entropy formulation for parabolic degenerate equations with general dirichlet boundary conditions and application to the convergence of fv methods this paper is devoted to the analysis and the approximation of parabolic hyperbolic degenerate problems defined on bounded domains with nonhomogeneous boundary conditions it consists of two parts the first part is devoted to the definition of an original notion of entropy solutions to the continuous problem which can be adapted to define a notion of measure valued solutions or entropy process solutions the uniqueness of such solutions is established in the second part the convergence of the finite volume method is proved this result relies on weak estimates and on the theorem of uniqueness of the first part it also entails the existence of a solution to the continuous problem

uniqueness and existence for anisotropic degenerate parabolic equations with boundary conditions on a bounded rectangle we study the comparison principle for anisotropic degenerate parabolic hyperbolic equations with initial and nonhomogeneous boundary conditions we prove a comparison theorem for any entropy sub and super solution which immediately deduces the l1 contractivity and therefore uniqueness of entropy solutions the method used here is based upon the kinetic formulation and the kinetic techniques developed by lions perthame and tadmor by adapting and modifying those methods to the case of dirichlet boundary problems for degenerate parabolic equations we can establish a comparison property moreover in the quasi isotropic case the existence of entropy solutions is proved

the solutions of a hyperbolic parabolic mixed type equation on half space domain abstract the solutions of the hyperbolic parabolic mixed type equation u t a u div b u in q t r n 0 t are considered where r n r n is half space a new kind of entropy solution to the equation is introduced the paper shows that the convection term div b u determines the explicit boundary value condition if b n 0 0 we can impose the general dirichlet boundary condition u x t 0 x t r n 0 t 0 t which is satisfied in a particular weak sense but if b n 0 0 then no boundary value condition is necessary the solution of the equation is free from any limitation of the boundary value condition

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

stability of hyperbolic parabolic mixed type equations with partial boundary condition abstract in this paper we are concerned with the hyperbolic parabolic mixed type equations with the non homogeneous boundary condition if it is degenerate on the boundary the part of the boundary whose boundary value should be imposed is determined by the entropy condition from the convection term if there is no convection term in the equation we show that the stability of solutions can be proved without any boundary condition if the equation is completely degenerate we show that the stability of solutions can be established just based on the partial boundary condition

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

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

nonhomogeneous dirichlet problems for degenerate parabolic hyperbolic equations the aim of the paper is to give a formulation for the initial boundary value problem of parabolic hyperbolic type in the case of nonhomogeneous boundary data a 0 here u u x t with x t q 0 t where is a bounded domain in n with smooth boundary and t 0 the function b is assumed to be nondecreasing allowing degeneration zones where b is constant is locally lipschitz continuous and g l 0 b after introducing the definition of an entropy solution to the above problem in the spirit of otto 14 we prove uniqueness of the solution in the proposed setting moreover we prove that the entropy solution previously defined can be obtained as the limit of solutions of regularized equations of nondegenerate parabolic type specifically the diffusion function b is approximated by functions b that are strictly increasing

computing with rewrite systems term rewriting systems that is sets of directed equations provide a paradigm of computation with particularly simple syntax and semantics rewrite systems may be used for straightforward computation by simplifying terms we show how in addition restricted forms of the knuth bendix completion procedure may be used to interpret logic programs written as a set of equivalence preserving rewrite rules we discuss verification issues and also illustrate the use of the full completion procedure to synthesize rewrite programs from specifications

homogeneous dirichlet problems for quasilinear anisotropic degenerate parabolic hyperbolic equations the aim of this paper is to prove the well posedness existence and uniqueness of entropy solutions to the general anisotropic degenerate parabolic hyperbolic equations with l initial data and homogeneous dirichlet boundary condition we use the doubling variables device to prove a comparison theorem which implies the uniqueness the existence of entropy solutions can be obtained by finding the limit of solutions for the regularized equation of strictly parabolic type

orderings for term rewriting systems abstract methods of proving that a term rewriting system terminates are presented they are based on the intuitive notion of simplification orderings orderings in which any term that is syntactically simpler than another is smaller than the other as a consequence of kruskal s tree theorem any nonterminating system must be self embedding in the sense that it allows for the derivation of some term from a simpler one thus termination is guaranteed if every rule in the system is a reduction in some simplification ordering most of the orderings that have been used for proving termination are indeed simplication orderings using this notion often allows for much easier proofs a particularly useful class of simplification orderings the recursive path orderings is defined examples of the use of simplication orderings in termination proofs are given

completely non clausal theorem proving the proof procedure we describe operates on quantifier free formulas of the predicate calculus which are not truth functionally normalized in any way the procedure involves a single inference rule called nc resolution and is shown to be complete completeness is also obtained for a simple restriction on the rule s application examples are given using nc resolution to derive a logic program from its specification and to execute a program specification in its original form

evolutionary weighted p laplacian with boundary degeneracy abstract the paper concerns the well posedness problem of an evolutionary weighted p laplacian with boundary degeneracy different from the classical theory for linear equations it is shown that the degenerate portion of the boundary should be decomposed into two parts the strongly degenerate boundary on which the equation exhibits hyperbolic characteristics and the weakly degenerate boundary on which the equation still exhibits parabolic characteristics we formulate reasonably the boundary value condition and establish the existence and uniqueness theorems

nonuniqueness and multi bump solutions in parabolic problems with the p laplacian abstract the validity of the weak and strong comparison principles for degenerate parabolic partial differential equations with the p laplace operator p is investigated for p 2 this problem is reduced to the comparison of the trivial solution 0 by hypothesis with a nontrivial nonnegative solution u x t the problem is closely related also to the question of uniqueness of a nonnegative solution via the weak comparison principle in this article realistic counterexamples to the uniqueness of a nonnegative solution the weak comparison principle and the strong maximum principle are constructed with a nonsmooth reaction function that satisfies neither a lipschitz nor an osgood standard uniqueness condition nonnegative multi bump solutions with spatially disconnected compact supports and zero initial data are constructed between sub and supersolutions that have supports of the same type

entropy solutions for nonlinear degenerate problems we consider a class of elliptic hyperbolic degenerate equations g u delta b u divg phi u f with dirichlet homogeneous boundary conditions and a class of elliptic parabolic hyperbolic degenerate equations g u t delta b u divg phi u f with homogeneous dirichlet conditions and initial conditions existence of entropy solutions for both problems is proved for nondecreasing continuous functions g and b vanishing at zero and for a continuous vectorial function satisfying rather general conditions comparison and uniqueness of entropy solutions are proved for g and b continuous and nondecreasing and for continuous

experimenting with process equivalence bloom b and a r meyer experimenting with process equivalence theoretical computer science 101 1992 223 237 distinctions between concurrent processes based on observable outcomes of computational experiments are examined the equivalence determined by a general class of experiments involving duplication of processes can be characterized by a notion of ready simulation resembling but strictly coarser than milner s bisimulation equivalence

a rationale for conditional equational programming abstract conditional equations provide a paradigm of computation that combines the clean syntax and semantics of lisp like functional programming with prolog like logic programming in a uniform manner for functional programming equations are used as rules for left to right rewriting for logic programming the same rules are used for conditional narrowing together rewriting and narrowing provide increased expressive power we discuss some aspects of the theory of conditional rewriting and the reasons underlying certain choices in designing a language based on them the most important correctness property a conditional rewriting program may posses is ground confluence this ensures that at most one value can be computed from any given variable free input term we give criteria for confluence reasonable conditions for ensuring the completeness of narrowing as an operational mechanism for solving goals are provided these results are then extended to handle rewriting with existentially quantified conditions and built in predicates some termination issues are also considered including the case of rewriting with higher order terms

a unification algorithm for associative commutative functions an important component of automated theorem proving systems are unification algorithms which fred most general substitutions which when apphed to two expressions make them equivalent functions which are associative and commutative such as the arithmetic addition and multiphcatton functions are often the subject of automated theorem proving an algorithm which unifies terms whose function is associative and commutauve is presented here termmaaon soundness and completeness of the algorithm have been proved for a subclass of the general case the algorithm is an efficient alternative to other methods of handling associative commutative functions

well posedness results for triply nonlinear degenerate parabolic equations we study well posedness of triply nonlinear degenerate elliptic parabolic hyperbolic problems of the kind b u t div a u u u f u t 0 u0 in a bounded domain with homogeneous dirichlet boundary conditions the nonlinearities b and are supposed to be continuous non decreasing and the nonlinearity a falls within the leray lions framework some restrictions are imposed on the dependence of a u u on u and also on the set where degenerates a model case is a u u f b u u u k u a0 u with a nonlinearity which is strictly increasing except on a locally finite number of segments and the nonlinearity a0 which is of the leray lions kind we are interested in existence uniqueness and stability of l entropy solutions for the parabolic hyperbolic equation b id we obtain a general continuous dependence result on data u0 f and nonlinearities b a similar result is shown for the degenerate elliptic problem which corresponds to the case of b 0 and general non decreasing surjective existence uniqueness and continuous dependence on data u0 f are shown in more generality for instance the assumptions b r r and the continuity of b 1 permit to achieve the well posedness result for bounded entropy solutions of this triply nonlinear evolution problem