Two of the more well known congruences for binomial coefficients modulo p, due to Gauss and Jacobi, are related to the representation of an odd prime (or an integer multiple of the odd prime) p as a sum of two squares (or an integer linear combination of two squares). These two congruences, along with many others, have been extended to analogues modulo p^2 and are well documented. More recently, J. Cosgrave and K. Dilcher have extended the congruences of Gauss and Jacobi to analogues modulo p^3. In this thesis we discuss their methods as well as the potential of applying them to similar congruences found in the literature.

A general expression for quantum transmission of non-interacting spinless electrons through models of a fully connected network of sites that can be regarded as a nanoparticle is obtained using matrix algebra. This matrix algebra method leads to the same results given by the Green’s function method without requiring the mathematical sophistication as required by the later. The model of the nanoparticle in this study comprises a single linear array of atoms that profile the input and output leads connected to a fully connected blob of atoms. A simple tight-binding Hamiltonian motivates the quantum transmission in the discrete lattice system. If there are n atoms in the nanoparticle, the methodology requires the inverse of a n × n matrix. The solution is obtained analytically for different cases: a single atom in the nanoparticle, a single dangle atom, n fully connected atoms in a meanield type cluster with symmetric input and output connections, and the most general case where the n fully connected atoms can be connected arbitrarily to the input and output leads. A numerical solution is also provided for the case where the intra-bonds among the atoms in the nanoparticle are varied (a case with notully connected atoms). The expression for the transmission coefficient thus obtained using the matrix method is compared with the transmission coefficients derived using the real space Renormalization Group method and the Green’s function method.

Throughout literature, one finds where numerous methodologies and models have been developed to predict the effect of surface roughness on a flat surface. Many of the models utilize a drag coefficient as one of the necessary parameters. In urban settings with groups of buildings, the drag coefficient on an individual obstacle would be determined by parameters like wind direction and the relative positioning of a building, in addition to Reynolds number and shape. Computational experiments were performed to simulate the fluid flow around a single row and two rows of “cube” obstacles. Based on dimensional analysis, the drag coefficient was formulated as a function of four input variables. The effect of these input variables on the drag coefficient was individually studied. Finally, using the central composite design method and the numerically obtained experiment data, a second-order mathematical model was devised for the drag coefficient as a function of the four input variables.

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Panel data analysis has become an important tool in applied econometrics and the respective statistical techniques are well described in several recent textbooks. However, for an analyst using these methods there remains the task of choosing a reasonable model for the behavior of the panel data. Of special importance is the choice between so-called fixed and random coefficient models. This choice can have a crucial effect on the interpretation of the analyzed phenomenon, which is demonstrated by an application on research and development spillover. (author's abstract) / Series: Forschungsberichte / Institut für Statistik

QC 351 A7 no. 48 / New materials of low thermal expansion are finding wide application. The expansion coefficient (a) is a function of temperature, and this function must be known for each material before its applicability can be assessed. A novel method for determining a, which is at once precise and easily implemented, has been devised. It is based on the dependence of mode frequencies in a Fabry-Perot interferometer on the mirror separation. The expansion sample is formed into an interferometer spacer with ends polished flat and parallel. Spherical mirrors are optically contacted to the ends, forming a confocal interferometer. The assembly is maintained at controlled temperatures in an environmental chamber. The two lowest -order transverse modes are probed by variable -frequency sidebands derived from a 633 -nm He- Ne laser by amplitude modulation. A change in sample temperature AT causes a change in interferometer length AL, which shifts the resonance frequencies by Av. Then a = (1 /AT) (AL /L) _ - (1 /AT)(iv /v). Thus, a can be measured with precision limited ultimately by the stability of the source laser, in practice 1:109 with presently available commercial lasers. For a sample of Owens -Illinois Cer -Vit, a has been measured at 10 temperatures in the range 3.0 to 32.4 °C, with a mean error of 2 x10-9 and a maximum error of 3 x10 -9. For a sample of Corning ULE silica, a has been measured at six temperatures in the same range, with a mean error of <1 x10 -9 and a maximum error of <1.3 x10 -9.

Un des points clés pour l’utilisation sur le long terme des SOEF et SOFC est l’étanchéité. Les solutions de scellement les plus répandues sont des matériaux rigides de type vitreux. Ils présentent l’inconvénient de se fissurer lorsqu’ils sont soumis à des cycles thermiques. Ceci est dû aux différences de CET entre les composants métalliques ou céramiques et les matériaux vitreux. L’autocicatrisation est une solution prometteuse pour pallier ce problème, deux mécanismes existent : intrinsèque et extrinsèque. L’autocicatrisation intrinsèque de matériaux vitreux est basée sur leur ramollissement. Nous avons développé une formulation de joints vitreux «pâteux» qui présente des propriétés autocicatrisantes, donc moins sensible aux différences de CET. La viscosité de ces verres a été estimée par microscopie de chauffage et leur stabilité a été caractérisée par DRX et microsonde de Castaing. L’autocicatrisation extrinsèque ne nécessite pas d’intervention extérieure. Elle est obtenue par ajout de particules cicatrisantes à la matrice vitreuse. Lors de l’apparition de fissures, les particules s’oxydent au contact de l’atmosphère pour former des oxydes qui donne in situ de nouveaux verres. Ces oxydes, fluides à la température de fonctionnement, s’écoulent dans la fissure, formant localement, par réaction, un nouveau verre. Des tests réalisés in situ par MEBE-HT ont permis de mettre en évidence le processus d’autocicatrisation. La formation de phases cicatrisantes a été caractérisée par microsonde de Castaing et par RMN. Un ensemble de caractérisations a été effectué afin de valider l’applicabilité de l’autocicatrisation extrinsèque aux joints de scellement vitreux. / A key point for using SOEC and SOFC in the long-term is the sealant. The most sealing solutions commonly used are rigid materials, particularly glassy seals. However, they have the disadvantage of cracking in operation when subjected to thermal cycles. This is mainly due to TEC differences between metal and ceramic components and glass materials. The self-healing is a promising solution to overcome this problem. Two mechanisms exist: intrinsic and extrinsic. The intrinsic self-healing of glassy materials is based on their softening at high temperature. We developed a formulation of viscous glass seal that exhibits self-healing properties at the operating temperature systems SOEC/SOFC. They are less sensitive to differences of TEC. The glass viscosity was estimated by hot stage microscopy and their stability under use condition has been characterized by XRD and Castaing microprobe. In contrast, the extrinsic self-healing requires no external intervention. It is obtained by the addition of healing particles in the glassy matrix. When cracks occur, the particles oxidize with atmosphere contact to form oxides and in-situ new glasses. We developed this extrinsic method from particles generating B2O3 and V2O5. These oxides, fluid at operating temperature 700-900°C, flow in the crack and form a new locally glass by reaction with glassy matrix. An in-situ test by HT-ESEM highlights the self-healing process. The formation of glass and crystal phases is characterized by Castaing microprobe and solid state NMR. A set of physico-chemical characterization was performed to validate extrinsic self-healing applicability in the SOEC/SOFC glassy seal.

Detta examensarbete har genomförts på ABB Power Products/Instrument Transformers i Ludvika. Syftet med arbetet har varit att undersöka temperaturexpansionen för en isolerande olja som använd i produkten Capacitor Voltage Transformer (CVT). Oljans egen temperatur­expansionskoefficient                          var redan känd, men arbetets syfte var att bestämma en Brutto­temperaturexpansionskoefficient, med andra ord undersöka hur mycket de ingående komponenterna i en CVT påverkar den totala volymexpansionen inuti enheten. För att bestämma bruttotemperaturexpansionskoefficienten har ett prov i klimatkammare utförts. Provet bestod av två aluminiumkärl vardera med ett monterat mätrör. Det ena kärlet fylldes med olja och det andra kärlet med olja och kondensatorinnehåll motsvarande inne­hållet i en CVT. Provobjekten utsattes sedan för en bestämd tid i temperaturer från -60°C till 65°C med intervall om 10°C. Under denna tid utfördes ett antal mätningar av oljevolymen i de båda provobjekten. Dessa mätningar användes sedan tillsammans med ett några korrektions­faktorer för att beräkna temperaturexpansionskoefficienten för både oljan (enbart för kontroll­erande syfte) och kondensatorinnehållet. Tillsammans ger dessa bruttotemperaturexpansions­koefficienten för alla möjliga volymkombinationer. Resultatet från provet har sedan analyserats och visualiserats i form av tabeller och diagram. Efter det beräknades temperaturexpansionskoefficient för oljan samt  för kondensator­innehållet vid olika temperaturer. Utifrån dessa värden beräknades de genomsnittliga temperatur­­expansionskoefficienterna för temperaturintervallet och slutligen angavs en formel för att bestämma bruttotemperaturexpansionskoefficienten för olika procentuella fördelningar mellan olja och kondensatorinnehåll. Till sist drogs slutsatser utifrånfrån resultatet. / This bachelor thesis was carried out at ABB Power Products/Instrument Transformers in Ludvika. The purpose of the work has been to examine the thermal expansion for an isolating oil that is used in the product Capacitor Voltage Transformer (CVT). The oil’s own expansion coefficient                          was already known, but the purpose of the work was to determine a gross thermal expansion coefficient, with other words to examine the amount of influense the other componets had on the volume expansion inside the unit. To decide the gross thermal expansion coefficient a test has been carried out in a climat chamer. To performe the test two aluminum tanks, each with a measurement tube mounted on the top were used. One of the tanks was filled with oil and the other one was filled with oil and capacitor contents corresponding tio the contents in a CVT. The two test objects were then exposed to temperatures reaching from -60°C to 65°C in steps of 10°C. During this time a number of volume measurements were performed on the two test objects. These measure­ments together with a few correction parameters were used to decide the thermal expansion coefficient for both the oil (only for control purpose) and the capacitor contents. Together these two gives the gross thermal expansion coefficient for all posible volume combinations. The results from the tests has been analised and visualised in the form of tables an diagrams. After that the thermal expansion coefficient for the oil and the capacitorcontents, for different temperatures was calculated. Bult on those calculations the avarege thermal expansion coeff­icient for the temperatureintervals was calculated, and finaly an modle for calculating the gross thermal expansion coefficient was created. At the end conclusions were drawn from the results.

During the drying of a surface, the liquid evaporation acts to keep the temperature relatively constant, due to evaporative cooling. As the drying nears completion the liquid film begins to break, exposing areas that are no longer cooled through evaporation, which begin to heat. Although this heating can be measured with an Infrared (IR) camera, the sensitivity is often not sufficient to recognize the point at which the film breaks. Complicating the measurement is the changing emissivity that commonly occurs as objects dry. The sensitivity and emissivity issues can be addressed by analyzing the temperature in the area of interest and computing the coefficient of variance (COV) of the temperature. This technique is compared to temperature and standard deviation measurements made with an IR camera and the COV technique is shown to be superior for determining when the liquid film breaks. The film breakage point is found to vary with temperature and material roughness in two industrially significant applications: the drying of wood flakes and the drying of polymer films. Film breakage in wood flakes is related to detrimental finished quality problems and also to emission problems. The rate at which an adhesive dries affects the roughness of the polymer film and subsequently, the bond strength. The COV technique is used to predict the roughness of the finished polymer film. Use of the COV technique allows the drying of a liquid film to be visualized in a way that has been previously unreported.