Thesis advisor: Martin Bridgeman / We begin this dissertation by studying the relationship between the Poincaré metric of a simply connected domain Ω ⊂ ℂ and the geometry of Dome(Ω), the boundary of the convex hull of its complement. Sullivan showed that there is a universal constant K[subscript]eq[subscript] such that one may find a conformally natural K[subscript]eq[subscript]-quasiconformal map from Ω to Dome(Ω) which extends to the identity on ∂Ω. Explicit upper and lower bounds on K[subscript]eq[subscript] have been obtained by Epstein, Marden, Markovic and Bishop. We improve upon these upper bounds by showing that one may choose K[subscript]eq[subscript] ≤ 7.1695. As part of this work, we provide stronger criteria for embeddedness of pleated planes. In addition, for Kleinian groups Γ where N = ℍ³/Γ has incompressible boundary, we give improved bounds for the average bending on the convex core of N and the Lipschitz constant for the homotopy inverse of the nearest point retraction. In the second part of this dissertation, we prove an extension of Basmajian's identity to n-Hitchin representations of compact bordered surfaces. For 3-Hitchin representations, we provide a geometric interpretation of this identity analogous to Basmajian's original result. As part of our proof, we demonstrate that for a closed surface, the Lebesgue measure on the Frenet curve of an n-Hitchin representation is zero on the limit set of any incompressible subsurface. This generalizes a classical result in hyperbolic geometry. In our final chapter, we prove the Bridgeman-Kahn identity for all finite volume hyperbolic n-manifolds with totally geodesic boundary. As part of this work, we correct a commonly referenced expression of the volume form on the unit tangent bundle of ℍⁿ in terms of the geodesic end point parametrization. / Thesis (PhD) — Boston College, 2016. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Mathematics.

Classification learning is dominated by systems which induce large numbers of small axis-orthogonal decision surfaces. This strongly biases such systems towards particular hypothesis types but there is reason believe that many domains have underlying concepts which do not involve axis orthogonal surfaces. Further, the multiplicity of small decision regions mitigates against any holistic appreciation of the theories produced by these systems, notwithstanding the fact that many of the small regions are individually comprehensible. This thesis investigates modeling concepts as large geometric structures in n-dimensional space. Convex hulls are a superset of the set of axis orthogonal hyperrectangles into which axis orthogonal systems partition the instance space. In consequence, there is reason to believe that convex hulls might provide a more flexible and general learning bias than axis orthogonal regions. The formation of convex hulls around a group of points of the same class is shown to be a usable generalisation and is more general than generalisations produced by axis-orthogonal based classifiers, without constructive induction, like decision trees, decision lists and rules. The use of a small number of large hulls as a concept representation is shown to provide classification performance which can be better than that of classifiers which use a large number of small fragmentary regions for each concept. A convex hull based classifier, CH1, has been implemented and tested. CH1 can handle categorical and continuous data. Algorithms for two basic generalisation operations on hulls, inflation and facet deletion, are presented. The two operations are shown to improve the accuracy of the classifier and provide moderate classification accuracy over a representative selection of typical, largely or wholly continuous valued machine learning tasks. The classifier exhibits superior performance to well-known axis-orthogonal-based classifiers when presented with domains where the underlying decision surfaces are not axis parallel. The strengths and weaknesses of the system are identified. One particular advantage is the ability of the system to model domains with approximately the same number of structures as there are underlying concepts. This leads to the possibility of extraction of higher level mathematical descriptions of the induced concepts, using the techniques of computational geometry, which is not possible from a multiplicity of small regions.

We describe our overall approach to building robot vision systems, and the conceptual systems architecture as a network of agents, which run in parallel, and cooperate to achieve the system’s goals. We present the current state of the 3D Feature-Based Tracker, a robot vision system for tracking and segmenting the 3D motion of objects using image input from a calibrated stereo pair of video cameras. The system runs in a multi-level cycle of prediction and verification or correction. The currently modelled 3D positions and velocities of the feature points are extrapolated a short time into the future to yield predictions of 3D position. These 3D predictions are projected into the two stereo views, and are used to guide a fast and highly focused visual search for the feature points. The image positions at which the features are re-acquired are back-projected in 3D space in order to update the 3D positions and velocities. At a higher level, features are dynamically grouped into clusters with common 3D motion. Predictions from the cluster level can be fed to the lower level to correct errors in the point-wise tracking.

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Η διπλωματική αυτή εργασία έχει ως αντικείμενο μελέτης εφαρμογές / διεργασίες στα ρομποτικά σμήνη. Οι εφαρμογές / διεργασίες που εξετάζονται είναι η καλυψιμότητα χώρου, η διατήρηση συνδεσιμότητας και τα παιχνίδια κυνηγού - κυνηγούμενου. Συγκεκριμένα, εξετάζονται δύο μέθοδοι καλυψιμότητας χώρου: κίνηση προς το κεντροειδές και κίνηση προς τη βάθμωση. Εισάγονται οι επικοινωνιακοί περιορισμοί και προτείνονται περιορισμοί κίνησης που εξασφαλίζουν διατήρηση N-hop συνδεσιμότητας, οι οποίοι έπειτα προσαρμόζονται στην καλυψιμότητα χώρου. Αναλύονται δύο τακτικές κυνηγού: οι έμμεσοι κυνηγοί και η τακτική περικύκλωσης, και προτείνονται δύο τακτικές για τους κυνηγούμενους. Η πρώτη αποτελεί μία παθητική αποφυγή των κυνηγών, ενώ η δεύτερη μια πιο δυναμική συνεργατική τακτική. Τέλος, προσομοιώσεις συνοδεύουν όλα τα παραπάνω μέρη, που επιβεβαιώνουν την αποτελεσματικότητά τους και προσφέρουν σύγκριση των διάφορων μεθόδων για την κάθε εφαρμογή. / This thesis addresses applications / tasks of robotic swarms. The applications / tasks presented are the area coverage, communication maintenance and pursuer - evasion games. Two methods for area coverage are examined; centroid and gradient movement. Communication constraints are considered and movement constraints are proposed, such that the swarm's N-hop connectivity is retained. Then the contraints are adapted for the area coverage problem. After that, two pursuing tactics are examined; the indirect pursuers and the blanket movement, and two evader tactics are proposed. The first is a passive pursuers avoidance, while the second is a more dynamic collaborative tactic. Finally, simulations accompanying the above sections, confirm the efficiency and offer a comparison between the different methods in each application.

H διατριβή διερευνά προβλήματα αποδοτικής οργάνωσης χωροταξικών δεδομένων, προτείνει συγκεκριμένες δενδρικές δομές για τη διαχείρισή τους και, τέλος, δίνει παραδείγματα χρήσης τους σε ειδικές περιοχές εφαρμογών. Το πρώτο κεφάλαιο ασχολείται με το γεωμετρικό πρόβλημα της εύρεσης των ισo-προσανατολισμένων ορθογωνίων που περικλείουν ένα query αντικείμενο που μπορεί να είναι ένα ισο-προσανατολισμένο ορθογώνιο είτε σημείο ή κάθετο / οριζόντιο ευθύγραμμο τμήμα. Για την επίλυσή του προτείνεται μια πολυεπίπεδη δενδρική δομή που βελτιώνει τις πολυπλοκότητες των προηγούμενων καλύτερων λύσεων. Το δεύτερο κεφάλαιο εξετάζει το πρόβλημα της ανάκτησης σημείων σε πολύγωνα. H προτεινόμενη γεωμετρική δομή είναι επίσης πολυεπίπεδη και αποδοτική όταν το query πολύγωνο έχει συγκεκριμένες ιδιότητες. Το τρίτο κεφάλαιο ασχολείται με την εφαρμογή δενδρικών δομών σε δύο βιομηχανικά προβλήματα. Το πρώτο αφορά στη μείωση της πολυπλοκότητας ανίχνευσης συγκρούσεων κατά την κίνηση ενός ρομποτικού βραχίονα σε μια επίπεδη σκηνή με εμπόδια. Ο αλγόριθμος επίλυσης κάνει χρήση μιας ουράς προτεραιότητας και μιας UNION-FIND δομής ενώ αξιοποιεί γνωστές δομές και αλγόριθμους της Υπολογιστικής Γεωμετρίας όπως υπολογισμός κυρτών καλυμμάτων, έλεγχος polygon inclusion, κλπ. Το δεύτερο πρόβλημα ασχολείται με το σχεδιασμό απαιτήσεων υλικών (MRP) σε ένα βιομηχανικό σύστημα παραγωγής. Για το σκοπό αυτό αναπτύχθηκε ένας MRP επεξεργαστής που χρησιμοποιεί διασυνδεμένες λίστες και εκτελείται στην κύρια μνήμη για να είναι αποδοτικός. Το τελευταίο κεφάλαιο εξετάζει το πρόβλημα του ελέγχου της παραγωγής και συγκεκριμένα της δρομολόγησης εργασιών. Στο πλαίσιο αυτό σχεδιάστηκε και υλοποιήθηκε ένα ευφυές σύστημα δρομολόγησης σε περιβάλλον ροής που συνδυάζει γνωσιακή τεχνολογία και προσομοίωση με on-line έλεγχο προκειμένου να υποστηρίξει το διευθυντή παραγωγής στη λήψη αποφάσεων. / Τhe dissertation examines problems of efficient organization of spatial data, proposes specific tree structures for their management, and finally, gives examples of their use in specific application areas. The first chapter is about the problem of finding the iso-oriented rectangles that enclose a query object which can be an iso-oriented rectangle either a point or a vertical / horizontal line segment. A multilevel tree structure is proposed to solve the problem which improves the complexities of the best previous known solutions. The second chapter examines the problem of point retrieval on polygons. The proposed geometric structure is also multileveled and efficient when the query polygon has specific properties. The third chapter is about the application of tree structures in two manufacturing problems. The first one concerns the reduction in the complexity of collision detection as a robotic arm moves on a planar scene with obstacles. For the solution a priority queue and a UNION-FIND structure are used, whereas known data structures and algorithms of Computational Geometry such as construction of convex hulls, polygon inclusion testing, etc. are applied. The second problem is about material requirements planning (MRP) in a manufacturing production system. To this end an MRP processor was developed, which uses linked lists and runs in main memory to retain efficiency. The last chapter examines the production control problem, and more specifically the job scheduling problem. In this context, an intelligent scheduling system was designed and developed for flow shop production control which combines knowledge-based technology and simulation with on-line control in order to support the production manager in decision making.