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Στην παρούσα Διπλωματική Εργασία θα ασχοληθούμε με το πασίγνωστο πρόβλημα της Παραγοντοποίησης Ακεραίων. Η προσέγγιση μας είναι από τη μεριά της Επιστήμης των Υπολογιστών και της Υπολογιστικής Πολυπλοκότητας. Προσπαθούμε να κατανοήσουμε γιατί η Παραγοντοποίηση Ακεραίων θεωρείται ένα υπολογιστικά δύσκολο πρόβλημα και να δούμε τις πρακτικές εφαρμογές και συνέπειες αυτής της θεώρησης. / In the present Diploma Thesis we are going to deal with the very well known problem of Integer-Factoring. Our approach is from the side of Computer Science and Computational Complexity. We are trying to understand why Integer-Factoring is widely considered as a hard problem to solve and see the practical applications of this consideration.

This thesis describes a parallel block-structured adaptive mesh refinement (AMR) method that is employed to solve some computational aeroacoustic problems with the aim of improving the computational efficiency. AMR adaptively refines and coarsens a computational mesh along with sound propagation to increase grid resolution only in the area of interest. While sharing many of the same features, there is a marked difference between the current and the established AMR approaches. Rather than low-order schemes generally used in the previous approaches, a high-order spatial difference scheme is employed to improve numerical dispersion and dissipation qualities. To use a high-order scheme with AMR, a number of numerical issues associated with fine-coarse block interfaces on an adaptively refined mesh, such as interpolations, filter and artificial selective damping techniques and accuracy are addressed. In addition, the asymptotic stability and the transient behaviour of a high-order spatial scheme on an adaptively refined mesh are also studied with eigenvalue analysis and pseudospectra analysis respectively. In addition, the fundamental AMR algorithm is simplified in order to make the work of implementation more manageable. Particular emphasis has been placed on solving sound radiation from generic aero-engine bypass geometry with mean flow. The approach of AMR is extended to support a body-fitted multi-block mesh. The radiation from an intake duct is modelled by the linearised Euler equations, while the radiation from an exhaust duct is modelled by the extended acoustic perturbation equations to suppress hydrodynamic instabilities generated in a sheared mean flow. After solving the near-field sound solution, the associated far-field sound directivity is estimated by solving the Ffowcs Williams-Hawkings equation. The overall results demonstrate the accuracy and the efficiency of the presented AMR method, but also reveal some limitations. The possible methods to avoid these limitations are given at the end of this thesis.

This study explores complexity metrics for business artifact process models described by Case Management Model and Notation (CMMN). Process models are usually described using Business Process Management (BPM), which is a relatively mature discipline with a large number of practitioners. Over the last few decades a new way of describing data intensive business processes has emerged in BPM literature, for which traditional BPM is no longer adequate. This emerging method, used to describe more flexible processes, is called business artifacts with Guard-Stage-Milestone (GSM). The work on GSM influenced CMMN, which was created to fill a market need for more flexible case management processes for knowledge workers. Complexity metrics have been developed for traditional BPM models, such as the Business Process Model and Notation (BPMN). However, traditional BPM is not suitable for describing GSM or CMMN process models. Therefore, complexity metrics developed for traditional process models may not be applicable to business artifact process models such as CMMN. This study addresses this gap by exploring complexity metrics for business artifact process models using CMMN. The findings of this study have practical implications for the CMMN standard and for the commercial products implementing CMMN. This research makes the following contributions: • The development of a formal description of CMMN using first-order logic. • An exploration of the relationship between CMMN and GSM and the development of transformation procedures between them. • A comparison between the method complexity of CMMN and other popular process methods, including BPMN, Unified Modeling Language (UML) Activity diagrams, and Event-driven Process Charts (EPC). • The creation of a systematic literature review of complexity metrics for process models, which was conducted in order to inform the creation of CMMN metrics. • The identification of a set of complexity metrics for the CMMN standard, which underwent theoretical and empirical validation. This research advances literature in the areas of method complexity, complexity metrics for process models, declarative processes, and research on CMMN by characterizing CMMN method complexity, identifying complexity metrics for CMMN, and exploring the relationship between CMMN and GSM. / School of Computing / Ph. D. (Computer Science)