Matrix measures and random walks

In this paper we study the connection between matrix measures and random walks with a tridiagonal block transition matrix. We derive sufficient conditions such that the blocks of the n-step transition matrix of the Markov chain can be represented as integrals with respect to a matrix valued spectral measure. Several stochastic properties of the processes are characterized by means of this matrix measure. In many cases this measure is supported in the interval [-1, 1]. The results are illustrated by several examples including random walks on a grid and the embedded chain of a queuing system. --Markov chain,block tridiagonal transition matrix,spectral measure,matrix measure,quasi birth and death processes,canonical moments

Variability in the chemistry of private drinking water supplies and the impact of domestic treatment systems on water quality.

Tap water from 497 properties using private water supplies, in an area of metalliferous and arsenic mineralisation (Cornwall, UK), was measured to assess the extent of compliance with chemical drinking water quality standards, and how this is influenced by householder water treatment decisions. The proportion of analyses exceeding water quality standards were high, with 65 % of tap water samples exceeding one or more chemical standards. The highest exceedances for health-based standards were nitrate (11 %) and arsenic (5 %). Arsenic had a maximum observed concentration of 440 µg/L. Exceedances were also high for pH (47 %), manganese (12 %) and aluminium (7 %), for which standards are set primarily on aesthetic grounds. However, the highest observed concentrations of manganese and aluminium also exceeded relevant health-based guidelines. Significant reductions in concentrations of aluminium, cadmium, copper, lead and/or nickel were found in tap waters where households were successfully treating low-pH groundwaters, and similar adventitious results were found for arsenic and nickel where treatment was installed for iron and/or manganese removal, and successful treatment specifically to decrease tap water arsenic concentrations was observed at two properties where it was installed. However, 31 % of samples where pH treatment was reported had pH < 6.5 (the minimum value in the drinking water regulations), suggesting widespread problems with system maintenance. Other examples of ineffectual treatment are seen in failed responses post-treatment, including for nitrate. This demonstrates that even where the tap waters are considered to be treated, they may still fail one or more drinking water quality standards. We find that the degree of drinking water standard exceedances warrant further work to understand environmental controls and the location of high concentrations. We also found that residents were more willing to accept drinking water with high metal (iron and manganese) concentrations than international guidelines assume. These findings point to the need for regulators to reinforce the guidance on drinking water quality standards to private water supply users, and the benefits to long-term health of complying with these, even in areas where treated mains water is widely available

Matrix measures and random walks

In this paper we study the connection between matrix measures and random walks with a tridiagonal block transition matrix. We derive sufficient conditions such that the blocks of the n-step transition matrix of the Markov chain can be represented as integrals with respect to a matrix valued spectral measure. Several stochastic properties of the processes are characterized by means of this matrix measure. In many cases this measure is supported in the interval [−1, 1]. The results are illustrated by several examples including random walks on a grid and the embedded chain of a queuing system

Constrained optimal discrimination designs for Fourier regression models

In this article, the problem of constructing efficient discrimination designs in a Fourier regression model is considered. We propose designs which maximize the power of the F-test,which discriminates between the two highest order models, subject to the constraints that the tests that discriminate between lower order models have at least some givenrelative power. A complete solution is presented in terms of the canonical moments of the optimal designs, and for the special case of equal constraints even more specific formulaeare availabl