Fractional-order operators: Boundary problems, heat equations

The first half of this work gives a survey of the fractional Laplacian (and related operators), its restricted Dirichlet realization on a bounded domain, and its nonhomogeneous local boundary conditions, as treated by pseudodifferential methods. The second half takes up the associated heat equation with homogeneous Dirichlet condition. Here we recall recently shown sharp results on interior regularity and on $L_p$-estimates up to the boundary, as well as recent H\"older estimates. This is supplied with new higher regularity estimates in $L_2$-spaces using a technique of Lions and Magenes, and higher $L_p$-regularity estimates (with arbitrarily high H\"older estimates in the time-parameter) based on a general result of Amann. Moreover, it is shown that an improvement to spatial $C^\infty$-regularity at the boundary is not in general possible.Comment: 29 pages, updated version, to appear in a Springer Proceedings in Mathematics and Statistics: "New Perspectives in Mathematical Analysis - Plenary Lectures, ISAAC 2017, Vaxjo Sweden

Smeared heat-kernel coefficients on the ball and generalized cone

We consider smeared zeta functions and heat-kernel coefficients on the bounded, generalized cone in arbitrary dimensions. The specific case of a ball is analysed in detail and used to restrict the form of the heat-kernel coefficients $A_n$ on smooth manifolds with boundary. Supplemented by conformal transformation techniques, it is used to provide an effective scheme for the calculation of the $A_n$. As an application, the complete $A_{5/2}$ coefficient is given.Comment: 23 pages, JyTe

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

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

Local regularity for fractional heat equations

We prove the maximal local regularity of weak solutions to the parabolic problem associated with the fractional Laplacian with homogeneous Dirichlet boundary conditions on an arbitrary bounded open set $\Omega\subset\mathbb{R}^N$. Proofs combine classical abstract regularity results for parabolic equations with some new local regularity results for the associated elliptic problems.Comment: arXiv admin note: substantial text overlap with arXiv:1704.0756

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

Wage returns to university disciplines in Greece: are Greek Higher Education degrees Trojan Horses?

This paper examines the wage returns to qualifications and academic disciplines in the Greek labour market. Exploring wage responsiveness across various degree subjects in Greece is interesting, as it is characterised by high levels of graduate unemployment, which vary considerably by field of study, and relatively low levels of wage flexibility. Using micro-data from recently available waves (2002-2003) of the Greek Labour Force Survey (LFS), the returns to academic disciplines are estimated by gender and public/private sector. Quantile regressions and cohort interactions are also used to capture the heterogeneity in wage returns across the various disciplines. The results show considerable variation in wage premiums across the fields of study, with lower returns for those that have a marginal role to play in an economy with a rising services/shrinking public sector. Educational reforms that pay closer attention to the future prospects of university disciplines are advocated

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

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