a quasilinear parabolic system arising in modelling of catalytic reactors a system of four quasilinear parabolic equations arising in modelling of catalytic reactors is studied the system is coupled in a nonstandard way we prove that the system has unique global solution the asymptotic behavior as t is studied

approaching 3 2 for the s t path tsp we show that there is a polynomial time algorithm with approximation guarantee 3 2 e for the s t path tsp for any fixed e 0 it is well known that wolsey s analysis of christofide algorithm also works for the s t path tsp with its natural lp relaxation except for the narrow cuts in which the lp solution has a value less than two a fixed optimum tour has either a single edge in a narrow cut then call the edge and the cut lonely or at least three then call the cut busy our algorithm guesses by dynamic programming lonely cuts and edges then we partition the instance into smaller instances and strengthen the lp requiring a value of at least three for busy cuts by setting up a k stage recursive dynamic program we can compute a spanning tree v s and an lp solution y such that o 2 k y is in the t join polyhedron where t is the set of vertices whose degree in s has the wrong parity

fast rate of formation of dead core for the heat equation with strong absorption and applications to fast blow up we consider the dead core problem for the semilinear heat equation with strong absorption ut uxx up with 0 p 1 and positive boundary values we investigate the dead core rate i e the rate at which the solution reaches its first zero surprisingly we find that the dead core rate is faster than the one given by the corresponding ode this stands in sharp contrast with known results for the related extinction quenching and blow up problems moreover we find that the dead core rate is actually quite unstable the ode rate can be recovered if the absorption term is replaced by a t x up for a suitable bounded uniformly positive function a t x

parabolic minkowski convolutions of solutions to parabolic boundary value problems abstract we introduce a new kind of convolution which is a sort of parabolic version of the classical supremal convolution of convex analysis this operation allows us to compare solutions of different parabolic problems in different domains as examples of applications of our main result we study the parabolic concavity of solutions to parabolic boundary value problems analyzing in particular the case of heat equation with an inhomogeneous term and with a nonlinear reaction term we also apply our technique to the study of the dead core problem obtaining new results about necessary conditions for the existence of a dead core and estimates of the dead core time proving some optimality of the ball

stability radius and internal versus external stability in banach spaces an evolution semigroup approach in this paper the theory of evolution semigroups is developed and used to provide a framework to study the stability of general linear control systems these include autonomous and nonautonomous systems modeled with unbounded state space operators acting on banach spaces this approach allows one to apply the classical theory of strongly continuous semigroups to time varying systems in particular the complex stability radius may be expressed explicitly in terms of the generator of an evolution semigroup examples are given to show that classical formulas for the stability radius of an autonomous hilbert space system fail in more general settings upper and lower bounds on the stability radius are proven for banach space systems in addition it is shown that the theory of evolution semigroups allows for a straightforward operator theoretic analysis of internal stability as determined by classical frequency domain and input output operators even for nonautonomous banach space systems in particular for the nonautonomous setting internal stability is shown to be equivalent to input output stability for stabilizable and detectable systems for the autonomous setting an explicit formula for the norm of input output operator is given

mathematical analysis of a chemical reaction with lumped temperature and strong absorption abstract we consider a mathematical model for the evolution of a single reactant and of the temperature in an isothermal catalyst the temperature is assumed to be spatially homogeneous and the absorption term is supposed to be nonlipschitzian existence and uniqueness of a regular solution is proved together with some bounds the steady state problem is also investigated and information about the dead core of the reactant are given

the finite element approximation of a coupled reaction diffusion problem with non lipschitz nonlinearities a coupled semilinear elliptic problem modelling an irreversible isothermal chemical reaction is introduced and discretised using the usual piecewise linear galerkin finite element approximation an interesting feature of the problem is that a reaction order of less than one gives rise to a dead core region initially one reactant is assumed to be acting as a catalyst and is kept constant it is shown that error bounds previously obtained for a scheme involving numerical integration can be improved upon by considering a quadratic regularisation of the nonlinear term this technique is then applied to the full coupled problem and optimal h 1 and l infty error bounds are proved in the absence of quadrature for a scheme involving numerical integration bounds similar to those obtained for the catalyst problem are shown to hold

large time behavior of the solution of the heat equation with nonlinear strong absorption abstract we give an estimate for the asymptotic behavior of the solution of a semilinear heat equation and of its free boundary

existence and geometric properties of solutions of a free boundary problem in potential theory let 0 g h l rn n 2 be two given density functions at least one of them bounded away from zero outside a compact set and g holder continuous we prove that for any compactly supported positive measure which is sufficiently concentrated e g has sufficiently high n 1 dimensional density there exists a bounded open set rn such that the newtonian potential of the measure hln ghn 1 agrees with that of outside some regularity of is obtained as well as several results on the geometry of example if h and g are constant then for any x the inward normal ray of at x if it exists intersects the closed convex hull of typeset by ams tex 1 2 bjorn gustafsson and henrik shahgholian

a singular reaction diffusion system modelling prey predator interactions invasion and co extinction waves abstract we consider a singular reaction diffusion system arising in modelling prey predator interactions in a fragile environment since the underlying odes system exhibits a complex dynamics including possible finite time quenching one first provides a suitable notion of global travelling wave weak solution then our study focusses on the existence of travelling waves solutions for predator invasion in such environments we devise a regularized problem to prove the existence of travelling wave solutions for predator invasion followed by a possible co extinction tail for both species under suitable assumptions on the diffusion coefficients and on species growth rates we show that travelling wave solutions are actually positive on a half line and identically zero elsewhere such a property arising for every admissible wave speeds

study of the blow up set by transformation abstract we obtain some new results on blow up for the heat equation with nonlinear sources by transformation into an absorption problem our work is related to that of kawohl and peletier who first developed such a transformation in some special cases here we provide a systematic method for constructing a suitable transformation by using the class of regularly varying functions through regularization we obtain a precise characterization of the blow up set as zeros of a certain steady state

compactly supported solutions to stationary degenerate diffusion equations we consider non negative solutions of the semilinear elliptic equation in rn with n 3 u a x uq b x up where 0 q a x sign changing a a a and b x 0 is non positive under appropriate growth assumption on a at infinity we prove that all solutions in d1 2 rn are compactly supported and their support is contained in a large ball with radius determined by a when 0 x rn a x 0 has several compact connected components we give conditions under which there may or may not exist solutions which vanish identically on one or more of the components of 0 for instance we introduce a positive parameter and replace a by a a we then show that for small all solutions have compact support and there exist solutions with supports in any combination of these connected components of 0 for large and p 1 the solution is unique and supported in all of 0 we also prove the existence of the limit of this solution which solves w a wq and lim x w x 0 the analysis is based on comparison arguments and a priori bounds

dead cores in nonlinear reaction diffusion systems on considere le systeme elliptique semilineaire u f u g v x u a x v k f u g v x v b x ou est un domaine borne dans r n 0 k 0 a x 0 b x 0 et max x a x max x b x 1

a bifurcation problem governed by the boundary condition ii we deal with positive solutions of u a x up in a bounded smooth domain omega subset mathbb r n subject to the boundary condition u v u a parameter p 1 we prove that this problem has a unique positive solution if and only if 0 0 and coincides with the first eigenvalue of an associated eigenvalue problem moreover we find the limit profile of the solution as 1

numerical methods for reaction diffusion problems with non differentiable kinetics we consider a class of steady state semilinear reaction diffusion problems with non differentiable kinetics the analytical properties of these problems have received considerable attention in the literature we take a first step in analyzing their numerical approximation we present a finite element method and establish error bounds which are optimal for some of the problems in addition we also discuss a finite difference approach numerical experiments for one and two dimensional problems are reported

the dead core for reaction diffusion equations with convection and its connection with the first exit time of the related markov diffusion process on considere un probleme de reaction diffusion avec convection dans une region bornee d r n on demontre l existence d une solution

compact solitary waves in linearly elastic chains with non smooth on site potential it was recently observed by saccomandi and sgura that one dimensional chains with nonlinear elastic interaction and regular on site potential can support compact solitary waves i e travelling solitary waves with strictly compact support in this paper we show that the same applies to chains with linear elastic interaction and an on site potential which is continuous but non smooth at minima some different features arise in particular the speed of compact solitary waves is not uniquely fixed by the equation we also discuss several generalizations of our findings

symmetrization techniques on unbounded domains application to a chemotaxis system on rn the authors study the parabolic elliptic system on rn u t u u v 0 v v u u 0 u0 a version of the mathematical model of chemotaxis proposed by keller and segel a differential inequality for the quantity s0u t d where u is the decreasing rearrangement of the solution u t with respect to the spatial variable is obtained as a consequence they obtain e g lp bounds of the solution u v on r2 and global in time existence of solutions under the condition r2u0 8 then the solution u v blows up in a finite time compared to the previous work of diaz diaz and nagai adv math sci appl 5 1995 no 2 659 680 mr1361010 96j 35246 where this problem has been considered on bounded domains of rn there are some additional technical difficulties connected with the regularity of the derivative u t

a new approach for analysis and synthesis of time varying systems in this paper new techniques are developed for the analysis of linear time varying ltv systems these lead to a formally simple treatment of robust control problems for ltv systems allowing methods more usually restricted to time invariant systems to be employed in the time varying case as an illustration of this methodology the so called h sub spl infin synthesis problem is solved for ltv systems

a variational approach for a class of singular perturbation problems and applications on etudie la limite quand e 0 des minimiseurs globaux des fonctionnelles du type je u e 2 2 u 2 dx f x u dx ou est un anneau ou une boucle de r n

nonuniqueness in the quenching problem this paper deals with nonnegative solutions of for u t delta u u q chi u 0 qquad rm in omega times 0 infty qquad qquad rm q with q in 0 1 and prescribed continuous dirichlet data b b x on it is proved that for n 6 there is a critical parameter q c in 0 1 with the following property if q q c then there exist at least two continuous weak solutions emanating from some explicitly known stationary solution w one that coincides with w and another one that satisfies u w but u not equiv w for n 6 and q q c or n 7 however such a second solution above w is impossible moreover it is shown that for n 6 q q c and any sufficiently small nonnegative boundary data b there exist initial values admitting at least two continuous weak solutions of q the final result asserts that for any n and q nonuniqueness for q holds at least for some boundary and initial data

sharpl error estimates for semilinear elliptic problems with free boundaries a numerical scheme to approximate a semilinear pde involving a singular maximal monotone graph is analyzed inl a preliminary regularization is combined with piecewise linear finite elements defined on a triangulation which is not assumed to be acute the discrete maximum principle is thus avoided sharp pointwise error estimates are derived for both the smoothing and the discretization procedures an optimal choice of the regularization parameter as a function of the mesh size leads to a sharp global rate of convergence these error estimates for solutions in conjunction with nondegeneracy properties of continuous problems provide sharp interface error estimates two model examples are discussed the obstacle problem and a combustion equation

quenching phenomena in strongly degenerate diffusion equations with strong absorption abstract we examine the degenerate parabolic equation u t u p u xx u q u 0 with boundary value u 1 in a bounded real interval where p 1 and q 1 p 1 when investigating the evolution of the dead core set u t 0 we find that the two regimes q p and q 1 p show completely different behavior finally we rule out the possibility of quenching in infinite time under certain assumptions on p q and the size of

asymptotic properties of a semilinear heat equation with strong absorption and small diffusion in this paper the authors study the asymptotic behaviour of solutions ue x t of the cauchy problems as e goes to zero ut e u up 0 x rn t 0 u x 0 u0 x x rn 0 0 u x 0 u0 x x rn it is proved under certain assumptions that ue x t u x t as e 0 uniformly on compact subsets of rn 0 and moreover a precise estimate is given local and global estimates for extinction time are also given the proofs are somewhat technical

moving boundaries in reaction diffusion systems with absorption consider the reaction diffusion system t u d 0 xx u ku 1 v 0 0 t u d 1 xx v ku 1 v with boundary and initial conditions u 0 t c 0 u t 0 t 0 x v 0 t 0 v t 0 u x 0 u 0 x v x 0 v 0 x where 0 u 0 x c 0 and 0 v 0 x 1 for x 0

better s t tours by gao trees we consider the s t path tsp given a finite metric space with two elements s and t we look for a path from s to t that contains all the elements and has minimum total distance we improve the approximation ratio for this problem from 1 599 to 1 566 like previous algorithms we solve the natural lp relaxation and represent an optimum solution x as a convex combination of spanning trees gao showed that there exists a spanning tree in the support of x that has only one edge in each narrow cut i e each cut c with x c 2 our main theorem says that the spanning trees in the convex combination can be chosen such that many of them are such gao trees simultaneously at all sufficiently narrow cuts

existence and uniqueness of solutions of nonlinear neumann problems soit d r n un domaine connexe borne lisse et on suppose qu en chaque point x d la normale exterieure n est bien definie soit a une fonction arbitraire continue de holder dans d on etudie les solutions positives du probleme de neumann non lineaire u a x g u 0 dans d u n 0 sur d ou g r r est une fonction de classe c r c 1 r 0 1

minimum weight two connected spanning networks we consider the problem of constructing a minimum weight two connected network spanning all the points in a setv we assume a symmetric nonnegative distance functiond defined onv v which satisfies the triangle inequality we obtain a structural characterization of optimal solutions specifically there exists an optimal two connected solution whose vertices all have degree 2 or 3 and such that the removal of any edge or pair of edges leaves a bridge in the resulting connected components these are the strongest possible conditions on the structure of an optimal solution since we also show thatany two connected graph satisfying these conditions is theunique optimal solution for a particular choice of canonical distances satisfying the triangle inequality we use these properties to show that the weight of an optimal traveling salesman cycle is at most 4 3 times the weight of an optimal two connected solution examples are provided which approach this bound arbitrarily closely in addition we obtain similar results for the variation of this problem where the network need only span a prespecified subset of the points

branches of positive and free boundary solutions for some singular quasilinear elliptic problems we study the existence and multiplicity of solutions strictly positive or nonnegative having a free boundary the boundary of the set where the solution vanishes of some one dimensional quasilinear problems of eigenvalue type with possibly singular nonlinear terms

finite element approximation of a model reaction diffusion problem with a non lipschitz nonlinearity given 0 andp 0 1 we consider the following problem findu such that begin gathered delta u lambda u p 0in omega hfill u 1on partial omega hfill end gathered where 2 is a smooth convex domain we prove optimalh 1 andl error bounds for the standard continuous piecewise linear galerkin finite element approximation in addition we analyse a more practical approximation using numerical integration on the nonlinear term finally we consider a modified nonlinear sor algorithm which is shown to be globally convergent for solving the algebraic system derived from the more practical approximation

some new results for second order boundary value problems abstract this paper presents existence results for a variety of singular second order boundary value problems of the form y t y y 0 here our nonlinear term is allowed to be singular at y 0 t 0 and or ft 1 the methods used are the topological transversality theorem which is based on the notion of an essential map and the arzela ascoli theorem

convexity of solutions of semilinear elliptic equations on considere le probleme elliptique suivant u f u dans u m sur ou est un domaine convexe de r 2 et m est soit une constante reelle soit on etablit que pour une bonne fonction monotone g t pout toute solution u on a g u est strictement convexe dans

critical exponents and lower bounds of blow up rate for a reaction diffusion system abstract this paper is concerned with a reaction diffusion system with absorption terms under dirichlet boundary conditions modelling the cooperative interaction of two diffusion biological species by constructing blow up sub solutions and bounded super solutions we obtain the optimal conditions on the exponent of reaction and absorption terms for the existence or nonexistence of global solutions moreover for a special case we derive the lower bound estimates of blow up rate

the formation of the dead core in parabolic reaction diffusion problems on donne des conditions pour l existence d une region a concentration de reactant nulle pour des problemes paraboliques non lineaires de reaction diffusion

asymptotic behavior of the solutions of the dead core problem u u pup 0 1 1 this equation originates from the study of a class of models for the reaction diffusion processes of a gas inside a chemical reactor 2 81 from the mathematical point of view the major interest of equation 1 1 lies in the competition of the linear diffusion and the nonlinear fast p 0 t 0 with initial and boundary conditions

on starshaped rearrangement and applications a radial symmetrization technique is investigated and new properties are proven the method transforms functions u into new functions u with starshaped level sets and is therefore called starshaped rearrangement this rearrangement is in general not equimeasurable a circumstance with some surprising consequences the method is then applied to certain variational and free boundary problems and yields new results on the geometrical properties of solutions to these problems in particular the lipschitz continuity of free boundaries can now be easily obtained in a new fashion introduction in 26 g szego introduced the concept of radial symmetrization this is a nonequimeasurable rearrangement which transforms functions u q r q c r2 with starshaped level sets qc x e flu x c into functions u with symmetric starshaped level sets later 2 3 15 16 22 his approach was generalized to transform functions with level sets containing the origin into starshaped and not necessarily symmetric level sets and to higher dimensions we call a set d c rn starshaped with respect to xo if x e d implies tx 1 t xo e d for every t e 0 1 we call a set d starshaped if it is starshaped with respect to the origin therefore a more appropriate name for this rearrangement might be starshaped rearrangement in particular since the notion of radial symmetrization can be easily confused with circular or spherical symmetrization or spherically symmetric rearrangement one of the standard results in the theory of rearrangements is that the capacity of a condenser decreases under rearrangement i e j vu x lpdx 1 vu x ipdx for p 1 where u 1 on ql u 0 on 9q and u o c q c q e 0 0 j ivu p x ipdx for p 1 provided q c rn is a bounded starshaped domain containing ue 0 and u 0 1 u e c0 1 q u 1 in u 0 and u 0 on iq another convenient feature of other rearrangements is their equimeasurability a rearrangement u u is called equimeasurable if the n dimensional lebesgue received by the editors march 26 1984 and in revised form september 24 1985 1980 mathematics subject classification primary 35j20 35r35 35b05 26d20 lthis research was supported by the deutsche forschungsgemeinschaft dfg and by e reid 1986 american mathematical society 0002 9947 86 1 00 25 per page

the evolution of travelling waves in fractional order autocatalysis with decay ii the initial boundary value problem in this paper we examine an initial boundary value problem for a singular reaction diffusion equation arising from a model of isothermal chemical autocatalysis with termination we consider the case where the reaction and termination orders m and n respectively are fractional that is 0 0 k being a parameter which measures the relative strength of the termination step to that of the autocatalysis we show that solutions approach the equilibrium state 1 k for k 1 we demonstrate that the problem exhibits solutions which decay to zero with contracting support in finite t finally for m n we show that traveling waves with semi infinite support occ

on the initial growth of interfaces in reaction diffusion equations with strong absorption we study the initial growth of the interfaces of non negative local solutions of the equation u t u m xx lambdau q when m greater than or equal to 1 and 0 c0 for some explicit c0 c0 lambda m q then the free boundary zeta t sup x u x t 0 is a heating front more precisely zeta t greater than or equal to at m q 2 1 q for any t small enough and for some a 0 if on the contrary u x 0 less than or equal to c x 2 m q with c 0 applications to solutions of the associated cauchy and dirichlet problems are also given

approaching an extinction point in one dimensional semilinear heat equations with strong absorption abstract this paper deals with the cauchy problem ut uxx up 0 0 u x 0 u0 x 0 and u x t 0 appear when t is close to t we describe here all possible asymptotic behaviours of solutions and interfaces near an extinction point as the extinction time is approached we also give conditions under which some of these behaviours actually occur

the evolution of travelling waves in fractional order autocatalysis with decay i permanent form travelling waves this paper considers a model isothermal autocatalytic reaction scheme with termination when the reaction orders m n are fractional i e 0 n and k in 0 k c where kc depends upon m and n it is further established that these wavefronts are of excitable rather than fisher kolmogorov type and have only semi infinite support that is the reaction is totally dormant ahead of the wavefront these features are in contrast to the case when m n geq 1 as studied by merkin needham and scott proc roy soc london ser a 424 1989 pp 187 209 merkin and needham j engrg math 23 1989 pp 343 356 and needham and merkin phil trans roy soc london ser a 337 1991 pp 261 274 it should be noted that the corresponding reaction diffusion equations are singular in the sense that the reaction and decay terms are not lipsch

h control with output feedback for time varying discrete systems h problems with state feedback are considered for time varying discrete systems first quadratic games related to h problems are studied a necessary and sufficient condition for the existence of a stabilizing controller such that the norm of the operator from the disturbance to the output is less than a prescribed number is given in terms of a riccati equation

survivable networks linear programming relaxations and the parsimonious property we consider the survivable network design problem the problem of designing at minimum cost a network with edge connectivity requirements as special cases this problem encompasses the steiner tree problem the traveling salesman problem and thek edge connected network design problem we establish a property referred to as the parsimonious property of the linear programming lp relaxation of a classical formulation for the problem the parsimonious property has numerous consequences for example we derive various structural properties of these lp relaxations we present some algorithmic improvements and we perform tight worst case analyses of two heuristics for the survivable network design problem

regularity for the fully nonlinear dead core problem we establish new geometric regularity estimates for reaction diffusion equations with strong absorption terms the model is given by a fully nonlinear elliptic equation with measurable coefficients and a mu holder continuous convection term f x d 2u f u the lack of lipschitz regularity of the map u mapsto f u allows the existence of plateaus i e nonnegative solutions may vanish identically within an a priori unknown region the dead core of the solution we prove that at any touching ground point z in partial u 0 solutions are varkappa mu differentiable for a sharp value varkappa mu ge 2 and in fact varkappa 1 infty the proof is based on a new flatness improvement method we apply this new regularity estimate to establish a liouville type theorem for entire solutions to dead core problems and also to obtain measure estimates on the touching ground boundary the results obtained in this article are new even for dead core problems ruled by linear equations

robust recursive estimation in the presence of heavy tailed observation noise a first order approximation is derived for the conditional prior distribution of the state of a discrete time stochastic linear dynamic system in the presence of contaminated normal observation noise this distribution is then used to derive a first order approximation of the conditional mean minimum variance estimator if the observation noise distribution can be represented as a member of the e contaminated normal neighborhood then the conditional prior is also to first order an analogous perturbation from a normal distribution whose first two moments are given by the kalman filter moreover the perturbation is itself of a special form combining distributions whose parameters are given by banks of parallel kalman filters and optimal smoothers

error bounds for balanced model reduction of linear time varying systems error bounds are developed for balanced truncation of linear time varying systems leading to an extension of the twice the sum of the tail formulas well known in the time invariant case the approach relies on an operator theoretic framework for analysis of linear time varying systems this provides a multivariable notion of frequency for such systems which are thus characterized by rational functions of many complex variables allowing the problem to be formulated in the linear fractional framework using a time varying version of standard necessary conditions for reduced order modeling based on convex operator inequalities we show that these error bounds for balanced truncation are related to the closest possible reduced order modeling error in a sense which parallels the time invariant case

regularity for degenerate evolution equations with strong absorption abstract in this manuscript we study geometric regularity estimates for degenerate parabolic equations of p laplacian type 2 p under a strong absorption condition p u u t 0 u q in t 0 t where 0 q 1 this model is interesting because it yields the formation of dead core sets i e regions where non negative solutions vanish identically we shall prove sharp and improved parabolic c regularity estimates along the set f 0 u t u 0 t the free boundary where p p 1 q 1 1 p 1 some weak geometric and measure theoretical properties as non degeneracy positive density porosity and finite speed of propagation are proved as an application we prove a liouville type result for entire solutions a specific analysis for blow up type solutions will be done as well the results are new even for dead core problems driven by the heat operator

methods for stability evaluation for linear time varying discrete time systems on finite time horizon this paper develops a method for determining stability of discrete time dt linear time varying ltv systems defined on a finite time horizon fth in our considerations we use a collection of stability definitions for linear time invariant lti and ltv systems which are defined on an infinite time horizon ith based on the analysis carried out and the introduced operator matrix notation we define four stability functions these functions allow examination of the stability of a system described by an ltv state space model moreover using these functions we introduce the time stability margin and making use of the operator matrix notation we propose a method of determining the stability on a finite time horizon the theoretical considerations are numerically verified on two examples of ltv systems with a variable degree of non stationarity depending on a parameter e several examples illustrate the application of the introduced concepts and definitions to examination of the system stability also de

sublinear elliptic problems with a hardy potential abstract in this paper we study the positive solutions of sub linear elliptic equations with a hardy potential which is singular at the boundary by means of ode techniques a fairly complete picture of the class of radial solutions is given local solutions with a prescribed growth at the boundary are constructed by means of contraction operators some of those radial solutions are then used to construct ordered upper and lower solutions in general domains by standard iteration arguments the existence of positive solutions is proved an important tool is the hardy constant

the strong maximum principle revisited in this paper we rst present the classical maximum principle due to e hopf together with an extended commentary and discussion of hopf s paper we emphasize the comparison technique invented by hopf to prove this principle which has since become a main mathematical tool for the study of second order elliptic partial dierential equations and has generated an enormous number of important applications while hopf s principle is generally understood to apply to linear equations it is in fact also crucial in nonlinear theories such as those under consideration here in particular we shall treat and discuss recent generalizations of the strong maximum principle and also the compact support principle for the case of singular quasilinear ellip tic dierential inequalities under generally weak assumptions on the quasilinear operators and the nonlinearities involved our principal interest is in necessary and sucient con ditions for the validity of both principles in exposing and simplifying earlier proofs of corresponding results and in extending the conclusions to wider classes of singular oper ators than previously considered the results have unexpected rami cations for other problems as will develop from the exposition e g i two point boundary value problems for singular quasilinear ordinary dierential equations sections 3 4 ii the exterior dirichlet boundary value problem section 5 iii the existence of dead cores and compact support solutions i e dead cores at in nity section 7 iv euler lagrange inequalities on a riemannian manifold section 9 v comparison and uniqueness theorems for solutions of singular quasilinear dierential inequalities section 10 the case of p regular elliptic inequalities is briesy considered in section 11

parametric conditions for stability of reduced order linear time varying control systems abstract in this paper we derive in a single framework parametric conditions encompassing those for both local and global bibo stability of a linear multivariable discrete time reduced order time varying control system these conditions indicate that the system will be bibo stable if the norm of the system parameter error matrix is bounded by an l 1 function superimposed on an l function

internal and external stability of linear time varying systems linear finite dimensional time varying systems are studied state variable representations of systems with a bounded input bounded output stability property which are uniformly stabilizable and detectable are shown to have their associated homogeneous state variable systems exponentially stable

from the ginzburg landau model to vortex lattice problems we introduce a coulombian renormalized energy w which is a logarithmic type of interaction between points in the plane computed by a renormalization we prove various of its properties such as the existence of minimizers and show in particular using results from number theory that among lattice configurations the triangular lattice is the unique minimizer its minimization in general remains open

short time behaviour of a singular solution to the heat equation with absorption on considere l evolution aux temps petits de l interface qui apparait dans le probleme aux limites u t u xx u p u o t 1 t 0 u x o 0 x 0 pour p 1 on construit une solution interieure d ordre zero consistante avec une interface mobile ou u est zero

a semilinear pde with free boundary abstract we study the semilinear problem u x u q 1 x u q 1 in b 1 from a regularity point of view for solutions and the free boundary u 0 here b 1 is the unit ball 1 q 2 and are lipschitz our main results concern local regularity analysis of solutions and their free boundaries one of the main difficulties encountered in studying this equation is classification of global solutions in dimension two we are able to present a fairly good analysis of global homogeneous solutions and hence a better understanding of the behavior of the free boundary in higher dimensions the problem becomes quite complicated but we are still able to state partial results e g we prove that if a solution is close to one dimensional solution in a small ball then in an even smaller ball the free boundary can be represented locally as two c 1 regular graphs u 0 and u 0 tangential to each other it is noteworthy that the above problem in contrast to the case q 1 introduces interesting and quite challenging features that are not encountered in the case q 1 e g one obtains homogeneous global solutions that are not one dimensional this complicates the analysis of the free boundary substantially

improving christofides algorithm for the s t path tsp we present a deterministic 1 5 2 approximation algorithm for the s t path tsp for an arbitrary metric given a symmetric metric cost on n vertices including two prespecified endpoints the problem is to find a shortest hamiltonian path between the two endpoints hoogeveen showed that the natural variant of christofides algorithm is a 5 3 approximation algorithm for this problem and this asymptotically tight bound in fact has been the best approximation ratio known until now we modify this algorithm so that it chooses the initial spanning tree based on an optimal solution to the held karp relaxation rather than a minimum spanning tree we prove this simple but crucial modification leads to an improved approximation ratio surpassing the 20 year old ratio set by the natural christofides algorithm variant our algorithm also proves an upper bound of 1 5 2 on the integrality gap of the path variant held karp relaxation the techniques devised in this article can be applied to other optimization problems as well these applications include improved approximation algorithms and improved lp integrality gap upper bounds for the prize collecting s t path problem and the unit weight graphical metric s t path tsp