Denna rapport är resultatet av ett examensarbete som utförts isamarbete med byggföretaget K360 i Uppsala. De arbetar medkakling av badrum i renovering och nybyggnation. En fallstudiehar besvarat forskningsfrågan “Hur kan byggföretag i Sverigegenom materialval och arbetssätt i samband med kakling avbadrum, minska sina koldioxidutsläpp och sin användning avfarliga ämnen?”. Materialen avser avjämningsmassa, fästmassa ochkakel. Litteraturstudierna bygger på rapporter och skrifter frånstatliga institutioner, kommuner, företag och tidningsartiklar. Dettycks saknas forskning om avjämning och fästmassa. Litteraturenberör klimatneutral betong, farliga ämnen, avfallshantering ochföretagskultur och visar följande. Cementen i betong ger högakoldioxidutsläpp. I klimatneutral betong har delar av cementenbytts ut mot bl.a. flygaska. Den gör att betongen eller avjämningentorkar långsamt. Cementindustrin forskar på metoder för attminska sina koldioxidutsläpp. En annan studie bygger på en teoriatt normaltorkande avjämning möjliggör klimatneutraltbetongbjälklag då den fungerar som fukt- och alkalispärr. Denempiriska studien beräknas färdig 2021. Farliga ämnen skaminskas och mycket farliga ämnen ska fasas ut. Dessa ärcancerframkallande, hormonstörande, svårnedbrytbara, kraftigtallergiframkallande och förändrande av arvsmassa ellerfortplantning. Försiktighetsprincipen bör tillämpas då det saknasforskning om många ämnens miljöpåverkan i kombination medandra. I cementbaserade fästmassor och avjämningsmassor kan detfinnas plasttillsatser som medför giftiga biocider. Det finnsnaturliga alternativ. Avfall bör i första hand minimeras, sedanåteranvändas, materialåtervinnas, energiutvinnas och sistdeponeras. I K360s miljöpolicy står att de värnar ommedarbetarnas delaktighet. Litteraturen stödjer delaktighet ochyrkesstolthet som drivkraft till ett företags kvalitetsutveckling därmiljö är en del. Beslut ska baseras på fakta som kräverkommunikation men K360 saknar idag forum för miljö- ochklimatfrågor. Sju kvalitativa intervjuer har genomförts medplattsättare och projektledare. Alla intervjuade anser att miljö- ochklimatfrågor är viktigt. Samtliga plattsättare strävar efter attminska spillet genom planering och beräkning av materialåtgång.Detta blir lättare med mer erfarenhet. För att effektivt kunnaanvända kakelplattorna krävs tid för planering och logiskttänkande. Tidsbrist, standarden för att kapa kakelplattor, mått ochutformning av badrummen utgör hinder för att förebygga spill.Möjligheterna för detsamma är att i projekteringsstadiet anpassatidsplan, badrummets mått och kakel. Allt material återvinns närmöjlighet ges. Beställarens krav och pris styr materialvalet. Andrafaktorer är låg vikt och att det fungerar med befintliga material.Plattsättare och projektledare uttrycker tilltro till att materialet ärkontrollerat för farliga ämnen. Överlag saknar de kunskap ommaterialens innehåll och möjligheten att påverka materialvaletupplevs vara litet. En produktjämförelse jämför fästmassornaCentro FK# 1000 som K360 använder mycket och Haga 325 Bio-Platten- und Fliesenkleber. Produkterna är likvärdiga ochcementbaserade. Centros innehåller plasttillsatser medan Hagasbara naturliga tillsatser. Diskussionen tar upp om ett riktatlitteratursökande kan ha lett till att källor missats och att källorfrån företag kan vara vinklat. Hänvisningar från kommunen ochoberoende källor stärker informationen. Vidare diskuteras omtilliten hos informanterna till materialkontrollerna är frånsägandeav ansvar men flera uttalanden pekar på starkt ansvarskännande.Slutsatsen från produktjämförelsen är att ett byte från Centro tillHaga skulle minska potentiellt farliga ämnen. Andra slutsatsersom dras är att det saknas klimatneutral avjämningsmassa påmarknaden men det kan komma att visa sig att normaltorkandeavjämning är att föredra. Företaget saknar idag forum för internkommunikation om miljöfrågor vilket gör att information fastnarhos enskilda individer. Rapporten rekommenderar K360 att avsättaresurser för att uppdatera sig om materialinnehåll ochmiljöutveckling, skapa forum och vägar för intern kunskaps- ocherfarenhetsöverföring, sprida information om material ocharbetssätt till beställare och producenter av material, fortsätta medcementbaserad avjämning och byta mot klimatneutral cement iframtiden.

No description available.

Trashandlitter discarded on the street is a large environmental issue in Sweden and across the globe. In Swedish cities alone it is estimated that 1.8 billion articles of trash are thrown to the street each year, constituting around 3 kilotons of waste. One avenue to combat this societal and environmental problem is to use robotics and AI. A robot could learn to detect trash in the wild and collect it in order to clean the environment. A key component of such a robot would be its computer vision system which allows it to detect litter and trash. Such systems are not trivially designed or implemented and have only recently reached high enough performance in order to work in industrial contexts. This master thesis focuses on creating and analysing such an algorithm by gathering data for use in a machine learning model, developing an object detection pipeline and evaluating the performance of that pipeline based on varying its components. Specifically, methods using hyperparameter optimisation, psuedolabeling and the preprocessing methods tiling and illumination normalisation were implemented and analysed. This thesis shows that it is possible to create an object detection algorithm with high performance using currently available state-of-the-art methods. Within the analysed context, hyperparameter optimisation did not significantly improve performance and psuedolabeling could only briefly be analysed but showed promising results. Tiling greatly increased mean average precision (mAP) for the detection of small objects, such as cigarette butts, but decreased the mAP for large objects and illumination normalisation improved mAPforimagesthat were brightly lit. Both preprocessing methods reduced the frames per second that a full detector could run at whilst psuedolabeling and hyperparameter optimisation greatly increased training times. / Skräp som slängs på marken har en stor miljöpåverkan i Sverige och runtom i världen. Enbart i Svenska städer uppskattas det att 1,8 miljarder bitar skräp slängs på gatan varje år, bestående av cirka 3 kiloton avfall. Ett sätt att lösa detta samhälleliga och miljömässiga problem är att använda robotik och AI. En robot skulle kunna lära siga att detektera skräp i utomhusmiljöer och samla in den för att på så sätt rengöra våra städer och vår natur. En nyckelkomponent av en sådan robot skulle vara dess system för datorseende som tillåter den att se och hitta skräp. Sådana system är inte triviala att designa eller implementera och har bara nyligen påvisat tillräckligt hög prestanda för att kunna användas i kommersiella sammanhang. Detta masterexamensarbete fokuserar på att skapa och analysera en sådan algoritm genom att insamla data för att använda i en maskininlärningsmodell, utveckla en objektdetekterings pipeline och utvärdera prestandan när dess komponenter modifieras. Specifikt analyseras metoderna pseudomarkering, hyperparameter optimering samt förprocesseringsmetoderna kakling och ljusintensitetsnormalisering. Examensarbetet visar att det är möjligt att skapa en objektdetekteringsalgoritm med hög prestanda med hjälp av den senaste tekniken på området. Inom det undersökta sammanhanget gav hyperparameter optimering inte någon större förbättring av prestandan och pseudomarkering kunde enbart ytligt analyseras men uppvisade preliminärt lovande resultat. Kakling förbättrade resultatet för detektering av små objekt, som cigarettfimpar, men minskade prestandan för större objekt och ljusintensitetsnormalisering förbättrade prestandan för bilder som var starkt belysta. Båda förprocesseringsmetoderna minskade bildhastigheten som en detektor skulle kunna köra i och psuedomarkering samt hyperparameter optimering ökade träningstiden kraftigt.

No description available.

The interaction between the study of geometric and analytic aspects of Riemannian manifolds and that of graphs is a very amazing subject. The study of synthetic curvature notions on graphs adds new contributions to this topic. In this thesis, we mainly study two kinds of synthetic curvature notions: the Ollivier-Ricci cuvature on locally finite graphs and the combinatorial curvature on infinite semiplanar graphs. In the first part, we study the Ollivier-Ricci curvature. As known in Riemannian geometry, a lower Ricci curvature bound prevents geodesics from diverging too fast on average. We translate this Riemannian idea into a combinatorial setting using the Olliver-Ricci curvature notion. Note that on a graph, the analogue of geodesics starting in different directions, but eventually approaching each other again, would be a triangle. We derive lower and upper Ollivier-Ricci curvature bounds on graphs in terms of number of triangles, which is sharp for instance for complete graphs. We then describe the relation between Ollivier-Ricci curvature and the local clustering coefficient, which is an important concept in network analysis introduced by Watts-Strogatz. Furthermore, positive lower boundedness of Ollivier-Ricci curvature for neighboring vertices imply the existence of at least one triangle. It turns out that the existence of triangles can also improve Lin-Yau\'s curvature dimension inequality on graphs and then produce an implication from Ollivier-Ricci curvature lower boundedness to the curvature dimension inequality. The existence of triangles prevents a graph from being bipartite. A finite graph is bipartite if and only if its largest eigenvalue equals 2. Therefore it is natural that Ollivier-Ricci curvature is closely related to the largest eigenvalue estimates. We combine Ollivier-Ricci curvature notion with the neighborhood graph method developed by Bauer-Jost to study the spectrum estimates of a finite graph. We can always obtain nontrivial estimates on a non-bipartite graph even if its curvature is nonpositive. This answers one of Ollivier\'s open problem in the finite graph setting. In the second part of this thesis, we study systematically infinite semiplanar graphs with nonnegative combinatorial curvature. Unlike the previous Gauss-Bonnet formula approach, we explore an Alexandrov approach based on the observation that the nonnegative combinatorial curvature on a semiplanar graph is equivalent to nonnegative Alexandrov curvature on the surface obtained by replacing each face by a regular polygon of side length one with the same facial degree and gluing the polygons along common edges. Applying Cheeger-Gromoll splitting theorem on the surface, we give a metric classification of infinite semiplanar graphs with nonnegative curvature. We also construct the graphs embedded into the projective plane minus one point. Those constructions answer a question proposed by Chen. We further prove the volume doubling property and Poincare inequality which make the running of Nash-Moser iteration possible. We in particular explore the volume growth behavior on Archimedean tilings on a plane and prove that they satisfy a weak version of relative volume comparison with constant 1. With the above two basic inequalities in hand, we study the geometric function theory of infinite semiplanar graphs with nonnegative curvature. We obtain the Liouville type theorem for positive harmonic functions, the parabolicity. We also prove a dimension estimate for polynomial growth harmonic functions, which is an extension of the solution of Colding-Minicozzi of a conjecture of Yau in Riemannian geometry.

Understanding the connection of the atomic structure and the physical properties of materials remains one of the elementary questions of condensed-matter physics. One research line in this quest started with the discovery of quasicrystals by Shechtman et al. in 1982. It soon became clear that these materials with their 5-, 8-, 10- or 12-fold rotational symmetries, which are forbidden according to classical crystallography, can be described in terms of mathematical models for nonperiodic tilings of a plane proposed by Penrose and Ammann in the 1970s. Due to the missing translational symmetry of quasicrystals, till today only finite, relatively small systems or periodic approximants have been investigated by means of numerical calculations and theoretical results have mainly been obtained for one-dimensional systems. In this thesis we study d-dimensional quasiperiodic models, so-called labyrinth tilings, with separable Hamiltonians in the tight-binding approach. This method paves the way to study higher-dimensional, quantum mechanical solutions, which can be directly derived from the one-dimensional results. This allows the investigation of very large systems in two and three dimensions with up to 10^10 sites. In particular, we contemplate the class of metallic-mean sequences. Based on this model we focus on the electronic properties of quasicrystals with a special interest on the connection of the spectral and dynamical properties of the Hamiltonian. Hence, we investigate the characteristics of the eigenstates and wave functions and compare these with the wave-packet dynamics in the labyrinth tilings by numerical calculations and by a renormalization group approach in connection with perturbation theory. It turns out that many properties show a qualitatively similar behavior in different dimensions or are even independent of the dimension as e.g. the scaling behavior of the participation numbers and the mean square displacement of a wave packet. Further, we show that the structure of the labyrinth tilings and their transport properties are connected and obtain that certain moments of the spectral dimensions are related to the wave-packet dynamics. Besides this also the photonic properties are studied for one-dimensional quasiperiodic multilayer systems for oblique incidence of light, and we show that the characteristics of the transmission bands are related to the quasiperiodic structure. / Eine der elementaren Fragen der Physik kondensierter Materie beschäftigt sich mit dem Zusammenhang zwischen der atomaren Struktur und den physikalischen Eigenschaften von Materialien. Eine Forschungslinie in diesem Kontext begann mit der Entdeckung der Quasikristalle durch Shechtman et al. 1982. Es stellte sich bald heraus, dass diese Materialien mit ihren laut der klassischen Kristallographie verbotenen 5-, 8-, 10- oder 12-zähligen Rotationssymmetrien durch mathematische Modelle für die aperiodische Pflasterung der Ebene beschrieben werden können, die durch Penrose und Ammann in den 1970er Jahren vorgeschlagen wurden. Aufgrund der fehlenden Translationssymmetrie in Quasikristallen sind bis heute nur endliche, relativ kleine Systeme oder periodische Approximanten durch numerische Berechnungen untersucht worden und theoretische Ergebnisse wurden hauptsächlich für eindimensionale Systeme gewonnen. In dieser Arbeit werden d-dimensionale quasiperiodische Modelle, sogenannte Labyrinth-Pflasterungen, mit separablem Hamilton-Operator im Modell starker Bindung betrachtet. Diese Methode erlaubt es, quantenmechanische Lösungen in höheren Dimensionen direkt aus den eindimensionalen Ergebnissen abzuleiten und ermöglicht somit die Untersuchung von sehr großen Systemen in zwei und drei Dimensionen mit bis zu 10^10 Gitterpunkten. Insbesondere betrachten wir dabei quasiperiodische Folgen mit metallischem Schnitt. Basierend auf diesem Modell befassen wir uns im Speziellen mit den elektronischen Eigenschaften der Quasikristalle im Hinblick auf die Verbindung der spektralen und dynamischen Eigenschaften des Hamilton-Operators. Hierfür untersuchen wir die Eigenschaften der Eigenzustände und Wellenfunktionen und vergleichen diese mit der Dynamik von Wellenpaketen in den Labyrinth-Pflasterungen basierend auf numerischen Berechnungen und einem Renormierungsgruppen-Ansatz in Verbindung mit Störungstheorie. Dabei stellt sich heraus, dass viele Eigenschaften wie etwa das Skalenverhalten der Partizipationszahlen und der mittleren quadratischen Abweichung eines Wellenpakets für verschiedene Dimensionen ein qualitativ gleiches Verhalten zeigen oder sogar unabhängig von der Dimension sind. Zudem zeigen wir, dass die Struktur der Labyrinth-Pflasterungen und deren Transporteigenschaften sowie bestimmte Momente der spektralen Dimensionen und die Dynamik der Wellenpakete in Beziehung zueinander stehen. Darüber hinaus werden auch die photonischen Eigenschaften für eindimensionale quasiperiodische Mehrschichtsysteme für beliebige Einfallswinkel untersucht und der Verlauf der Transmissionsbänder mit der quasiperiodischen Struktur in Zusammenhang gebracht.

Efficient memory usage is crucial for data-intensive applications as a smaller memory footprint ensures better cache performance and allows one to run a larger problem size given a axed amount of main memory. The solutions found by existing techniques for automatic storage optimization for arrays in a new loop-nests, which minimize the storage requirements for the arrays, are often far from good or optimal and could even miss nearly all storage optimization potential. In this work, we present a new automatic storage optimization framework and techniques that can be used to achieve intra-array as well as inter-array storage reuse within a new loop-nests with a pre-determined schedule. Over the last two decades, several heuristics have been developed for achieving complex transformations of a new loop-nests using the polyhedral model. However, there are no comparably strong heuristics for tackling the problem of automatic memory footprint optimization. We tackle the problem of storage optimization for arrays by formulating it as one of ending the right storage partitioning hyperplanes: each storage partition corresponds to a single storage location. Statement-wise storage partitioning hyperplanes are determined that partition a unit end global array space so that values with overlapping live ranges are not mapped to the same partition. Our integrated heuristic for exploiting intra-array as well as inter-array reuse opportunities is driven by a fourfold objective function that not only minimizes the dimensionality and storage requirements of arrays required for each high-level statement, but also maximizes inter-statement storage reuse. We built an automatic polyhedral storage optimizer called SMO using our storage partitioning approach. Storage reduction factors and other results we report from SMO demon-strate the e activeness of our approach on several benchmarks drawn from the domains of image processing, stencil computations, high-performance computing, and the class of tiled codes in general. The reductions in storage requirement over previous approaches range from a constant factor to asymptotic in the loop blocking factor or array extents { the latter being a dramatic improvement for practical purposes. As an incidental and related topic, we also studied the problem of polyhedral compilation of graphical data programs. While polyhedral techniques for program transformation are now used in several proprietary and open source compilers, most of the research on poly-herald compilation has focused on imperative languages such as C, where the computation is species in terms of statements with zero or more nested loops and other control structures around them. Graphical data ow languages, where there is no notion of statements or a schedule specifying their relative execution order, have so far not been studied using a powerful transformation or optimization approach. The execution semantics and ref-eventual transparency of data ow languages impose a di errant set of challenges. In this work, we attempt to bridge this gap by presenting techniques that can be used to extract polyhedral representation from data ow programs and to synthesize them from their equivalent polyhedral representation. We then describe Polyglot, a framework for automatic transformation of data ow programs that we built using our techniques and other popular research tools such as Clan and Pluto. For the purpose of experimental evaluation, we used our tools to compile LabVIEW, one of the most widely used data ow programming languages. Results show that data ow programs transformed using our framework are able to outperform those compiled otherwise by up to a factor of seventeen, with a mean speed-up of 2.30 while running on an 8-core Intel system.

Understanding the connection of the atomic structure and the physical properties of materials remains one of the elementary questions of condensed-matter physics. One research line in this quest started with the discovery of quasicrystals by Shechtman et al. in 1982. It soon became clear that these materials with their 5-, 8-, 10- or 12-fold rotational symmetries, which are forbidden according to classical crystallography, can be described in terms of mathematical models for nonperiodic tilings of a plane proposed by Penrose and Ammann in the 1970s. Due to the missing translational symmetry of quasicrystals, till today only finite, relatively small systems or periodic approximants have been investigated by means of numerical calculations and theoretical results have mainly been obtained for one-dimensional systems. In this thesis we study d-dimensional quasiperiodic models, so-called labyrinth tilings, with separable Hamiltonians in the tight-binding approach. This method paves the way to study higher-dimensional, quantum mechanical solutions, which can be directly derived from the one-dimensional results. This allows the investigation of very large systems in two and three dimensions with up to 10^10 sites. In particular, we contemplate the class of metallic-mean sequences. Based on this model we focus on the electronic properties of quasicrystals with a special interest on the connection of the spectral and dynamical properties of the Hamiltonian. Hence, we investigate the characteristics of the eigenstates and wave functions and compare these with the wave-packet dynamics in the labyrinth tilings by numerical calculations and by a renormalization group approach in connection with perturbation theory. It turns out that many properties show a qualitatively similar behavior in different dimensions or are even independent of the dimension as e.g. the scaling behavior of the participation numbers and the mean square displacement of a wave packet. Further, we show that the structure of the labyrinth tilings and their transport properties are connected and obtain that certain moments of the spectral dimensions are related to the wave-packet dynamics. Besides this also the photonic properties are studied for one-dimensional quasiperiodic multilayer systems for oblique incidence of light, and we show that the characteristics of the transmission bands are related to the quasiperiodic structure. / Eine der elementaren Fragen der Physik kondensierter Materie beschäftigt sich mit dem Zusammenhang zwischen der atomaren Struktur und den physikalischen Eigenschaften von Materialien. Eine Forschungslinie in diesem Kontext begann mit der Entdeckung der Quasikristalle durch Shechtman et al. 1982. Es stellte sich bald heraus, dass diese Materialien mit ihren laut der klassischen Kristallographie verbotenen 5-, 8-, 10- oder 12-zähligen Rotationssymmetrien durch mathematische Modelle für die aperiodische Pflasterung der Ebene beschrieben werden können, die durch Penrose und Ammann in den 1970er Jahren vorgeschlagen wurden. Aufgrund der fehlenden Translationssymmetrie in Quasikristallen sind bis heute nur endliche, relativ kleine Systeme oder periodische Approximanten durch numerische Berechnungen untersucht worden und theoretische Ergebnisse wurden hauptsächlich für eindimensionale Systeme gewonnen. In dieser Arbeit werden d-dimensionale quasiperiodische Modelle, sogenannte Labyrinth-Pflasterungen, mit separablem Hamilton-Operator im Modell starker Bindung betrachtet. Diese Methode erlaubt es, quantenmechanische Lösungen in höheren Dimensionen direkt aus den eindimensionalen Ergebnissen abzuleiten und ermöglicht somit die Untersuchung von sehr großen Systemen in zwei und drei Dimensionen mit bis zu 10^10 Gitterpunkten. Insbesondere betrachten wir dabei quasiperiodische Folgen mit metallischem Schnitt. Basierend auf diesem Modell befassen wir uns im Speziellen mit den elektronischen Eigenschaften der Quasikristalle im Hinblick auf die Verbindung der spektralen und dynamischen Eigenschaften des Hamilton-Operators. Hierfür untersuchen wir die Eigenschaften der Eigenzustände und Wellenfunktionen und vergleichen diese mit der Dynamik von Wellenpaketen in den Labyrinth-Pflasterungen basierend auf numerischen Berechnungen und einem Renormierungsgruppen-Ansatz in Verbindung mit Störungstheorie. Dabei stellt sich heraus, dass viele Eigenschaften wie etwa das Skalenverhalten der Partizipationszahlen und der mittleren quadratischen Abweichung eines Wellenpakets für verschiedene Dimensionen ein qualitativ gleiches Verhalten zeigen oder sogar unabhängig von der Dimension sind. Zudem zeigen wir, dass die Struktur der Labyrinth-Pflasterungen und deren Transporteigenschaften sowie bestimmte Momente der spektralen Dimensionen und die Dynamik der Wellenpakete in Beziehung zueinander stehen. Darüber hinaus werden auch die photonischen Eigenschaften für eindimensionale quasiperiodische Mehrschichtsysteme für beliebige Einfallswinkel untersucht und der Verlauf der Transmissionsbänder mit der quasiperiodischen Struktur in Zusammenhang gebracht.

The interaction between the study of geometric and analytic aspects of Riemannian manifolds and that of graphs is a very amazing subject. The study of synthetic curvature notions on graphs adds new contributions to this topic. In this thesis, we mainly study two kinds of synthetic curvature notions: the Ollivier-Ricci cuvature on locally finite graphs and the combinatorial curvature on infinite semiplanar graphs. In the first part, we study the Ollivier-Ricci curvature. As known in Riemannian geometry, a lower Ricci curvature bound prevents geodesics from diverging too fast on average. We translate this Riemannian idea into a combinatorial setting using the Olliver-Ricci curvature notion. Note that on a graph, the analogue of geodesics starting in different directions, but eventually approaching each other again, would be a triangle. We derive lower and upper Ollivier-Ricci curvature bounds on graphs in terms of number of triangles, which is sharp for instance for complete graphs. We then describe the relation between Ollivier-Ricci curvature and the local clustering coefficient, which is an important concept in network analysis introduced by Watts-Strogatz. Furthermore, positive lower boundedness of Ollivier-Ricci curvature for neighboring vertices imply the existence of at least one triangle. It turns out that the existence of triangles can also improve Lin-Yau\''s curvature dimension inequality on graphs and then produce an implication from Ollivier-Ricci curvature lower boundedness to the curvature dimension inequality. The existence of triangles prevents a graph from being bipartite. A finite graph is bipartite if and only if its largest eigenvalue equals 2. Therefore it is natural that Ollivier-Ricci curvature is closely related to the largest eigenvalue estimates. We combine Ollivier-Ricci curvature notion with the neighborhood graph method developed by Bauer-Jost to study the spectrum estimates of a finite graph. We can always obtain nontrivial estimates on a non-bipartite graph even if its curvature is nonpositive. This answers one of Ollivier\''s open problem in the finite graph setting. In the second part of this thesis, we study systematically infinite semiplanar graphs with nonnegative combinatorial curvature. Unlike the previous Gauss-Bonnet formula approach, we explore an Alexandrov approach based on the observation that the nonnegative combinatorial curvature on a semiplanar graph is equivalent to nonnegative Alexandrov curvature on the surface obtained by replacing each face by a regular polygon of side length one with the same facial degree and gluing the polygons along common edges. Applying Cheeger-Gromoll splitting theorem on the surface, we give a metric classification of infinite semiplanar graphs with nonnegative curvature. We also construct the graphs embedded into the projective plane minus one point. Those constructions answer a question proposed by Chen. We further prove the volume doubling property and Poincare inequality which make the running of Nash-Moser iteration possible. We in particular explore the volume growth behavior on Archimedean tilings on a plane and prove that they satisfy a weak version of relative volume comparison with constant 1. With the above two basic inequalities in hand, we study the geometric function theory of infinite semiplanar graphs with nonnegative curvature. We obtain the Liouville type theorem for positive harmonic functions, the parabolicity. We also prove a dimension estimate for polynomial growth harmonic functions, which is an extension of the solution of Colding-Minicozzi of a conjecture of Yau in Riemannian geometry.

Treemaps are a popular space-filling visualization of hierarchical data that maps an attribute of a datum, or a data aggregate, to a proportional amount of area. Assuming a rectangular treemap consisting of nested rectangles (also called tiles), there are multiple possible valid tiling arrangements. A common criterion for optimization is aspect ratio. Nevertheless, treemaps usually consist of multiple rectangles, so the aspect ratios need be aggregated. The basic definition of aspect ratio (width divided by height) cannot be meaningfully aggregated. Given this, a definition of aspect ratio that does not differentiate height from width was suggested. This definition allows for meaningful aggregation, but only as long as there are no large differences in the data distribution, and the target aspect ratio is 1:1. Originally, a target aspect ratio of 1:1 was deemed to be axiomatically ideal. Currently, perceptual studies have found an aspect ratio of 1:1 to lead to the largest area estimation error. However, with any other target this definition of aspect ratio cannot be meaningfully aggregated. This thesis suggests a correction that can be applied to the current metric and would allow it to be meaningfully aggregated even when there are large value differences in the data. Furthermore, both the uncorrected and corrected metrics can be generalized for any target (i.e. targets other than 1:1). Another issue with current evaluation techniques is that algorithm fitness is evaluated through Monte Carlo trials. In this method, synthetic data is generated and then aggregated to generate a single final result. However, tiling algorithm performance is dependant on data distribution, so a single aggregateresult cannot generalize overall performance. The alternative suggested in this thesis is visual cluster analysis, which should hold more general predictive power.All of the above is put into practice with an experiment. In the experiment, a new family of tiling algorithms, based on criteria derived from the results of the perceptual tests in literature,is compared to the most popular tiling algorithm, Squarify. The results confirm that there are indeed vast but consistent value fluctuations for different normal distributions. At least for a target aspect ratio of 1.5, the new proposed algorithms are shown to perform better than Squarify for most use cases in terms of aspect ratio.