The Zeros of Riemann Zeta Partial Sums Yield Solutions to f(x) + f(2x) + · · · + f(nx) = 0

This paper proves that every zero of any n th , n ≥ 2, partial sum of the Riemann zeta function provides a vector space of basic solutions of the functional equation f(x)+f(2x)+⋯+f(nx)=0,x∈R . The continuity of the solutions depends on the sign of the real part of each zero.

Reduction of tobacco smoke components yield in commercial cigarette brands by addition of HUSY, NaY and Al-MCM-41 to the cigarette rod

The effect of two zeolites, HUSY, NaY and a mesoporous synthesized Al-MCM-41 material on the smoke composition of ten commercial cigarettes brands has been studied. Cigarettes were prepared by mixing the tobacco with the three powdered materials, and the smoke obtained under the ISO conditions was analyzed. Up to 32 compounds were identified and quantified in the gas fraction and 80 in the total particulate matter (TPM) condensed in the cigarettes filters and in the traps located after the mouth end of the cigarettes. Al-MCM-41 is by far the best additive, providing the highest reductions of the yield for most compounds and brands analyzed. A positive correlation was observed among the TPM and nicotine yields with the reduction obtained in nicotine, CO, and most compounds with the three additives. The amount of ashes in additive free basis increases due to the coke deposited on the solids, especially with Al-MCM-41. Nicotine is reduced with Al-MCM-41 by an average of 34.4% for the brands studied (49.5% for the brand where the major reduction was obtained and 18.5 for the brand behaving the worst). CO is reduced by an average of 18.6% (ranging from 10.3 to 35.2% in the different brands).

An adaptive algorithm for clustering cumulative probability distribution functions using the Kolmogorov–Smirnov two-sample test

This paper proposes an adaptive algorithm for clustering cumulative probability distribution functions (c.p.d.f.) of a continuous random variable, observed in different populations, into the minimum homogeneous clusters, making no parametric assumptions about the c.p.d.f.’s. The distance function for clustering c.p.d.f.’s that is proposed is based on the Kolmogorov–Smirnov two sample statistic. This test is able to detect differences in position, dispersion or shape of the c.p.d.f.’s. In our context, this statistic allows us to cluster the recorded data with a homogeneity criterion based on the whole distribution of each data set, and to decide whether it is necessary to add more clusters or not. In this sense, the proposed algorithm is adaptive as it automatically increases the number of clusters only as necessary; therefore, there is no need to fix in advance the number of clusters. The output of the algorithm are the common c.p.d.f. of all observed data in the cluster (the centroid) and, for each cluster, the Kolmogorov–Smirnov statistic between the centroid and the most distant c.p.d.f. The proposed algorithm has been used for a large data set of solar global irradiation spectra distributions. The results obtained enable to reduce all the information of more than 270,000 c.p.d.f.’s in only 6 different clusters that correspond to 6 different c.p.d.f.’s.