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Abstract. This book is intended to serve both as an introduction and a reference to spectral and inverse spectral theory of Jacobi operators (i.e., second order symmetric difference operators) and applications of these theories to the Toda and Kac-van Moerbeke hierarchy.Starting from second order difference equations we move on to self-adjoint operators and develop discrete Weyl-Titchmarsh-Kodaira theory, covering all classical aspects like Weyl m-functions, spectral functions, the moment problem, inverse spectral theory, and uniqueness results. Next, we investigate some more advanced topics like locating the essential, absolutely continuous, and discrete spectrum, subordinacy, oscillation theory, trace formulas, random operators, almost periodic operators, (quasi-)periodic operators, scattering theory, and spectral deformations.Then, the Lax approach is used to introduce the Toda hierarchy and its modified counterpart, the Kac-van Moerbeke hierarchy. Uniqueness and existence theorems for the initial value problem, solutions in terms of Riemann theta functions, the inverse scattering transform, Bäcklund transformations, and soliton solutions are discussed.Corrections and complements to this book are available from:

A real-time recording setup combining exhaled breath volatile organic compound (VOC) measurements by proton transfer reaction-mass spectrometry (PTR-MS) with hemodynamic and respiratory data is presented. Continuous automatic sampling of exhaled breath is implemented on the basis of measured respiratory flow: a flow-controlled shutter mechanism guarantees that only end-tidal exhalation segments are drawn into the mass spectrometer for analysis. Exhaled breath concentration profiles of two prototypic compounds, isoprene and acetone, during several exercise regimes were acquired, reaffirming and complementing earlier experimental findings regarding the dynamic response of these compounds reported by Senthilmohan et al (2000 Redox Rep. 5 151-3) and Karl et al (2001 J. Appl. Physiol. 91 762-70). While isoprene tends to react very sensitively to changes in pulmonary ventilation and perfusion due to its lipophilic behavior and low Henry constant, hydrophilic acetone shows a rather stable behavior. Characteristic (median) values for breath isoprene concentration and molar flow, i.e., the amount of isoprene exhaled per minute are 100 ppb and 29 nmol min(-1), respectively, with some intra-individual day-to-day variation. At the onset of exercise breath isoprene concentration increases drastically, usually by a factor of ∼3-4 within about 1 min. Due to a simultaneous increase in ventilation, the associated rise in molar flow is even more pronounced, leading to a ratio between peak molar flow and molar flow at rest of ∼11. Our setup holds great potential in capturing continuous dynamics of non-polar, low-soluble VOCs over a wide measurement range with simultaneous appraisal of decisive physiological factors affecting exhalation kinetics. In particular, data appear to favor the hypothesis that short-term effects visible in breath isoprene levels are mainly caused by changes in pulmonary gas exchange patterns rather than fluctuations in endogenous synthesis.

Abstract. We apply the method of nonlinear steepest descent to compute the longtime asymptotics of the Camassa-Holm equation for decaying initial data, completing previous results by A. Boutet de Monvel and D. Shepelsky.

Human breath contains a myriad of endogenous volatile organic compounds (VOCs) which are reflective of ongoing metabolic or physiological processes. While research into the diagnostic potential and general medical relevance of these trace gases is conducted on a considerable scale, little focus has been given so far to a sound analysis of the quantitative relationships between breath levels and the underlying systemic concentrations. This paper is devoted to a thorough modeling study of the end-tidal breath dynamics associated with isoprene, which serves as a paradigmatic example for the class of low-soluble, blood-borne VOCs. Real-time measurements of exhaled breath under an ergometer challenge reveal characteristic changes of isoprene output in response to variations in ventilation and perfusion. Here, a valid compartmental description of these profiles is developed. By comparison with experimental data it is inferred that the major part of breath isoprene variability during exercise conditions can be attributed to an increased fractional perfusion of potential storage and production sites, leading to higher levels of mixed venous blood concentrations at the onset of physical activity. In this context, various lines of supportive evidence for an extrahepatic tissue source of isoprene are presented. Our model is a first step towards new guidelines for the breath gas analysis of isoprene and is expected to aid further investigations regarding the exhalation, storage, transport and biotransformation processes associated with this important compound.