homogenization of fully nonlinear uniformly elliptic and parabolic partial differential equations in stationary ergodic media we study the homogenization of fully nonlinear uniformly elliptic and parabolic second order partial differential equations in stationary ergodic media c 2004 wiley periodicals inc

integral representation results for energies defined on stochastic lattices and application to nonlinear elasticity this article is devoted to the study of the asymptotic behavior of a class of energies defined on stochastic lattices under polynomial growth assumptions we prove that the energy functionals f varepsilon stored in the deformation of an varepsilon scaling of a stochastic lattice converge to a continuous energy functional when varepsilon goes to zero in particular the limiting energy functional is of integral type and deterministic if the lattice is ergodic we also generalize to systems and nonlinear settings well known results on stochastic homogenization of discrete elliptic equations as an application of the main result we prove the convergence of a discrete model for rubber towards the nonlinear theory of continuum mechanics we finally address some mechanical properties of the limiting models such as frame invariance isotropy and natural states

well posedness of feedback systems insights into exact robustness analysis and approximate computations this paper establishes a framework for robust stability analysis of linear time invariant uncertain systems the uncertainty is assumed to belong to an arbitrary subset of complex matrices the concept used here is well posedness of feedback systems leading to necessary and sufficient conditions for robust stability based on this concept some insights into exact robust stability conditions are given in particular frequency domain and state space conditions for well posedness are provided in terms of hermitian form inequalities it is shown that these inequalities can be interpreted as small gain conditions with a generalized class of scalings given by linear fractional transformations lft using the lft scaled small gain condition in the state space setting the duality is established between the h sub spl infin norm condition with frequency dependent scalings and the parameter dependent lyapunov condition connections to the existing results including the structured singular value and the integral quadratic constraints are also discussed finally we show that our well posedness conditions can be used to give a less conservative yet computable bound on the real structured singular value this result is illustrated by numerical examples

rates of convergence for the homogenization of fully nonlinear uniformly elliptic pde in random media we establish a logarithmic type rate of convergence for the homogenization of fully nonlinear uniformly elliptic second order pde in strongly mixing media with similar i e logarithmic decorrelation rate the proof consists of two major steps the first which is actually the only place in the paper where probability plays a role establishes the rate for special quadratic data using the methodology developed by the authors and wang to study the homogenization of nonlinear uniformly elliptic pde in general stationary ergodic random media the second is a general argument based on the new notion of viscosity solutions which is introduced in this paper that shows that rates known for quadratic can be extended to general data as an application of this we also obtain here rates of convergence for the homogenization in periodic and almost periodic environments the former is algebraic while the latter depends on the particular equation

stability of polynomials under coefficient perturbation let the real polynomial a 0 x n a 1 x n 1 a n 1 x a n be stable and let the real numbers b k c k geq 0 0 leq k leq n be given we present a simple determinant criterion for finding the largest t 0 geq 0 such that the polynomial alpha 0 x n alpha 1 x n 1 alpha n 1 x alpha n is stable for all alpha k in a k b k t 0 a k c k t 0 cup a k 0 leq k leq n several further observations allow us to reduce the computational cost considerably

stability margin calculation for systems with structured time delay uncertainties a method of calculating a stability margin of systems with uncertain time delay elements is presented this method utilizes the bilinear transformation of time delay elements to obtain a multilinear function in order to compute the stability margin by applying the mapping theory proposed for the robust stability test

random parking euclidean functionals and rubber elasticity we study subadditive functions of the random parking model previously analyzed by the second author in particular we consider local functions s of subsets of mathbb r d and of point sets that are almost subadditive in their first variable denoting by the random parking measure in mathbb r d and by r the random parking measure in the cube qr r r d we show under some natural assumptions on s that there exists a constant overline s in mathbb r such that lim r to infty frac s q r xi q r lim r to infty frac s q r xi r q r overline s almost surely if zeta mapsto s q r zeta is the counting measure of zeta in qr then we retrieve the result by the second author on the existence of the jamming limit the present work generalizes this result to a wide class of almost subadditive functions in particular classical euclidean optimization problems as well as the discrete model for rubber previously studied by alicandro cicalese and the first author enter this class of functions in the case of rubber elasticity this yields an approximation result for the continuous energy density associated with the discrete model at the thermodynamic limit as well as a generalization to stochastic networks generated on bounded sets

fast construction of robustness degradation function we develop a fast algorithm to construct the robustness degradation function which describes quantitatively the relationship between the proportion of systems guaranteeing the robustness requirement and the radius of the uncertainty set this function can be applied to predict whether a controller design based on an inexact mathematical model will perform satisfactorily when implemented on the true system

method of robust stability analysis with highly structured uncertainties this paper presents a method of analyzing the stability of linear time invariant interconnected systems with uncertainties each subsystem is single input single output and its vector locus lies inside a polygon at each frequency it is shown that the convex closure of the image of the cartesian product of these polygons under the mapping phi det i fh agrees with the convex closure of the image of the vertices of these polygons under the mapping from this result the image can be estimated by a finite dumber of calculations a sufficient stability condition is obtained by applying this result to the multivariable nyquist stability criterion

the nose poincare method for constant temperature molecular dynamics we present a new extended phase space method for constant temperature canonical ensemble molecular dynamics our starting point is the hamiltonian introduced by nose to generate trajectories corresponding to configurations in the canonical ensemble using a poincare time transformation we construct a hamiltonian system with the correct intrinsic timescale and show that it generates trajectories in the canonical ensemble our approach corrects a serious deficiency of the standard change of variables nose hoover dynamics which yields a time reversible system but simultaneously destroys the hamiltonian structure a symplectic discretization method is presented for solving the nose poincare equations the method is explicit and preserves the time reversal symmetry in numerical experiments it is shown that the new method exhibits enhanced stability when the temperature fluctuation is large extensions are presented for nose chains holonomic constraints and rigid bodies

homogenization and enhancement of the g equation in random environments we study the homogenization of a g equation which is advected by a divergence free stationary vector field in a general ergodic random environment we prove that the averaged equation is an anisotropic deterministic g equation and we give necessary and sufficient conditions in order to have enhancement since the problem is not assumed to be coercive it is not possible to have uniform bounds for the solutions in addition as we show the associated minimal first passage time function does not satisfy in general the uniform integrability condition which is necessary to apply the sub additive ergodic theorem we overcome these obstacles by i establishing a new reachability controllability estimate for the minimal function and ii constructing for each direction and almost surely a random sequence which has both a long time averaged limit due to the sub additive ergodic theorem and stays in the same sense asymptotically close to the minimal time

stochastic homogenization of the bending plate model abstract we use the notion of stochastic two scale convergence introduced in 32 to solve the problem of stochastic homogenization of the elastic plate in the bending regime

variational integrators for electric circuits in this contribution we develop a variational integrator for the simulation of stochastic and multiscale electric circuits when considering the dynamics of an electric circuit one is faced with three special situations 1 the system involves external control forcing through external controlled voltage sources and resistors 2 the system is constrained via the kirchhoff current kcl and voltage laws kvl 3 the lagrangian is degenerate based on a geometric setting an appropriate variational formulation is presented to model the circuit from which the equations of motion are derived a time discrete variational formulation provides an iteration scheme for the simulation of the electric circuit dependent on the discretization the intrinsic degeneracy of the system can be canceled for the discrete variational scheme in this way a variational integrator is constructed that gains several advantages compared to standard integration tools for circuits in particular a comparison to bdf methods which are usually the method of choice for the simulation of electric circuits shows that even for simple lcr circuits a better energy behavior and frequency spectrum preservation can be observed using the developed variational integrator

higher order linearization and regularity in nonlinear homogenization we prove large scale c regularity for solutions of nonlinear elliptic equations with random coefficients thereby obtaining a version of the statement of hilbert s 19th problem in the context of homogenization the analysis proceeds by iteratively improving three statements together i the regularity of the homogenized lagrangian l ii the commutation of higher order linearization and homogenization and iii large scale c 0 1 type regularity for higher order linearization errors we consequently obtain a quantitative estimate on the scaling of linearization errors a liouville type theorem describing the polynomially growing solutions of the system of higher order linearized equations and an explicit heterogenous analogue of the taylor series for an arbitrary solution of the nonlinear equations with the remainder term optimally controlled these results give a complete generalization to the nonlinear setting of the large scale regularity theory in homogenization for linear elliptic equations

stochastic homogenization of nonconvex unbounded integral functionals with convex growth we consider the well trodden ground of the problem of the homogenization of random integral functionals when the integrand has standard growth conditions the qualitative theory is well understood when it comes to unbounded functionals that is when the domain of the integrand is not the whole space and may depend on the space variable there is no satisfactory theory in this contribution we develop a complete qualitative stochastic homogenization theory for nonconvex unbounded functionals with convex growth we first prove that if the integrand is convex and has p growth from below with p d the dimension then it admits homogenization regardless of growth conditions from above this result that crucially relies on the existence and sublinearity at infinity of correctors is also new in the periodic case in the case of nonconvex integrands we prove that a similar homogenization result holds provided that the nonconvex integrand admits a two sided estimate by a convex integrand the domain of which may depend on the space variable that itself admits homogenization this result is of interest to the rigorous derivation of rubber elasticity from polymer physics which involves the stochastic homogenization of such unbounded functionals

structure preserving algorithms for the two dimensional sine gordon equation with neumann boundary conditions abstract this paper presents two kinds of strategies to construct structure preserving algorithms with homogeneous neumann boundary conditions for the sine gordon equation while most existing structure preserving algorithms are only valid for zero or periodic boundary conditions the first strategy is based on the conventional second order central difference quotient but with a cell centered grid while the other is established on the regular grid but incorporated with summation by parts sbp operators both the methodologies can provide conservative semi discretizations with different forms of hamiltonian structures and the discrete energy however utilizing the existing sbp formulas schemes obtained by the second strategy can directly achieve higher order accuracy while it is not obvious for schemes based on the cell centered grid to make accuracy improved easily further combining the implicit midpoint method and the scalar auxiliary variable sav approach we construct symplectic integrators and linearly implicit energy preserving schemes for the two dimensional sine gordon equation respectively extensive numerical experiments demonstrate their effectiveness with the homogeneous neumann boundary conditions

stochastic homogenisation of free discontinuity problems in this paper we study the stochastic homogenisation of free discontinuity functionals assuming stationarity for the random volume and surface integrands we prove the existence of a homogenised random free discontinuity functional which is deterministic in the ergodic case moreover by establishing a connection between the deterministic convergence of the functionals at any fixed realisation and the pointwise subadditive ergodic theorem by akcoglou and krengel we characterise the limit volume and surface integrands in terms of asymptotic cell formulas

robustness analysis of polynomials with polynomial parameter dependency using bernstein expansion this paper considers the robust stability verification of polynomials with coefficients depending polynomially on parameters varying in given intervals two algorithms are presented both rely on the expansion of a multivariate polynomial into bernstein polynomials the first one is an improvement of the so called bernstein algorithm and checks the hurwitz determinant for positivity over the parameter set the second one is based on the analysis of the value set of the family of polynomials and profits from the convex hull property of the bernstein polynomials numerical results to real world control problems are presented showing the efficiency of both algorithms

homogenization of multiparameter integrals where is a bounded domain in rn u rm 0 and w rn rmn 0 is supposed to be 0 1 periodic with respect to the 7rst variable x rn the distinguishing feature of 1 here is that the integrand is permitted to depend on a vector of parameters 1 k r with k 1 we are interested in the asymptotic behavior of 1 as 0 and 0 integral functionals of this type appear in the mathematical modeling of cellular composite materials when n m 3 1 can be interpreted as the stored strain energy of an elastic and heterogeneous material u being a deformation or displacement 7eld when m 1 u may be a di erence of potential in a condenser in any case the medium under consideration is composed of several materials which are periodically distributed at the microscopic scale given by in the applications we often deal with

robust stability of polynomials with multilinear parameter dependence abstract the problem is studied of testing for stability a class of real polynomials in which the coefficients depend on a number of variable parameters in a multilinear way we show that the testing for real unstable roots can be achieved by examining the stability of a finite number of corner polynomials obtained by setting parameters at their extreme values while checking for unstable complex roots normally involves examining the real solutions of up to m 1 simultaneous polynomial equations where m is the number of parameters when m 2 this is an especially simple task

nonlinear stochastic homogenization and ergodic theory on etudie l homogeneisation stochastique non lineaire d une classe de fonctionnelles integrales on relie le probleme a la theorie ergodique

conservative integrators for a toy model of weak turbulence abstract weak turbulence is a phenomenon by which a system generically transfers energy from low to high wave numbers while persisting for all finite time it has been conjectured by bourgain that the 2d defocusing nonlinear schrodinger equation nls on the torus has this dynamic and several analytical and numerical studies have worked towards addressing this point in the process of studying the conjecture colliander keel staffilani takaoka and tao introduced a toy model dynamical system as an approximation of nls which has been subsequently studied numerically in this work we formulate and examine several numerical schemes for integrating this model equation the model has two invariants and our schemes aim to conserve at least one of them we prove convergence in some cases and our numerical studies show that the schemes compare favorably to others such as trapezoidal rule and fixed step fourth order runge kutta the preservation of the invariants is particularly important in the study of weak turbulence as the energy transfer tends to occur on long time scales

homogenization of a hele shaw problem in periodic and random media we investigate the homogenization limit of a free boundary problem with space dependent free boundary velocities the problem under consideration has a well known obstacle problem transformation formally obtained by integrating with respect to the time variable by making rigorous the link between these two problems we are able to derive an explicit formula for the homogenized free boundary velocity and we establish the uniform convergence of the free boundaries

γ convergence and homogenisation for a class of degenerate functionals abstract this paper is on convergence for degenerate integral functionals related to homogenisation problems in the heisenberg group here both the rescaling and the notion of invariance or periodicity are chosen in a way motivated by the geometry of the heisenberg group without using special geometric features these functionals would be neither coercive nor periodic so classic results do not apply all the results apply to the more general case of carnot groups

stochastic reinforcement problems and ergodic theory we study the asymptotic behaviour of dirichlet problems in domains of r 2 bounded by thin layers whose thickness is given by means of an assigned ergodic random function using a capacitary method together with ergodic theorems for additive and superadditive processes we are able to characterise the limit problem precisely

regularity and stochastic homogenization of fully nonlinear equations without uniform ellipticity we prove regularity and stochastic homogenization results for certain degenerate elliptic equations in nondivergence form the equation is required to be strictly elliptic but the ellipticity may oscillate on the microscopic scale and is only assumed to have a finite d th moment where d is the dimension in the general stationary ergodic framework we show that the equation homogenizes to a deterministic uniformly elliptic equation and we obtain an explicit estimate of the effective ellipticity which is new even in the uniformly elliptic context showing that such an equation behaves like a uniformly elliptic equation requires a novel reworking of the regularity theory we prove deterministic estimates depending on averaged quantities involving the distribution of the ellipticity which are controlled in the macroscopic limit by the ergodic theorem we show that the moment condition is sharp by giving an explicit example of an equation whose ellipticity has a finite p th moment for every p less than d but for which regularity and homogenization break down while these results are new even for linear equations we prove them in the fully nonlinear context in probabilistic terms our homogenization results correspond to quenched invariance principles for controlled diffusion processes in random media including linear diffusions as well as diffusions controlled by one controller or two competing players

quasistatic droplets in randomly perforated domains we consider the hele shaw problem in a randomly perforated domain with zero neumann boundary conditions a homogenization limit is obtained as the characteristic scale of the domain goes to zero specifically we prove that the solutions as well as their free boundaries converge uniformly to those corresponding to a homogeneous and anisotropic hele shaw problem set in mathbb r d the main challenge when deriving the limit lies in controlling the oscillations of the free boundary this is overcome first by extending de giorgi nash moser type estimates to perforated domains and second by proving the almost sure non degenerate growth of the solution near its free boundary

homogenization of polycrystalline aggregates the effective elastic properties of a polycrystalline material depend on the single crystal elastic constants of the crystallites comprising the polycrystal and on the manner in which the crystallites are arranged in this paper we apply the techniques of homogenization to put the problem of determining effective elastic constants in a precise mathematical framework that permits us to derive an expression for the effective elasticity tensor we also study how the homogenized elasticity tensor changes as the probability characterizing the ensemble changes under the assumption that the field of orientations of the crystallographic axes of the crystallites is an independent random field we show that our theory is compatible with the formulation used in texture analysis in particular we are able to prove that the physical assumption made by 10 in his study of weakly textured polycrystals holds true in addition we establish some elementary bounds on the material constants that characterize the effective elasticity tensor of weakly textured orthorhombic aggregates of cubic crystallites

explicit symplectic integrators for solving nonseparable hamiltonians by exploiting the error functions of explicit symplectic integrators for solving separable hamiltonians i show that it is possible to develop explicit time reversible symplectic integrators for solving nonseparable hamiltonians of the product form the algorithms are unusual in that they are of fractional orders

symplectic maps for approximating polynomial hamiltonian systems we study how to approximate polynomial hamiltonian systems by composition of symplectic maps recently a number of methods preserving the symplectic character have appeared however they are not completely satisfactory because in general they are computationally expensive very difficult to obtain or their accuracy is relatively low the efficiency of a numerical method depends on both its computational cost and its accuracy polynomial hamiltonians are separable in exactly solvable parts and this can be done in many different ways here we study how to find a separation for the hamiltonian in a small number of cheaply computed terms since the proposed methods depend on some free parameters we also indicate how to choose these parameters in order to improve the accuracy without increasing the computational cost

brief positive polynomial matrices and improved lmi robustness conditions recently several new lmi conditions for stability of linear systems have been proposed introducing additional slack variables to reduce the gap between conservative convex quadratic stability conditions and intractable non convex robust stability conditions in this paper we show that these improved lmi conditions can be derived with the help of some basic results on positive polynomial matrices the approach allows us to derive in a unifying way results in the state space and polynomial frameworks applications to robust stability analysis and robust stabilization of systems with multi linear parametric uncertainty are fully described

the additive structure of elliptic homogenization one of the principal difficulties in stochastic homogenization is transferring quantitative ergodic information from the coefficients to the solutions since the latter are nonlocal functions of the former in this paper we address this problem in a new way in the context of linear elliptic equations in divergence form by showing that certain quantities associated to the energy density of solutions are essentially additive as a result we are able to prove quantitative estimates on the weak convergence of the gradients fluxes and energy densities of the first order correctors under blow down which are optimal in both scaling and stochastic integrability the proof of the additivity is a bootstrap argument completing the program initiated in armstrong et al commun math phys 347 2 315 361 2016 using the regularity theory recently developed for stochastic homogenization we reduce the error in additivity as we pass to larger and larger length scales in the second part of the paper we use the additivity to derive central limit theorems for these quantities by a reduction to sums of independent random variables in particular we prove that the first order correctors converge in the large scale limit to a variant of the gaussian free field

a generalization of kharitonov s four polynomial concept for robust stability problems with linearly dependent coefficient perturbations kharitonov s four polynomial concept is generalized to the case of linearly dependent coefficient perturbations and more general zero location regions to this end a specially constructed scalar function of a scalar variable is instrumental to the robustness analysis the present work is motivated by two fundamental limitations of kharitonov s theorem namely 1 the theorem only applies to polynomials with independent coefficient perturbations and 2 it only applies to zeros in the left hand plane

mesoscopic higher regularity and subadditivity in elliptic homogenization we introduce a new method for obtaining quantitative results in stochastic homogenization for linear elliptic equations in divergence form unlike previous works on the topic our method does not use concentration inequalities such as poincare or logarithmic sobolev inequalities in the probability space and relies instead on a higher ck k 1 regularity theory for solutions of the heterogeneous equation which is valid on length scales larger than a certain specified mesoscopic scale this regularity theory which is of independent interest allows us to in effect localize the dependence of the solutions on the coefficients and thereby accelerate the rate of convergence of the expected energy of the cell problem by a bootstrap argument the fluctuations of the energy are then tightly controlled using subadditivity the convergence of the energy gives control of the scaling of the spatial averages of gradients and fluxes that is it quantifies the weak convergence of these quantities which yields by a new multiscale poincare inequality quantitative estimates on the sublinearity of the corrector

homogenized dynamics of stochastic partial differential equations with dynamical boundary conditions a microscopic heterogeneous system under random influence is considered the randomness enters the system at physical boundary of small scale obstacles as well as at the interior of the physical medium this system is modeled by a stochastic partial differential equation defined on a domain perforated with small holes obstacles or heterogeneities together with random dynamical boundary conditions on the boundaries of these small holes

robustness analysis with correlated highly structured complex uncertainties it is shown that the results reported by the author ibid vol 35 no 3 pp 350 352 mar 1990 on robustness analysis with highly structured independent perturbations is directly extendable to the case of correlated perturbations a direct consequence of this extension is that the phase sensitive structured singular value and the multiloop phase margin can be computed exactly

explicit symplectic approximation of nonseparable hamiltonians algorithm and long time performance explicit symplectic integrators have been important tools for accurate and efficient approximations of mechanical systems with separable hamiltonians this article proposes for arbitrary hamiltonians similar integrators which are explicit of any even order symplectic in an extended phase space and with pleasant long time properties they are based on a mechanical restraint that binds two copies of phase space together using backward error analysis kolmogorov arnold moser theory and additional multiscale analysis an error bound of o t l is established for integrable systems where t l and are respectively the long simulation time step size integrator order and some binding constant for nonintegrable systems with positive lyapunov exponents such an error bound is generally impossible but satisfactory statistical behaviors were observed in a numerical experiment with a nonlinear schrodinger equation

stochastic homogenization of the keller segel chemotaxis system abstract in this paper we focus on the keller segel chemotaxis system in a random heterogeneous domain we assume that the corresponding diffusion and chemotaxis coefficients are given by stationary ergodic random fields and apply stochastic two scale convergence methods to derive the homogenized macroscopic equations in establishing our results we also derive a priori estimates for the keller segel system that rely only on the boundedness of the coefficients in particular no differentiability assumption on the diffusion and chemotaxis coefficients for the chemotactic species is required finally we prove the convergence of a periodization procedure for approximating the homogenized macroscopic coefficients

when is the multiaffine image of a cube a convex polygon abstract we give two simple sufficient conditions under which the multiaffine image in the complex plane of an m dimensional cube is a convex polygon a third condition which in some generic sense is necessary and sufficient is then obtained our conditions involve checking the locations of the image of the vertices of the cube these results help determine whether a parameterized family of polynomials th stable

exact calculation of the multiloop stability margin a mapping theorem by l a zadeh and c a desoer 1963 serves as the basis for an algorithm that computes the stability margin k sub m of diagonally perturbed multivariable feedback systems without conservatism the stability margin determination not only verifies system stability but also quantifies how much further the plan uncertainties can be extended before instability occurs the essence of the computational approach is the realization that the true image of a given domain can be approximated with arbitrary accuracy by first subdividing the domain into the requisite number of subdomains and then forming the union of the convex hulls of their images the technique is illustrated in an example with a plant model having three uncertainties in general the method is applicable to either siso or mimo lti systems whose uncertainties can be modeled with noninteracting coefficients in the open loop transfer function representations

robust stability for time delay systems the edge theorem and graphical tests the robust stability problem is discussed for a class of uncertain delay systems where the characteristic equations involve a polytope p of quasi polynomials i e polynomials in one complex variable and exponential powers of the variable given a set d in the complex plane the goal is to find a constructive technique to verify whether all roots of every quasi polynomial in p belong to d that is to verify the d stability of p first it is demonstrated by counterexample that kharitonov s theorem does not hold for general delay systems next it is shown that under a mild assumption on the set d a polytope of quasi polynomials is d stable if and only if the edges of the polytope are d stable this extends the edge theorem for the d stability of a polytope of polynomials the third result gives a constructive graphical test for checking the d stability of a polytope of quasi polynomials which is especially simple when the set d is the open left half plane an application is given to demonstrate the power of the results

homogenization of the poisson equation with dirichlet conditions in random perforated domains we consider a sequence of open sets o e contained in a fixed bounded open set o of r n n 3 which vary randomly with e 0 the corresponding distribution function is given by an ergodic measure preserving dynamical system in such a way that o o e is a union of closed sets of size e n n 2 and the distance between them of order e for this sequence o e we study the asymptotic behavior of the solutions of the poisson equation with dirichlet conditions on o e similarly to the classical cioranescu murat result for the deterministic problem we show the existence of a new term of zero order in the limit equation we emphasize the fact that this new term is deterministic

a functional version of the birkhoff ergodic theorem for a normal integrand a variational approach in this paper we prove a new version of the birkhoff ergodic theorem bet for random variables depending on a parameter alias integrands this involves variational convergences namely epigraphical hypographical and uniform convergence and requires a suitable definition of the conditional expectation of integrands we also have to establish the measurability of the epigraphical lower and upper limits with respect to the field of invariant subsets from the main result applications to uniform versions of the bet to sequences of random sets and to the strong consistency of estimators are briefly derived

homogenization of the free boundary velocity in this paper we investigate some free boundary problems with space dependent free boundary velocities based on maximum principle type arguments we show the uniform convergence of the solutions in the homogenization limit the main step is to show the uniqueness of the limiting free boundary velocity which turns out to be a continuous function of the normal direction of the free boundary

geometric integration for particle accelerators this paper is a very personal view of the field of geometric integration in accelerator physics a field where often work of the highest quality is buried in lost technical notes or even not published one has only to think of simon van der meer nobel prize work on stochastic cooling unpublished in any refereed journal so i reconstructed the relevant history of geometrical integration in accelerator physics as much as i could by talking to collaborators and using my own understanding of the field the reader should not be too surprised if this account is somewhere between history science and perhaps even fiction

the two scale convergence method applied to generalized besicovitch spaces the twoscale convergence method has proved to be a very useful tool for dealing with periodic homogenization problems in the present paper we develop this theory to generalized besicovitch spaces

computation of algebraic combinations of uncertainty value sets a robust analysis approach based on the computation of boundaries of uncertainty value sets is presented the idea is to use the strongly connected subsystem separation and the tree structured decomposition tsd concept to extend the class of uncertainty structures to be considered to this end we compute the exact boundaries of regions produced by operations spl times spl divide among boundaries of simply connected value sets in polynomial time and without incrementing the total amount of input data parametric dynamic or general multiply connected value set descriptions may be accommodated

lipschitz regularity for elliptic equations with random coefficients we develop a higher regularity theory for general quasilinear elliptic equations and systems in divergence form with random coefficients the main result is a large scale l type estimate for the gradient of a solution the estimate is proved with optimal stochastic integrability under a one parameter family of mixing assumptions allowing for very weak mixing with non integrable correlations to very strong mixing for example finite range of dependence we also prove a quenched l2 estimate for the error in homogenization of dirichlet problems the approach is based on subadditive arguments which rely on a variational formulation of general quasilinear divergence form equations

homogenization of one phase stefan type problems in periodic and random media we investigate the homogenization of stefan type problems with oscillating diusion coecients both cases of periodic and random stationary ergodic medium are considered the proof relies on the coincidence of viscosity solutions and weak solutions which are the time derivatives of the solutions of an obstacle problem for the stefan problem this coincidence result is of independent interest