We derive dispersion estimates for solutions of the one-dimensional discrete perturbed Dirac equation. To this end we develop basic scattering theory and establish a limiting absorption principle for discrete perturbed Dirac operators.
Volatile organic compounds emitted by a human body form a chemical signature capable of providing invaluable information on the physiological status of an individual and, thereby, could serve as signs-of-life for detecting victims after natural or man-made disasters. In this review a database of potential biomarkers of human presence was created on the basis of existing literature reports on volatiles in human breath, skin emanation, blood, and urine. Approximate fluxes of these species from the human body were estimated and used to predict their concentrations in the vicinity of victims. The proposed markers were classified into groups of different potential for victim detection. The major classification discriminants were the capability of detection by portable, real-time analytical instruments and background levels in urban environment. The data summarized in this review are intended to assist studies on the detection of humans via chemical analysis and accelerate investigations in this area of knowledge.
We consider the stability of the periodic Toda lattice (and slightly more generally of the algebro-geometric finite-gap lattice) under a short range perturbation. We prove that the perturbed lattice asymptotically approaches a modulated lattice. More precisely, let $g$ be the genus of the hyperelliptic curve associated with the unperturbed solution. We show that, apart from the phenomenon of the solitons travelling on the quasi-periodic background, the $n/t$-pane contains $g+2$ areas where the perturbed solution is close to a finite-gap solution in the same isospectral torus. In between there are $g+1$ regions where the perturbed solution is asymptotically close to a modulated lattice which undergoes a continuous phase transition (in the Jacobian variety) and which interpolates between these isospectral solutions. In the special case of the free lattice ($g=0$) the isospectral torus consists of just one point and we recover the known result. Both the solutions in the isospectral torus and the phase transition are explicitly characterized in terms of Abelian integrals on the underlying hyperelliptic curve. Our method relies on the equivalence of the inverse spectral problem to a matrix Riemann--Hilbert problem defined on the hyperelliptic curve and generalizes the so-called nonlinear stationary phase/steepest descent method for Riemann--Hilbert problem deformations to Riemann surfaces.
We survey a selection of Fritzs principal contributions to the field of spectral theory and, in particular, to Schroedinger operators.
Isothermal rebreathing has been proposed as an experimental technique for estimating the alveolar levels of hydrophilic volatile organic compounds (VOCs) in exhaled breath. Using the prototypic test compound acetone we demonstrate that the end-tidal breath profiles of such substances during isothermal rebreathing show characteristics that contradict the conventional pulmonary inert gas elimination theory due to Farhi. On the other hand, these profiles can reliably be captured by virtue of a previously developed mathematical model for the general exhalation kinetics of highly soluble, blood-borne VOCs, which explicitly takes into account airway gas exchange as major determinant of the observable breath output. This model allows for a mechanistic analysis of various rebreathing protocols suggested in the literature. In particular, it clarifies the discrepancies between in vitro and in vivo blood-breath ratios of hydrophilic VOCs and yields further quantitative insights into the physiological components of isothermal rebreathing.
Isoprene is one of the most abundant endogenous volatile organic compounds (VOCs) contained in human breath and is considered to be a potentially useful biomarker for diagnostic and monitoring purposes. However, neither the exact biochemical origin of isoprene nor its physiological role are understood in sufficient depth, thus hindering the validation of breath isoprene tests in clinical routine. Exhaled isoprene concentrations are reported to change under different clinical and physiological conditions, especially in response to enhanced cardiovascular and respiratory activity. Investigating isoprene exhalation kinetics under dynamical exercise helps to gather the relevant experimental information for understanding the gas exchange phenomena associated with this important VOC. A first model for isoprene in exhaled breath has been developed by our research group. In the present paper, we aim at giving a concise overview of this model and describe its role in providing supportive evidence for a peripheral (extrahepatic) source of isoprene. In this sense, the results presented here may enable a new perspective on the biochemical processes governing isoprene formation in the human body.
In a recent paper we have considered the long time asymptotics of the periodic Toda lattice under a short range perturbation and we have proved that the perturbed lattice asymptotically approaches a modulated lattice. In the present paper we capture the higher order asymptotics, at least away from some resonance regions. In particular we prove that the decay rate is $O(t^{-1/2})$. Our proof relies on the asymptotic analysis of the associated Riemann-Hilbert factorization problem, which is here set on a hyperelliptic curve. As in previous studies of the free Toda lattice, the higher order asymptotics arise from local Riemann-Hilbert factorization problems on small crosses centered on the stationary phase points. We discover that the analysis of such a local problem can be done in a chart around each stationary phase point and reduces to a Riemann--Hilbert factorization problem on the complex plane. This result can then be pulled back to the hyperelliptic curve.
Recommended standardized procedures for determining exhaled lower respiratory nitric oxide and nasal nitric oxide have been developed by task forces of the European Respiratory Society and the American Thoracic Society. These recommendations have paved the way for the measurement of nitric oxide to become a diagnostic tool for specific clinical applications. It would be desirable to develop similar guidelines for the sampling of other trace gases in exhaled breath, especially volatile organic compounds (VOCs) which reflect ongoing metabolism. The concentrations of water-soluble, blood-borne substances in exhaled breath are influenced by: (i) breathing patterns affecting gas exchange in the conducting airways; (ii) the concentrations in the tracheo-bronchial lining fluid; (iii) the alveolar and systemic concentrations of the compound. The classical Farhi equation takes only the alveolar concentrations into account. Real-time measurements of acetone in end-tidal breath under an ergometer challenge show characteristics which cannot be explained within the Farhi setting. Here we develop a compartment model that reliably captures these profiles and is capable of relating breath to the systemic concentrations of acetone. By comparison with experimental data it is inferred that the major part of variability in breath acetone concentrations (e.g., in response to moderate exercise or altered breathing patterns) can be attributed to airway gas exchange, with minimal changes of the underlying blood and tissue concentrations. Moreover, it is deduced that measured end-tidal breath concentrations of acetone determined during resting conditions and free breathing will be rather poor indicators for endogenous levels. Particularly, the current formulation includes the classical Farhi and the Scheid series inhomogeneity model as special limiting cases.
We apply the method of nonlinear steepest descent to compute the long-time asymptotics of the periodic (and slightly more generally of the quasi-periodic finite-gap) Toda lattice for decaying initial data in the soliton region. In addition, we show how to reduce the problem in the remaining region to the known case without solitons.
We apply the method of nonlinear steepest descent to compute the long-time asymptotics of the Toda lattice for decaying initial data in the soliton region. In addition, we point out how to reduce the problem in the remaining region to the known case without solitons.
