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### The number of rooted 2-connected triangular maps on the projective plane

Zhi-Cheng Gao
1991 Journal of combinatorial theory. Series B (Print)
Let s0 and pii be the number of rooted loopless near-triangular maps on the sphere and projective plane, respectively, having i vertices and root face valency j.  ...  The 2-connected triangular maps on the projective plane for n = 3,4 are shown in Fig. 4.1 . From Table I , we observe that P,,~ is even iff n is a power of 2.  ...

### Exact enumeration of rooted 3-connected triangular maps on the projective plane

Zhicheng Gao, Jianyu Wang
2004 Discrete Applied Mathematics
We study the relation between rooted 3-connected triangular maps and rooted 2-connected triangular maps on the projective plane.  ...  We then use this relation to derive a simple parametric expression for the generating function of rooted 3-connected triangular maps on the projective plane.  ...  Let P(x) be the generating function of rooted 3-connected triangular maps on the projective plane, where x marks the total number of vertices.  ...

### The asymptotic number of rooted 2-connected triangular maps on a surface

Zhi Cheng Gao
1992 Journal of combinatorial theory. Series B (Print)
maps on the projective plane [4, 5] ), Bender and Wormald obtained the asymptotic number of rooted 2-connected maps on general surfaces [3] .  ...  INTRODUCTION A (rooted) triangular map on a surface is a (rooted) map on the surface [2] such that each face has valency three; a (rooted) near-triangular map on a surface is a (rooted) map on the surface  ...

### An optimal algorithm to generate rooted trivalent diagrams and rooted triangular maps

Samuel Alexandre Vidal
2010 Theoretical Computer Science
We shall describe and analyze an algorithm giving an exhaustive list of rooted trivalent diagrams of a given size (number of edges), the list being non-redundant in that no two diagrams of the list are  ...  A trivalent diagram is a connected, two-colored bipartite graph (parallel edges allowed but not loops) such that every black vertex is of degree 1 or 3 and every white vertex is of degree 1 or 2, with  ...  Comments and suggestions from the anonymous referees helped a lot to improve the clarity of the paper and the quality of presentation.  ...

### Freeform texture representation and characterisation based on triangular mesh projection techniques

Hussein S. Abdul-Rahman, Shan Lou, Wenhan Zeng, Xiangqian Jiang, Paul J. Scott
2016 Measurement (London)
Different projection algorithms have been adopted from the mesh projection techniques found in the field of computer graphics.  ...  This paper, however, shed the light on how to represent the texture of freeform non-Euclidean surfaces before calculating the parameters.  ...  Acknowledgement: The  ...

### Chromatic sums of rooted triangulations on the projective plane

Zhaoxiang Li, Yanpei Liu
2005 Journal of Applied Mathematics and Computing
In this paper we study the chromatic sum functions for rooted nonseparable near-triangular maps on the projective plane. A chromatic sum equation for such maps is obtained.  ...  Let S and P be the set of all rooted nonseparable near-triangular maps on the plane and the projective plane respectively.  ...  a set of maps on the projective plane, denoted it by L.  ...

### Page 3586 of Mathematical Reviews Vol. , Issue 92g [page]

1992 Mathematical Reviews
Palmer (1-MIS) 92g:05101 05C30 05A15 05C10 57M99 Gao, Zhi-Cheng (1-UCSD) The number of rooted 2-connected triangular maps on the projective plane. J. Combin. Theory Ser.  ...  B 53 (1991), no. 1, 130-142. in this paper the rooted loopless triangular maps on the projec- tive plane are enumerated by number of vertices.  ...

### A new view of combinatorial maps by Smarandache's notion [article]

Linfan Mao
2005 arXiv   pre-print
On a geometrical view, the conception of map geometries are introduced, which is a nice model of the Smarandache geometries, also new kind of and more general intrinsic geometry of surface.  ...  Results convinced one that map geometries are Smarandache geometries and their enumertion are obtained.  ...  sum of n, n ≥ 1, tori: a 1 b 1 a −1 1 b −1 1 a 2 b 2 a −1 2 b −1 2 · · · a n b n a −1 n b −1 n ; (Q n ) The connected sum of n, n ≥ 1, projective planes: a 1 a 1 a 2 a 2 · · · a n a n .  ...

### Page 7614 of Mathematical Reviews Vol. , Issue 2004j [page]

2004 Mathematical Reviews
The authors prove that a 5-connected graph in the projective plane contains a complete graph on 6 vertices as a minor if and only if it has face width at least 3.  ...  A plane tree is a finite rooted unlabeled loop-free connected graph in which the set of children of every vertex is endowed with a total order.  ...

### Compact symmetric spaces, triangular factorization, and Poisson geometry [article]

Arlo Caine
2006 arXiv   pre-print
It is shown that the action is Hamiltonian and the momentum map is computed using triangular factorization.  ...  Let X be a simply connected compact Riemannian symmetric space, let U be the universal covering group of the identity component of the isometry group of X, and let \g denote the complexification of the  ...  In this coordinate, the projection of π U (25) is given by π = −2iIm(w)(1 + |w| 2 ) ∂ ∂w ∧ ∂ ∂w . (27) w-plane z-plane SO (2) S(U(1) × U(1)) Im(w) = 0 |z| = 1 Figure 3 : The symplectic foliation for  ...

### How tangents solve algebraic equations, or a remarkable geometry of discriminant varieties

Gabriel Katz
2003 Expositiones mathematicae
We also connect the geometry of the Vi~te map Vd: Adot--~ Ad~oer, given by the elementary symmetric polynomials, with the tangents to the discfiminant varieties {Dd,~}.  ...  Let Dr, k denote the discriminant variety of degree d polynomials in one variable with at least one of its roots being of multiplicity > k, We prove that the tangent cones to Dd, k span Dr, k_ ~ thus,  ...  Thus, over the complex numbers, the plane curve 1rx(7)3) C ]P~ is inscribed in the triangular projective horizon. A similar property holds in the real case when P E L43.  ...

### Submaps of maps. I. General 0–1 laws

Edward A Bender, Zhi-Cheng Gao, L.Bruce Richmond
1992 Journal of combinatorial theory. Series B (Print)
If P is not a map on the projective plane, we need not require that the maps in A' be rooted.  ...  At present, this is available only for 2-and 3-connected rooted maps in the plane ( [7] and [S], respectively). The case of all rooted maps on the plane can also be done.  ...

### Measuring the Similarity of Map Projections

Waldo R. Tobler
1986 The American Cartographer
The departure of one map projection from another can be calculated from the lengths of the vectors connecting corresponding points.  ...  This allows one to decide the relative similarity of two map projections and suggests a procedure for finding a projection close to a given one but with different properties.  ...  For the complete set of map points the root mean square distance between the projections is given by n { ∑(x in -x ib ) 2 + (y ia -y ib ) 2 } ½ i=1 Differences of map scale, orientation, or origin are  ...

### An Optimal Algorithm to Generate Pointed Trivalent Diagrams and Pointed Triangular Maps [article]

Samuel Alexandre Vidal
2009 arXiv   pre-print
We shall describe and analyze an algorithm giving an exhaustive list of rooted trivalent diagrams of a given size (number of edges), the list being non-redundant in that no two diagrams of the list are  ...  A trivalent diagram is a connected, two-colored bipartite graph (parallel edges allowed but not loops) such that every black vertex is of degree 1 or 3 and every white vertex is of degree 1 or 2, with  ...  Order one hundred and twenty development of the generating seriesM 3 (t) giving the number of connected unrooted unlabeled triangular maps with n arcs, thus n/2 undirected edges and n/3 triangular faces  ...

### Page 6813 of Mathematical Reviews Vol. , Issue 2000j [page]

2000 Mathematical Reviews
We show that a triangulation G of the sphere, the projective plane, the torus or the Klein bottle is 1-loosely tight if and only if both the independence number and the diameter of G do not exceed 2.  ...  The complex A(G,,), where G, is the complete bidirected graph on n vertices, is the basis for Pitman’s elegant proof of Cayley’s formula n"~' for the number of rooted trees on an n-element set [see J.  ...
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