The Feynman-Kac formula and applications to PDEs

The thesis presents a probabilistic approach to the theory of semigroups of operators, with particular attention to the Markov and Feller semigroups. The first goal of this work is the proof of the fundamental Feynman-Kac formula, which gives the solution of certain parabolic Cauchy problems, in terms of the expected value of the initial condition computed at the associated stochastic diffusion processes. The second target is the characterization of the principal eigenvalue of the generator of a semigroup with Markov transition probability function and of second order elliptic operators with real coefficients not necessarily self-adjoint. The thesis is divided into three chapters. In the first chapter we study the Brownian motion and some of its main properties, the stochastic processes, the stochastic integral and the Itô formula in order to finally arrive, in the last section, at the proof of the Feynman-Kac formula. The second chapter is devoted to the probabilistic approach to the semigroups theory and it is here that we introduce Markov and Feller semigroups. Special emphasis is given to the Feller semigroup associated with the Brownian motion. The third and last chapter is divided into two sections. In the first one we present the abstract characterization of the principal eigenvalue of the infinitesimal generator of a semigroup of operators acting on continuous functions over a compact metric space. In the second section this approach is used to study the principal eigenvalue of elliptic partial differential operators with real coefficients. At the end, in the appendix, we gather some of the technical results used in the thesis in more details. Appendix A is devoted to the Sion minimax theorem, while in appendix B we prove the Chernoff product formula for not necessarily self-adjoint operators.

Uncovering the dynamics of cardiac systems using stochastic pacing and frequency domain analyses

Alternans of cardiac action potential duration (APD) is a well-known arrhythmogenic mechanism which results from dynamical instabilities. The propensity to alternans is classically investigated by examining APD restitution and by deriving APD restitution slopes as predictive markers. However, experiments have shown that such markers are not always accurate for the prediction of alternans. Using a mathematical ventricular cell model known to exhibit unstable dynamics of both membrane potential and Ca2+ cycling, we demonstrate that an accurate marker can be obtained by pacing at cycle lengths (CLs) varying randomly around a basic CL (BCL) and by evaluating the transfer function between the time series of CLs and APDs using an autoregressive-moving-average (ARMA) model. The first pole of this transfer function corresponds to the eigenvalue (λalt) of the dominant eigenmode of the cardiac system, which predicts that alternans occurs when λalt≤−1. For different BCLs, control values of λalt were obtained using eigenmode analysis and compared to the first pole of the transfer function estimated using ARMA model fitting in simulations of random pacing protocols. In all versions of the cell model, this pole provided an accurate estimation of λalt. Furthermore, during slow ramp decreases of BCL or simulated drug application, this approach predicted the onset of alternans by extrapolating the time course of the estimated λalt. In conclusion, stochastic pacing and ARMA model identification represents a novel approach to predict alternans without making any assumptions about its ionic mechanisms. It should therefore be applicable experimentally for any type of myocardial cell.

Effective Length K-Factors For Flexural Buckling Strengths Of Web Members In Open Web Steel Joists

Open web steel joists are designed in the United States following the governing specification published by the Steel Joist Institute. For compression members in joists, this specification employs an effective length factor, or K-factor, in confirming their adequacy. In most cases, these K-factors have been conservatively assumed equal to 1.0 for compression web members, regardless of the fact that intuition and limited experimental work indicate that smaller values could be justified. Given that smaller K-factors could result in more economical designs without a loss in safety, the research presented in this thesis aims to suggest procedures for obtaining more rational values. Three different methods for computing in-plane and out-of-plane K-factors are investigated, including (1) a hand calculation method based on the use of alignment charts, (2) computational critical load (eigenvalue) analyses using uniformly distributed loads, and (3) computational analyses using a compressive strain approach. The latter method is novel and allows for computing the individual buckling load of a specific member within a system, such as a joist. Four different joist configurations are investigated, including an 18K3, 28K10, and two variations of a 32LH06. Based on these methods and the very limited number of joists studied, it appears promising that in-plane and out-of-plane K-factors of 0.75 and 0.85, respectively, could be used in computing the flexural buckling strength of web members in routine steel joist design. Recommendations for future work, which include systematically investigating a wider range of joist configurations and connection restraint, are provided.

Data-driven estimates of the number of clusters in multivariate time series

An important problem in unsupervised data clustering is how to determine the number of clusters. Here we investigate how this can be achieved in an automated way by using interrelation matrices of multivariate time series. Two nonparametric and purely data driven algorithms are expounded and compared. The first exploits the eigenvalue spectra of surrogate data, while the second employs the eigenvector components of the interrelation matrix. Compared to the first algorithm, the second approach is computationally faster and not limited to linear interrelation measures.

Spectral bounds and basis results for non-self-adjoint pencils, with an application to Hagen-Poiseuille flow

We obtain eigenvalue enclosures and basisness results for eigen- and associated functions of a non-self-adjoint unbounded linear operator pencil A−λBA−λB in which BB is uniformly positive and the essential spectrum of the pencil is empty. Both Riesz basisness and Bari basisness results are obtained. The results are applied to a system of singular differential equations arising in the study of Hagen–Poiseuille flow with non-axisymmetric disturbances.

Early Eocene benthic foraminifera of DSDP Hole 48-401

Within the last decade, several early Eocene hyperthermals have been detected globally. These transient warming events have mainly been characterized geochemically - using stable isotopes, carbonate content measurements or XRF core scanning - yet detailed micropaleontological records are sparse, limiting our understanding of the driving forces behind hyperthermals and of the contemporaneous paleoceanography. Here, detailed geochemical and quantitative benthic foraminiferal records are presented from lower Eocene pelagic sediments of Deep Sea Drilling Project Site 401 (Bay of Biscay, northeast Atlantic). In calcareous nannofossil zone NP11, several clay-enriched levels correspond to negative d13C and d18O bulk-rock excursions with amplitudes of up to ~0.75 per mil, suggesting that significant injections of 12C-enriched greenhouse gasses and small temperature rises took place. Coeval with several of these hyperthermal events, the benthic foraminiferal record reveals increased relative abundances of oligotrophic taxa (e.g. Nuttallides umbonifera) and a reduction in the abundance of buliminid species followed by an increase of opportunistic taxa (e.g. Globocassidulina subglobosa and Gyroidinoides spp.). These short-lived faunal perturbations are thought to be caused by reduced seasonality of productivity resulting in a decreased Corg flux to the seafloor. Moreover, the sedimentological record suggests that an enhanced influx of terrigenous material occurred during these events. Additionally, the most intense d13C decline (here called level d) gives rise to a small, yet pronounced long-term shift in the benthic foraminiferal composition at this site, possibly due to the reappraisal of upwelling and the intensification of bottom water currents. These observations imply that environmental changes during (smaller) hyperthermal events are also reflected in the composition of deep-sea benthic communities on both short (<100 kyr) and longer time scales. We conclude that the faunal patterns of the hyperthermals observed at Site 401 strongly resemble those observed in other deep-sea early Paleogene hyperthermal deposits, suggesting that similar processes have driven them.

Magnetic measurements on ODP holes 178-1095A, 178-1095B and 178-1101A

We have determined the azimuth of bottom-current flow in drift deposit sediments recovered at ODP Sites 1095 and 1101, Antarctic Peninsula, using paleomagnetic reorientation of anisotropy of magnetic susceptibility (AMS) ellipsoids. A total of 38 cores from the two ODP sites have been measured, providing spatial and directional information on the physical record of the ACC (Antarctic Circumpolar Current) in the Plio-Pleistocene. Declination and inclination of the paleomagnetic vector of each core segment were used to reorient the AMS principal axes to the geographic coordinates. The cores were reoriented using the measured direction of the characteristic remanent magnetization (ChRM) with respect to a common reference line for the core, from which we are able to determine the orientation of the paleocurrent flow for Sites 1095 (Drift 7) and 1101 (Drift 4) relative to the geographic coordinates. Both sites have paleocurrent directions trending ~NW-SE, which in the former locality are parallel to a sediment wave field. Our study shows that a combination of magnetic fabric analysis and paleomagnetism allows deep-sea sedimentary fabric to be used as a long-term proxy of bottom-current flow history.

Paleomagnetic and rock magnetic properties of IODP Expedition 318 cores

The Integrated Ocean Drilling Program Expedition 318 to the Wilkes Land margin of Antarctica recovered a sedimentary succession ranging in age from lower Eocene to the Holocene. Excellent stratigraphic control is key to understanding the timing of paleoceanographic events through critical climate intervals. Drill sites recovered the lower and middle Eocene, nearly the entire Oligocene, the Miocene from about 17 Ma, the entire Pliocene and much of the Pleistocene. The paleomagnetic properties are generally suitable for magnetostratigraphic interpretation, with well-behaved demagnetization diagrams, uniform distribution of declinations, and a clear separation into two inclination modes. Although the sequences were discontinuously recovered with many gaps due to coring, and there are hiatuses from sedimentary and tectonic processes, the magnetostratigraphic patterns are in general readily interpretable. Our interpretations are integrated with the diatom, radiolarian, calcareous nannofossils and dinoflagellate cyst (dinocyst) biostratigraphy. The magnetostratigraphy significantly improves the resolution of the chronostratigraphy, particularly in intervals with poor biostratigraphic control. However, Southern Ocean records with reliable magnetostratigraphies are notably scarce, and the data reported here provide an opportunity for improved calibration of the biostratigraphic records. In particular, we provide a rare magnetostratigraphic calibration for dinocyst biostratigraphy in the Paleogene and a substantially improved diatom calibration for the Pliocene. This paper presents the stratigraphic framework for future paleoceanographic proxy records which are being developed for the Wilkes Land margin cores. It further provides tight constraints on the duration of regional hiatuses inferred from seismic surveys of the region.