Suppose G is a graph with n vertices and m edges. Let n¡¬ be the maximum number of vertices in an induced bipartite subgraph of G and let m¡¬ be the maximum number of edges in a spanning bipartite subgraph of G. Then b(G) = m¡¬/m is called the bipartite density of G, and b∗(G) = n¡¬/n is called the bipartite ratio of G. It is proved in [18] that if G is a 2-connected triangle-free subcubic graph, then apart from seven exceptional graphs, we have b(G) ≥ 17/21. If G is a 2-connected triangle-free subcubic graph, then b∗(G) ≥ 5/7 provided that G is not the Petersen graph and not the dodecahedron. These two results are consequences of a more technical result which is proved by induction: If G is a 2-connected triangle-free subcubic graph with minimum degree 2, then G has an induced bipartite subgraph H with |V (H)| ≥ (5n3 + 6n2 + ǫ(G))/7, where ni = ni(G) are the number of degree i vertices of G, and ǫ(G) ∈ {−2,−1, 0, 1}. To determine ǫ(G), four classes of graphs G1, G2, G3 and F-cycles are onstructed. For G ∈ Gi, we have ǫ(G) = −3 + i and for an F-cycle G, we have ǫ(G) = 0. Otherwise, ǫ(G) = 1. To construct these graph classes, eleven graph operations are used. This thesis studies the structural property of graphs in G1, G2, G3. First of all, a computer algorithm is used to generate all the graphs in Gi for i = 1, 2, 3. Let P be the set of 2-edge connected subcubic triangle-free planar graphs with minimum degree 2. Let G¡¬ 1 be the set of graphs in P with all faces of degree 5, G¡¬2 the set of graphs in P with all faces of degree 5 except that one face has degree 7, and G¡¬3 the set of graphs in P with all faces of degree 5 except that either two faces are of degree 7 or one face is of degree 9. By checking the graphs generated by the computer algorithm, it is easy to see that Gi ⊆ G¡¬i for i = 1, 2, 3. The main results of this thesis are that for i = 1, 2, Gi = G¡¬i and G¡¬3 = G3 ¡åR, where R is a set of nine F-cycles.

Absolute quantification of I-131 activity in tumors and normal tissue is essential for internal dose estimates and is one of the greatest challenges in contemporary Nuclear Medicine. Therefore, for obtaining more accurate results, the most suitable and clinical applicable method, for both patients and scientists, for the quantification of I-131 has to be found. In the theoretical part of this project, the two methods for the quantitation are explained and analyzed as well as the factors that affect the obtained clinical image followed by the corrections that must be applied for gaining a more discrete image. For the planar technique, two methods for scatter and septal penetration correction were used applying two different matrix sizes specifically, 256x256 and 512x512 matrix sizes. On the other hand, for the SPECT technique, a 64x64 matrix size was used and a comparison between an auto-ROI and a same pattern of ROI segmentation that was produced manually for the purpose of this project, was performed. Furthermore, for patient specific dosimetry the total absorbed dose for the thyroid gland was calculated. Figures and tables for planar and SPECT technique were produced for each matrix size along with the final curve of the absorbed dose of the patient. For the planar technique it was observed that the method proposed by Anne Larsson, according to statistical analysis, gives results with lower statistical errors than the method proposed by Macey. In addition, 512x512 matrix size for the planar technique leads to lower statistical errors when compared with the results from using the 256x256 matrix size. Moreover, for the SPECT technique, the auto-sum that the software of the camera provides gives exactly the same results compared with the manual-sum. Finally, the total thyroid absorbed dose was 45,0472 Gy a result very close to the ICRU proposed dose value. / Η απόλυτη ποσοτικοποίηση της ενεργότητας του I-131 σε καρκινικούς και σε φυσιολογικούς ιστούς είναι απαραίτητη για την εκτίμηση της εσωτερικής δόσης και αποτελεί μία από τις μεγαλύτερες προκλήσεις της σύγχρονης Πυρηνικής Ιατρικής. Για τον λόγο αυτό, για την απόκτηση ακριβέστερων αποτελεσμάτων, θα πρέπει να βρεθεί η καταλληλότερη μέθοδος ποσοτικοποίησης, που να είναι εύχρηστη τόσο για τον ασθενή όσο και για το επιστημονικό προσωπικό των νοσοκομείων. Στο θεωρητικό μέρος της παρούσας εργασίας, αναλύονται οι δύο μέθοδοι ποσοτικοποίησης καθώς και οι παράγοντες που επηρεάζουν την ποιότητα της εικόνας ακολουθούμενοι από τους διορθωτικούς παράγοντες για την απόκτηση μιας ευκρινέστερης κλινικής εικόνας. Όσον αφορά την Planar τεχνική, περιγράφονται δύο μέθοδοι για διόρθωση της σκεδαζόμενης ακτινοβολίας καθώς επίσης και του septal penetration στην τελική εικόνα, χρησιμοποιώντας δύο διαφορετικά μεγέθη μήτρας, συγκεκριμένα, 256x256 και 512x512. Για την τεχνική SPECT, το μέγεθος μήτρας που χρησιμοποιήθηκε ήταν 64x64 και στην συνέχεια έγινε σύγκριση της περιοχής ενδιαφέροντος (ROI) που έδωσε το λογισμικό της κάμερας με την περιοχή ενδιαφέροντος που σχεδιάστηκε χειροκίνητα για την συγκεκριμένη μελέτη. Επίσης, υπολογίστηκε η συνολική απορροφούμενη δόση στον θυρεοειδή αδένα ενός ασθενούς. Κατασκευάστηκαν διαγράμματα τόσο για την planar όσο και για την SPECT τεχνική και για τα δύο μεγέθη μήτρας καθώς και η καμπύλη απορροφούμενης δόσης του ασθενούς. Παρατηρήθηκε ότι στην planar τεχνική, η μέθοδος που προτείνεται από την Anne Larsson, δίνει καλύτερα αποτελέσματα και μικρότερα σφάλματα συγκρινόμενη με την μέθοδο που προτείνεται από τον Macey. Επιπλέον, παρατηρήθηκε ότι το 512x512 μέγεθος μήτρας δίνει πιο ευκρινή εικόνα απ’ ότι το 256x256 μέγεθος μήτρας. Επίσης, με την SPECT τεχνική, παρατηρείται ότι χρησιμοποιώντας το αυτόματο άθροισμα των κρούσεων που δίνεται από το λογισμικό της κάμερας προκύπτουν τα ίδια ακριβώς αποτελέσματα με την χρήση του αθροίσματος των κρούσεων όταν αυτό γίνεται χειροκίνητα. Τέλος, η τελική απορροφούμενη δόση του ασθενούς υπολογίστηκε 45,0472 Gy, ένα αποτέλεσμα που δεν αποκλείει από τις τιμές που προτείνονται από το ICRU.

The central topic of this thesis is Schnyder's Theorem. Schnyder's Theorem provides a characterization of planar graphs in terms of their poset dimension, as follows: a graph G is planar if and only if the dimension of the incidence poset of G is at most three. One of the implications of the theorem is proved by giving an explicit mapping of the vertices to R^2 that defines a straightline embedding of the graph. The other implication is proved by introducing the concept of normal labelling. Normal labellings of plane triangulations can be used to obtain a realizer of the incidence poset. We present an exposition of Schnyder’s theorem with his original proof, using normal labellings. An alternate proof of Schnyder’s Theorem is also presented. This alternate proof does not use normal labellings, instead we use some structural properties of a realizer of the incidence poset to deduce the result. Some applications and a generalization of one implication of Schnyder’s Theorem are also presented in this work. Normal labellings of plane triangulations can be used to obtain a barycentric embedding of a plane triangulation, and they also induce a partition of the edge set of a plane triangulation into edge disjoint trees. These two applications of Schnyder’s Theorem and a third one, relating realizers of the incidence poset and canonical orderings to obtain a compact drawing of a graph, are also presented. A generalization, to abstract simplicial complexes, of one of the implications of Schnyder’s Theorem was proved by Ossona de Mendez. This generalization is also presented in this work. The concept of order labelling is also introduced and we show some similarities of the order labelling and the normal labelling. Finally, we conclude this work by showing the source code of some implementations done in Sage.

This thesis deals with planar drawings of planar graphs such that each interior face has a prescribed area. Our work is divided into two main sections. The rst one deals with straight-line drawings and the second one with orthogonal drawings. For straight-line drawings, it was known that such drawings exist for all planar graphs with maximum degree 3. We show here that such drawings exist for all planar partial 3-trees, i.e., subgraphs of a triangulated planar graph obtained by repeatedly inserting a vertex in one triangle and connecting it to all vertices of the triangle. Moreover, vertices have rational coordinates if the face areas are rational, and we can bound the resolution. For orthogonal drawings, we give an algorithm to draw triconnected planar graphs with maximum degree 3. This algorithm produces a drawing with at most 8 bends per face and 4 bends per edge, which improves the previous known result of 34 bends per face. Both vertices and bends have rational coordinates if the face areas are rational.

This dissertation studies the colored Jones polynomial of knots and links, colored by representations of simple Lie algebras, and the stability of its coefficients. Chapter 1 provides an explicit formula for the second plethysm of an arbitrary representation of sl3. This allows for an explicit formula for the colored Jones polynomial of the trefoil, and more generally, for T(2,n) torus knots. This formula for the sl3 colored Jones polynomial of T(2,n)$ torus knots makes it possible to verify the Degree Conjecture for those knots, to efficiently compute the sl3 Witten-Reshetikhin-Turaev invariants of the Poincare sphere, and to guess a Groebner basis for the recursion ideal of the sl3 colored Jones polynomial of the trefoil. Chapter 2 gives a formulation of a stability conjecture for the coefficients of the colored Jones polynomial of a knot, colored by irreducible representations in a fixed ray of a simple Lie algebra. The conjecture is verified for all torus knots and all simple Lie algebras of rank 2. Chapter 3 supplies an efficient method to compute those q-series that come from planar graphs (i.e., reduced Tait graphs of alternating links) and compute several terms of those series for all graphs with at most 8 edges. In addition, a graph-theory proof of a theorem of Dasbach-Lin which identifies the coefficient of q^k in those series for k=0,1,2 in terms of polynomials on the number of vertices, edges and triangles of the graph is given. Chapter 4 provides a study of the structure of the stable coefficients of the Jones polynomial of an alternating link.The first four stable coefficients are identified with polynomial invariants of a (reduced) Tait graph of the link projection. A free polynomial algebra of invariants of graphs whose elements give invariants of alternating links is introduced which strictly refines the first four stable coefficients. It is conjectured that all stable coefficients are elements of this algebra, and experimental evidence for the fifth and sixth stable coefficient is given. The results are illustrated in tables of all alternating links with at most 10 crossings and all irreducible planar graphs with at most 6 vertices.

In this thesis we present a method for detecting planar targets in real-time on power-constrained, or low-powered, hand-held devices such as mobile phones. We adopt the feature recognition (also referred to as feature matching) approach and employ fast-to-compute local feature descriptors to establish point correspondences. To obtain a satisfactory localization accuracy, most local feature descriptors seek a transformation of the input intensity patch that is invariant to various geometric and photometric deformations. Generally, such transformations are computationally intensive, hence are not ideal for real-time applications on limited hardware platforms. On the other hand, descriptors which are fast to compute are typically limited in their ability to provide invariance to a vast range of deformations. To address these shortcomings, we have developed a learning-based approach which can be applied to any local feature descriptor to increase the system’s robustness to both affine and perspective deformations. The motivation behind applying a learning-based approach is to transfer as much of the computational burden (as possible) onto an offline training phase, allowing a reduction in cost during online matching. The approach comprises of identifying keypoints which remain stable under artificially induced perspective transformations, extracting the corresponding feature vectors, and finally aggregating the feature vectors of coincident keypoints to obtain the final descriptors. We strictly focus on objects which are planar, thus allowing us to synthesize images of the object in order to capture the appearance of keypoint patches under several perspectives.

The goal of the research of this Dissertation is using actuation redundancy to reduce backlash in parallel manipulators (PMs.) Initially, 3-RRR and 3-RPR PM layouts where 3 is the number of branches, R is a revolute joint and P is a prismatic joint, are introduced. Actuated joints will later be underlined in the PM desciptions. A method for determining PM working area for rotated payload platforms, based on a mechanism inversion, is presented. Force solutions for non-redundantly actuated 3-RRR, 3-RRR, 3-RPR and 3-RPR PMs are formulated in terms of screw coordinates. The reciprocal product of screw coordinates is demonstrated to be invarient under changes in reference location and orientation. As examples, the PMs execute basic circle, logarithmic spiral and arc displacement and force trajectories. All non-redundantly-actuated PMs, encounter two backlash-prone zero-actuator-output configurations when executing any of the trajectories. Therefore, non-redundantly actuated PMs are found inadequate for precision applications. Force-uncertainties, where PMs cannot sustain or apply forces in uncertain directions, are examined. For typically actuated 3-RRR and 3-RPR PMs, force uncertainties are identified using screw system arguments based on the existance of 3 actuated forces forming degenerate (rank = 2) planar pencils of forces. These degenerate force pose make arbitrary force and moment application impossible and cause singularities in the force solutions. The working area of the 3-RRR PM is found compatible with all trajectories. This compatibility is due to zero minimum branch length being possible with the limitless angular displacements possible with stacked R joints. In comparison, the 3-RPR PM with minimum joint lengthes imposed on the P joints, has a smaller working area, and is not compatible with any of the trajectories. A P joint modification allowing relative length minimums of zero and a compatible working area identical to the 3-RRR PM, is considered. To address inadequacies, symmetric actuation-redundant 3-RRR and 3-RPR PMs are considered. Pseudo (right Moore-Penrose) inverse of the 3×6 ARS (associated reciprocal screw) matrix is considered to solve for the required actuation. This solution, while providing a minimum 2-norm of the vector of required actuator outputs, does not reduce backlash-prone configurations with all actuators still having two backlash-prone zero-output configurations. An algorithm for reducing backlash, using MATLAB’s constrained optimization routine FMINCON is applied. Minimizing the 2-norm of the vector of actuator outputs, subject to the backlash-free constraint of having outputs ≥ 0 or ≤ 0 depending on the initial values, is considered. Actuators providing the best conditioned ARS matices are utilized for the particular solutions. / Graduate

Endothelial cells (ECs) respond to flow but the exact mechanism producing alignment is not completely understood. We characterized EC alignment in microfluidic channels, 4 mm wide by 350 um high, to generate shear of 20 dynes / cm2 across the cell surface. In microchannels, ECs aligned perpendicular under flow. Analytical tools were developed to quantify nuclear alignment at 67% for human umbilical vein endothelial cells (HUVECs); cell elongation under shear flow shifted aspect ratio from 2.41 to 2.86. We next sought to probe the mechanism through which ECs communicate during realignment. The planar cell polarity (PCP) signaling pathway is involved in cell organization and coordination during development. A number of genes are known to affect the formation and organization of cellular structures through PCP signaling in human ECs. Higher expression of Vangl1 and Dvl1 proteins did not alter cell reorganization; knockdown of Vangl1 expression decreased EC alignment.

Endothelial cells (ECs) respond to flow but the exact mechanism producing alignment is not completely understood. We characterized EC alignment in microfluidic channels, 4 mm wide by 350 um high, to generate shear of 20 dynes / cm2 across the cell surface. In microchannels, ECs aligned perpendicular under flow. Analytical tools were developed to quantify nuclear alignment at 67% for human umbilical vein endothelial cells (HUVECs); cell elongation under shear flow shifted aspect ratio from 2.41 to 2.86. We next sought to probe the mechanism through which ECs communicate during realignment. The planar cell polarity (PCP) signaling pathway is involved in cell organization and coordination during development. A number of genes are known to affect the formation and organization of cellular structures through PCP signaling in human ECs. Higher expression of Vangl1 and Dvl1 proteins did not alter cell reorganization; knockdown of Vangl1 expression decreased EC alignment.

During neuronal development, the axonal and dendritic projections are polarized and oriented along specific body axis. To further explore the molecular basis of neuritogenesis in vivo, we used the nematode Caenorhabditis elegans as a developmental model and performed a forward genetic screen to identify genes that specify the polarity of neurite outgrowth. We examined the VC4 and VC5 neurons, members of the six VC motor neurons using the Pcat-1::gfp transgene cyIs4. The VC motor neurons are ventrally located neurons that extend two processes. VC1, VC2, VC3, and VC6 extend axons along the anterior-posterior (A/P) axis; VC4 and VC5 extend axons around the vulva along a mediolateral left-right (L/R) axis perpendicular to the A/P axis. We identified and showed that vang-1/Van Gogh, a core component of planar cell polarity (PCP) signalling pathway, acts cell-autonomously in VC4 and VC5 neurons and non-autonomously from the epithelial cells to restrict neurite formation along the A/P axis. vang-1 mutant animals display ectopic neurites along the A/P axis. Using a candidate gene approach, we further identified and revealed two additional core members of PCP signalling, Prickle (PRKL-1) and Dishevelled (DSH-1), to play a role in A/P-directed neurite suppression. We also showed prkl-1 and dsh-1 genetically interact with vang-1 and VANG-1 is required to suppress A/P-directed neurite outgrowth from larval stage 4 to adulthood. Overexpression of VANG-1 results in a loss-of-function (lof) phenotype, suggesting that an appropriate level of VANG-1 activity is important. Additionally, vang-1/prkl-1, and dsh-1 may interact in parallel pathways. Our findings implicate PCP genes to play a previously unidentified role in maintaining polarized neuronal morphology by inhibiting neuronal outgrowth responses to environmental cues.