ON THE SHEAF OF GENERALIZED FUNDAMENTAL GROUP AND COVERING SPACE

ＯＮ ＴＨＥ ＳＨＥＡＦ ＯＦ ＧＥＮＥＲＡＬＩＺＥＤ ＦＵＮＤＡＭＥＮＴＡＬ ＧＲＯＵＰ ＡＮＤ ＣＯＶＥＲＩＮＧ ＳＰＡＣＥ￥ＧＡＮＤＡＮＹＡＮＡｂｓｔｒａｃｔ：Ｔｈｅａｕｔｈｏｒｃｏｎｓｔｒｕｃｔｓｔｈｅｓｈｅａｆｏｆｇｅｎｅｒａｌｉｚｅｄｆｕ...     

REGULARITY OF THE FREE BOUNDARY IN ELECTROCHEMICAL MACHINING PROBLEM

ＲＥＧＵＬＡＲＩＴＹＯＦＴＨＥＦＲＥＥＢＯＵＮＤＡＲＹＩＮＥＬＥＣＴＲＯＣＨＥＭＩＣＡＬＭＡＣＨＩＮＩＮＧＰＲＯＢＬＥＭＹｕａｎＧｕａｎｇｗｅｉ（ＩｎｓｔｉｔｕｔｅｏｆＡｐｐｌｉｅｄＰｈｙｓｉｃｓａｎｄＣｏｍｐｕｔａｔｉｏｎａｌＭａｔｈｅｍａｔｉｃｓ...     

A NEW LAPLACIAN COMPRISON THEOREM AND THE ESTIMATE OF EIGENVALUES

Ａ ＮＥＷ ＬＡＰＬＡＣＩＡＮ ＣＯＭＰＲＩＳＯＮ ＴＨＥＯＲＥＭ ＡＮＤ ＴＨＥ ＥＳＴＩＭＡＴＥ ＯＦ ＥＩＧＥＮＶＡＬＵＥＳ ￥ＤＩＮＧＱＩＮＧＡｂｓｔｒａｃｔ：ＴｈｉｓｐａｐｅｒｅｓｔａｂｌｉｓｈｅｓａｎｅｗＬａｐｌａｃｉａｎｃｏｍｐａｒｉｓ...     

STABILITY OF MAJORLY EFFICIENT POINTS AND SOLUTIONS IN MULTIOBJECTIVE PROGRAMMING

ＳＴＡＢＩＬＩＴＹＯＦＭＡＪＯＲＬＹＥＦＦＩＣＩＥＮＴＰＯＩＮＴＳＡＮＤＳＯＬＵＴＩＯＮＳＩＮＭＵＬＴＩＯＢＪＥＣＴＩＶＥＰＲＯＧＲＡＭＭＩＮＧＧＵＧＵＯＨＵＡＡＮＤＨＵＹＵＤＡ（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓａｎｄＭｅｃｈａｎｉｃｓ...     

SUBJECTIVE RADE-OFF RATE METHOD OF MULTIOBJECTIVE DECISION-MAKING

ＳＵＢＪＥＣＴＩＶＥＲＡＤＥ－ＯＦＦＲＡＴＥＭＥＴＨＯＤＯＦＭＵＬＴＩＯＢＪＥＣＴＩＶＥＤＥＣＩＳＩＯＮ－ＭＡＫＩＮＧ￥ＷａｎｇＸｉａｎｊｉａ（王先甲）（ＷｕｈａｎＵｎｉｖｅｒｓｉｔｙｏｆＨｙｄｒａｕｌｉｃａｎｄＥｌｅｃｔｒｉｃＥｎｇｉｎｅｅｒｉｎｇ...     

ON THE CONVERGENCE OF PARALLEL BFGS METHOD

ＯＮＴＨＥＣＯＮＶＥＲＧＥＮＣＥＯＦＰＡＲＡＬＬＥＬＢＦＧＳＭＥＴＨＯＤＣｈｅｎＺｈｏｎｇＦｅｉＰｕｓｈｅｎｇ（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＷｕｈａｎＵｎｉｖｅｒｓｉｔｙ，Ｗｕｈａｎ４３００７２，Ｃｈｉｎａ．）ＺｈｏｕＹｕｎｃａｉ...     

ON THE REVERSE ORDER LAW (AB)D=BDAD

1　ＩｎｔｒｏｄｕｃｔｉｏｎＩｔｉｓｗｅｌｌ ｋｎｏｗｎｔｈａｔｔｈｅＤｒａｚｉｎｉｎｖｅｒｓｅｈａｓｂｅｅｎｗｉｄｅｌｙａｐｐｌｉｅｄｔｏｔｈｅｔｈｅｏｒｙｏｆｆｉｎｉｔｅＭａｒｋｏｖｃｈａｉｎｓａｎｄｓｉｎｇｕｌａｒｄｉｆｆｅｒｅｎｔｉａｌａｎｄｄｉｆｆｅｒｅｎｃｅｅｑｕａｔｉｏｎｓ[2 ] .Ｉｎａｃｌａｓｓｉｃｐａｐｅｒ [3],Ｇｒｅｖｉｌｌｅｇａｖｅｎｅｃｅｓｓａｒｙａｎｄｓｕｆｆｉｃｉｅｎｔｃｏｎｄｉｔｉｏｎｓｆｏｒｔｈｅｔｗｏｔｅｒｍｒｅ ｖｅｒｓｅｏｒｄｅｒｌａｗ (ＡＢ) + =Ｂ+ Ａ+ ｔｏｈｏ…     

THE DIMENSION OF A CLASS OF BIVARIATE SPLINE SPACES

ＴＨＥＤＩＭＥＮＳＩＯＮＯＦＡＣＬＡＳＳＯＦＢＩＶＡＲＩＡＴＥＳＰＬＩＮＥＳＰＡＣＥＳ￥ＧＡＯＪＵＮＢＩＮＡｂｓｔｒａｃｔ：ＷｅｅｓｔａｂｌｉｓｈｔｈｅｄｉｍｅｎｓｉｏｎｆｏｒｍｕｌａｏｆｔｈｅｓｐａｃｅｏｆＣｒｂｉｖａｒｉａｔｅｐｉｅｃｅｗｉｓｅｐ...     

WHEN CAN THE STABLE ALGEBRA DETERMINE THE STRUCTURE OF A C^＊-ALGEBRA ？

ＷＨＥＮＣＡＮＴＨＥＳＴＡＢＬＥＡＬＧＥＢＲＡＤＥＴＥＲＭＩＮＥＴＨＥＳＴＲＵＣＴＵＲＥＯＦＡＣ－ＡＬＧＥＢＲＡ？￥ＷＵＬＩＡＮＧＳＥＮＡｂｓｔｒａｃｔ：ＬｅｔＡａｎｄＢｂｅＣ－ａｌｇｅｂｒａｓ．ＳｕｐｐｏｓｅｔｈａｔＫｉｓｔｈｅａｌｇｅｂｒａｏｆａ...     

THE DISCRETE DYNAMICS FOR COMPETITIVE POPULATIONS OF LOTKA-VOLTERRA TYPE

ＴＨＥＤＩＳＣＲＥＴＥＤＹＮＡＭＩＣＳＦＯＲＣＯＭＰＥＴＩＴＩＶＥＰＯＰＵＬＡＴＩＯＮＳＯＦＬＯＴＫＡ－ＶＯＬＴＥＲＲＡＴＹＰＥＬＩＵＬＡＩＦＵＡＮＤＧＯＮＧＢＩＮＡｂｓｔｒａｃｔ：Ｔｈｅｄｉｓｃｒｅｔｅｄｙｎａｍｉｃｓｆｏｒｃｏｍｐｅｔｉｔｉｏｎｐ...     

图的不可定向最大亏格

本文证明了:对于任何一个有圈连通图G,其不可定向最大亏格为这里,α₀,α₁分别为G的顶点和边的数目.从而,也解决了图的不可定向嵌入的存在性问题.     

The bondage number of graphs on topological surfaces and Teschner’s conjecture

The bondage number of a graph is the smallest number of its edges whose removal results in a graph having a larger domination number. We provide constant upper bounds for the bondage number of graphs on topological surfaces, and improve upper bounds for the bondage number in terms of the maximum vertex degree and the orientable and non-orientable genera of graphs. Also, we present stronger upper bounds for graphs with no triangles and graphs with the number of vertices larger than a certain threshold in terms of graph genera. This settles Teschner’s Conjecture in affirmative for almost all graphs. As an auxiliary result, we show tight lower bounds for the number of vertices of graphs 2-cell embeddable on topological surfaces of a given genus.     

一个不可定向曲面的极小禁用子图的构造

曲面S的一个极小禁用子图是这样的一个图,它的任何一个顶点的度都不小于3,它不能嵌入在S上,但是删去任何一条边后得到的图能嵌入在S上.本文给出了四种构造一个不可定向曲面的极小禁用子图的方式,即粘合一个顶点,一个图的边被其它的图替换,粘合两个顶点,将一个图放在另一个图的一个曲面嵌入的面内.     

不可定向曲面上的最大亏格嵌入和最小亏格嵌入

研究了不可定向曲面上最大亏格嵌入的估计数,得到了几类图的指数级不可定向最大亏格嵌入的估计数的下界.利用电流图理论,证明了完全图K_(12s)在不可定向曲面上至少有2~(3s-1)个最小亏格嵌入;完全图K_(12s+3)在不可定向曲面上至少有2~(2s)个最小亏格嵌入;完全图K_(12s+7)在不可定向曲面上至少有2~(2s+1)个最小亏格嵌入.     

The Semi-Arc Automorphism Group of a Graph with Application to Map Enumeration

A map is a connected topological graph cellularly embedded in a surface. For a given graph Γ, its genus distribution of rooted maps and embeddings on orientable and non-orientable surfaces are separately investigated by many researchers. By introducing the concept of a semi-arc automorphism group of a graph and classifying all its embeddings under the action of its semi-arc automorphism group, we find the relations between its genus distribution of rooted maps and genus distribution of embeddings on orientable and non-orientable surfaces, and give some new formulas for the number of rooted maps on a given orientable surface with underlying graph a bouquet of cycles B_n, a closed-end ladder L_n or a Ringel ladder R_n. A general scheme for enumerating unrooted maps on surfaces(orientable or non-orientable) with a given underlying graph is established. Using this scheme, we obtained the closed formulas for the numbers of non-isomorphic maps on orientable or non-orientable surfaces with an underlying bouquet B_n in this paper.     

On the genus of the semidirect product of ℤ9 by ℤ3

We show that the non-commutative semidirect product Γ of ?₉ by ?₃ has orientable genus 4. In other words, some Cayley graph of Γ embeds in an orientable surface of genus 4 (Euler characteristic ?6), but no Cayley graph of Γ embeds in an orientable surface of genus less than 4 (Euler characteristic greater than ?6). We also show that some Cayley graph of Γ embeds in a (non-orientable) surface of Euler characteristic ?3, but no Cayley graph of Γ embeds in a surface of Euler characteristic greater than ?3. Γ is the first known example of a group whose orientable Euler characteristic and non-orientable Euler characteristic differ by more than 1. Our results also complete the determination of the orientable genus of each group of order less than 32.     

A characterization of the minimalbasis of the torus

D. König asks the interesting question in [7] whether there are facts corresponding to the theorem of Kuratowski which apply to closed orientable or non-orientable surfaces of any genus. Since then this problem has been solved only for the projective plane ([2], [3], [8]). In order to demonstrate that König’s question can be affirmed we shall first prove, that every minimal graph of the minimal basis of all graphs which cannot be embedded into the orientable surface f of genusp has orientable genusp+1 and non-orientable genusq with 1≦q≦2p+2. Then let f be the torus. We shall derive a characterization of all minimal graphs of the minimal basis with the nonorientable genusq=1 which are not embeddable into the torus. There will be two very important graphs signed withX ₈ andX ₇ later. Furthermore 19 graphsG ₁,G ₂, ...,G ₁₉ of the minimal basisM(torus, >₄) will be specified. We shall prove that five of them have non-orientable genusq=1, ten of them have non-orientable genusq=2 and four of them non-orientable genusq=3. Then we shall point out a method of determining graphs of the minimal basisM(torus, >₄) which are embeddable into the projective plane. Using the possibilities of embedding into the projective plane the results of [2] and [3] are necessary. This method will be called saturation method. Using the minimal basisM(projective plane, >₄) of [3] we shall at last develop a method of determining all graphs ofM(torus, >₄) which have non-orientable genusq≧2. Applying this method we shall succeed in characterizing all minimal graphs which are not embeddable into the torus. The importance of the saturation method will be shown by determining another graphG ₂₀≠G ₁,G ₂, ...,G ₁₉ ofM(torus, >₄).     

A note on strong embeddings of maximal planar graphs on non-orientable sufraces

Abstract. In this paper, it is shown that for every maximal planar graph     

Total embedding distributions of Ringel ladders

The total embedding distributions of a graph consists of the orientable embeddings and non-orientable embeddings and are known for only a few classes of graphs. The orientable genus distribution of Ringel ladders is determined in [E.H. Tesar, Genus distribution of Ringel ladders, Discrete Mathematics 216 (2000) 235–252] by E.H. Tesar. In this paper, using the overlap matrix, we obtain nonhomogeneous recurrence relation for rank distribution polynomial, which can be solved by the Chebyshev polynomials of the second kind. The explicit formula for the number of non-orientable embeddings of Ringel ladders is obtained. Also, the orientable genus distribution of Ringel ladders is re-derived.     

Every connected,locally connected graph is upper embeddable

In this Note it is proved that every connected, locally connected graph is upper embeddable. Moreover, a lower bound for the maximum genus of the square of a connected graph is given.