Karhunen-Lo\`eve expansion for a generalization of Wiener bridge

We derive a Karhunen-Lo\`eve expansion of the Gauss process $B_t - g(t)\int_0^1 g'(u)\,d B_u$, $t\in[0,1]$, where $(B_t)_{t\in[0,1]}$ is a standard Wiener process and $g:[0,1]\to R$ is a twice continuously differentiable function with $g(0) = 0$ and $\int_0^1 (g'(u))^2\,d u =1$. This process is an important limit process in the theory of goodness-of-fit tests. We formulate two special cases with the function $g(t)=\frac{\sqrt{2}}{\pi}\sin(\pi t)$, $t\in[0,1]$, and $g(t)=t$, $t\in[0,1]$, respectively. The latter one corresponds to the Wiener bridge over $[0,1]$ from $0$ to $0$.Comment: 25 pages, 1 figure. The appendix is extende

Transformation Products and Human Metabolites of Triclocarban and Triclosan in Sewage Sludge Across the United States

Removal of triclocarban (TCC) and triclosan (TCS) from wastewater is a function of adsorption, abiotic degradation, and microbial mineralization or transformation, reactions that are not currently controlled or optimized in the pollution control infrastructure of standard wastewater treatment. Here, we report on the levels of eight transformation products, human metabolites, and manufacturing byproducts of TCC and TCS in raw and treated sewage sludge. Two sample sets were studied: samples collected once from 14 wastewater treatment plants (WWTPs) representing nine states, and multiple samples collected from one WWTP monitored for 12 months. Time-course analysis of significant mass fluxes (α = 0.01) indicate that transformation of TCC (dechlorination) and TCS (methylation) occurred during sewage conveyance and treatment. Strong linear correlations were found between TCC and the human metabolite 2′-hydroxy-TCC (<i>r</i> = 0.84), and between the TCC-dechlorination products dichlorocarbanilide (DCC) and monochlorocarbanilide (<i>r</i> = 0.99). Mass ratios of DCC-to-TCC and of methyl-triclosan (MeTCS)-to-TCS, serving as indicators of transformation activity, revealed that transformation was widespread under different treatment regimes across the WWTPs sampled, though the degree of transformation varied significantly among study sites (α = 0.01). The analysis of sludge sampled before and after different unit operation steps (i.e., anaerobic digestion, sludge heat treatment, and sludge drying) yielded insights into the extent and location of TCC and TCS transformation. Results showed anaerobic digestion to be important for MeTCS transformation (37–74%), whereas its contribution to partial TCC dechlorination was limited (0.4–2.1%). This longitudinal and nationwide survey is the first to report the occurrence of transformation products, human metabolites, and manufacturing byproducts of TCC and TCS in sewage sludge