). Beneficial HLA variants were primarily found in slow/standard progressors in the URF group, whereas they predominated in long-term non-progressors/survivors in the remaining cohort (p = 0.0377). The URFs may sometimes spread and become circulating recombinant forms (CRFs) of HIV and local CRF fragments were over-represented in the URF sequences (p < 0.0001). Collectively, our results suggest that CTL-responses associated with beneficial HLA variants likely drive the outgrowth of URFs that might reduce the positive effect of these CTL responses on disease progression

Dynamically generated embeddings of spacetime

We discuss how embeddings in connection with the Campbell-Magaard (CM) theorem can have a physical interpretation. We show that any embedding whose local existence is guaranteed by the CM theorem can be viewed as a result of the dynamical evolution of initial data given in a four-dimensional spacelike hypersurface. By using the CM theorem, we establish that for any analytic spacetime, there exist appropriate initial data whose Cauchy development is a five-dimensional vacuum space into which the spacetime is locally embedded. We shall see also that the spacetime embedded is Cauchy stable with respect these the initial data.Comment: (8 pages, 1 figure). A section on Cauchy Stability of the embedding was added. (To appear in Class. Quant. Grav.

Rapid evidence review to inform safe return to campus in the context of coronavirus disease 2019 (COVID-19)

Background: Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) is transmitted predominantly through the air in crowded and unventilated indoor spaces among unvaccinated people. Universities and colleges are potential settings for its spread. Methods: An interdisciplinary team from public health, virology, and biology used narrative methods to summarise and synthesise evidence on key control measures, taking account of mode of transmission. Results: Evidence from a wide range of primary studies supports six measures. Vaccinate (aim for > 90% coverage and make it easy to get a jab). Require masks indoors, especially in crowded settings. If everyone wears well-fitting cloth masks, source control will be high, but for maximum self-protection, respirator masks should be worn. Masks should not be removed for speaking or singing. Space people out by physical distancing (but there is no “safe” distance because transmission risk varies with factors such as ventilation, activity levels and crowding), reducing class size (including offering blended learning), and cohorting (students remain in small groups with no cross-mixing). Clean indoor air using engineering controls—ventilation (while monitoring CO2 levels), inbuilt filtration systems, or portable air cleaners fitted with high efficiency particulate air [HEPA] filters). Test asymptomatic staff and students using lateral flow tests, with tracing and isolating infectious cases when incidence of coronavirus disease 2019 (COVID-19) is high. Support clinically vulnerable people to work remotely. There is no direct evidence to support hand sanitising, fomite controls or temperature-taking. There is evidence that freestanding plastic screens, face visors and electronic air-cleaning systems are ineffective. Conclusions: The above six evidence-based measures should be combined into a multi-faceted strategy to maximise both student safety and the continuation of in-person and online education provision. Staff and students seeking to negotiate a safe working and learning environment should collect data (e.g. CO2 levels, room occupancy) to inform conversations

Four conjectures in Nonlinear Analysis

In this chapter, I formulate four challenging conjectures in Nonlinear Analysis. More precisely: a conjecture on the Monge-Amp\`ere equation; a conjecture on an eigenvalue problem; a conjecture on a non-local problem; a conjecture on disconnectedness versus infinitely many solutions.Comment: arXiv admin note: text overlap with arXiv:1504.01010, arXiv:1409.5919, arXiv:1612.0819