Minmax and maxmin of repeated games with incomplete information

For a class of 2-Person 0-sum repeated games with incomplete information,Aumann/Masch1er [1967] andStearns [1967] have given a necessary and sufficient condition for the existence of v_∞ (the value of the infinitely repeated game).Mertens/Zamir [1971] andMertens [1971/72] have given the formula (and thus proved the existence) of \(\mathop {\lim }\limits_{n \to \infty } \) v _n , the limit of the values of the games withn repetitions, for a much larger class of games than that treated byAumann/Maschler andSteams. In this paper we extend the Aumann-Maschler-Stearns results to the larger family of games studied byMertens [1971/72].     

The maximal variation of a bounded martingale

Let \(\chi _0^n = \left\{ {X_t } \right\}_0^n \) be a martingale such that 0≦X_i≦1;i=0, …,n. For 0≦p≦1 denote by ? _p ⁿ the set of all such martingales satisfying alsoE(X₀)=p. Thevariation of a martingale χ ₀ ⁿ is denoted byV ₀ ⁿ and defined by \(V(\chi _0^n ) = E\left( {\sum {_{l = 0}^{n - 1} } \left| {X_{l + 1} - X_l } \right|} \right)\) . It is proved that $$\mathop {\lim }\limits_{n \to \infty } \left\{ {\mathop {Sup}\limits_{x_0^n \in \mathcal{M}_p^n } \left[ {\frac{1}{{\sqrt n }}V(\chi _0^n )} \right]} \right\} = \phi (p)$$ , where ?(p) is the well known normal density evaluated at itsp-quantile, i.e. $$\phi (p) = \frac{1}{{\sqrt {2\pi } }}\exp ( - \frac{1}{2}\chi _p^2 ) where \int_{ - \alpha }^{x_p } {\frac{1}{{\sqrt {2\pi } }}\exp ( - \frac{1}{2}\chi ^2 )} dx = p$$ . A sequence of martingales χ ₀ ⁿ ,n=1,2, … is constructed so as to satisfy \(\lim _{n \to \infty } (1/\sqrt n )V(\chi _0^n ) = \phi (p)\) .     

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.     

Lateral confined epitaxy of GaN layers on Si substrates

We developed a novel, simple procedure for achieving lateral confined epitaxy (LCE). This procedure enables the growth of uncracked GaN layers on a Si substrate, using a single, continuous metalorganic chemical vapor deposition (MOCVD) run. The epitaxial growth of GaN is confined to mesas, defined by etching into the Si substrate prior to the growth. The LCE-GaN layers exhibit improved morphological and optical properties compared to the plain GaN-on-Si layers grown in the same MOCVD system. By performing a set of LCE growth runs on mesas of varying lateral dimensions, we specified the crack-free range of GaN on Si as 14.0±0.3 μm.     

Zero-sum sequential games with incomplete information

Repeated zero-sum two-person games of incomplete information on one side are considered. If the one-shot game is played sequentially, the informed player moving first, it is proved that the value of then-shot game is constant inn and is equal to the concavification of the game in which the informed player disregards his extra information. This is a strengthening ofAumann andMaschler's results for simultaneous games. Optimal strategies for both players are constructed explicitly.     

Investigation of a nanostrip patch antenna in optical frequencies

This is the first report and investigation of a patch antenna in optical frequency range. Variety of plasmonic nanoantenna reported so far is good at enhancing the local field intensity of light by orders of magnitude. However, their far-field radiation efficiency is very poor. The proposed patch antenna emits a directional beam with high efficacy in addition to enhancing the intensity of near field. The nano-patch antenna (NPA) consists of a square patch of gold film of dimension 480 nm², placed on a substrate of dielectric constant \( \varepsilon_{\text{r}} \)  = 3.9 and thickness 150 nm with a ground plane of gold film of dimension 1,080 nm². The NPA resonates at 210 THz and has gain nearly 2 dB and radiation efficiency 45.18 %. The NPA might be useful in variety of applications such as optical communication, nano-photonics, biosensing, and spectroscopy.     

Nanosecond pulsed laser ablation of silicon in liquids

Laser fluence and laser shot number are important parameters for pulse laser based micromachining of silicon in liquids. This paper presents laser-induced ablation of silicon in liquids of the dimethyl sulfoxide (DMSO) and the water at different applied laser fluence levels and laser shot numbers. The experimental results are conducted using 15 ns pulsed laser irradiation at 532 nm. The silicon surface morphology of the irradiated spots has an appearance as one can see in porous formation. The surface morphology exhibits a large number of cavities which indicates as bubble nucleation sites. The observed surface morphology shows that the explosive melt expulsion could be a dominant process for the laser ablation of silicon in liquids using nanosecond pulsed laser irradiation at 532 nm. Silicon surface’s ablated diameter growth was measured at different applied laser fluences and shot numbers in both liquid interfaces. A theoretical analysis suggested investigating silicon surface etching in liquid by intense multiple nanosecond laser pulses. It has been assumed that the nanosecond pulsed laser-induced silicon surface modification is due to the process of explosive melt expulsion under the action of the confined plasma-induced pressure or shock wave trapped between the silicon target and the overlying liquid. This analysis allows us to determine the effective lateral interaction zone of ablated solid target related to nanosecond pulsed laser illumination. The theoretical analysis is found in excellent agreement with the experimental measurements of silicon ablated diameter growth in the DMSO and the water interfaces. Multiple-shot laser ablation threshold of silicon is determined. Pulsed energy accumulation model is used to obtain the single-shot ablation threshold of silicon. The smaller ablation threshold value is found in the DMSO, and the incubation effect is also found to be absent.     

NMR study of hydrogen diffusion in uranium hydride

The diffusion of hydrogen in uranium hydride is studied employing the NMR technique. From measurements of spin-spin relaxation time T₂, the activation energy for hydrogen diffusion in β-UH₃ is determined to be $\text{E}{}_{\mathrm{a}}\text{= (19.25 ± 0.4)}\text{kcal}\text{mole}$ and the preexponential factor to be A₀ ≈ 5 × 10¹⁴ Hz. It is shown that these results are in fair agreement with spin-lattice relaxation time T₁ data. Assuming that hydrogen diffusion proceeds via vacancies whose concentration is temperature dependent, it is concluded that E_a is the sum of the energies of vacancy formation and barrier height, and that A₀ contains an entropy change factor. Using vacancy concentration data calculated by Libowitz, we estimate the barrier height energy to be E_b ≈7 kcal/mole. Using a value for the frequency of hydrogen vibration v₀ determined from inelastic neutron scattering by Rush et al., we estimate the entropy change due to vacancy formation and the hydrogen atom jump to be about $\text{S}\text{k}{}_{\mathrm{B}}\text{≈3}$. Similar measurements on samples containing less hydrogen than is needed to compose stoichiometric UH₃, show that the rate of diffusion is enhanced by the presence of excess metal in the sample. The jump frequency at 500°K in UH₃ is found to be approximately 10⁶ Hz while for the two-phase samples of H/U = 2.8 and 2.5, it is larger by a factor of about 3 and 3.5, respectively.     

On a repeated game without a recursive structure

The solution is given here for the infinitely repeated two-person zero-sum games of incomplete information characterized by 2×2 games, with information matrices $\left( {{*{20}c} a & b \\ b & b \\ } \right)$\left( {\begin{array}{*{20}c} a & b \\ b & b \\ \end{array} } \right) for the first game and $\left( {{*{20}c} b & b \\ b & a \\ } \right)$\left( {\begin{array}{*{20}c} b & b \\ b & a \\ \end{array} } \right) for the second game.