Abstract: Considering the Schrödinger equation $\Delta_x u = i\partial{u}/\partial{t}$, we have a solution $u$ on the form $$u(x, t)= (2\pi)^{-n} \int_{\RR} {e^{i x\cdot \xi}e^{it|\xi|^2}\widehat{f}(\xi)}\, d \xi, x \in \RR, t \in \mathbf{R}$$ where $f$ belongs to the Sobolev space. It was shown by Sjögren and Sjölin, that assuming $\gamma : \mathbf{R}_+ \rightarrow \mathbf{R}_+ $ being a strictly increasing function, with $\gamma(0) = 0$ and $u$ and $f$ as above, there exists an $f \in H^{n/2} (\RR)$ such that $u$ is continuous in $\{ (x, t); t>0 \}$ and $$\limsup_{(y,t)\rightarrow (x,0),|y-x|<\gamma (t), t>0} |u(y,t)|= + \infty$$ for all $x \in \RR$. This theorem was proved by choosing $$\widehat{f}(\xi )=\widehat{f_a}(\xi )= | \xi | ^{-n} (\log | \xi |)^{-3/4} \sum_{j=1}^{\infty} \chi _j(\xi)e^{- i( x_{n_j} \cdot \xi + t_j | \xi | ^a)}, \, a=2,$$ where $\chi_j$ is the characteristic function of shells $S_j$ with the inner radius rapidly increasing with respect to $j$. The purpose of this essay is to explain the proof given by Sjögren and Sjölin, by first showing that the theorem is true for $\gamma (t)=t$, and to investigate the result when we use $$S^a f_a (x, t)= (2 \pi)^{-n}\int_{\RR} {e^{i x\cdot \xi}e^{it |\xi|^a}\widehat{f_a}(\xi)}\, d \xi$$ instead of $u$.

The samples this thesis investigated were m-plane nitrides films grown on m-plane sapphire by plasma-assisted molecular beam epitaxy (PAMBE). Scanning electron microscopy (SEM) images revealed the surface morphologies of the films and thicknesses of the films were measured by cross-sectional scanning electron microscopy. Then we used electron back-scattered diffraction (EBSD) and X-ray diffraction (XRD) to check the growth orientation of the films. The m-plane nitrides films have the anisotropic optical properties were due to the growth orientation of the films. The films are under anisotropic stress since they were grown along m-axis and hence change the electron band structure (EBS), which resulted in anisotropic optical property. We studied the polarization properties of the luminescence at 15 K and 300 K by polarization dependent photoluminescence (PL) and calculated the degree of polarization. And then measured the strain of the m-plane nitrides films by micro-Raman spectroscopy, discussed the degree of polarization and stress. The degree of polarization larger as the anisotropic stress of the film increased.

Cocaine (CA) exposure during pregnancy causes long-lasting negative effects on fetal brain development and eventually results in motor dysfunction or changes in learning and memory performance. £^-amino-butyric acid (GABA) is the primary inhibitory neurotransmitter in the adult brain and undergo a switch from excitatory to inhibitory during early postnatal period. The excitatory/inhibitory switch is resulted in the relative temporal expression of K+-Cl- cotransporter 2 (KCC2). GABA is the neurotransmitter in the rat was born from excitement to inhibition and until the growth of thirty days have completely inhibitory. Here we test the effect of CA prenatal exposure on the expression of KCC2 in prefrontal cortex (recognition), hippocampus (memory), VTA (reward) and nucleus accumbens (reward). Protein expression profile of control or prenatal CA treated groups were evaluated by western blot in 2 days interval from postnatal day (PND) 8 to 30. The expression of KCC2 was time-dependently enhanced from PND 8 and reaches its maximal expression around PND 18 in prenatal CA exposure groups. The time-dependent profile of KCC2 expression in prefrontal cortex and NAc was significantly delayed in prenatal CA exposure group. We then correlate the KCC2 expression and the cocaine sensitivity by locomotor activity assay. We found group A shows a higher sensitivity to cocaine than group B in control rats. Surprisingly, group A of prenatal cocaine reduce the sensitive to cocaine to a similar extend like group B in control rats, suggesting prenatal exposure of cocaine might enhance the KCC2 expression. Furthermore, age range of A group (PND 22~27) and B group (PND 29~34) to repeated cocaine exposure resulted in up-regulation of KCC2 expression in B group earlier than A group. We found that the KCC2 expressions of repeated cocaine exposure in B group were higher than A group. In other words, in the B group, the inhibitory effect of GABA was significant and the locomotor activity was relatively slow. Therefore, the A group was more easy be cocaine addiction than B group. We next explore the signaling mechanism underlying cocaine exposure-induced KCC2 expression inhibition. Brain slices were incubated with cocaine with or without dopamine receptor antagonists and KCC2 expression was evaluated by western blot. Either SCH23390 (dopamine D1-receptor inhibitor) or eticlopride (dopamine D2-receptor inhibitor) significantly hamper the inhibition of KCC2 expression by cocaine in normal slices. However, only D2 antagonist eticlopride but not SCH23390 is effective reverse cocaine-induced KCC2 expression inhibition. Overall, results from our current studies provide a further insight into the molecular mechanism of cocaine-induced synaptic modification.

The dissertation focuses on the analysis, through combined analytical and numerical techniques, of the partial differential equations arising from a new approach to modeling brittle fracture, based on extension of continuum mechanics to the nanoscale. The main part of this work deals with the analysis of several fracture models. Integral transform methods are used to reduce the problem to a Cauchy singular, linear integro-differential equation. It is shown that ascribing constant surface tension to the fracture surfaces and using the appropriate crack surface boundary condition, given by the jump momentum balance, leads to a sharp crack opening profile at the crack tip, in contrast to the classical theory of brittle fracture. However, such a model still predicts singular crack tip stress. For this reason a modified model is studied, where the surface excess property is responsive to the curvature of the fracture surfaces. It is shown that curvature-dependent surface tension, together with boundary conditions in the form of the jump momentum balance, leads to bounded stresses and a cusp-like opening profile at the crack tip. Further, an alternative approach, based on asymptotic analysis, which is suitable to apply in cases when the model includes a mutual body force correction term, is considered. The nonlinear nonlocal problem, resulting from the proposed model, is simplified which allows us to approximate the crack opening profile and derive asymptotic forms for the cleavage stress in a neighborhood of the crack tip. Finally, two possible fracture criteria, in the context of the new theory, are discussed. The first one is an energy based fracture criterion. Classically the energy release rate arises due to singular fields, whereas in the case of the modeling approach adopted here, a notion analogous to the energy release rate arises through a different mechanism, associated to the rate of working of the surface excess properties at the crack tip. Due to the fact that the proposed modeling approach allows us to fully resolve the stress in a neighborhood of the crack tip, without the customary singularity, a second fracture criterion, based on crack tip stress, is possible.

Studying dynamics of magnetization relaxation in excited magnetic materials is important both for understanding the rates and pathways of magnetization relaxation and for the potential use in spin-based electronics and data storage devices in the future. Previous studies have demonstrated that the size of nanocrystals is an important factor for energy relaxation in quantum dots and metal nanoparticles. Since magnetization relaxation is one of energy relaxation pathways, the size of nanocrystals may be also an important factor for magnetization relaxation in nanoscale magnetic materials. The goal of this study is to have a better understanding of magnetization relaxation in nanoscale magnetic materials. In particular, we focused on the correlation between the nanocrystal size and the rates of spin-lattice relaxation (SLR), a magnetization relaxation pathway, in magnetic nanocrystals. The size-dependent magnetization relaxation rate after optically induced demagnetization in colloidal Fe3O4 nanocrystals was measured by using time-resolved Faraday rotation (FR). Fe3O4 nanocrystals were chosen as the model system to study the correlation between the size of nanocrystals and the rates of SLR due to the wellestablished synthetic procedure of making nanocrystals with various sizes and narrow size dispersion. Faster SLR rates were observed in smaller Fe3O4 nanocrystals. The results suggested the surface of nanocrystals have higher efficiency of SLR than the interior region by using a simple model to analyze the SLR rates of Fe3O4 nanocrystals with various sizes. Higher efficiency of SLR at the surface may be due to the stronger spin-orbit coupling at the surface relative to the interior region. In addition to magnetization dynamics studies, the effect of oxidation on static FR in iron oxide nanocrystals (between Fe3O4 and y-Fe2O3) was studied. The results indicated FR signal is linearly correlated to the strength of optical transition between Fe2 and Fe3 in Fe3O4 for a given size of nanocrystals.

Due to the impressive capabilities of human visual processing, interactive visualization methods have become essential tools for scientists to explore and analyze large, complex datasets. However, traditional approaches do not account for the increased size or latency of data retrieval when interacting with these often remote datasets. In this dissertation, I discuss two novel design paradigms, based on accepted models of the information visualization process and graphics hardware pipeline, that are appropriate for interactive visualization of large remote datasets. In particular, I discuss novel solutions aimed at improving the performance of interactive visualization systems when working with large numeric datasets and large terrain (elevation and imagery) datasets by using data reduction and asynchronous retrieval of view-prioritized data, respectively. First I present a modified version of the standard information visualization model that accounts for the challenges presented by interacting with large, remote datasets. I also provide the details of a software framework implemented using this model and discuss several different visualization applications developed within this framework. Next I present a novel technique for leveraging the hardware graphics pipeline to provide asynchronous, view-prioritized data retrieval to support interactive visualization of remote terrain data. I provide the results of statistical analysis of performance metrics to demonstrate the effectiveness of this approach. Finally I present the details of two novel visualization techniques, and the results of evaluating these systems using controlled user studies and expert evaluation. The results of these qualitative and quantitative evaluation mechanisms demonstrate improved visual analysis task performance for large numeric datasets.

Spinal cord injury, acute pain, and morphine tolerance are important issues in the clinical practice. A primary injury to the spinal cord causes both morphological and biochemical changes with initiation of the devastating secondary pathophysiological pathways that ultimately destroy CNS cells and cause degeneration of nerve fibers. Tissue injury is associated with sensitization of nociceptors and subsequent changes in the excitability of central neurons, known as central sensitization. Nociceptor sensitization and central sensitization are believed to underlie the development of primary and secondary hyperalgesia, respectively. The most efficacious drugs used to relieve pain are the opioid analgesics. Chronic administration leads to the development of tolerance. Tolerance is manifested as a decreased potency of the drug, so that progressively larger doses must be administered to achieve a given level of analgesia. The processes underlying opioid tolerance still need to be elucidated. Recently, it is found calpain-Cdk5 (cyclin-dependent kinase-5)-p35 pathway modulation implicated in neuroprotection, acute nociceptive response, and morphine analgesia. In this thesis, we evaluate calpain inhibitor-MDL28170 and Cdk5 inhibitor-roscovitine against rat spinal cord hemisection, formalin-induced acute nociceptive responses, and chronic morphine tolerance. We found calpain-Cdk5-p35 pathway inhibition could protect spinal cord hemisection and subsequent neurodegeneration, inhibit formalin-induced flinch response involving DARPP-32 (dopamine and c-AMP regulated phosphoprotein, MW=32 kDa) phosphorylation, and reverse right shifted morphine dose-response curve with upregulated ED50 (50% of effective dose) reduction. Taken together, calpain-Cdk5-p35 pathway inhibition is useful in the management of spinal cord injury, acute inflammatory pain, and attenuate morphine tolerance development with further clinical application.

Cyclin-dependent kinase-5 (Cdk5) was identified as a serine/threonine kinase that plays an important role in neuronal development. Association with one of the neuronal activators, p35 or p39, is required for Cdk5 to elicit its diverse effects in the nervous system, such as neurite outgrowth. In addition to these, increasing evidence suggests that Cdk5 also plays an important role in cocaine addiction, neurotransmitter release, NMDA receptor phosphorylation. This thesis is divided into three parts which deals with the effects of Cdk5 inhibitor¡Ðroscovitine on the morphine tolerance development, acute inflammatory pain, and pilocarpine-induced seizure respectively. The first part explored the effect of Cdk5 inhibitior¡Ðroscovitine on the morphine antinociceptive tolerance development. Delta FosB activation is involved in morphine tolerance. Cyclin-dependent kinase- 5 (Cdk5) is found to be the downstream target of delta FosB. We examined the effects of the potent selective Cdk5 inhibitor¡Ðroscovitine on the development of antinociceptive tolerance of morphine. Tolerance was induced by continuous infusion of morphine 5 µg/hr intrathecally (i.t.) for 5 days. The effect of co-administration of roscovitine 1 µg/hr i.t. for 5 days was also examined. Roscovitine co-administration enhanced the antinociceptive effect of morphine in morphine tolerant rats. It also shift the morphine antinociceptive dose¡Ðresponse curve to the left during morphine tolerance induction, and reduced the increase in the ED50 of morphine two-fold. Collectively, these findings suggest Cdk5 modulation may be involved in the development of morphine tolerance and its inhibitor will enhance antinociceptive effect. The second part discussed the roscovitine effect on acute inflammatory pain. Formalin injected into the rat hind paw will evoke flinching (consisting of an elevation and shrinking back of the injected paw), a reliable parameter of pain behavior. The nociceptive response to formalin occurs in a biphasic pattern: there isan initial acute period (phase 1), and after a short period of remission, phase 2 begins and consists of a longer period (1 hour) of sustained activity. The initial response was initially attributed to a direct algogenic effect of formalin, whereas phase 2 was associated with the central sensitization. In this study, the Cdk5 inhibitor¡Ðroscovitine was injected intrathecally to elucidate the mechanism of Cdk5 activation during formalin-induced hyperalgesia. The 50 ul of 5% formalin solution was used as the noxious stimulant. The rats were injected with 0, 50, 100, and 200ug roscovitine intrathecally thirty minutes before hind paw formalin injection. Intrathecal 200ug roscovitine injection attenuates the phase I flinch response. And intrathecal 50, 100, and 200ug roscovitine injection suppress phase II flinch response effectively. Roscovitine administration could effectively suppress the formalin-induced flinch behavior. This implies the activation of Cdk5 plays an important role in the sensitization after nociceptive stimulation. The third part focus on the roscovitine effect on the pilocarpine induced seizure. Pilocarpine temporal lobe epilepsy model is widely used. Chronic electroconvulsive therapy could upregulate Cdk5 activity. Cdk5 inhibitor¡Ðroscovitine could suppress NMDA induced long-term potentiation in hippocampal slice. Intracerebroventricular injection of 100£gg roscovitine 30 min before pilocarpine-induced epilepsy could significantly decrease the seizure-induced mortality ( 11% in roscovitine group VS 77% in control group). The escape latency, spatial memory impairment, in the pilocarpine-induced seizure group is significant longer than the roscovitine pretreatment group in the Morris water maze test after one month (p¡Õ0.05). It is concluded Cdk5 may play an important in the pathogenesis of epilepsy. Therefore, Cdk5 inhibition may become another way for the epilepsy treatment.

This research is focused on development and enhancement of the model of fluid flow in a formation with stress-dependent permeability. Several typical axi-symmetrical problems of fluid flow in a multi-layered reservoir with account for wellbore storage and skin have been solved numerically. The permeability was assumed to be a function of the vertical deformation of the reservoir. This deformation is the result of changing stress-strain state in the elastic system, comprised of the reservoir itself and the surrounding rock mass. The change in the stress-strain state of the system is induced by pressure change in the layers of the reservoir. Numerical results qualitatively agree with observed field behavior. Such behavior includes (1) deviation of an inflow performance curve from the straight-line relationship at pressures above bubble-point pressure, (2) time- and rate-dependence of well-testing derivative, (3) asymmetry of processes of production and of injection, and (4) inconsistent results between drawdown and buildup, or injection and falloff tests. Based on the results, a procedure to estimate the parameters of the suggested permeability model is proposed.

This dissertation discusses two independent topics: a resource-constrained shortest-path problem (RCSP) and a literature review on scheduling problems involving sequence-dependent setup (SDS) times (costs). RCSP is often used as a subproblem in column generation because it can be used to solve many practical problems. This dissertation studies RCSP with multiple resource constraints on an acyclic graph, because many applications involve this configuration, especially in column genetation formulations. In particular, this research focuses on a dynamic RCSP since, as a subproblem in column generation, objective function coefficients are updated using new values of dual variables at each iteration. This dissertation proposes a pseudo-polynomial solution method for solving the dynamic RCSP by exploiting the special structure of an acyclic graph with the goal of effectively reoptimizing RCSP in the context of column generation. This method uses a one-time “preliminary” phase to transform RCSP into an unconstrained shortest path problem (SPP) and then solves the resulting SPP after new values of dual variables are used to update objective function coefficients (i.e., reduced costs) at each iteration. Network reduction techniques are considered to remove some nodes and/or arcs permanently in the preliminary phase. Specified techniques are explored to reoptimize when only several coefficients change and for dealing with forbidden and prescribed arcs in the context of a column generation/branch-and-bound approach. As a benchmark method, a label-setting algorithm is also proposed. Computational tests are designed to show the effectiveness of the proposed algorithms and procedures. This dissertation also gives a literature review related to the class of scheduling problems that involve SDS times (costs), an important consideration in many practical applications. It focuses on papers published within the last decade, addressing a variety of machine configurations - single machine, parallel machine, flow shop, and job shop - reviewing both optimizing and heuristic solution methods in each category. Since lot-sizing is so intimately related to scheduling, this dissertation reviews work that integrates these issues in relationship to each configuration. This dissertation provides a perspective of this line of research, gives conclusions, and discusses fertile research opportunities posed by this class of scheduling problems. since, as a subproblem in column generation, objective function coefficients are updated using new values of dual variables at each iteration. This dissertation proposes a pseudo-polynomial solution method for solving the dynamic RCSP by exploiting the special structure of an acyclic graph with the goal of effectively reoptimizing RCSP in the context of column generation. This method uses a one-time