Tensor products of contexts and complete lattices

Although the categoryCLC of complete lattices and complete homomorphisms does not possess arbitrary coproducts, we show that the tensor product introduced by Wille has the universal property of coproducts for so-called distributing families of morphisms (and only for these). As every family of morphisms into a completely distributive lattice is distributing, this includes the known fact that in the category of completely distributive lattices, arbitrary coproducts exist and coincide with the tensor products. Since the definition of tensor products is based on the notion of contexts and their concept lattices, many results on tensor products extend from complete lattices to contexts. Thus we introduce two kinds of tensor products for arbitrary families of contexts, a partial and a complete one, and establish universal properties of these tensor products.Presented by B. Jonsson.     

Real hypersurfaces of quaternionic projective space satisfying\nabla _{U_i } A = 0

We classify real hypersurfaces of quaternionic projective space satisfying , i=1,2,3.Dedicated to Prof. Nikolaus Stephanidis on his 65th birthday.Research partially supported by DGICYT Grant PS87-0115-CO3-02.     

On approximation by bivariate incomplete polynomials

In this paper we prove that given certain convex domains Δ on the plane, ε>0, andf∈C(Δ) such thatf=0 on θ²Δ={(θ² x,θ² y):(x,y)?Δ} (0<θ<1), a polynomialp(x, y) of the form $$p(x,y) = \sum\limits_{\theta n \leqslant k + l \leqslant n} {a_{kl} x^k y^l }$$ exists such that ∥f?p∥ _C(Δ) ≤ε. The admissible convex domains include triangles and parallelograms with a vertex at the origin and sections of unit disk.     

The diagnonal multivariate natural exponential families and their classification

A natural exponential family (NEF)F in ? ⁿ ,n>1, is said to be diagonal if there existn functions,a ₁,...,a _n , on some intervals of ?, such that the covariance matrixV _F (m) ofF has diagonal (a ₁(m ₁),...,a _n (m _n )), for allm=(m ₁,...,m _n ) in the mean domain ofF. The familyF is also said to be irreducible if it is not the product of two independent NEFs in ? ^k and ? ^n-k , for somek=1,...,n?1. This paper shows that there are only six types of irreducible diagonal NEFs in ? ⁿ , that we call normal, Poisson, multinomial, negative multinomial, gamma, and hybrid. These types, with the exception of the latter two, correspond to distributions well established in the literature. This study is motivated by the following question: IfF is an NEF in ? ⁿ , under what conditions is its projectionp(F) in ? ^k , underp(x ₁,...,x _n )∶=(x ₁,...,x _k ),k=1,...,n?1, still an NEF in ? ^k ? The answer turns out to be rather predictable. It is the case if, and only if, the principalk×k submatrix ofV _F (m ₁,...,m _n ) does not depend on (m _k+1,...,m _n ).     

Retrieval and chaos in extremely dilutedQ-Ising neural networks

Using a probabilistic approach, the deterministic and the stochastic parallel dynamics of aQ-Ising neural network are studied at finiteQ and in the limitQ. Exact evolution equations are presented for the first time-step. These formulas constitute recursion relations for the parallel dynamics of the extremely diluted asymmetric versions of these networks. An explicit analysis of the retrieval properties is carried out in terms of the gain parameter, the loading capacity, and the temperature. The results for theQ network are compared with those for theQ=3 andQ=4 models. Possible chaotic microscopic behavior is studied using the time evolution of the distance between two network configurations. For arbitrary finiteQ the retrieval regime is always chaotic. In the limitQ the network exhibits a dynamical transition toward chaos.