Volume 57, Issue 1

We determine the Boolean transformations F: B n → B n which have unique fixed points.

In this paper we study the L p-discrepancy of digitally shifted Hammersley point sets. While it is known that the (unshifted) Hammersley point set (which is also known as Roth net) with N points has L p-discrepancy (p an integer) of order (log N)/N, we show that there always exists a shift such that the digitally shifted Hammersley point set has L p-discrepancy (p an even integer) of order $$\sqrt {\log N} /N$$ which is best possible by a result of W. Schmidt. Further we concentrate on the case p = 2. We give very tight lower and upper bounds for the L 2-discrepancy of digitally shifted Hammersley point sets which show that the value of the L 2-discrepancy of such a point set mostly depends on the number of zero coordinates of the shift and not so much on the position of these.

This work has been divided in two parts. In the first part we give a sufficient conditions on an HNN extension of a free group to be cyclic subgroup seperable. In the second part we define just subgroup separability on a split extension of special groups which is actually on holomorph.

In probability theory, each random variable f can be viewed as channel through which the probability p of the original probability space is transported to the distribution p f, a probability measure on the real Borel sets. In the realm of fuzzy probability theory, fuzzy probability measures (equivalently states) are transported via statistical maps (equivalently, fuzzy random variables, operational random variables, Markov kernels, observables). We deal with categorical aspects of the transportation of (fuzzy) probability measures on one measurable space into probability measures on another measurable spaces. A key role is played by D-posets (equivalently effect algebras) of fuzzy sets.

In some situations estimates of unknown parameters must be corrected by additional measurements. It is in principle no problem to calculate the corrected estimates, however, it is of more interest to find formulae for correction itself. The formulae enable us to design an additional experiment and to judge its usefulness. The aim of the paper is to find such formulae for several situations.

The problem of computing the mean squared error (MSE) of the best linear predictor (BLP) in finite discrete spectrum with an additive white noise models(FDSWNMs) for an observed time series is considered. This is done under the assumption that the corresponding vectors in models for finite observation of this time series are not orthogonal.