uniqueness in wloc1 p x ω and continuity up to portions of the boundary of positive solutions for a strongly singular elliptic problem abstract in this paper we consider issues about existence and uniqueness of w l o c 1 p x solutions and its continuity up to portions of the boundary for the elliptic equation p x u c x d x x u x under a general sense of zero boundary condition for a smooth bounded domain r n where x and x may change their signals in multiple sub regions of this domain or on its boundary we also prove a comparison principle for w l o c 1 p x sub and super solutions for a related problem in addition we present a kind of compatibility condition involving the trio c to obtain solution still with zero boundary in the sense of the trace on portions of the boundary some of our results are new even for the classical laplacian operator

on the spaces lp x ω and wm p x ω in this paper we present some basic results on the generalized lebesgue spaces l p x omega and generalized lebesgue sobolev spaces w m w p x omega these results provide the necessary framework for the study of variational problems and elliptic equations with non standard p x growth conditions c 2001 academic press

nonlinear three point third order boundary value problems abstract in this paper we consider three point boundary value problems for third order differential equations we provide simple sufficient conditions in order to obtain a priori bounds on solutions we then rely on the topological transversality theorem of granas or on schaefer s theorem to prove existence of solutions

three point boundary value problems for ordinary differential equations by matching solutions motivational results for solution matching were proven first in the text by bailey et al l where they dealt with solutions of two point problems for y f x y y by matching solutions of initial value problems then in 1973 barr sherman 2 adapted the arguments of bailey et al to match solutions of two point boundary value problems for third order equations to obtain solutions of three point problems in the same paper they showed that this result could be generalized to equations of arbitrary order yielding solutions of l 2 barr miletta 3 matched solutions of k point problems for nth order linear equations and obtained solutions of n point conjugate type boundary value problems in 1978 moorti garner 4 extended the results of 2 to boundary value problems similar to l 3 except p y were restricted

eigenvalues of p x laplacian dirichlet problem abstract this paper studies the eigenvalues of the p x laplacian dirichlet problem div u p x 2 u u p x 2 u in u 0 on where is a bounded domain in rn and p x is a continuous function on such that p x 1 we show that the set of eigenvalues is a nonempty infinite set such that sup we present some sufficient conditions for inf 0 and for inf 0 respectively

local boundedness of general minimizers with nonstandard growth abstract we prove the local boundedness of local minimizers of f x u u d x where f is subject to the general structural conditions z p x b x y r x g x f x y z z p x b x y r x g x

regularity results for a class of functionals with non standard growth we consider the integral functional under non standard growth assumptions that we call p x type namely we assume that a relevant model case being the functional under sharp assumptions on the continuous function p x 1 we prove regularity of minimizers energies exhibiting this growth appear in several models from mathematical physics

existence uniqueness and iterative methods for third order boundary value problems abstract in this paper we shall provide necessary and sufficient conditions for the existence and uniqueness of solutions of third order nonlinear differential equations satisfying three point boundary conditions for the linear case we propose a constructive method which is a variation of the method of chasing for the nonlinear problems sufficient conditions are provided to ensure the convergence of a general class of iterative methods several examples are also included

approximate counting uniform generation and rapidly mixing markov chains the paper studies effective approximate solutions to combinatorial counting and uniform generation problems using a technique based on the simulation of ergodic markov chains it is shown that for self reducible structures almost uniform generation is possible in polynomial time provided only that randomised approximate counting to within some arbitrary polynomial factor is possible in polynomial time it follows that for self reducible structures polynomial time randomised algorithms for counting to within factors of the form 1 n are available either for all fl e r or for no fi e r a substantial part of the paper is devoted to investigating the rate of convergence of finite ergodic markov chains and a simple but powerful characterisation of rapid convergence for a broad class of chains based on a structural property of the underlying graph is established finally the general techniques of the paper are used to derive an almost uniform generation procedure for labelled graphs with a given degree sequence which is valid over a much wider range of degrees than previous methods this in turn leads to randomised approximate counting algorithms for these graphs with very good

regularity results for stationary electro rheological fluids we prove regularity results for weak solutions to systems modelling electro rheological fluids in the stationary case as proposed in 27 31 a particular case of the system we consider is

boundedness of solutions for an elliptic equation with non standard growth abstract under non standard growth conditions following moser s iteration technique we shall prove boundedness of solutions for an equation of fourth order of the following kind 1 2 1 d a x 2 u f in

on three point boundary value problems associated with third order differential equations introduction in this paper we shall be concerned with the existence and uniqueness of solutions to three point boundary value problems associated with the differential equation y f x y y s y 1 1 where f x y z w is assumed to be continuous on a subset of r4 initial value problems associated with 1 1 exist and are unique in the interval x1 x a monotonicity restriction on f x y z w ensures that the following boundary value problems y j x y y y y xj y y xj y y x m i 1 2 12i y j x y y y y x y y x m2 2 2 9 y x y i 421 le3i

approximating probabilistic inference in bayesian belief networks is np hard abstract it is known that exact computation of conditional probabilities in belief networks is np hard many investigators in the ai community have tacitly assumed that algorithms for performing approximate inference with belief networks are of polynomial complexity indeed special cases of approximate inference can be performed in time polynomial in the input size however we have discovered that the general problem of approximating conditional probabilities with belief networks like exact inference resides in the np hard complexity class we develop a complexity analysis to elucidate the difficulty of approximate probabilistic inference more specifically we show that the existence of a polynomial time relative approximation algorithm for major classes of problem instances implies that np p we present our proof and explore the implications of the result

the computational complexity of probabilistic inference using bayesian belief networks research note abstract bayesian belief networks provide a natural efficient method for representing probabilistic dependencies among a set of variables for these reasons numerous researchers are exploring the use of belief networks as a knowledge representation in artificial intelligence algorithms have been developed previously for efficient probabilistic inference using special classes of belief networks more general classes of belief networks however have eluded efforts to develop efficient inference algorithms we show that probabilistic inference using belief networks is np hard therefore it seems unlikely that an exact algorithm can be developed to perform probabilistic inference efficiently over all classes of belief networks this result suggests that research should be directed away from the search for a general efficient probabilistic inference algorithm and toward the design of efficient special case average case and approximation algorithms

an optimal approximation algorithm for bayesian inference abstract approximating the inference probability pr x x e e in any sense even for a single evidence node e is np hard this result holds for belief networks that are allowed to contain extreme conditional probabilities that is conditional probabilities arbitrarily close to 0 nevertheless all previous approximation algorithms have failed to approximate efficiently many inferences even for belief networks without extreme conditional probabilities we prove that we can approximate efficiently probabilistic inference in belief networks without extreme conditional probabilities we construct a randomized approximation algorithm the bounded variance algorithm that is a variant of the known likelihood weighting algorithm the bounded variance algorithm is the first algorithm with provably fast inference approximation on all belief networks without extreme conditional probabilities from the bounded variance algorithm we construct a deterministic approximation algorithm using current advances in the theory of pseudorandom generators in contrast to the exponential worst case behavior of all previous deterministic approximations the deterministic bounded variance algorithm approximates inference probabilities in worst case time that is subexponential 2 log n d for some integer d that is a linear function of the depth of the belief network

sobolev embedding theorems for spaces wk p x ω this paper gives a sobolev type embedding theorem for the generalized lebesgue sobolev space wk p x where is an open domain in rn n 2 with cone property and p x is a lipschitz continuous function defined on satisfying 1 p p nk the main result can be stated as follows for any measurable function q x x formula with p x q x p x np x n kp x there exists a continuous embedding from wk p x to lq x

global l estimates and holder continuity of weak solutions to elliptic equations with the general nonstandard growth conditions abstract we study the elliptic equations in divergence form with the general nonstandard growth conditions involving lebesgue measurable functions on in this paper we prove global l estimates and holder continuity of weak solutions for these equations our results improve the one in giusti 2003 even for the case when p x p and q x q are constants

existence of solutions for p x laplacian dirichlet problem this paper presents several sufficient conditions for the existence of solutions for the dirichlet problem of p x laplacian div u p x 2 u f x u x u 0 x especially an existence criterion for infinite many pairs of solutions for the problem is obtained the discussion is based on the theory of the spaces lp x and w01 p x

eigenvalues of a general class of boundary value problem with derivative dependent nonlinearity we consider a general class of boundary value problem bvp comprising the differential equation y m t f t y t y t y t y q t t 0 1 where 1 q m 1 and 0 together with multi point boundary conditions y 0 y 0 y 0 y q 1 0 0 a i y q t j y q 1 t j y m 1 t j 0 j r 0 1 i m q where 0 t 0 t 1 t r 1 particular cases of this general bvp include the well known abel gontscharoff focal m p sturm liouville and complementary lidstone bvps it is noted that bvps with derivative dependent nonlinear terms are less investigated in the literature due to technical difficulty in this paper a new technique is developed to characterize the eigenvalues so that the bvp has a positive solution explicit eigenvalue intervals are also established we include several well known examples in the literature to illustrate the results obtained

local holder regularity of the gradients for the elliptic p x laplacian equation abstract in this paper we obtain the interior holder regularity of the gradients of weak solutions for the elliptic p x laplacian equation div a u u p x 2 2 a u div f p x 2 f under some proper assumptions on the holder continuous functions p f and a

approximating the permanent a randomised approximation scheme for the permanent of a 0 1s presented the task of estimating a permanent is reduced to that of almost uniformly generating perfect matchings in a graph the latter is accomplished by simulating a markov chain whose states are the matchings in the graph for a wide class of 0 1 matrices the approximation scheme is fully polynomial i e runs in time polynomial in the size of the matrix and a parameter that controls the accuracy of the output this class includes all dense matrices those that contain sufficiently many 1 s and almost all sparse matrices in some reasonable probabilistic model for 0 1 matrices of given density for the approach sketched above to be computationally efficient the markov chain must be rapidly mixing informally it must converge in a short time to its stationary distribution a major portion of the paper is devoted to demonstrating that the matchings chain is rapidly mixing apparently the first such result for a markov chain with genuinely c

random generation of combinatorial structures from a uniform distribution abstract the class of problems involving the random generation of combinatorial structures from a uniform distribution is considered uniform generation problems are in computational difficulty intermediate between classical existence and counting problems it is shown that exactly uniform generation of efficiently verifiable combinatorial structures is reducible to approximate counting and hence is within the third level of the polynomial hierarchy natural combinatorial problems are presented which exhibit complexity gaps between their existence and generation and between their generation and counting versions it is further shown that for self reducible problems almost uniform generation and randomized approximate counting are inter reducible and hence of similar complexity

bvp s with odd differences of gaps in boundary conditions for nth order ode s by matching solutions matching of solutions of boundary value problems plays an important role in interface problems we are concerned with the existence and uniqueness of solutions to boundary value problems on an interval a c for the nth order ordinary differential equation y n f x y y y n 1 for n 3 by matching solutions on a b with solutions on b c in this paper we consider a general case where the gap in boundary conditions at b is odd different monotonicity conditions on f are imposed for distinct cases that arise

approximating probabilistic inference in bayesian belief networks a belief network comprises a graphical representation of dependencies between variables of a domain and a set of conditional probabilities associated with each dependency unless rho np an efficient exact algorithm does not exist to compute probabilistic inference in belief networks stochastic simulation methods which often improve run times provide an alternative to exact inference algorithms such a stochastic simulation algorithm d bnras which is a randomized approximation scheme is presented to analyze the run time belief networks are parameterized by the dependence value d sub xi which is a measure of the cumulative strengths of the belief network dependencies given background evidence xi this parameterization defines the class of f dependence networks the run time of d bnras is polynomial when f is a polynomial function thus the results prove the existence of a class of belief networks for which inference approximation is polynomial and hence provably faster than any exact algorithm

overview of differential equations with non standard growth abstract differential equations with non standard growth have been a very active field of investigation in recent years in this survey we present an overview of the field as well as several of the most important results we consider both existence and regularity questions finally we provide a comprehensive list of papers published to date

a new modification of the adomian decomposition method for solving boundary value problems for higher order nonlinear differential equations abstract in this paper we propose a new modified recursion scheme for the resolution of multi order and multi point boundary value problems for nonlinear ordinary and partial differential equations by the adomian decomposition method adm our new approach including duan s convergence parameter provides a significant computational advantage by allowing for the acceleration of convergence and expansion of the interval of convergence during calculations of the solution components for nonlinear boundary value problems in particular for such cases when one of the boundary points lies outside the interval of convergence of the usual decomposition series we utilize the boundary conditions to derive an integral equation before establishing the recursion scheme for the solution components thus we can derive a modified recursion scheme without any undetermined coefficients when computing successive solution components whereas several prior recursion schemes have done so this modification also avoids solving a sequence of nonlinear algebraic equations for the undetermined coefficients fraught with multiple roots which is required to complete calculation of the solution by several prior modified recursion schemes using the adm

global boundedness and holder continuity of quasiminimizers with the general nonstandard growth conditions abstract in this paper we study the integral functionals with the general nonstandard growth we prove the boundedness and holder continuity of quasiminimizers of these functionals our results for quasiminimizers improve variable exponent case and generalize constant exponent case studied in direct methods in the calculus of variations 2003 by giusti