A discrete analogue of the Witten Laplacian on the n-dimensional integer lattice is considered. After rescaling of the operator and the lattice size we analyze the tunnel effect between different wells, providing sharp asymptotics of the low-lying spectrum. Our proof, inspired by work of B. Helffer, M. Klein and F. Nier in continuous setting, is based on the construction of a discrete Witten complex and a semiclassical analysis of the corresponding discrete Witten Laplacian on 1-forms. The result can be reformulated in terms of metastable Markov processes on the lattice. / In dieser Arbeit wird auf dem n-dimensionalen Gitter der ganzen Zahlen ein Analogon des Witten-Laplace-Operatoren eingeführt. Nach geeigneter Skalierung des Gitters und des Operatoren analysieren wir den Tunneleffekt zwischen verschiedenen Potentialtöpfen und erhalten vollständige Aymptotiken für das tiefliegende Spektrum. Der Beweis (nach Methoden, die von B. Helffer, M. Klein und F. Nier im Falle des kontinuierlichen Witten-Laplace-Operatoren entwickelt wurden) basiert auf der Konstruktion eines diskreten Witten-Komplexes und der Analyse des zugehörigen Witten-Laplace-Operatoren auf 1-Formen. Das Resultat kann im Kontext von metastabilen Markov Prozessen auf dem Gitter reformuliert werden und ermöglicht scharfe Aussagen über metastabile Austrittszeiten.

A high spatial resolution acoustic directivity acquisition system (ADAS) has been developed to acquire anechoic measurements of the far field radiation of musical instruments that are either remote controlled or played by musicians. Building upon work performed by the BYU Acoustic Research Group in the characterization of loudspeaker directivity, one can rotate a musical instrument with sequential azimuthal angle increments under a fixed semicircular array of microphones while recording repeated notes or sequences of notes. This results in highly detailed and instructive directivity data presented in the form of high-resolution balloon plots. The directivity data and corresponding balloon plots may be shown to vary as functions of time or frequency. This thesis outlines the development of a prototype ADAS and its application to different sources including loudspeakers, a concert grand piano, trombone, flute, and violin. The development of a method of compensating for variations in the played amplitude at subsequent measurement positions using a near-field reference microphone and Frequency Response Functions (FRF) is presented along with the results of its experimental validation. This validation involves a loudspeaker, with known directivity, to simulate a live musician. It radiates both idealized signals and anechoic recordings of musical instruments with random variations in amplitude. The concept of coherence balloon maps and surface averaged coherence are introduced as tools to establish directivity confidence. The method of creating composite directivities for musical instruments is also introduced. A composite directivity comes from combining the directivities of all played partials to approximate what the equivalent directivity from a musical instrument would be if full spectral excitation could be used. The composite directivities are derived from an iterative averaging process that uses coherence as an inclusion criterion. Sample directivity results and discussions of experimental considerations of the piano, trombone, flute, and violin are presented. The research conducted is preliminary and will be further developed by future students to expand and refine the methods presented here.

The topic of this thesis is a diffusion process on a potential landscape which is given by a smooth Hamiltonian function in the regime of small noise. The work provides a new proof of the Eyring-Kramers formula for the Poincaré inequality of the associated generator of the diffusion. The Poincaré inequality characterizes the spectral gap of the generator and establishes the exponential rate of convergence towards equilibrium in the L²-distance. This result was first obtained by Bovier et. al. in 2004 relying on potential theory. The presented approach in the thesis generalizes to obtain also asymptotic sharp estimates of the constant in the logarithmic Sobolev inequality. The optimal constant in the logarithmic Sobolev inequality characterizes the convergence rate to equilibrium with respect to the relative entropy, which is a stronger distance as the L²-distance and slightly weaker than the L¹-distance. The optimal constant has here no direct spectral representation. The proof makes use of the scale separation present in the dynamics. The Eyring-Kramers formula follows as a simple corollary from the two main results of the work: The first one shows that the associated Gibbs measure restricted to a basin of attraction has a good Poincaré and logarithmic Sobolev constants providing the fast convergence of the diffusion to metastable states. The second main ingredient is a mean-difference estimate. Here a weighted transportation distance is used. It contains the main contribution to the Poincaré and logarithmic Sobolev constant, resulting from exponential long waiting times of jumps between metastable states of the diffusion.

For the last ten years, DVD-R media have played an important role in the storage of large amounts of digital data throughout the world. During this time it was assumed that the DVD-R was as long-lasting and stable as its predecessor, the CD-R. Several reports have surfaced over the last few years questioning the DVD-R's ability to maintain many of its claims regarding archival quality life spans. These reports have shown a wide range of longevity between the different brands. While some DVD-Rs may last a while, others may result in an early and unexpected failure. Compounding this problem is the lack of information available for consumers to know the quality of the media they own. While the industry works on devising a standard for labeling the quality of future media, it is currently up to the consumer to pay close attention to their own DVD-R archives and work diligently to prevent data loss. This research shows that through accelerated aging and the use of logistic regression analysis on data collected through periodic monitoring of disc read-back errors it is possible to accurately predict unrecoverable failures in the test discs. This study analyzed various measurements of PIE errors, PIE8 Sum errors, POF errors and jitter data from three areas of the disc: the whole disc, the region of the disc where it first failed as well as the last half of the disc. From this data five unique predictive equations were produced, each with the ability to predict disc failure. In conclusion, the relative value of these equations for end-of-life predictions is discussed.