Gegenstand der vorliegenden Dissertation ist das Verhalten sogenannter Selbstüberschneidungslokalzeiten $\\|\\ell_t\\|_p^p$ einer zeitstetigen Irrfahrt $(S_r)_r$ auf dem $d$-dimensionalen Gitter $\\Z^d$. Dabei ist für $p>1$ die Funktion $\\ell_t$ definiert durch $$ \\ell_t(z):=\\int_{0}^{t}\\1_{\\{S_r=z\\}}\\,\\d r\\nonumber $$ und bezeichnet die Aufenthaltsdauer der Irrfahrt bis zum Zeitpunkt $t\\in(0,\\infty)$ im Punkt $z\\in\\Z^d$. Ziel ist es, ein Prinzip großer Abweichungen zu entwickeln, d.h. das Hauptaugenmerk liegt auf dem asymptotischen Verhalten der Wahrscheinlichkeit, dass die Selbstüberschneidungslokalzeiten von ihrem Erwartungswert in erheblichem Maße nach oben abweichen. Mit anderen Worten; es soll das asymptotische Verhalten von $$ \\log\\P(\\|\\ell_t\\|_p^p\\geq r^p_t) $$ genau bestimmt werden, wobei $r_t^p\\in(0,\\infty)$ schneller als der Erwartungswert $\\E[\\|\\ell_t\\|_p^p]$ gegen unendlich streben soll. Dieses Verhalten kann dabei durch $t$, $r_t$ und eine gewisse Variationsformel beschrieben werden. Es wird sich herausstellen, dass es zwei Fälle zu betrachten gilt, in denen sich das probabilistisch beste Verhalten stark unterscheidet; die genaue Position des Phasenübergangs hängt dabei von den Parametern $p$ und $d$ ab. Im Vorgriff auf die Resultate kann man festhalten, dass die nötigen Selbstüberschneidungen in kleinen Dimensionen (im sogenannten subkritischen Fall) über einen großen Bereich erfolgen, aufgrund dessen bei der mathematischen Modellierung eine Reskalierung erforderlich ist. In hohen Dimensionen (dem sogenannten superkritischen Fall) ist dies nicht nötig, da die erforderlichen Selbstüberschneidungen innerhalb eines begrenzten Intervalles erfolgen. Das Interesse an der Untersuchung entstand unter anderem aus der Verbindung zu Modellen der statistischen Mechanik (parabolisches Anderson Modell) und zur Variationsanalysis. In der Vergangenheit wurde eine Vielzahl an Methoden benutzt, um dieses Problem zu lösen. In der vorliegenden Dissertation soll die sogenannte Momentenmethode bestmöglich ausgereizt werden und es wird gezeigt, welche Ergebnisse damit möglich sind.

Transit agencies seeking to improve transit service delivery are increasingly considering the deployment of transit signal priority (TSP). However, the impact of TSP on transit service and on the general traffic stream is a function of many factors, including intersection geometry, signal timings, traffic demands, TSP strategies and parameters, transit vehicle headways, timing when transit vehicles arrive at the intersection, etc. Previous studies have shown that depending on these factors, the net impact of TSP in terms of vehicle or person delay can be positive or negative. Furthermore, due to financial constraints, transit agencies are often able to deploy TSP at only a portion of all of the candidate intersections. Consequently, there is a need to estimate the impact of TSP prior to implementation in order to assist in determining at which intersections TSP should be deployed. Currently, the impacts of TSP are often estimated using microscopic simulation models. However, the application of these models is resource intensive and requires specialized expertise that is often not available in-house to transit agencies. In this thesis, an analytical model was proposed for estimating the delay impacts of green extension and early green (red truncation) TSP strategies. The proposed model is validated with analytical model reported in the literature and microscopic simulation model. This is followed by model sensitivity analysis. A software module is developed using the proposed model. The usefulness of the model is illustrated through its application to estimate the TSP performance. Finally, a prioritization is conducted on sixteen intersections with different geometric and operational traffic strategies. The overall results indicate that the proposed model is suitable for both estimating the pre-deployment and post-deployment TSP performance. The proposed model is suitable for implementation within a spreadsheet and requires considerably less effort, and less technical expertise, to apply than a typical micro-simulation model and therefore is a more suitable tool for transit agencies to use for prioritising TSP deployment.

This paper presents the theory of Noetherian filtrations, an important concept in commutative algebra. The paper describes many aspects of the theory of these objects, presenting basic results, examples and applications. In the study of Noetherian filtrations, a few other important concepts are introduced such as Rees algebras, essential powers filtrations, and filtrations on modules. Basic results on these are presented as well. This thesis discusses at length how Noetherian filtrations relate to important constructions in commutative algebra, such as graded rings and modules, dimension theory and associated primes. In addition, the paper presents an original proof of the finiteness of the intersection algebra of principal ideals in a UFD. It concludes by discussing possible applications of this result to other areas of commutative algebra.

Darbe pristatyti sukurti ir ištirti algoritmai operacijoms su plokščiosiomis geometrinėmis figūromis. Tai figūrų sujungimas, sankirta ir atėmimas. Šios operacijos yra labai svarbios ir labai plačiai naudojamos grafinėse projektavimo, technologinio paruošimo sistemose. Algoritmai operacijoms su plokščiosiomis geometrinėmis figūromis yra vertinami pagal šiuos kriterijus: skaičiavimų laikas, rezultatų (gautų figūrų) kokybė. Šio algoritmo išskirtinė savybė yra ta, kad figūrų viršūnės gali būti jungiamos lankais, kvadratinėmis bei kubinėmis bezjė kreivėmis. Šitame darbe nagrinėjamas algoritmas operacijoms su geometrinėmis figūromis, kuris paremtas šluojančios tiesės principu. Horizontali šluojanti tiesė juda per įvykių taškus, kuriuos sudaro pradinių duomenų viršūnės bei segmentų tarpusavio susikirtimo taškai. Suskaičiuojamos šluojančios tiesės ir duomenų segmentų susikirtimo aibės kiekviename įvykio taške. Iš šių susikirtimo aibių yra suformuojami intervalai, kurių pagalba yra konstruojami galutiniai rezultatai. Operacijų rezultatai buvo lyginami su skenavimo algoritmo rezultatais bei su Java2D API paketo rezultatais (Java2D buvo naudojama tik rezultatų kokybei palyginti). Atlikus eksperimentinius tyrimus buvo nustatyti, kad darbe pasiūlyti sprendimai pateikia geresnius rezultatus. Detalesnę tyrimų analizę galima rasti eksperimentinėje darbo dalyje. / An algorithm for polygon set-operations (union, intersection, difference) was introduced. In CAD system these operations are very important and widely used. The most important aspects are speed and quality of results. Previously used and analyzed polygon operations was based on grid calculations. Thus this strategy takes big amount of time to perform operation and generates not exact results. The major idea in presented algorithm for polygon operations was to use sweeplines through calculated event points (all vertexes and intersection points between data segments). For each sweepline the intervals information is generated (depending on amount of up and down directions from crossed segments). Afterward these intervals are connected between each other to construct the final polygon. As additional feature of this algorithm is support of arcs, quadratic and cubic bezier curves, which connects polygon vertexes. The results of this algorithm were compared with results of scanline (grid) based algorithm and Java2D API (this API is not able to process big amount of data, thus this is used only to check the correctness of result for polygon operations). After implementation and experiments it is obvious that this algorithm works and provides better results (speed and quality of resulting polygons). The detailed comparison and analysis of polygons set-operations is presented in experimental part of this paper.

This thesis reports the use of Paramics microscopic simulation software to model the differences between the performance of ten Christchurch intersections under the existing New Zealand road rule which requires left turning vehicles to give way to vehicles turning right into the same road, and a changed rule that would see the right turning vehicle have priority. Previous research concerning this issue is reviewed and the history of the existing road rule and recent moves to change it are discussed. At each of the ten intersections a range of traffic volume combinations was assessed and the journey times and queue lengths were compared. The ten intersections represent a range of different layouts and forms of control including give way signs, stop signs and traffic signals. The impact of a rule change on the use of shared lanes at intersections using a Paramics model of the Christchurch Central Business District, as developed for the Christchurch City Council, is also reported. Conclusions are drawn about which types of intersections and traffic volume combinations are likely to be affected by a rule change. The features of intersections that contribute to this susceptibility are identified and conclusions drawn about whether positive or negative effects are likely. It is concluded that there is no compelling intersection performance reason why the rule could not be changed. The successful implication of such change would require a review of the road network to identify critical intersections. Some monitoring and mitigation measures may also be required.

The initial attempt was to find efficient technique to identify shadow plylgons in the shadow-volume based shadow generation algorithm. It was observed that shadows correspond to loops of ridge edges (REs). By identifying all the non-overlapping RE loops of a 3D object, one finds all the shadow polygons and, consequently, all the shadows it generates on other objects as well as shadows it generates on itself. This, however, requires extensive edge-edge intersection tests.It was subsequently realized that by storing the angular representations of the RE looks in a look up table, one can avoid the need of decomposing RE loops into non-overlapping loops and, consequently, the need of performation extensive edge-edge intersection tests. Actually, by building the look up table in a way similar to the bucket-sorted edge table of the standard scan-line method, one can use the table in the scan conversion process to mark the pixels that are in shadow directly, without the need of performing any ray-polygon intersection tests as required in the shadow-volume based shadow generation algorithm. Hence, one gets a new shadow generation technique without the need of performing expensive tests.

This paper presents the theory of Noetherian filtrations, an important concept in commutative algebra. The paper describes many aspects of the theory of these objects, presenting basic results, examples and applications. In the study of Noetherian filtrations, a few other important concepts are introduced such as Rees algebras, essential powers filtrations, and filtrations on modules. Basic results on these are presented as well. This thesis discusses at length how Noetherian filtrations relate to important constructions in commutative algebra, such as graded rings and modules, dimension theory and associated primes. In addition, the paper presents an original proof of the finiteness of the intersection algebra of principal ideals in a UFD. It concludes by discussing possible applications of this result to other areas of commutative algebra.

We consider p independent Brownian motions in ℝd. We assume that p ≥ 2 and p(d- 2) < d. Let ℓt denote the intersection measure of the p paths by time t, i.e., the random measure on ℝd that assigns to any measurable set A ⊂ ℝd the amount of intersection local time of the motions spent in A by time t. Earlier results of Chen derived the logarithmic asymptotics of the upper tails of the total mass ℓt(ℝd) as t →∞. In this paper, we derive a large-deviation principle for the normalised intersection measure t-pℓt on the set of positive measures on some open bounded set B ⊂ ℝd as t →∞ before exiting B. The rate function is explicit and gives some rigorous meaning, in this asymptotic regime, to the understanding that the intersection measure is the pointwise product of the densities of the normalised occupation times measures of the p motions. Our proof makes the classical Donsker-Varadhan principle for the latter applicable to the intersection measure. A second version of our principle is proved for the motions observed until the individual exit times from B, conditional on a large total mass in some compact set U ⊂ B. This extends earlier studies on the intersection measure by König and Mörters.

In this thesis, we consider real matrix functions that depend on two parameters and study the problem of how to detect and approximate parameters' values where the singular values coalesce. We prove several results connecting the existence of coalescing points to the periodic structure of the smooth singular values decomposition computed around the boundary of a domain enclosing the points. This is further used to develop algorithms for the detection and approximation of coalescing points in planar regions. Finally, we present techniques for continuing curves of coalescing singular values of matrices depending on three parameters, and illustrate how these techniques can be used to locate coalescing singular values of complex-valued matrices depending on three parameters.

Increasing demand for road freight has lead to a widespread adoption of more-productive large freight vehicles (LFVs), such as B-Doubles, by Australia's road freight industry. Individual LFVs have a greater potential to impact traffic efficiency through their greater length and poorer longitudinal performance. However, this is offset to an extent as fewer vehicles are required to perform a given freight task on a tonne-km basis. This research has developed a means of characterising the effects that large freight vehicles have on the performance of an urban arterial corridor managed by signalised intersections. A corridor-level microsimulation model was developed from first principles, which modelled the longitudinal performance of individual vehicles to a greater accuracy than most existing traffic simulation software does. The model was calibrated from traffic counts and GPS-equipped chase car surveys conducted on an urban arterial corridor in Brisbane's southern suburbs. The model was applied to various freight policy and traffic management scenarios, including freight vehicle mode choice, lane utilisation and traffic signal settings; as well as the effectiveness of green time extension for approaching heavy vehicles. Benefits were able to be quantified in terms of reduced travel times and stop rates for both heavy and light vehicles in urban arterial corridors.