The Kari-Culik tilings are formed from a set of 13 Wang tiles that tile the plane only aperiodically. They are the smallest known set of Wang tiles to do so and are not as well understood as other examples of aperiodic Wang tiles. We show that a certain subset of the Kari-Culik tilings, namely those whose rows can be interpreted as Sturmian sequences (rotation sequences), is minimal with respect to the Z^2 action of translation. We give a characterization of this space as a skew product as well as explicit bounds on the waiting time between occurrences of m × n configurations. / Graduate / 0405

Embedded/real-time systems are becoming ubiquitous in today's world and their pervasive nature is increasing with the advent of cyber-physical systems. Providing temporal guarantees is paramount in such systems. Most of the normal operation in real-time systems is modelled using periodic tasks. Event-driven behaviour is modelled using aperiodic jobs. To ensure an acceptable Quality of Service for aperiodic jobs without jeopardizing safety of periodic tasks, aperiodic servers were introduced [2], [3]. Aperiodic servers are used to reserve a quota for the execution of aperiodic jobs. However, they do not take into account, cache-related delays that the execution of aperiodic jobs could impose on periodic tasks, thereby making their use in systems with caches unsafe. In this thesis, we introduce Cache Related Delay Servers to solve this problem. Statically, every periodic task's worst-case execution time includes a pre-determined delay quota for delay caused by aperiodic jobs. During system operation, the aperiodic server is allowed to execute only if periodic jobs that may be affected by it have sufficient delay quota to accommodate its execution. Otherwise, the priority of the aperiodic server is temporarily decreased to the level of the lowest-priority periodic job with insufficient quota, thereby ensuring safe execution of periodic tasks.

Quasicrystals are long-range ordered and yet non-periodic. This interplay results in a wealth of intriguing physical phenomena, such as the inheritance of topological properties from higher dimensions, self-similarity, and the presence of non-trivial structure on all scales. The concept of aperiodic order has been extensively studied in mathematics and geometry, exemplified by the celebrated Penrose tiling. However, the understanding of physical quasicrystals (the vast majority of them are intermetallic compounds) is still incomplete owing to their complexity, regarding both growth processes and stability. Ultracold atoms in optical lattices offer an ideal, yet untested environment for investigating quasicrystals. Optical lattices, i.e. standing waves of light, allow the defect-free formation of a large variety of potential landscapes, including quasiperiodic geometries. In recent years, optical lattices have become one of the most successful tools in the large-scale quantum simulation of condensed-matter problems. This study presents the first experimental realisation of a two-dimensional quasicrystalline potential for ultracold atoms, based on an eightfold symmetric optical lattice. It is aimed at bringing together the fields of ultracold atoms and quasicrystals - and the more general concept of aperiodic order. The first part of this thesis introduces the theoretical aspects of aperiodic order and quasicrystalline structure. The second part comprises a detailed account of the newly designed apparatus that has been used to produce quantum-degenerate gases in quasicrystalline lattices. The third and final part summarises the matter-wave diffraction experiments that have been performed in various lattice geometries. These include one- and two-dimensional simple cubic lattices, one-dimensional quasiperiodic lattices, as well as two-dimensional quasicrystalline lattices. The striking self-similarity of this quasicrystalline structure has been directly observed, in close analogy to Shechtman's very first discovery of quasicrystals using electron diffraction. In addition, an in-depth study of the diffraction dynamics reveals the fundamental differences between periodic and quasicrystalline lattices, in excellent agreement with ab initio theory. The diffraction dynamics on short timescales constitutes a continuous-time quantum walk on a homogeneous four-dimensional tight-binding lattice. On the one hand, these measurements establish a novel experimental platform for investigating quasicrystals proper. On the other hand, ultracold atoms in quasicrystalline optical lattices are worth studying in their own right: Possible avenues include the observation many-body localisation and Bose glasses, as well as the creation of topologically non-trivial systems in higher dimensions.

In this thesis a new image-based texturing method has been developed. This new method allows users to synthesize tiled textures that can be mapped to any quadrilateral mesh without discontinuity or singularity. An interface has been developed that allows user control over out put textures. Three methods have been included in the interface to create a periodic looking texture for 3D models and two methods have been developed to create wallpaper images (repeating textures on a 2D surface). Using these texturing methods, texturing problems are simplified, and more time can be spent solving artistic problems.

In this thesis a new image-based texturing method has been developed. This new method allows users to synthesize tiled textures that can be mapped to any quadrilateral mesh without discontinuity or singularity. An interface has been developed that allows user control over out put textures. Three methods have been included in the interface to create a periodic looking texture for 3D models and two methods have been developed to create wallpaper images (repeating textures on a 2D surface). Using these texturing methods, texturing problems are simplified, and more time can be spent solving artistic problems.

No description available.

Real-time systems require a-priori temporal guarantees. While most of the normal operation in such a system is modeled using time-driven, hard-deadline sporadic tasks, event-driven behavior is modeled using aperiodic jobs with soft or no deadlines. To provide good Quality-of- Service for aperiodic jobs in the presence of sporadic tasks, aperiodic servers were introduced. Aperiodic servers act as a sporadic task and reserve a quota periodically to serve aperiodic jobs. The use of aperiodic servers in systems with caches is unsafe because aperiodic servers do not take into account, the indirect cache-related preemption delays that the execution of aperiodic jobs might impose on the lower-priority sporadic tasks, thus jeopardizing their safety. To solve this problem, we propose an enhancement to the aperiodic server that we call a Cache Delay Server. Here, each lower-priority sporadic task is assigned a delay quota to accommodate the cache-related preemption delay imposed by the execution of aperiodic jobs. Aperiodic jobs are allowed to execute at their assigned server priority only when all the active lower-priority sporadic tasks have a sufficient delay quota to accommodate it. Simulation results demonstrate that a Cache Delay Server ensures the safety of sporadic tasks while providing acceptable Quality-of-Service for aperiodic jobs. We propose a Integer Linear Program based approach to calculate delay quotas for sporadic tasks within a task set where Cache Delay Servers have been pre-assigned. We then propose algorithms to determine Cache Delay Server characteristics for a given sporadic task set. Finally, we extend the Cache Delay Server concept to multi-core architectures and propose approaches to schedule aperiodic jobs on appropriate Cache Delay Servers. Simulation results demonstrate the effectiveness of all our proposed algorithms in improving aperiodic job response times while maintaining the safety of sporadic task execution.

No description available.

No description available.

No description available.