Although there is no natural internal product for hermitian forms over an algebra with involution of the first kind, we describe how tomultiply two ε-hermitian forms to obtain a quadratic form over the base field. This allows to define a commutative graded ring structure by taking together bilinear forms and ε-hermitian forms, which we call the mixedWitt ring of an algebra with involution. We also describe a less powerful version of this construction for unitary involutions, which still defines a ring, but with a grading over Z instead of the Klein group. We first describe a general framework for defining graded rings out of monoidal functors from monoidal categories with strong symmetry properties to categories of modules. We then give a description of such a strongly symmetric category Brₕ(K, ι) which encodes the usual hermitian Morita theory of algebras with involutions over a field K. We can therefore apply the general framework to Brₕ(K, ι) and theWitt group functors to define our mixed Witt rings, and derive their basic properties, including explicit formulas for products of diagonal forms in terms of involution trace forms, explicit computations for the case of quaternion algebras, and reciprocity formulas relative to scalar extensions. We intend to describe in future articles further properties of those rings, such as a λ-ring structure, and relations with theMilnor conjecture and the theory of signatures of hermitian forms.

Making gradient in the microstructure of metals is proven to be a superior method for improving their mechanical properties. In this research, we 3D print, characterize and evaluate the mechanical properties of 316L Stainless Steel with a gradient in their microstructure. During 3D printing, the gradient in the microstructure is created by tailoring the processing parameters (hatch spacing, scanning speed, and laser power and scanning speed) of the Selective Laser Melting (SLM). The Materials with Graded Microstructure (MGMs) are characterized by optical and scanning electron microscopy (SEM). Image processing framework is utilized to reveal the distribution of cells and melt pools shapes and sizes in the volume of the material when the processing parameters change. It is shown that the laser power, scanning speed and the hatch spacing have a more significant effect on the size and shape of cells and melt pools compared to the speed. Multiple Dog bones are 3D printed with a microstructure that has smaller features (cells and melt polls) at the edges of the structure compared to the center. Tensile and fatigue tests are performed and compared for samples with constant and graded microstructures. / Master of Science / The mechanical performance of Selective Laser Melting (SLM) fabricated materials is an important topic in research. Strengthening the performance of these materials can be achieved through implementing a gradient within the microstructure, referred to as Materials with Graded Microstructure (MGMs). A complicated microstructure can weaken the microstructure, and this can be resolved by optimizing the microstructure during SLM 3D printing, in which the processing parameters are tailored. In this study, the mechanical properties of these MGMs were characterized and evaluated. The gradient in these materials were created by modifying SLM process parameters (scanning speed, hatch spacing, and laser power and scanning speed) during the build. Optical and scanning electron microscopy (SEM) was used to characterize these the microstructure of these MGMs, and image processing was used to examine the distribution of cells and melt pools characteristics throughout the region where the processing parameters changed. This investigation shows that laser power, scanning speed, and hatch spacing have a direct effect on the size and shape of the cells and melt pools, compared to scanning speed, which shows an effect on melt pools. Dog bone structures are 3-D printed with a graded microstructure that has small cells and melt pools at the edges, compared to the center, by changing the laser power and scanning speed. Tensile and fatigue analysis are performed and compared for samples with constant and graded microstructures, which reveal that the mechanical properties of the MGMs perform similar to the parameter at the edges, but differently in fracture mechanics.

Composite materials and hybrid materials such as fiber-metal laminates, and functionally graded materials are increasingly common in application in aerospace structures. However, adhesive bonding of dissimilar materials makes these materials susceptible to delamination. The use of integrated Sinc methods for predicting interlaminar failure in laminated composites and hybrid material systems was examined. Because the Sinc methods first approximate the highest-order derivative in the governing equation, the in-plane derivatives of in-plane strain needed to obtain interlaminar stresses by integration of the equilibrium equations of 3D elasticity are known without post-processing. Interlaminar stresses obtained with the Sinc method based on Interpolation of Highest derivative were compared for the first-order and third-order shear deformable theories, the refined zigzag beam theory and the higher-order shear and normal deformable beam theory. The results indicate that the interlaminar stresses by the zigzag theory compare well with those obtained by a 3D finite element analysis, while the traditional equivalent single layer theories perform well for some laminates. The philosophy of the Sinc method based on Interpolation of Highest Derivative was extended to create a novel weak form based approach called the Integrated Local Petrov-Galerkin Sinc Method. The Integrated Local Petrov-Galerkin Sinc Method is easily utilized for boundary-value problem on non-rectangular domains as demonstrated for analysis of elastic and elastic-plastic plane-stress panels with elliptical notches. The numerical results showed excellent accuracy compared to similar results obtained with the finite element method. The Integrated Local Petrov-Galerkin Sinc Method was used to analyze interlaminar debonding of composite and fiber-metal laminated beams. A double-cantilever beam and a fixed-ratio mixed mode beam were analyzed using the Integrated Local Petrov-Galerkin Sinc Method and the results were shown to correlate well with those by the finite element method. An adaptive Sinc point distribution technique was implemented for the delamination analysis which significantly improved the methods accuracy for the present problem. Delamination of a GLARE, plane-strain specimen was also analyzed using the Integrated Local Petrov-Galerkin Sinc Method. The results correlate well with 2D, plane-strain analysis by the finite element method, including interlaminar stresses obtained by through-the-thickness integration of the equilibrium equations of 3D elasticity. / Ph. D.

Advances in wireless technology have recently led to the potential for higher data rates and greater functionality. Wireless home and business networks and 3G and 4G cellular phone systems are promising technologies striving for market acceptance, requiring low-cost, low-power, and compact solutions. One approach to meet these demands is system-on-a-chip (SoC) integration, where RF/analog and digital circuitry reside on the same chip, creating a mixed-signal environment. Concurrently, there is tremendous incentive to utilize Si-based technologies to leverage existing fabrication and design infrastructure and the corresponding economies of scale. While the SoC approach is attractive, it presents major challenges for circuit designers, particularly in the design of monolithic voltage controlled oscillators (VCOs). VCOs are important components in the up or downconversion of RF signals in wireless transceivers. VCOs must have very low phase noise and spurious emissions, and be extremely power efficient to meet system requirements. To meet these specifications, VCOs require high-quality factor (Q) tank circuits and reduction of noise from active devices; however, the lack of high-quality monolithic inductors, along with low noise transistors in traditional Si technologies, has been a limiting factor. This thesis presents the design, characterization, and comparison of three monolithic 3-4 GHz VCOs and an integrated 5-6 GHz VCO with tunable polyphase outputs. Each VCO is designed around a differential -G_{M} core with an LC tank circuit. The circuits exploit two Si-based device technologies: Silicon Germanium (SiGe) Heterojunction Bipolar Transistors (HBTs) for a cross-coupled collectors circuit and Graded-Channel MOS (GC-MOS) transistors for a complementary (CMOS) implementation. The circuits were fabricated using the Motorola 0.4 μm CDR1 SiGe BiCMOS process, which consists of four interconnected metal layers and a thick copper (10 μm) metal bump layer for improved inductive components. The VCO implementations are targeted to meet the stringent phase noise specifications for the GSM/EGSM 3G cellular standard. The specifications state that the VCO output cannot exceed -162 dBc/Hz sideband noise at 20 MHz offset from the carrier. Simultaneously, oscillators must be designed to address other system level effects, such as feed-through of the local oscillator (LO). LO feed-through directly results in self-mixing in direct conversion receivers, which gives rise to unwanted corrupting DC offsets. Therefore, a system-level strategy is employed to avoid such issues. For example, multiplying the oscillator frequency by two or four times can help avoid self-mixing during downconversion by moving the LO out of the bandwidth of the RF front-end. Meanwhile, direct conversion or low-IF (intermediate frequency) receiver architectures utilize in-phase and quadrature (I/Q) downconversion signal recovery and image rejection. Any imbalance between the I and Q channels can result in an increase in bit-error-rate (BER) and/or decrease in the image rejection ratio (IRR). To compensate for such an imbalance, an integrated tunable polyphase filter is implemented with a VCO. Control voltages between the differential I and Q channels can be individually controlled to help compensate for I/Q mismatches. This thesis includes an introduction to design flow and layout strategies for oscillator implementations. A detailed comparison of the advantages and disadvantages of the SiGe HBTs and GC-MOS device in 3-4 GHz VCOs is presented. In addition, an overview of full-wave electromagnetic characterization of differential dual inductors is given. The oscillators are characterized for tuning range, output power, and phase noise. Finally, new measurement techniques for the 5-6 GHz VCO with a tunable polyphase filter are explored. A comparison between the time and frequency approaches is also offered. / Master of Science

In this dissertation, we study the optical property of 2D graded photonic super-crystals (GPSCs) for photon management. We focused primarily on manipulation and control of light by using the newly discovered GPSCs which present great opportunity for electromagnetic wave control in photonic devices. The GPSC has been used to explore the superior capability of improving the light extraction efficiency of OLEDs. The enhancement of extraction efficiency has been explained in term of destructive interference of surface plasmon resonance and out-coupling of surface plasmon through phase matching provided by GPSC and verified by e-field intensity distributions. A large light extraction efficiency up to 75% into glass substrate has been predicted through simulation. We also study the light trapping enhancement in GPSCs. Broadband, wide incident angle, and polarization independent light trapping enhancement is achieved in silicon solar cells patterned with the GPSCs. In addition, novel 2D GPSCs were fabricated using holographic lithography through the interference lithography by two sets of multiple beams arranged in a cone geometry using a spatial light modulator (SLM). Finally, we also report a fabrication of GPSCs with a super-cell size of 12a×12a by using e-beam lithography. Diffraction pattern from GPSCs reveals unique diffraction properties. In an application aspect, light emitting diode arrays can be replaced by a single light emitting diode shinning onto the diffraction pattern for a uniform fluorescence.

Functionally graded materials (FGMs) are inhomogeneous materials in which the material properties vary with respect to space. Research has been done by scientific community in developing techniques like photothermal radiometry (PTR) to measure the thermal conductivity and volumetric heat capacity of FGMs. One of the problems involved in the technique is to solve the inverse problem, i.e., estimating the thermal properties after the frequency scan has been obtained. The present work involves finding the unknown thermal conductivity and volumetric heat capacity of the FGMs by using finite volume method. By taking the flux entering the sample as periodic and solving the discretized 1-D thermal wave field equation at a frequency domain, one can obtain the complex temperatures at the surface of the sample for each frequency. These complex temperatures when solved for a range of frequencies gives the phase vs frequency scan which can then be compared to original frequency scan obtained from the PTR experiment by using a residual function. Brute force and gradient descent optimization methods have been implemented to estimate the unknown thermal conductivity and volumetric specific heat of the FGMs through minimization of the residual function. In general, the spatial composition profile of the FGMs can be approximated by using a smooth curve. Three functional forms namely Arctangent curve, Hermite curve, and Bezier curve are used in approximating the thermal conductivity and volumetric heat capacity distributions in the FGMs. The use of Hermite and Bezier curves gives the flexibility to control the slope of the curve i.e. the thermal property distribution along the thickness of the sample. Two-layered samples with constant thermal properties and three layered samples in which one of the layer has varying thermal properties with respect to thickness are considered. The program is written in Fortran and several test runs are performed. Results obtained are close to the original thermal property values with some deviation based on the stopping criteria used in the gradient descent algorithm. Calculating the gradients at each iteration takes considerable amount of time and if these gradient values are already available, the problem can be solved at a faster rate. One of the methods is extending automatic differentiation to complex numbers and calculating the gradient values ahead; this is left for future work.

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Gegenstand dieser Arbeit ist die Entwicklung und Untersuchung eines Aufbereitungsprozesses zur Gewinnung von Bitumen aus kanadischem Athabasca Ölsand, der im Tagebau gewonnen wurde. Es wird ein mechanisch-thermisches Verfahren zur Fest-Flüssig-Trennung eingesetzt. Dabei handelt es sich um vier Schritte: (1) Suspendierung des Ölsandes mit den organischen Lösungsmitteln, Toluol und n-Heptan, (2) Filterkuchenbildung, (3) Waschung des Filterkuchens mit Wechsel der Waschflüssigkeiten (gradierte Waschung) und (4) Dampfbeaufschlagung. Der Prozess stellt eine Alternative zur herkömmlichen Heißwasser-extraktion des Ölsandes dar. Die Nachteile der Heißwasserextraktion sind ökologische Probleme, ein hoher Energie- und Frischwasserbedarf. Die Ziele des Alternativprozesses sind die Minimierung des Wasser- und Energiebedarfs, Vermeidung schädlicher Abfallstoffe sowie die Maximierung der Bitumenausbeute. Als Produkte sollen feststofffreies Bitumen und rückstandsfreier, deponierbarer Feststoff gewonnen werden.

Dans cette thèse on étudie certaines constructions géométriques qui apparaissent naturellement dans le contexte des modèles sigma, leur jaugeage et supersymétrisation. La thèse comprend trois parties. La première partie (chapitres 1 et 2) contient des faits issus de la géométrie différentielle classique et de la géométrie graduée nécessaires pour comprendre les résultats clés de la thèse. On survole la géométrie liée aux variétés de Poisson et variétés symplectiques. On généralise ces notions aux variétés de Dirac et variétés n-plectiques, et établit leur liens avec les algebroïdes de Courant. Le langage principal utilisé dans la thèse pour la description mathématique des modèles sigma – c'est la géométrie graduée – on définit donc des bases de calcul sur les supervariétés et variétés graduées ainsi que les notions des Q-structures et des variétés multigraduées. La deuxième partie (chapitres 3 et 4) a pour but d’interpréter géométriquement l'invariance de jauge de certains modèles sigma. On établit la relation entre les symétries de modèle sigma de Dirac, et comme cas particulier de modèle sigma de Poisson (tordu), avec les sous-algèbres des sections d'algebroïde de Courant. On généralise la notion de cohomologie équivariante, ce qui permet d'obtenir les modèles sigma avec le groupe des symétries prescrit, en particulier on construit les groupes nécessaires pour les modèles sigma mentionnés. La troisième partie (chapitre 5) adresse l'extension graduée des modèles sigma (comme en supersymétrisation). Ceci est en fait lié auxstructures géométriques qui peuvent être définies sur l'espace des applications entre les variétés multigraduées / In this thesis we study some geometric constructions appearing naturally in the context of sigma models, their gauging and supersymmetrization. The thesis consists of three parts. The first part (chapters 1 and 2) contains facts coming from classical differential geometry and graded geometry, they are needed to understand the main results of the thesis. We review the geometric constructions related to Poisson and symplectic manifolds. We generalize these notions to Dirac and n-plectic manifolds and establish the links with Courant algebroids. The main language used in the thesis for mathematical description of the sigma models is the graded geometry - we thus define the basis of calculus on supermanifolds and graded manifolds, as well as describe the notions of Q-structures and multigraded manifolds. The main goal of the second part (chapters 3 and 4) is to interpret geometrically the gauge invariance of some sigma models. We establish the relation of the symmetries of the Dirac sigma model, and as a particular case of the (twisted) Poisson sigma model, with the subalgebra of sections of Courant algebroid. We generalize the notion of equivariant cohomology, that permits to recover the sigma models with a prescribed group of gauge symmetries. In particular we construct the necessary groups for the mentioned sigma models. The third part (chapter 5) addresses the graded extension of the sigma models (like in supersymmetrization). It is in fact related to the geometric structures that can be defined on the space of maps between multigraded manifolds.

La présente thèse porte sur le développement d'une théorie d'algèbre linéaire, de géométrie et d'analyse basée sur les algèbres (Z2)n-commutatives, c'est-à-dire des algèbres (Z2)n-graduées associatives unitaires satisfaisant ab = (-1)<deg(a),deg(b)>ba, pour tout couple d'éléments homogènes a, b de degrés deg(a), deg(b) où <.,.> est le produit scalaire usuel). Cette généralisation de la supergéométrie a de nombreuses applications : en mathématiques (l'algèbre de Deligne des superformes différentielles, l'algèbre des quaternions et les algèbres de Clifford en sont des exemples) et même en physique (paraparticules). Dans ce travail, les notions de trace et de (super)déterminant pour des matrices à coefficients dans une algèbre gradué-commutative sont définies et étudiés. Une attention particulière est portée au cas des algèbres de Clifford : ce point de vue gradué fournit une nouvelle approche au problème classique du « bon » déterminant pour des matrices à coefficient non-commutatifs (quaternioniques). En outre, nous entreprenons l'étude de la géométrie différentielle (Z2)n-graduée. Privilégiant l'approche par les espaces annelés, les (Z2)n-supervariétés sont définies en choisissant l'algèbre (Z2)n-commutative des séries formelles en variables graduées comme modèle pour le faisceau de fonctions. Les résultats les plus marquants ainsi obtenus sont : le Berezinien gradué et son interprétation cohomologique (essentielle pour établir une théorie de l'intégration) ; le théorème des morphismes, attestant qu'on peut rétablir un morphisme entre (Z2)n-supervariétés à partir de sa seule expression sur les coordonnées ; le théorème de Batchelor-Gawedzki pour les (Z2)n-supervariétés lisses / The present thesis deals with a development of linear algebra, geometry and analysis based on (Z2)n-superalgebras ; associative unital algebras which are (Z2)n-graded and graded-commutative, i.e. statisfying ab=(-1)<deg(a),deg(b)>ba, for all homogeneous elements a, b of respective degrees deg(a), deg(b) in (Z2)n (<.,.> denoting the usual scalar product). This generalization widens the range of applications of supergeometry to many mathematical structures (quaternions and more generally Clifford algebras, Deligne algebra of superdifferential forms, higher vector bundles) and appears also in physics (for describing paraparticles) proving its worth and relevance. In this dissertation, we first focus on (Z2)n-superalgebra theory ; we define and characterize the notions of trace and (super)determinant of matrices over graded-commutative algebras. Special attention is given to the case of Clifford algebras, where our study gives a new approach to treat the classical problem of finding a “good” determinant for matrices with noncommuting (quaternionic) entries. Further, we undertake the study of (Z2)n-graded differential geometry. Privileging the ringed space approach, we define (smooth) (Z2)n-supermanifolds modeling their algebras of functions on the (Z2)n-commutative algebra of formal power series in graded variables, and develop the theory along the lines of supergeometry. Notable results are : the graded Berezinian and its cohomological interpretation (essential to establish integration theory) ; the theorem of morphism, which states that a morphism of (Z2)n-supermanifolds can be recovered from its coordinate expression ; Batchelor-Gawedzki theorem for (Z2)n-supermanifolds