Colorful Strips

We study the following geometric hypergraph coloring problem: given a planar point set and an integer k, we wish to color the points with k colors so that any axis-aligned strip containing sufficiently many points contains all colors. We show that if the strip contains at least 2k − 1 points, such a coloring can always be found. In dimension d, we show that the same holds provided the strip contains at least k(4 ln k + ln d) points. We also consider the dual problem of coloring a given set of axis-aligned strips so that any sufficiently covered point in the plane is covered by k colors. We show that in dimension d the required coverage is at most d(k − 1) + 1. This complements recent impossibility results on decomposition of strip coverings with arbitrary orientations. From the computational point of view, we show that deciding whether a three-dimensional point set can be 2-colored so that any strip containing at least three points contains both colors is NP-complete. This shows a big contrast with the planar case, for which this decision problem is easy.     

Helly Numbers of Polyominoes

We define the Helly number of a polyomino P as the smallest number h such that the h-Helly property holds for the family of symmetric and translated copies of P on the integer grid. We prove the following: (i) the only polyominoes with Helly number 2 are the rectangles, (ii) there does not exist any polyomino with Helly number 3, (iii) there exist polyominoes of Helly number k for any k ≠ 1, 3.     

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 ${f(x) + f(2x) + \cdots + f(nx) = 0, x \in \mathbb{R}}$ . The continuity of the solutions depends on the sign of the real part of each zero.     

Preliminary assessment of the performance of oyster shells and chitin materials as adsorbents in the removal of saxitoxin in aqueous solutions

ABSTRACT: BACKGROUND: This study evaluated the adsorption capacity of the natural materials chitin and oyster shell powder (OSP) in the removal of saxitoxin (STX) from water. Simplified reactors of adsorption were prepared containing 200 mg of adsorbents and known concentrations of STX in solutions with pH 5.0 or 7.0, and these solutions were incubated at 25 [DEGREE SIGN]C with an orbital shaker at 200 RPM. The adsorption isotherms were evaluated within 48 hours, with the results indicating a decrease in STX concentrations in different solutions (2[EN DASH]16 mug/L). The kinetics of adsorption was evaluated at different contact times (0[EN DASH]4320 min) with a decrease in STX concentrations (initial concentration of 10 mug/L). The sampling fractions were filtered through a membrane (0.20 mum) and analyzed with high performance liquid chromatography to quantify the STX concentration remaining in solution. RESULTS: Chitin and OSP were found to be efficient adsorbents with a high capacity to remove STX from aqueous solutions within the concentration limits evaluated (> 50 % over 18 h). The rate of STX removal for both adsorbents decreased with contact time, which was likely due to the saturation of the adsorbing sites and suggested that the adsorption occurred through ion exchange mechanisms. Our results also indicated that the adsorption equilibrium was influenced by pH and was not favored under acidic conditions. CONCLUSIONS: The results of this study demonstrate the possibility of using these two materials in the treatment of drinking water contaminated with STX. The characteristics of chitin and OSP were consistent with the classical adsorption models of linear and Freundlich isotherms. Kinetic and thermodynamic evaluations revealed that the adsorption process was spontaneous (DeltaGads < 0) and favorable and followed pseudo-second-order kinetics.     

Final-state polarization in <Emphasis Type="Italic">B</Emphasis><Stack><Subscript><Emphasis Type="Italic">s</Emphasis></Subscript><Superscript>0</Superscript></Stack> decays

Certain B _s ⁰→V ₁ V ₂ decays (V _i is a vector meson) can be related by flavor SU(3) symmetry to corresponding B _d ⁰→V ₃ V ₄ decays. In this paper, we show that the final-state polarization can be predicted in the B _s ⁰ decay, assuming polarization measurements of the B _d ⁰ decay. This can be done within the scenario of penguin annihilation (PA), which has been suggested as an explanation of the unexpectedly large transverse polarization in B→φ K ^*. PA is used to estimate the breaking of flavor SU(3) symmetry in pairs of decays. Two of these for which PA makes a reasonably precise prediction of the size of SU(3) breaking are (B _s ⁰→φ φ,B _d ⁰→φ K ^0*) and ( ). The polarization measurement in the B _d ⁰ decay can be used to predict the transverse polarization in the B _s ⁰ decay and will allow for testing of PA as well as the other assumptions in the analysis.