Robust game theory

We present a distribution-free model of incomplete-information games, both with and without private information, in which the players use a robust optimization approach to contend with payoff uncertainty. Our ``robust game' model relaxes the assumptions of Harsanyi's Bayesian game model, and provides an alternative distribution-free equilibrium concept, which we call ``robust-optimization equilibrium,' to that of the ex post equilibrium. We prove that the robust-optimization equilibria of an incomplete-information game subsume the ex post equilibria of the game and are, unlike the latter, guaranteed to exist when the game is finite and has bounded payoff uncertainty set. For arbitrary robust finite games with bounded polyhedral payoff uncertainty sets, we show that we can compute a robust-optimization equilibrium by methods analogous to those for identifying a Nash equilibrium of a finite game with complete information. In addition, we present computational results. The research of the author was partially supported by a National Science Foundation Graduate Research Fellowship and by the Singapore-MIT Alliance. The research of the author was partially supported by the Singapore-MIT Alliance.     

Self-phase modulation in slow-wave structures: A comparative numerical analysis

Self-phase modulation effects in 1D optical slow-wave structures made of Fabry–Pérot cavities coupled by Distributed Bragg Reflectors (DBRs) are discussed. The nonlinear response of the structure is investigated by a comparative analysis of several numerical methods operating either in time or frequency-domain. Time-domain methods include two Finite-Difference Time-Domain approaches, respectively, optimized to compensate for numerical dispersion and to model nonlinearity at any order. In the frequency-domain an efficient method based on a numerical integration of Maxwell’s equations and an iterative nonlinear extension of the Eigenmode Expansion method are discussed. A Nonlinear Equivalent Circuit of DBRs is also presented as a useful model to reduce computational efforts. Numerical results show that bistable effects and self-pulsing phenomena can occur when either the optical power or the number of coupled cavities of the structure are sufficiently increased.     

A Remark on the Estimate of a Determinant by Minami

In the context of the Anderson model, Minami proved a Wegner type bound on the expectation of 2 × 2 determinant of Green’s functions. We generalize it so as to allow for a magnetic field, as well as to determinants of higher order.      

Herman-Wallis factor to improve thermometric accuracy of vibrational coherent anti-Stokes Raman spectra of H2

The Herman-Wallis factor is a molecular parameter that measures the influence of centrifugal force on the intensity of spectral lines. Understandably, the effect is significant for very light molecules that necessarily have large couplings between vibrational and rotational degrees of freedom. Although known, the conceptual basis of the Herman-Wallis factor are nevertheless not clearly established in the literature. Over the years, different approaches have been proposed to explain the corrections to spectral line-strengths and, recently, an experimental study has demonstrated that Q-branch Raman transitions of H₂ are highly sensitive to the theoretical model employed to determine the Herman-Wallis factor. In this paper, this fact is used to analyze the consequences on thermometry based on coherent anti-Stokes Raman scattering (CARS) designed to probe H₂ molecules in combustion studies. It is found that the different Herman-Wallis factors lead to relative thermometric disagreements from several tens up to hundreds of degrees. This analysis could explain why H₂ CARS thermometry has been considered less reliable than thermometric predictions based on CARS of more common molecules such as N₂, O₂ and others. In particular, it is remarked that unreliable expressions of Herman-Wallis factors have been used so far to interpret Q-branch H₂ CARS experiments.     

Optimization of the Hermitian and Skew-Hermitian Splitting Iteration for Saddle-Point Problems

We study the asymptotic rate of convergence of the alternating Hermitian/skew-Hermitian iteration for solving saddle-point problems arising in the discretization of elliptic partial differential equations. By a careful analysis of the iterative scheme at the continuous level we determine optimal convergence parameters for the model problem of the Poisson equation written in div-grad form. We show that the optimized convergence rate for small mesh parameter h is asymptotically 1–O(h ^1/2). Furthermore we show that when the splitting is used as a preconditioner for a Krylov method, a different optimization leading to two clusters in the spectrum gives an optimal, h-independent, convergence rate. The theoretical analysis is supported by numerical experiments.This revised version was published online in October 2005 with corrections to the Cover Date.     

Investigation of ion-selective electrodes with neutral ionophores and ionic sites by EIS. I. Theory

Models of an ion selective electrode involving an ionophore and mobile sites in a membrane are proposed. The first model, called the phase boundary potential model, supposed thermodynamic equilibrium; it allows the concentrations of the various species to be calculated. Then, a kinetic model, which takes into account the ionic transfer at the membrane|solution interfaces, was derived. The impedance of the membrane was calculated. It shows that a membrane with nernstian behavior shows only one capacitive loop in the impedance diagram, which is related to the conductivity and dielectric properties of the material of the membrane. Non-nernstian behavior is related to slow ionic transfer at the membrane|solution interfaces or/and transport limitation of the species in the membrane. Finite rate constants of the ionic transfer lead to a capacitive loop in the middle frequency range, whereas finite rate transport leads to a diffusional impedance in the low frequency range.     

Mean value extension theorems and microlocal analysis

We use microlocal analysis to prove new mean value theorems for harmonic functions on harmonic manifolds and for solutions to more general differential equations. The equations we consider all satisfy spherical mean value equalities, at least locally. Microlocal analysis and the mean value property in a small set allows us to show that the solution to the differential equation in a small set is also a solution in a much larger set.

     

Molecular weight distribution in emulsion polymerization. I. The homopolymer case

A model for evaluating the instantaneous degree of polymerization distribution of homopolymers produced in emulsion, based on the mathematics of the Markov chains, is developed. The model accounts for any number of active chains per particle, as well as for the two fundamental mechanisms of chain termination: mono- and bi-molecular, both by combination and by disproportionation. The core of the model is the so called subprocessmain process treatment, which allows us to correctly evaluate the degree of polymerization of the chains growing in the polymer particles, by distinguishing between the events experienced by the polymer chain which imply a change of its degree of polymerization (subject transitions) and those which imply only a change in the particle state (environment transitions). This is obtained by properly defining the one-step transition probability matrix of the relevant Markov process. Once this is done, the evaluation of the distribution of the degrees of polymerization reduces to a few simple operations among matrices. Explicit expressions for the instantaneous probability density functions and the relative cumulative distributions are obtained. The application of such relationships is facilitated by the numerical procedures reported in the Appendices. The results of the model developed in this work are in agreement with those of earlier models in the range of parameter values of practical interest. In the limit of very low molecular weights, only the model developed in this work provides the correct answer. Moreover, a much more significant result is its applicability to the case of emulsion copolymerization, as it is shown in Part II.