On the measure of the spectrum for the almost Mathieu operator

We obtain partial results on the conjecture that for the almost Mathieu operator at irrational frequency, , the measure of the spectrum,S(, , )=|4–2|. For ||2 we show that if _n is rational and irrational, then .Dedicated to Res Jost and Arthur WightmanResearch partially supported by U.S. NSF grant number DMS-8801918 and by BSF under grant number 88-00325     

The coexistence problem for the discrete Mathieu operator

We solve the coexistence problem for the periodic discrete Mathieu operator in all parametric cases. The main tool in the proof will be Bezout's theorem for projective plane curves. As an additional result we obtain the gap opening and gap growth powers for this operator.     

Cournot equilibrium uniqueness via demi-concavity

A family of oligopolies that possess a unique equilibrium was identified in the second authors doctoral dissertation. For such a family, it is therein specified a class of functions-economically related to the price function of a Cournot oligopoly – that satisfy a particular type of quasi-concavity. The first part of the present article (i) conceptualizes that type of quasi-concavity by introducing the notion of demi-concavity, (ii) considers two possible variants and (iii) provides some calculus properties. The second part, by relying on the results on demi-concavity, proves a Cournot equilibrium uniqueness theorem which is new for the journal literature and subsumes various results thereof. A third part shows an example that illustrates the novelty of the result.     

A simplified asymptotic theory for ocean surface electromagnetic wave scattering

The normalized radar cross-section (NRCS) expression of the Local Curvature Approximation (LCA-1) is derived to first order. The polarization sensitivity of this model is compared to the Kirchhoff Approximation (KA), Two-Scale Model (TSM), Small Slope Approximation (SSA-1) and Small Perturbation Method (SPM-1) to first order in the backscattering configuration. Analytical comparisons and numerical simulations show that LCA-1 and TSM could be rewritten with the same formulation and that their polarization sensitivities are comparable. Comparisons with experimental data acquired in C- and Ku-band reveal that the polarization sensitivities of these models are not adequate. However, the NRCS azimuth modulation predicted by LCA-1 is found to be dependent on polarization and sea surface roughness. This property of the LCA-1 model yields to an azimuth modulation for the polarization ratio. Based on the surface curvature correction concept, a simplified electromagnetic model is proposed. The curvature correction is restricted to the resonant wave-number of the sea roughness spectrum. This is found to reproduce the polarization ratio given by experimental data versus incidence angle and wind speed.     

Predicted Doppler shifts induced by ocean surface wave displacements using asymptotic electromagnetic wave scattering theories

Sea surface motions can produce different measured Doppler shifts with respect to instrumental configurations (incidence angle, electromagnetic wavelength, polarization). Under Gaussian statistics for the sea surface elevation and in the general framework of asymptotic theories for ocean surface electromagnetic wave scattering, Doppler shifts can be predicted. The small-slope, Kirchhoff, local curvature and resonant curvature approximations are compared in the backscatter configuration. Predicted Doppler shifts for Kirchhoff and small-slope approximations in co-polarized configuration are insensitive to the polarization state. On the other hand, the local and resonant curvature solutions, through a phase perturbation formalism, yield to significant differences between co-polarized predicted Doppler shifts. Comparisons with data are shown to confirm the polarization and wind speed sensitivities.     

Analysis of Displacement Variances in Stochastic Nonuniform Flows by Means of a First Order Analytical Model and Comparison with Monte-Carlo Simulations

Flow, particle displacement and particle arrival time statistics in 2D nonuniform flows, without microdispersion, are studied both theoretically, in the framework of stochastic modelling, and numerically by means of Monte-Carlo simulations.Turning, radial convergent and dipolar flow fields are considered. These three types of flow are numerically investigated in heterogeneous media with different levels of heterogeneity. Monte-Carlo simulations show (1) that the scale separation hypothesis, frequently used in fluid mechanics, is justified for one-point flow statistics; and (2) how displacement variances, and consequently the dispersivities defined as their spatial derivatives, depend on the type and the amplitude of flow nonuniformity: in none of the investigated cases does the assumption of scale separation hold for displacement, except in the turning flow when the spatial scale associated to the nonuniformity is much greater than the correlation scale of transmissivity.The theoretical approach of displacement and arrival time statistics relies on the analysis of particle trajectory. Displacement variances expressions are derived by the perturbation method for each type of flow and for different approximation orders. The proposed expressions of displacement variances are, on the whole, in good agreement with the numerical results. On the other hand, the uniform flow approximation – commonly used for the interpretation of tracer experiments – chosen such as to satisfy the mean arrival time to the pumping well, gives the best prediction of the breakthrough curves.     

Interconnected games and international environmental problems,II

Some features of international environmental problems are considered. A basic problem is to induce countries to adopt a cooperative approach. One of the instruments to induce countries to cooperate is an exchange of concessions in fields of relative strengths, such as swapping trade concessions for cooperation on international environmental problems. This instrument will be modelled in this paper with tensor games. Both tradeoff and non-tradeoff tensor games will be addressed, with emphasis on tradeoff tensor games with linear strict weights. The relationship between the Pareto equilibria of a non-tradeoff tensor game and the Nash equilibria of the associated tradeoff tensor games will be studied. Due to structural similarities between tensor games and repeated multiple objective games, some attention will also be paid to the latter. Relationships between objects related to Folk theorems for the tradeoff tensor game with completely additive weights and the corresponding objects for its constituting isolated games will be studied. Since many international environmental problems have prisoners' dilemma characteristics, it is analyzed how interconnection may enhance cooperation in prisoners' dilemma games.     

The Hotelling bi-matrix game

We study the pure equilibrium set for a specific symmetric finite game in strategic form, referred to as the Hotelling bi-matrix game. General results that guarantee non-emptiness of this set (for all parametric values) do not seem to exist. We prove non-emptiness by determining the pure equilibrium set. In this proof so-called demi-modality properties of the conditional payoff functions play an important role.     

ON A LESS KNOWN NASH EQUILIBRIUM UNIQUENESS RESULT

We generalize a less known Nash equilibrium uniqueness result for games in strategic form. Its power is illustrated by applying it to a Public Goods Game, a Homogeneous Cournot Oligopoly Game and a Formal Transboundary Pollution Game.     

On the uniqueness of Cournot equilibrium in case of concave integrated price flexibility

We consider a class of homogeneous Cournot oligopolies with concave integrated price flexibility and convex cost functions. We provide new results about the semi-uniqueness and uniqueness of (Cournot) equilibria for the oligopolies that satisfy these conditions. The condition of concave integrated price flexibility is implied by (but does not imply) the log-concavity of a continuous decreasing price function. So, our results generalize previous results for decreasing log-concave price functions and convex cost functions. We also discuss the particular type of quasi-concavity that characterizes the conditional revenue and profit functions of the firms in these oligopolies and we point out an error of the literature on the equilibrium uniqueness in oligopolies with log-concave price functions. Finally, we explain how the condition of concave integrated price flexibility relates to other conditions on the price and aggregate revenue functions usually considered in the literature, e.g., their concavity.