The Wnt signaling network has critical roles in embryonic development and is implicated in human disease. One of the outputs of the Wnt network, called the planar cell polarity (PCP) pathway, regulates tissue polarity and directs cell migration. Core PCP components (Frizzled, Dishevelled, Prickle, Vangl, Celsr) localize asymmetrically in polarized cells and establish polarity across the tissue through protein interactions between adjacent cells. The core PCP component activate tissue-specific "effectors" which translate the signal into morphological changes. PCP is related to several disease conditions, including neural tube defects, cystic kidney disease, and cance metastasis. However, mechanisms of the PCP underlying physiological and disease-related conditions are not well understood. Here, I explore functions of the core PCP component Pk, and its relationship to disease, in the zebrafish model system. Mutations in Pk1 and Pk2 have been identified in human progressive myoclonic epilepsy patients. Pk coodinate cell movement, neuronal migration and axonal outgrowth during embryonic development. Yet, how dysfunctions of pk relates to epilepsy is unknown. Here, I show that knockdown of pk1a sensitizes the zebrafish larva to convulsant drug. To model the defects in central nervous system, I examine neurogenesis in the retina and find that both pk1a and pk2 are required for proper dendritic outgrowth in the retinal inner plexiform layer. Furthermore, I characterize the epilepsy-related mutant forms of Pk1a and Pk2. The mutant Pk1a forms show reduced ability to suppress the retinal neurogenesis defects compared to the wild-type, as well as differential ubiquitination levels. Pk2 mutant forms also show differential activities in overexpression assays and seemingly more stable proteins relative to the wild-type. Taken together, pk1a and pk2 may contribute to epilepsy by affecting neuronal patterning and thus signal processing. Another aspect of PCP function has been implicated in cilia and cilia-related disorders, also called ciliopathy. PCP effectors have been shown to modulate ciliogenesis and core PCP proteins (Vang and Dvl) regulate cilia orientation. On the other hand, cilia are not required for PCP signaling, especially asymmetric core PCP protein localization. These findings leave open the question what is the precise relationship between PCP and cilia. The Bardet Biedl Syndrome (BBS) is a type of ciliopathy that leads to obesity, retinitis pigmentosa, polydactyly, mental retardation and other symptons. A subset of BBS genes share similar knockdown phenotype in cell migration as seen in PCP knockdown embryos. Shared pehnotypes have led some to proposethat PCP and bbs genes may interact. Yet a direct relationship has yet to be established. I examine the interaction between pk2 and a central Bbs gene, bbs7. By analyzing shared phenotypes in double knockdown embryos, I find no synergistic interaction between the two, suggesting they act in distinct pathways. Bbs regulate ciliary trafficking and in zebrafish, knockdown of bbs genes leads to delayed retrograde melanosome transport. Interestingly, I find knockdown of pk2 suppresses this retrograde transport delay. Additionally, pk2 knockdown embryos show a delay in anterograde melanosome transport. These findings highlight a new role for pk2 in intracellular transport and clarifies the relationship between PCP and BBS. In summary, my work here strengthens the link between pk mutations and human epilepsy and identifies functions of pk in retinal neurogenesis and in intracellular transport. To what extent the role of neurogenesis and intracellular transport are related is worth future study. Yet, this new information provides insights into potential mechanisms of epilepsy and the relationship between PCP and BBS.

In this thesis, we study the relationship between the geometric shape of a region in the plane, and certain probabilistic information about the behavior of Brownian particles inside the region. The probabilistic information is contained in the function h(r), called the harmonic measure distribution function. Consider a domain Ω in the plane, and fix a basepoint z0. Imagine lining the boundary of this domain with fly paper and releasing a million fireflies at the basepoint z0. The fireflies wander around inside this domain randomly until they hit a wall and get stuck in the fly paper. What fraction of these fireflies are stuck within a distance r of their starting point z0? The answer is given by evaluating our h-function at this distance; that is, it is given by h(r). In more technical terms, the h-function gives the probability of a Brownian first particle hitting the boundary of the domain Ω within a radius r of the basepoint z0. This function is dependent on the shape of the domain Ω, the location of the basepoint z0, and the radius r. The big question to consider is: How much information does the h-function contain about the shape of the domain’s boundary? It is known that an h-function cannot uniquely determine a domain, but is it possible to construct a domain that generates a given hfunction? This is the question we try to answer. We begin by giving some examples of domains with their h-functions, and then some examples of sequences of converging domains whose corresponding h-functions also converge to the h-function. In a specific case, we prove that artichoke domains converge to the wedge domain, and their h-functions also converge. Using another class of approximating domains, circle domains, we outline a method for constructing bounded domains from possible hfunctions f(r). We prove some results about these domains, and we finish with a possible for a proof of the convergence of the sequence of domains constructed.

A graph is planar if it can be drawn on a piece of paper such that no two edges cross. The smallest complete and complete bipartite graphs that are not planar are K5 and K{3,3}. A biplanar graph is a graph whose edges can be colored using red and blue such that the red edges induce a planar subgraph and the blue edges induce a planar subgraph. In this thesis, we determine the smallest complete and complete bipartite graphs that are not biplanar.

Over the past three years I have engaged in a search for a form of moving image that would serve as a medium to contain, express and communicate my concerns and ideas. My initial investigations led me to motion graphics but as my understanding of moving image broadened I came to the conclusion that the models I was examining did not fit this genre, they are something new and do not have a definition. There are conflicting ideas about what the term motion graphics means. For the purpose of clarity, I adopted Matt Frantz's definition as a start point: "designed, non-narrative, non-figurative based visuals that change over time." Motion graphics is often considered a component of a larger moving image work or a filler element between two larger works. For example, the opening moments of a film or television programme, or a swirling abstract that forms a background for an interstitial between programmes. I require a description of a moving image type of that can be used as a guide to making work. Research into the field of moving image work made by designers was conducted with grounded theory employed as the principle methodology. This research has revealed a moving image type that I refer to as "graphic film". During the past year I have identified its key characteristics. I have explored and tested the boundaries of this new genre by constructing graphic film and comparing it to previously defined forms of moving image. The outcome of this project is a comprehensive description of what graphic film is and its ten primary characteristics. This project can serve as a guide for other graphic designers who wish to make work of this type.

Les composants passifs, représentent actuellement une butée importante en terme de volume occupé, de pertes et de faisabilité de l'intégration. De nouvelles structures électromagnétiques proposent de marier les trois composants habituellement rencontrés dans les convertisseurs sous la forme d'un seul composant baptisé LCT, assurant simultanément trois fonctionnalités : inductance-condensateur-transformateur. Nous avons dimensionné deux structures intégrées de prototypes LCT (bobiné & planar), employant, tous deux, une nouvelle topologie de circuit magnétique. En nous basant sur ces réalisations de composants passifs intégrées ainsi que sur une réalisation discrète, diverses approches sont proposées pour mieux estimer les pertes dans ces composants afin de pouvoir réaliser une étude comparative sur chacun des dispositifs. En particulier, la caractérisation fine du LCT en vue d'obtenir les éléments d'un schéma équivalent constitue un moyen d'atteindre ce résultat.

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.

Given a set of points in the Euclidean plane, we are interested in its triangulations, i.e., the maximal sets of non-overlapping triangles with vertices in the given points whose union is the convex hull of the point set. With respect to the area of the triangles in a triangulation, several optimality criteria can be considered. We study two of them. The MaxMin area triangulation is the triangulation of the point set that maximizes the area of the smallest triangle in the triangulation. Similarly, the MinMax area triangulation is the triangulation that minimizes the area of the largest area triangle in the triangulation. In the case when the point set is in a convex position, we present algorithms that construct MaxMin and MinMax area triangulations of a convex polygon in $O(n^2log{n})$ time and $O(n^2)$ space. These algorithms are based on dynamic programming. They use a number of geometric properties that are established within this work, and a variety of data structures specific to the problems. Further, we study polynomial time computable approximations to the optimal area triangulations of general point sets. We present geometric properties, based on angular constraints and perfect matchings, and use them to evaluate the approximation factor and to achieve triangulations with good practical quality compared to the optimal ones. These results open new direction in the research on optimal triangulations and set the stage for further investigations on optimization of area.

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.

Given a set of points in the Euclidean plane, we are interested in its triangulations, i.e., the maximal sets of non-overlapping triangles with vertices in the given points whose union is the convex hull of the point set. With respect to the area of the triangles in a triangulation, several optimality criteria can be considered. We study two of them. The MaxMin area triangulation is the triangulation of the point set that maximizes the area of the smallest triangle in the triangulation. Similarly, the MinMax area triangulation is the triangulation that minimizes the area of the largest area triangle in the triangulation. In the case when the point set is in a convex position, we present algorithms that construct MaxMin and MinMax area triangulations of a convex polygon in $O(n^2log{n})$ time and $O(n^2)$ space. These algorithms are based on dynamic programming. They use a number of geometric properties that are established within this work, and a variety of data structures specific to the problems. Further, we study polynomial time computable approximations to the optimal area triangulations of general point sets. We present geometric properties, based on angular constraints and perfect matchings, and use them to evaluate the approximation factor and to achieve triangulations with good practical quality compared to the optimal ones. These results open new direction in the research on optimal triangulations and set the stage for further investigations on optimization of area.