Selection Lemmas for various geometric objects

Selection lemmas are classical results in discrete geometry that have been well studied and have applications in many geometric problems like weak epsilon nets and slimming Delaunay triangulations. Selection lemma type results typically show that there exists a point that is contained in many objects that are induced (spanned) by an underlying point set. In the first selection lemma, we consider the set of all the objects induced (spanned) by a point set $P$. This question has been widely explored for simplices in $\mathbb{R}^d$, with tight bounds in $\mathbb{R}^2$. In our paper, we prove first selection lemma for other classes of geometric objects. We also consider the strong variant of this problem where we add the constraint that the piercing point comes from $P$. We prove an exact result on the strong and the weak variant of the first selection lemma for axis-parallel rectangles, special subclasses of axis-parallel rectangles like quadrants and slabs, disks (for centrally symmetric point sets). We also show non-trivial bounds on the first selection lemma for axis-parallel boxes and hyperspheres in $\mathbb{R}^d$. In the second selection lemma, we consider an arbitrary $m$ sized subset of the set of all objects induced by $P$. We study this problem for axis-parallel rectangles and show that there exists an point in the plane that is contained in $\frac{m^3}{24n^4}$ rectangles. This is an improvement over the previous bound by Smorodinsky and Sharir when $m$ is almost quadratic

Matching points with disks with a common intersection

We consider matchings with diametral disks between two sets of points R and B. More precisely, for each pair of matched points p in R and q in B, we consider the disk through p and q with the smallest diameter. We prove that for any R and B such that |R|=|B|, there exists a perfect matching such that the diametral disks of the matched point pairs have a common intersection. In fact, our result is stronger, and shows that a maximum weight perfect matching has this property

Sunflowers in Set Systems of Bounded Dimension

Given a family $\mathcal F$ of $k$-element sets, $S_1,\ldots,S_r\in\mathcal F$ form an {\em $r$-sunflower} if $S_i \cap S_j =S_{i'} \cap S_{j'}$ for all $i \neq j$ and $i' \neq j'$. According to a famous conjecture of Erd\H os and Rado (1960), there is a constant $c=c(r)$ such that if $|\mathcal F|\ge c^k$, then $\mathcal F$ contains an $r$-sunflower. We come close to proving this conjecture for families of bounded {\em Vapnik-Chervonenkis dimension}, VC-dim$(\mathcal F)\le d$. In this case, we show that $r$-sunflowers exist under the slightly stronger assumption $|\mathcal F|\ge2^{10k(dr)^{2\log^{*} k}}$. Here, $\log^*$ denotes the iterated logarithm function. We also verify the Erd\H os-Rado conjecture for families $\mathcal F$ of bounded {\em Littlestone dimension} and for some geometrically defined set systems

Living in Ilopango’s Shadow: Using a Figurative Analysis of Grave Goods Excavated from Ciudad Nuevo Cuscatlán in El Salvador To Investigate the Community’s Relationship to the Ilopango Volcano

The purpose of this thesis is to determine what can be learned about the people living near the Ilopango Volcano during the Classic period of Mesoamerican history by conducting a figurative analysis of the living subjects depicted on the ceramics and figurines excavated from the area, to determine what, if any, relationship the people there had with the volcano. I examine the frontier Mesoamerican village of Ciudad Nuevo Cuscatlán by analyzing grave goods found in the mortuary burials located in the area and determining the meaning of the figural subjects on each using established diagnostic elements from experts in the field. I then put these subjects and their meanings into a pantheon of beliefs to determine if a holistic view of their subject matter reveals a connection to the volcano. The contents of five burial features and a single figurine from a pile of backdirt were selected for analysis, resulting in 30 artifacts. Utilizing the iconography on the artifacts, nine subjects were identified as being representations of life: birds, breath or wind, crocodiles, God N or Pawahtuun, humans, monkeys, serpents, spiders, and toads. These subjects were then used as a window into the spiritual pantheon of the people who lived in the area. The results found that the people who repopulated the area after the Tierra Blanca Joven eruption of the Ilopango Volcano had a spiritual pantheon focused on the central figure of Pawahtuun who was either a representation of the Ilopango Volcano, or who had control over it. They also practiced divinatory rituals to connect with Pawahtuun and other deities to interpret of predict the future activity of the volcano. The people living at CNC also participated in some of the artistic traditions from larger population centers in the region and had some artistic traditions which were unique to this community

Кинетика и морфологиjа депозициjе честица на хетерогеним површинама

Monolayer and multilayer thin film formation on solid or liquid surfaces is a growing multidisciplinary area of research of great interest for new and emerging technologies in photonics, microelectronics, nanotechnology, plasmonics, biosensors, bio-medical devices, etc. Adsorption and deposition (irreversible adsorption) of colloids, proteins, and other bio-materials on solid/liquid interfaces are of large significance for many practical and natural processes such as filtration, paper-making, chromatography, separation of proteins, viruses, bacteria, pathological cells, immunological assays, thrombosis, biofouling, biomineralization, etc. Controlled adsorption of colloid particles on sites of nanometric scale can also be exploited for direct visualization of surface features. The Random Sequential Adsorption (RSA) model is one of the basic models used to describe the irreversible formation of monolayer deposits of microscopic and mesoscopic particles. Inter-particle interactions are approximated classically with the hard-core exclusion model, which means that overlaps between the particles are not allowed. Particles can only adsorb if they are in direct contact with the substrate. This feature ensures a monolayer deposition. Irreversible adsorption means that the adsorbed particle stays permanently fixed to the substrate and diffusion or desorption processes are not allowed. Previously adsorbed particles block a certain area of the substrate for new adsorptions and consequently, the system becomes jammed. Heterogeneities of a substrate impose further limitations on the positions of adsorbed particles. Our aim is to quantify structural changes in the jammed state that are introduced by different patterns of substrate heterogeneities. We use the RSA approach to analyze the deposition of identical spherical particles of a fixed radius on non-uniform flat surfaces covered by rectangular cells. Two different types of patterns are of interest: randomly positioned cells and square lattice centred cells. In the first part of the dissertation, the configuration of the cells (heterogeneities) was produced by performing RSA simulations to a prescribed coverage fraction θ (cell) 0 . Adsorption was assumed to occur if the particle (projected) centre lies within a rectangular cell area, i.e., if the sphere touches one of the cells. The jammed-state properties of the model were studied for different values of the cell size α (comparable with the adsorbing particle size) and density θ (cell) 0 . Numerical simulations were carried out to investigate adsorption kinetics, jamming coverage, and structure of coverings. Structural properties of the jammed-state coverings were analyzed in terms of the radial distribution function g(r) and distribution of the Delaunay ‘free’ volumes P(v). It was demonstrated that adsorption kinetics and the jamming coverage decreased significantly, at a fixed density θ (cell) 0 , when the cell size α increased. The predictions following our calculation suggest that the porosity (pore volumes) of the deposited monolayer can be controlled by the size and shape of landing cells, and by the anisotropy of the cell deposition procedure. v The second direction of research in this thesis analyses the adsorption of spherical particles of a fixed diameter on nonuniform surfaces covered by square cells arranged in a square lattice pattern. To characterize such a pattern two dimensionless parameters are used: the cell size α and the cellcell separation β, measured in terms of the particle diameter d0. We focus on the kinetics of the deposition process in the case when no more than a single disk can be placed onto any square cell (α < 1/ √ 2 ≈ 0.707). We find that the asymptotic approach of the coverage fraction θ(t) to the jamming limit θJ is algebraic if the parameters α and β satisfy the simple condition, β +α/2 < 1. If this condition is not satisfied, the late time kinetics of the deposition process is not consistent with the power-law behaviour. However, if the geometry of the pattern approaches “noninteracting conditions” (β > 1), when adsorption on each cell can be decoupled, the approach of the coverage fraction θ(t) to θJ becomes closer to the exponential law. Consequently, changing the pattern parameters in the present model allows for interpolating the deposition kinetics between the continuum limit and the lattice-like behaviour. Structural properties of the jammed-state coverings are studied in terms of the radial distribution function g(r) and the spatial distribution of particles inside the cell. Various, non-trivial spatial distributions are observed depending on the geometry parameters of the pattern.Формирање jеднослоjних и вишеслоjних танких филмова на чврстим и течним површинама jе растућа мултидисциплинарна област истраживања од великог интереста у фотоници, микроелектроници, нанотехнологиjама, плазмоници, за биосензоре, биомедицинске уређаjе, итд. Адсорпциjа и депозициjа (иреверзибилна адсорпциjа) колоида, протеина и других биоматериjала на чврстим/течним површинама су од велике важности за многе практичне и природне процесе као што су филтрациjа, производња папира, хроматографиjа, сепарациjа протеина, вируса, бактериjа и патолошких ћелиjа, имунолошки тестови, тромбоза, биоминерализациjа, итд. Контролисана адсорпциjа колоидних честица на структурама на нанометарскоj скали се такође могу искористит за директну визуализациjу структурних карактеристика. Модел случаjне секвенциjалне адсорпциjе (RSA модел) jе jедан од основних модела за описивање формирања jеднослоjних депозита мезоскопских честица. Међучестична интеракциjа jе апроксимирана класичним моделом крутих тела, што значи да jе забрањено међусобно преклапање честица. Честице се могу адсорбовати jедино ако су у директном контакту са супстратом. Ова особина доводи до формирања jеднослоjних депозита. Поjам иреверзибилна адсорпциjа поразумева да су адсорбоване честице траjно причвршћене за подлогу, а процеси дифузиjе или десорпциjе су забрањени. Претходно адсорбоване честице блокираjу одређени део подлоге за адсорпциjу нових честица што доводи до загушења система. Нехомогеност супстрата намеће додатна ограничења на позициjе адсорбованих честица. Наш циљ jе да квантификуjемо структурне промене загушеног стања настале услед разлличитих хетерогених образаца на адсорбуjоћоj подлози. Користимо RSA приступ за анализу депозициjе идентичних сферних честица на нехомогене равне површине покривене правоугаоним ћелиjама. Од интереса су два различита типа распореда: случаjно распоређене ћелиjе и ћелиjе распоређене у чворовима квадратне решетке. У првом делу истраживања у оквиру ове тезе, конфигурациjа ћелиjа се формира помоћу RSA симулациjе док се не постигне жељена покривеност супстрата θ (cell) 0 . До адсорпциjе долази ако (проjектовани) центар честице лежи унутар правоугаоне ћелиjе, тj. ако сферна честица додируjе неку од ћелиjа. Особине загушеног стања су изучаване за различите вредности величине ћелиjа α (упоредивих са величином честице) и различите густине ћелиjа θ (cell) 0 . Извршене су нумеричке симулациjе како би истражили кинетику адсорпциjе, покривеност у загушењу и структуру депозита. Структурне особине загушеног стања анализиране су помоћу парне корелационе функциjе g(r) и дистрибуциjе Делонеjевих слободних површина. Резултати наших симулациjа сугеришу да се контрола порозности jеднослоjног депозита може постићи подешавањем величине, облика и ориjентациjе прихватних ћелиjа. vii Други правац истраживања у тези jе анализа адсорпциjе сферних честица фиксног пречника на хетерогеним површинама прекривеним квадратним ћелиjама распоређеним у чворове квадратне решетке. За карактеризциjу овог шаблона користимо два бездимензиона параметра: величину квадратне ћелиjе α и размак између две суседне ћелиjе β. За jединицу мере користимо пречник адсорбуjућих ћелиjа d0. У фокусу истраживања jе кинетика процеса депозициjе у случаjу када било коjа прихватна ћелиjа може да адсорбуjе наjвише jедну честицу (α < √ 2/2). Покривеност θ(t) асимптотски тежи граничноj вредности θJ по алгебарском закону ако параметри α и β задовољаваjу услов β + α/2 < 1. Ако оваj услов ниjе испуњен, кинетика касне фазе процеса депозициjе ниjе конзистента са степеном законитошћу. Ипак, како се геометриjа подлоге приближава неинтерагуjућем режиму (β > 1), асимптотски прилаз покривености се приближава експоненциjалноj законости. Сходно томе, промена параметара патерна субстрата у овом моделу омогућуjе интерполациjу између два гранична случаjа адсорпциjе на континууму и на квадратноj решетки. За изучавање структурних особина загушеног стања користимо парну корелациону функциjу g(r) и просторну дистрибуциjу честица унутар ћелиjа. Примећене су разноврсне нетривиjалне просторне дистрибуциjе у зависности од геометриjе патерна подлоге

Strong Hanani-Tutte for the Torus

If a graph can be drawn on the torus so that every two independent edges cross an even number of times, then the graph can be embedded on the torus

Working With Incremental Spatial Data During Parallel (GPU) Computation

Central to many complex systems, spatial actors require an awareness of their local environment to enable behaviours such as communication and navigation. Complex system simulations represent this behaviour with Fixed Radius Near Neighbours (FRNN) search. This algorithm allows actors to store data at spatial locations and then query the data structure to find all data stored within a fixed radius of the search origin. The work within this thesis answers the question: What techniques can be used for improving the performance of FRNN searches during complex system simulations on Graphics Processing Units (GPUs)? It is generally agreed that Uniform Spatial Partitioning (USP) is the most suitable data structure for providing FRNN search on GPUs. However, due to the architectural complexities of GPUs, the performance is constrained such that FRNN search remains one of the most expensive common stages between complex systems models. Existing innovations to USP highlight a need to take advantage of recent GPU advances, reducing the levels of divergence and limiting redundant memory accesses as viable routes to improve the performance of FRNN search. This thesis addresses these with three separate optimisations that can be used simultaneously. Experiments have assessed the impact of optimisations to the general case of FRNN search found within complex system simulations and demonstrated their impact in practice when applied to full complex system models. Results presented show the performance of the construction and query stages of FRNN search can be improved by over 2x and 1.3x respectively. These improvements allow complex system simulations to be executed faster, enabling increases in scale and model complexity