In this article we present a simple and elegant algebraic proof of Pascal's hexagon theorem which requires only knowledge of basics on conic sections without theory of projective transformations. Also, we provide an efficient algorithm for finding an equation of the conic containing five given points and a criterion for verification whether a set of points is a subset of the conic.

Background and aims: Benzodiazepine (BDZ) abuse highly prevalent among former heroin addicts, currently in methadone maintenance treatment. Discontinuation of BDZ abuse is accompanied by sleep disturbances. We evaluated the effectiveness of melatonin in attenuating sleep difficulties in a BDZ withdrawal program. Methods: Patients in a managed BDZ withdrawal program entered a double blind crossover control study with melatonin or placebo: 6 weeks one arm, one week washout, 6 weeks other arm. Urine BDZ, the self reported Pittsburgh Sleep Quality Index (PSQI) and the Center for Epidemiologic Studies Depression (CES-D, mood) questionnaire were administered at baseline, and after 6, 8 and 13 weeks. Results: Eighty patients were randomly assigned into two arms. Both groups (n¼40) had similar baseline PSQI (13.8AE3.8) and CES-D (1.5AE0.6) scores, which correlated (R¼0.4, p¼0.001). Sixty one patients (77.5% in "melatonin-first" and 75% in "placebo first") finished 6 weeks, showing similar BDZ discontinuation rate 11/31 and 11/30 respectively. PSQI scores were significantly lower (better sleep) in the 22 patients who discontinued BDZ (8.9AE4.4) than in 39 with urine BDZ (11.2AE4.2 p¼0.04). Interaction between study groups and BDZ groups showed that sleep quality in patients who continued abusing BDZ improved more in the "melatonin first" group than in the "placebo first" group, with no differences in sleep quality improvement in patients who stopped BDZ (F¼4.3, p¼0.04). Conclusions: Most improvement in sleep quality was attributed to BDZ discontinuation. Although melatonin did not enhance BDZ discontinuation, it improved sleep quality, especially in patients who did not stop BDZ.

In the paper [1] Andy Zucker and Chris Lambie-Hanson proved the consistency result for some coloring principle for the products of polish spaces by at most countable many colors. This principle easy implies the Halpern and Läuchli's theorem. The aim of this paper is to generalize this consistency result about uncountable sets of colors. The proof presented here differs than the proof presented in [1].

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