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Elevators are a necessary component of the modern urban and suburban life. The guide rails the car and counterweight move on are the most sensitive parts when it comes to de-habilitating damage that can be caused by an earthquake. Conventional sensors are becoming obsolete in sensing for today's multistory buildings because they don't monitor the structural health of the guide rails. This sensing task falls into the fiber sensing niche market because of a fiber sensor's ability to be multiplexed. Previous work by Taplin and Jackson showed demodulation of the interference spectrum of two Fabry-Perot cavities using Fourier analysis. The goal of this research is to use Fourier analysis to demodulate the spectrum of four multiplexed extrinsic Fabry-Perot fiber interferometers for strain measurements. Comparisons of fiber, foil, and theoretical strains are made. Also, experiments showing the system's air-gap stability and crosstalk are provided. / Master of Science

Fourier spectral and pseudospectral methods are used in numerical modeling of ionospheric processes, namely macroscopic evolution of naturally and artificially created ionospheric density irregularities. The simulation model consists of two-dimensional electrostatic nonlinear fluid plasma equations that describe the plasma evolution. The spectral and pseudospectral methods are used to solve the spatial dependence of these self-consistent equations. They are chosen over the widely used finite difference and finite element techniques since spectral methods are straightforward to implement on nonlinear equations. They are at least as accurate as finite difference simulations. A potential equation solver is developed to solve the nonlinear potential equation iteratively. Time integration is accomplished using a combination of leapfrog and leapfrog-trapezoidal methods. A FORTRAN program is developed to implement the simulation model. All calculations are performed in the Fourier domain. The simulation model is tested by considering three types of problems. This is accomplished by specifying an initial density (Pedersen conductivity) profile that represents slab model density, density enhancement (due to releases such as barium), or density depletion (due to late times effects of electron attachment material releases) in the presence of a neutral wind. The evolution of the irregularities is monitored and discussed. The simulation results agree with similar results obtained using finite difference methods. A comparison is made between the ionospheric depletion and enhancement problems. Our results show that, given the same parameters and perturbation level, the depletion profiles bifurcate much faster than that of the enhancement. We argue that this is due to the larger growth rate in the E X B interchange instability of the density depletion case. / Master of Science

"Blind" inversion of the effects of a given operator on a signal is an extremely difficult task that has no easy solutions. However,. Dr. Hany Farid has published several works that each individua:lly appear to achieve exactly this seemingly impossible result. In this work, we contribute a comprehensive overview of the published applications of blind process inversion, as well as provide the generalized form of the algorithms and requirements that are found in each of these applications, thereby formulating and explaining a general framework for blind process inversion using Farid's Algorithm. Additionally, we explain the knowledge required to derive the ROSA-based cost function on which Farid's Algorithm depends. As our primary contribution, we analyze the algorithmic complexity of this cost function based on the way it is currently, naively calculated, and derive a new algorithm to compute this cost function that has greatly reduced algorithmic complexity. Finally, we suggest an additional application of Farid's Algorithm to the problem of blindly estimating true camera response functions from a single image.

Diese Dissertation behandelt Fragen aus der stochastischen Analysis und der Finanzmathematik, die sich unter dem Begriff der Robustheit zusammenfassen lassen. Zunächst betrachten wir finanzmathematische Modelle mit Arbitragemöglichkeiten. Wir identifizieren die Abwesenheit von Arbitragemöglichkeiten der ersten Art (NA1) als minimale Eigenschaft, die in jedem finanzmathematischen Modell gelten muss, und zeigen, dass (NA1) äquivalent zur Existenz eines dominierenden lokalen Martingalmaßes ist. Als Beispiel für Prozesse, die (NA1) erfüllen, studieren wir stetige lokale Martingale, die darauf bedingt werden nie Null zu treffen. Anschließend verwenden wir eine modellfreie Version der (NA1) Eigenschaft, die es erlaubt, qualitative Eigenschaften von “typischen Preistrajektorien” zu beschreiben. Hier konstruieren wir ein pfadweises Itô-Integral. Dies deutet an, dass sich typische Preispfade als rough-path-Integratoren verwenden lassen. Nun entwickeln wir mittels Fourierentwicklungen einen alternativen Zugang zur rough-path-Theorie. Wir zerlegen das Integral in drei Operatoren mit verschiedenen Eigenschaften. So wird offensichtlich, dass Integratoren mit der Regularität der Brownschen Bewegung mit ihrer Lévy-Fläche versehen werden müssen, um ein pfadweise stetiges Integral zu erhalten. Daraufhin bemerken wir, dass die Integration zweier Funktionen gegeneinander äquivalent dazu ist, eine Funktion mit der Ableitung einer anderen (im Allgemeinen eine Distribution) zu multiplizieren. In höheren Dimensionen ist das Multiplikationsproblem jedoch allgemeiner. Wir verwenden Littlewood-Paley-Theorie, um unseren Fourier-Zugang zur rough-path-Theorie auf Funktionen mehrdimensionaler Variablen zu erweitern. Wir konstruieren einen Operator, der für Funktionen mit dem punktweisen Produkt übereinstimmt und in einer geeigneten Topologie stetig ist. Nun lassen sich stochastische partielle Differentialgleichungen lösen, die bisher aufgrund von Nichtlinearitäten nicht zugänglich waren. / This thesis deals with various problems from stochastic analysis and from mathematical finance that can best be summarized under the common theme of robustness. We begin by studying financial market models with arbitrage opportunities. We identify the weak notion of absence of arbitrage opportunities of the first kind (NA1) as the minimal property that every sensible asset price model should satisfy, and we prove that (NA1) is equivalent to the existence of a dominating local martingale. As examples of processes that satisfy (NA1) but do not admit equivalent local martingale measures, we study continuous local martingales conditioned not to hit zero. We continue by working with a model free formulation of the (NA1) property, which permits to describe qualitative properties of “typical asset price trajectories”. We construct a pathwise Itô integral for typical price paths. Our results indicate that typical price paths can be used as integrators in the theory of rough paths. Next, we use a Fourier series expansion to develop an alternative approach to rough path integration. We decompose the integral into three components with different behavior. Then it is easy to see that integrators with the regularity of the Brownian motion must be equipped with their Lévy area to obtain a pathwise continuous integral operator. We now note that integrating two functions against each other is equivalent to multiplying one with the derivative of the other, which will in general only be a distribution. In higher index dimensions however, the multiplication problem is more general. We use Littlewood-Paley theory to extend our Fourier approach from rough path integrals to multiplying functions of a multidimensional index. We construct an operator which agrees with the usual product for smooth functions, and which is continuous in a suitable topology. We apply this to solve stochastic partial differential equations that were previously difficult to access due to nonlinearities.

<p>Multi-linear operators play an important role in analysis due to their multiple connections with and applications to other mathematical areas such as ergodic theory, elliptic regularity, and other problems in partial differential equations.</p> <p>Within the area of multi-linear operators, powerful methods were developed originating from the problem of the almost everywhere convergence of Fourier series. Indeed, in their work, Carleson and Fefferman lay the foundation of time-frequency analysis. By further refining their methods, M. Lacey and C. Thiele proved the boundedness of the classical bilinear Hilbert transform for a suitable range of Hölder indices.</p> <p>In this thesis, we consider the general boundedness properties of the bilinear Hilbert transform and the sub-bilinear maximal function along a suitable family of curves.</p> <p>In the first part of our work, we present a short proof of the maximal boundedness range for the sub-bilinear maximal function along non-flat curves, giving a unified treatment of both the singular and the maximal operators.</p> <p>In the second part, we discuss the boundedness of these operators along hybrid curves. This work aims to present a unified perspective that treats the case obtained by joining the zero-curvature features of the operators along flat curves with the non-zero curvature features along non-flat curves.</p>

Thesis(M. Tech.) - Central University of Technology, Free State, 2004 / Growing markets leads to an increase in production. In these modern industries, weight measurement is of high priority. Weight measurement instrumentation is used for quality control, as well as for effective process control. Ineffective instrumentation with inaccurate data will influence the production process and profit margins negatively. Experimental data is gathered from an angled load cell, placed as a crossover between two conveyer belts. A weight measurement instrument with the ability to acquire accurate measurement of individual, moving parts is produced with the aid of DSP techniques. This was accomplished by analyzing the frequency spectrum for the undesirable signals with the use of Wavelets transformations (WT) and Fourier transformations (FT). After these undesired signals were identified a digital filter was designed to remove the undesired signals. Repetition of performance is achieved by the automatic zeroing of the instrument after every individual measurement. This weight measurement instrumentation also has the ability to store data consisting of the amount of objects and their individual weights. This instrument can also determine the material of which an object is made of. This is done by calculating the friction coefficient. This function has the ability to effectively identify between iron and rubber components irrespective of their mass or area.

The discrete Fourier transform has become an essential tool in the analysis of digital signals. Applications have become widespread since the discovery of the Fast Fourier Transform and the rise of personal computers. The field of digital signal processing is an exciting intersection of mathematics, statistics, and electrical engineering. In this study we aim to gain understanding of the mathematics behind algorithms that can extract chord information from recorded music. We investigate basic music theory, introduce and derive the discrete Fourier transform, and apply Fourier analysis to audio files to extract spectral data.

ITC/USA 2010 Conference Proceedings / The Forty-Sixth Annual International Telemetering Conference and Technical Exhibition / October 25-28, 2010 / Town and Country Resort & Convention Center, San Diego, California / Prognostic technology uses a series of algorithms, combined forms a prognostic-based inference engine (PBIE) for the identification of deterministic behavior embedded in completely normal appearing telemetry from fully functional equipment. The algorithms used to define normal behavior in the PBIE from which deterministic behavior is identified can be adapted to quantify normal spacecraft telemetry behavior while in orbit about a moon or planet or during interplanetary travel. Time-series analog engineering data (telemetry) from orbiting satellites and interplanetary spacecraft are defined by harmonic and non-harmonic influences, which shape it behavior. Spectrum analysis can be used to understand and quantify the fundamental behavior of spacecraft analog telemetry and relate the behavior's frequency and phase to its time-series behavior through Fourier analysis.

ITC/USA 2009 Conference Proceedings / The Forty-Fifth Annual International Telemetering Conference and Technical Exhibition / October 26-29, 2009 / Riviera Hotel & Convention Center, Las Vegas, Nevada / Prognostic technology uses a series of algorithms, combined forms a prognostic-based inference engine (PBIE) for the identification of deterministic behavior embedded in completely normal appearing telemetry from fully functional equipment. The algorithms used to define normal behavior in the PBIE from which deterministic behavior is identified can be adapted to quantify normal spacecraft telemetry behavior while in orbit about a moon or planet or during interplanetary travel. Time-series analog engineering data (telemetry) from orbiting satellites and interplanetary spacecraft are defined by harmonic and non-harmonic influences which shape it behavior. Spectrum analysis can be used to understand and quantify the fundamental behavior of spacecraft analog telemetry and relate the behavior's frequency and phase to its time-series behavior through Fourier analysis.