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Enumeration of three kinds of rooted maps on the Klein bottle
Wenzhong Liu,Yanpei Liu 한국전산응용수학회 2007 Journal of applied mathematics & informatics Vol.24 No.1
In this paper, 2 essential rooted maps on the Klein bottle arecounted and an explicit expression with the size as a parameter is given.Further, the numbers of singular maps and the maps with one vertex onthe Klein bottle are derived.AMS Mathematics Subject Classication: 05C45.Key words and phrases : 2 essential map, singular map, enufunction.1. IntroductionEnumeration of non-planar maps has been initiated in the 1960s by Brown [3].On the object, some scholars did some inuential works such as Bender et al. [2],Arqu`es [1], Gao [4] and Liu [79], only name a few. In this paper, enumeratingformulae of three kinds of rooted maps on the Klein bottle are obtained byinvestigating g essential maps with the small genera which play an importantrole in the study of general maps on surfaces.The article begins with some denitions. Terms without description can beseen in [7-9].LetX = {x1,x2 · ,xn} be a set andK = {1,β,γ}, the Klein group,i.e.,α2 = β2 = γ2 = 1, where γ= = . F o rx ∈X, Kx = {x,αx,βx,γx}.Ifthe permutationP on X (X)=n.i=1Kxi = X satises the following conditions:for anyl ∈N and any x ∈ X, Plx .=αx, αP = P1 and the group ψτgenerated byτ= {α,β,γ} is transitive onX , and thenM =(X,P) is called amap.If a map M = (X,P) has an element, theroot denoted by r = r(M ), inXmarked beforehand, thenM is called arooted mapand the marked edge is calledthe root-edgeofM . Likewise theroot-vertexand the root-faceare clear. In thisReceived November 24, 2005.Corresponding author.1 Supported by NNSFC under Grant 603730308 and 10571013.c. 2007 Korean Society for Computational & Applied Mathematics and Korean SIGCAM .411412 Wenzhong Liu and Yanpei Liupaper, maps always mean rooted . ForM = (X,P), the root, the root-vertex,the root-edge and the root-face ofM are denoted by r(M ),vr(M ),er(M ) andfr(M ) respectively.A surfaceΣ here is a compact close 2 manifolds. Anorientable (nonori-
ENUMERATION OF THREE KINDS OF ROOTED MAPS ON THE KLEIN BOTTLE
Liu, Wenzhong,Liu, Yanpei 한국전산응용수학회 2007 Journal of applied mathematics & informatics Vol.24 No.1
In this paper, $\tilde{2}$-essential rooted maps on the Klein bottle are counted and an explicit expression with the size as a parameter is given. Further, the numbers of singular maps and the maps with one vertex on the Klein bottle are derived.
A cache placement algorithm based on comprehensive utility in big data multi-access edge computing
( Yanpei Liu ),( Wei Huang ),( Li Han ),( Liping Wang ) 한국인터넷정보학회 2021 KSII Transactions on Internet and Information Syst Vol.15 No.11
The recent rapid growth of mobile network traffic places multi-access edge computing in an important position to reduce network load and improve network capacity and service quality. Contrasting with traditional mobile cloud computing, multi-access edge computing includes a base station cooperative cache layer and user cooperative cache layer. Selecting the most appropriate cache content according to actual needs and determining the most appropriate location to optimize the cache performance have emerged as serious issues in multi-access edge computing that must be solved urgently. For this reason, a cache placement algorithm based on comprehensive utility in big data multi-access edge computing (CPBCU) is proposed in this work. Firstly, the cache value generated by cache placement is calculated using the cache capacity, data popularity, and node replacement rate. Secondly, the cache placement problem is then modeled according to the cache value, data object acquisition, and replacement cost. The cache placement model is then transformed into a combinatorial optimization problem and the cache objects are placed on the appropriate data nodes using tabu search algorithm. Finally, to verify the feasibility and effectiveness of the algorithm, a multi-access edge computing experimental environment is built. Experimental results show that CPBCU provides a significant improvement in cache service rate, data response time, and replacement number compared with other cache placement algorithms.
CHROMATIC SUMS OF ROOTED TRIANGULATIONS ON THE PROJECTIVE PLANE
LI, ZHAOXIANG,LIU, YANPEI 한국전산응용수학회 2005 Journal of applied mathematics & informatics Vol.18 No.1
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.
ZHAOXIANG LI,YANPEI LIU 한국전산응용수학회 2007 Journal of applied mathematics & informatics Vol.23 No.1
A map is bisingular if each edge is either a loop or an isthmus (i.e., on the boundary of the same face). In this paper we study the number of rooted bisingular maps on the sphere and the torus, and we also present formula for such maps with four parameters: the root-valency,the number of isthmus, the number of planar loops and the number of essential loops.
Li, Zhaoxiang,Liu, Yanpei 한국전산응용수학회 2007 Journal of applied mathematics & informatics Vol.23 No.1
A map is bisingular if each edge is either a loop or an isthmus (i.e., on the boundary of the same face). In this paper we study the number of rooted bisingular maps on the sphere and the torus, and we also present formula for such maps with four parameters: the root-valency, the number of isthmus, the number of planar loops and the number of essential loops.
CHROMATIC SUMS OF NONSEPARABLE SIMPLE MAPS ON THE PLANE
Li, Zhaoxiang,Liu, Yanpei 한국전산응용수학회 2003 Journal of applied mathematics & informatics Vol.12 No.1
In this paper we study the chromatic sum functions for rooted nonseparable simple maps on the plane. The chromatic sum function equation for such maps is obtained . The enumerating function equation of such maps is derived by the chromatic sum equation of such maps. From the chromatic sum equation of such maps, the enumerating function equation of rooted nonseparable simple bipartite maps on the plane is also derived.
ENUMERATION OF LOOPLESS MAPS ON THE PROJECTIVE PLANE
Li, Zhaoxiang,Liu, Yanpei 한국전산응용수학회 2002 Journal of applied mathematics & informatics Vol.10 No.1
In this paper we study the rooted loopless maps on the sphere and the projective plane with the valency of root-face and the number of edges as parameters. Explicit expression of enumerating function is obtained for such maps on the sphere and the projective plane. A parametric expression of the generating function is obtained for such maps on the projective plane, from which asymptotic evaluations are derived.