surmounting fluctuating barriers a simple model in discrete time the escape rate is calculated for a piecewise linear map in the presence of two additive weak noises gaussian white noise thermal noise and ornstein uhlenbeck noise barrier fluctuations a transition state theory yields the exact escape rate in the weak noise limit by including the dominant finite noise corrections we find very good agreement with numerical simulations of particular interest is the behavior of the escape rate k ensuremath tau as a function of the correlation time ensuremath tau of the ornstein uhlenbeck noise we show that the qualitative behavior of k ensuremath tau at any fixed thermal noise intensity is the same as in the absence of thermal noise whether k ensuremath tau exhibits a local maximum resonant activation is determined by the detailed ensuremath tau dependence of the colored noise intensity e g resonant activation is found at a correlation time of order 1 if the integral autocorrelation of the colored noise is kept ensuremath tau independent but not if the variance is kept ensuremath tau independent in the continuous time limit neither of these cases shows resonant activation

polynomially bounded error estimates for trapezoidal rule convolution quadrature abstract we clarify the dependence with respect to the time variable of some estimates about the convergence of the trapezoidal rule based convolution quadrature method applied to hyperbolic problems this requires a careful investigation of the article of lehel banjai where the first convergence estimates were introduced and of some technical results from a classical paper of christian lubich

theoretical aspects of the application of convolution quadrature to scattering of acoustic waves in this paper we address several theoretical questions related to the numerical approximation of the scattering of acoustic waves in two or three dimensions by penetrable non homogeneous obstacles using convolution quadrature cq techniques for the time variable and coupled boundary element method finite element method for the space variable the applicability of cq to waves requires polynomial type bounds for operators related to the operator s 2 in the right half complex plane we propose a new systematic way of dealing with this problem both at the continuous and semidiscrete in space cases we apply the technique to three different situations scattering by a group of sound soft and hard obstacles by homogeneous and non homogeneous obstacles

effect of asymmetry on stochastic resonance and stochastic resonance induced by multiplicative noise and by mean field coupling in the paper we investigate the effect of asymmetry of the potential on stochastic resonance sr for a model with an asymmetric bistable potential and driven by additive noise the signal to noise ratio snr for a model with a monostable potential and driven by additive and multiplicative noises and the snr for a mean field coupled model with infinite globally coupling oscillators driven by additive noises it is shown that for the first model the asymmetry of the potential can weaken the phenomenon of sr for the second and third models a sr induced by multiplicative noise and a different one caused by mean field coupling are found

rapid solution of the wave equation in unbounded domains in this paper we propose and analyze a new fast method for the numerical solution of time domain boundary integral formulations of the wave equation we employ lubich s convolution quadrature method for the time discretization and a galerkin boundary element method for the spatial discretization the coefficient matrix of the arising system of linear equations is a triangular block toeplitz matrix possible choices for solving the linear system arising from the above discretization include the use of fast fourier transform fft techniques and the use of data sparse approximations by using fft techniques the computational complexity can be reduced substantially while the storage cost remains unchanged and is typically high using data sparse approximations the gain is reversed i e the computational cost is approximately unchanged while the storage cost is substantially reduced the method proposed in this paper combines the advantages of these two approaches first the discrete convolution related to the block toeplitz system is transformed into the discrete fourier image thereby arriving at a decoupled system of discretized helmholtz equations with complex wave numbers a fast data sparse e g fast multipole or panel clustering method can then be applied to the transformed system additionally significant savings can be achieved if the boundary data are smooth and time limited in this case the right hand sides of many of the helmholtz problems are almost zero and hence can be disregarded finally the proposed method is inherently parallel we analyze the stability and convergence of these methods thereby deriving the choice of parameters that preserves the convergence rates of the unperturbed convolution quadrature we also present numerical results which illustrate the predicted convergence behavior

mean curvature flow on ricci solitons we study monotonic quantities in the context of combined geometric flows in particular focusing on ricci solitons as the ambient space we consider solutions of the heat type equation integrated over embedded submanifolds evolving by the mean curvature flow and we study their monotonicity properties this is part of an ongoing project with magni and mantegazza which will treat the case of non solitonic backgrounds arxiv 0911 5130v1

fluctuational transitions across different kinds of fractal basin boundaries we study fluctuational transitions in discrete and continuous dynamical systems that have two coexisting attractors in phase space separated by a fractal basin boundary which may be either locally disconnected or locally connected theoretical and numerical evidence is given to show that in each case the transition occurs via a unique accessible point on the boundary both in discrete systems and in flows the complicated structure of the escape paths inside the locally disconnected fractal basin boundary is determined by a hierarchy of homoclinic points the interrelation between the mechanism of transitions and the hierarchy is illustrated by consideration of fluctuational transitions in dynamical systems demonstrating fractal fractal basin boundary metamorphosis at some value of a control parameter the most probable escape path from an attractor which can be either regular or chaotic is found for each type of boundary using both statistical analysis of fluctuational trajectories and the hamiltonian theory of fluctuations

an eigenvalue problem related to the non linear σ model analytical and numerical results an eigenvalue problem relevant for the non linear sigma model with singular metric is considered we prove the existence of a non degenerate pure point spectrum for all finite values of the size r of the system in the infrared ir regime large r the eigenvalues admit a power series expansion around the ir critical point r we compute high order coefficients and prove that the series converges for all finite values of r in the ultraviolet uv limit the spectrum condenses into a continuum spectrum with a set of residual bound states the spectrum agrees nicely with the central charge computed by the thermodynamic bethe ansatz method

nonequilibrium phase transitions induced by multiplicative noise we review a mean field analysis and give the details of a correlation function approach for spatially distributed systems subject to multiplicative noise white in space and time we confirm the existence of a pure noise induced reentrant nonequilibrium phase transition in the model introduced in c van den broeck et al phys rev lett 73 3395 1994 give an intuitive explanation of its origin and present extensive simulations in dimension d52 the observed critical properties are compatible with those of the ising universality class s1063 651x 97 08704 7

stochastic resonance in linear systems subject to multiplicative and additive noise exact expressions have been found for the first two moments and the correlation function for an overdamped linear system subject to an external periodic field as well as to multiplicative and additive noise stochastic resonance is absent for gaussian white noise however when the multiplicative noise has the form of an asymmetric dichotomous noise the signal to noise ratio snr becomes a nonmonotonic function of the correlation time and the asymmetry of noise moreover the snr turns out to be a nonmonotonic function of the frequency of the external field as well as strongly depending on the strength of the cross correlation between multiplicative and additive noise

on the multistep time discretization of linear initial boundary value problems and their boundary integral equations convergence estimates in terms of the data are shown for multistep methods applied to non homogeneous linear initial boundary value problems similar error bounds are derived for a new class of time discrete and fully discrete approximation schemes for boundary integral equations of such problems e g for the single layer potential equation of the wave equation in both cases the results are obtained from convergence and stability estimates for operational quadrature approximations of convolutions these estimates which are also proved here depend on bounds of the laplace transform of the distributional convolution kernel outside the stability region scaled by the time stepsize and on the smoothness of the data

properties of resonant activation phenomena the phenomenon of resonant activation of a brownian particle over a fluctuating barrier is revisited we discuss the important distinctions between barriers that can fluctuate among up and down configurations and barriers that are always up but that can fluctuate among different heights a resonance as a function of the barrier fluctuation rate is found in both cases but the nature and physical description of these resonances is quite distinct the nature of the resonances the physical basis for the resonant behavior and the importance of boundary conditions are discussed in some detail we obtain analytic expressions for the escape time over the barrier that explicitly capture the minima as a function of the barrier fluctuation rate and show that our analytic results are in excellent agreement with numerical results s1063 651x 98 11304 1

runge kutta convolution quadrature for operators arising in wave propagation an error analysis of runge kutta convolution quadrature is presented for a class of non sectorial operators whose laplace transform satisfies besides the standard assumptions of analyticity in a half plane re s 0 and a polynomial bound operatorname o s mu 1 there the stronger polynomial bound operatorname o s mu 2 in convex sectors of the form operatorname arg s leq pi 2 theta for 0 the order of convergence of the runge kutta convolution quadrature is determined by 2 and the underlying runge kutta method but is independent of 1 time domain boundary integral operators for wave propagation problems have laplace transforms that satisfy bounds of the above type numerical examples from acoustic scattering show that the theory describes accurately the convergence behaviour of runge kutta convolution quadrature for this class of applications our results show in particular that the full classical order of the runge kutta method is attained away from the scattering boundary

stochastic resonance in a bistable system subject to multiplicative and additive noise the stochastic resonance sr phenomenon in a bistable system under the simultaneous action of multiplicative and additive noise and periodic signal is studied by using the theory of signal to noise ratio snr in the adiabatic limit two cases have been considered the case of no correlations between multiplicative and additive noise and the case of correlations between two noises the expressions of the snr for both cases are obtained the effects of intensity of multiplicative and additive noise and the intensity of the correlations between noises on the snr are discussed for both cases respectively it is found that the existence of a maximum in the snr is the identifying characteristic of the sr phenomenon in the case of no correlations between multiplicative and additive noise the snr is independent of the initial condition of the system however the snr is not only dependent on the intensity of correlations between noises but also on the initial condition of the system in the presence of correlations between two noises

a matrix li yau hamilton estimate for kahler ricci flow in this paper we prove a new matrix li yau hamilton lyh estimate for kahler ricci flow on manifolds with nonnegative bisectional curvature the form of this new lyh estimate is obtained by the interpolation consideration originated in ch by chow this new inequality is shown to be connected with perelman s entropy formula through a family of differential equalities in the rest of the paper we show several applications of this new estimate and its corresponding estimate for linear heat equation these include a sharp heat kernel comparison theorem generalizing the earlier result of li and tian lt a manifold version of stoll s theorem st on the characterization of algebraic divisors and a localized monotonicity formula for analytic subvarieties which sharpens the bishop volume comparison theorem motivated by the connection between the heat kernel estimate and the reduced volume monotonicity of perelman p we prove a sharp lower bound heat kernel estimate for the time dependent heat equation which is in a certain sense dual to perelman s monotonicity of the reduced volume as an application of this new monotonicity formula we show that the blow down limit of a certain type long time solution is a gradient expanding soliton we also illustrate the connection between the new lyh estimate and the hessian comparison theorem of fin on the forward reduced distance local monotonicity formulae on entropy and forward reduced volume are also derived

noisy one dimensional maps near a crisis i weak gaussian white and colored noise we study one dimensional single humped maps near the boundary crisis at fully developed chaos in the presence of additive weak gaussian white noise by means of a new perturbation like method the quasi invariant density is calculated from the invariant density at the crisis in the absence of noise in the precritical regime where the deterministic map may show periodic windows a necessary and sufficient condition for the validity of this method is derived from the quasi invariant density we determine the escape rate which has the form of a scaling law and compares excellently with results from numerical simulations we find that deterministic transient chaos is stabilized by weak noise whenever the maximum of the map is of orderz 1 finally we extend our method to more general maps near a boundary crisis and to multiplicative as well as colored weak gaussian noise within this extended class of noises and for single humped maps with any fixed orderz 0 of the maximum in the scaling law for the escape rate both the critical exponents and the scaling function are universal

rates and mean first passage times the relation between mean first passage times t and transition rates f in noisy dynamical systems with metastahle states is investigated it is shown that the inverse mean first passage time to the separatrix of the noiseless system may deviate from twice the rate not only because in general the deterministic separatrix is not the locus in the state space from which a noisy trajectory goes to either side with equal probability a further cause of a deviation from the often assumed relation ft 1 2 between rates and mean first passage times is given if the noisy dynamics is discontinuous i e shows jumps with finite probability then the value of the splitting probability at the separatrix does not fix the value of ft since the system need not visit the separatrix during a transition from one to the other side most important for discontinuous processes the deviation from the ft rule survives even in the weak noise limit a mathematical relation for the product of the rate and the mean first passage time is proposed for markovian processes and numerically confirmed for a particular one dimensional noisy map

runge kutta methods for parabolic equations and convolution quadrature we study the approximation properties of runge kutta time discretizations of linear and semilinear parabolic equations including incompressible navier stokes equations we derive asymptotically sharp error bounds and relate the temporal order of convergence which is generally noninteger to spatial regularity and the type of boundary conditions the analysis relies on an interpretation of runge kutta methods as convolution quadratures in a different context these can be used as efficient computational methods for the approximation of convolution integrals and integral equations they use the laplace transform of the confolution kernal via a discrete operational calculus 23 refs 2 tabs

convolution quadrature and discretized operational calculus ii operational quadrature rules are applied to problems in numerical integration and the numerical solution of integral equations singular integrals power and logarithmic singularities finite part integrals multiple timescale convolution volterra integral equations wiener hopf integral equations frequency domain conditions which determine the stability of such equations can be carried over to the discretization

transparent boundary conditions for wave propagation in fractal trees convolution quadrature approach in this work we propose high order transparent boundary conditions for the weighted wave equation on a fractal tree with an application to the modeling of sound propagation in a human lung this article follows the recent work 29 dedicated to the mathematical analysis of the corresponding problem and the construction of low order absorbing boundary conditions the method proposed in this article consists in constructing the exact trans parent boundary conditions for the semi discretized problem in the spirit of the convolution quadrature method developed by ch lubich we analyze the stability and convergence of the method and propose an efficient algorithm for its implementation the exposition is concluded with numerical experiments

multistep and multistage convolution quadrature for the wave equation algorithms and experiments we describe how a time discretized wave equation in a homogeneous medium can be solved by boundary integral methods the time discretization can be a multistep runge kutta or a more general multistep multistage method the resulting convolutional system of boundary integral equations belongs to the family of convolution quadratures of lubich the aim of this work is twofold it describes an efficient robust and easily parallelizable method for solving the semidiscretized system the resulting algorithm has the main advantages of time stepping methods and of fourier synthesis at each time step a system of linear equations with the same system matrix needs to be solved yet computations can easily be done in parallel the computational cost is almost linear in the number of time steps and only the laplace transform of the time domain fundamental solution is needed the new aspect of the algorithm is that all this is possible without ever explicitly constructing the weights of the convolution quadrature this approach also readily allows the use of modern data sparse techniques to efficiently perform computation in space we investigate theoretically and numerically to which extent hierarchical matrix mathcal h matrix techniques can be used to speed up the space computation the second aim of this article is to perform a series of large scale 3d experiments with a range of multistep and multistage time discretization methods the backward difference formula of order 2 bdf2 the trapezoid rule and the 3 stage radau iia methods are investigated in detail one of the conclusions of the experiments is that the radau iia method often performs overwhelmingly better than the linear multistep methods especially for problems with many reflections yet in connection with hyperbolic problems bdfs have so far been predominant in the literature on convolution quadrature

transitions and transport for a spatially periodic stochastic system with locally coupled oscillators in this paper with a special model we investigate the spatially periodic stochastic system with locally coupled oscillators subject to a constant force f a nonequilibrium second order phase transition is found when f 0 this phase transition is reentrant when the additive noise is weak with varying the constant force f a continuous or discontinuous transition between the states with positive and negative mean fields m 0 and m 0 is observed which is not a phase transition the mean field or current sometimes exhibits hysteresis as a function of f with the variation of the force f when hysteresis of the mean field or current versus f appears a nonzero probability current with definite direction will occur at the point f 0 the correlation between the additive and multiplicative noises has an effect on the transitions and the transport

a fully discrete bem fem scheme for transient acoustic waves abstract we study a symmetric bem fem coupling scheme for the scattering of transient acoustic waves by bounded inhomogeneous anisotropic obstacles in a homogeneous field an incident wave in free space interacts with the obstacles and produces a combination of transmission and scattering the transmitted part of the wave is discretized in space by finite elements while the scattered wave is reduced to two fields defined on the boundary of the obstacles and is discretized in space with boundary elements we choose a coupling formulation that leads to a symmetric system of integro differential equations the retarded boundary integral equations are discretized in time by convolution quadrature and the interior field is discretized in time with the trapezoidal rule we show that the scattering problem generates a c 0 group of isometries in a hilbert space and use associated estimates to derive stability and convergence results we provide numerical experiments and simulations to validate our results and demonstrate the flexibility of the method

escape through an unstable limit cycle resonant activation we consider a brownian particle acted on by a linear conservative force a nonlinear frictional force and multiplicative colored and additive white noises the frictional force can be negative when the external energy supply is large enough we numerically calculate the mean first passage time mfpt for the particle to escape from an unstable limit cycle and find resonant activation i e the mfpt first decreases followed by a rise after passing through a minimum with increasing noise correlation time tau for a fixed noise variance for fixed noise strength of the multiplicative noise the mfpt increases linearly with tau this is in sharp contrast to the case of fluctuations of nonlinear potentials in which the mfpt first increases nonlinearly before reaching a limiting value our model could be useful for understanding some biological processes