scholarly journals On a conjecture concerning helly circle graphs

2003 ◽  
Vol 23 (1) ◽  
pp. 221-229 ◽  
Author(s):  
Guillermo Durán ◽  
Agustín Gravano ◽  
Marina Groshaus ◽  
Fábio Protti ◽  
Jayme L. Szwarcfiter
Keyword(s):  
Systems Of Equations ◽  
Intersection Graphs ◽  
Circle Graph ◽  
Circle Graphs ◽  
Analytic Tool ◽  
Two Systems ◽  
Cartesian Plane ◽  
Definition Of ◽  
Straight Lines

We say that G is an e-circle graph if there is a bijection between its vertices and straight lines on the cartesian plane such that two vertices are adjacent in G if and only if the corresponding lines intersect inside the circle of radius one. This definition suggests a method for deciding whether a given graph G is an e-circle graph, by constructing a convenient system S of equations and inequations which represents the structure of G, in such a way that G is an e-circle graph if and only if S has a solution. In fact, e-circle graphs are exactly the circle graphs (intersection graphs of chords in a circle), and thus this method provides an analytic way for recognizing circle graphs. A graph G is a Helly circle graph if G is a circle graph and there exists a model of G by chords such that every three pairwise intersecting chords intersect at the same point. A conjecture by Durán (2000) states that G is a Helly circle graph if and only if G is a circle graph and contains no induced diamonds (a diamond is a graph formed by four vertices and five edges). Many unsuccessful efforts - mainly based on combinatorial and geometrical approaches - have been done in order to validate this conjecture. In this work, we utilize the ideas behind the definition of e-circle graphs and restate this conjecture in terms of an equivalence between two systems of equations and inequations, providing a new, analytic tool to deal with it.

Reconciling hospital-acquired complications and CHADx+ in Victorian coded hospital data

2018 ◽  
Vol 48 (2) ◽  
pp. 76-86 ◽  
Author(s):  
Jennie Shepheard ◽  
Elsa Lapiz ◽  
Carla Read ◽  
Terri J Jackson
Keyword(s):  
Quality Improvement ◽  
Adverse Events ◽  
Classification Systems ◽  
Hospital Inpatient ◽  
Hospital Data ◽  
Two Systems ◽  
Icd 10 ◽  
Definition Of ◽  
Hospital Acquired ◽  
Coded Hospital Data

Background: The Council of Australian Governments has focused the attention of health service managers and state health departments on a list of hospital-acquired complications (HACs) proposed as the basis of funding adjustments for poor quality of hospital inpatient care. These were devised for the Australian Commission on Safety and Quality in Health Care as a subset of their earlier classification of hospital-acquired complications (CHADx) and designed to be used by health services to monitor safety performance for their admitted patients. Objective: To improve uptake of both classification systems by clarifying their purposes and by reconciling the ICD-10-AM code sets used in HACs and the Victorian revisions to the CHADx system (CHADx+). Method: Frequency analysis of individual clinical codes with condition onset flag (COF 1) included in both classification systems using the Victorian Admitted Episodes Dataset for 2014/2015 ( n = 2,623,275 separations). Narrative description of the resulting differences in definition of “adverse events” embodied in the two systems. Results: As expected, a high proportion of ICD-10-AM codes used in the HACs also appear in CHADx+, and given the wider scope of CHADx+, it uses a higher proportion of all COF 1 diagnoses than HACs (82% vs. 10%). This leads to differing estimates of rates of adverse events: 2.12% of cases for HACs and 11.13% for CHADx+. Most CHADx classes (70%) are not covered by the HAC system; discrepancies result from the exclusion from HACs of several major CHADx+ groups and from a narrower definition of detailed HAC classes compared with CHADx+. Case exclusion criteria in HACs (primarily mental health admissions) resulted in a very small proportion of discrepancies (0.13%) between systems. Discussion: Issues of purpose and focus of these two Australian systems, HACs for clinical governance and CHADx+ for local quality improvement, explain many of the differences between them, and their approach to preventability, and risk stratification. Conclusion: A clearer delineation between these two systems using routinely coded hospital data will assist funders, clinicians, quality improvement professionals and health information managers to understand discrepancies in case identification between them and support their different information needs.

History of Air Maps and Charts

1998 ◽  
Vol 51 (2) ◽  
pp. 203-215 ◽  
Author(s):  
Philip Steele
Keyword(s):  
Nineteenth Century ◽  
Ordnance Survey ◽  
History Of ◽  
Minor Significance ◽  
Aerial Navigation ◽  
The Difference ◽  
Definition Of ◽  
Straight Lines ◽  
The Hill

There is no generally accepted definition of the difference between a map and a chart. A widespread feeling probably exists favouring the old saying that maps are to look at and charts to work on. It is true that the term ‘aeronautical chart’ gained a general currency over alternative terms as contact flying gave way to aerial navigation. But, in this paper, the terms ‘map’ and ‘chart’ will be used as seems appropriate to each occasion, without attempt to conform to any particular definition.We can get an idea of what was available to the earliest aviators by looking at an Ordnance Survey reprint of one of their nineteenth century maps (Fig. 1). They are printed in one colour only, black on white. By far the predominant feature is the hill shading. Quite gentle hills are hachured with a heaviness which tends to obscure both natural features like rivers, lakes and woodlands and man-made constructions such as towns and villages, roads, canals and railways. Hills are, of course, very important features to those on the ground, since they limit the extent to which other features can be seen. To the soldier, the significance of high ground is self-evident, and it was principally for the ordnance requirements of soldiers that these maps had been developed. But when men began to view the ground from the air, the perspective changed. Hills appeared flattened out and, provided that you knew the height of the tallest in the area and were sure none would impede your take-off or landing, were of minor significance. Lakes and woods, though, were spread out before you in their distinctive shapes, while railway lines and canals presented bold straight lines and curves, and rivers their unique courses, to your view. The need was for new kinds of maps which would give due prominence to such features.

scholarly journals A Characterization of Circle Graphs in Terms of Multimatroid Representations

10.37236/6992 ◽  
2020 ◽  
Vol 27 (1) ◽  
Author(s):  
Robert Brijder ◽  
Lorenzo Traldi
Keyword(s):  
Simple Graph ◽  
Rank Function ◽  
Binary Matroid ◽  
Circle Graph ◽  
Circle Graphs ◽  
Isotropic System

The isotropic matroid $M[IAS(G)]$ of a looped simple graph $G$ is a binary matroid equivalent to the isotropic system of $G$. In general, $M[IAS(G)]$ is not regular, so it cannot be represented over fields of characteristic $\neq 2$. The ground set of $M[IAS(G)]$ is denoted $W(G)$; it is partitioned into 3-element subsets corresponding to the vertices of $G$. When the rank function of $M[IAS(G)]$ is restricted to subtransversals of this partition, the resulting structure is a multimatroid denoted $\mathcal{Z}_{3}(G)$. In this paper we prove that $G$ is a circle graph if and only if for every field $\mathbb{F}$, there is an $\mathbb{F}$-representable matroid with ground set $W(G)$, which defines $\mathcal{Z}_{3}(G)$ by restriction. We connect this characterization with several other circle graph characterizations that have appeared in the literature.

My Equations Are the Same as Yours!

2012 ◽  
pp. 259-273
Author(s):  
M Badger ◽  
C J Sangwin
Keyword(s):  
Computer Algebra ◽  
Computer Algebra System ◽  
System Of Equations ◽  
Systems Of Equations ◽  
Computer Aided ◽  
Two Systems

In this chapter we explain how computer aided assessment (CAA) can automatically assess an answer that consists of a system of equations. In particular, we will use a computer algebra system (CAS) and Buchberger’s Algorithm to establish when two systems of equations are the “same.”

Linear Functions with Two Points of Intersection?

1995 ◽  
Vol 88 (5) ◽  
pp. 376-378
Author(s):  
Michael A. Contino
Keyword(s):  
Euclidean Space ◽  
Graphing Calculator ◽  
Linear Functions ◽  
First Year ◽  
Systems Of Equations ◽  
College Mathematics ◽  
Algebra Students ◽  
Straight Lines

Of course two straight lines in Euclidean space cannot intersect in more than one point unless they are the same line and intersect everywhere—or can they? Follow this problem on the graphing calculator, however, and the surprising twist that gives this article its name will be seen. The material covered should be readily accessible to first-year-algebra students who have studied systems of equations, but it also contains valuable lessons for college mathematics professors who have been easily deceived by its apparent simplicity and familiarity.

scholarly journals Box and Segment Intersection Graphs with Large Girth and Chromatic Number

2021 ◽  
Author(s):  
James Davies
Keyword(s):  
Chromatic Number ◽  
Intersection Graphs ◽  
Large Girth ◽  
Straight Lines

We prove that there are intersection graphs of axis-aligned boxes in R3 and intersection graphs of straight lines in R3 that have arbitrarily large girth and chromatic number.

Formal systems of constructive mathematics

1956 ◽  
Vol 21 (1) ◽  
pp. 63-75
Author(s):  
M. H. Löb
Keyword(s):  
Constructive Mathematics ◽  
Formal Systems ◽  
Two Systems ◽  
Universal Quantification ◽  
Definition Of

In the present paper we investigate in some detail two systems closely related to those contained in [4]. The system K1 of [4] is simplified by eliminating conjunction and disjunction, giving rise to a system Σ whose primitives are two individuals, concatenation, identity, existential and limited universal quantification. Any relation expressible in Σ may be expressed in the form (Eα1)(β)α1(Eα2)…(Eαn)(γ = δ). A further reduction of primitives leads to a system Σμ, which differs from Σ by containing the operator μ (i.e. “the first string, such that” with respect to a certain ordering) in place of the quantifiers. Reasons are given for restricting Myhill's definition of constructive ideas to those expressible in Σμ.Acquaintance with [4], [5], [6] is assumed. We deviate from [4] by dropping concatenation-signs and parentheses in chain matrices from the official notation, and, furthermore, by using Greek and German letters as metalinguistic variables standing for chain matrices and chains, respectively, while retaining small Roman letters (excluding a, b) for use as variables of the systems.

Graphic-Analytical Researches of Involution

2017 ◽  
Vol 5 (1) ◽  
pp. 3-11 ◽  
Author(s):  
Графский ◽  
O. Grafskiy ◽  
Усманов ◽  
A. Usmanov ◽  
Холодилов ◽  
...  
Keyword(s):  
Cross Ratio ◽  
Rectangular Coordinate System ◽  
Projective Transformations ◽  
Coordinate Axis ◽  
Graphic Editor ◽  
Applied Informatics ◽  
The Cross ◽  
Pass Through ◽  
Definition Of ◽  
Straight Lines

Known projective transformations, namely their private types such as harmonism and involution are considered. It is known that projective transformations are collinear, at their performance the order, the cross ratio of fours of elements (on a straight line — the cross ratio of four points, in a bunch of straight lines — the cross ratio of this bunch’s four straight lines, this property (invariant) is similarly preserved for a bunch of planes, i.e. in considering of first step forms) is preserved. At a constructive approach to such transformations there are some ways for definition of position for corresponding elements which students use when studying discipline "Affine and projective geometry" on preparation profiles 09.03.01 — “CAD Systems" and 09.03.03 — “Applied Informatics in Design”. The received constructions are checked by analytical calculations, proceeding from known dependences for harmonism and involutions. In such a case results both for a range of points, and for a bunch of straight lines which pass through these points are analytically compared. The provided computational and graphic work contains three sections: prospects, harmonism and involution, and is carried out by students on individual options with application of the graphic editor Microsoft Visio or the graphic package CO MPAS voluntary. In the present paper some constructions in definition of corresponding points in elliptic and hyperbolic involution are considered, some of these constructions are published for the first time. Besides, a proposition has been formulated: in a rectangular coordinate system the work for coordinates of two points related to a circle intersection with one coordinate axis is equal to the product for coordinates of two other points related to this circle intersection with the other coordinate axis. This proposition is fairly for imaginary points of circle intersection with coordinate axes as well.

Shape Features Detection Based on Hough Transform in Images

2014 ◽  
Vol 644-650 ◽  
pp. 1104-1106 ◽  
Author(s):  
Guang Li Chu ◽  
Yan Jie Wang
Keyword(s):  
Hough Transform ◽  
Template Matching ◽  
Traditional Method ◽  
Detection Method ◽  
Feature Points ◽  
Shape Features ◽  
Line Segments ◽  
Definition Of ◽  
Straight Lines ◽  
And Storage

Hough transform as an effective graphics target detection method can detect straight lines, circles, ellipses, parabolas and many other analytical graphics. The discretization of space, as well as the calculation of the process make Hough transform have some limitations, such as poor detection results because of high-intensity noise, a large amount of calculation, large demand of storage resources and so on. This paper analyzes the Hough Transform voting process and points out that the accumulation with 1 in the method is unreasonable. The paper proposed a Hough transform based on template matching via the modification of the definition of the traditional method. In this method, each parameter unit identifies a template in image space. The feature points according with the conditions can be searched by the template actively. The method takes the number of feature points as the value of parameter unit and takes the record of the coordinates of line segment endpoints. So line segments can be detected and storage resources can be saved.

scholarly journals Note on Gauss' Proof of the Reciprocity of Parallelism

1913 ◽  
Vol 32 ◽  
pp. 15-18
Author(s):  
D. M. Y. Sommerville
Keyword(s):  
Unique Line ◽  
Definition Of ◽  
Straight Lines

The proposition that if AA′‖BB′ then BB′‖AA′ appears at first sight so simple that it might be regarded as almost intuitive. This is because we already think of parallelism as a symmetrical relationship between two straight lines, in accordance with Euclid's definition of parallels as “straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction.” If we take along with this definition Euclid's fifth postulate, or Playfair's equivalent, it defines a unique line through a given point parallel to a given line; but, without the postulate, it cannot be assumed to define more than a class of lines, and a stricter definition is required.

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