Recently, a multi-resolution, hierarchical version of the hexagon mesh was introduced. In order for computers to display this mesh of hexagons, it must be triangulated. This thesis describes an algorithm to triangulate the multi-resolution, hierarchical hexagon mesh. The triangulation provides good triangle strips, which result in efficient rendering of the hexagon mesh, and well-proportioned triangles, which avoid rendering artifacts. This algorithm was designed for the rendering of hexagon meshes on the globe, though it currently does not support the pentagons of the truncated icosahedron. / Thesis (Master, Computing) -- Queen's University, 2009-01-21 12:11:02.532

All-optical devices allow improvements in the speed of optical communication and computation systems by avoiding the conversion between the optical and electronic domains. The focus of this thesis is the experimental investigation of a new type of all-optical switch that is based on the control of optical patterns formed by nonlinear interactions between light and matter. The all-optical switch consists of a pair of light beams that counterpropagate through warm rubidium vapor. These beams induce a nonlinear optical instability that gives rise to mirrorless parametric self-oscillation and generates light in the state of polarization that is orthogonal to that of the pump beams. In the far-field, the generated light forms patterns consisting of two or more spots. To characterize this instability, I observe experimentally the amount of generated power and the properties of the generated patterns as a function of pump beam intensity, frequency, and size. Near an atomic resonance, the instability has a very low threshold: with less than 1~mW of total pump power, >3~$\mu$W of power is generated. To apply this system to all-optical switching, I observe that the orientation of the generated patterns can be controlled by introducing a symmetry-breaking perturbation to the system. A perturbation in the form of a weak switch beam injected into the nonlinear medium is suitable for controlling the orientation of the generated patterns. The device operates as a switch where each state of the pattern orientation corresponds to a state of the switch, and spatial filtering of the generated pattern defines the output ports of the device. Measurements of the switch response show that it can be actuated by as few as 600~photons. For a switch beam with 1/e field radius $w_0=185\,\mu$m, 600 photons correspond to $5.4\times10^{-4}$ photons/\lambdasquared which is comparable to the best reported results from all-optical switches based on electromagnetically-induced transparentcy (EIT). This approach to all-optical switching operates at very low light levels and exhibits cascadability and transistorlike response. Furthermore, the sensitivity is comparable to switches using cold-atom EIT or cavity quantum-electrodynamics techniques but is achieved with a simpler system, requiring only one optical frequency and occurring in warm atomic vapor. I develop a numerical model for the switch that exhibits patterns that rotate in the presence of a weak applied optical field. Results from this model, and from my experiment, show that the switch response time increases as the input power decreases. I propose that this increase is due to critical slowing down (CSD). Mapping the pattern orientation to a simple one-dimensional system shows that CSD can account for the observed increase in response time at low input power. The ultimate performance of the device is likely limited by CSD and I conclude that the minimum number of photons capable of actuating the switch is between 400 and 600 photons. / Dissertation

In 1948 Besicovitch proved that an affine image of a regular hexagon may be inscribed into an arbitrary planar convex body. We prove Besicovitch's result using a variational approach based on special approximation by triangles and generalize the Besicovitch theorem to a certain new class of hexagons. We survey the results on the Banach-Mazur distance between different classes of convex bodies. We hope that our generalization of the Besicovitch theorem may become useful for estimation of the Banach-Mazur distance between planar convex bodies. We examined our special approximation by triangles in some specific cases, and it showed a noticeable improvement in comparison with known general methods. We also consider the Banach-Mazur distance between a simplex and an arbitrary convex body in the three-dimensional case. Using the idea of an inscribed simplex of maximal volume, we obtain a certain related algebraic optimization problem that provides an upper estimate.

In 1948 Besicovitch proved that an affine image of a regular hexagon may be inscribed into an arbitrary planar convex body. We prove Besicovitch's result using a variational approach based on special approximation by triangles and generalize the Besicovitch theorem to a certain new class of hexagons. We survey the results on the Banach-Mazur distance between different classes of convex bodies. We hope that our generalization of the Besicovitch theorem may become useful for estimation of the Banach-Mazur distance between planar convex bodies. We examined our special approximation by triangles in some specific cases, and it showed a noticeable improvement in comparison with known general methods. We also consider the Banach-Mazur distance between a simplex and an arbitrary convex body in the three-dimensional case. Using the idea of an inscribed simplex of maximal volume, we obtain a certain related algebraic optimization problem that provides an upper estimate.

In geometry we will consider n-dimensional generalizations of the Power of a Point Theorem and of Pascal's Hexagon Theorem. In generalizing the Power of a Point Theorem, we will consider collections of cones determined by the intersections of an (n-1)-sphere and a pair of hyperplanes. We will then use these constructions to produce an n-dimensional generalization of Pascal's Hexagon Theorem, a classical plane geometry result which states that "Given a hexagon inscribed in a conic section, the three pairs of continuations of opposite sides meet on a straight line." Our generalization of this theorem will consider a pair of n-simplices intersecting an (n-1)-sphere, and will conclude with the intersections of corresponding faces lying in a hyperplane. In graph theory we will explore the interaction between zero forcing and cut-sets. The color change rule which lies at the center of zero forcing says "Suppose that each of the vertices of a graph are colored either blue or white. If u is a blue vertex and v is its only white neighbor, then u can force v to change to blue." The concept of zero forcing was introduced by the AIM Minimum Rank - Special Graphs Work Group in 2007 as a way of determining bounds on the minimum rank of graphs. Later, Darren Row established results concerning the zero forcing numbers of graphs with a cut-vertex. We will extend his work by considering graphs with arbitrarily large cut-sets, and the collections of components they yield, to determine results for the zero forcing numbers of these graphs.

Denna hexagon och diamantformade låg- och mellanstadie skola har 14 skolklasser(varav 2 förskoleklasser), kapacitet för ca. 400 elever och 50 anställda.Jag ville försöka skapa en korridorslös skola med mycket ljus och öppenhet inuti. Varje innervägg är icke bärande, vilket gör att man på sikt kan göra om rummen enlig behov.  Formen kommer dels ifrån att området har olika typer av huskroppar och dels ifrån att jag inte ville jobba med traditionella fyrkantiga rum. Efter arbete med teknikmodeller och undersökning av programmets olika rumsstorlekar kom jag fram till ett hexagoniskt grid (utifrån basrummens storlek på 60 m2) med ett liksidigt triangulärt grid inbakat där varje triangel är 10 m2. Detta började då likna mer ett diamantgrid, därav namnet på projektet. Varje hörn av 60 m2 hexagonerna har bärande pelare och i taket går bärandebalkar samt kl-skivor. För att leka mera med byggnadens diamantgrids former har även taken vinklats snett uppåt. Detta ger mer ljus in, via takfönster till basrummen och rum längre in i byggnaden. Hexagon och diamantgridet fortsätter ut på skolgården i placering av aktivitet och träd.På grund av höjdskillnader på tomten hoppar byggnaden 1.5 meter uppåt på norra sidan. Idrottshallen har jag valt att placera nere i tomtens södra hörn för att skapa en stor skolgård med en mindre fotbollsplan. I huvudbyggnaden finns det terasser utomhus mot norr på våning två och tre för att utöka skolgårdsytan och för att ge byggnaden fler öppningar utåt. Stora matsalens hexagonform fortsätter uppåt och är öppet alla tre våningar. Här går också byggnadens huvudtrappa som länkar alla våningar med varandra. Ovanför denna är taket glasat eftersom byggnaden är bred och detta ger in ljus till kommunikationsutrymmena på alla våningar i byggnaden.Byggnaden är helt i trä (bortsett från betong i mark och i installations golvet), har frånluft i trapphusen och möjlighet till långsiktig omformning av rum enligt behov och aktivitet. Byggnaden strävar efter att vara ekologiskt hållbar genom val av material, funktion och användning. / Denna hexagon och diamantformade låg- och mellanstadie skola har 14 skolklasser(varav 2 förskoleklasser), kapacitet för ca. 400 elever och 50 anställda.Jag ville försöka skapa en korridorslös skola med mycket ljus och öppenhet inuti. Varje innervägg är icke bärande, vilket gör att man på sikt kan göra om rummen enlig behov.  Formen kommer dels ifrån att området har olika typer av huskroppar och dels ifrån att jag inte ville jobba med traditionella fyrkantiga rum. Efter arbete med teknikmodeller och undersökning av programmets olika rumsstorlekar kom jag fram till ett hexagoniskt grid (utifrån basrummens storlek på 60 m2) med ett liksidigt triangulärt grid inbakat där varje triangel är 10 m2. Detta började då likna mer ett diamantgrid, därav namnet på projektet. Varje hörn av 60 m2 hexagonerna har bärande pelare och i taket går bärandebalkar samt kl-skivor. För att leka mera med byggnadens diamantgrids former har även taken vinklats snett uppåt. Detta ger mer ljus in, via takfönster till basrummen och rum längre in i byggnaden. Hexagon och diamantgridet fortsätter ut på skolgården i placering av aktivitet och träd.På grund av höjdskillnader på tomten hoppar byggnaden 1.5 meter uppåt på norra sidan. Idrottshallen har jag valt att placera nere i tomtens södra hörn för att skapa en stor skolgård med en mindre fotbollsplan. I huvudbyggnaden finns det terasser utomhus mot norr på våning två och tre för att utöka skolgårdsytan och för att ge byggnaden fler öppningar utåt. Stora matsalens hexagonform fortsätter uppåt och är öppet alla tre våningar. Här går också byggnadens huvudtrappa som länkar alla våningar med varandra. Ovanför denna är taket glasat eftersom byggnaden är bred och detta ger in ljus till kommunikationsutrymmena på alla våningar i byggnaden.Byggnaden är helt i trä (bortsett från betong i mark och i installations golvet), har frånluft i trapphusen och möjlighet till långsiktig omformning av rum enligt behov och aktivitet. Byggnaden strävar efter att vara ekologiskt hållbar genom val av material, funktion och användning.

The discovery of integrability in planar N=4 super Yang-Mills and ABJM has enabled a precise study of AdS/CFT. In the past decade integrability has been successfully applied to the spectrum of anomalous dimensions, which can now be obtained at any value of the coupling. However, in order to solve conformal field theories one also needs to understand their structure constants. Recently, there has been great progress in this direction with the all-loop proposal of Basso, Komatsu and Vieira. But there is still much to understand, as it is not yet possible to use that formalism to find structure constants of short operators at strong coupling. It is important to study wrapping corrections and resum them as the TBA did for the spectrum. It is also crucial to obtain perturbative data that can be used to check if the all-loop proposal is correct or if there are new structures that need to be unveiled. In this thesis we compute several structure constants of short operators at strong coupling, including the structure constant of Konishi with half-BPS operators. Still at strong coupling, we find a relation between the building blocks of superstring amplitudes and the tensor structures allowed by conformal symmetry. We also consider the case of extremal correlation functions and the relation of their poles to mixing with double-trace operators. We also study three-point functions at weak coupling. We take the OPE limit in a four-point function of half-BPS operators in order to shed some light on the structure of five-loop wrapping corrections of the Hexagon form factors. Finally, we take the first steps in the generalization of the Hexagon programme to other theories. We find the non-extremal setup in ABJM and the residual symmetry that it preserves, which we use to fix the two-particle form factor and constrain the four-particle hexagon. Finally, we find that the Watson equations hint at a dressing phase that needs to be further investigated.

Wireless networks become more and more important in modern information systems as the last mile/meter solutions, thanks to the flexibility of mobile access to facilitate Internet access anytime, anywhere. Given the limited resources, e.g., spectrum and energy supplies, to meet the ever increasing demand for wireless data services, new approaches are beckoned to enhance the spectrum and energy efficiency. We investigate this problem from three important aspects, digital modulation, clock synchronization and concurrent transmission scheduling. The contributions of this dissertation are four-fold. First, we employ the cross-layer design to explore the spatial diversity and broadcast nature of wireless links and propose a novel network modulation scheme that can superpose the information bits of different priorities into one symbol. It offers a new dimension to improve the network throughput since we can flexibly configure the transmission according to the channels among transceivers. Moreover, it is compatible with the main-stream hardware and we just need a software upgrade to implement the idea. Second, we propose modulation schemes based on hexagonal tiling, which is known to be the most compact way of two-dimensional regular tiling. In order to fully utilize the advantage of hexagonal constellation, we employ the non-binary error controlcoding since the number of constellation points of hexagonal constellation is not necessarily to be an integer power-of-two. The feasibility of these new modulation schemes is verified by the prototype system based on the software defined radio platform USRP2 and GNU Radio. Third, to facilitate a wide range of wireless communications technologies and protocols, clock synchronization among several wireless devices is a fundamental requirement. We investigated this problem by tracing to the source of clock desynchronization, which is the clock skew. However, as shown by measurement results, the clock skew is not constant and related to the working temperature. We propose a novel clock skew estimation algorithm that can leverage the temperature information to accurately estimate the clock skew. Based on the estimation results, we propose a clock synchronization scheme that can directly remove the clock skew according to the working temperature. Fourth, the traditional time-sharing based scheduling schemes usually schedule one transmission within certain area. The emerging broadband wireless devices can dynamically adjust the transmitted data rate according to the received signal to interference and noise ratio (SINR). Allowing concurrent transmissions may be more efficient, while optimal scheduling problem for concurrent transmissions is an NP-hard problem. We propose simple yet effective heuristic algorithms that can significantly improve the system throughput with moderate computational complexity. / Graduate / 0544 / yangzhe2007@gmail.com

The aim of the diploma thesis is design and assessment of steel structure of greenhouse in botanical garden in Ostrava. The building is designed as a hexagon with a diameter 50 m, the highest point of the construction is 32,907 m. Whole structure will be made of hollow round tubes of different profiles. The support system consist of columns and truss girders. The bracing of the structure is in three levels. Main construction material is steel S355 J0+AR. The covering of the roofs and walls is made of glass panels ISOTHERM.

European Paper Wasps (Polistes dominula) are social insects that build round, symmetrical nests. Current models indicate that these wasps develop colonies by following simple heuristics based on nest stimuli. Computer simulations can model wasp behavior to imitate natural nest building. This research investigated various building heuristics through a novel Markov-based simulation. The simulation used a hexagonal grid to build cells based on the building rule supplied to the agent. Nest data was compared with natural data and through visual inspection. Larger nests were found to be less compact for the rules simulated.