On four-sheeted polynomial mappings of ℂ2. I. The case of an irreducible ramification curve

The paper is devoted to the Jacobian Conjecture: a polynomial mappingf²² with a constant nonzero Jacobian is polynomially invertible. The main result of the paper is as follows. There is no four-sheeted polynomial mapping whose Jacobian is a nonzero constant such that after the resolution of the indeterminacy points at infinity there is only one added curve whose image is not a point and does not belong to infinity.Translated fromMatematicheskie Zametki, Vol. 64, No. 6, pp. 847–862, December, 1998.The authors are grateful to A. G. Vitushkin and P. Cassou-Nogues for useful discussions.This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-01218. The work of the second author was done under the financial support of DGICYT (Spain), grant No. SAB95-0502.     

On Lower Bounds For Covering Codes

We study lower bounds on K(n,R), the minimum number of codewords of any binary code of length n such that the Hamming spheres of radius R with center at codewords cover the Hamming space . We generalize Honkala's idea toobtain further improvements only by using some simple observationsof Zhang's result. This leads to nineteen improvements of thelower bound on K(n,R) within the range of .     

广义有限元方法──一类新的逼近空间

本文提出了一类新的有限元空间,它把每个节点只有一个广义位移的Ｌａｇｒａｎｇｅ型插值空间推广为每个节点具有任意多个广义位移的任何级数展开式。它包括了解析法和Ｂａｂｕｓｋａ的Ｐ－ｖｅｒｓｉｏｎ方法。由于它保持了空间的协调性,并且全部自由度都定义在节点上,故程序实现简单,易被传统的有限元软件接纳。本文还给出了这一新空间的误差估计。      

Oriented walk double covering and bidirectional double tracing

An oriented walk double covering of a graph G is a set of oriented closed walks, that, traversed successively, combined will have traced each edge of G once in each direction. A bidirectional double tracing of a graph G is an oriented walk double covering that consists of a single closed walk. A retracting in a closed walk is the immediate succession of an edge by its inverse. Every graph with minimum degree 2 has a retracting free oriented walk double covering and every connected graph has a bidirectional double tracing. The minimum number of closed walks in a retracting free oriented walk double covering of G is denoted by c(G). The minimum number of retractings in a bidirectional double tracing of G is denoted by r(G). We shall prove that for all connected noncycle graphs G with minimum degree at least 2, r(G) = c(G) − 1. The problem of characterizing those graphs G for which r(G) = 0 was raised by Ore. Thomassen solved this problem by relating it to the existence of certain spanning trees. We generalize this result, and relate the parameters r(G), c(G) to spanning trees of G. This relation yields a polynomial time algorithm to determine the parameters c(G) and r(G). © 1998 John Wiley & Sons, Inc. J. Graph Theory 29: 89–102, 1998     

Independent coverings and orthogonal colourings

In this paper, two open conjectures are disproved. One conjecture regards independent coverings of sparse partite graphs, whereas the other conjecture regards orthogonal colourings of tree graphs. A relation between independent coverings and orthogonal colourings is established. This relation is applied to find independent coverings of some sparse partite graphs. Additionally, a degree condition providing the existence of an independent covering in the case where the graph has a square number of vertices is found.     

Optimum super-simple mixed covering arrays of type <Emphasis Type="Italic">a</Emphasis><Superscript>1</Superscript><Emphasis Type="Italic">b</Emphasis><Superscript><Emphasis Type="Italic">k</Emphasis>−1</Superscript>

A mixed covering array (MCA) of type (_v 1, v ₂,..., v _k ), denoted by MCA_λ (N; t, k, (v ₁, v ₂,..., v _k )), is an N × k array with entries in the i-th column from a set _{V i} of v _i symbols and has the property that each N × t sub-array covers all the t-tuples at least λ times, where 1 ≤ i ≤ k. An MCA _λ (N; t, k, (_v 1, v ₂,..., v _k )) is said to be super-simple, if each of its N × (t + 1) sub-arrays contains each (t + 1)-tuple at most once. Recently, it was proved by Tang, Yin and the author that an optimum super-simple MCA of type (a, b, b,..., b) is equivalent to a mixed detecting array (DTA) of type (a, b, b,..., b) with optimum size. Such DTAs can be used to generate test suites to identify and determine the interaction faults between the factors in a component-based system. In this paper, some combinatorial constructions of optimum super-simple MCAs of type (a, b, b,..., b) are provided. By employing these constructions, some optimum super-simple MCAs are then obtained. In particular, the spectrum across which optimum super-simple MCA₂(2b ²; 2, 4, (a, b, b, b))′s exist, is completely determined, where 2 ≤ a ≤ b.     

Cubic semisymmetric graphs of order 8p~3

A regular edge-transitive graph is said to be semisymmetric if it is not vertex-transitive.By Folkman [J.Combin.Theory 3(1967),215-232],there is no semisymmetric graph of order 2p or 2p 2 for a prime p,and by Malni et al.[Discrete Math.274(2004),187-198],there exists a unique cubic semisymmetric graph of order 2p 3,the so called Gray graph of order 54.In this paper,it is shown that there is no connected cubic semisymmetric graph of order 4p 3 and that there exists a unique cubic semisymmetric graph of orde...     

覆盖族生成拓扑的约简

本文研究了在覆盖族产生的拓扑不变的条件下覆盖族的约简问题.利用拓扑学理论讨论覆盖广义粗糙集的约简理论,给出计算约简的方法,丰富了覆盖广义粗糙集理论.     

Weighted covering numbers of convex sets

In this paper we define the notions of weighted covering number and weighted separation number for convex sets, and compare them to the classical covering and separation numbers. This sheds new light on the equivalence of classical covering and separation. We also provide a formula for computing these numbers via a limit of classical covering numbers in higher dimensions.