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Doctor of Philosophy / We consider the problem of minimising the eigenvalues of the Laplacian with Robin boundary conditions $\frac{\partial u}{\partial \nu} + \alpha u = 0$ and generalised Wentzell boundary conditions $\Delta u + \beta \frac{\partial u}{\partial \nu} + \gamma u = 0$ with respect to the domain $\Omega \subset \mathbb R^N$ on which the problem is defined. For the Robin problem, when $\alpha > 0$ we extend the Faber-Krahn inequality of Daners [Math. Ann. 335 (2006), 767--785], which states that the ball minimises the first eigenvalue, to prove that the minimiser is unique amongst domains of class $C^2$. The method of proof uses a functional of the level sets to estimate the first eigenvalue from below, together with a rearrangement of the ball's eigenfunction onto the domain $\Omega$ and the usual isoperimetric inequality. We then prove that the second eigenvalue attains its minimum only on the disjoint union of two equal balls, and set the proof up so it works for the Robin $p$-Laplacian. For the higher eigenvalues, we show that it is in general impossible for a minimiser to exist independently of $\alpha > 0$. When $\alpha < 0$, we prove that every eigenvalue behaves like $-\alpha^2$ as $\alpha \to -\infty$, provided only that $\Omega$ is bounded with $C^1$ boundary. This generalises a result of Lou and Zhu [Pacific J. Math. 214 (2004), 323--334] for the first eigenvalue. For the Wentzell problem, we (re-)prove general operator properties, including for the less-studied case $\beta < 0$, where the problem is ill-posed in some sense. In particular, we give a new proof of the compactness of the resolvent and the structure of the spectrum, at least if $\partial \Omega$ is smooth. We prove Faber-Krahn-type inequalities in the general case $\beta, \gamma \neq 0$, based on the Robin counterpart, and for the ``best'' case $\beta, \gamma > 0$ establish a type of equivalence property between the Wentzell and Robin minimisers for all eigenvalues. This yields a minimiser of the second Wentzell eigenvalue. We also prove a Cheeger-type inequality for the first eigenvalue in this case.

Doctor of Philosophy / We consider the problem of minimising the eigenvalues of the Laplacian with Robin boundary conditions $\frac{\partial u}{\partial \nu} + \alpha u = 0$ and generalised Wentzell boundary conditions $\Delta u + \beta \frac{\partial u}{\partial \nu} + \gamma u = 0$ with respect to the domain $\Omega \subset \mathbb R^N$ on which the problem is defined. For the Robin problem, when $\alpha > 0$ we extend the Faber-Krahn inequality of Daners [Math. Ann. 335 (2006), 767--785], which states that the ball minimises the first eigenvalue, to prove that the minimiser is unique amongst domains of class $C^2$. The method of proof uses a functional of the level sets to estimate the first eigenvalue from below, together with a rearrangement of the ball's eigenfunction onto the domain $\Omega$ and the usual isoperimetric inequality. We then prove that the second eigenvalue attains its minimum only on the disjoint union of two equal balls, and set the proof up so it works for the Robin $p$-Laplacian. For the higher eigenvalues, we show that it is in general impossible for a minimiser to exist independently of $\alpha > 0$. When $\alpha < 0$, we prove that every eigenvalue behaves like $-\alpha^2$ as $\alpha \to -\infty$, provided only that $\Omega$ is bounded with $C^1$ boundary. This generalises a result of Lou and Zhu [Pacific J. Math. 214 (2004), 323--334] for the first eigenvalue. For the Wentzell problem, we (re-)prove general operator properties, including for the less-studied case $\beta < 0$, where the problem is ill-posed in some sense. In particular, we give a new proof of the compactness of the resolvent and the structure of the spectrum, at least if $\partial \Omega$ is smooth. We prove Faber-Krahn-type inequalities in the general case $\beta, \gamma \neq 0$, based on the Robin counterpart, and for the ``best'' case $\beta, \gamma > 0$ establish a type of equivalence property between the Wentzell and Robin minimisers for all eigenvalues. This yields a minimiser of the second Wentzell eigenvalue. We also prove a Cheeger-type inequality for the first eigenvalue in this case.

Thesis (M.A.)--University of Montana, 2008. / Title from title screen. Description based on contents viewed Oct. 27, 2008. Includes bibliographical references (p. 73-75).

Thesis (M.A.)--Northern Michigan University, 2007. / Essays, journal entries, and short stories. Bibliography: leaf 51.

Made available in DSpace on 2016-06-02T20:27:40Z (GMT). No. of bitstreams: 1 5896.pdf: 735982 bytes, checksum: 92ffc34627bda308fd23db0bfc679a10 (MD5) Previous issue date: 2014-05-23 / Financiadora de Estudos e Projetos / First we briefly discuss the spectral properties of the Robin Laplacian in a straight and infinitely long planar strip with Robin coupling parameter changing signs. Assuming that such parameter is a constant outside a compact set, we find the essential spectrum, and also derive a sufficient condition that guarantees the existence of bound states. Next, we have studied the Laplacian in some thin curved domains in the plane and space, with a particular type of Robin boundary conditions and cross-sections; such conditions are nonhomogeneous, in the sense that the Robin coupling parameter depends on the curvature of the reference curve. We derive, when the diameter of the cross sections tends to zero, effective operators by means of a kind of convergence of the resolvents. With such modeling framework, the main novelties are that the curvature contributes with a repulsive potential whereas the torsion (in the spatial case) plays no role to effective operators. / Iniciamos com uma breve análise das propriedades espectrais do laplaciano de Robin em uma faixa reta infinita, com parâmetro de acoplamento de Robin mudando de sinal. Assumindo que tal parâmetro é uma constante fora de um compacto, encontramos o espectro essencial, e também derivamos uma condição suficiente que garante a existência de estados ligados. Em seguida, temos nos dedicado a estudar o laplaciano em alguns domínios finos e encurvados no plano e no espaço, com um tipo particular de condições de contorno de Robin e de seções transversais; tais condições são não homogêneas, no sentido de que o parâmetro de acoplamento de Robin depende da curvatura da curva de referência. Obtivemos, quando o diâmetro das seções transversais tendem a zero, operadores efetivos por meio de um tipo de convergência em norma dos resolventes. Com essa estrutura de modelagem, as principais novidades são que a curvatura contribui com um potencial repulsivo enquanto a torção (no caso espacial) não desempenha nenhum papel para os operadores efetivos.

Det här arbetet behandlar min prestationsångest inom låtskrivningsprocessen. Mitt syfte var att undersöka hur jag kan få en mer glädjefylld, kravlös process i mitt låtskrivande. Detta gjordes med hjälp av en genomgång av relevant litteratur inom ämnet samt skapandet av sex egenskrivna låtar med enkel produktion. I låtskapandet användes Robin Fredericks Song Starter teknik ur boken Song Starters: 365 Lyric, Melody, &amp; Chord Ideas to Kickstart Your Songwriting, samt en metod som liknar det Gary Ewer kallar för ”Speedwriting” på sin blogg www.secretsofsongwriting.com och som gick ut på att skriva en låt inom en timme. Mitt resultat visar att den mentala inställningen spelar stor roll för hur mycket plats prestationsångesten får i min skaparprocess. För att få en mer kravlös process bör jag bland annat lägga fokus på processen snarare än på resultatet, inte se misstag som misslyckanden utan som en nödvändig del av processen, tillåta mig själv att leka och experimentera samt ta mig tid till pauser och miljöombyten. Dessutom är det viktigt att jag vågar släppa taget om prestationsångesten och inte ser dess avsaknad som ett tecken på att jag inte tar uppgiften på allvar. Song Starters hjälpte mig att minska prestationskraven i början av skapandeprocessen medan entimmeslåtar minskade prestationskraven under hela processen. Jag upptäckte att det inte är den första brainstormingperioden i låtskrivarprocessen som är den mest kritiska fasen för mig när det gäller prestationsångest, utan snarare perioden när jag ska värdera om mitt material är bra nog för att bli en låt. Dessutom verkar jag gilla mina låtar mer när de har mognat ett tag än när jag precis har skrivit dem och därför är det viktigt att jag inte dömer mina låtar för snabbt.

With the increasing demand of computation capabilities, many-core processors are gain-ing more and more attention. As a communication subsystem many-core processors, Network-on-Chip (NoC) draws a lot of attention in the related research fields. A NoC is used to deliver messages among different cores. For many applications, timeliness is of great importance, especially when the application has hard real-time requirements. Thus, the worst-case end-to-end delays of all the messages passing through a NoC should be concerned. Unfortunately, there is no existing analysis tool that can support multiple NoC architectures as well as provide a user-friendly interface.This thesis focuses on a wormhole switched NoC using different arbitration policies which are Fixed Priority (FP) and Round Robin (RR) respectively. FP based arbitration policy includes distinct and shared priority based arbitration policies. We have developed a timing analysis tool targeting the above NoC designs. The Graphical User Interface (GUI) in the tool can simplify the operation of users. The tool takes characteristics of flow sets as input, and returns results regarding the worst-case end-to-end delay of each flow. These results can be used to assist the design of real-time applications on the corre-sponding platform.A number of experiments have been generated to compare different arbitration mecha-nisms using the developed tool. The evaluation focuses on the effect of different param-eters including the number of flows and the number of virtual-channels in a NoC, and the number of hops of each flow. In the first set of experiment, we focus on the schedulabil-ity ratio achieved by different arbitration policies regarding the number of flows. The sec-ond set of experiments focus on the comparison between NoCs with different number of virtual-channels. In the last set of experiments, we compare different arbitration mecha-nisms with respect to the worst-case end-to-end latencies.