We present a numerical inversion method for generating random variates from continuous distributions when only the density function is given. The algorithm is based on polynomial interpolation of the inverse CDF and Gauss-Lobatto integration. The user can select the required precision which may be close to machine precision for smooth, bounded densities; the necessary tables have moderate size. Our computational experiments with the classical standard distributions (normal, beta, gamma, t-distributions) and with the noncentral chi-square, hyperbolic, generalized hyperbolic and stable distributions showed that our algorithm always reaches the required precision. The setup time is moderate and the marginal execution time is very fast and the same for all distributions. Thus for the case that large samples with fixed parameters are required the proposed algorithm is the fastest inversion method known. Speed-up factors up to 1000 are obtained when compared to inversion algorithms developed for the specific distributions. This makes our algorithm especially attractive for the simulation of copulas and for quasi-Monte Carlo applications. (author´s abstract) / Series: Research Report Series / Department of Statistics and Mathematics

IMRT uses non-uniform beam intensities within a radiation field to provide patient-specific dose shaping, resulting in a dose distribution that conforms tightly to the planning target volume (PTV). Unavoidable geometric uncertainty arising from patient repositioning and internal organ motion can lead to lower conformality index (CI), a decrease in tumor control probability (TCP) and an increase in normal tissue complication probability (NTCP). The CI of the IMRT plan depends heavily on steep dose gradients between the PTV and organ at risk (OAR). Geometric uncertainties reduce the planned dose gradients and result in a less steep or “blurred” dose gradient. The blurred dose gradients can be maximized by constraining the dose objective function in the static IMRT plan or by reducing geometric uncertainty during treatment with corrective verification imaging. Internal organ motion and setup error were evaluated simultaneously for 118 individual patients with implanted fiducials and MV electronic portal imaging (EPI). The Gaussian PDF is patient specific and group standard deviation (SD) should not be used for accurate treatment planning for individual patients. Frequent verification imaging should be employed in situations where geometric uncertainties are expected. The dose distribution including geometric uncertainties was determined from integration of the convolution of the static dose gradient with the PDF. Local maximum dose gradient (LMDG) was determined via optimization of dose objective function by manually adjusting DVH control points or selecting beam numbers and directions during IMRT treatment planning. EUDf is a useful QA parameter for interpreting the biological impact of geometric uncertainties on the static dose distribution. The EUDf has been used as the basis for the time-course NTCP evaluation in the thesis. Relative NTCP values are useful for comparative QA checking by normalizing known complications (e.g. reported in the RTOG studies) to specific DVH control points. For prostate cancer patients, rectal complications were evaluated from specific RTOG clinical trials and detailed evaluation of the treatment techniques. Treatment plans that did not meet DVH constraints represented additional complication risk. Geometric uncertainties improved or worsened rectal NTCP depending on individual internal organ motion within patient.

IMRT uses non-uniform beam intensities within a radiation field to provide patient-specific dose shaping, resulting in a dose distribution that conforms tightly to the planning target volume (PTV). Unavoidable geometric uncertainty arising from patient repositioning and internal organ motion can lead to lower conformality index (CI), a decrease in tumor control probability (TCP) and an increase in normal tissue complication probability (NTCP). The CI of the IMRT plan depends heavily on steep dose gradients between the PTV and organ at risk (OAR). Geometric uncertainties reduce the planned dose gradients and result in a less steep or “blurred” dose gradient. The blurred dose gradients can be maximized by constraining the dose objective function in the static IMRT plan or by reducing geometric uncertainty during treatment with corrective verification imaging. Internal organ motion and setup error were evaluated simultaneously for 118 individual patients with implanted fiducials and MV electronic portal imaging (EPI). The Gaussian PDF is patient specific and group standard deviation (SD) should not be used for accurate treatment planning for individual patients. Frequent verification imaging should be employed in situations where geometric uncertainties are expected. The dose distribution including geometric uncertainties was determined from integration of the convolution of the static dose gradient with the PDF. Local maximum dose gradient (LMDG) was determined via optimization of dose objective function by manually adjusting DVH control points or selecting beam numbers and directions during IMRT treatment planning. EUDf is a useful QA parameter for interpreting the biological impact of geometric uncertainties on the static dose distribution. The EUDf has been used as the basis for the time-course NTCP evaluation in the thesis. Relative NTCP values are useful for comparative QA checking by normalizing known complications (e.g. reported in the RTOG studies) to specific DVH control points. For prostate cancer patients, rectal complications were evaluated from specific RTOG clinical trials and detailed evaluation of the treatment techniques. Treatment plans that did not meet DVH constraints represented additional complication risk. Geometric uncertainties improved or worsened rectal NTCP depending on individual internal organ motion within patient.

Next generation data centres are expected to support both advance resource reservation and on-demand access, but the system performance for such a computing environment has not been well-investigated. A reservation request is characterized by a start time, duration, and resource requirement. Discrete event simulation is used to study the performance characteristics of reservation systems. The basic strategy is to accept a request if resources are available and reject the request otherwise. The performance metrics considered are resource utilization and blocking probability. Results showing the impact of input parameters on these performance metrics are presented. It is found that the resource utilization is quite low. Two strategies that can be used to improve the performance for advance reservation are evaluated. The first strategy allows the start time to be delayed up to some maximum value, while the second allows the possibility of non-uniform resource allocation over the duration of the reservation. Simulation results showing the performance improvements of these two strategies are presented. Resources not used by advance reservation are used to support on-demand access. The performance metrics of interest is the mean response time. Simulation results showing the impact of resource availability and its variation over time on the mean response time are presented. These results provide valuable insights into the performance of systems with time-varying processing capacity. They can also be used to develop guidelines for the non-uniform resource allocation strategy for advance reservation in case the reserved resources are used for interactive access.

The minimum set cover problem is, without question, among the most ubiquitous and well-studied problems in computer science. Its theoretical hardness has been fully characterized--logarithmic approximability has been established, and no sublogarithmic approximation exists unless P=NP. However, the gap between real-world instances and the theoretical worst case is often immense--many covering problems of practical relevance admit much better approximations, or even solvability in polynomial time. Simple combinatorial or geometric structure can often be exploited to obtain improved algorithms on a problem-by-problem basis, but there is no general method of determining the extent to which this is possible. In this thesis, we aim to shed light on the relationship between the structure and the hardness of covering problems. We discuss several measures of structural complexity of set cover instances and prove new algorithmic and hardness results linking the approximability of a set cover problem to its underlying structure. In particular, we provide: - An APX-hardness proof for a wide family of problems that encode a simple covering problem known as Special-3SC. - A class of polynomial dynamic programming algorithms for a group of weighted geometric set cover problems having simple structure. - A simplified quasi-uniform sampling algorithm that yields improved approximations for weighted covering problems having low cell complexity or geometric union complexity. - Applications of the above to various capacitated covering problems via linear programming strengthening and rounding. In total, we obtain new results for dozens of covering problems exhibiting geometric or combinatorial structure. We tabulate these problems and classify them according to their approximability.

We study the relationship of dominance for sequences and trees in Banach spaces. In the context of sequences, we prove that domination of weakly null sequences is a uniform property. More precisely, if $(v_i)$ is a normalized basic sequence and $X$ is a Banach space such that every normalized weakly null sequence in $X$ has a subsequence that is dominated by $(v_i)$, then there exists a uniform constant $C\geq1$ such that every normalized weakly null sequence in $X$ has a subsequence that is $C$-dominated by $(v_i)$. We prove as well that if $V=(v_i)_{i=1}^\infty$ satisfies some general conditions, then a Banach space $X$ with separable dual has subsequential $V$ upper tree estimates if and only if it embeds into a Banach space with a shrinking FDD which satisfies subsequential $V$ upper block estimates. We apply this theorem to Tsirelson spaces to prove that for all countable ordinals $\alpha$ there exists a Banach space $X$ with Szlenk index at most $\omega^{\alpha \omega +1}$ which is universal for all Banach spaces with Szlenk index at most $\omega^{\alpha\omega}$.

In some antenna applications, having high directivity while keeping the antenna dimensions small is desired, which can be obtained by use of superdirective arrays. Superdirective arrays have been popular in academic world since a superdirective array provides higher directivity than the uniformly excited antenna array of same length. In this thesis, superdirective arrays are investigated by making high precision numerical computations. Superdirective array element excitations, array factors and directivities are inspected for different number of elements. Superdirective array pattern and directivity features are compared to uniformly excited array pattern and directivities. Superdirective array tolerance is investigated by examination of array element excitation sensitivities. Bandwidth of superdirective arrays is also inspected. Multiple Precision Toolbox is used during numerical computations in Matlab.

The aim of this study is to design a mixer with appropriate parts for uniform treatment of the material in household microwave ovens which can not be achieved with the turntable. The designed mixer&rsquo / s performance was tested by the help of color and surface temperature values. In the design of the mixer primarily mixing in the vertical and radial directions were sought and for this purpose blades and wings for directing the material especially in these directions were present. The rotational motion of the mixer was provided by a shaft actuated by the motor of the turntable where the motor was replaced by a speed adjustable one. Couscous macaroni beads wetted with CoCl2 solution were dried for processing in the microwave oven. The initial color values of the samples were L*= 52.0&plusmn / 0.35, a*= 8.8&plusmn / 0.21 and b*= 14.1&plusmn / 0.11 . The studied parameters were microwave power level (10%, 40%, 67% and 100% ), processing time (60,90, 120 sec), speed of rotation of the mixer (5,10,15 rpm) , location (4up, 4bt, 6up, 6bt) for the cases of with and without the mixer. v The macaroni beads were well arranged in a mixing container and then put into the microwave oven for operation. Same parameters with coloring experiments were used for the surface temperature determination. After operation the container was photographed by an IR camera. Whether the designed mixer was present or not, average a* and b* values decreased while temperature increased . All these values were significantly affected by the time and power increase. The L* value became an insignificant parameter to decide for the performance Location of the particles in the container appeared as a significant parameter affecting the a*, b* and temperature values without the mixer whereas, with the use of the mixer it became an insignificant parameter indicating uniform energy distribution. Speed of rotation of the mixer was a significant parameter for both cases. However, the color values obtained did not show the same trend with mixer which it showed without mixer. It is concluded that the designed mixer is effective in providing homogeneity of the product by providing sufficient mixing in the container hence the particles can receive about equal energy. Keywords: Microwave oven, particulate solids, mixing, mixer design, testing performance, uniform treatment

Bryant and Eastern Canyon systems are located on the northwest Gulf of Mexico, and they are characterized by a very complex sedimentological history related to glacioeustatic cycles, river discharges, and interactions of depositional and halokinetic processes. Both canyon systems were active during the low sea-level stand of Oxygen Isotope Stage 6, and provided the pathways for the transport of enormous amounts of sediments on the continental slope and abyssal plain of the northwest Gulf of Mexico. Right after their abandonment, at the beginning of Stage 5, salt diapirs encroached into the canyons, and resulted in their transformation into a network of intraslope basins. The transformation of the canyons resulted in the generation of massive sediment failures. The mid-shelf (Stages 4 and 3) to shelf edge (Stage 2) lowering of the sea-level during the last glacial episode resulted in: 1) extensive river-sourced deposits on the outer shelf and/or upper continental slope that contributed in a seaward mobilization of the underlying salt masses, and 2) the generation of numerous gravity flows and turbidity currents on the outer shelf/upper continental slope. The seaward mobilization of the salt masses resulted in the oversteepening of the flanks of the basins, and consequently in the generation of numerous and massive sediment failures. The turbidity currents were confined on the intraslope basins of the upper continental slope, depositing their coarsest material. However, their most diluted upper and end members were able to continue their downslope propagation depositing characteristic fine-grained turbidites. The frequency of the turbidity currents was highly increased during the last glacial maximum (Stage 2), and three short melt-water pulses centered at 30.5, 36, and 52 ky B.P. The last deglaciation event is characterized by the development of a major melt water event that resulted in the deposition of distinct organic rich sediments, similar to the sapropels of the Eastern Mediterranean. At about 11 ky B.P. the melt water discharges of the North America switched from Mississippi River to St Lawrence Seaway, causing the domination of hemipelagic sedimentation on the continental slope of the northwest Gulf of Mexico.

Al-Mg alloys spontaneously infiltrate into porous ceramic preform in a nitrogenous atmosphere above 750 °C with Mg either pre-alloyed or introduced at the interface to initiate the process. The governing process variables are temperature, alloy composition, atmosphere and particle size of the porous preform. The present study investigates the flow kinetics of Al-Mg melts into porous Al2O3 preforms as a function of particle size of the preform from the standpoint of a physical phenomena fluid flow through a non-uniform capillary. Pressureless infiltration involves two major stages: (a) initiation associated with an incubation period and, (b) continuation where the melt infiltrates the preform. Long (~1 hr) and irreproducible incubation periods are typically observed in the Al- Mg/Al2O3 system when the samples are slowly heated in N2 atmosphere. Such lengthy periods prior to infiltration also lead to excessive Mg loss from the system. In order to accurately measure infiltration rates during the continuation stage, the incubation period was minimized by upquenching samples in air under self-sealing conditions. Interrupted experiments reveal that infiltration occurs within 5 mins. Different phenomena are expected to dictate the capillary rise kinetics through the porous ceramic post-incubation (more specifically, retard the melt movement) (a) triple-point ridging of the melt meniscus on the alumina surface (meniscus pinning) (b) interfacial reaction limited wetting and infiltration (c) pore size and distribution of the porous ceramic (d) melt (Al-Mg) / atmosphere (N2) reaction to form products inside the pore space (decrease in permeability) (e) time-dependent loss of Mg from the system (time-dependent contact angle) Some of the above phenomena viz., fluid flow inside the porous medium and chemical reaction of the melt with the reinforcement are invariably coupled in a complex manner. The contribution of each phenomenon to the kinetics of infiltration (a) and (e) was investigated separately. Triple-line ridging Al sessile drops on alumina substrate spread 4-5 orders of magnitude slower than that predicted by hydrodynamic equilibrium. The melt is pinned by ridges leading to spreading rates of 0.4-4 mm/hr in contrast to viscous drag controlled spreading rates of 1-10 mm/sec. In order to detect ridging in the Al-Mg/Al2O3 reactive couple, uniform Al2O3 capillaries were infiltrated. Experiments were conducted under sealed configuration with metal on both sides of the capillary and Mg turnings at the interface. The uniform capillary itself was placed inside an alumina preform and the assembly upquenched to 800-900 °C to minimize evaporative loss of Mg. Examination of the inner walls of the capillary after leaching away the infiltrated metal shows rough, granular features on the polycrystalline Al2O3 surface. No continuous ridges were seen. EDS of the granular phase suggested stoichiometry of spinel, MgAl2O4, formed as a result of the reaction between the melt and the capillary. From interrupted experiments the average infiltration rate inside the uniform capillary was calculated to be in the ballpark range of 2-6 µm/sec (which is a lower limit to the meniscus velocity), an order of magnitude faster than the spreading rates observed during triple-line ridging (0.1 – 1 µm/sec) indicating that the melt front pinning was not the operative mechanism for influencing infiltration kinetics. Pore size distribution of porous medium Additionally, infiltration was found to be faster in uniform channels (fractures in a preform, annular spaces and aligned pores in freeze-cast preforms) compared to the randomly packed bed itself. The effect of pore size on infiltration kinetics was studied by varying the particle size of the packed bed. Experiments were conducted for two systems (a) non-reactive liquid polyethylene glycol PEG 600 (b) reactive Al-Mg melts into packed alumina beds as a function of particle size and temperature. The PEG 600 / Al2O3 ‘model’ system was used to benchmark the effect of pore size and distribution of the particle bed on flow kinetics from a purely physical standpoint. Typically, a Washburn type of ‘parabolic’ kinetics was observed for the non-reactive couple and the ‘effective’ hydrodynamic radius, reff was extracted. (For a uniform capillary, reff and the physical radius of the capillary are the same). Surprisingly, the ‘Washburn’ radius was found to be 1-2 orders of magnitude smaller than the average pore size and even smaller than the minimum average pore size of the compact. The ‘Washburn’ radii for infiltration of Al-Mg melts was a further order of magnitude smaller than the corresponding values for infiltration of non-reactive PEG 600 through the same packed beds. Non-uniform capillary model To predict the infiltration kinetics through porous media, a sinusoidal capillary model was developed based on the pore size distribution. The input parameters for the model were the average pore neck size and average pore bulge size, which were extracted from the experimentally measured pore size distribution. The flow was assumed to be quasi-steady state and laminar. Hagen-Poiseuille’s equation was employed to calculate the total pressure drop, which was equated with the instantaneous pressure drop across the meniscus. The meniscus velocity within the non-uniform capillary was solved numerically based on the instantaneous pressure drop. The infiltration profile for the sinusoidal capillary displayed jumps associated rise in the narrow segments of the profile while the rise through the broad segment was considerably slow. The overall infiltration profile could be fitted by a parabolic Washburn-type equation. The ‘effective’ hydrodynamic radius of such a sinusoidal capillary was found to be 2-3 orders of magnitude smaller than the average capillary size and even smaller than the narrowest opening of the sinusoidal capillary. The overall kinetics was limited by flow through the broad segment of the profile where the capillary driving force is the lowest coupled with a large viscous retarding force due to the narrow feeding segment thereby leading to extremely slow flow rates. The calculated ‘effective’ radius of the sinusoidal capillary (reff = 0.03 µm) based on the pore size distribution of the 25-37 µm (1.4-10.8 µm) packed bed was similar to the experimentally observed ‘effective’ radius for flow in the non-reactive couple (reff = 0.06 µm) implying good agreement between experiments and modeling. The model was extended for the case of pressure infiltration of Al melts into SiC & TiC compacts reported in the literature, under conditions where chemical reactions are negligible. A good agreement to within a factor of 4 between the observed kinetics and the ones predicted by the current model is observed. In order to understand the origin of this ‘unphysical’ radius dictating capillary rise, the physics of flow through a stepped capillary was analysed. The kinetics of flow through the wide segment could be expressed by an ‘effective’ drodynamic radius r 4min based on geometrical parameters of the stepped capillary as: reff= r3max (Wetting situation) where rminand rmax are the radii of the narrow and broad segments of the capillary. The ‘effective’ radius from the above equation matched well with the numerically derived ‘effective’ radius for flow through the stepped capillary. A r 2 similar expression for flow under applied pressure was derived as: reff= min rmax (non- wetting situation) which is strictly correct for large values of applied pressure. Chemical reactions influencing infiltration kinetics: Upquenched samples (time-dependent contact angle due to Mg loss) The previous investigation of fluid flow in porous media from a purely physical standpoint reveals the dominant role of the pore size and distribution in the porous medium in controlling infiltration kinetics. This however, is accurate only if chemical factors are minimized. In case of the upquenched experiments for the Al-Mg/Al2O3 system, the ‘effective’ radius was determined to be an order of magnitude smaller than that for the PEG 600/Al2O3 couple implying additional chemical factors influencing flow kinetics in this reactive system. Experiments with Mg turnings mixed with the powder bed shows faster infiltration compared to the ones where the entire Mg was placed at the interface showing that local availability of Mg was responsible for slower infiltration kinetics. Diminishing Mg at the melt front, leads to increase of surface tension and increase in contact angle. This was modeled by incorporating a kinetics (time-dependent) contact angle into the sinusoidal capillary model developed for non-reactive infiltration. The infiltration kinetics was found to be retarded in the case of a kinetic contact angle. Thus, both flow retardation through a packed bed and time-dependent variations of contact angle due to Mg loss from the system are responsible for slow pressureless infiltration kinetics of Al-Mg melts inside Al2O3 preforms. The infiltration kinetics predicted by the sinusoidal capillary model thus defines an upper envelope to the rate of infiltration and subsequent composite formation for such a process governed by fluid flow; all other factors if present in effect, retard the kinetics further. Samples processed in N2 atmosphere (reduced permeability due to AlN formation) The more practical case of composite fabrication (PRIMEXTM process) by pressureless infiltration of Al-Mg melts in a flowing N2 containing atmosphere was also examined. The kinetics of infiltration of Al-Mg melts in a flowing N2-H2 atmosphere (pO2 ~ 10-20atm) for different particle sizes of the packed bed was investigated. A large scatter in the infiltrated heights was observed and the absolute infiltration rates could not be established. Moreover, incubation periods were seen to range from 1-2 hours for different particle sizes. Post-incubation, the infiltration kinetics for a wide range of particle sizes was found to be approximately an order of magnitude slower than that for the upquenched samples. Microstructural investigations of the etched samples revealed significant AlN formation at the start of the composite near the preform/billet interface. This reduced the cross-sectional area available for melt flow and possibly led to long incubation periods encountered in the process. AlN formation was also detected in the matrix on the particle surfaces as well as in the interior of the matrix. This reduced the permeability of the compact and increased the hydrodynamic resistance for flow through the porous compact leading to slower infiltration kinetics. Thus both AlN formation in the matrix and Mg loss from the melt retard capillary flow of the melt through the porous ceramic over and above the intrinsic hydrodynamic resistance for flow through the packed bed. Role of atmosphere on the pressureless infiltration process The role of atmosphere in promoting the pressureless infiltration process was examined by using different processing atmospheres such as vacuum, N2-H2 and Ar and combinations thereof. It is known that the pressureless infiltration of Al melts into porous Al2O3 preforms requires both N2 and a critical level of Mg in the system. Samples heated under vacuum and Ar to 900 °C under open conditions did not infiltrate. Rather these showed discoloration related to the formation of MgAl2O4 on the particle surface due to reduction of Al2O3 by Mg vapour. Moreover, samples heated in Ar upto 500 °C followed by heating up in N2-H2 till 900 °C did not infiltrate indicating irreversible changes. Interestingly enough, if the samples were heated in vacuum upto 700 °C followed by N2-H2 at 900 °C, infiltration was observed. Dewetted regions of the compact were seen too adjacent to the preform-billet interface. This indicated a minimum critical partial pressure of N2, which promotes infiltration. From an analysis of the different interfacial energies and their dependence on atmosphere, it was concluded that either an increase in the solid-vapour interfacial energy (~ 10%) or a decrease in the solid-liquid interfacial energy (~ 10%) would lead to a decrease in the contact angle, θ, by 10°, large enough to ensure wettability and infiltration in certain atmospheres. It was also established that Mg infiltrates into porous Al2O3 both in N2-H2 as well as Ar under sealed conditions. So the presence of a minimum partial pressure of N2 favouring wettability was specific to the Al-Mg/Al2O3 system. (pl see the original document for formulas)