gamma convergence of gradient flows with applications to ginzburg landau we present a method to prove convergence of gradient flows of families of energies that converge to a limiting energy it provides lower bound criteria to obtain the convergence that correspond to a sort of c1 order convergence of functionals we then apply this method to establish the limiting dynamical law of a finite number of vortices for the heat flow of the ginzburg landau energy in dimension 2 retrieving in a different way the existing results for the case without magnetic field and obtaining new results for the case with magnetic field 2004 wiley periodicals inc

slow motion manifolds far from the attractor in multistable reaction diffusion equations abstract we consider a scalar reaction diffusion equation with multistable nonlinearity with a particular symmetry by reduction to a family of transmission problems in r and by contraction arguments a manifold close to an invariant manifold formed by functions exhibiting a pattern of transition layers is constructed an approximation for the associated vector field is also provided this shows that the motion on those manifolds is exponentially slow as in the well known case of the bistable equation however in opposition to the bistable case some of these manifolds are far from the attractor since these manifolds correspond to metastable patterns this shows the importance of the transient motion toward the attractor and the importance of these manifolds in organizing that motion it is also shown that by a suitable perturbation we can obtain new equilibria on those manifolds

dynamics of one and two dimensional fronts in a bistable equation with time delayed global feedback propagation failure and control mechanisms we study the evolution of fronts in a bistable equation with time delayed global feedback in the fast reaction and slow diffusion regime this equation generalizes the hodgkin grafstein and allen cahn equations we derive a nonlinear equation governing the motion of fronts which includes a term with delay in the one dimensional case this equation is linear we study the motion of one and two dimensional fronts finding a much richer dynamics than for the previously studied cases without time delayed global feedback we explain the mechanism by which localized fronts created by inhibitory global coupling loose stability in a hopf bifurcation as the delay time increases we show that for certain delay times the prevailing phase is different from that corresponding to the system in the absence of global coupling numerical simulations of the partial differential equation are in agreement with the analytical predictions

a novel approach to exponential stability of nonlinear systems with time varying delays in this paper the stability of nonlinear systems with time varying delays is investigated by means of the concepts of generalized dahlquist constant generalized relative dahlquist constant and relative minimal lipschitz constant in detail two sufficient conditions are derived for the exponential stability of nonlinear systems with time varying delays and the exponential decay of the solutions is also estimated compared with some existing results our stability conditions are less conservative some examples are given to illustrate the effectiveness of the obtained results

threshold dynamics for the piecewise constant mumford shah functional we propose an efficient algorithm for minimizing the piecewise constant mumford shah functional of image segmentation it is based on the threshold dynamics of merriman bence and osher for evolving an interface by its mean curvature we show that a very fast minimization can be achieved by alternating the solution of a linear parabolic partial differential equation and simple thresholding

the far field of a spherical array of point dipoles in this paper the far field of a spherical array of point dipoles is related to the array currents in an elegant manner by invoking the properties of the associated legendre functions the inverse problem viz determining the array currents from the far field data is solved in a recursive manner thereby precluding the need for a matrix inversion

geometric optics approach to reaction diffusion equations the speed of propagation of traveling waves of reaction diffusion equations is calculated by an asymptotic method similar to geometrical optics this also works for reaction diffusion waves in random media

nonparabolic dissipative systems modeling the flow of liquid crystals we study a simplified system which retains most of the interesting mathematical properties of the original ericksen leslie equations for the flow of liquid crystals this is a coupled nonparabolic dissipative dynamic system we derive several energy laws which enable us to prove the global existence of the weak solutions and the classical solutions we also discuss uniqueness and some stability properties of the system 1995 john wiley sons inc

generalized analytical technique for the synthesis of unequally spaced arrays with linear planar cylindrical or spherical geometry an effective method for optimizing the performance of a fixed current distribution uniformly spaced antenna array has been to adjust its element positions to provide performance improvement in comparison with the default uniform structure this approach yields performance improvements such as smaller sidelobe levels or beamwidth values additionally it provides practical advantages such as reductions in size weight and number of antenna elements the objective of this paper is to describe a unified mathematical approach to nonlinear optimization of multidimensional array geometries the approach utilizes a class of limiting properties of sinusoidal bessel or legendre functions that are dictated by the array geometry addressed the efficacy of the method is demonstrated by its generalized application to synthesis of rectangular cylindrical and spherical arrays the unified mathematical approach presented below is a synthesis technique founded on the mathematical transformation of the desired field pattern followed by the application of limiting forms of the transformation and resulting in the development of a closed form expression for the element positions the method offers the following advantages over previous techniques such as direct nonlinear optimization or genetic algorithms first it is not an iterative searching algorithm and provides element spacing values directly in a single run of the algorithm thereby saving valuable cpu time and memory storage second it permits the array designer to place practical constraints on the array geometry e g the minimum maximum spacing between adjacent elements

robust stabilization of linear systems with norm bounded time varying uncertainty abstract in this paper robust stabilization of a class of linear systems with norm bounded time varying uncertainties is considered it is shown that for this class of uncertain systems quadratic stabilizability via linear control is equivalent to the existence of a positive definite symmetric matrix solution to a parameter dependent riccati equation also a construction for the stabilizing feedback law is given in terms of the solution to the riccati equation

generation motion and thickness of transition layers for a nonlocal allen cahn equation abstract we investigate the behavior as e 0 of the nonlocal allen cahn equation u t u 1 e 2 f u e u where f u 0 is of the bistable type given a rather general initial datum u 0 that is independent of e we perform a rigorous analysis of both the generation and the motion of the interface and obtain a new estimate for its thickness more precisely we show that the solution develops a steep transition layer within the time scale of order e 2 ln e and that the layer obeys the law of motion that coincides with the limit problem within an error margin of order e

metastability and stability of patterns in a convolution model for phase transitions where e is a small positive parameter and f is bistable having only three zeros 1 a 1 and is balanced i e r 1 1 f u du 1 4 0 see fig 1 for a typical initial datum j which crosses a several times at the beginning the solution evolves as if the diffusion term euxx were not there and in the time scale o ej n ej the solution gets close to 1 or 1 in intervals separated by thin layers of thickness at the order of o ej n ej 17 28 at this moment a spatial pattern is generated it persists on a time scale o ece but is eventually destroyed typically u x t 1 u x t 1 uniformly for x on 1 20 1 as t 1 it is possible that u x t converges to a or a nonconstant steady state as t 1 but this is an unlikely event because all these steady states are unstable 16 27 research supported in part by the national science foundation and bor leqsf

the dynamics of nucleation for the cahn hilliard equation when a constant metastable solution of the cahn hilliard equation is subjected to a spatially localized large amplitude perturbation a transition process may be triggered leading to a globally stable stationary solution in one space dimension the existence and instability of a third stationary solution with the same mass is proved a spike like solution called a canonical nucleus within the class of solutions which are even with respect to the center of the spike it has a one dimensional unstable manifold in addition the process of nucleation by formal arguments using two space scales and two timescales is described the last stage in the process can be approximated by a nonlinear stefan free boundary problem

on the slowness of phase boundary motion in one space dimension we study the limiting behavior of the solution of u t e 2 u xx u 3 u 0 a x b with a neumann boundary condition or an appropriate dirichlet condition the analysis is based on energy methods we assume that the initial data has a transition layer structure i e ue 1 except near finitely many transition points we show that in the limit as e 0 the solution maintains its transition layer structure and the transition points move slower than any power of e

convergence of the parabolic ginzburg landau equation to motion by mean curvature for the complex parabolic ginzburg landau equation we prove that asymptotically vorticity evolves according to motion by mean curvature in brakke s weak formulation the only assumption is a natural energy bound on the initial data in some cases we also prove convergence to enhanced motion in the sense of ilmanen

existence and regularity of mean curvature flow with transport term in higher dimensions given an initial c 1 hypersurface and a time dependent vector field in a sobolev space we prove a time global existence of a family of hypersurfaces which start from the given hypersurface and which move by the velocity equal to the mean curvature plus the given vector field we show that the hypersurfaces are c 1 for a short time and even after some singularities occur almost everywhere c 1 away from the higher multiplicity region

the singular limit of the allen cahn equation and the fitzhugh nagumo system abstract we consider an allen cahn type equation of the form u t u e 2 f e x t u where e is a small parameter and f e x t u f u e g e x t u a bistable nonlinearity associated with a double well potential whose well depths can be slightly unbalanced given a rather general initial data u 0 that is independent of e we perform a rigorous analysis of both the generation and the motion of interface more precisely we show that the solution develops a steep transition layer within the time scale of order e 2 ln e and that the layer obeys the law of motion that coincides with the formal asymptotic limit within an error margin of order e this is an optimal estimate that has not been known before for solutions with general initial data even in the case where g e 0 next we consider systems of reaction diffusion equations of the form u t u e 2 f e u v v t d v h u v which include the fitzhugh nagumo system as a special case given a rather general initial data u 0 v 0 we show that the component u develops a steep transition layer and that all the above mentioned results remain true for the u component of these systems

on robust stabilization for neutral delay differential systems with parametric uncertainties and its application this paper considers the problem of robust stabilization for uncertain neutral delay differential systems based on the lyapunov method a delay dependent criterion for determining the stability of systems is obtained in terms of matrix inequalities which can be easily solved by efficient convex optimization algorithms then the method presented is extended to systems with controller gain variations that is the design problem for the nonfragile controller is considered a numerical example is included to illustrate the design procedure of the proposed method

slow motion of gradient flows abstract we present sufficient conditions on an energy landscape in order for the associated gradient flow to exhibit slow motion or dynamic metastability the first condition is a weak form of convexity transverse to the so called slow manifold n the second condition is that the energy restricted to n is lipschitz with a constant 1 one feature of the abstract result that makes it of broader interest is that it does not rely on maximum principles as an application we give a new proof of the exponentially slow motion of transition layers in the one dimensional allen cahn equation the analysis is more nonlinear than previous work it relies on the nonlinear convexity condition or energy energy dissipation inequality although we do use the maximum principle for convenience in the application we believe it may be removed with additional work our result demonstrates that a broad class of initial data relaxes with an exponential rate into a neighborhood of the slow manifold where it is then trapped for an exponentially long time

the effects of obstacle size on periodic travelling waves in oscillatory reaction diffusion equations many natural populations undergo multi year cycles and field studies have shown that these can be organized into periodic travelling waves ptws mathematical studies have shown that large scale landscape obstacles represent a natural mechanism for wave generation here we investigate how the amplitude and wavelength of the selected waves depend on the obstacle size we firstly consider a large circular obstacle in an infinite domain for a reaction diffusion system of l u type we use perturbation theory to derive a leading order approximation to the wave generated by the obstacle this shows the dependence of the wave properties on both parameter values and obstacle size we find that the limiting values of the amplitude and wavelength are approached algebraically with distance from the obstacle edge rather than exponentially in the case of a flat boundary we use our results to predict the properties of waves generated by a large circular obstacle for an oscillatory predator prey system via a reduction of the predator prey model to normal form close to hopf bifurcation our predictions compare well with numerical simulations we also discuss the implications of these results for wave stability and briefly investigate the effects of obstacles with elliptical geometries

the generation and propagation of internal layers 4 c2u f u x 7 1 1 is studied where f is of bistable type for each x here bistable means that f vanishes for exactly three values of u and fu is negative at the two outer ones the prototype is f u u3 when f is independent of x the equation has a dynamical theory which is pretty well understood its most striking feature in that case is the existence of travelling wave solutions which are globally stable this is also true for the heterogeneous equation 1 l although such travelling wave solutions have up to now only been studied by formal asymptotics they are not travelling waves in the strictest sense but are rather frontal structures confined to thin zones of width o e which move with variable velocity starting with smooth arbitrary initial data us which do not depend on e the formation of these fronts at certain locations can be predicted and their approximate subsequent motion can be determined by solving an ordinary differential equation this heuristic analysis is explained in section 2 the purpose of this paper is to provide the heuristics with a rigorous footing in fact we obtain detailed estimates for the error made by approximating the exact solution by the function constructed in section 2 for simplicity we treat the case when there is only a single front although multiple fronts may clearly occur the process under consideration consists of two parts proceeding under two different time scales which are the initial generation of a layer followed by its propagation the analysis describing the first part is the more involved if comparison functions based on the ordinary differential equation in section 2 are used the analysis simplifies but weaker results are obtained

development of interfaces in ℝ n consider the reaction diffusion equation in n u t h 2 u u 0 where is the derivative of a bistable even potential and h is a small parameter if the initial data have a smooth noncritical zero set we prove that an interface appears in time o log h 1 and that the solution stays close to it for at least time o 1 h

linear quadratic optimal control of hereditary differential systems infinite dimensional riccati equations and numerical approximations recent theory of infinite dimensional riccati equations is applied to the linear quadratic optimal control problem for hereditary differential systems and it is shown that for most such problems the operator solutions of the riccati equations are of trace class i e nuclear with special attention to trace norm convergence an abstract approximation theory is developed and applied to a particular approximation scheme numerical examples are given problems on both finite and infinite time intervals are studied for both the hereditary system and the approximating systems in the infinite time problem characteristic equations are derived for the closed loop eigenvalues and formulas for the corresponding eigenvectors are given

fast reaction slow diffusion and curve shortening the reaction diffusion problem u 1 varepsilon delta u varepsilon 1 v n u quad u x 0 varepsilon g x quad partial n u 0 text on partial omega for a vector u x t varepsilon is considered in a domain omega in r m an asymptotic solution is constructed for varepsilon small it shows that at each x u tends quickly to a minimum of v u when v has several minima when u tends to a piecewise constant function boundary layer expansions are constructed around the resulting surfaces of discontinuity or fronts each front is found to move along its normal with a constant velocity determined by the discontinuity v in v across it when v 0 the front s normal velocity is varepsilon kappa where kappa is its mean curvature the motion of fronts in this manner is studied for arcs in the plane which are normal to partial omega at their endpoints and for fronts that are closed curves it is shown a front can shrink to a point in

front propagation and phase field theory the connection between the weak theories for a class of geometric equations and the asymptotics of appropriately rescaled reaction diffusion equations is rigorously established two different scalings are studied in the first the limiting geometric equation is a first order equation in the second it is a generalization of the mean curvature equation intrinsic definitions for the geometric equations are obtained and uniqueness under a geometric condition on the initial surface is proved in particular in the case of the mean curvature equation this condition is satisfied by surfaces that are strictly starshaped that have positive mean curvature or that satisfy a condition that interpolates between the positive mean curvature and the starshape conditions

slow motion for the 1d swift hohenberg equation abstract the goal of this paper is to study the behavior of certain solutions to the swift hohenberg equation on a one dimensional torus t combining results from convergence and ode theory it is shown that solutions corresponding to initial data that is l 1 close to a jump function v remain close to v for large time this can be achieved by regarding the equation as the l 2 gradient flow of a second order energy functional and obtaining asymptotic lower bounds on this energy in terms of the number of jumps of v

impulsive stabilization of second order nonlinear delay differential systems modeling of many dynamic systems results in second order delay differential systems in this paper we prove that two kinds of delay systems without impulses can be stabilized by proper impulsive control respectively our results have improved and generated some of the known results found in the literature we also give an example to illustrate the obtained results

generation and propagation of interfaces for reaction diffusion equations abstract this paper is concerned with the asymptotic behavior as e 0 of the solution ue of the reaction diffusion equation in r n r u t 1 2 0 where is the derivative of a bistable potential we show that if the initial data u 0 has values in both domains of attraction of the potential then an interface will develop in a short time o e 2 ln e we also show that if the wells of the potential are of equal depth then this interface will propagate with normal velocity equal to its mean curvature our result is valid as long as the interface remains smooth if the initial interface is compact and n 2 then the interface will disappear in a finite time but not if n 1 in case the depths of the wells are not equal then in order to obtain get reasonable results we must work on the scaled time s t e slower time scale in this scale we show that the interface moves with a constant speed proportional to the difference of the depths of the two wells along the normal towards the domain of the deeper well this result is valid for all s 0 and does not actually depend on the regularity of the interface we also extend the above results to the homogeneous second initial boundary value problem in case the depths of the wells are equal and the initial interface is orthogonal to the boundary we prove that the interface moves with normal velocity equal to its mean curvature provided that there is a family of hypersurfaces which moves according to their mean curvature and intersect the boundary orthogonally

evaporation of droplets in the two dimensional ginzburg landau equation abstract we consider the problem of coarsening in two dimensions for the real scalar ginzburg landau equation this equation has exactly two stable stationary solutions the constant functions 1 and 1 we assume most of the initial condition is in the 1 phase with islands of 1 phase we use invariant manifold techniques to prove that the boundary of a circular island moves according to allen cahn curvature motion law we give a criterion for non interaction of two arbitrary interfaces and a criterion for merging of two nearby interfaces

phase transitions and generalized motion by mean curvature we study the limiting behavior of solutions to appropriately rescaled versions of the allen cahn equation a simplified model for dynamic phase transitions we rigorously establish the existence in the limit of a phase antiphase interface evolving according to mean curvature motion this assertion is valid for all positive time the motion interpreted in the generalized sense of evans spruck and chen giga goto after the onset of geometric singularities

the effect of a singular perturbation on nonconvex variational problems we study the effect of a singular perturbation on certain nonconvex variational problems the goal is to characterize the limit of minimizers as some perturbation parameter 0 the technique utilizes the notion of convergence of variational problems developed by de giorgi the essential idea is to identify the first nontrivial term in an asymptotic expansion for the energy of the perturbed problem in so doing one characterizes the limit of minimizers as the solution of a new variational problem for the cases considered here these new problems have a simple geometric nature involving minimal surfaces and geodesics

robust output feedback constrained controllability of uncertain linear time varying systems abstract this paper introduces the concept of robust output feedback constrained controllability for a class of linear uncertain systems the uncertainty in the system is assumed to have a deterministic character and satisfy a certain integral quadratic constraint a necessary and sufficient condition for robust output feedback controllability is given in terms of the existence of suitable solutions to a riccati differential equation of the game type and a dynamic programming equation

metastable patterns in solutions of ut ϵ2uxx f u we consider the equation in question on the interval 0 x 1 having neumann boundary conditions with f u f u where f is a double well energy density with equal minima at u 1 the only stable states of the system are patternless constant solutions but given two phase initial data a pattern of interfacial layers typically forms far out of equilibrium the ensuing nonlinear relaxation process is extremely slow patterns persist for exponentially long times proportional to exp a l where a f 1 and l is the minimum distance between layers physically a tiny potential jump across a layer drives its motion we prove the existence and persistence of these metastable patterns and characterise accurately the equations governing their motion the point of view is reminiscent of center manifold theory a manifold parametrising slowly evolving states is introduced a neighbourhood is shown to be normally attracting and the parallel flow is characterised to high relative accuracy proofs involve a detailed study of the dirichlet problem spectral gap analysis and energy estimates

improved exponential stability for time varying systems with nonlinear delayed perturbations in this paper a new sufficient delay dependent exponential stability condition for a class of linear time varying systems with nonlinear delayed perturbations is derived by using an improved lyapunov krasovskii functional the proposed exponential stability conditions are formulated in terms of the solution of lyapunov differential equations the approach allows for computation of the bounds that characterize the exponential stability rate of convergence of the solution compared with existing results our conditions are shown to be less conservative numerical examples are given to illustrate the effectiveness of the obtained conditions

the heat equation shrinks embedded plane curves to round points soit c 0 s 1 r 2 une courbe lisse plongee dans le plan alors c s 1 0 t r 2 existe en satisfaisant c t k n ou k est la courbure de c et n est son vecteur unite normal entrant c t est lisse pour tout t il converge vers un point quand t t et sa forme limite quand t t est un cercle rond avec convergence dans norme c

motion by curvature of planar curves with end points moving freely on a line this paper deals with the motion by curvature of planar curves having end points moving freely along a line with fixed contact angles to this line we first prove the existence and uniqueness of self similar shrinking solution then we show that the curve shrinks to a point in a self similar manner if initially the curve is a graph

technical communique static output feedback stabilization an ilmi approach in this note the static output feedback stabilization problem is addressed using the linear matrix inequality technique a necessary and sufficient condition for static output feedback stabilizability for linear time invariant systems is derived in the form of a matrix inequality the extension of the result to h control is studied an iterative lmi ilmi algorithm is proposed to compute the feedback gain numerical examples are employed to demonstrate the effectiveness and the convergence of the algorithm

phase field method to optimize dielectric devices for electromagnetic wave propagation we discuss a phase field method for shape optimization in the context of electromagnetic wave propagation the proposed method has the same functional capabilities as the level set method for shape optimization the first advantage of the method is the simplicity of computation since extra operations such as re initialization of functions are not required the second is compatibility with the topology optimization method due to the similar domain representation and the sensitivity analysis structural shapes are represented by the phase field function defined in the design domain and this function is optimized by solving a time dependent reaction diffusion equation the artificial double well potential function used in the equation is derived from sensitivity analysis we study four types of 2d or 2 5d axisymmetric optimization problems two are the classical problems of photonic crystal design based on the bloch theory and photonic crystal wave guide design and two are the recent topics of designing dielectric left handed metamaterials and dielectric ring resonators

diffuse interface approximations of osmosis free boundary problems free boundary problems based on mass conservation and surface tension with application in osmotic swelling are the topic of this contribution phase field approximations of such models are introduced in order to numerically investigate properties of the solutions formal justification of the proposed approximations is provided by matched asymptotic expansions supported by numerical tests reproducing the convergence for shrinking interface thickness

asymptotic stabilization of a class of continuous time linear periodic systems abstract in this paper the problem of the asymptotic stabilization of a class of continuous time linear periodic systems is considered first under a simplifying assumption necessary conditions involving the concept of left eigenvector are derived for the asymptotic stabilization of continuous time linear periodic systems belonging to such a class then under the same simplifying assumption conditions are given under which a static state feedback control law can be designed in an iterative manner so that the closed loop system is stable

near field analysis of spherical loop arrays in a recent paper see ibid vol 42 p473 7 april 1994 the authors presented the development of an algorithm for the reconstruction of element currents of a spherical loop array given the far field pattern data in an azimuthal plane and demonstrated the validity of the method with several computational examples in this letter it is shown that the same algorithm can be applied to the reconstruction of the spherical loop array currents from the near field data by application of spherical wave expansion of the electric field integral of the array the main feature of the reconstruction algorithm is that the array currents are obtained recursively through the solution of a triangular equation set

global stabilization for linear continuous time varying systems in this paper stabilization problem via static output feedback controls for linear time varying systems is investigated based on the lyapunov function techniques we show that for linear time varying systems the global null controllability guarantees the output feedback stabilization the verifiable stabilizability conditions and output feedback control design are stated the result can be applicable to the output feedback stabilizability of a class of nonlinear time varying systems numerical examples illustrated the conditions are given

motion by anisotropic mean curvature as sharp interface limit of an inhomogeneous and anisotropic allen cahn equation we consider the spatially inhomogeneous and anisotropic reaction diffusion equation ut m x 1 div m x ap x u e 2 f u involving a small parameter e 0 and a bistable nonlinear term whose stable equilibria are 0 and 1 we use a finsler metric related to the anisotropic diffusion term and work in relative geometry we prove a weak comparison principle and perform an analysis of both the generation and the motion of interfaces more precisely we show that within the time scale of order e2 ln e the unique weak solution u e develops a steep transition layer that separates the regions u e 0 and u e 1 then on a much slower time scale the layer starts to propagate consequently as e 0 the solution u e converges almost everywhere a e to 0 in t and 1 in t where t and t are sub domains of separated by an interface t whose motion is driven by its anisotropic mean curvature we also prove that the thickness of the transition layer is of order e

the heat equation shrinking convex plane curves soient m et m des varietes de riemann et f m m une application reguliere si m est une courbe convexe plongee dans le plan r 2 l equation de la chaleur contracte m a un point

a singularly perturbed reaction diffusion problem with inhomogeneous environment reaction diffusion problems have been used to describe pattern formation in developmental biology and material science i study the existence and stability of a singularly perturbed reaction diffusion problem with inhomogeneous environment in one dimensional space domain and also high dimensional space domain

static output feedback control for periodically time varying systems most control designs for periodically time varying systems use either full state feedback or observer based state feedback in this paper it is shown that static output feedback is sufficient for the exponential stabilization of a periodical system under both the controllability and observability assumptions in fact by incorporating a new generalized hold function in the control design one is able to arbitrarily shift all the poincare exponents of the periodical system most importantly the control signal is guaranteed to be continuous in time while the control signal from previous designs may be discontinuous

onset and structure of interfaces in a kawasaki glauber interacting particle system we investigate the spatial structure of typical configurations of a reaction diffusion spin system kawasaki glauber model following the noise induced escape from an unstable spatially homogeneous state after the escape the system will be locally in a stationary phase but will display a globally nontrivial spatial behavior characterized by large clusters of the two different phases the system can be spatially rescaled according to the typical linear dimension of the clusters and on this space scale regions of the opposite phases are separated by smooth hyper surfaces called interfaces the location of the interfaces is determined by means of the zero level set of the trajectories of a gaussian random field this paper is devoted primarily to the characterization of the structure which appears on a finer scale the hydrodynamical one at the interface a better understanding of the dynamics of the escape especially in its last and nonlinear stage leads to substantial improvements of the results in 7 12

stability and stabilization of switched linear dynamic systems with time delay and uncertainties this paper considers the problem of exponential stability and stabilization of switched linear time delay systems the system parameter uncertainties are time varying and unknown but norm bounded the delay in the system states is also time varying by using an improved lyapunov krasovskii functional a switching rule for the exponential stability and stabilization is designed in terms of the solution of riccati type equations the approach allows for computation of the bounds that characterize the exponential stability rate of the solution numerical examples are given to illustrate the results

multiclass data segmentation using diffuse interface methods on graphs we present two graph based algorithms for multiclass segmentation of high dimensional data on graphs the algorithms use a diffuse interface model based on the ginzburg landau functional related to total variation and graph cuts a multiclass extension is introduced using the gibbs simplex with the functional s double well potential modified to handle the multiclass case the first algorithm minimizes the functional using a convex splitting numerical scheme the second algorithm uses a graph adaptation of the classical numerical merriman bence osher mbo scheme which alternates between diffusion and thresholding we demonstrate the performance of both algorithms experimentally on synthetic data image labeling and several benchmark data sets such as mnist coil and webkb we also make use of fast numerical solvers for finding the eigenvectors and eigenvalues of the graph laplacian and take advantage of the sparsity of the matrix experiments indicate that the results are competitive with or better than the current state of the art in multiclass graph based segmentation algorithms for high dimensional data