on flows associated to sobolev vector fields in wiener spaces an approach a la diperna lions in this paper we extend the diperna lions theory of flows associated to sobolev vector fields to the case of cameron martin valued vector fields in wiener spaces e having a sobolev regularity the proof is based on the analysis of the continuity equation in e and on uniform gaussian commutator estimates in finite dimensional spaces

tractable reasoning via approximation abstract problems in logic are well known to be hard to solve in the worst case two different strategies for dealing with this aspect are known from the literature language restriction and theory approximation in this paper we are concerned with the second strategy our main goal is to define a semantically well founded logic for approximate reasoning which is justifiable from the intuitive point of view and to provide fast algorithms for dealing with it even when using expressive languages we also want our logic to be useful to perform approximate reasoning in different contexts we define a method for the approximation of decision reasoning problems based on multivalued logics our work expands and generalizes in several directions ideas presented by other researchers the major features of our technique are 1 approximate answers give semantically clear information about the problem at hand 2 approximate answers are easier to compute than answers to the original problem 3 approximate answers can be improved and eventually they converge to the right answer 4 both sound approximations and complete ones are described the method we propose is flexible enough to be applied to a wide range of reasoning problems in our research we considered approximation of several decidable problems with different worstcase complexity involving both propositional and first order languages in particular we defined approximation techniques for propositional logic fragments of first order logic concept description languages and modal logic in our research we also addressed the issue of representing the knowledge of a reasoner with limited resources and how to use such a knowledge for approximate reasoning purposes

uniqueness of continuous solutions for bv vector fields we consider a vector field whose coefficients are functions of bounded variation with a bounded divergence we prove the uniqueness of continuous solutions for the cauchy problem

reductions for non clausal theorem proving this paper presents the tas methodology as a new framework for generating non clausal automated theorem provers we present a complete description of the atp for classical propositional logic named tas d but the ideas which make use of implicants and implicates can be extended in a natural manner to first order logic and non classical logics the method is based on the application of a number of reduction strategies on subformulas in a rewrite system style in order to reduce the complexity of the formula as much as possible before branching specifically we introduce the concept of complete reduction and extensions of the pure literal rule and ofthe collapsibility theorems these strategies allow to limit the size ofthe search space in addition tas d is a syntactical countermodel construction as an example of the power of tas d we study a class of formulas which has linear proofs in the number of branchings when either resolution or dissolution with factoring is applied when applying our method to these formulas we get proofs without branching in addition some experimental results are reported copyright 2001 elsevier science b v

flows for non smooth vector fields with subexponentially integrable divergence abstract in this paper we study flows associated to sobolev vector fields with subexponentially integrable divergence our approach is based on the transport equation following diperna lions 17 a key ingredient is to use a quantitative estimate of solutions to the cauchy problem of transport equation to obtain the regularity of density functions

bounds for the extremal eigenvalues of a class of symmetric tridiagonal matrices with applications abstract we consider a class of symmetric tridiagonal matrices which may be viewed as perturbations of toeplitz matrices the toeplitz structure is destroyed since two elements on each off diagonal are perturbed based on a careful analysis we derive sharp bounds for the extremal eigenvalues of this class of matrices in terms of the original data of the given matrix in this way we also obtain a lower bound for the smallest singular value of certain matrices some numerical results indicate that our bounds are extremely good

equality conditions for lower bounds on the smallest singular value of a bidiagonal matrix abstract several lower bounds have been proposed for the smallest singular value of a square matrix such as johnson s bound brauer type bound li s bound and ostrowski type bound in this paper we focus on a bidiagonal matrix and investigate the equality conditions for these bounds we show that the former three bounds give strict lower bounds if all the bidiagonal elements are non zero for the ostrowski type bound we present an easily verifiable necessary and sufficient condition for the equality to hold

existence and uniqueness of maximal regular flows for non smooth vector fields in this paper we provide a complete analogy between the cauchy lipschitz and the diperna lions theories for ode s by developing a local version of the diperna lions theory more precisely we prove the existence and uniqueness of a maximal regular flow for the diperna lions theory using only local regularity and summability assumptions on the vector field in analogy with the classical theory which uses only local regularity assumptions we also study the behaviour of the ode trajectories before the maximal existence time unlike the cauchy lipschitz theory this behaviour crucially depends on the nature of the bounds imposed on the spatial divergence of the vector field in particular a global assumption on the divergence is needed to obtain a proper blow up of the trajectories

flows associated with irregular r d vector fields abstract this work consists on the study of flows associated with non smooth r d vector fields namely concerning existence and uniqueness for almost every initial condition it is also proved that the flows avoid some special compact sets

solving satisfiability via boltzmann machines boltzmann machines bms are proposed as a computational model for the solution of the satisfiability sat problem in the propositional calculus setting conditions that guarantee consensus function maxima for configurations of the bm associated with solutions to the satisfaction problem are given experimental results that show a linear behavior of bms solving the satisfiability problem are presented and discussed

new methods for 3 sat decision and worst case analysis abstract we prove the worst case upper bound 1 5045 n for the time complexity of 3 sat decision where n is the number of variables in the input formula introducing new methods for the analysis as well as new algorithmic techniques we add new 2 and 3 clauses called blocked clauses generalizing the extension rule of extended resolution our methods for estimating the size of trees lead to a refined measure of formula complexity of 3 clause sets and can be applied also to arbitrary trees

ordinary differential equations transport theory and sobolev spaces we obtain some new existence uniqueness and stability results for ordinary differential equations with coefficients in sobolev spaces these results are deduced from corresponding results on linear transport equations which are analyzed by the method of renormalized solutions

further lower bounds for the smallest singular value abstract in an earlier paper of the first author gersgorin s theorem was used in a novel way to give a simple lower bound for the smallest singular value of a general complex matrix that lower bound was stronger than previous published bounds here we use three variants of gersgorin s theorem in a similar way to give further lower bounds each of the new bounds is more complicated but generally stronger than the pure gersgorin based bound the three new bounds are mutually noncomparable

a satisfiability tester for non clausal propositional calculus an algorithm for satisfiability testing in the propositional calculus with a worst case running time that grows at a rate less than 2 25 e l is described where l can be either the length of the input expression or the number of occurrences of literals i e leaves in it this represents a new upper bound on the complexity of non clausal satisfiability testing the performance is achieved by using lemmas concerning assignments and pruning that preserve satisfiability together with choosing a good variable upon which to recur for expressions in clause form it is shown that the davis putnam procedure satisfies the same upper bound

lower bounds for the smallest singular value abstract let a aij be an n n complex matrix suppose that g a the undirected graph of a has no isolated vertex let e be the set of edges of g a we prove that the smallest singular value of a n satisfies n min ij i j e where gij ai aj ai aj 2 ri ci rj cj 1 2 2 with ai aii and ri ci are the ith deleted absolute row sum and column sum of a respectively the result simplifies and improves that of johnson and szulc n mini j ij see 1

continuity equations and ode flows with non smooth velocity in this paper we review many aspects of the well posedness theory for the cauchy problem for the continuity and transport equations and for the ordinary differential equation ode in this framework we deal with velocity fields that are not smooth but enjoy suitable weak differentiability assumptions we first explore the connection between the partial differential equation pde and the ode in a very general non smooth setting then we address the renormalization property for the pde and prove that such a property holds for sobolev velocity fields and for bounded variation velocity fields finally we present an approach to the ode theory based on quantitative estimates

convergence properties of optimization algorithms for the sat problem the satisfiability sat problem is a basic problem in computing theory presently an active area of research on sat problem is to design efficient algorithms to find a solution for a satisfiable conjunctive normal form cnf formula a new formulation the universal sat problem model which transforms the sat problem on boolean space into an optimization problem on real space has been developed j gu 1988 1992 1994 many optimization techniques such as the steepest descent method newton s method and the coordinate descent method can be used to solve the universal sat problem we prove that when the initial solution is sufficiently close to the optimal solution the steepest descent method has a linear convergence ratio spl beta 1 newton s method has a convergence ratio of order two and the convergence ratio of the steepest descent method is approximately 1 spl beta m for the universal sat problem with m variables an algorithm based on the coordinate descent method for the universal sat problem is also presented experimental results show that this algorithm is more efficient than some previous ones in finding a solution for certain classes of the satisfiable cnf formulas

estimates and regularity results for the diperna lions flow in this paper we derive new simple estimates for ordinary differential equations with sobolev coefficients these estimates not only allow to recover some old and recent results in a simple direct way but they also have some new interesting corollaries

bounds for the spectral radius and the largest singular value abstract we derive an increasing sequence of lower bounds for the spectral radius of a matrix with real spectrum and progressively improved bounds for the largest singular value of a complex matrix we also find estimates for the rank of normal matrices with real spectrum and for the rank of normal nonnegative matrices including some sufficient condition for such matrices to be invertible