On the Duality of Semiantichains and Unichain Coverings

We study a min-max relation conjectured by Saks and West: For any two posets $P$ and $Q$ the size of a maximum semiantichain and the size of a minimum unichain covering in the product $P\times Q$ are equal. For positive we state conditions on $P$ and $Q$ that imply the min-max relation. Based on these conditions we identify some new families of posets where the conjecture holds and get easy proofs for several instances where the conjecture had been verified before. However, we also have examples showing that in general the min-max relation is false, i.e., we disprove the Saks-West conjecture.Comment: 10 pages, 3 figure

2Laboratoire Charles Coulomb

I show here that due to the fact that time has a logarithmic increase as a function of the radius of curvature of our threedimensional universe, the rate of expansion of the universe is accelerating. The instantaneous pressure within this universe is negative (as predicted by the literature) because black holes lead to a leak of matter and light from our universe. Though, the total energy of the universe is increasing. The transition from a still fourdimensional universe with no physical laws to our threedimensional curved universe is due to Heisenberg’s uncertainty principle

A consistent test for bilinear time series

This paper presents a non-classical test for some bilinear models with an error process which have zero conditional median. The consistence of such a test is obtained using the asymptotic separation of the sequences of probability laws defined by each hypothesis to be tested. An exponential decay of the level rate of convergence is also established. A simulation study is done to illustrate the behaviour of the power and level functions in small moderate samplesAvailable from Departamento de Matematica, Universidade de Coimbra, 3000 Coimbra, Portugal / FCT - Fundação para o Ciência e a TecnologiaSIGLEPTPortuga