the dirichlet problem for nonlinear second order elliptic equations iii functions of the eigenvalues of the hessian on etudie le probleme de dirichlet dans un domaine borne de r n a frontiere lisse f d 2 u dans u sur

equational inference canonical proofs and proof orderings we describe the application of proof orderings a technique for reasoning about inference systems to various rewrite based theorem proving methods including refinements of the standard knuth bendix completion procedure based on critical pair criteria huet s procedure for rewriting modulo a congruence ordered completion a refutationally complete extension of standard completion and a proof by consistency procedure for proving inductive theorems

a technique for establishing completeness results in theorem proving with equality it is proved that an automatic theorem proving system consisting of resolution paramodulation factoring equality reversal simplification and subsumption removal is complete in first order logic with equality when restricted to equality units the system is similar to the knuth bendix procedure for deriving consequences from equalities however our proofs of completeness are restricted to the case in which the ordering on words terms or atoms that is required in this type of process is order isomorphic to the positive integers the completeness of resolution and paramodulation without the functionally reflexive axioms is a simple corollary of our result the methods used are based upon the familiar ideas associated with semantic trees and should be helpful in showing that other theorem proving systems with equality are complete

a superposition oriented theorem prover abstract this paper describes a theorem proving procedure which combines the approach of resolution with that of rewriting the basic theoretical result is the completeness of a strong restriction of paramodulation for locking resolution procedures its oriented character suggests to consider the restricted paramodulation as a form of superposition the rewriting operation this is achieved by means of a new formalism of clauses named equational clauses in which each literal is converted into an equation thereby superposition on equational clauses is shown to embody not only paramodulation but also binary resolution so clausal superposition will build up our major rule of inference in addition term simplification is incorporated in our procedure as well as subsumption experimental results and potential applications for our theorem prover are lastly reported

semantic confluence tests and completion methods we present semantic methods for showing that a term rewriting system is confluent we also present methods for completing a given term rewriting system to obtain an equivalent confluent system these methods differ from the well known and widely studied knuth bendix method in that they emphasize semantics rather than syntax also they often require more user interaction than the purely syntactic knuth bendix method the concept of ground confluence is discussed methods for demonstrating ground confluence are also given we give decision procedures for some sub problems that arise in this method

existence and convexity of local solutions to degenerate hessian equations abstract in this work we prove the existence of convex solutions to the following k hessian equation s k u k y g y u d u in the neighborhood of a point y 0 u 0 p 0 r n r r n where g c g y 0 u 0 p 0 0 k c is nonnegative near y 0 k y 0 0 and rank d y 2 k y 0 n k 1

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

proof by consistency abstract advances of the past decade in methods and computer programs for showing consistency of proof systems based on first order equations have made it feasible in some settings to use proof by consistency as an alternative to conventional rules of inference musser described the method applied to proof of properties of inductively defined objects refinements of this inductionless induction method were discussed by kapur goguen huet and hullot huet and oppen lankford dershowitz paul and more recently by jouannaud and kounalis as well as by kapur narendran and zhang this paper gives a very general account of proof by consistency and inductionless induction and shows how previous results can be derived simply from the general theory new results include a theorem giving characterizations of an unambiguity property that is key to applicability of proof by consistency and a theorem similar to the birkhoff s completeness theorem for equational proof systems but concerning inductive proof

existence uniqueness and construction of rewrite systems the construction of term rewriting systems specifically by the knuth bendix completion procedure is considered we look for conditions that might ensure the existence of a finite canonical rewriting system for a given equational theory and that might guarantee that the completion procedure will find it we define several notions of equivalence between rewriting systems in the ordinary and modulo case and examine uniqueness of systems and the need for backtracking in implementing completion

a fourier transform approach to christoffel s problem we use fourier transform techniques to provide a new approach to berg s solution of the christoffel problem this leads to an explicit description of berg s spherical kernel and to new regularity properties of the associated integral transform

closed weingarten hypersurfaces in semi riemannian manifolds the existence of closed hypersurfaces of prescribed curvature in semi riemannian manifolds is proved provided there are barriers

convexity of level sets of minimal graph on space form with nonnegative curvature abstract for the minimal graph defined on a convex ring in the space form with nonnegative curvature we obtain the regularity and the strict convexity about its level sets by the continuity method

a hopf theorem for non constant mean curvature and a conjecture of a d alexandrov we prove a uniqueness theorem for immersed spheres of prescribed non constant mean curvature in homogeneous three manifolds in particular this uniqueness theorem proves a conjecture by a d alexandrov about immersed spheres of prescribed weingarten curvature in mathbb r 3 for the special but important case of prescribed mean curvature as a consequence we extend the classical hopf uniqueness theorem for constant mean curvature spheres to the case of immersed spheres of prescribed antipodally symmetric mean curvature in mathbb r 3

convergence of an engquist osher scheme for a multi dimensional triangular system of conservation laws we consider a multi dimensional triangular system of conservation laws this system arises as a model of three phase flow in porous media and includes multi dimensional conservation laws with discontinuous coefficients as a special case the system is neither strictly hyperbolic nor symmetric we propose an engquist osher type scheme for this system and show that the approximate solutions generated by the scheme converge to a weak solution numerical examples are also presented

automatic proofs by induction in theories without constructors abstract inductionless induction consists of using pure equational reasoning for proving the validity of an equation in the initial algebra of a set of equational axioms which would normally require some kind of induction under given hypotheses the equation is valid iff adding it to the set of axioms does not result in an inconsistency this inconsistency can be found by the knuth bendix completion algorithm provided that the signature of the algebra is split into free constructors and defined symbols which must be completely defined in terms of constructors this is the base of the so called inductive completion algorithm of huet and hullot two key concepts inductive reducibility and inductive co reducibility allow us to extend these techniques in various directions incomplete specifications nonfree constructors no constructors specified equational term rewriting systems the method is adapted for proving the consistency property of an enrichment of a specification by new operators and new equations in addition we get also a simple algorithm to exhibit a set of constructors of a specification finally inductive co reducibility is reduced to inductive reducibility and an algorithm for deciding inductive reducibility is given for left linear term rewriting systems

the dirichlet problem for nonlinear second order elliptic equations i monge ampegre equation on considere le probleme de dirichlet pour des equations elliptiques non lineaires pour une fonction reelle u definie dans la fermeture d un domaine borne dans r n avec une frontiere c

abstract canonical presentations solving goals like proving properties deciding word problems or resolving constraints is much easier with some presentations of the underlying theory than with others typically what have been called completion processes in particular in the study of equational logic involve finding appropriate presentations of a given theory to more easily solve a given class of problems we provide a general proof theoretic setting that relies directly on the fundamental concept of good that is normal form proofs itself defined using well founded orderings on proof objects this foundational framework allows for abstract definitions of canonical presentations and very general characterizations of saturation and redundancy criteria

towards a foundation of completion procedures as semidecision procedures completion procedures originated from the seminal work of knuth and bendix are well known as procedures for generating confluent rewrite systems i e decision procedures for equational theories in this paper we present a new abstract framework for the utilization of completion procedures as semidecision procedures for theorem proving the key idea in our approach in that a semidecision process should be target oriented i e keep into account the target theorem to be proved for the inference rules of a completion procedure we present target oriented schemes of contraction inference rules i e inference rules that delete sentences which are redundant for proving the target for the search plan we give a target oriented definition offairness according to which not all critical pairs need to be considered we prove that our notion of fairness together with the refutational completeness of the inference rules is sufficient for a completion procedure to be a semidecision procedure by relaxing the requirement of considering all critical pairs our target oriented framework should be more suitable for designing efficient procedures for theorem proving the generation of decision procedures is included as a special side effect and all the results of the classical approach to completion are re obtained in our framework the application of completion to disprove inductive conjectures i e the so called inductionless induction method is also covered as a semidecision process finally we present according to our framework some equational completion procedures based on unfailing knuth bendix completion