Discrete Multiscale Analysis: A Biatomic Lattice System

We discuss a discrete approach to the multiscale reductive perturbative method and apply it to a biatomic chain with a nonlinear interaction between the atoms. This system is important to describe the time evolution of localized solitonic excitations. We require that also the reduced equation be discrete. To do so coherently we need to discretize the time variable to be able to get asymptotic discrete waves and carry out a discrete multiscale expansion around them. Our resulting nonlinear equation will be a kind of discrete Nonlinear Schr\"odinger equation. If we make its continuum limit, we obtain the standard Nonlinear Schr\"odinger differential equation

Almost free splitters

Let R be a subring of the rationals. We want to investigate self splitting R-modules G that is Ext_R(G,G)=0 holds. For simplicity we will call such modules splitters. Our investigation continues math.LO/9910159. In math.LO/9910159, we answered an open problem by constructing a large class of splitters. Classical splitters are free modules and torsion-free, algebraically compact ones. In math.LO/9910159 we concentrated on splitters which are larger then the continuum and such that countable submodules are not necessarily free. The `opposite' case of aleph_1-free splitters of cardinality less or equal to aleph_1 was singled out because of basically different techniques. This is the target of the present paper. If the splitter is countable, then it must be free over some subring of the rationals by a result of Hausen. We can show that all aleph_1-free splitters of cardinality aleph_1 are free indeed

Modeling the effect of anisotropic pressure on tokamak plasmas normal modes and continuum using fluid approaches

Extending the ideal MHD stability code MISHKA, a new code, MISHKA-A, is developed to study the impact of pressure anisotropy on plasma stability. Based on full anisotropic equilibrium and geometry, the code can provide normal mode analysis with three fluid closure models: the single adiabatic model (SA), the double adiabatic model (CGL) and the incompressible model. A study on the plasma continuous spectrum shows that in low beta, large aspect ratio plasma, the main impact of anisotropy lies in the modification of the BAE gap and the sound frequency, if the q profile is conserved. The SA model preserves the BAE gap structure as ideal MHD, while in CGL the lowest frequency branch does not touch zero frequency at the resonant flux surface where $m+nq=0$, inducing a gap at very low frequency. Also, the BAE gap frequency with bi-Maxwellian distribution in both model becomes higher if $p_\perp > p_\parallel$ with a q profile dependency. As a benchmark of the code, we study the m/n=1/1 internal kink mode. Numerical calculation of the marginal stability boundary with bi-Maxwellian distribution shows a good agreement with the generalized incompressible Bussac criterion [A. B. Mikhailovskii, Sov. J. Plasma Phys 9, 190 (1983)]: the mode is stabilized(destabilized) if $p_\parallel < p_\perp (p_\parallel > p_\perp)$

Martingales and arbitrage: a new look

This paper addresses the equivalence between the absence of arbitrage and the existence of equivalent martingale measures. The equivalence will be established under quite weak assumptions since there are no conditions on the set of trading dates (it may be finite or infinite, with bounded or unbounded horizon, etc.) or on the trajectories of the price process (for instance, they do not have to be right-continuous). Besides we will deal with arbitrage portfolios rather than free-lunches. The concept of arbitrage is much more intuitive than the concept of free lunch and has more clear economic interpretation. Furthermore it is more easily tested in theoretical models or practical applications. In order to overcome the usual mathematical difficulties arising when dealing with arbirage strategies, the set of states of nature will be widened by drawing on projective systems of Radon probability measures, whose projective limit will be the martingale measure. The existence of densities between the "real" probabilities and the "risk-neutral" probabilities will be guaranteed by introducing the concept of "projective equivalence". Hence some classical counter-examples will be solved and a complete characterization of the absence of arbitrage will be provided in a very general framework