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Compressed sensing (CS) is an emerging technique that exploits the properties of a sparse or compressible signal to efficiently and faithfully capture it with a sampling rate far below the Nyquist rate. The primary goal of compressed sensing is to achieve the best signal recovery with the least number of samples. To this end, two research directions have been receiving increasing attention: customizing the measurement matrix to the signal of interest and optimizing the reconstruction algorithm. In this thesis, contributions in both directions are made in the Bayesian setting for compressed sensing. The work presented in this thesis focuses on the approximate message passing (AMP) schemes, a new class of recovery algorithm that takes advantage of the statistical properties of the CS problem. First of all, a complete sample distortion (SD) framework is presented to fundamentally quantify the reconstruction performance for a certain pair of measurement matrix and recovery scheme. In the SD setting, the non-optimality region of the homogeneous Gaussian matrix is identified and the novel zeroing matrix is proposed with an improved performance. With the SD framework, the optimal sample allocation strategy for the block diagonal measurement matrix are derived for the wavelet representation of natural images. Extensive simulations validate the optimality of the proposed measurement matrix design. Motivated by the zeroing matrix, we extend the seeded matrix design in the CS literature to the novel modulated matrix structure. The major advantage of the modulated matrix over the seeded matrix lies in the simplicity of its state evolution dynamics. Together with the AMP based algorithm, the modulated matrix possesses a 1-D performance prediction system, with which we can optimize the matrix configuration. We then focus on a special modulated matrix form, designated as the two block matrix, which can also be seen as a generalization of the zeroing matrix. The effectiveness of the two block matrix is demonstrated through both sparse and compressible signals. The underlining reason for the improved performance is presented through the analysis of the state evolution dynamics. The final contribution of the thesis explores improving the reconstruction algorithm. By taking the signal prior into account, the Bayesian optimal AMP (BAMP) algorithm is demonstrated to dramatically improve the reconstruction quality. The key insight for its success is that it utilizes the minimum mean square error (MMSE) estimator for the CS denoising. However, the prerequisite of the prior information makes it often impractical. A novel SURE-AMP algorithm is proposed to address the dilemma. The critical feature of SURE-AMP is that the Stein’s unbiased risk estimate (SURE) based parametric least square estimator is used to replace the MMSE estimator. Given the optimization of the SURE estimator only involves the noisy data, it eliminates the need for the signal prior, thus can accommodate more general sparse models.

Sixty years after the works of Wigner and Dyson, Random Matrix Theory still remains a very active and challenging area of research, with countless applications in mathematical physics, statistical mechanics and beyond. In this thesis, we focus on rotationally invariant models where the requirement of independence of matrix elements is dropped. Some classical examples are the Jacobi and Wishart-Laguerre (or chiral) ensembles, which constitute the core of the present work. The Wishart-Laguerre ensemble contains covariance matrices of random data, and represents a very important tool in multivariate data analysis, with recent applications to finance and telecommunications. We will first consider large deviations of the maximum eigenvalue, providing new analytical results for its large N behavior, and then a power-law deformation of the classical Wishart-Laguerre ensemble, with possible applications to covariance matrices of financial data. For the Jacobi matrices, which arise naturally in the quantum conductance problem, we provide analytical formulas for quantities of interest for the experiments.

Typescript (photocopy). / Digitized by Kansas Correctional Industries

A Ni/TiC/C metal matrix composite (MMC) has been processed using the laser engineered net shaping (LENS) process from commercially available powders with a Ni-3Ti-20C (atomic %) composition. This processing route produces the in-situ formation of homogeneously distributed eutectic and primary titanium carbide and graphite precipitates throughout the Ni matrix. The composite exhibits promising tribological properties when tested in dry sliding conditions with a low steady state coefficient of friction (CoF) of ~0.1 and lower wear rates in comparison to LENS deposited pure Ni. The as deposited and tribologically worn composite has been characterized using Auger electron spectroscopy, scanning electron microscopy (SEM), X-ray diffraction, high resolution transmission electron microscopy (HRTEM) with energy dispersive spectroscopy (EDS), dual beam focused ion beam SEM (FIB/SEM) serial sectioning and Vickers micro-hardness testing. The evolution of subsurface stress states and precipitate motion during repeated sliding contact has been investigated using finite element analysis (FEA). The results of FIB/SEM serial sectioning, HRTEM, and Auger electron spectroscopy in conjunction with FEA simulations reveal that the improved tribological behavior is due to the in-situ formation of a low interfacial shear strength amorphous carbon tribofilm that is extruded to the surface via refined Ni grain boundaries.

Recent work has shown that fast matrix multiplication algorithms can be constructed by embedding the two input matrices into a group algebra, applying a generalized discrete Fourier transform, and performing the multiplication in the Fourier basis. Developing an embedding that yields a matrix multiplication algorithm with running time faster than naive matrix multiplication leads to interesting combinatorial problems in group theory. The crux of such an embedding, after a group G has been chosen, lies in finding a triple of subsets of G that satisfy a certain algebraic relation. I show how the process of finding such subsets can in some cases be greatly simplified by considering the action of the group G on an appropriate set X. In particular, I focus on groups acting on regularly branching trees.

In resent years the rate of biologics (proteins, cytokines and growth-factors) as newly registered drugs has steadily risen. The greatest challenge for pharmaceutical biologics poses its arrival at the desired target location due to e.g. proteolytic and pH dependent degradation, plasma protein binding, insolubility etc. Therefore, advanced drug delivery systems, where biologics are site directed immobilized to carriers mimicking endogenous storage sites such as the extra cellular matrix can enormously assist the application and consequently the release of exogenous administered pharmaceutical biologics. We have resorted to the fibroblast growth factor 2/ heparansulfate/ fibroblast growth factor bindingprotein 1 system as a model. Phase I deals with the selection and subcloning of a wild type murine FGF-2 construct into the bacterial pHis-Trx vector system for high yields of expression and fast, feasible purification measurements. This first step enables the provision of mFGF-2, which plays a pivotal part as a growth factor in the wound healing process as well as the vascularization of tumors, for future investigations. Therefore, the correct expression of mFGF-2 was monitored via MALDI-MS and SDS-PAGE, whereas the proper folding of the tertiary beta-trefoil structure was assessed by fluorescence spectroscopy. The MTT assay allowed us to ensure that the bioactivity was comparable to sourced FGF-2. In the last step, the purity; a requirement for future binding- and protein-protein interaction assays was monitored chromatographically (RP-HPLC). In addition, a formulation for freeze-drying was developed to ensure protein stability and integrity over a period of 60 days. Altogether, the bacterial expression and purification proved to be suitable, leading to bioactive and stable production of mFGF-2. In Phase II the expression, purification and characterization of FGFBP1, as the other key partner in the FGF-2/ HS/ FGFBP1 system is detailed. As FGFBP1 exhibits a complex tertiary structure, comprised of five highly conserved disulfide bonds and presumably multiple glycosylation sites, a eukaryotic expression was used. Human embryonic kidney cells (HEK 293F) as suspension cells were transiently transfected with DNA-PEI complexes, leading to expression of Fc-tagged murine FGFBP1. Different PEI to DNA ratios and expression durations were investigated for optimal expression yields, which were confirmed by western blot analysis and SDS-PAGE. LC-MS/MS analysis of trypsin and elastase digested FGFBP1 gave first insights of the three O-glycosylation sites. Furthermore, the binding protein was modified by inserting a His6-tag between the Fc-tag (for purification) and the binding protein itself to enable later complexation with radioactive 99mTc as radio ligand to track bio distribution of administered FGFBP1 in mice. Overall, expression, purification and characterization of mFGFBP1 variants were successful with a minor draw back of instability of the tag free binding protein. Combining the insights and results of expressed FGF-2 as well as FGFBP1 directed us to the investigation of the interaction of each partner in the FGF-2/ HS/ FGFBP1 system as Phase III. Thermodynamic behavior of FGF-2 and low molecular weight heparin (enoxaparin), as a surrogate for HS, under physiological conditions (pH 7.4) and pathophysiological conditions, similar to hypoxic, tumorous conditions (acidic pH) were monitored by means of isothermal titration calorimetry. Buffer types, as well as the pH influences binding parameters such as stoichiometry (n), enthalpy (ΔH) and to some extent the dissociation constant (KD). These findings paved the way for kinetic binding investigations, which were performed by surface plasmon resonance assays. For the first time the KD of full length FGFBP1 and FGF-2 was measured. Furthermore the binding behavior of FGF-2 to FGFBP1 in the presence of various heparin concentrations suggest a kinetic driven release of bound FGF-2 by its chaperone FGFBP1. Having gathered multiple data on the FGF-2 /HS /FGFBP1 system mainly in solution, our next step in Phase IV was the development of a test system for immobilized proteins. With the necessity to better understand and monitor the cellular effects of immobilized growth factors, we decorated glass slides in a site-specific manner with an RGD-peptide for adhesion of cells and via the copper(I)-catalyzed-azide-alkyne cycloaddition (CuAAC) a fluorescent dye (a precursor for modified proteins for click chemistry). Human osteosarcoma cells were able to grow an the slides and the fluorescence dye was immobilized in a biocompatible way allowing future thorough bioactivity assay such as MTT-assays and phospho-ERK-assays of immobilized growth factors. / In den letzten Jahren ist der Anteil an Biologika (Proteine, Zytokine und Wachstumsfaktoren), die neu zugelassen wurden, kontinuierlich angestiegen. Die größte Herausforderung für Biopharmazeutika stellt das Erreichen des gewünschten Wirkortes dar, aufgrund von beispielsweise enzymatischen und pH abhängigem Abbau, Plasmaproteinbindung, und niedriger Löslichkeit. Daher können moderne Wirkstoffträgersysteme, in denen Biologika ortsspezifisch an Träger immobilisiert sind und endogene Aufbewahrungsorte nachahmen, wie zum Beispiel die extrazelluläre Matrix, die Anwendung enorm erleichtern und folglich auch die Freisetzung von Biopharmazeutika, die exogen verabreicht wurden ermöglichen. Wir haben uns auf das Fibroblasten-Wachstumsfaktor 2/ Heparansulfat/ Fibroblasten-Wachstumsfaktor Bindungsprotein 1 (FGF-2/ HS/ FGFBP1) System als Modell gestützt. Abschnitt I handelt von der Auswahl und der Subklonierung von einem wildtypischen, murinen FGF-2 Konstrukt in ein bakterielles pHis-Trx Vektorsystem – für hohe Ausbeuten bei der Expression und für eine schnell durchführbare Aufreinigung. Dieser erste Schritt ermöglicht die Bereitstellung von mFGF-2 für zukünftige Untersuchungen, das eine zentrale Rolle als Wachstumsfaktor im Wundheilungsprozess spielt, genauso wie bei der Gefäßversorgung von Tumoren. Daher wurde die richtige Expression von mFGF-2 durch MALDI-MS und SDS-PAGE überwacht, wobei die korrekte Faltung der tertiären beta-Faltblatt-Struktur durch Fluoreszenzmikroskopie ausgewertet wurde. Mit dem MTT Test konnten wir gewährleisten, dass die Bioaktivität von mFGF-2 und dem käuflich erworbenen FGF-2 übereinstimmen. Im letzten Schritt wurde die Reinheit, die eine wichtige Voraussetzung für künftige Bindungs- und Protein-Protein-Wechselwirkung-Untersuchungen darstellt, chromatographisch (RP-HPLC) überwacht. Des Weiteren wurde eine Formulierung zur Gefriertrocknung entwickelt, um die Stabilität und Unversehrtheit des Proteins über 60 Tage sicherzustellen. Insgesamt erwiesen sich die bakterielle Expression und Aufreinigung als geeignet und führten zur Herstellung von bioaktivem und stabilem mFGF-2. Im Abschnitt II wird die Expression, Aufreinigung und Charakterisierung von FGFBP1 genau beschrieben, ein ebenso wichtiger Partner im FGF-2/HS/FGFBP1-System. Da FGFBP1 eine komplexe tertiäre Struktur aufweist, die aus fünf hochkonservierten Disulfidbrücken und vermutlich einigen Glykosylierungsstellen besteht, wurde ein eukaryotisches Expressionssystem angewendet. Menschliche embryonale Nierenzellen (HEK 293F) wurden als Suspensionszellen transient mit DNA-PEI Komplexen transfiziert und führten zur Expression von Fc markiertem mausartigem FGFBP1. Verschiedene PEI:DNA Verhältnisse wurden untersucht, sowie die Dauer der Expression variiert, um die Ausbeute der Expression zu optimieren. Die Ergebnisse wurden mittels Western Blot und SDS-PAGE bestätigt. Die LC-MS/MS Messung, des mit Trypsin und Elastase verdautem FGFBP1, ergaben erste Erkenntnisse über die drei O-Glykosylierungsstellen. Des Weiteren wurde das Fusionsprotein durch Einschub eines His6-tags zwischen den Fc-tag und FGFBP1 erweitert, um dem Protein die Eigenschaft zu geben, mehrwertige Kationen zu komplexieren. 99mTc, als radioaktiver Ligand, kann in späteren Untersuchungen die Verteilung im Gewebe, anhand von Mäusen darstellen. Die Expression, Aufreinigung, sowie die Charakterisierung von den murinen FGFBP1 Varianten war erfolgreich, jedoch erwies sich das Bindungsprotein ohne Fc-Teil als weniger stabil. Durch Zusammenfassung der Ergebnisse und Erkenntnisse der beiden exprimierten Proteine mFGF-2 und FGFBP1, konnten wir die Wechselwirkungen der Partner im FGF-2 /HS / FGFBP1 Systems untereinander im Abschnitt III untersuchen. Isotherme Titrationskalorimetrie wurde herangezogen, um das thermodynamische Verhalten von FGF-2 mit niedermolekularem Heparin (Enoxaparin), als Stellvertreter für HS, unter physiologischen Bedingungen (pH 7,4) und pathologischer Bedingungen (saurer pH, hervorgerufen durch Sauerstoffmangel im Tumorgewebe) aufzuklären. Sowohl die Art des Puffersystems als auch der pH Wert beeinflussen die Bindungsparameter: Enthalpie (DH), die Stöchiometrie (n) und zu einem gewissen Teil sogar die Dissoziationkonstante (KD). Die thermodynamischen Untersuchungen ermöglichten im folgenden Schritt, mittels Oberflächen Plasmon Resonanz (SPR), kinetische Bindungsverhältnisse zu ermitteln. Zum ersten Mal wurde dank der SPR Dissoziationskonstanten von FGFBP1 und FGF-2 erfasst. Da alle bisherigen Erkenntnisse des FGF-2/ HS/ FGFBP1 Systems hauptsächlich von in Lösung gebrachten Partnern gewonnen wurden, erfolgte im Abschnitt IV die Entwicklung eines Testsystems für immobilisierte Proteine. Durch die zwingende Notwendigkeit zelluläre Effekte von immobilisierten Wachstumsfaktoren zu verstehen und zu beobachten, haben wir Objektträger aus Glas, in einem ortsspezifischem Verfahren mit einem RGD-Peptid, für die Zellanhaftung und mittels 1,3-Dipolare Cycloaddition einen Fluoreszenzfarbstoff (ein Vorläufer für „clickbare“, modifizierte Proteine) dekoriert. Menschliche Knochenkrebszellen waren in der Lage auf diesen Objektträgern zu wachsen und der Farbstoff konnte in einem nicht zytotoxischen Verfahren spezifisch an der Oberfläche immobilisiert werden. Dieses Testsystem erlaubt in Zukunft ausführliche Bioaktivitäts- und Proliferationsstudien von immobilisierten Proteinen durchzuführen.