large deviation principle for benedicks carleson quadratic maps since the pioneering works of jakobson and benedicks carleson and others it has been known that a positive measure set of quadratic maps admit invariant probability measures absolutely continuous with respect to lebesgue these measures allow one to statistically predict the asymptotic fate of lebesgue almost every initial condition estimating fluctuations of empirical distributions before they settle to equilibrium requires a fairly good control over large parts of the phase space we use the sub exponential slow recurrence condition of benedicks carleson to build induced markov maps of arbitrarily small scale and associated towers to which the absolutely continuous measures can be lifted these various lifts together enable us to obtain a control of recurrence that is sufficient to establish a level 2 large deviation principle for the absolutely continuous measures this result encompasses dynamics far from equilibrium and thus significantly extends presently known local large deviations results for quadratic maps

pointwise estimates and stability for dispersive diffusive shock waves we study the stability and pointwise behavior of perturbed viscous shock waves for a general scalar conservation law with constant diffusion and dispersion along with the usual lax shocks such equations are known to admit undercompressive shocks we unify the treatment of these two cases by introducing a new wave tracking method based on instantaneous projection giving improved estimates even in the lax case another important feature connected with the introduction of dispersion is the treatment of a non sectorial operator an immediate consequence of our pointwise estimates is a simple spectral criterion for stability in all l p norms p 1 for the lax case and p 1 for the undercompressive case our approach extends immediately to the case of certain scalar equations of higher order and would also appear suitable for extension to systems

effective properties of composite materials with periodic microstructure a computational approach abstract this study reviews several problems which are specific of composites with periodic microstructure composed of linear or nonlinear constituents the theoretical background of the method is recalled first two different families of numerical methods are considered to solve the problem the first is based on the finite element method the concept of macroscopic degrees of freedom is presented the implementation of periodicity conditions is discussed a general framework permitting either a strain or stress control is proposed the second numerical method is based on fast fourier transforms it considers first the problem of a homogeneous linear reference material undergoing a nonhomogeneous periodic eigenstrain the solution of this problem is based on the explicit form of the periodic green s function of the reference medium the relative merits of the two methods are compared and several examples are discussed both methods give very comparable results on test examples and their domains of applications appear to be complementary

the gap lemma and geometric criteria for instability of viscous shock profiles an obstacle in the use of evans function theory for stability analysis of traveling waves occurs when the spectrum of the linearized operator about the wave accumulates at the imaginary axis since the evans function has in general been constructed only away from the essential spectrum a notable case in which this difficulty occurs is in the stability analysis of viscous shock profiles here we prove a general theorem the gap lemma concerning the analytic continuation of the evans function associated with the point spectrum of a traveling wave into the essential spectrum of the wave this allows geometric stability theory to be applied in many cases where it could not be applied previously we demonstrate the power of this method by analyzing the stability of certain undercompressive viscous shock waves a necessary geometric condition for stability is determined in terms of the sign of a certain melnikov integral of the associated viscous profile this sign can easily be evaluated numerically we also compute it analytically for solutions of several important classes of systems in particular we show for a wide class of systems that homoclinic solitary waves are linearly unstable confirming these as the first known examples of unstable viscous shock waves we also show that strong heteroclinic undercompressive waves are sometimes unstable similar stability conditions are also derived for lax and overcompressive shocks and for nn

asymptotic behavior near planar transition fronts for the cahn hilliard equation abstract we consider the asymptotic behavior of perturbations of planar wave solutions arising in the cahn hilliard equation in space dimensions d 2 such equations are well known to arise in the study of spinodal decomposition a phenomenon in which rapid cooling of a homogeneously mixed binary alloy causes separation to occur resolving the mixture into regions in which one component or the other is dominant with these regions separated by steep transition layers a critical feature of the cahn hilliard equation in one space dimension is that the linear operator that arises upon linearization of the equation about a standing wave solution has essential spectrum extending onto the imaginary axis a feature that is known to complicate the step from spectral to nonlinear stability the analysis of planar waves in multiple space dimensions is further complicated by the fact that the leading eigenvalue for this linearized operator leading in the case of stability moves into the negative real half plane with cubic scaling 3 where r d 1 denotes a fourier variable associated with spatial components transverse to the planar wave under the assumption of spectral stability described in terms of an appropriate evans function we develop detailed asymptotics for perturbations from planar wave solutions establishing asymptotic stability for initial perturbations decaying at an appropriate algebraic rate in an l 1 norm of the transverse variables

transition to longitudinal instability of detonation waves is generically associated with hopf bifurcation to time periodic galloping solutions we show that transition to longitudinal instability of strong detonation solutions of reactive compressible navier stokes equations is generically associated with hopf bifurcation to nearby time periodic galloping or pulsating solutions in agreement with physical and numerical observation in the process we determine readily numerically verifiable stability and bifurcation conditions in terms of an associated evans function and obtain the first complete nonlinear stability result for strong detonations of the reacting navier stokes equations in the limit as amplitude hence also heat release goes to zero the analysis is by pointwise semigroup techniques introduced by the authors and collaborators in previous works

random homogenization analysis in linear elasticity based on analytical bounds and estimates in this work random homogenization analysis of heterogeneous materials is addressed in the context of elasticity where the randomness and correlation of components properties are fully considered and random effective properties together with their correlation for the two phase heterogeneous material are then sought based on the analytical results of homogenization in linear elasticity when the randomness of bulk and shear moduli the volume fraction of each constituent material and correlation among random variables are considered simultaneously formulas of random mean values and mean square deviations of analytical bounds and estimates are derived from random factor method results from the random factor method and the monte carlo method are compared with each other through numerical examples and impacts of randomness and correlation of random variables on the random homogenization results are inspected by two methods moreover the correlation coefficients of random effective properties are obtained by the monte carlo method the random factor method is found to deliver rapid results with comparable accuracy to the monte carlo approach

ion acoustic shock in a collisional plasma abstract the paper is concerned with the propagation of ion acoustic shock waves in a collision dominated plasma whose equations of motion are described by the one dimensional isothermal navier stokes poisson system for ions with the electron density determined by the boltzmann relation the main results include three parts a we establish the existence and uniqueness of a small amplitude smooth traveling wave by solving a 3 d ode in terms of the center manifold theorem b we study the shock structure in a specific asymptotic regime where the viscosity coefficient and the shock strength are proportional to e and the debye length is proportional to e 1 2 with two parameters e and and show that in the limit e 0 shock profiles obtained in a can be approximated by the profiles of kdv burgers uniformly for 0 0 with some 0 0 the proof is based on the suitable construction of the kdv burgers shock profiles together with the delicate analysis of a linearized variable coefficient system in exponentially weighted sobolev spaces involving parameters e and c we also prove the large time asymptotic stability of traveling waves under suitably small smooth zero mass perturbations note that the ions temperature is allowed to be zero in parts a and b but necessarily required to be strictly positive in the proof of part c

asymptotic behavior near transition fronts for equations of generalized cahn hilliard form we consider the asymptotic behavior of perturbations of standing wave solutions arising in evolutionary pde of generalized cahn hilliard form in one space dimension such equations are well known to arise in the study of spinodal decomposition a phenomenon in which the rapid cooling of a homogeneously mixed binary alloy causes separation to occur resolving the mixture into its two components with their concentrations separated by sharp transition layers motivated by work of bricmont kupiainen and taskinen 5 we regard the study of standing waves as an interesting step toward understanding the dynamics of these transitions a critical feature of the cahn hilliard equation is that the linear operator that arises upon linearization of the equation about a standing wave solution has essential spectrum extending onto the imaginary axis a feature that is known to complicate the step from spectral to nonlinear stability under the assumption of spectral stability described in terms of an appropriate evans function we develop detailed asymptotics for perturbations from standing wave solutions establishing phase asymptotic orbital stability for initial perturbations decaying with appropriate algebraic rate

nonlinear stability of an undercompressive shock for complex burgers equation though there is strong numerical evidence for the stability of undercompressive shocks their stability has not been verified analytically in particular the energy methods used to analyze stability of standard shocks do not apply here we present the first proof of stability for a particular undercompresive shock the real burgers shock considered as a solution of the complex burgers equation our analysis is by direct calculation of the green s function for the linearized equations combined with pointwise estimates of nonlinear effects a benefit of this method is to obtain fairly detailed information about the solution includingl1 behavior and rates of decay in different regions of space

vertex method for computing functions of fuzzy variables abstract this paper is to introduce a new approach the vertex method for computing functions of fuzzy variables the method is based on the cut concept and interval analysis the vertex method can avoid abnormality due to the discretization technique on the variables domain and the widening of the resulting function value set due to multi occurrence of variables in the functional expression by conventional interval analysis methods the algorithm is very easy to implement and can be applied to many practical problems some examples are given to illustrate the applications

l1 asymptotic behavior of compressible isentropic viscous 1 d flow we study the large time behavior in l1 of the compressible isentropic viscous 1 d flow under the assumption that the initial data are smooth and small we show that the solutions are approximated by the solutions of a parabolic system and in turn by diffusion waves which are solutions of burgers equations decay rates in l1 are obtained our method is based on the study of pointwise properties in the physical space of the fundamental solution to the linearized system 1994 john wiley sons inc

nonlocalized modulation of periodic reaction diffusion waves the whitham equation in a companion paper we established nonlinear stability with detailed diffusive rates of decay of spectrally stable periodic traveling wave solutions of reaction diffusion systems under small perturbations consisting of a nonlocalized modulation plus a localized l1 perturbation here we determine time asymptotic behavior under such perturbations showing that solutions consist of a leading order of a modulation whose parameter evolution is governed by an associated whitham averaged equation

stability of planar shock fronts for multidimensional systems of relaxation equations abstract we investigate stability of multidimensional planar shock profiles of a general hyperbolic relaxation system whose equilibrium model is a system under the necessary assumption of spectral stability and a standard set of structural conditions that are known to hold for many physical systems our main result generalizing the work of kwon and zumbrun in the scalar relaxation case is to establish the bounds on the green s function for the linearized equation and obtain nonlinear l 2 asymptotic behavior sharp decay rate of perturbed weak shock profiles to establish green s function bounds we use the semigroup approach in the low frequency regime and use the energy method for the high frequency bounds separately for the system equilibrium case the analysis of the linearized equation is complicated due to glancing phenomena we treat this difficulty similarly as in the inviscid and viscous systems under the constant multiplicity condition

large deviations for nonuniformly hyperbolic systems we obtain large deviation estimates for a large class of nonuniformly hyperbolic systems namely those modelled by young towers with summable decay of correlations in the case of exponential decay of correlations we obtain exponential large deviation estimates given by a rate function in the case of polynomial decay of correlations we obtain polynomial large deviation estimates and exhibit examples where these estimates are essentially optimal in contrast with many treatments of large deviations our methods do not rely on thermodynamic formalism hence for holder observables we are able to obtain exponential estimates in situations where the space of equilibrium measures is not known to be a singleton as well as polynomial estimates in situations where there is not a unique equilibrium measure

spectral stability of periodic wave trains of the korteweg de vries kuramoto sivashinsky equation in the korteweg de vries limit we study the spectral stability of a family of periodic wave trains of the korteweg de vries kuramoto sivashinsky equation tv v xv 3v 2 xv 4 xv 0 0 in the korteweg de vries limit 0 a canonical limit describing small amplitude weakly unstable thin lm ow more precisely we carry out a rigorous singular perturbation analysis reducing the problem of spectral stability in this limit to the validation of a set of three conditions each of which have been numerically analyzed in previous studies and shown to hold simultaneously on a non empty set of parameter space the main technical diculty

on nonlinear stability of general undercompressive viscous shock waves we study the nonlinear stability of general undercompressive viscous shock waves previously the authors showed stability in a special case when the shock phase shift can be determined a priori from the total mass of the perturbation using new pointwise methods by examining time invariants associated with the linearized equations we can now overcome a new difficulty in the general case namely nonlinear movement of the shock we introduce a coordinate transformation suitable to treat this new aspect and demonstrate our method by analyzing a model system of generic type we obtain sharp pointwise bounds andlp behavior of the solution for allp 1 p

nonlinear stability of time periodic viscous shocks in order to understand the nonlinear stability of many types of time periodic travelling waves on unbounded domains one must overcome two main difficulties the presence of embedded neutral eigenvalues and the time dependence of the associated linear operator this problem is studied in the context of time periodic lax shocks in systems of viscous conservation laws using spatial dynamics and a decomposition into separate floquet eigenmodes it is shown that the linear evolution for the time dependent operator can be represented using a contour integral similar to that of the standard time independent case by decomposing the resulting green s distribution the leading order behavior associated with the embedded eigenvalues is extracted sharp pointwise bounds are then obtained which are used to prove that time periodic lax shocks are linearly and nonlinearly stable under the necessary conditions of spectral stability and minimal multiplicity of the translational eigenvalues the latter conditions hold for example for small oscillation time periodic waves that emerge through a supercritical hopf bifurcation from a family of time independent lax shocks of possibly large amplitude

analysis of uncertain structural systems using interval analysis the imprecision or uncertainty present in many engineering analysis design problems can be modeled using probabilistic fuzzy or interval methods this work considers the modeling of uncertain structural systems using interval analysis by representing each uncertain input parameter as an interval number a static structural analysis problem can be expressed in the form of a system of linear interval equations in addition to the direct and gaussian elimination based solution approaches a combinatorial approach based on an exhaustive combination of the extreme values of the interval numbers and an inequality hased method are presented for finding the solution of interval equations the range or interval of the solution vector response parameters is found to increase with increasing size of the problem in all of the methods an interval truncation approach is proposed to limit the growth of intervals of response parameters so that realistic and accurate solutions can be obtained in the presence of large amounts of uncertainty numerical examples are presented to illustrate the computational aspects of the methods and also to indicate the importance of the truncation approach in practical problems the utility of interval methods in predicting the extreme values of the response parameters of structures is discussed

on the stability of traveling waves in weighted l spaces abstract i present new techniques and results concerning the stability of travelling waves to semilinear parabolic equations i am concerned with those problems in which the essential spectrum of the linearization about the wave contains 0 in these circumstances the stability results of d henry in lecture notes in mathematics vol 840 springer verlag new york 1981 and p bates and c jones dynamics rep 2 1989 1 38 are not applicable as the spectral information does not yield an exponential time decay estimate on the semigroup i show that under certain relatively easily computable conditions the wave is stable in polynomially weighted l spaces furthermore the rate at which the perturbation decays to the wave as t depends on the growth rate of the polynomial near infinity the method of proof uses the semigroup of the linearized equation with solutions of the full problem expressed by the variation of constants formula estimates for the semigroup are derived through a new technique for estimating the resolvent

conditional stability of unstable viscous shock waves in compressible gas dynamics and mhd extending our previous work in the strictly parabolic case we show that a linearly unstable lax type viscous shock solution of a general quasilinear hyperbolic parabolic system of conservation laws possesses a translation invariant center stable manifold within which it is nonlinearly orbitally stable with respect to small l 1 h 3 perturbations converging time asymptotically to a translate of the unperturbed wave that is for a shock with p unstable eigenvalues we establish conditional stability on a codimension p manifold of initial data with sharp rates of decay in all l p for p 0 we recover the result of unconditional stability obtained by mascia and zumbrun the main new difficulty in the hyperbolic parabolic case is to construct an invariant manifold in the absence of parabolic smoothing

long time stability of multi dimensional noncharacteristic viscous boundary layers we establish long time stability of multi dimensional noncharacteristic boundary layers of a class of hyperbolic parabolic systems including the compressible navier stokes equations with inflow outflow boundary conditions under the assumption of strong spectral or uniform evans stability evans stability has been verified for small amplitude layers by gues metivier williams and zumbrun for large amplitude layers it may be efficiently checked numerically as done in the one dimensional case by costanzino humpherys nguyen and zumbrun

conditional stability of unstable viscous shocks abstract continuing a line of investigation initiated by texier and zumbrun on dynamics of viscous shock and detonation waves we show that a linearly unstable lax type viscous shock solution of a semilinear strictly parabolic system of conservation laws possesses a translation invariant center stable manifold within which it is nonlinearly orbitally stable with respect to small l 1 h 2 perturbations converging time asymptotically to a translate of the unperturbed wave that is for a shock with p unstable eigenvalues we establish conditional stability on a codimension p manifold of initial data with sharp rates of decay in all l p for p 0 we recover the result of unconditional stability obtained by howard mascia and zumbrun

a copula based perturbation stochastic method for fiber reinforced composite structures with correlations abstract under most circumstances there are correlations among random variables for composite structures and these correlations can have a considerable effect on the reliability analysis result consequently a copula based perturbation stochastic method is proposed to amend the disability of traditional perturbation stochastic method on correlation problems as a statistical tool copula function meets the intrinsic demand of fiber reinforced composite structures on variable s correlation and allows the choice of joint cumulative distribution functions cdfs to be separate from the marginal cdfs with the aid of change of variable technique copula is introduced straightly into the framework to describe the relevance without any special transformation besides the probability curves and reliability index of multi variables of gaussian and non gaussian can be achieved easily furthermore in order to significantly improve the efficiency and capability to resist the mesh distortion compared with the traditional stochastic fem strain matrices are reconstructed by the edge based smoothing technique the overall uncertain condition coupled by stochastic fiber orientations material parameters and external loads is analyzed several examples are provided to verify the validity of the method

pointwise green function bounds for shock profiles of systems with real viscosity following the pointwise semigroup approach of zh mz 1 we establish sharp pointwise green function bounds and consequent linearized stability for viscous shock profiles of general hyperbolic parabolic systems of conservation laws of dissipative type under the necessary assumptions z 1 z 3 z 4 of spectral stability i e stable point spectrum of the linearized operator about the wave transversality of the connecting profile and hyperbolic stability of the corresponding ideal shock of the associated inviscid system with no additional assumptions on the structure or strength of the shock these bounds are used in a companion paper mz 2 to establish nonlinear stability of small amplitude lax shocks of symmetrizable hyperbolic parabolic systems

nonlinear stability of periodic traveling wave solutions of systems of viscous conservation laws in the generic case abstract extending previous results of oh zumbrun and johnson zumbrun we show that spectral stability implies linearized and nonlinear stability of spatially periodic traveling wave solutions of viscous systems of conservation laws for systems of generic type removing a restrictive assumption that wave speed be constant to first order along the manifold of nearby periodic solutions key to our analysis is a nonlinear cancellation estimate observed by johnson and zumbrun along with a detailed understanding of the whitham averaged system the latter motivates a careful analysis of the bloch perturbation expansion near zero frequency and suggests factoring out an appropriate translational modulation of the underlying wave allowing us to derive the sharpened low frequency estimates needed to close the nonlinear iteration arguments

pointwise green s function bounds for multidimensional scalar viscous shock fronts we determine detailed pointwise bounds for the green s function of the linearized operator about a multidimensional scalar viscous shock front these extend the pointwise semigroup methods introduced by howard and zumbrun in the one dimensional case to multidimensions sharpening lp estimates obtained by goodman and miller using a weighted norm approach moreover our results apply to shocks of arbitrary strength as previous results did not as described in a companion paper the bounds we obtain are sufficient to give a straightforward treatment of the nonlinear lp asymptotic behavior of the front under small perturbation the analysis of the multidimensional case involves several new features not found in the one dimensional case concerned with the geometry of propagating signals

spectral stability of ideal gas shock layers extending recent results in the isentropic case we use a combination of asymptotic ode estimates and numerical evans function computations to examine the spectral stability of shock wave solutions of the compressible navier stokes equations with ideal gas equation of state our main results are that in appropriately rescaled coordinates the evans function associated with the linearized operator about the wave i converges in the large amplitude limit to the evans function for a limiting shock profile of the same equations for which internal energy vanishes at one end state and ii has no unstable positive real part zeros outside a uniform ball thus the rescaled eigenvalue ode for the set of all shock waves augmented with the nonphysical limiting case form a compact family of boundary value problems that can be conveniently investigated numerically an extensive numerical evans function study yields one dimensional spectral stability independent of amplitude for gas constant in 1 2 3 and ratio of heat conduction to viscosity coefficient within 0 2 5 1 4 1 47 for air other values may be treated similarly but were not considered the method of analysis extends also to the multi dimensional case a direction that we shall pursue in a future work

pointwise green s function estimates toward stability for multidimensional fourth order viscous shock fronts abstract for the case of multidimensional viscous conservation laws with fourth order smoothing only we develop detailed pointwise estimates on the green s function for the linear fourth order convection equation that arises upon linearization of the conservation law about a viscous planar wave solution as in previous analyses in the case of second order smoothing our estimates are sufficient to establish that spectral stability implies nonlinear stability though the full development of this result will be considered in a companion paper

a survey of non probabilistic uncertainty treatment in finite element analysis the objective of this paper is to critically review the emerging non probabilistic approaches for uncertainty treatment in finite element analysis the paper discusses general theoretical and practical aspects of both the interval and fuzzy finite element analysis first the applicability of the non probabilistic concepts for numerical uncertainty analysis is discussed from a theoretical viewpoint the necessary conditions for a useful application of the non probabilistic concepts are determined and are proven to be complementary rather than competitive to the classical probabilistic approach the second part of the paper focuses on numerical aspects of the interval finite element method it describes two principal strategies for the implementation i e the anti optimisation and the interval arithmetic approach and gives a state of the art of the interval finite element algorithms available from literature it is shown how the application of the interval arithmetic approach to the classical finite element procedure can result in a severe overestimation of the uncertainty on the output and the main sources of this conservatism are identified a numerical example in the final part of the paper illustrates the capabilities of the different strategies on an eigenfrequency analysis of a built up benchmark structure

a stability index for detonation waves in majda s model for reacting flow abstract using evans function techniques we develop a stability index for weak and strong detonation waves analogous to that developed for shock waves in siam j math anal 32 2001 929 commun pure appl math 51 7 1998 797 yielding useful necessary conditions for stability here we carry out the analysis in the context of the majda model a simplified model for reacting flow the method is extended to the full navier stokes equations of reacting flow in g lyng one dimensional stability of detonation waves doctoral thesis indiana university 2002 g lyng k zumbrun stability of detonation waves preprint 2003 the resulting stability condition is satisfied for all nondegenerate i e spatially exponentially decaying weak and strong detonations of the majda model in agreement with numerical experiments of siam j sci statist comput 7u 1986 1059 and analytical results of commun math phys 204 3 1999 551 commun math phys 202 3 1999 547 for a related model of majda and rosales we discuss also the role in the znd limit of degenerate subalgebraically decaying weak detonation and for a modified bump type ignition function deflagration profiles as discussed in siam j math anal 24 1993 968 siam j appl math 55 1995 175 for the full equations

interval and subinterval homogenization based method for determining the effective elastic properties of periodic microstructure with interval parameters abstract the main issue of this paper is to present an interval homogenization based method ihm for the estimation of the effective elastic tensor for microscopic material properties with interval uncertainty based on the finite element method fem for homogenization analysis the interval effective elastic tensor is expanded by the first or second order taylor series for obtaining its upper and lower bounds in order to guarantee the accuracy of ihm the subinterval technique is introduced into the ihm for estimating the effective elastic tensor numerical examples of a unit rectangular voids cell and a unit particle filled polymers cell are implemented through comparing the numerical results with the results obtained by the monte carlo simulation the effectiveness and the efficiency of the ihm for homogenization analysis with interval uncertainty are verified the influence of the microscopic interval uncertainty in material properties on a homogenized macroscopic elastic property of an inhomogeneous material has also been investigated

spectral analysis of θ periodic schrodinger operators and applications to periodic waves abstract assuming a symmetric matrix valued potential we consider the associated periodic schrodinger operators h on intervals 0 p where in our applications p denotes the period of a stationary periodic solution to a nonlinear evolutionary pde we relate the morse index of h to certain maslov indices and apply our theory to operators obtained when allen cahn equations and systems are linearized about stationary periodic solutions

the static displacement and the stress analysis of structures with bounded uncertainties using the vertex solution theorem abstract in practical engineering on account of physical imperfections model inaccuracies and system complexities almost all structures have physical and geometrical uncertainties in some degrees in this paper we focus on the static response problem and the stress distribution of structures with bounded uncertainties in terms of the vertex notation in interval analysis a new mathematical proof of the vertex solution theorem which is used to determine the supremum and the infimum of the static responses of structures with bounded uncertainties is given then we extend this theorem for calculating the interval stress and the deformation distributions which was only used to determine the supremum and the infimum of the static responses of structures with bounded uncertainties the basic idea of the vertex solution theorem is to convert the interval linear equations into systems of equivalent deterministic linear equations the resulting deterministic linear equations are then solved by using the familiar techniques like the gauss elimination method or the gauss seidel iteration method another advantage of this method is that the increase in number of the uncertain parameters in a structure element will not result in the increase in number of the uncertain elements in the stiffness matrix for practical engineering problem with bounded uncertainties the chief thing is to avoid the disadvantages of large calculation and much runtime of the vertex solution in this paper we present the parallel arithmetic of the vertex solution theorem which can be used in large scale computations in practical engineering to avoid much runtime on the other hand we can use the symmetrical characteristic of the stiffness matrix and the substructure method to reduce the computational effort in practical engineering the numerical results of the typical example presented by deif calculated by the proposed vertex solution theorem show that the width between the supremum and the infimum on the static response yielded by the vertex solution theorem is tighter than those produced by the kcn s method presented by koyluoglu cakmak and nielson the numerical examples of the two dimensional shell with an edge fixed and the two dimensional shell with surrounding fixed are used to illustrate the computational feasibility of the vertex solution theorem

stability of undercompressive shock profiles abstract using a simplified pointwise iteration scheme we establish nonlinear phase asymptotic orbital stability of large amplitude lax undercompressive overcompressive and mixed under overcompressive type shock profiles of strictly parabolic systems of conservation laws with respect to initial perturbations u 0 x e 0 1 x 3 2 in c 0 e 0 sufficiently small under the necessary conditions of spectral and hyperbolic stability together with transversality of the connecting profile this completes the program initiated by zumbrun and howard in k zumbrun p howard pointwise semigroup methods and stability of viscous shock waves indiana univ math j 47 4 1998 741 871 extending to the general undercompressive case results obtained for lax and overcompressive shock profiles in a szepessy z xin nonlinear stability of viscous shock waves arch ration mech anal 122 1993 53 103 t p liu pointwise convergence to shock waves for viscous conservation laws comm pure appl math 50 11 1997 1113 1182 k zumbrun p howard pointwise semigroup methods and stability of viscous shock waves indiana univ math j 47 4 1998 741 871 k zumbrun refined wave tracking and nonlinear stability of viscous lax shocks methods appl anal 7 2000 747 768 m r raoofi l 1 asymptotic behavior of perturbed viscous shock profiles thesis indiana univ 2004 c mascia k zumbrun pointwise green s function bounds and stability of relaxation shocks indiana univ math j 51 4 2002 773 904 c mascia k zumbrun stability of small amplitude shock profiles of symmetric hyperbolic parabolic systems comm pure appl math 57 7 2004 841 876 c mascia k zumbrun pointwise green s function bounds for shock profiles with degenerate viscosity arch ration mech anal 169 3 2003 177 263 c mascia k zumbrun stability of large amplitude shock profiles of hyperbolic parabolic systems arch ration mech anal 172 1 2004 93 131 c mascia k zumbrun stability of large amplitude shock profiles of general relaxation systems siam j math anal in press and for special weakly coupled respectively scalar diffusive dispersive undercompressive profiles in t p liu k zumbrun nonlinear stability of an undercompressive shock for complex burgers equation comm math phys 168 1 1995 163 186 t p liu k zumbrun on nonlinear stability of general undercompressive viscous shock waves comm math phys 174 2 1995 319 345 respectively p howard k zumbrun pointwise estimates for dispersive diffusive shock waves arch ration mech anal 155 2000 85 169 in particular together with spectral results of k zumbrun dynamical stability of phase transitions in the p system with viscosity capillarity siam j appl math 60 2000 1913 1924 our results yield nonlinear stability of large amplitude undercompressive phase transitional profiles near equilibrium of slemrod s model m slemrod admissibility criteria for propagating phase boundaries in a van der waals fluid arch ration mech anal 81 4 1983 301 315 for van der waal gas dynamics or elasticity with viscosity capillarity

pointwise nonlinear stability of nonlocalized modulated periodic reaction diffusion waves abstract in this paper extending previous results of 2 we obtain pointwise nonlinear stability of periodic traveling reaction diffusion waves assuming spectral linearized stability under nonlocalized perturbations more precisely we establish pointwise estimate of nonlocalized modulational perturbation under a small initial perturbation consisting of a nonlocalized modulation plus a localized perturbation decaying algebraically

pointwise stability estimates for periodic traveling wave solutions of systems of viscous conservation laws abstract in the previous paper 9 we showed time asymptotic behavior with detailed decaying rates of perturbations of periodic traveling reaction diffusion waves under small initial perturbations with a gaussian rate and an algebraic rate here we establish pointwise nonlinear stability up to an appropriate modulation of periodic traveling waves of systems of viscous conservation laws under small algebraic decaying initial data similar to the reaction diffusion equations by using bloch decomposition we start with pointwise bounds on the green function of the linearized operator about underlying solutions

stability of bright solitary wave solutions to perturbed nonlinear schrodinger equations abstract the propagation of pulses in ideal nonlinear optical fibers without loss is governed by the nonlinear schrodinger equation nls when considering realistic fibers one must examine perturbed nls equations with the particular perturbation depending on the physical situation that is being modeled a common example is the complex ginzburg landau equation cgl which is a dissipative perturbation it is known that some of the stable bright solitons of the nls survive a dissipative perturbation such as the cgl given that a wave persists it is then important to determine its stability with respect to the perturbed nls a major difficulty in analyzing the stability of solitary waves upon adding dissipative terms is that eigenvalues may bifurcate out of the essential spectrum since the essential spectrum of the nls is located on the imaginary axis such eigenvalues may lead to an unstable wave in fact eigenvalues can pop out of the essential spectrum even if the unperturbed problem has no eigenvalue embedded in the essential spectrum here we present a technique which can be used to track these bifurcating eigenvalues as a consequence we are able to locate the spectrum for bright solitary wave solutions to various perturbed nonlinear schrodinger equations and determine precise conditions on parameters for which the waves are stable in addition we show that a particular perturbation the parametrically forced nls equation supports stable multi bump solitary waves the technique for tracking eigenvalues which bifurcate from the essential spectrum is very general and should therefore be applicable to a larger class of problems than those presented here

stability of noncharacteristic boundary layers in the standing shock limit we investigate one and multi dimensional stability of noncharacteristic boundary layers in the limit approaching a standing planar shock wave u x1 x 1 0 obtaining necessary conditions of i weak stability of the limiting shock ii weak stability of the constant layer u u lim z z and iii nonnegativity of a modified lopatinski determinant similar to that of the inviscid shock case for lax 1 shocks we obtain equally simple sufficient conditions for p shocks p 1 the situation appears to be more complicated using these results we determine the stability of certain gas dynamical boundary layers generalizing earlier work of serre zumbrun and costanzino humphreys nguyen zumbrun

stability of periodic solutions of conservation laws with viscosity pointwise bounds on the green function we investigate stability of periodic traveling wave solutions of systems of conservation laws with viscosity within the abstract evans function framework established by r a gardner our main result generalizing the work of zumbrun and howard in the traveling front or pulse setting is to establish sharp pointwise bounds on the green function for the linearized evolution equations provided that an appropriate evans function condition applies to the linearized operator about the wave this condition is equivalent to a spectral stability criterion introduced by schneider in the context of periodic reaction diffusion waves an immediate consequence is that strong spectral stability in the sense of schneider implies linearized l1 l p asymptotic stability for all p 1 on the other hand we show that the strict version of schneider s condition generically fails in the conservation law setting leading to complicated new metastable behavior reminiscent of that seen for degenerate neutrally stable families in the traveling front or pulse case our results apply also to the reaction diffusion setting sharpening at the linearized level results obtained by schneider using weighted norm and bloch decomposition methods

stability of multidimensional viscous shocks for symmetric systems with variable multiplicities we establish long time stability of multi dimensional viscous shocks of a general class of symmetric hyperbolic parabolic systems with variable multiplicities notably including the equations of compressible magnetohydrodynamics mhd in dimensions d 2 this extends the existing result established by k zumbrun for systems with characteristics of constant multiplicity to the ones with variable multiplicity yielding the first such a stability result for fast mhd shocks at the same time we are able to drop a technical assumption on structure of the so called glancing set that was necessarily used in previous analyses the key idea to the improvements is to introduce a new simple argument for obtaining a l l resolvent bound in low frequency regimes by employing the recent construction of degenerate kreiss symmetrizers by o gues g metivier m williams and k zumbrun thus at the low frequency resolvent bound level our analysis gives an alternative to the earlier pointwise green s function approach of k zumbrun high frequency solution operator bounds have been previously established entirely by nonlinear energy estimates

continued fraction and decimal expansions of an irrational number abstract for an irrational number x and n 1 we denote by k n x the exact number of partial quotients in the continued fraction expansion of x given by the first n decimals of x and p n x q n x the nth convergent of x let x lim inf n log q n x n x lim sup n log q n x n we prove that lim sup n k n x n log 10 2 x lim inf n k n x n log 10 2 x this result significantly strengthens the results of g lochs and c faivre

nonlinear modulational stability of periodic traveling wave solutions of the generalized kuramoto sivashinsky equation abstract in this paper we consider the spectral and nonlinear stabilities of periodic traveling wave solutions of a generalized kuramoto sivashinsky equation in particular we resolve the long standing question of nonlinear modulational stability by demonstrating that spectrally stable waves are nonlinearly stable when subject to small localized integrable perturbations our analysis is based upon detailed estimates of the linearized solution operator which are complicated by the fact that the necessarily essential spectrum of the associated linearization intersects the imaginary axis at the origin we carry out a numerical evans function study of the spectral problem and find bands of spectrally stable periodic traveling waves in close agreement with previous numerical studies of frisch she thual bar nepomnyashchy chang demekhin kopelevich and others carried out by other techniques we also compare predictions of the associated whitham modulation equations which formally describe the dynamics of weak large scale perturbations of a periodic wave train with numerical time evolution studies demonstrating their effectiveness at a practical level for the reader s convenience we include in an appendix the corresponding treatment of the swift hohenberg equation a nonconservative counterpart of the generalized kuramoto sivashinsky equation for which the nonlinear stability analysis is considerably simpler together with numerical evans function analyses extending spectral stability analyses of mielke and schneider

antioptimization of structures with large uncertain but non random parameters via interval analysis non probabilistic interval modeling of uncertain but non random parameters for structures is developed in this paper for antioptimization analysis consisting in determining the least favorable responses the uncertain but non random parameter is considered to be a deterministic variable belonging to a set modeled as an interval the least favorable static displacement bound estimation for structures with uncertain but non random parameters is transformed into solving interval linear equations for small interval parameters the width of interval being small the uncertainties of interval parameters are treated as the perturbed quantities around the midpoint of interval parameters by means of the interval matrix central notation and the natural interval extension interval perturbation method for estimating the static displacement bound of structures with interval parameters was presented in the recent study by qiu et al 1 for large interval parameters a subinterval perturbation method for estimating the static displacement bound of structures with interval parameters is put forward in the study the numerical results show that a subinterval perturbation method yields tighter bounds than those yielded by the interval perturbation method

stability of isentropic navier stokes shocks in the high mach number limit by a combination of asymptotic ode estimates and numerical evans function calculations we establish stability of viscous shock solutions of the isentropic compressible navier stokes equations with law pressure i in the limit as mach number m goes to infinity for any 1 proved analytically and ii for m 2 500 gamma in 1 2 5 or m 13 000 gamma in 2 5 3 demonstrated numerically this builds on and completes earlier studies by matsumura nishihara and barker humpherys rudd zumbrun establishing stability for low and intermediate mach numbers respectively indicating unconditional stability independent of shock amplitude of viscous shock waves for law gas dynamics in the range gamma in 1 3 other values may be treated similarly but have not been checked numerically the main idea is to establish convergence of the evans function in the high mach number limit to that of a pressureless or infinitely compressible gas with additional upstream boundary condition determined by a boundary layer analysis recall that low mach number behavior is formally incompressible

stable viscosity matrices for systems of conservation laws abstract a natural class of appropriate viscosity matrices for strictly hyperbolic systems of conservation laws in one space dimension u 1 f u x 0 u r m is studied these matrices are admissible in the sense that small amplitude shock wave solutions of the hyperbolic system are shown to be limits of smooth traveling wave solutions of the parabolic system u t f u x v du x x as ifv 0 if d is in this class the class is determined by a linearized stability requirement the cauchy problem for the equation u 1 f u 0 u x vdu xx should be well posed in l 2 uniformly in v as v 0 previous examples of inadmissible viscosity matrices are accounted for through violation of the stability criterion

boundary layer stability in real vanishing viscosity limit in the previous paper 20 an evans function machinery for the study of boundary layer stability was developed there the analysis was restricted to strongly parabolic perturbations that is to an approximation of the form ut f u x b u ux x 1 with an elliptic matrix b however real models like the navier stokes approximation of the euler equations for a gas flow involve incompletely parabolic perturbations b is not invertible in general

stable viscosities and shock profiles for systems of conservation laws abstract many equations of mathematical physics take the form of nonlinear hyperbolic systems of conservation laws with small dissipative effects neglected typically smooth solutions must develop discontinuities shocks in finite time re incorporating dissipation helps select those discontinuities which are physically relevant for this purpose many different sorts of dissipation will do in particular the physical viscosity is typically degenerate and not convenient in this paper the author provide an understanding of what high order viscosity terms smooth the physical discontinuities a natural class of admissible viscosity terms is determined based on a simple linearized stability criterion in addition they determine a class of degenerate second order viscosity terms of physical type which are admissible these results are applied to the equations of compressible fluid dynamics to determine what conditions ensure the existence of the shock layer with viscosity and heat conduction this should be of interest to others interested in general equations of state for compressible fluids such as those investigating phase transitions

global well posedness of the cauchy problem for the navier stokes equations of nonisentropic flow with discontinuous initial data abstract we prove the global existence uniqueness and continuous dependence on initial data for discontinuous solutions of the navier stokes equations for nonisentropic compressible flow in one space dimension a great deal of information is obtained concerning the qualitative behavior of the solution including an analysis of the convection and evolution of jump discontinuities the derivation of sharp rates of smoothing and the l asymptotic behavior

random continued fractions levy constant and chernoff type estimate abstract given a stochastic process a n n 1 taking values in natural numbers the random continued fractions are defined as a 1 a 2 a n analogously to the continued fraction expansion of real numbers assume that a n n 1 is ergodic and the expectation e log a 1 we give a levy type metric theorem which covers that of real case presented by levy in 1929 moreover a corresponding chernoff type estimate is obtained under the conditions that a n n 1 is mixing and for each 0 t 1 e a 1 t

galloping instability of viscous shock waves abstract motivated by physical and numerical observations of time oscillatory galloping spinning and cellular instabilities of detonation waves we study poincare hopf bifurcation of traveling wave solutions of viscous conservation laws the main difficulty is the absence of a spectral gap between oscillatory modes and essential spectrum preventing standard reduction to a finite dimensional center manifold we overcome this by direct lyapunov schmidt reduction using detailed pointwise bounds on the linearized solution operator to carry out a nonstandard implicit function construction in the absence of a spectral gap the key computation is a space time stability estimate on the transverse linearized solution operator reminiscent of duhamel estimates carried out on the full solution operator in the study of nonlinear stability of spectrally stable traveling waves

diffusive stability of spatial periodic solutions of the swift hohenberg equation we are interested in the nonlinear stability of the eckhaus stable equilibria of the swift hohenberg equation on the infinite line with respect to small integrable perturbations the difficulty in showing stability for these stationary solutions comes from the fact that their linearizations possess continuous spectrum up to zero the nonlinear stability problem is treated with renormalization theory which allows us to show that the nonlinear terms are irrelevant i e that the fully nonlinear problem behaves asymptotically as the linearized one which is under a diffusive regime

efficient computation of analytic bases in evans function analysis of large systems in evans function computations of the spectra of asymptotically constant coefficient linearized operators of large systems a problem that becomes important is the efficient computation of global analytically varying bases for invariant subspaces of the limiting coefficient matrices in the case that the invariant subspace is spectrally separated from its complementary invariant subspace we propose an efficient numerical implementation of a standard projection based algorithm of kato for which the key step is the solution of an associated sylvester problem this may be recognized as the analytic cousin of a c k algorithm developed by dieci and collaborators based on orthogonal projection rather than eigenprojection as in our case for a one dimensional subspace it reduces essentially to an algorithm of bridges derks and gottwald based on path finding and continuation methods

deviations of trajectory averages and the defect in pesin s formula for anosov diffeomorphisms a large deviation theorem at the donsker varadhan level three is obtained for the convergence of trajectory averages of anosov diffeomorphisms it is possible to provide an explicit description of the rate function

random continued fractions with beta hypergeometric distribution in a recent paper statist probab lett 78 2008 1711 1721 it has been shown that certain random continued fractions have a density which is a product of a beta density and a hypergeometric function 2f1 in the present paper we fully exploit a formula due to thomae j reine angew math 87 1879 26 73 in order to generalize substantially the class of random continuous fractions with a density of the above form this involves the design of seven particular graphs infinite paths on them lead to random continued fractions with an explicit distribution a careful study about the set of five real parameters leading to a beta hypergeometric distribution is required relying on almost forgotten results mainly due to felix klein

diffusive stability against nonlocalized perturbations of planar wave trains in reaction diffusion systems abstract planar wave trains are traveling wave solutions whose wave profiles are periodic in one spatial direction and constant in the transverse direction in this paper we investigate the stability of planar wave trains in reaction diffusion systems we establish nonlinear diffusive stability against perturbations that are bounded along a line in r 2 and decay exponentially in the distance from this line our analysis is the first to treat spatially nonlocalized perturbations that do not originate from a phase modulation we also consider perturbations that are fully localized and establish nonlinear stability with better decay rates suggesting a trade off between spatial localization of perturbations and temporal decay rate our stability analysis utilizes pointwise estimates to exploit the spatial structure of the perturbations the nonlocalization of perturbations prevents the use of damping estimates in the nonlinear iteration scheme instead we track the perturbed solution in two different coordinate systems

universal grey system theory for analysis of uncertain structural systems the uncertainty present in many structural systems is commonly modeled using probabilistic fuzzy or interval approaches depending on the nature of uncertainty present the probabilistic approach

pointwise asymptotic behavior of modulated periodic reaction diffusion waves abstract by working with the periodic resolvent kernel and the bloch decomposition we establish pointwise bounds for the green function of the linearized equation associated with spatially periodic traveling waves of a system of reaction diffusion equations with our linearized estimates together with a nonlinear iteration scheme developed by johnson zumbrun we obtain l p behavior p 1 of a nonlinear solution to a perturbation equation of a reaction diffusion equation with respect to initial data in l 1 h 2 recovering and slightly sharpening results obtained by schneider using weighted energy and renormalization techniques we obtain also pointwise nonlinear estimates with respect to two different initial perturbations u 0 e 0 e x 2 m u 0 h 2 e 0 and u 0 e 0 1 x r r 2 u 0 h 2 e 0 respectively e 0 0 sufficiently small and m 1 sufficiently large showing that behavior is that of a heat kernel these pointwise bounds have not been obtained elsewhere and do not appear to be accessible by previous techniques

stability of undercompressive viscous shock profiles of hyperbolic parabolic systems abstract extending to systems of hyperbolic parabolic conservation laws results of howard and zumbrun for strictly parabolic systems we show for viscous shock profiles of arbitrary amplitude and type that necessary spectral evans function conditions for linearized stability established by mascia and zumbrun are also sufficient for linearized and nonlinear phase asymptotic stability yielding detailed pointwise estimates and sharp rates of convergence in l p 1 p

modified sub interval perturbation finite element method for 2d acoustic field prediction with large uncertain but bounded parameters abstract based on the sub interval perturbation analysis a modified sub interval perturbation finite element method is proposed to determine the bounds of sound pressure in the 2d acoustic field with large uncertain but bounded parameters in the proposed method the inversion of the invertible sub interval dynamic stiffness matrix is approximated by a modified approximate interval value sherman morrison woodbury formula to overcome the drawbacks arising from the dependency phenomenon of parameters and the unpredictable effect of neglecting the higher order terms in neumann series the modified sub interval perturbation procedures are implemented in a numerical finite element framework numerical results on a 2d acoustic tube and a 2d acoustic cavity of a car with large uncertain but bounded parameters evidence the remarkable accuracy and effectiveness of the proposed method

composite material design of two dimensional structures using the homogenization design method summary composite materials of two dimensional structures are designed using the homogenization design method the composite material is made of two or three dierent material phases designing the composite material consists of nding a distribution of material phases that minimizes the mean compliance of the macrostructure subject to volume fraction constraints of the constituent phases within a unit cell of periodic microstructures at the start of the computational solution the material distribution of the microstructure is represented as a pure mixture of the constituent phases as the iteration procedure unfolds the component phases separate themselves out to form distinctive interfaces the eective material properties of the articially mixed materials are dened by the interpolation of the constituents the optimization problem is solved using the sequential linear programming method both the macrostructure and the microstructures are analysed using the nite element method in each iteration step several examples of optimal topology design of composite material are presented to demonstrate the validity of the present numerical algorithm copyright 2001 john wiley sons ltd

stability of large amplitude viscous shock profiles of hyperbolic parabolic systems we establish nonlinear l1 h3 lp orbital stability 2 p with sharp rates of decay of large amplitude lax type shock profiles for a class of symmetric hyperbolic parabolic systems including compressible gas dynamics and magnetohydrodynamics mhd under the necessary conditions of strong spectral stability i e a stable point spectrum of the linearized operator about the wave transversality of the profile and hyperbolic stability of the associated ideal shock this yields in particular together with the spectral stability results of 50 the nonlinear stability of arbitrarily large amplitude shock profiles of isentropic navier stokes equations for a gamma law gas as 1 the first complete large amplitude stability result for a shock profile of a system with real i e partial viscosity a corresponding small amplitude result was established in 53 54 for general systems of kawashima class by a combination of kawashima type energy estimates and pointwise green function bounds where the small amplitude assumption was used only to close the energy estimates here under the mild additional assumption that hyperbolic characteristic speeds relative to the shock are not only nonzero but of a common sign we close the estimates instead by use of a goodman type weighted norm 25 26 designed to control estimates in the crucial hyperbolic modes