具有某种断面的半群的研究进展

本文综述了几类具有特殊断面的半群的近期研究结果。在介绍逆半群和正则半群的一般结构之后，概述了具有逆断面的正则半群的结构和同余格的研究成果。总结了作为逆断面的推广的可裂断面，纯正断面，正则^*－断面和恰当断面。提出了可以进一步研究的重要的问题。     

Concerning seven and eight mutually orthogonal Latin squares

In this paper, three new direct Mutually Orthogonal Latin Squares (MOLS) constructions are presented for 7 MOLS(24), 7 MOLS(75) and 8 MOLS(36); then using recursive methods, several new constructions for 7 and 8 MOLS are obtained. These reduce the largest value for which 7 MOLS are unknown from 780 to 570, and the largest odd value for which 8 MOLS are unknown from 1935 to 419. © 2003 Wiley Periodicals, Inc.     

HYPERFINITE TRANSVERSAL THEORY. II

We continue the investigation of validity of Hall's theorem in the case of the Loeb space of an internal, uniformly distributed, hyperfinite measure space initiated in1992 by the author. Some new classes of graphs are introduced for which the measure theoretic version of Hall's theorem still holds.

     

A Three‐Factor Product Construction for Mutually Orthogonal Latin Squares

It is well known that mutually orthogonal latin squares, or MOLS, admit a (Kronecker) product construction. We show that, under mild conditions, “triple products” of MOLS can result in a gain of one square. In terms of transversal designs, the technique is to use a construction of Rolf Rees twice: once to obtain a coarse resolution of the blocks after one product, and next to reorganize classes and resolve the blocks of the second product. As consequences, we report a few improvements to the MOLS table and obtain a slight strengthening of the famous theorem of MacNeish.     

A class of differentiable toral maps which are topologically mixing

We show that on the 2-torus there exists a open set of regular maps such that every map belonging to is topologically mixing but is not Anosov. It was shown by Mañé that this property fails for the class of toral diffeomorphisms, but that the property does hold for the class of diffeomorphisms on the 3-torus . Recently Bonatti and Diaz proved that the second result of Mañé is also true for the class of diffeomorphisms on the -torus ().

     

Stability of two player game structures

We have extended a two player game-theoretical model proposed by V. Gurvich [To theory of multi-step games, USSR Comput. Math and Math. Phys. 13 (1973)] and H. Moulin [The Strategy of Social Choice, North Holland, Amsterdam, 1983]: All the considered game situations are framed by the same game structure. The structure determines the families of potential decisions of the two players, as well as the subsets of possible outcomes allowed by pairs of such choices. To be a solution of a game, a pair of decisions has to determine a (pure) functional equilibrium of the situational pair of payoff mappings which transforms the realized outcome into real-valued rewards of the players. Accordingly we understand that a structure is stable, if it admits functional equilibria for all possible game situations; and that it is complete, if every situation that only partitions the potential outcomes, is dominated by one of the players. We have generalized and strengthened a theorem by V. Gurvich [Equilibrium in pure strategies, Soviet Math. Dokl. 38 (1989)], proving that a proper structure is stable iff it is complete. Additional results provide game-theoretical insight that focuses the inquiry on the complexity of the stability decision problem; in particular, for coherent structures.These results also have combinatorial importance because every structure is characterized by a pair of hypergraphs [C. Berge, Graphes et Hypergraphes, Dunod, 1970] over a common ground set. The structure is dual (complete/coherent) iff the clutter of one hypergraph equals (includes/is included in) the blocker of the other one. So, for non-void coherent structures, the stability decision problem is equivalent to the much studied subexponential [M.L. Fredman, L. Khachiyan, On the complexity of dualization of monotone disjunctive normal forms, J. Algorithms 21 (1996)] hypergraph duality decision problem.     

Helly-Type Theorems for Line Transversals to Disjoint Unit Balls

We prove Helly-type theorems for line transversals to disjoint unit balls in ℝ ^d . In particular, we show that a family of n≥2d disjoint unit balls in ℝ ^d has a line transversal if, for some ordering ≺ of the balls, any subfamily of 2d balls admits a line transversal consistent with ≺. We also prove that a family of n≥4d−1 disjoint unit balls in ℝ ^d admits a line transversal if any subfamily of size 4d−1 admits a transversal. Andreas Holmsen was supported by the Research Council of Norway, prosjektnummer 166618/V30. Otfried Cheong and Xavier Goaoc acknowledge support from the French-Korean Science and Technology Amicable Relationships program (STAR).     

关于矩阵的复符号模式类的行列式值域的一个猜想

研究矩阵的复符号模式类的行列式值域是与复符号非异阵的刻画有关的一个问题.文中部分解决了邵嘉裕先生等人提出的一个猜想.     

On the Helly Number for Line Transversals to Parallel Triangles

We prove two results about the problem of finding the Helly number for line transversals to a family of parallel triangles in the plane: (1) If each three triangles of a family of parallel right triangles are intersected by an ascending (or a descending) line, then there is an ascending (or a descending) line that intersects all