Index set

Abstract: A vague set is a set of objects, each of which has a grade of membership whose value is a continuous subinterval of

Abstract: A version of Dirichlet's box argument asserts that given a positive integer a and any a2 +1 objects x0 , x1 , . . ., xa 2, there are always a+1 distinct indices v (0 < v < a 2) such that the corresponding ad-1 objects x,, are either all equal to each other or mutually different from each other . This proposition can be restated as follows . Let N be an index set of more than a 2 elements and let, for each element v of N, X v be a one-element set . Then there is a subset N' of N having more than a elements, such that all intersections X, X, corresponding to distinct elements μ, v of N' have the same value . In this note we investigate extensions of this principle to cases when the sets X„ are of any prescribed cardinal b . Both a and bare given cardinals, finite or infinite . In the case of finite a and b we obtain estimates for the number which corresponds to a2 in Dirichlet's case, and we show that when at least one of a and b is infinite then av+ 1 is the best possible value of that number . We introduce some definitions ;' . A system E1 : Y v (v e N) of sets Y,, where v ranges over the index set N, is said to contain the system 10 : N1,. (μ c 31) if, for every μo of 111, the set X., occurs in 'r''1 at least as often as in Eo , i .e . if

Abstract: This paper is concerned with the mapping of cyclic loop algorithms into special-purpose VLSI arrays. The mapping procedure is based on the mathematical transformations of index sets and data dependence vectors. Necessary and sufficient conditions for the existence of valid transformations are given for algorithms with constant data dependences. Two examples of different algorithms are given to illustrate the proposed mapping procedure; first is the LU decomposition of a matrix which leads to constant data dependence vectors, and secondly is the dynamic programming which leads to dependences which are functions on the index set and are more difficult to be mapped into VLSI arrays.

Abstract: Summary. In geostatistics it is common practice to assume that the underlying spatial process is stationary and isotropic, i.e. the spatial distribution is unchanged when the origin of the index set is translated and under rotation about the origin. However, in environmental problems, such assumptions are not realistic since local influences in the correlation structure of the spatial process may be found in the data. The paper proposes a Bayesian model to address the anisot- ropy problem. Following Sampson and Guttorp, we define the correlation function of the spatial process by reference to a latent space, denoted by D, where stationarity and isotropy hold. The space where the gauged monitoring sites lie is denoted by G. We adopt a Bayesian approach in which the mapping between G and D is represented by an unknown function d(·). A Gaussian process prior distribution is defined for d(·). Unlike the Sampson–Guttorp approach, the mapping of both gauged and ungauged sites is handled in a single framework, and predictive inferences take explicit account of uncertainty in the mapping. Markov chain Monte Carlo methods are used to obtain samples from the posterior distributions. Two examples are discussed: a simulated data set and the solar radiation data set that also was analysed by Sampson and Guttorp.