existence of sign changing solutions for a nonlinear elliptic problem on rn with a hardy term abstract let n 3 1 p n 2 n 2 and 0 n 2 2 4 we study the existence of multiple sign changing solutions of u h 1 r n u u u x 2 u p 1 u in r n which are not radially symmetric

stochastic boundary value problems a white noise functional approach we give a program for solving stochastic boundary value problems involving functionals of multiparameter white noise as an example we solve the stochastic schrodinger equation ie391 1 wherev is a positive noisy potential we represent the potentialv by a white noise functional and interpret the product of the two distribution valued processesv andu as a wick productv u such an interpretation is in accordance with the usual interpretation of a white noise product in ordinary stochastic differential equations the solutionu will not be a generalized white noise functional but can be represented as anl 1 functional process

uniqueness and global markov property for euclidean fields the case of general polynomial interactions we give a general method for proving uniqueness and global markov property for euclidean quantum fields the method is based on uniform continuity of local specifications proved by using potential theoretical tools and exploitation of a suitable fkg order structure we apply this method to give a proof of uniqueness and global markov property for the gibbs states and to study extremality of gibbs states also in the case of non uniqueness in particular we prove extremality for 2 4 also in the case of non uniqueness and global markov property for weak coupling 2 4 which solves a long standing problem uniqueness and extremality holds also at any point of differentiability of the pressure with respect to the external magnetic field

stochastic calculus with anticipating integrands we study the stochastic integral defined by skorohod in 24 of a possibly anticipating integrand as a function of its upper limit and establish an extended ito formula we also introduce an extension of stratonovich s integral and establish the associated chain rule in all the results the adaptedness of the integrand is replaced by a certain smoothness requirement

solving maximum entropy sampling problems using factored masks we present a practical approach to anstreicher and lee s masked spectral bound for maximum entropy sampling and we describe favorable results that we have obtained with a branch and bound algorithm based on our approach by representing masks in factored form we are able to easily satisfy a semidefiniteness constraint moreover this representation allows us to restrict the rank of the mask as a means for attempting to practically incorporate second order information

spectral analysis of fractional kinetic equations with random data we present a spectral representation of the mean square solution of the fractional kinetic equation also known as fractional diffusion equation with random initial condition gaussian and non gaussian limiting distributions of the renormalized solution of the fractional in time and in space kinetic equation are described in terms of multiple stochastic integral representations

positive solutions for elliptic equations involving critical sobolev exponents and hardy terms with neumann boundary conditions abstract in this paper neumann problem for nonlinear elliptic equations with critical sobolev exponents and hardy terms is studied by variational method based on the variant of the mountain pass theorem of ambrosetti and rabinowitz without ps condition we prove the existence of positive solutions

scaling limits of solutions of the heat equation for singular non gaussian data limiting distributions of the parabolically rescaled solutions of the heat equation with singular non gaussian initial data with long range dependence are described in terms of their multiple stochastic integral representations

generalized stochastic integrals and the malliavin calculus the paper first reviews the skorohod generalized stochastic integral with respect to the wiener process over some general parameter space t and it s relation to the malliavin calculus as the adjoint of the malliavin derivative some new results are derived and it is shown that every sufficiently smooth process ut t t can be decomposed into the sum of a malliavin derivative of a wiener functional and a process whose generalized integral over t vanishes using the results on the generalized integral the bismut approach to the malliavin calculus is generalized by allowing non adapted variations of the wiener process yielding sufficient conditions for the existence of a density which is considerably weaker than the previously known conditions

interaction between the geometry of the boundary and positive solutions of a semilinear neumann problem with critical nonlinearity abstract we consider the problem u u un 2 n 2 u 0 in u v 0 on where is a bounded smooth domain in r n n 3 we show that for large least energy solutions of the above problem have a unique maximum point p on and the limit points of p as are contained in the set of the points of maximum mean curvature we also prove that if has k peaks then the equation has at least k solutions for large

an exact algorithm for maximum entropy sampling we study the experimental design problem of selecting a most informative subset having prespecified size from a set of correlated random variables the problem arises in many applied domains such as meteorology environmental statistics and statistical geology in these applications observations can be collected at different locations and possibly at different times information is measured by entropy in the gaussian case the problem is recast as that of maximizing the determinant of the covariance matrix of the chosen subset we demonstrate that this problem is np hard we establish an upper bound for the entropy based on the eigenvalue interlacing property and we incorporate this bound in a branch and bound algorithm for the exact solution of the problem we present computational results for estimated covariance matrices that correspond to sets of environmental monitoring stations in the united states

positive solutions for neumann elliptic problems involving critical hardy sobolev exponent with boundary singularities abstract in this paper we consider the existence of positive solution for some semilinear elliptic equations with neumann boundary condition involving a critical hardy sobolev exponent and hardy terms with boundary singularities using mountain pass lemma without ps condition and the strong maximum principle we get the existence of a positive solution

a linear integer programming bound for maximum entropy sampling we introduce a new upper bound for the maximum entropy sampling problem our bound is described as the solution of a linear integer program the bound depends on a partition of the underlying set of random variables for the case in which each block of the partition has bounded cardinality we describe an efficient dynamic programming algorithm to calculate the bound for the very special case in which the blocks have no more than two elements we describe an efficient algorithm for calculating the bound based on maximum weight matching this latter formulation has particular value for local search procedures that seek to find a good partition we relate our bound to recent bounds of hoffman lee and williams finally we report on the results of some computational experiments

generalized brownian functionals and the solution to a stochastic partial differential equation abstract a class of generalized wiener functionals related to those of hida and watanabe is introduced and the malliavin calculs associated with these functionals is developed these notions are applied to the derivation of a solution to the stochastic partial differential equation y t ly y where l is a second order partial differential operator in the n space variables denotes a white noise in the n 1 space time variables y denotes a skorohod type stochastic integral and is a non random function of the n 1 independent variables

stochastic heat equation driven by fractional noise and local time the aim of this paper is to study the d dimensional stochastic heat equation with a multiplicative gaussian noise which is white in space and has the covariance of a fractional brownian motion with hurst parameter h 0 1 in time two types of equations are considered first we consider the equation in the ito skorohod sense and later in the stratonovich sense an explicit chaos expansion for the solution is obtained on the other hand the moments of the solution are expressed in terms of the exponential moments of some weighted intersection local time of the brownian motion

weighted stochastic sobolev spaces and bilinear spdes driven by space time white noise in this paper we develop basic elements of malliavin calculus on a weightedl2 this class of generalized wiener functionals is a hilbert space it turns out to be substantially smaller than the space of hida distributions while large enough to accommodate solutions of bilinear stochastic pdes as an example we consider a stochastic advection diffusion equation driven by space time white noise in rd it is known that ford 1 this equation has no solutions inl2 in contrast it is shown in the paper that in an appropriately weightedl2 there is a unique solution to the stochastic advection diffusion equation for anyd 1 in addition we present explicit formulas for the hermite fourier coefficients in the wiener chaos expansion of the solution

multiple positive solutions for robin problem involving critical weighted hardy sobolev exponents with boundary singularities abstract this paper deals with the existence of positive solutions for robin elliptic problems involving critical weighted hardy sobolev exponents with boundary singularities using the caffarelli kohn nirenberg inequalities and variational methods we prove the existence and multiplicity of positive solutions

existence and multiplicity of positive solutions for elliptic equation with critical weighted hardy sobolev exponents and boundary singularities abstract we study a semilinear elliptic equation involving critical weighted hardy sobolev exponents with boundary singularities the existence and multiplicity of positive solutions are established our method relies upon ekeland s variational principle and mountain pass lemma

existence and multiplicity of positive solutions for some hardy sobolev critical elliptic equation with boundary singularities abstract in this paper by ekeland s variational principle and strong maximum principle we consider the existence and multiplicity of positive solutions for some semilinear elliptic equation involving critical hardy sobolev exponents and hardy terms with boundary singularities

the effect of domain topology on the number of positive solutions for singular elliptic problems involving the caffarelli kohn nirenberg inequalities abstract in this paper we consider the critical singular equation involving the caffarelli kohn nirenberg inequalities of the type p 1 div x 2 a u u x 2 1 a u q 2 u x d d u p 2 u x b p x u 0 x u 0 x here is a bounded domain with smooth boundary in r n and contains 0 in its interior 0 a 2 n 2 2 2 n 3 a b a 1 a d a 1 p p a b 2 n n 2 1 a b d d a d 2 n n 2 1 a d is a positive parameter and 2 q d by lusternik schnirelmann category theory we prove that the problem p 1 has at least cat positive solutions

constrained maximum entropy sampling a fundamental experimental design problem is to select a most informative subset having prespecified size from a set of correlated random variables instances of this problem arise in many applied domains such as meteorology environmental statistics and statistical geology in these applications observations can be collected at different locations and possibly at different times information is measured by entropy practical situations have further restrictions on the design space for example budgetary limits geographical considerations as well as legislative and political considerations may restrict the design space in a complicated manner using techniques of linear algebra combinatorial optimization and convex optimization we develop upper and lower bounds on the optimal value for the gaussian case we describe how these bounds can be integrated into a branch and bound algorithm for the exact solution of these design problems finally we describe how we have implemented this algorithm and we present computational results for estimated covariance matrices corresponding to sets of environmental monitoring stations in the ohio valley of the united states

multiple positive solutions for semilinear elliptic equations with critical weighted hardy sobolev exponents abstract the existence and multiplicity of positive solutions are obtained for a class of semilinear elliptic equations with critical weighted hardy sobolev exponents and the concave convex nonlinearity by variational methods and some analysis techniques

convergence to spdes in stratonovich form we consider the perturbation of parabolic operators of the form t p x d by large amplitude highly oscillatory spatially dependent potentials modeled as gaussian random fields the amplitude of the potential is chosen so that the solution to the random equation is affected by the randomness at the leading order we show that when the dimension is smaller than the order of the elliptic pseudo differential operator p x d the perturbed parabolic equation admits a solution given by a duhamel expansion moreover as the correlation length of the potential vanishes we show that the latter solution converges in distribution to the solution of a stochastic parabolic equation with multiplicative noise that should be interpreted in the stratonovich sense the theory of mild solutions for such stochastic partial differential equations is developed

using continuous nonlinear relaxations to solve constrained maximum entropy sampling problems we consider a new nonlinear relaxation for the constrained maximum entropy sampling problem the problem of choosing the s x s principal submatrix with maximal determinant from a given n x n positive definite matrix subject to linear constraints we implement a branch and bound algorithm for the problem using the new relaxation the performance on test problems is far superior to a previous implementation using an eigenvalue based relaxation a parallel implementation of the algorithm exhibits approximately linear speed up for up to 8 processors and has successfully solved problem instances that were heretofore intractable

random non linear wave equations smoothness of the solutions we show existence and uniqueness for the solution of a onedimensional wave equation with non linear random forcing then we give sufficient conditions for the solution at a given time and a given point to have a density and for this density to be smooth the proof uses the extension of the malliavin calculus to the two parameters wiener functionals