A class of well-posed parabolic final value problems

This paper focuses on parabolic final value problems, and well-posedness is proved for a large class of these. The clarification is obtained from Hilbert spaces that characterise data that give existence, uniqueness and stability of the solutions. The data space is the graph normed domain of an unbounded operator that maps final states to the corresponding initial states. It induces a new compatibility condition, depending crucially on the fact that analytic semigroups always are invertible in the class of closed operators. Lax--Milgram operators in vector distribution spaces constitute the main framework. The final value heat conduction problem on a smooth open set is also proved to be well posed, and non-zero Dirichlet data are shown to require an extended compatibility condition obtained by adding an improper Bochner integral.Comment: 16 pages. To appear in "Applied and numerical harmonic analysis"; a reference update. Conference contribution, based on arXiv:1707.02136, with some further development

The hybrid spectral problem and Robin boundary conditions

The hybrid spectral problem where the field satisfies Dirichlet conditions (D) on part of the boundary of the relevant domain and Neumann (N) on the remainder is discussed in simple terms. A conjecture for the C_1 coefficient is presented and the conformal determinant on a 2-disc, where the D and N regions are semi-circles, is derived. Comments on higher coefficients are made. A hemisphere hybrid problem is introduced that involves Robin boundary conditions and leads to logarithmic terms in the heat--kernel expansion which are evaluated explicitly.Comment: 24 pages. Typos and a few factors corrected. Minor comments added. Substantial Robin additions. Substantial revisio

Media framing of women’s football during the Covid-19 pandemic

This article examines British media coverage of women’s association football during the coronavirus (Covid-19) pandemic, to identify how the media framed the women’s game and how these frames could shape the public perceptions of it. Through a database search of British-based news coverage of women’s football, 100 news articles were identified in the first six months after the start of the pandemic. A thematic analysis was conducted, and five dominant frames were detected in the context of Covid-19: 1) financial precariousness of women’s football; 2) the commercial prioritisation of men’s football; 3) practical consideration of the sport (e.g., alterations to national and international competitions); 4) debating the future of women’s football; and 5) concern for players (e.g., welfare, uncertain working conditions). These frames depart from the past trivialisation and sexualisation of women’s sport, demonstrate the increased visibility of women’s football, and shift the narrative towards the elite stratum of the game. Most of this reporting was by women journalists, while men were shown to write less than women about women’s football. This research advocates continued diversification of the sports journalism workforce to dissolve the hegemonic masculine culture that still largely dominates the industry

Strong ellipticity and spectral properties of chiral bag boundary conditions

We prove strong ellipticity of chiral bag boundary conditions on even dimensional manifolds. From a knowledge of the heat kernel in an infinite cylinder, some basic properties of the zeta function are analyzed on cylindrical product manifolds of arbitrary even dimension.Comment: 16 pages, LaTeX, References adde

Schrödinger operators with δ and δ′-potentials supported on hypersurfaces

Self-adjoint Schrödinger operators with δ and δ′-potentials supported on a smooth compact hypersurface are defined explicitly via boundary conditions. The spectral properties of these operators are investigated, regularity results on the functions in their domains are obtained, and analogues of the Birman–Schwinger principle and a variant of Krein’s formula are shown. Furthermore, Schatten–von Neumann type estimates for the differences of the powers of the resolvents of the Schrödinger operators with δ and δ′-potentials, and the Schrödinger operator without a singular interaction are proved. An immediate consequence of these estimates is the existence and completeness of the wave operators of the corresponding scattering systems, as well as the unitary equivalence of the absolutely continuous parts of the singularly perturbed and unperturbed Schrödinger operators. In the proofs of our main theorems we make use of abstract methods from extension theory of symmetric operators, some algebraic considerations and results on elliptic regularity

Multiple reflection expansion and heat kernel coefficients

We propose the multiple reflection expansion as a tool for the calculation of heat kernel coefficients. As an example, we give the coefficients for a sphere as a finite sum over reflections, obtaining as a byproduct a relation between the coefficients for Dirichlet and Neumann boundary conditions. Further, we calculate the heat kernel coefficients for the most general matching conditions on the surface of a sphere, including those cases corresponding to the presence of delta and delta prime background potentials. In the latter case, the multiple reflection expansion is shown to be non-convergent.Comment: 21 pages, corrected for some misprint

Forehead reflectance photoplethysmography to monitor heart rate: preliminary results from neonatal patients

Around 5%–10% of newborn babies require some form of resuscitation at birth and heart rate (HR) is the best guide of efficacy. We report the development and first trial of a device that continuously monitors neonatal HR, with a view to deployment in the delivery room to guide newborn resuscitation. The device uses forehead reflectance photoplethysmography (PPG) with modulated light and lock-in detection. Forehead fixation has numerous advantages including ease of sensor placement, whilst perfusion at the forehead is better maintained in comparison to the extremities. Green light (525 nm) was used, in preference to the more usual red or infrared wavelengths, to optimize the amplitude of the pulsatile signal. Experimental results are presented showing simultaneous PPG and electrocardiogram (ECG) HRs from babies (n = 77), gestational age 26–42 weeks, on a neonatal intensive care unit. In babies ≥32 weeks gestation, the median reliability was 97.7% at ±10 bpm and the limits of agreement (LOA) between PPG and ECG were +8.39 bpm and −8.39 bpm. In babies <32 weeks gestation, the median reliability was 94.8% at ±10 bpm and the LOA were +11.53 bpm and −12.01 bpm. Clinical evaluation during newborn deliveries is now underway

Heat Kernel Expansion for Semitransparent Boundaries

We study the heat kernel for an operator of Laplace type with a $\delta$-function potential concentrated on a closed surface. We derive the general form of the small $t$ asymptotics and calculate explicitly several first heat kernel coefficients.Comment: 16 page

New Kernels in Quantum Gravity

Recent work in the literature has proposed the use of non-local boundary conditions in Euclidean quantum gravity. The present paper studies first a more general form of such a scheme for bosonic gauge theories, by adding to the boundary operator for mixed boundary conditions of local nature a two-by-two matrix of pseudo-differential operators with pseudo-homogeneous kernels. The request of invariance of such boundary conditions under infinitesimal gauge transformations leads to non-local boundary conditions on ghost fields. In Euclidean quantum gravity, an alternative scheme is proposed, where non-local boundary conditions and the request of their complete gauge invariance are sufficient to lead to gauge-field and ghost operators of pseudo-differential nature. The resulting boundary conditions have a Dirichlet and a pseudo-differential sector, and are pure Dirichlet for the ghost. This approach is eventually extended to Euclidean Maxwell theory.Comment: 19 pages, plain Tex. In this revised version, section 5 is new, section 3 is longer, and the presentation has been improve

Existence of global strong solutions to a beam-fluid interaction system

We study an unsteady non linear fluid-structure interaction problem which is a simplified model to describe blood flow through viscoleastic arteries. We consider a Newtonian incompressible two-dimensional flow described by the Navier-Stokes equations set in an unknown domain depending on the displacement of a structure, which itself satisfies a linear viscoelastic beam equation. The fluid and the structure are fully coupled via interface conditions prescribing the continuity of the velocities at the fluid-structure interface and the action-reaction principle. We prove that strong solutions to this problem are global-in-time. We obtain in particular that contact between the viscoleastic wall and the bottom of the fluid cavity does not occur in finite time. To our knowledge, this is the first occurrence of a no-contact result, but also of existence of strong solutions globally in time, in the frame of interactions between a viscous fluid and a deformable structure