DOCSLIB.ORG

UC Berkeley UC Berkeley Electronic Theses and Dissertations

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

UC Berkeley UC Berkeley Electronic Theses and Dissertations Title Algebraic Matroids in Applications Permalink https://escholarship.org/uc/item/1tq3k5bz Author Rosen, Zvi Publication Date 2015 Peer reviewed|Thesis/dissertation eScholarship.org Powered by the California Digital Library University of California Algebraic Matroids in Applications by Zvi H Rosen A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Mathematics in the Graduate Division of the University of California, Berkeley Committee in charge: Professor Bernd Sturmfels, Chair Professor Lior Pachter Professor Yun S. Song Spring 2015 Algebraic Matroids in Applications Copyright 2015 by Zvi H Rosen 1 Abstract Algebraic Matroids in Applications by Zvi H Rosen Doctor of Philosophy in Mathematics University of California, Berkeley Professor Bernd Sturmfels, Chair Algebraic matroids are combinatorial objects defined by the set of coordinates of an algebraic variety. These objects are of interest whenever coordinates hold significance: for instance, when the variety describes solution sets for a real world problem, or is defined using some combinatorial rule. In this thesis, we discuss algebraic matroids, and explore tools for their computation. We then delve into two applications that involve algebraic matroids: probability matrices and tensors from statistics, and chemical reaction networks from biology. i In memory of my grandparents יצחק אהרN וחנה רייזל דייוויס ז"ל! Isaac and Ann Davis יצחק יוסP ושרה רוזN ז"ל! Isaac and Sala Rosen whose courage and perseverance through adversity will inspire their families for generations. ii Contents Contents ii List of Figures iv List of Tables v 1 Introduction 1 1.1 Summary of Main Results . 1 1.2 Examples . 2 1.3 Definitions, Axioms, and Notation . 5 2 Computation 10 2.1 Symbolic Algorithm . 11 2.2 Linear Algebra . 11 2.3 Sample Computations for Applications . 15 3 Statistics: Joint Probability Matroid 25 3.1 Completability of Partial Probability Matrix . 28 3.2 Semialgebraic Description . 35 3.3 Completion Algorithms . 39 3.4 Algebraic Matroids . 45 3.5 Generalizations . 47 4 Biology: Chemical Reaction Matroid 50 4.1 From Biology to Algebra . 53 4.2 Ideals, Varieties, and Nine Points . 57 4.3 Multistationarity and its Discriminant . 59 4.4 Algebraic Matroids and Parametrizations . 62 4.5 Polyhedral Geometry . 66 4.6 Parameter Estimation . 68 4.7 From Algebra to Biology . 73 5 Further Directions 76 iii Bibliography 79 iv List of Figures 1.1 Example of linear matroid. 3 1.2 Example of graphic matroid. 4 2.1 Schlegel diagram for the affine representation of M(PL4) . 16 2.2 Circuit degree frequency for M(PL4). ....................... 18 2.3 Base degree frequency for M(PL4).......................... 18 2.4 Non-cobases of M(Imix)................................ 19 2.5 Base degrees for M(Imix)............................... 19 2.6 Affine representation of the MAP kinase matroid. 21 2.7 Base of M(Gr(3; 6)) with base degree 7....................... 22 2.8 Circuit degree frequency for M(Gr(3; 6)). ..................... 23 2.9 Circuit of M(Gr(3; 6)) with degree 12........................ 23 2.10 Non-Pappus matroid. 23 3.1 Completable probability masks of 2 × 2 matrices with diagonal entries observed. 29 3.2 Completable probability masks of 3 × 3 matrices with diagonal entries observed. 31 3.3 Region defined by (3.11) inside the cube [0; 2]3. .................. 36 3.4 Solution curves for c = 1=9; 1=10; 1=16; 1=36; 1=64, and 1=150. 43 4.1 Graphic rep. of Mone, and affine rep. of the rank 4 component of Mpar. 70 4.2 Schematic diagram for systems biology research. 73 v List of Tables 2.1 Commands to compute algebraic matroids using matroids.m2 .......... 12 3.2 Algorithm for completing probability matrix projections. 41 4.1 The 19 species in the Wnt shuttle model. 55 4.2 The 31 reactions in the Wnt shuttle model. 56 4.3 Frequencies for the sampling schemes. 61 4.4 Frequencies for testing robustness scheme. 61 4.5 The 951 circuit polynomials, by numbers of unknowns xi and kj. 64 4.6 Reducing the steady state equations to the 2092 bases of base degree 1 . 65 vi Acknowledgments אוֹדה ה' בכל לבי אספרה כל נפלאותיK; (תהליM ט;ב)! I thank the Lord with my whole heart . (Psalms 9:2) Working towards a doctorate can come with unexpected challenges. The process involves learning a great deal about one’s area of research, as well as learning a lot about yourself. I have been blessed to have many supporters who encouraged me to pursue a career in math- ematics. Amitai Bin-Nun encouraged me to pursue research when I was an undergraduate, helping me apply to summer programs; he continues providing insightful advice when I call. Jessica Sidman gave me my first experience in math research, getting me hooked on the ex- perience; she also continues to be a great advisor. Many professors from my undergraduate training, especially Dennis DeTurck and Andreea Nicoara, taught me beautiful mathematics and helped me apply to graduate school. My mathematical life in Berkeley included terrific teaching experiences with Per Persson and Slobodan Simic. My graduate research experiences were fun and enriching thanks to all of my coauthors, especially Elizabeth Gross, Heather Harrington, and Kaie Kubjas, with whom I wrote the work forming much of this thesis. I also want to extend my gratitude to Carina Curto, Alex Fink, Heather Harrington, and Shmuel Onn for their help in the process of applying to postdoctoral grants and jobs. I owe a great debt to Bernd Sturmfels for taking me on as a student. It has been a privilege to benefit from his mathematical insight and to participate in the community of great thinkers that surrounds him. I also want to thank him for going through multiple drafts of all my writing including this dissertation, and providing detailed feedback to improve it. He has demonstrated tremendous patience and support, and I am incredibly grateful. The professors who served on the committee for my qualifying examination and my disser- tation were Mark Haiman, Lior Pachter, Satish Rao, Yun Song, and Dan-Virgil Voiculescu. Their willingness to participate is highly appreciated. The Berkeley math department has an incredible staff who make the lives of the graduate student population much easier. In par- ticular, I want to extend my appreciation to Barb Waller, Marsha Snow, and Judie Filomeo. I also spent a semester at the Max Planck Institute for Mathematics in Bonn, Germany, and Anke Völzmann was incredibly helpful in locating resources. Moving to Berkeley, CA from my home on the East Coast was a daunting enterprise in itself and I thank the members of Congregation Beth Israel, particularly Rabbi Yonatan and Frayda Cohen and Avraham and Ruchama Burrell whose hospitality and kindness to me has been nothing short of extraordinary. I also want to thank Maharat Victoria Sutton and Adam Brelow, Zeev and Tamara Neumeier, Naaman and Meechal Kam, and Maayan and Elishav Rabinovich for being so welcoming and generous. My group of friends, particularly Ali Austerlitz, Arabella Bangura, Benjamin Epstein, Boaz Haberman, Amalya Lehmann, Matty Lichtenstein, Shivaram Lingamneni and Jonathan Thirman made Berkeley feel like home, and gave me tremendous support and friendship. I also want to thank Eitan Adler, Yonatan Cantor, Jonathan Eskreis-Winkler, Elie Friedman, Miki Friedmann, Yoni Halpern, vii Ariella Meisel, Noam Pratzer, David Pruwer, and Mordechai Treiger whose support from afar kept me going. I could never have made it this far without the love and support of my family: my mother and father especially, for always making themselves available in times of need, and for serving as living examples for me. My siblings and my brothers-in-law have also been unfailingly kind and welcoming whenever I have needed their help. It is a wonderful gift to have family that always picks up the phone. 1 Chapter 1 Introduction Both chapters 1 and 2 are based on material from Computing Algebraic Matroids [48]. The definitions and introductory examples are brought to aid exposition, but similar examples can be found in standard texts such as [45] and [56]. Algebraic matroids have a surprisingly long history. They were studied as early as the 1930’s by van der Waerden, in his textbook Moderne Algebra [55, Chapter VIII], and MacLane in one of his earliest papers on matroids [35]. The topic lay dormant until the 70’s and 80’s, when a number of papers about representability of matroids as algebraic ma- troids were published: notably by Ingleton and Main [23], Dress and Lovasz [11], the thesis of Piff [46], and extensively by Lindström ([32, 33, 30, 31], among others). In recent years, the algebraic matroids of toric varieties found application (e.g. in [43]); however, they have been primarily confined to that setting. Renewed interest in algebraic matroids comes from the field of matrix completion, start- ing with [26], where the set of entries of a low-rank matrix are the ground set of the algebraic matroid associated to the determinantal variety. In applied algebra in general, coordinates typically carry real-world significance, and the matroid has inherent interest as the depen- dence structure among those quantities. Even for varieties arising in pure mathematics, distinguished coordinates may have combinatorial meaning, in which case the matroid also provides insight. 1.1 Summary of Main Results The purpose of this dissertation is to provide the relevant tools for computation of algebraic matroids and to actually use them in practice. The remainder of Chapter 1 introduces matroids to the reader and defines technical language that will be used in the thesis. We prove some basic results where the proof is particularly instructive or relevant.
Recommended publications
  • Duality Respecting Representations and Compatible Complexity

    Duality Respecting Representations and Compatible Complexity

    Duality Respecting Representations and Compatible Complexity Measures for Gammoids Immanuel Albrecht FernUniversit¨at in Hagen, Department of Mathematics and Computer Science, Discrete Mathematics and Optimization D-58084 Hagen, Germany Abstract We show that every gammoid has special digraph representations, such that a representation of the dual of the gammoid may be easily obtained by reversing all arcs. In an informal sense, the duality notion of a poset applied to the digraph of a special representation of a gammoid commutes with the operation of forming the dual of that gammoid. We use these special representations in order to define a complexity measure for gammoids, such that the classes of gammoids with bounded complexity are closed under duality, minors, and direct sums. Keywords: gammoids, digraphs, duality, complexity measure 2010 MSC: 05B35, 05C20, 06D50 A well-known result due to J.H. Mason is that the class of gammoids is closed under duality, minors, and direct sums [1]. Furthermore, it has been shown by D. Mayhew that every gammoid is also a minor of an excluded minor for the class of gammoids [2], which indicates that handling the class of all gammoids may get very involved. In this work, we introduce a notion of complexity for gammoids which may be used to define subclasses of gammoids with bounded complexity, that still have the desirable property of being closed under duality, minors, and direct sums; yet their representations have a more limited number arXiv:1807.00588v2 [math.CO] 27 Jun 2020 of arcs than the general class of gammoids. 1. Preliminaries In this work, we consider matroids to be pairs M = (E, I) where E is a finite set and I is a system of independent subsets of E subject to the usual axioms ([3], Sec.
  • Hardware Support for Non-Photorealistic Rendering

    Hardware Support for Non-Photorealistic Rendering

    Hardware Support for Non-photorealistic Rendering Ramesh Raskar MERL, Mitsubishi Electric Research Labs (ii) Abstract (i) Special features such as ridges, valleys and silhouettes, of a polygonal scene are usually displayed by explicitly identifying and then rendering ‘edges’ for the corresponding geometry. The candidate edges are identified using the connectivity information, which requires preprocessing of the data. We present a non- obvious but surprisingly simple to implement technique to render such features without connectivity information or preprocessing. (iii) (iv) At the hardware level, based only on the vertices of a given flat polygon, we introduce new polygons, with appropriate color, shape and orientation, so that they eventually appear as special features. 1 INTRODUCTION Figure 1: (i) Silhouettes, (ii) ridges, (iii) valleys and (iv) their combination for a polygonal 3D model rendered by processing Sharp features convey a great deal of information with very few one polygon at a time. strokes. Technical illustrations, engineering CAD diagrams as well as non-photo-realistic rendering techniques exploit these features to enhance the appearance of the underlying graphics usually not supported by rendering APIs or hardware. The models. The most commonly used features are silhouettes, creases rendering hardware is typically more suited to working on a small and intersections. For polygonal meshes, the silhouette edges amount of data at a time. For example, most pipelines accept just consists of visible segments of all edges that connect back-facing a sequence of triangles (or triangle soups) and all the information polygons to front-facing polygons. A crease edge is a ridge if the necessary for rendering is contained in the individual triangles.
  • The Minimum Cost Query Problem on Matroids with Uncertainty Areas Arturo I

    The Minimum Cost Query Problem on Matroids with Uncertainty Areas Arturo I

    The minimum cost query problem on matroids with uncertainty areas Arturo I. Merino Department of Mathematical Engineering and CMM, Universidad de Chile & UMI-CNRS 2807, Santiago, Chile [email protected] José A. Soto Department of Mathematical Engineering and CMM, Universidad de Chile & UMI-CNRS 2807, Santiago, Chile [email protected] Abstract We study the minimum weight basis problem on matroid when elements’ weights are uncertain. For each element we only know a set of possible values (an uncertainty area) that contains its real weight. In some cases there exist bases that are uniformly optimal, that is, they are minimum weight bases for every possible weight function obeying the uncertainty areas. In other cases, computing such a basis is not possible unless we perform some queries for the exact value of some elements. Our main result is a polynomial time algorithm for the following problem. Given a matroid with uncertainty areas and a query cost function on its elements, find the set of elements of minimum total cost that we need to simultaneously query such that, no matter their revelation, the resulting instance admits a uniformly optimal base. We also provide combinatorial characterizations of all uniformly optimal bases, when one exists; and of all sets of queries that can be performed so that after revealing the corresponding weights the resulting instance admits a uniformly optimal base. 2012 ACM Subject Classification Mathematics of computing → Matroids and greedoids Keywords and phrases Minimum spanning tree, matroids, uncertainty, queries Funding Work supported by Conicyt via Fondecyt 1181180 and PIA AFB-170001. 1 Introduction We study fundamental combinatorial optimization problems on weighted structures where the numerical data is uncertain but it can be queried at a cost.
  • The Tutte Polynomial of Some Matroids Arxiv:1203.0090V1 [Math

    The Tutte Polynomial of Some Matroids Arxiv:1203.0090V1 [Math

    The Tutte polynomial of some matroids Criel Merino∗, Marcelino Ram´ırez-Iba~nezy Guadalupe Rodr´ıguez-S´anchezz March 2, 2012 Abstract The Tutte polynomial of a graph or a matroid, named after W. T. Tutte, has the important universal property that essentially any mul- tiplicative graph or network invariant with a deletion and contraction reduction must be an evaluation of it. The deletion and contraction operations are natural reductions for many network models arising from a wide range of problems at the heart of computer science, engi- neering, optimization, physics, and biology. Even though the invariant is #P-hard to compute in general, there are many occasions when we face the task of computing the Tutte polynomial for some families of graphs or matroids. In this work we compile known formulas for the Tutte polynomial of some families of graphs and matroids. Also, we give brief explanations of the techniques that were use to find the for- mulas. Hopefully, this will be useful for researchers in Combinatorics arXiv:1203.0090v1 [math.CO] 1 Mar 2012 and elsewhere. ∗Instituto de Matem´aticas,Universidad Nacional Aut´onomade M´exico,Area de la Investigaci´onCient´ıfica, Circuito Exterior, C.U. Coyoac´an04510, M´exico,D.F.M´exico. e-mail:[email protected]. Supported by Conacyt of M´exicoProyect 83977 yEscuela de Ciencias, Universidad Aut´onoma Benito Ju´arez de Oaxaca, Oaxaca, M´exico.e-mail:[email protected] zDepartamento de Ciencias B´asicas Universidad Aut´onoma Metropolitana, Az- capozalco, Av. San Pablo No. 180, Col. Reynosa Tamaulipas, Azcapozalco 02200, M´exico D.F.
  • Automatic Workflow for Roof Extraction and Generation of 3D

    Automatic Workflow for Roof Extraction and Generation of 3D

    International Journal of Geo-Information Article Automatic Workflow for Roof Extraction and Generation of 3D CityGML Models from Low-Cost UAV Image-Derived Point Clouds Arnadi Murtiyoso * , Mirza Veriandi, Deni Suwardhi , Budhy Soeksmantono and Agung Budi Harto Remote Sensing and GIS Group, Bandung Institute of Technology (ITB), Jalan Ganesha No. 10, Bandung 40132, Indonesia; [email protected] (M.V.); [email protected] (D.S.); [email protected] (B.S.); [email protected] (A.B.H.) * Correspondence: arnadi_ad@fitb.itb.ac.id Received: 6 November 2020; Accepted: 11 December 2020; Published: 12 December 2020 Abstract: Developments in UAV sensors and platforms in recent decades have stimulated an upsurge in its application for 3D mapping. The relatively low-cost nature of UAVs combined with the use of revolutionary photogrammetric algorithms, such as dense image matching, has made it a strong competitor to aerial lidar mapping. However, in the context of 3D city mapping, further 3D modeling is required to generate 3D city models which is often performed manually using, e.g., photogrammetric stereoplotting. The aim of the paper was to try to implement an algorithmic approach to building point cloud segmentation, from which an automated workflow for the generation of roof planes will also be presented. 3D models of buildings are then created using the roofs’ planes as a base, therefore satisfying the requirements for a Level of Detail (LoD) 2 in the CityGML paradigm. Consequently, the paper attempts to create an automated workflow starting from UAV-derived point clouds to LoD 2-compatible 3D model.
  • A Unified Algorithmic Framework for Multi-Dimensional Scaling

    A Unified Algorithmic Framework for Multi-Dimensional Scaling

    A Unified Algorithmic Framework for Multi-Dimensional Scaling † ‡ Arvind Agarwal∗ Jeff M. Phillips Suresh Venkatasubramanian Abstract In this paper, we propose a unified algorithmic framework for solving many known variants of MDS. Our algorithm is a simple iterative scheme with guaranteed convergence, and is modular; by changing the internals of a single subroutine in the algorithm, we can switch cost functions and target spaces easily. In addition to the formal guarantees of convergence, our algorithms are accurate; in most cases, they converge to better quality solutions than existing methods, in comparable time. We expect that this framework will be useful for a number of MDS variants that have not yet been studied. Our framework extends to embedding high-dimensional points lying on a sphere to points on a lower di- mensional sphere, preserving geodesic distances. As a compliment to this result, we also extend the Johnson- 2 Lindenstrauss Lemma to this spherical setting, where projecting to a random O((1=" ) log n)-dimensional sphere causes "-distortion. 1 Introduction Multidimensional scaling (MDS) [23, 10, 3] is a widely used method for embedding a general distance matrix into a low dimensional Euclidean space, used both as a preprocessing step for many problems, as well as a visualization tool in its own right. MDS has been studied and used in psychology since the 1930s [35, 33, 22] to help visualize and analyze data sets where the only input is a distance matrix. More recently MDS has become a standard dimensionality reduction and embedding technique to manage the complexity of dealing with large high dimensional data sets [8, 9, 31, 6].
  • Lecture 0: Matroid Basics

    Lecture 0: Matroid Basics

    Parameterized Algorithms using Matroids Lecture 0: Matroid Basics Saket Saurabh The Institute of Mathematical Sciences, India and University of Bergen, Norway. ADFOCS 2013, MPI, August 04-09, 2013 Kruskal's Greedy Algorithm for MWST Let G = (V; E) be a connected undirected graph and let ≥0 w : E ! R be a weight function on the edges. Kruskal's so-called greedy algorithm is as follows. The algorithm consists of selecting successively edges e1; e2; : : : ; er. If edges e1; e2; : : : ; ek has been selected, then an edge e 2 E is selected so that: 1 e=2f e1; : : : ; ekg and fe; e1; : : : ; ekg is a forest. 2 w(e) is as small as possible among all edges e satisfying (1). We take ek+1 := e. If no e satisfying (1) exists then fe1; : : : ; ekg is a spanning tree. Kruskal's Greedy Algorithm for MWST Let G = (V; E) be a connected undirected graph and let ≥0 w : E ! R be a weight function on the edges. Kruskal's so-called greedy algorithm is as follows. The algorithm consists of selecting successively edges e1; e2; : : : ; er. If edges e1; e2; : : : ; ek has been selected, then an edge e 2 E is selected so that: 1 e=2f e1; : : : ; ekg and fe; e1; : : : ; ekg is a forest. 2 w(e) is as small as possible among all edges e satisfying (1). We take ek+1 := e. If no e satisfying (1) exists then fe1; : : : ; ekg is a spanning tree. It is obviously not true that such a greedy approach would lead to an optimal solution for any combinatorial optimization problem.
  • Branch-Depth: Generalizing Tree-Depth of Graphs

    Branch-Depth: Generalizing Tree-Depth of Graphs

    Branch-depth: Generalizing tree-depth of graphs ∗1 †‡23 34 Matt DeVos , O-joung Kwon , and Sang-il Oum† 1Department of Mathematics, Simon Fraser University, Burnaby, Canada 2Department of Mathematics, Incheon National University, Incheon, Korea 3Discrete Mathematics Group, Institute for Basic Science (IBS), Daejeon, Korea 4Department of Mathematical Sciences, KAIST, Daejeon, Korea [email protected], [email protected], [email protected] November 5, 2020 Abstract We present a concept called the branch-depth of a connectivity function, that generalizes the tree-depth of graphs. Then we prove two theorems showing that this concept aligns closely with the no- tions of tree-depth and shrub-depth of graphs as follows. For a graph G = (V, E) and a subset A of E we let λG(A) be the number of vertices incident with an edge in A and an edge in E A. For a subset X of V , \ let ρG(X) be the rank of the adjacency matrix between X and V X over the binary field. We prove that a class of graphs has bounded\ tree-depth if and only if the corresponding class of functions λG has arXiv:1903.11988v2 [math.CO] 4 Nov 2020 bounded branch-depth and similarly a class of graphs has bounded shrub-depth if and only if the corresponding class of functions ρG has bounded branch-depth, which we call the rank-depth of graphs. Furthermore we investigate various potential generalizations of tree- depth to matroids and prove that matroids representable over a fixed finite field having no large circuits are well-quasi-ordered by restriction.
  • A Dual Fano, and Dual Non-Fano Matroidal Network Stephen Lee Johnson California State University - San Bernardino, 002445637@Coyote.Csusb.Edu

    A Dual Fano, and Dual Non-Fano Matroidal Network Stephen Lee Johnson California State University - San Bernardino, [email protected]

    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CSUSB ScholarWorks California State University, San Bernardino CSUSB ScholarWorks Electronic Theses, Projects, and Dissertations Office of Graduate Studies 6-2016 A Dual Fano, and Dual Non-Fano Matroidal Network Stephen Lee Johnson California State University - San Bernardino, [email protected] Follow this and additional works at: http://scholarworks.lib.csusb.edu/etd Part of the Other Mathematics Commons Recommended Citation Johnson, Stephen Lee, "A Dual Fano, and Dual Non-Fano Matroidal Network" (2016). Electronic Theses, Projects, and Dissertations. Paper 340. This Thesis is brought to you for free and open access by the Office of Graduate Studies at CSUSB ScholarWorks. It has been accepted for inclusion in Electronic Theses, Projects, and Dissertations by an authorized administrator of CSUSB ScholarWorks. For more information, please contact [email protected]. A Dual Fano, and Dual Non-Fano Matroidal Network A Thesis Presented to the Faculty of California State University, San Bernardino In Partial Fulfillment of the Requirements for the Degree Master of Arts in Mathematics by Stephen Lee Johnson June 2016 A Dual Fano, and Dual Non-Fano Matroidal Network A Thesis Presented to the Faculty of California State University, San Bernardino by Stephen Lee Johnson June 2016 Approved by: Dr. Chris Freiling, Committee Chair Date Dr. Wenxiang Wang, Committee Member Dr. Jeremy Aikin, Committee Member Dr. Charles Stanton, Chair, Dr. Corey Dunn Department of Mathematics Graduate Coordinator, Department of Mathematics iii Abstract Matroidal networks are useful tools in furthering research in network coding. They have been used to show the limitations of linear coding solutions.
  • Geant4 Integration Status

    Geant4 Integration Status

    Geant4 integration status ● All CMS shapes integrated: box, tube, trapezoid, cone, polycone, and polyhedra – stress-tested with Geant4 test job FullCMS, a few warnings printed multiple times, ~ all of them coming from polycone or polyhedron special cases. – several fixes uploaded, job still crashes using VecGeom shapes (runs to the end using Geant4 or USolids shapes) ● ...and a few more: Trd, torus2, paraboloid, orb, sphere – integrated, but not stress-tested yet, since not used in FullCMS geant4 test job. G. Lima – GeantV weekly meeting – 2015/10/27 1 During code sprint ● With Sandro, fixed few more bugs with the tube (point near phi- section surface) and polycone (near a ~vertical section) ● Changes in USolids shapes, to inherit from specialized shapes instead of “simple shapes” ● Crashes due to Normal() calculation returning (0,0,0) when fully inside, but at the z-plane between sections z Both sections need to be checked, and several possible section outcomes Q combined → normals added: +z + (-z) = (0,0,0) G. Lima – GeantV weekly meeting – 2015/10/27 2 Geant4 integration status ● What happened since our code sprint? – More exceptions related to Inside/Contains inconsistencies, triggered at Geant4 navigation tests ● outPoint, dir → surfPoint = outPoint + step * dir ● Inside(surfPoint) → kOutside (should be surface) Both sections need to be checked, several possible section outcomes combined, e.g. in+in = in surf + out = surf, etc... P G. Lima – GeantV weekly meeting – 2015/10/27 3 Geant4 integration status ● What happened after our code sprint? – Normal() fix required a similar set of steps → code duplication ● GenericKernelContainsAndInside() ● ConeImplementation::NormalKernel() (vector mode) ● UnplacedCone::Normal() (scalar mode) – First attempt involved Normal (0,0,0) and valid=false for all points away from surface --- unacceptable for navigation – Currently a valid normal is P always provided, BUT it is not always the best one (a performance priority choice) n G.
  • Arxiv:2008.03027V2 [Math.CO] 19 Nov 2020 an Optimal Way

    Arxiv:2008.03027V2 [Math.CO] 19 Nov 2020 an Optimal Way

    A Whitney type theorem for surfaces: characterising graphs with locally planar embeddings Johannes Carmesin University of Birmingham November 20, 2020 Abstract We prove that for any parameter r an r-locally 2-connected graph G embeds r-locally planarly in a surface if and only if a certain ma- troid associated to the graph G is co-graphic. This extends Whitney's abstract planar duality theorem from 1932. 1 Introduction A fundamental question in Structural Graph Theory is how to embed graphs in surfaces. There are two main lines of research. Firstly, in the specific embedding problem we have a fixed surface and are interested in embedding a given graph in that surface. Mohar proved the existence of a linear time algorithm for this problem, solving the al- gorithmic aspect of this problem [16]. This proof was later simplified by Kawarabayashi, Mohar and Reed [13]. Secondly, in the general embedding problem, we are interested in finding for a given graph a surface so that the graph embeds in that surface in arXiv:2008.03027v2 [math.CO] 19 Nov 2020 an optimal way. Usually people used minimum genus as the optimality criterion [13, 15, 17]. However, in 1989 Thomassen showed that with this interpretation, the problem would be NP-hard [24]. Instead of minimum genus, here we use local planarity as our optimality criterion { and provide a polynomial algorithm for the general embedding problem. While this discrepancy in the algorithmic complexity implies that maxi- mally locally planar embeddings cannot always be of minimum genus, Thomassen 1 showed that they are of minimum genus if all face boundaries are shorter than non-contractible cycles of the embedding [25].
  • A Dual Fano, and Dual Non-Fano Matroidal Network

    A Dual Fano, and Dual Non-Fano Matroidal Network

    California State University, San Bernardino CSUSB ScholarWorks Electronic Theses, Projects, and Dissertations Office of aduateGr Studies 6-2016 A Dual Fano, and Dual Non-Fano Matroidal Network Stephen Lee Johnson California State University - San Bernardino Follow this and additional works at: https://scholarworks.lib.csusb.edu/etd Part of the Other Mathematics Commons Recommended Citation Johnson, Stephen Lee, "A Dual Fano, and Dual Non-Fano Matroidal Network" (2016). Electronic Theses, Projects, and Dissertations. 340. https://scholarworks.lib.csusb.edu/etd/340 This Thesis is brought to you for free and open access by the Office of aduateGr Studies at CSUSB ScholarWorks. It has been accepted for inclusion in Electronic Theses, Projects, and Dissertations by an authorized administrator of CSUSB ScholarWorks. For more information, please contact [email protected]. A Dual Fano, and Dual Non-Fano Matroidal Network A Thesis Presented to the Faculty of California State University, San Bernardino In Partial Fulfillment of the Requirements for the Degree Master of Arts in Mathematics by Stephen Lee Johnson June 2016 A Dual Fano, and Dual Non-Fano Matroidal Network A Thesis Presented to the Faculty of California State University, San Bernardino by Stephen Lee Johnson June 2016 Approved by: Dr. Chris Freiling, Committee Chair Date Dr. Wenxiang Wang, Committee Member Dr. Jeremy Aikin, Committee Member Dr. Charles Stanton, Chair, Dr. Corey Dunn Department of Mathematics Graduate Coordinator, Department of Mathematics iii Abstract Matroidal networks are useful tools in furthering research in network coding. They have been used to show the limitations of linear coding solutions. In this paper we examine the basic information on network coding and matroid theory.