Helly-type theorems for hollow axis-aligned boxes

A hollow axis-aligned box is the boundary of the cartesian product of compact intervals in . We show that for , if any of a collection of hollow axis-aligned boxes have non-empty intersection, then the whole collection has non-empty intersection; and if any of a collection of hollow axis-aligned rectangles in have non-empty intersection, then the whole collection has non-empty intersection. The values for and for are the best possible in general. We also characterize the collections of hollow boxes which would be counterexamples if were lowered to , and to , respectively.

     

Broadcasting, Scalability, and Reconfigurability Aspects in an All-Optical Network Architecture

All-optical networks are a sound answer to the huge data traffic demand expected for the forthcoming information society. Interesting networking aspects are the broadcasting, scalability, and reconfiguration in an all-optical environment. These issues are addressed in the present article for an all-optical network called MATRIX, which is time slotted and uses packet switching. The architecture enables the all-optical interconnection of a large number of nodes with both a small number of wavelengths and wavelength continuity based on a consequent exploitation of wavelength-division multiplexing and space-division multiplexing (i.e., multiple fibers per cable are used). For an efficient use of the resources, each node is equipped with multiple transmitters and receivers, respectively. Two different broadcasting schemes are investigated. An analytical model is developed, and the corresponding simulation results are in good agreement. It is further shown how different networks of arbitrary size can be interconnected to larger all-optical network clusters, thereby providing scalability not only with respect to the number of nodes but also to the geographical extension. Finally, wavelength reassignments allow change of optical paths through the network in case of changing traffic patterns or network failures in order to best exploit the available network resources. The total number of possible configurations is determined by the theory of Latin Squares.     

A simple mathematical model for seasonal planktonic blooms

We show how the inclusion of the defense strategy by different species can alter the prediction of simple models. One of the defense strategy by the phytoplankton population against their grazer is the release of toxic chemicals. In turn the zooplankton population reduces there predation rate over toxin producing phytoplankton (TPP) to protect themselves from those toxic chemicals. Thus, when the level of toxicity is high, the grazing pressure is low and when the level of toxicity is low or when the toxin is absent, the grazing pressure is high. Here we have considered a TPP–zooplankton system where the rate of toxin liberation and the predation rate vary with zooplankton abundance. We observe that our proposed model has the potential to show different dynamical behaviour that are similar to that seen in real‐world situations. Further, we consider three different functional forms for the distribution of the toxins and compare them using latin hypercube sampling technique and found that the functional forms seem to have no effect in determining the final outcome of the system. Copyright © 2009 John Wiley & Sons, Ltd.     

均匀设计抽样的一、二阶矩

（７）中给出并研究了均匀设计抽样（ＵＤＳ）及随机化均匀设计（ＲＵＤ）的一些优良性质，作者给出该设计和抽样的一、二阶矩。     

On the range of influences in back-circulant Latin squares

In this note we obtain a large lower bound for the index of a certain critical set in the back-circulant Latin squares of odd order. This resolves in the negative a conjecture of Fitina, Seberry and Chaudhry [Back-circulant Latin square and the influence of a set, Austral. J. Combin. 20 (1999) 163-180].     

When is a partial Latin square uniquely completable, but not its completable product?

Let P and Q be uniquely completable partial Latin squares. It is an open problem to determine necessary and sufficient conditions so that the completable product P⊗Q is also uniquely completable. So far, only a few specific examples of P have been given such that the completable product of P with itself (P⊗P) does not have a unique completion. In this paper, we find a whole class of such partial Latin squares.     

两两正交拉丁方最大数目的新上界

记D（x）是使得TD（x，n）存在的最小的数．本文给出D（x）的一个上界．     

More on regular linear spaces

We extend the enumeration of regular linear spaces in 1 to at most 19 points. In addition, one of the 5 missing cases in the previous list is settled. The number of regular linear spaces of type (15|2¹⁵,3³⁰) is 10,177,328. © 2005 Wiley Periodicals, Inc. J Combin Designs.     

D[X2m]的下界估计与X2m矩阵的快速构造

本文利用有向图理论研究了Xn矩阵的特性，给出了X2m矩阵类的快速构造方法，证明了拉丁方矩阵D[X2m]数目的下界估计是：2m(2m)! ∑^mi=2[(2m)!]^2/П^ij=1Kj!П^rj=1bj!。     

Routing of 2-D switching networks by their embedding into cubes

The proposed all-optical 2-D switching networks are (i) M×N-gon prism switches (M2, N3) and (ii) 3-D grids of any geometry N3. For the routing we assume (1) the projection of the spatial architectures onto plane graphs (2) the embedding of the latter guest graphs into (in)complete host hypercubes (N=4) and generally, into N-cube networks (N3) and (3) routing by means of the cube algorithms of the host. By the embedding mainly faulty cubes (synonyms: injured cubes, incomplete cubes) arise which complicate the routing and analysis. The application of N-cube networks (i) extend the hypercube principles to any N3 (ii) increase the number of plane host graphs and (iii) reduce the incompleteness of the host cubes. Several different embeddings of the intersection graphs (IGs) of 2-D switching networks and several different routings are explained for N=4 and 6 by various examples. By the expansion of the grids (enlargement) internal waveguides (WGs) and internal switches are introduced which interact with the switches of the original 3-D grid without increasing the number of stages (NS). The embeddings by expansion apply to interconnection networks whereas dilation-2 embeddings (dilation ≡ distance of the nearest-neighbour nodes of the guest graph at the host) are rather suitable for the emulation of algorithms. Concepts for fault-tolerant routing and algorithm mapping are briefly explained.