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A nonlinear, nonlocal cochlear model of the transmission line type is studied in order to capture the multitone interactions and resulting tonal suppression effects. The model can serve as a module for voice signal processing, and is a one-dimensional (in space) damped dispersive nonlinear PDE based on the mechanics and phenomenology of hearing. It describes the motion of the basilar membrane (BM0 in the cochlea driven by input pressure waves. Both elastic dampling and selective longitudinal fluid damping are present. The forner is nonlinear and nonlocal in BM displacement, and plays a kep role in capturing tonal interactions. The latter is active only near the exit boundary (helicotrema), and is built in to damp out the remaining long waves. The initial boundary value problem is numerically solved with a semi-implicit second order finite difference method. Solutions reach a multi-frequency quai-steady state. Numerical results are shown on two tone suppression. Suppression effects among three tones are demonstrated by showing how the response magnitudes of the fixed two tones are reduced as we vary the third tone in frequency and amplitude. We observe qualitative agreement of our model solutions with exisiting cat auditory neural data. The model is thus a simple and efficient processing tool for voice signals.
We prove some regularity for viscosity solutions to strongly degenerate parabolic semilinear problems. These results apply to a specific model used for pricing Mortgage-Backed Securities and allow a complete justification of the existence of the proposed risk-neutral-measure.
We review the euclidean path-integral formalism in connection with the one-dimensional non-relativistic particle. The configurations which allow construction of a semiclassical approximation classify themselves into either topological (instantons) and non-topological (bounces) solutions. The quantum amplitudes consist of an exponential associated with the classical contribution as well as the energy eigenvalues of the quadratic operators at issue can be written in closed form due to the shape-invariance property. Accordingly, we resort to the zeta-function method to compute the functional determinants in a systematic way. The effect of the multi-instantons configurations is also carefully considered. To illustrate the instanton calculus in a relevant model, we go to the double-wall potential. The second popular case is the periodic-potential where the initial levels split into bands. The quantum decay rate of the metastable states in a cubic model is evaluated by means of the bounce-like solution.
We construct elementary examples of systems of hyperbolic equations having solutions which blow up in finite time. We explicitly describe the system, initial data and solution. First, we exhibit a 3x3 system with compactly supported data which blows up in finite time. The solutions blows up in amplitude (Linfinity] norm) on an entire interval, so there is no possibilty of continuing the solution beyond the blowup time. We then consider a system of two Burger equations which are coupled through linear boundary conditions. We record the interesting observation that although the IBVP with a single boundary condition is globally well-posed, when two boundary conditiond are used on a finite domain, the IBVP is ill-posed. Because waves are reflected back into the domain, multiple interactions combine to give blowup in finite time, for arbitrarily small initial data. We conclude that some global integral or energy condition must be imposed in order to expect stability of solutions to IBVPs on compact domains. Finally, we show that the presence of shocks is not necessary, by exhibiting solutions which are continuous in the nonlinear fields. However, our solutions do contain discontinuities in the linearly degenerate field.
We use the method of characteristics to prove the short-time existence of smooth solutions of the unsteady inviscid Prandtl equations, and present a simple explicit solution that forms a singularity in finite time. We give numerical and asymptotic solutions which indicate that this singularity persists for nonmonotone solutions of the viscous Prandtl equations. We also solve the linearization of the inviscid Prandtl equation about shear flow. We show that the resulting problem is weakly, but not strongly, well-posed, and that it has an unstable continuous spectrum when the shear flow has a critical point, in contrast with the behavior of the linearized Euler equations.
We present a new formulaiton of the incompressible Navier-Stokes equation in terms of an auxiliary field that differs from the velocity by a gauge transformation. The gauge freedom allows us to assign simple and specific boundary conditions for both the auxiliary field and the gauge field, thus eliminating the issue of pressure boundary condition in the usual primitive variable formulation. The resulting dynamic and kinematic equations can then be solved by standard methods for heat and Poisson equations. A normal mode analysis suggests that in contrast to the classical projection method, the gauge method does not suffer from the problem of numerical boundary layers. Thus the subtleties on the spatial discretization for the projection methods are removed. Consequently, the projection step in the gauge method can be accomplished by standard Poissin solves. We demonstrate the efficiency and accuracy of the gauge method by several numerical examples, including the flow past cylinder.
From molecular dynamics simulations on immiscible flows, we find the relative slipping between the fluids and the solid wall everywhere to follow the generalized Navier boundary condition, in which the amounnt of slipping is proportional to the sum of the tangential viscous stress and the uncompensated Young stress. The latter arises fron deviation of the fluid-fluid interface from its static configuration. We give a continuum formulaiton of the immiscible flow hydrodynamics, comprising the generalized Navier boundary condition, the Navier-Strokes equation, and the Cahn-Hilliard interfacial and velocity profiles matching quantitatively with those from the molecular dynamics simulations.
In this paper, we introduce reversion conditions for stochastic models. Also we prove that if the models satisfy reversion conditions and the market prices of risks are bounded, then the final-value problem of general two-factor financial derivative equations on rectangular domains has a unique solution. For such problems we can obtain their numerical solutions without using any artificial conditions. Examples show that if the singularity-seperating method and extrapolation techniques are used, then very good solutions can be obtained even on very coarse meshes.
In this paper, a spectral method is formulated as a numerical solution for the stochastic Ginzburg-Landau equation driven by space-time white noise. The rates of pathwise convergence and convergence in expectation in Sobolev spaces are given based on the convergence rates of the spectral approximation for the stochastic convolution. The analysis can be generalized to other spectral methods for stochastic PDEs driven by additive noises, provided the regularity condition for the noises.
The string method is an efficent numerical method for finding transition paths and transition rates in metastable systems. The dynamics of the string are governed by a Hamilton-Jacobi type of equation. We construct a stable and high order numerical scheme to estimate the first order spatial derivatives, or the tangent vectors in the equation. The construction is based on the idea of the upwind scheme and the essentially nonoscillatory scheme (ENO). Numerical examples demonstrate the improvement of the accuracy by the new scheme.
Numerical schemes are presented for dynamical systems with multiple time-scales. Two classes of mehtods are discussed, depending on the time interval which the evolution of the slow variables in the system is sought. On rather short time intervals, the slow variables satisfy ordinary differential equations. On longer time intervals, however, fluctuations become important, and stochastic differential equations are obtained. In both cases, the numerical methods compute the evolution of the slow variables without having to derive explicitly the effective equations beforehand; rather, the coefficients entering these equations are obtained on the fly using simulations of appropriate auxiliary systems.