The Nicolas and Robin inequalities with sums of two squares

In 1984, G. Robin proved that the Riemann hypothesis is true if and only if the Robin inequality σ(n) < e ^γ n log log n holds for every integer n > 5040, where σ(n) is the sum of divisors function, and γ is the Euler–Mascheroni constant. We exhibit a broad class of subsets of the natural numbers such that the Robin inequality holds for all but finitely many . As a special case, we determine the finitely many numbers of the form n = a ² + b ² that do not satisfy the Robin inequality. In fact, we prove our assertions with the Nicolas inequality n/φ(n) < e ^γ log log n; since σ(n)/n < n/φ(n) for n > 1 our results for the Robin inequality follow at once.      

Locally anisotropic toposes

This paper continues the investigation of isotropy theory for toposes. We develop the theory of isotropy quotients of toposes, culminating in a structure theorem for a class of toposes we call locally anisotropic. The theory has a natural interpretation for inverse semigroups, which clarifies some aspects of how inverse semigroups and toposes are related.     

Computation and efficiency of potential function minimizers of combinatorial congestion games

Mathematical Programming - We study the computation and efficiency of pure Nash equilibria in combinatorial congestion games, where the strategies of each player i are given by the binary vectors...     

Photostabilizing of bisphenol A polycarbonate by using UV-absorbers and self protective block copolymers based on resorcinol polyarylate blocks

Bisphenol A polycarbonate degrades due to sunlight, humidity and oxygen. In this study two possible techniques to stabilize the polymer were compared, i.e. blending of UV-absorbers (UVAs) into the polymer or using block copolymers based on resorcinol polyarylates. Combination of different analysis techniques shows that the protection by UVAs is not as good as by the resorcinol polyarylate block copolymers. The block copolymer rearranges itself through a photo-Fries rearrangement within hours into a UV-absorbing top layer. Two different block compositions were studied, and the copolymer with the highest concentration of resorcinol polyarylate groups shows the best protection.     

DFT study on H2O activation by stepped and planar Rh surfaces

In this research the effect of steps (lower coordinated surface atoms) and the presence of pre-adsorbed oxygen on the activation energy of water are studied with DFT. Without oxygen water activation is found to be structure insensitive. When oxygen is adsorbed on the surface and acts as the acceptor for the hydrogen at the step edge, the barrier will decrease significantly.     

Magnetically induced decrease in droplet contact angle on nanostructured surfaces

We report a magnetic technique for altering the apparent contact angle of aqueous droplets deposited on a nanostructured surface. Polymeric tubes with embedded superparamagnetic magnetite (Fe(3)O(4)) nanoparticles were prepared via layer-by-layer deposition in the 800 nm diameter pores of polycarbonate track-etched (PCTE) membranes. Etching away the original membrane yields a superparamagnetic film composed of mostly vertical tubes attached to a rigid substrate. We demonstrate that the apparent contact angle of pure water droplets deposited on the nanostructured film is highly sensitive to the ante situm strength of an applied magnetic field, decreasing linearly from 117 ± 1.3° at no applied field to 105 ± 0.4° at an applied field of approximately 500 G. Importantly, this decrease in contact angle did not require an inordinately strong magnetic field: a 15° decrease in contact angle was observed even with a standard alnico bar magnet. We interpret the observed contact angle behavior in terms of magnetically induced conformation changes in the film nanostructure, and we discuss the implications for reversibly switching substrates from hydrophilic to hydrophobic via externally tunable magnetic fields.