Maximum-Sized Matroids with No Minors Isomorphic to U2,5, F7, F−7 or P7

2001 ◽  
Vol 10 (6) ◽  
pp. 531-542
Author(s):  
STEFAN T. MECAY
Keyword(s):  
Fano Plane ◽  
Point Line

Let [Mscr ] be the class of simple matroids which do not contain the 5-point line U2,5, the Fano plane F7, the non-Fano plane F−7, or the matroid P7 as minors. Let h(n) be the maximum number of points in a rank-n matroid in [Mscr ]. We show that h(2) = 4, h(3) = 7, and h(n) = (n+12) for n [ges ] 4, and we also find all the maximum-sized matroids for each rank.

scholarly journals Veldkamp spaces: From (Dynkin) diagrams to (Pauli) groups

2017 ◽  
Vol 14 (05) ◽  
pp. 1750080
Author(s):  
Metod Saniga ◽  
Frédéric Holweck ◽  
Petr Pracna
Keyword(s):  
Incidence Structure ◽  
Polar Space ◽  
Exceptional Point ◽  
Specific Point ◽  
Fano Plane ◽  
Magic Square ◽  
Dynkin Diagrams ◽  
Pauli Matrix ◽  
Matrix Formula ◽  
Point Line

Regarding a Dynkin diagram as a specific point-line incidence structure (where each line has just two points), one can associate with it a Veldkamp space. Focusing on extended Dynkin diagrams of type [Formula: see text], [Formula: see text], it is shown that the corresponding Veldkamp space always contains a distinguished copy of the projective space PG[Formula: see text]. Proper labeling of the vertices of the diagram (for [Formula: see text]) by particular elements of the two-qubit Pauli group establishes a bijection between the 15 elements of the group and the 15 points of the PG[Formula: see text]. The bijection is such that the product of three elements lying on the same line is the identity and one also readily singles out that particular copy of the symplectic polar space [Formula: see text] of the PG[Formula: see text] whose lines correspond to triples of mutually commuting elements of the group; in the latter case, in addition, we arrive at a unique copy of the Mermin–Peres magic square. In the case of [Formula: see text], a more natural labeling is that in terms of elements of the three-qubit Pauli group, furnishing a bijection between the 63 elements of the group and the 63 points of PG[Formula: see text], the latter being the maximum projective subspace of the corresponding Veldkamp space; here, the points of the distinguished PG[Formula: see text] are in a bijection with the elements of a two-qubit subgroup of the three-qubit Pauli group, yielding a three-qubit version of the Mermin–Peres square. Moreover, save for [Formula: see text], each Veldkamp space is also endowed with some exceptional point(s). Interestingly, two such points in the [Formula: see text] case define a unique Fano plane whose inherited three-qubit labels feature solely the Pauli matrix [Formula: see text].

scholarly journals Twisted cubic and point-line incidence matrix in $$\mathrm {PG}(3,q)$$

2021 ◽  
Vol 89 (10) ◽  
pp. 2211-2233 ◽  
Author(s):  
Alexander A. Davydov ◽  
Stefano Marcugini ◽  
Fernanda Pambianco
Keyword(s):  
Incidence Matrix ◽  
Twisted Cubic ◽  
Point Line

scholarly journals A Multi-Feature Fusion Slam System Attaching Semantic Invariant to Points and Lines

Sensors ◽  
2021 ◽  
Vol 21 (4) ◽  
pp. 1196
Author(s):  
Gang Li ◽  
Yawen Zeng ◽  
Huilan Huang ◽  
Shaojian Song ◽  
Bin Liu ◽  
...  
Keyword(s):  
Feature Matching ◽  
Feature Fusion ◽  
Error Function ◽  
Line Segments ◽  
Cumulative Error ◽  
Localization And Mapping ◽  
Tracking Process ◽  
Line Features ◽  
Point Line ◽  
Segment Data

The traditional simultaneous localization and mapping (SLAM) system uses static points of the environment as features for real-time localization and mapping. When there are few available point features, the system is difficult to implement. A feasible solution is to introduce line features. In complex scenarios containing rich line segments, the description of line segments is not strongly differentiated, which can lead to incorrect association of line segment data, thus introducing errors into the system and aggravating the cumulative error of the system. To address this problem, a point-line stereo visual SLAM system incorporating semantic invariants is proposed in this paper. This system improves the accuracy of line feature matching by fusing line features with image semantic invariant information. When defining the error function, the semantic invariant is fused with the reprojection error function, and the semantic constraint is applied to reduce the cumulative error of the poses in the long-term tracking process. Experiments on the Office sequence of the TartanAir dataset and the KITTI dataset show that this system improves the matching accuracy of line features and suppresses the cumulative error of the SLAM system to some extent, and the mean relative pose error (RPE) is 1.38 and 0.0593 m, respectively.

Point Cloud Georeferencing Based on Fixed Control Primitives of Point, Line, and Surface

2021 ◽  
Vol 58 (8) ◽  
pp. 0828001
Author(s):  
王家晖 Wang Jiahui ◽  
姚吉利 Yao Jili ◽  
赵雪莹 Zhao Xueying ◽  
胡信志 Hu Xinzhi ◽  
赵猛 Zhao Meng
Keyword(s):  
Point Cloud ◽  
Point Line

Characterization and Coordination of Point-line Trajectories

2004 ◽  
Vol 127 (3) ◽  
pp. 502-505 ◽  
Author(s):  
Kwun-Lon Ting ◽  
Yi Zhang ◽  
Ruj Bunduwongse
Keyword(s):  
Orientation Relationship ◽  
Complete Characterization ◽  
Free Form ◽  
Directed Line ◽  
Free Form Surface ◽  
Point Line ◽  
Line Trajectory ◽  
The Relationship ◽  
Line Direction

A point-line refers to a rigid combination of a directed line and an endpoint along the line. To trace a point-line trajectory, one must control not only the trajectory of the endpoint (the directrix) but also the direction of the point-line (the indicatrix). This paper addresses three issues on point-line trajectories. First of all, by considering the relationship between the point trajectory and the corresponding point-line direction, it offers the complete characterization of point-line trajectories. It presents the coordination of the point-line axis with a free point trajectory and also with a point trajectory constrained on a free form surface. The issue of maintaining an invariant orientation relationship between the point-line axis and the point trajectory is also addressed.

Study on the Conversion Method of Three-Dimensional Shape Prediction Model of BGA Solder Joints

2013 ◽  
Vol 706-708 ◽  
pp. 1693-1696
Author(s):  
Hua Bin Zhao ◽  
De Jian Zhou
Keyword(s):  
Solder Joints ◽  
Three Dimensional ◽  
Ball Grid Array ◽  
Analysis Software ◽  
Surface Evolver ◽  
Element Analysis ◽  
Shape Prediction ◽  
Dimensional Shape ◽  
Point Line ◽  
Three Dimensional Shape

In the study of three-dimensional shape prediction of SMT solder joints, the software Surface Evolver has been widely applied as a quick and accurate effective tool for the prediction of solder joints shape. But the model it builds is not able to be directly imported into any finite element analysis software like ANSYS, and even after the import it still needs a lot of time to mend the import model. For this issue, to predict of the solder joints shape of ball grid array (BGA), the implement programs of three conversion methods of point-line-area method, axisymmetric method and infinitesimal method are given. By comparison, axisymmetric method and infinitesimal method are more suitable for the shape conversion of BGA solder joints.

scholarly journals PEMANFAATAN AUTOGRAPH SEBAGAI MEDIA PEMBELAJARAN MATEMATIKA DENGAN MENERAPKAN MODEL PEMBELAJARAN BERBASIS MASALAH (PBM) AUTOGRAPH USE AS A LEARNING MEDIA WITH MATH APPLYING THE MODEL OF LEARNING BASED PROBLEM (LBP)

2016 ◽  
Vol 2 (1) ◽  
Author(s):  
Nailul Himmi Hasibuan
Keyword(s):  
Problem Based Learning ◽  
Thinking Skills ◽  
Critical Thinking Skills ◽  
Teaching Mathematics ◽  
Conic Sections ◽  
Mathematical Communication ◽  
Learning And Teaching ◽  
Learning Mathematics ◽  
Problem Solving Process ◽  
Point Line

Tulisan ini mengkaji tentang pemanfaatan autograph sebagai sumber belajar matematika yang dikombinasikan dengan Pembelajaran Berbasis Masalah (PBL). Pembelajaran berbasis masalah adalah suatu pembelajaran yang dimulai dengan masalah dunia nyata sehingga siswa agar dapat mengkonstruksi pengetahuannya sendiri dengan kemampuan berpikir yang dimilikinya, dengan mengacu pada lima langkah pokok, yaitu: orientasi siswa pada masalah, mengorganisir siswa untuk belajar, membimbing penyelidikan individual maupun kelompok, mengembangkan dan menyajikan hasil karya dan menganalisis dan mengevaluasi proses pemecahan masalah. Autograph adalah software software yang sangat serbaguna dan dinamis sebagai media pembelajaran untuk belajar dan mengajar matematika tingkat menengah berupa menggambar titik, ruas garis, vektor, garis, poligon, irisan kerucut, dan kurva dua dimensi. Dengan menerapkan model pembelajaran berbasis masalah yang memanfaatkan autograph dapat meningkatakan pemahaman konsep, penalaran, kemampuan berpikir kritis dan komunikasi matematis siswa. Kata Kunci: Pembelajaran Matematika, PBL, dan  AutographThis paper examines the use of autographs as a source of learning mathematics combined with Problem Based Learning (PBL). Problem-based learning is a learning that begins with real world issues so that students in order to construct his own knowledge with the ability to think it has, with reference to the five main steps, namely: orientation of students on the issue, organizing students to learn, guiding the investigation of individual and group, develop and present the work and analyze and evaluate the problem-solving process. Autograph software is software that is very versatile and dynamic as a learning medium for learning and teaching mathematics in the form of a mid-level draw point, line segment, vector, line, polygon, conic sections, and two-dimensional curve. By implementing problem based learning model that utilizes autographs can increase the understanding of concepts, reasoning, critical thinking skills and students' mathematical communication.                          Keywords: Learning Mathematics, PBL, and Autograph

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