On repeated games with general information function

For a class of repeated two-person zero-sum games with incomplete information it was proved byAumann andMaschler that limν _n exists,ν _n being the value of the game withn repetitions. If the players know at each stage the moves done by both players at all previous stages,Aumann andMaschler could prove that the error termδ _n=¦ν _n — limν _n ¦ satisfiesδ _n≤c/√n for somec>0. It was then shown byZamir that this bound is the lowest possible. In this paper it is shown that if previous moves are not always announced,δ _n may be of higher order of magnitude e.g.δ _n≥c/n ^1/3 for somec>0. New upper bounds forδ _n are given for two classes of games.     

Ageing and fatigue of lead titanate and lead zirconate titanate thin ferroelectric films

The phenomena of ageing and fatigue have been experimentally investigated in lead titanate and lead zirconate titanate thin ferroelectric films for samples on different substrates and with different materials of the measuring electrodes. A certain broadening of the dielectric permittivity peak is observed for the films on the silicon substrate after a year keeping without external actions. The lead titanate films on corundum substrates did not demonstrate visible changes in structural and dielectric parameters in the course of this time. In the course of repeated cycling the reduction of switching polarization in the lead titanate and lead zirconate titanate films on silicon substrates takes place at considerably greater number of cycles as compared to the same films on corundum substrates under the identical conditions. The above changes of dielectric and switching characteristics can be relevant to the changes in the domain structure of the materials under investigation in the process of their ageing and repeated switching. The reason for the acceleration of the ageing processes in the films on corundum substrates could be either an increase in the absolute magnitude of the switching field or an increase of the internal bias field, that facilitate the migration of oxygen vacancies in the films with the perovskite structure to the electrode–ferroelectric surface with the consequent fixation of domain walls.     

First example of a taxane-derived propellane in Taxus canadensis needles

The first example of a propellane isolated from the needles of a yew is reported; the biogenesis from a putative taxane precursor is proposed.     

On the model for the diffusion of hydrogen in titanium hydride

The diffusion parameters of hydrogen in the gamma phase of titanium hydride are discussed in the light of nuclear magnetic resonance measurements and inelastic neutron scattering data. A comparison with a previously proposed electrostatic model for diffusion is made. It is shown that the electrostatic model is not satisfactory in explaining hydrogen diffusion in titanium. An alternate potential well system based on inelastic neutron scattering data is shown to be consistent with nuclear magnetic resonance measurements. The model can in principle be used for estimating the tunneling contribution to diffusion. The potential well shape is consistent with the notion that for hydrogen diffusion in titanium, the activation energy is given by the difference between the ground state and the well height, in contrast to the case of hydrogen diffusion in niobium where the activation energy is less than this value.     

Separation and determination of saturated very-long-chain free fatty acids in plasma of patients with adrenoleukodystrophy using solid-phase extraction and high-performance liquid chromatographic analysis

An improved method was developed for the isolation of very-long-chain free fatty acids (VLCFFAs) in plasma and their separation and determination by high-performance liquid chromatography (HPLC). The method includes sample clean-up using solid-phase extraction, fluorophoric labelling of the FFAs and reversed-phase HPLC separation. The solid-phase extraction was carried out with aminopropyl-bonded phase columns. The FFAs were then derivatized with 9-anthryldiazomethane (fluorescent) reagent and separated by HPLC on an RP-18 column with methanol as the mobile phase. Using this method, the concentrations of C20:0, C22:0, C24:0 and C26:0 were determined in the plasma of five adrenoleukodystrophy (ALD) patients, one obligatory heterozygote, four healthy male volunteers and one child with cerebral leukodystrophy but without any other ALD symptoms. Statistically significant differences were found in the levels of C24 and C26 and in the ratios C24/C22 and C26/C22 in ALD patients and in normal controls. The values were higher in patients with X-ALD. This method therefore provides a rapid and accurate procedure for the laboratory confirmation of X-ALD.     

Determination of inorganic sulfur species in highly alkaline solutions by liquid chromatography with polarographic detection

Indirect anodic detection of sulfide, polysulfide, sulfite and thiosulfate can be achieved at a dropping mercury electrode based on the anodic oxidation of mercury to produce stable products of Hg(II). Results for sulfide (S^2?) and polysulfide (S^2?_x) ions in alkaline media are emphasized. The maximum value of x in S^2?_x was found to be 5, which is in agreement with the literature. Values of diffusion coefficients for the various species of S^2?_x (D_x; x = 1–5) were found to decrease dramatically as x is increase, e.g., D₁/D₂ = 3.6 and D₁/D₅ = 32.4. Mixing solutions of S^2?₅ with S^2? is not followed by rapid re-equilibration to produce a polysulfide with average x<5. It is concluded that polarography alone is not sufficient for quantifying S^2? and S^2?_x in their mixture. High-performance anion-chromatography with polarographic detection was examined for separation of alkaline mixtures of inorganic sulfur compounds. Separations were achieved within ca. 12 min. Polysulfide (S^2?_x) is concluded to undergo dissociation to S^2? and S° on the anion-exchange column and no cathodic detection peak is obtained which is characteristic of the polarographic reduction of S^2?_x. A negative peak in the cathodic baseline current for reduction of dissolved oxygen at E <0.0 V just prior to the peak for S₂O^2?₃ is attributed to the effect of eluted S°. The retention time of the S° peak is independent of the concentration of nitrate in the mobile phase, which is consistent with a non-ionic interaction between S° and the anion-exchange separator. The technique does not detect sulfate.     

The normal distribution and repeated games

For a reperated zero-sum two-person game with incomplete information discussed byZamir, it is proved here that \(\mathop {\lim }\limits_{n \to \infty } \sqrt n v_n (p) = \phi (p)\) whereφ (p) is the normal density function evaluated at itsp-quantile (i.e. \(\phi (p) = \frac{1}{{\sqrt {2\pi } }}e^{ - ({1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2})x^2 } p\) where \(\frac{1}{{\sqrt {2\pi } }}\mathop {\smallint ^p }\limits_{ - \infty }^x e^{ - ({1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2})x^2 } dx = p\) . Here for 0?p?1, (p, 1 ?p) is the a priori probability distribution on two states of nature, the actual state of nature is known to the maximizer but not to the minimizer.v _n (p) is the minimax value of the game withn stages.