On Hausdorff dimension of the set of closed orbits for a cylindrical transformation

We deal with Besicovitch's problem of existence of discrete orbits for transitive cylindrical transformations $T_\varphi:(x,t)\mapsto(x+\alpha,t+\varphi(x))$ where $Tx=x+\alpha$ is an irrational rotation on the circle \T and \varphi:\T\to\R is continuous, i.e.\ we try to estimate how big can be the set D(\alpha,\varphi):=\{x\in\T:|\varphi^{(n)}(x)|\to+\infty\text{as}|n|\to+\infty\}. We show that for almost every $\alpha$ there exists $\varphi$ such that the Hausdorff dimension of $D(\alpha,\varphi)$ is at least $1/2$. We also provide a Diophantine condition on $\alpha$ that guarantees the existence of $\varphi$ such that the dimension of $D(\alpha,\varphi)$ is positive. Finally, for some multidimensional rotations $T$ on \T^d, $d\geq3$, we construct smooth $\varphi$ so that the Hausdorff dimension of $D(\alpha,\varphi)$ is positive.Comment: 32 pages, 1 figur

Ergodic directions for billiards in a strip with periodically located obstacles

We study the size of the set of ergodic directions for the directional billiard flows on the infinite band $\R\times [0,h]$ with periodically placed linear barriers of length $0<\lambda<h$. We prove that the set of ergodic directions is always uncountable. Moreover, if $\lambda/h\in(0,1)$ is rational the Hausdorff dimension of the set of ergodic directions is greater than 1/2. In both cases (rational and irrational) we construct explicitly some sets of ergodic directions.Comment: The article is complementary to arXiv:1109.458

Sex determination based on the analysis of a contemporary Polish population’s palatine bones: a computed tomography study of 1,200 patients

Background: The aims of the present study were to assess whether the hard palate reveals any measurable sex-related differences, and to create a mathematical model which would differentiate between males and females using hard palate measurements alone. Materials and methods: The present study was conducted on 1,200 archived sinus computed tomography (CT) scans. Each cranial measurement was taken twice by the same observer, and in cases of any discrepancies, the mean of the two values was recorded. Twenty per cent of randomly chosen samples were re-measured by an observer who did not partake in assessing the samples the first time. Logistic regression was used to derivate two mathematical formulas which would calculate the probability of a skull being male. Results: The studied group comprised 1,200 head CT’s (627 female; 52.3%). The mean age of the group was 43.5 ± 17.4 years — no age difference between sexes was noted (p = 0.37). All of the performed measurements were significantly (p &lt; 0.0001) larger in males than in females. The mathematical formula based on the “orale-spina nasalis posterior” (O-SNP) distance alone had a reliability rate of 68.35%. The equation based on the depth of the right greater palatine canal (GPC), the O-SNP distance and the anterior width of the palatal arch (AWPA) had a reliability rate of 78.37%. Conclusions: The most prominent sexually dimorphic parameters were the O-SNP, the GPC depth and the AWPA. The mathematical models presented in the current study can be used to successfully distinguish between sexes during forensic examination.

Ergodic infinite group extensions of geodesic flows on translation surfaces

We show that generic infinite group extensions of geodesic flows on square tiled translation surfaces are ergodic in almost every direction, subject to certain natural constraints. Recently K. Fr\c{a}czek and C. Ulcigrai have shown that certain concrete staircases, covers of square-tiled surfaces, are not ergodic in almost every direction. In contrast we show the almost sure ergodicity of other concrete staircases. An appendix provides a combinatorial approach for the study of square-tiled surfaces

Determination of nitrogen dioxide, sulfur dioxide, ozone, and ammonia in ambient air using the passive sampling method associated with ion chromatographic and potentiometric analyses

Concentrations of nitrogen dioxide (NO2), sulfur dioxide (SO2), ozone (O3), and ammonia (NH3) were determined in the ambient air of Al-Ain city over a year using the passive sampling method associated with ion chromatographic and potentiometric detections. IVL samplers were used for collecting nitrogen and sulfur dioxides whereas Ogawa samplers were used for collecting ozone and ammonia. Five sites representing the industrial, traffic, commercial, residential, and background regions of the city were monitored in the course of this investigation. Year average concentrations of ≤59.26, 15.15, 17.03, and 11.88 μg/m3 were obtained for NO2, SO2, O3, and NH3, respectively. These values are lower than the maxima recommended for ambient air quality standards by the local environmental agency and the world health organization. Results obtained were correlated with the three meteorological parameters: humidity, wind speed, and temperature recorded during the same period of time using the paired t test, probability p values, and correlation coefficients. Humidity and wind speed showed insignificant effects on NO2, SO2, O3, and NH3 concentrations at 95% confidence level. Temperature showed insignificant effects on the concentrations of NO2 and NH3 while significant effects on SO2 and O3 were observed. Nonlinear correlations (R2 ≤ 0.722) were obtained for the changes in measured concentrations with changes in the three meteorological parameters. Passive samplers were shown to be not only precise (RSD ≤ 13.57) but also of low cost, low technical demand, and expediency in monitoring different locations