Determinants of Dirac operators with local boundary conditions

We study functional determinants for Dirac operators on manifolds with boundary. We give, for local boundary conditions, an explicit formula relating these determinants to the corresponding Green functions. We finally apply this result to the case of a bidimensional disk under bag-like conditions.Comment: standard LaTeX, 24 pages. To appear in Jour. Math. Phy

Realizations of Differential Operators on Conic Manifolds with Boundary

We study the closed extensions (realizations) of differential operators subject to homogeneous boundary conditions on weighted L_p-Sobolev spaces over a manifold with boundary and conical singularities. Under natural ellipticity conditions we determine the domains of the minimal and the maximal extension. We show that both are Fredholm operators and give a formula for the relative index.Comment: 41 pages, 1 figur

Evolution of size-dependent flowering in a variable environment: construction and analysis of a stochastic integral projection model

Understanding why individuals delay reproduction is a classic problem in evolutionary biology. In plants, the study of reproductive delays is complicated because growth and survival can be size and age dependent, individuals of the same size can grow by different amounts and there is temporal variation in the environment. We extend the recently developed integral projection approach to include size- and age-dependent demography and temporal variation. The technique is then applied to a long-term individually structured dataset for Carlina vulgaris, a monocarpic thistle. The parameterized model has excellent descriptive properties in terms of both the population size and the distributions of sizes within each age class. In Carlina, the probability of flowering depends on both plant size and age. We use the parameterized model to predict this relationship, using the evolutionarily stable strategy approach. Considering each year separately, we show that both the direction and the magnitude of selection on the flowering strategy vary from year to year. Provided the flowering strategy is constrained, so it cannot be a step function, the model accurately predicts the average size at flowering. Elasticity analysis is used to partition the size- and age-specific contributions to the stochastic growth rate, λs. We use λs to construct fitness landscapes and show how different forms of stochasticity influence its topography. We prove the existence of a unique stochastic growth rate, λs, which is independent of the initial population vector, and show that Tuljapurkar's perturbation analysis for log(λs) can be used to calculate elasticities

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

Ellipticity Conditions for the Lax Operator of the KP Equations

The Lax pseudo-differential operator plays a key role in studying the general set of KP equations, although it is normally treated in a formal way, without worrying about a complete characterization of its mathematical properties. The aim of the present paper is therefore to investigate the ellipticity condition. For this purpose, after a careful evaluation of the kernel with the associated symbol, the majorization ensuring ellipticity is studied in detail. This leads to non-trivial restrictions on the admissible set of potentials in the Lax operator. When their time evolution is also considered, the ellipticity conditions turn out to involve derivatives of the logarithm of the tau-function.Comment: 21 pages, plain Te

Improved estimation of glomerular filtration rate (GFR) by comparison of eGFRcystatin C and eGFRcreatinine

Objective. GFR-prediction equations based upon cystatin C and creatinine have better diagnostic performance in estimating GFR than equations based upon only one of the two markers. The present work concerns in what way a comparison between separate estimations of GFR based upon cystatin C (eGFR(cystatin C)) or creatinine (eGFR(creatinine)) can be used to evaluate the diagnostic performance of a combined cystatin C-and creatinine-based estimation of GFR. Methods. The difference between eGFR(cystatin C) and eGFR(creatinine) was compared with measured GFR (iohexol clearance) and a combined cystatin C- and creatinine-based estimation of GFR in a Swedish-Caucasian cohort of 857 adult patients. Results. A difference between eGFR(cystatin C) and eGFR(creatinine) of >= 40% indicated a markedly reduced diagnostic performance of the combined cystatin C- and creatinine-based estimation of GFR. Conclusion. Comparison of the agreement between eGFR(cystatin C) and eGFR(creatinine) can be used to evaluate the diagnostic performance of combined cystatin C-and creatinine-based estimations of GFR. If 'threshold values' for discordance are exceeded, it must be considered whether the clinical context requires the use of an invasive gold standard method to measure GFR. In some clinical contexts either creatinine or cystatin C are known to be invalidated as markers of GFR and in these situations the use of only the cystatin C-or the creatinine-based GFR estimate should be considered when the 'threshold values' are exceeded

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

Spectral asymmetry for bag boundary conditions

We give an expression, in terms of boundary spectral functions, for the spectral asymmetry of the Euclidean Dirac operator in two dimensions, when its domain is determined by local boundary conditions, and the manifold is of product type. As an application, we explicitly evaluate the asymmetry in the case of a finite-length cylinder, and check that the outcome is consistent with our general result. Finally, we study the asymmetry in a disk, which is a non-product case, and propose an interpretation.Comment: Some minor changes. To appear in Journal of Physics A: Mathematical and Genera

Quantum Control at the Boundary

We present a scheme for controlling the state of a quantum system by modifying the boundary conditions. This constitutes an infinite-dimensional control problem. We provide conditions for the existence of solutions of the dynamics and prove that this system is approximately controllable

A Survey on the Krein-von Neumann Extension, the corresponding Abstract Buckling Problem, and Weyl-Type Spectral Asymptotics for Perturbed Krein Laplacians in Nonsmooth Domains

In the first (and abstract) part of this survey we prove the unitary equivalence of the inverse of the Krein--von Neumann extension (on the orthogonal complement of its kernel) of a densely defined, closed, strictly positive operator, $S\geq \varepsilon I_{\mathcal{H}}$ for some $\varepsilon >0$ in a Hilbert space $\mathcal{H}$ to an abstract buckling problem operator. This establishes the Krein extension as a natural object in elasticity theory (in analogy to the Friedrichs extension, which found natural applications in quantum mechanics, elasticity, etc.). In the second, and principal part of this survey, we study spectral properties for $H_{K,\Omega}$, the Krein--von Neumann extension of the perturbed Laplacian $-\Delta+V$ (in short, the perturbed Krein Laplacian) defined on $C^\infty_0(\Omega)$, where $V$ is measurable, bounded and nonnegative, in a bounded open set $\Omega\subset\mathbb{R}^n$ belonging to a class of nonsmooth domains which contains all convex domains, along with all domains of class $C^{1,r}$, $r>1/2$.Comment: 68 pages. arXiv admin note: extreme text overlap with arXiv:0907.144