An improved bound on the number of point-surface incidences in three dimensions

We show that $m$ points and $n$ smooth algebraic surfaces of bounded degree in $\mathbb{R}^3$ satisfying suitable nondegeneracy conditions can have at most $O(m^{\frac{2k}{3k-1}}n^{\frac{3k-3}{3k-1}}+m+n)$ incidences, provided that any collection of $k$ points have at most O(1) surfaces passing through all of them, for some $k\geq 3$. In the case where the surfaces are spheres and no three spheres meet in a common circle, this implies there are $O((mn)^{3/4} + m +n)$ point-sphere incidences. This is a slight improvement over the previous bound of $O((mn)^{3/4} \beta(m,n)+ m +n)$ for $\beta(m,n)$ an (explicit) very slowly growing function. We obtain this bound by using the discrete polynomial ham sandwich theorem to cut $\mathbb{R}^3$ into open cells adapted to the set of points, and within each cell of the decomposition we apply a Turan-type theorem to obtain crude control on the number of point-surface incidences. We then perform a second polynomial ham sandwich decomposition on the irreducible components of the variety defined by the first decomposition. As an application, we obtain a new bound on the maximum number of unit distances amongst $m$ points in $\mathbb{R}^3$.Comment: 17 pages, revised based on referee comment

Geometric algorithms for geographic information systems

A geographic information system (GIS) is a software package for storing geographic data and performing complex operations on the data. Examples are the reporting of all land parcels that will be flooded when a certain river rises above some level, or analyzing the costs, benefits, and risks involved with the development of industrial activities at some place. A substantial part of all activities performed by a GIS involves computing with the geometry of the data, such as location, shape, proximity, and spatial distribution. The amount of data stored in a GIS is usually very large, and it calls for efficient methods to store, manipulate, analyze, and display such amounts of data. This makes the field of GIS an interesting source of problems to work on for computational geometers. In chapters 2-5 of this thesis we give new geometric algorithms to solve four selected GIS problems.These chapters are preceded by an introduction that provides the necessary background, overview, and definitions to appreciate the following chapters. The four problems that we study in chapters 2-5 are the following: Subdivision traversal: we give a new method to traverse planar subdivisions without using mark bits or a stack. Contour trees and seed sets: we give a new algorithm for generating a contour tree for d-dimensional meshes, and use it to determine a seed set of minimum size that can be used for isosurface generation. This is the first algorithm that guarantees a seed set of minimum size. Its running time is quadratic in the input size, which is not fast enough for many practical situations. Therefore, we also give a faster algorithm that gives small (although not minimal) seed sets. Settlement selection: we give a number of new models for the settlement selection problem. When settlements, such as cities, have to be displayed on a map, displaying all of them may clutter the map, depending on the map scale. Choices have to be made which settlements are selected, and which ones are omitted. Compared to existing selection methods, our methods have a number of favorable properties. Facility location: we give the first algorithm for computing the furthest-site Voronoi diagram on a polyhedral terrain, and show that its running time is near-optimal. We use the furthest-site Voronoi diagram to solve the facility location problem: the determination of the point on the terrain that minimizes the maximal distance to a given set of sites on the terrain

Doctor of Philosophy

dissertationKernel smoothing provides a simple way of finding structures in data sets without the imposition of a parametric model, for example, nonparametric regression and density estimates. However, in many data-intensive applications, the data set could be large. Thus, evaluating a kernel density estimate or kernel regression over the data set directly can be prohibitively expensive in big data. This dissertation is working on how to efficiently find a smaller data set that can approximate the original data set with a theoretical guarantee in the kernel smoothing setting and how to extend it to more general smooth range spaces. For kernel density estimates, we propose randomized and deterministic algorithms with quality guarantees that are orders of magnitude more efficient than previous algorithms, which do not require knowledge of the kernel or its bandwidth parameter and are easily parallelizable. Our algorithms are applicable to any large-scale data processing framework. We then further investigate how to measure the error between two kernel density estimates, which is usually measured either in L1 or L2 error. In this dissertation, we investigate the challenges in using a stronger error, L ∞ (or worst case) error. We present efficient solutions for how to estimate the L∞ error and how to choose the bandwidth parameter for a kernel density estimate built on a subsample of a large data set. We next extend smoothed versions of geometric range spaces from kernel range spaces to more general types of ranges, so that an element of the ground set can be contained in a range with a non-binary value in [0,1]. We investigate the approximation of these range spaces through ϵ-nets and ϵ-samples. Finally, we study coresets algorithms for kernel regression. The size of the coresets are independent of the size of the data set, rather they only depend on the error guarantee, and in some cases the size of domain and amount of smoothing. We evaluate our methods on very large time series and spatial data, demonstrate that they can be constructed extremely efficiently, and allow for great computational gains

LIPIcs, Volume 258, SoCG 2023, Complete Volume

LIPIcs, Volume 258, SoCG 2023, Complete Volum

Planar Nef polyhedra and generic higher-dimensional geometry

We present two generic software projects that are part of the software library CGAL. The first part described the design of a geometry kernel for higher-dimensional Euclidian geometry and the interaction with application programs. We describe software structures, interface concepts, and their models that are based on cooordinate representation, number types, and memory layout. In the higher-dimensional software kernel the interaction between linear algebra and geometric objects and primitves is one important facet. In the actual design our users can replace number types, representation types, and the traits classes that inflate kernel functionality into our current application programs: higher-dimensional convex hulls and Delaunay tedrahedralisations. In the second part we present the realization of planar Nef polyhedra. The concept of Nef polyhedra subsumes all kinds of rectilinear polyhedral subdivisions and is therefore of general applicability within a geometric software library. The software is based on the theory of extended points and segments that allows us to reuse classical algorithmic solutions like plane sweep to realize binary operations of Nef polyhedra.Wir präsentieren zwei Softwareprojekte, die Teil der Softwarebibliothek CGAL sind. Der erste Teil beschreibt den Entwurf eines Geometriekerns für höherdimensionale euklidische Geometrie und dessen Interaktion mit Anwendungsprogrammen. Wir beschreiben die Softwarestruktur, die auf der Herausarbeitung von Schnittstellenkonzepten und ihren Modellen hinsichtlich Koordinatenrepräsentation, Zahlentypen und Speicherablage beruht. Dabei spielt im Höherdimensionalen die Interaktion zwischen linearer Algebra und den entsprechenden geometrischen Objekten und primitiven Operationen eine wesentliche Rolle. Unser Entwurf erlaubt das Auswechseln von Zahlentypen, Repräsentations- und Traitsklassen bei der Berechnung von d-dimensionalen konvexen Hüllen und Delaunay-Simplexzerlegungen. Im zweiten Teil stellen wir die Realisierung von planaren Nef-Polyedern vor. Das Konzept der Nef-Polyeder umfasst alle linear-polyedrisch begrenzten Unterteilungen. Wir beschreiben ein Softwaremodul das umfassende Funktionalität zur Verfügung stellt. Als theoretische Grundlage des Entwurfs dient die Theorie erweiterter Punkte und Segmente, die es uns erlaubt, vorhandene Algorithmen wie z.B. Plane-Sweep zur Realisierung binärer Operationen von Nef-Polyedern zu nutzen

Large bichromatic point sets admit empty monochromatic 4-gons

We consider a variation of a problem stated by Erd˝os and Szekeres in 1935 about the existence of a number fES(k) such that any set S of at least fES(k) points in general position in the plane has a subset of k points that are the vertices of a convex k-gon. In our setting the points of S are colored, and we say that a (not necessarily convex) spanned polygon is monochromatic if all its vertices have the same color. Moreover, a polygon is called empty if it does not contain any points of S in its interior. We show that any bichromatic set of n ≥ 5044 points in R2 in general position determines at least one empty, monochromatic quadrilateral (and thus linearly many).Postprint (published version

Algebraic Topology for Data Scientists

This book gives a thorough introduction to topological data analysis (TDA), the application of algebraic topology to data science. Algebraic topology is traditionally a very specialized field of math, and most mathematicians have never been exposed to it, let alone data scientists, computer scientists, and analysts. I have three goals in writing this book. The first is to bring people up to speed who are missing a lot of the necessary background. I will describe the topics in point-set topology, abstract algebra, and homology theory needed for a good understanding of TDA. The second is to explain TDA and some current applications and techniques. Finally, I would like to answer some questions about more advanced topics such as cohomology, homotopy, obstruction theory, and Steenrod squares, and what they can tell us about data. It is hoped that readers will acquire the tools to start to think about these topics and where they might fit in.Comment: 322 pages, 69 figures, 5 table

Brillouin light scattering study of linear and nonlinear spin waves in continuous and patterned magnetic thin films

2014 Summer.This thesis focuses on the use of the Brillouin light scattering (BLS) technique to measure spin waves or magnons in thin films. BLS is an experimental technique that measures the inelastically scattered light from photon-magnon interactions. Broadly, three different experiments are presented in this thesis: the measurements of spin wave properties in iron cobalt (FeCo), yttrium iron garnet (YIG), and microstructures involving Permalloy (Ni80Fe20) and cobalt nickel (CoNi). First, conventional backward scattering BLS was used to measure the spin waves in a set of Fe65Co35 films that were provided by Seagate Technologies. By fitting the spin wave frequencies that were measured as a function of the external magnetic field and film thickness, the quantum mechanical parameter responsible for short range order, known as the exchange parameter, was determined. Second, nonlinear spin waves were measured in YIG using conventional forward scattering BLS with time resolution. Two nonlinear three wave processes were observed, namely, the three magnon splitting and confluence. The nonlinear power threshold, the saturation magnetization, and the film thickness were determined independently using network analyzer measurements. The spin wave group velocities were determined from the space- and time-resolved BLS data and compared to calculations from the dispersion relations. Back calculations showed the location where the three magnon splitting process took place. Lastly, spin waves in Permalloy and CoNi microstrips were measured using a recently developed micro-BLS. The micro-BLS, with a spatial resolution of 250 nm, allows for measuring the effects on the lateral confinement of spin waves in microstrips. The confinement of spin waves led to modifications to the dispersion relations, which were compared against the spin wave frequencies obtained from the micro-BLS. The Permalloy experiments shows non-reciprocity in surface spin wave modes with opposite wavevectors and provides a quantitative measure of the difference in excitation efficiency between the surface spin wave and the backward volume spin wave modes. Measurements were also conducted in the Permalloy microstrips at zero external magnetic field, showing evidence that propagating spin waves can be observed by exploiting the effects of shape anisotropy. Finally, preliminary measurements were done on CoNi microstrips with perpendicular anisotropy. A magnetic signal was detected, however further investigation will be needed to determine the exact origin of the observed signal and to definitively answer the question as to whether or not BLS can be used to measure spin waves in perpendicularly magnetized films. Overall, the experiments and results presented in this thesis show that BLS is a useful tool for measuring spin wave properties in magnetic thin films