set inversion via interval analysis for nonlinear bounded error estimation abstract in the context of bounded error estimation one is interested in characterizing the set of all the values of the parameters to be estimated that are consistent with the data in the sense that the errors between the data and model outputs fall within prior bounds while the problem can be considered as solved when the model output is linear in the parameters the situation is far less advanced in the general nonlinear case in this paper the problem of nonlinear bounded error estimation is viewed as one of set inversion an original algorithm is proposed based upon interval analysis that makes it possible to characterize the feasible set for the parameters by enclosing it between internal and external unions of boxes the convergence of the algorithm is proved and the algorithm is applied to two test cases the results obtained are compared with those provided by signomial analysis

removable lateral singularities of semilinear parabolic pdes and besov capacities suppose 1 2 lis a second order elliptic differential operator in rdanddis a bounded smooth domain in rd let q r dand let be a compact set on the lateral boundary of q we prove that is a removable lateral singularity for the equationu lu u in q if and only if cap1 2 0 where cap stands for the besov capacity on the boundary

on the existence of positive solutions for semilinear elliptic equations with neumann boundary conditions we give sufficient conditions for the existence of positive solutions to some semilinear elliptic equations in unbounded lipschitz domainsd d d 3 having compact boundary with nonlinear neumann boundary conditions on the boundary ofd for this we use an implicit probabilistic representation schauder s fixed point theorem and a recently proved sobolev inequality forw 1 2 d special cases include equations arising from the study of pattern formation in various models in mathematical biology and from problems in geometry concerning the conformal deformation of metrics

a priori error analysis and spring arithmetic error analysis is defined by the following concern bounding the output variation of a nonlinear function with respect to a given variation of the input variables this paper investigates this issue in the framework of interval analysis the classical way of analyzing the error is to linearize the function around the point corresponding to the actual input but this method is local and not reliable both drawbacks can be easily circumvented by a combined use of interval arithmetic and domain splitting however because of the underlying linearization a standard interval algorithm leads to a pessimistic bound and even simply fails i e returns an infinite error in case of singularity we propose an original nonlinear approach where intervals are used in a more sophisticated way through the so called springs this new structure allows one to represent an infinite set of intervals constrained by their midpoints and their radius the output error is then calculated with a spring arithmetic in the same way as the image of a function is calculated with interval arithmetic our method is illustrated on two examples including an application of geopositioning

diffusion versus absorption in semilinear elliptic equations we study the limit behaviour of a sequence of singular solutions of a nonlinear elliptic equation with a strongly degenerate absorption term at the boundary of the domain we give sharp conditions on the rate of degeneracy in terms of the distance to the boundary in order the pointwise singularity not to propagate along the boundary

cone paths for the planar brownian snake for the brownian path valued process of le gall or brownian snake in bbb r 2 the times at which the process is a cone path are considered as a function of the size of the cone and the terminal position of the path the results show that the paths for the path valued process have local properties unlike those of a standard brownian motion

the brownian snake and solutions of δu u2 in a domain we investigate the connections between the path valued process called the brownian snake and nonnegative solutions of the partial differential equation u u2 in a domain of d in particular we prove two conjectures recently formulated by dynkin the first one gives a complete characterization of the boundary polar sets which correspond to boundary removable singularities for the equation u u2 the second one establishes a one to one correspondence between nonnegative solutions that are bounded above by a harmonic function and finite measures on the boundary that do not charge polar sets this correspondence can be made explicit by a probabilistic formula involving a special class of additive functionals of the brownian snake our proofs combine probabilistic and analytic arguments an important role is played by a new version of the special markov property which is of independent interest

a system of elliptic equations arising in chern simons field theory we prove the existence of topological vortices in a relativistic self dual abelian chem simons theory with two higgs particles and two gauge fields through a study of a coupled system of two nonlinear elliptic equations over r 2 we present two approaches to prove existence of solutions on bounded domains via minimization of an indefinite functional and via a fixed point argument we then show that we may pass to the full r 2 limit from the bounded domain solutions to obtain a topological solution in r 2

brief guaranteed parameter bounding for nonlinear models with uncertain experimental factors in the context of bounded error estimation it is customary to assume that the error between the model output and output data should lie between some known prior bounds in this paper it is also assumed that the factors characterizing the experiments that have been carried out e g measurement times are uncertain with known prior bounds an algorithm based on interval analysis is used to characterize the set of all values of the parameter vector to be estimated that are consistent with these hypotheses this is performed in a guaranteed way even when the model output is a nonlinear function of the parameters and factors characterizing the experiments

on optimal solution of interval linear equations for interval linear algebraic systems bf a x bf b we consider the problem of component wise evaluation of the set sigma exists exists bf a bf b a 1 b mid a in bf a b in bf b formed by all solutions of ax b when a and b vary independently in bf a and bf b respectively an iterative pss algorithm is introduced that computes optimal exact componentwise estimates of sigma exists exists and its convergence is proved under fairly general conditions on the interval system we introduce the concept of a sequentially guaranteeing algorithm as a reasonable compromise between the requirements for the interval result to be guaranteed and to be obtained in a practically acceptable time

boundary value problems with measures for elliptic equations with singular potentials we study the boundary value problem with radon measures for nonnegative solutions of lvu u vu 0 in a bounded smooth domain when v is a locally bounded nonnegative function introducing some specific capacity we give sufficient conditions on a radon measure on so that the problem can be solved we study the reduced measure associated to this equation as well as the boundary trace of positive solutions in appendix a a ancona solves a question raised by m marcus and l veron concerning the vanishing set of the poisson kernel of lv for an important class of potentials v

constraint reasoning based on interval arithmetic the tolerance propagation approach abstract interval constraint satisfaction interval labeling systems have traditionally been based on local waltz filtering techniques that cannot in general determine global solutions in contrast this paper documents a related technique tolerance propagation tp that generalizes the idea of numerical exact value propagation into interval propagation in tp consistency techniques based on the topology of the constraint net can be combined with techniques of interval arithmetic in a new fruitful way in particular by tp it is possible to determine global solutions for interval constraint satisfaction problems with arbitrary accuracy and without losing all attractions of simple local computations

semilinear fractional elliptic equations involving measures abstract we study the existence of weak solutions to e u g u in a bounded regular domain in r n n 2 which vanish in r n where denotes the fractional laplacian with 0 1 is a radon measure and g is a nondecreasing function satisfying some extra hypotheses when g satisfies a subcritical integrability condition we prove the existence and uniqueness of weak solution for problem e for any measure in the case where is a dirac measure we characterize the asymptotic behavior of the solution when g r r k 1 r with k supercritical we show that a condition of absolute continuity of the measure with respect to some bessel capacity is a necessary and sufficient condition in order e to be solved

nonlinear boundary value problems relative to harmonic functions abstract we study the problem of finding a function u verifying u 0 in under the boundary condition u n g u on where r n is a smooth domain n is the normal unit outward vector to is a measure on and g a continuous nondecreasing function we give sufficient condition on g for this problem to be solvable for any measure when g r r p 1 r p 1 we give conditions in order an isolated singularity on to be removable we also give capacitary conditions on a measure in order the problem with g r r p 1 r to be solvable for some we also study the isolated singularities of functions satisfying u 0 in and u n g u 0 on 0

the essence of constraint propagation we show that several constraint propagation algorithms also called local consistency consistency enforcing waltz filtering or narrowing algorithms are instances of algorithms that deal with chaotic iteration to this end we propose a simple abstract framework that allows us to classify and compare these algorithms and to establish in a uniform way their basic properties

on a class of semilinear fractional elliptic equations involving outside dirac data abstract the purpose of this article is to give a complete study of the weak solutions of the fractional elliptic equation 0 1 u u p 0 in b 1 e n u 0 in r n b 1 e n where p 0 with 0 1 denotes the fractional laplacian operator in the principle value sense b 1 e n is the unit ball centered at e n 0 0 1 in r n with n 2 and 0 is the dirac mass concentrated at the origin we prove that problem eq 0 1 admits a unique weak solution when p 1 2 n moreover if in addition p n 2 n 2 the weak solution vanishes as 1 we also show that problem eq 0 1 does not have any weak solution when p 0 1 2 n these results are very surprising since they are in total contradiction with the classical setting i e u u p 0 in b 1 e n u 0 in r n b 1 e n for which it has been proved that there are no solutions for p n 1 n 1

le nombre de solutions de certains problemes semi lineaires elliptiques abstract in this paper we compute the exact number of solutions for certain semi linear elliptic problems in the first part we derive some complementary results to a theorem of ambrosetti and prodi the global behaviour of bifurcating branches is then studied for some classes of problems in the context of a concave or a convex nonlinearity in the second part we consider the branch of positive solutions lastly in the third part in dimension one for an autonomous equation we obtain a precise description of all the solutions to these problems the proofs rely on simple eigenvalue comparison arguments

branching particle systems and superprocesses we start from a model of a branching particle system with immigration and with death rate and branching mechanism depending on time and location then we consider a limit case when the mass of particles and their life times are small and their density is high this way we construct a measure valued process x t which we call a superprocess replacing the underlying markov process xi t by the corresponding historical process xi leq t we construct a measure valued process m t in functional spaces which we call a historical superprocess the moment functions for superprocesses are evaluated linear positive additive functionals are studied they are used to construct a continuous analog of a random tree obtained by stopping every particle at a time depending on its path say at the first exit time from a domain a related special markov property for superprocesses is proved which is useful for applications to certain nonlinear partial differential equations the concluding section is devoted to a survey of the literature and the terminology on markov processes used in the paper is explained in the appendix

on anisotropic singularities for semi linear elliptic equations in r2 abstract in this note we study weak solutions of 1 u g u 8 0 x 2 in r 2 g u x 0 as x where g s 2 e s 1 e s 1 is an odd increasing and bounded function and 0 x 2 is defined in the sense of distribution 0 x 2 0 x 2 for c c 1 r 2 we obtain that problem 1 admits a unique x 2 odd weak solution which has anisotropic singularity 4 x 2 x 2 near the origin

schrodinger operators with leray hardy potential singular on the boundary we study the kernel function of the operator u l u u x 2 u in a bounded smooth domain r n such that 0 where n 2 4 is a constant we show the existence of a poisson kernel vanishing at 0 and a singular kernel with a singularity at 0 we prove the existence and uniqueness of weak solutions of l u 0 in with boundary data k 0 where is a radon measure on 0 k r and show that this boundary data corresponds in a unique way to the boundary trace of positive solution of l u 0 in

continuous state feedback guaranteeing uniform ultimate boundedness for uncertain dynamic systems we consider a dynamic system containing uncertain elements only the set of possible values of these uncertainties is known based on this information a class of state feedback controls is proposed in order to guarantee uniform ultimate boundedness of every system response within an arbitrarily small neighborhood of the zero state these feedback controls are continuous functions of the state

on states of exit measures for superdiffusions we consider the exit measures of l superdiffusions 1 2 from a bounded smooth domain d in r d by using analytic results about solutions of the corresponding boundary value problem we study the states of the exit measures abraham and le gall investigated earlier this problem for a special case l and 2 also as an application of these analytic results we give a different proof for the critical hausdorff dimension of boundary polarity established earlier by le gall under more restrictive assumptions and by dynkin and kuznetsov for general situations

regular transition functions and regular superprocesses the class of regular markov processes is very close to the class of right processes studied by meyer getoor and others we say that a transition function p is regular if it is the transition function of a well defined regular markov process a characterization of regular transition functions is given which implies that if p is regular then the dawson watanabe and the flemingviot supertransition functions over p belong to the same class

boundary singularities of solutions to elliptic viscous hamilton jacobi equations abstract we study the boundary value problem with measures for e1 u g u 0 in a bounded domain in r n satisfying e2 u on and prove that if g l 1 1 t 2 n 1 n d t is nondecreasing e1 e2 can be solved with any positive bounded measure when g r r q with q 1 we prove that any positive function satisfying e1 admits a boundary trace which is an outer regular borel measure not necessarily bounded when g r r q with 1 q q c n 1 n we prove the existence of a positive solution with a general outer regular borel measure as boundary trace and characterize the boundary isolated singularities of positive solutions when g r r q with q c q 2 we prove that a necessary condition for solvability is that must be absolutely continuous with respect to the bessel capacity c 2 q q q we also characterize boundary removable sets for moderate and sigma moderate solutions

brief paper stability and resonance conditions of elementary fractional transfer functions elementary fractional transfer functions are studied in this paper some basic properties of elementary transfer functions of the first kind are recalled then two main results are presented regarding elementary fractional transfer functions of the second kind written in a canonical form and characterized by a commensurate order a pseudo damping factor and a natural frequency first stability conditions are established in terms of the pseudo damping factor and the commensurate order as a corollary to matignon s stability theorem they extend the previous result into conditions that are simpler to check then resonance conditions are established numerically in terms of the commensurate order and the pseudo damping factor and give interesting information on the frequency behavior of fractional systems it is shown that elementary transfer functions of the second kind might have up to two resonant frequencies moreover three abaci are given allowing to determine the pseudo damping factor and the commensurate order for respectively a desired normalized gain at each resonance a desired phase at each resonance and a desired normalized first or second resonant frequency

semilinear fractional elliptic equations with gradient nonlinearity involving measures we study the existence of solutions to the fractional elliptic equation e1 delta alpha u epsilon g nabla u nu in a bounded regular domain omega of r n n ge2 subject to the condition e2 u 0 in omega c where epsilon 1 or 1 delta alpha denotes the fractional laplacian with alpha in 1 2 1 nu is a radon measure and g r mapsto r is a continuous function we prove the existence of weak solutions for problem e1 e2 when g is subcritical furthermore the asymptotic behavior and uniqueness of solutions are described when nu is dirac mass g s s p p geq 1 and epsilon 1

on the equation δu eu 1 0 with measures as boundary data if is a bounded domain in mathbb r n we study conditions on a radon measure on for solving the equation u e u 1 0 in with u on the conditions are expressed in terms of orlicz capacities

the dirichlet elliptic problem involving regional fractional laplacian in this paper we study the solutions of elliptic equations involving regional fractional laplacian e u f in a bounded regular domain in rn n 2 with c2 boundary subject to dirichlet boundary g on where 12 1 and the operator denotes the regional fractional laplacian we prove that when g 0 problem e admits a unique weak solution under the hypotheses that f l2 f l1 dx and f m where x dist x 2 1 and m is a space of all radon measures satisfying d finally we provide an integration by parts formula for the classical solution of e with boundary data g

singular solutions of some nonlinear elliptic equations abstract we give a classification of isolated singularities of any solution of the equation u u u q 1 0 if 0 is the singular point we prove that there exist two types of singularities when 1 n n 2 and that we have essentially the following in 0 i either u x l q n x 2 q 1 with l q n 2 q 1 2q q 1 n 1 q 1 ii or u x c x where c is any constant and the fundamental solution of the laplacian in r n

some properties of the exit measure for super brownian motion we consider the exit measure of super brownian motion with a stable branching mechanism of a smooth domain d of d we derive lower bounds for the hitting probability of small balls for the exit measure and upper bounds in the critical dimension this completes results given by sheu 22 and generalizes the results of abraham and le gall 2 because of the links between exits measure and partial differential equations those results imply bounds on solutions of elliptic semi linear pde we also give the hausdorff dimension of the support of the exit measure and show it is totally disconnected in high dimension eventually we prove the exit measure is singular with respect to the surface measure on d in the critical dimension our main tool is the subordinated brownian snake introduced by bertoin le gall and le jan 4

hardy sobolev inequalities with singularities on non smooth boundary hardy constant and extremals part i influence of local geometry abstract let be a domain of r n n 3 the classical caffarelli kohn nirenberg inequality rewrites as the following inequality for any s 0 2 and any n 2 2 4 there exists a constant k s 0 such that h s u 2 s x s d x 2 2 s k s u 2 u 2 x 2 d x for all u d 1 2 the completion of c c for the relevant norm when 0 is an interior point the range n 2 2 4 for cannot be improved moreover the optimal constant k s is independent of and there is no extremal for h s but when 0 the situation turns out to be drastically different since the geometry of the domain impacts the range of s for which h s holds the value of the optimal constant k s the existence of extremals for h s when is smooth the problem was tackled by ghoussoub robert 2017 where the role of the mean curvature was central in the present paper we consider nonsmooth domain with a singularity at 0 modeled on a cone we show how the local geometry induced by the cone around the singularity influences the value of the hardy constant on when is small we introduce a new geometric object at the conical singularity that generalizes the mean curvature this allows to get extremals for h s the case of larger values for will be dealt in the forthcoming paper cheikh ali 2018 as an intermediate result we prove the symmetry of some solutions to singular pdes that has an interest on its own

super brownian motion path properties and hitting probabilities sample path properties of super brownian motion including a one sided modulus of continuity and exact hausdorff measure function of the range and closed support are obtained analytic estimates for the probability of hitting balls lead to upper bounds on the hausdorff measure of the set of k multiple points and a sufficient condition for a set to be polar

boundary singularities of solutions of semilinear elliptic equations with critical hardy potentials we study the boundary behaviour of positive functions u satisfying e gd u frac xk d 2 x u g u 0 in a bounded domain gw of bbr n where 0 xk leq frac 1 4 g is a continuous nonndecreasing function and d is the distance function to prt gw we first construct the martin kernel associated to the the linear operator cl xk gd frac xk d 2 x and give a general condition for solving equation e with any radon measure gm for boundary data when g u u q 1 u we show the existence of a critical exponent q c q c n xk 1 whith the following properties when 0 q q c any measure is eligible for solving e with gm for boundary data if q geq q c a necessary and sufficient condition is expressed in terms of the absolute continuity of gm with respect to some besov capacity the same capacity characterizes the removable compact boundary sets at end any positive solution f gd u frac xk d 2 x u u q 1 u 0 with q 1 admits a boundary trace which is a positive outer regular borel measure when 1 q q c we prove that to any positive outer regular borel measure we can associate a positive solutions of f with this boundary trace

path processes and historical superprocesses a superprocessx over a markov process can be obtained by a passage to the limit from a branching particle system for which describes the motion of individual particles the historical process hat xi for is the process whose state at timet is the path of over time interval 0 t the superprocess hat x over hat xi the historical superprocess over reflects not only the particle distribution at any fixed time but also the structure of family trees the principal property of a historical process hat xi is that hat xi s is a function of hat xi t for alls t every process with this property is calleda path process we develop a theory of superprocesses over path processes whose core is the integration with respect to measure functionals by applying this theory to historical superprocesses we construct the first hitting distributions and prove a special markov property for superprocesses

the boundary trace and generalized boundary value problem for semilinear elliptic equations with coercive absorption in the first part of the paper we establish the existence of a boundary trace for positive solutions of the equation u g x u 0 in a smooth domain n for a general class of positive nonlinearities this class includes every space independent monotone increasing g which satisfies the keller osserman condition as well as degenerate nonlinearities g q of the form g q x u d x u q 1u with 2 and q 1 the boundary trace is given by a positive regular borel measure which may blow up on compact sets in the second part we concentrate on the family of nonlinearities g q determine the critical value of the exponent q for fixed 2 and discuss a positive solutions with an isolated singularity for subcritical nonlinearities and b the boundary value problem for u g q x u 0 with boundary data given by a positive regular borel measure possibly unbounded we show that in the subcritical case the problem possesses a unique solution for every such measure 2003 wiley periodicals inc

isolated singularities of some semilinear elliptic equations abstract let be an open subset of r n n 3 containing 0 we consider the solutions of u x g u x f x in 0 where g is nondecreasing and f is bounded and we study the possible singularities at 0 when u x o x 1 n we prove that u is isotropic near 0 and show that either it is a c1 function in removable singularity or x n 2 u x c c 0 weak singularity or x n 2 u x strong singularity we also characterize the g s for which solutions with a weak singularity exist and improve a previous removability result of h brezis and l veron arch rational mech anal 23 1979 153 166

stabilization of nonlinear large scale uncertain dynamical systems abstract the notions of ultimate stabilizability and asymptotical stabilizability by local feedback strategies for nonlinear large scale systems are introduced under some structural conditions on the interconnected subsystems we obtain the ultimate stabilizability and the asymptotical stabilizability of the system by local feedback strategies

superdiffusions and parabolic nonlinear differential equations we establish connections between superdiffusion processes and one class of nonlinear parabolic differential equations analytic results due to brezis and friedman baras and pierre and others are used to investigate the graphs of superdiffusions a survey of the literature and general comments are presented in section 4

on differential pursuit games a nonlinear differential pursuit game with a general closed terminal set is studied notions of various terminabilities are introduced these notions include the notions of asymptotic stability stabilizability and reachability of control systems by using lyapunov s direct method some sufficient conditions for these terminabilities of the game are obtained as a special case a linear differential pursuit game with the terminal set as a subspace is discussed as well

on the existence of weak solutions of semilinear elliptic equations and systems with hardy potentials abstract let r n n 3 be a bounded c 2 domain and x dist x put l 2 with 0 in this paper we provide various necessary and sufficient conditions for the existence of weak solutions to l u u p in u on where 0 p 0 and are measures on and respectively we then establish existence results for the system l u v p in l v u p in u v on where 1 p 0 p 0 and are measures on and are measures on we also deal with elliptic systems where the nonlinearities are more general

thick set inversion abstract this paper deals with the set inversion problem x f 1 y in the case where the function f r n r m and the set y are both uncertain the uncertainty is treated under the form of intervals more precisely for all x f x is inside the box f x and the uncertain set y is bracketed between an inner set y and an outer set y the introduction of new tools such as thick intervals and thick boxes will allow us to propose an efficient algorithm that handles the uncertainty of sets in an elegant and efficient manner some elementary test cases that cannot be handled easily and properly by existing methods show the efficiency of the approach

sur la mesure de sortie du super mouvement brownien we study some properties of the exit measure of super brownian motion from a smooth domaind inr d in particular we give precise estimates for the probability that the exit measure gives a positive mass to a small ball on the boundary as an application we compute the hausdorff dimension of the support of the exit measure in dimension 2 we prove that the exit measure is absolutely continuous with respect to the lebesgue measure on the boundary in connection with dynkin s work our results give some information on the behavior of solutions of u u 2 ind and are related to the characterization of removable singularities at the boundary as a consequence of our estimates we give a sufficient condition for the uniqueness of the positive solution of u u 2 ind that tends to on an open subseto of d and to 0 on the complement in d of the closure ofo our proofs use the path valued process studied in l2 l3

a probabilistic approach to one class of nonlinear differential equations we establish connections between positive solutions of one class of nonlinear partial differential equations and hitting probabilities and additive functionals of superdiffusion processes as an application we improve results on superprocesses by using the recent progress in the theory of removable singularities for differential equations

reduced measures on the boundary we study the existence of solutions of the nonlinear problem delta u g u 0 in omega u mu on partial derivative omega where mu is a and g r r is a nondecreasing continuous function with g t 0 for all t 0 problem 0 1 admits a solution for every mu epsilon l 1 partial derivative omega but this need not be the case when p is a general bounded measure we introduce a concept of reduced measure mu in the spirit of brezis et al ann math stud to appear this is the closest measure to mu for which 0 1 admits a solution

on the supports of measure valued critical branching brownian motion on etudie la taille globale et la localisation des supports des etats du processus de markov a valeur de mesure qui est un analogue du mouvement brownien ramifie critique classique sur l espace euclidien a d dimensions

trace on the boundary for solutions of nonlinear differential equations let l be a second order elliptic differential operator in rd with no zero order terms and let e be a bounded domain in rd with smooth boundary e we say that a function h is l harmonic if lh 0 in e every positive l harmonic function has a unique representation h x j x y i dy e where k is the poisson kernel for l and z is a finite measure on e we call z the trace of h on me our objective is to investigate positive solutions of a nonlinear equation lu u in e for 1 2 we associate with every solution u a pair if i where f is a closed subset of e and zis a radon measure on 0 e f we call f zi the trace of u on me f is empty if and only if u is dominated by an l harmonic function we call such solutions moderate a moderate solution is determined uniquely by its trace in general many solutions can have the same trace we establish necessary and sufficient conditions for a pair f z to be a trace and we give a probabilistic formula for the maximal solution with a given trace

an elliptic semilinear equation with source term and boundary measure data the supercritical case we give new criteria for the existence of weak solutions to equation with source term begin align delta u u q text in omega u sigma text on partial omega end align where q 1 omega is a either a bounded smooth domain or mathbb r n and sigma in mathfrak m partial omega is a nonnegative radon measure on partial omega in particular one of the criteria is expressed in terms of some bessel capacities on partial omega we also give a sufficient condition for the existence of weak solutions to equation with source mixed term begin align delta u u q 1 1 u nabla u q 2 text in omega u sigma text on partial omega end align where q 1 q 2 geq 0 q 1 q 2 1 q 2 2 sigma in mathfrak m partial omega is a radon measure on partial omega

a new class of stabilizing controllers for uncertain dynamical systems this paper is concerned with the problem of designing a stabilizing controller for a class of uncertain dynamical systems the vector of uncertain parameters q cdot is time varying and its values q t lie within a prespecified bounding set q in r p furthermore no statistical description of q cdot is assumed and the controller is shown to render the closed loop system practically stable in a so called guaranteed sense that is the desired stability properties are assured no matter what admissible uncertainty q cdot is realized within the perspective of previous research in this area this paper contains one salient feature the class of stabilizing controllers which we characterize is shown to include linear controllers when the nominal system happens to be linear and time invariant in contrast in much of the previous literature see for example 1 2 7 and 9 a linear system is stabilized via nonlinear control another feature of this paper is the fact that the

very weak solutions with boundary singularities for semilinear elliptic dirichlet problems in domains with conical corners let rn be a bounded lipschitz domain with a cone like corner at 0 we prove existence of at least two positive unbounded very weak solutions of the problem u up in u 0 on which have a singularity at 0 for any p slightly bigger that the generalized brezis turner exponent p on an example of a planar polygonal domain the actual size of the p interval on which the existence result holds is computed the solutions are found variationally as perturbations of explicitly constructed singular solutions in cones this approach also makes it possible to find numerical approximations of the two very weak solutions on following a gradient flow of a suitable functional and using the mountain pass algorithm two dimensional examples are presented

removable singularities for lu ψ u and orlicz capacities suppose l is a second order elliptic differential operator in rd and let 1 baras and pierre have proved in 1984 that is removable for lu u if and only if its bessel capacity cap2 0 we extend this result to a general equation lu u where u is an increasing convex function subject to 2 and 2 conditions namely we prove that is removable for lu u if and only if its orlicz capacity is zero that is the integral b dx x y 2 d dy is equal to 0 or for every measure concentrated on where b stands for any ball containing

constraint propagation with interval labels abstract constraint propagation is often used in ai systems to perform inference about quantities this paper studies one particular kind of constraint propagation where quantities are labelled with signs or with intervals and these labels are propagated through recorded constraints we review the uses of such inference schemes in ai systems of various kinds and evaluate their strengths and weaknesses in particular we determine the completeness and running time of constraint propagation for various kinds of labels and constraints

nonlinear parabolic p d e and additive functionals of superdiffusions suppose that e is an arbitrary domain in r d l is a second order elliptic differential operator in s r x e and s e is the extremal part of the martin boundary for the corresponding diffusion let 1 2 we investigate a boundary value problem u r lu u in s u v on s e u 0 on x e involving two measures and for the existence of a solution we give sufficient conditions in terms of a martin capacity and necessary conditions in terms of hitting probabilities for an l superdiffusion x if a solution exists then it can be expressed by an explicit formula through an additive functional a of x an l superdiffusion is a branching measure valued process a natural linear additive nla functional a of x is determined uniquely by its potential h defined by the formula p a 0 h r x dr dx for all m the determining set of a every potential h is an exit rule for and it has a unique decomposition into extremal exit rules if and are measures which appear in this decomposition then can be replaced by an integral equation u r x p r x t dy u t y ds h r x where p r x t dy is the transition function of we prove that h is the potential of a nla functional if and only if has a solution u moreover u r x log p r x e a 0 by applying these results to homogeneous functionals of time homogeneous superdiffusions we get a stronger version of theorems proved in an earlier publication the foundation for our present investigation is laid by a general theory developed in the accompanying paper

semilinear elliptic equations with subcritical absorption in lipschitz domains abstract let be a bounded lipschitz domain in r n consider the equation u g x u 0 g c r and g x positive and increasing on r x we say that g is subcritical at y if has a solution u k y with boundary trace k y k 0 for a large family of functions g we establish existence and stability results for boundary value problems for with data given by measures concentrated on the set of subcritical points in addition we describe the precise asymptotic behavior of u k y at y some related results have been obtained in marcus and veron 2011 when g t t q in the case that g satisfies the keller osserman condition we prove if u is a positive solution with strong singularity at y i e u is not dominated by a harmonic function then u lim k u k y finally we extend estimates related to the keller osserman condition that are well known in the case of c 2 domains when g is independent of the space variable to lipschitz domains and a large class of functions g including cases where g x t 0 or g x t as x

on the evadable sets of differential evasion games abstract the notions of evasion strategies and evadable sets are introduced for a nonlinear differential game with information delay and results about the evadable sets and the strict evadability of the game are obtained a linear case is treated at the end