rayleigh wave scattering by two adhering elastic wedges we consider the problem of wave propagation in a composite bi material elastic wedge with traction free faces the wedge is composed of two distinct isotropic elastic solids which are perfectly bonded along their common face the angles of each wedge are essentially arbitrary satisfying 0 up to 2 2 1 2 2 up to 360 in particular we solve the problem of reflection transmission and diffraction of an incoming rayleigh wave on one of the wedge faces the problem is treated by the method of sommerfeld integrals whose spectral functions are defined through singular integral equations contour integration and analytic continuation we first follow the maliuzhinetz method and derive functional equations for the spectral functions we convert the maliuzhinetz functional equations into well posed singular integral equations which are shown to be suitable for effective inversion the solution of the integral equations explicitly determines the sommerfeld amplitudes inside some strips of the complex plane then explicit recursive formulas analytically continue the amplitudes to a larger region of the complex plane in a finite number of steps which correspond to multiple reflections from the boundaries the integral equations admit clear physical interpretation in terms of fermat9s reflection principle and their right hand sides are directly related to the boundary values of incident waves it is shown how our previous solutions on homogeneous wedges are recovered from the bi material wedge case we also show how several degenerate cases such as a liquid solid wedge and a liquid liquid wedge can be treated in appropriate limits finally the integral equations are solved numerically for particular material combinations and wedge angles to show the usefulness of the solution the results are plotted graphically and shown to be in general agreement with relevant special cases where prior results are available

fractional differential equations and lyapunov functionals abstract we consider a scalar fractional differential equation write it as an integral equation and construct several lyapunov functionals yielding qualitative results about the solution it turns out that the kernel is convex with a singularity and it is also completely monotone as is the resolvent kernel while the kernel is not integrable the resolvent kernel is positive and integrable with an integral value of one these kernels give rise to essentially different types of lyapunov functionals it is to be stressed that the lyapunov functionals are explicitly given in terms of known functions and they are differentiated using leibniz s rule the results are readily accessible to anyone with a background of elementary calculus

singular integral equations liapunov functionals and resolvents abstract this paper together with a recent paper by the second author on convex singular kernels establishes a base for further investigation of mildly singular equations with liapunov theory we study the two nonlinear scalar integral equations x t a t 0 t d t s x s g s x s d s and z t a t 0 t d t s g s z s d s where d has a singularity at t s the first equation is decomposed into three other simpler equations we then construct a liapunov functional for each of the equations which will yield l p properties of the solutions

resolvents and solutions of singular volterra integral equations with separable kernels the volterra integral equation 1 x t a t 0 tb t s x s dswith a kernel of the form b t s p t q s is investigated where a p and q are functions that are defined a e on an interval 0 t and are measurable the main result of this paper states that if qa is lebesgue integrable on 0 t the sign of b t s does not change for almost all t s and if there is a function f that is continuous on 0 t except possibly at countably many points with b t t f t a e on 0 t then the function x defined by 2 x t a t 0 tr t s a s ds where 3 r t s b t s e s t b u u d u solves 1 a e on 0 t three diverse examples illustrate the efficacy of using 2 and 3 to calculate solutions of 1 two of the examples involve singular kernels the solution of one of them is nowhere continuous on 0 t

lp solutions of singular integro differential equations abstract we study a variety of scalar integro differential equations with singular kernels including linear nonlinear and resolvent equations the first result involves a type of existence theorem which uses a fixed point mapping defined by the integro differential equation itself and produces a unique solution with a continuous derivative in a very simple way we then construct a liapunov functional yielding qualitative properties of solutions the work answers questions raised by volterra in 1928 by levin in 1963 and by grimmer and seifert in 1975 previous results had produced bounded solutions from bounded perturbations our results mainly concern integrable solutions from integrable perturbations

a block by block method for volterra integro differential equations with weakly singular kernel the theory of a block by block method for solving volterra integro differential equations with continuous kernels see makroglou 14 5d is adapted to volterra integrodifferential equations with weakly singular kernels and a rate of convergence is given

stability analysis of product t methods for abel integral equations of the second kind stability analysis of product methods for abel integral equations of the second kind based on the test equation y t 1 lambda int limits 0 1 frac y s t s alpha ds t geqq 0 0 0 and we investigate whether this property is inherited by numerical approximations toy

solutions to range dependent benchmark problems by the finite difference method an explicit second order finite difference scheme has been used to solve the elastic wave equation in the time domain solutions are presented for the perfect wedge the lossless penetrable wedge and the plane parallel waveguide that have been proposed as benchmarks by the acoustical society of america good agreement with reference solutions is obtained if the media is discretized at 20 gridpoints per wavelength there is a major discrepancy up to 20 db in reference source level because the reference solutions are normalized to the source strength at 1 m in the model but the finite difference solutions are normalized to the source strength at 1 m in a homogeneous medium the finite difference method requires computational times between 10 and 20 h on a super minicomputer without an array processor the method has the advantage of providing phase information and when run for a pulse source of providing insight into the evolution of the wave field and energy partitioning more complex models including velocity gradients and strong lateral heterogeneities can be solved with no additional computational effort the method has also been formulated to include shear wave effects

reasoning about temporal relations a maximal tractable subclass of allen s interval algebra we introduce a new subclass of allen s interval algebra we call ord horn subclass which is a strict superset of the pointisable subclass we prove that reasoning in the ord horn subclass is a polynomial time problem and show that the path consistency method is sufficient for deciding satisfiability further using an extensive machine generated case analysis we show that the ord horn subclass is a maximal tractable subclass of the full algebra assuming p np in fact it is the unique greatest tractable subclass amongst the subclasses that contain all basic relations

variable transformations in the numerical solution of second kind volterra integral equations with continuous and weakly singular kernels extensions to fredholm integral equations abstract a number of techniques that use variable transformations in numerical integration have been developed recently cf sidi numerical integration iv h brass g hammerlin eds birkhauser basel 1993 pp 359 373 laurie j comput appl math 66 1996 337 344 the use of these transformations resulted in increasing the order of convergence of the trapezoidal and the midpoint quadrature rule in this paper the application of variable transformation techniques of sidi and laurie type to the numerical solution of second kind volterra integral equations with continuous and weakly singular kernels is considered since the transformations are such that the end points of integration need not be used as mesh points the methods introduced can be used for vie with both continuous and weakly singular kernel in a uniform way the methods have also the advantages of simplicity of application and of achieving high order of convergence the application of the idea to fredholm integral equations with continuous and weakly singular equations is also considered numerical results are included and they verify the expected increased order of convergence they were obtained by using the trapezoidal formula for the evaluation of the transformed integrals

fractional linear multistep methods for abel volterra integral equations of the second kind fractional powers of linear multistep methods are suggested for the numerical solution of weakly singular volterra integral equations the proposed methods are convergent of the order of the underlying multistep method also in the generic case of solutions which are not smooth at the origin the stability properties stability region a stability a a stability are closely related to those of the underlying linear multistep method

mathematical programming methods in the numerical solution of volterra integral and integro differential equations with weakly singular kernel abstract nonlinear volterra integral and integro differential equations with weakly singular kernel are considered and solved numerically using nonlinear mathematical programming methods based on minimax approximations in both cases polynomial and multiquadric approximation are used

solution of a class of volterra integral equations with singular and weakly singular kernels a class of volterra integral equations are solved in this paper where the involved kernel may be weakly singular singular and hypersingular for various cases existence and uniqueness of such volterra integral equations are established furthermore by reducing such equations to ordinary differential equations analytic solutions have been determined explicitly when restricting the solutions to certain special function classes the number of suitable solutions is given

resolvents and solutions of weakly singular linear volterra integral equations abstract the structure of the resolvent r t s for a weakly singular matrix function b t s is determined where b t s is the kernel of the linear volterra vector integral equation e x t a t 0 t b t s x s d s and a t is a given continuous vector function using contraction mappings in a banach space of continuous vector functions with an exponentially weighted norm we show that when b t s satisfies certain integral conditions r t s has the form r t s b t s r 1 t s where r 1 t s is the unique continuous solution of the integral equation r 1 t s b 1 t s s t b t u r 1 u s d u and b 1 t s is defined by b 1 t s s t b t u b u s d u as examples the formulas of resolvents for a couple of weakly singular kernels of practical interest are derived we also obtain conditions under which a weakly singular integral equation e has a unique continuous solution x t and show that it can be expressed in terms of r t s by x t a t 0 t r t s a s d s finally we show that there are parallel results for an alternative resolvent r t s and examine when it and r t s are equivalent

a guide to completeness and complexity for modal logics of knowledge and belief abstract we review and re examine possible worlds semantics for propositional logics of knowledge and belief with three particular points of emphasis 1 we show how general techniques for finding decision procedures and complete axiomatizations apply to models for knowledge and belief 2 we show how sensitive the difficulty of the decision procedure is to such issues as the choice of modal operators and the axiom system and 3 we discuss how notions of common knowledge and distributed knowledge among a group of agents fit into the possible worlds framework as far as complexity is concerned we show among other results that while the problem of deciding satisfiability of an s5 formula with one agent is np complete the problem for many agents is pspace complete adding a distributed knowledge operator does not change the complexity but once a common knowledge operator is added to the language the problem becomes complete for exponential time

product integration for volterra integral equations of the second kind with weakly singular kernels we introduce a new numerical approach for solving volterra integral equations of the second kind when the kernel contains a mild singularity we give a convergence result we also present numerical examples which show the performance and efficiency of our method

a survey of recent advances in the numerical treatment of volterra integral and integro differential equations abstract the present survey paper samples recent advances in the numerical analysis of volterra integral equations of the first and second kind and of integro differential equations including equations with weakly singular kernels except for some important earlier references the discussion focuses on the development which has taken place during the last dozen years a fairly extensive bibliography selected to be representative rather than comprehensive complements the paper

reasoning about infinite computations we investigate extensions of temporal logic by connectives defined by finite automata on infinite words we consider three different logics corresponding to three different types of acceptance conditions finite looping and repeating for the automata it turns out however that these logics all have the same expressive power and that their decision problems are all pspace complete we also investigate connectives defined by alternating automata and show that they do not increase the expressive power of the logic or the complexity of the decision problem

combining deduction and model checking into tableaux and algorithms for converse pdl this paper presents a prefixed tableaux calculus for propositional dynamic logic with converse based on a combination of different techniques such as prefixed tableaux for modal logics and model checkers for calculus we prove the correctness and completeness of the calculus and illustrate its features we also discuss the transformation of the tableaux method naively nexptime into an exptime algorithm

numerical solution of non linear volterra integral equations this paper deals with non linear volterra integral equations of the type y x f x 0x h t x y t y x dt convergence criteria are given in the same sense of the maximum and ca norms for the numerical solution of this type of volterra integral equation several numerical methods are compared

diffraction by a highly contrast transparent wedge the problem of a plane wave diffraction by a highly contrast transparent wedge of an arbitrary opening is studied with the aid of the sommerfeld integrals it is reduced to a system of coupled maliuzhinets equations by use of the theory of s integrals the system of functional equations is transformed to a system of linear equations in a banach space in the case of the high contrast of a material inside the wedge in comparison with that in the wedge s exterior the linear equations are solved by means of perturbation theory convergence of the corresponding neumann series is proved singularities of the integrands in the sommerfeld integrals are investigated application of the steepest descent method leads to the determination of the reflected transmitted and diffracted waves expressions for the diffraction coefficients are also represented

enhancement schemes for constraint processing backjumping learning and cutset decomposition abstract researchers in the areas of constraint satisfaction problems logic programming and truth maintenance systems have suggested various schemes for enhancing the performance of the backtracking algorithm this paper defines and compares the performance of three such schemes backjumping learning and cycle cutset the backjumping and the cycle cutset methods work best when the constraint graph is sparse while the learning scheme mostly benefits problem instances with dense constraint graphs an integrated strategy is described which utilizes the distinct advantages of each scheme experiments show that in hard problems the average improvement realized by the integrated scheme is 20 25 higher than any of the individual schemes

numerical solutions of volterra integral equations with a solution dependent delay on demontre l existence et l unicite des solutions pour l equation ainsi qu une limitation a priori de la solution on definit une solution numerique et on demontre que la solution numerique converge vers la solution de l equation integrale

numerical solutions of range dependent benchmark problems in ocean acoustics accurate numerical solutions are presented for benchmark problems associated with propagation in a wedge with different bottom boundary conditions as well as in a plane parallel waveguide with changing profile with range the benchmark problems are solved with three different general purpose numerical codes 1 a coupled mode code providing a full spectrum two way backscatter included solution of the wave equation 2 a finite difference pe code providing a wide angle solution of the one way parabolic approximation of the wave equation and 3 a split step pe code providing a more narrow angle solution of the one way parabolic approximation of the wave equation only the two way coupled mode results can be considered accurate solutions of the benchmark problems

deciding expressive description logics in the framework of resolution we present a decision procedure for the description logic shiq based on the basic superposition calculus and show that it runs in exponential time for unary coding of numbers to derive our algorithm we extend basic superposition with a decomposition inference rule which transforms conclusions of certain inferences into equivalent but simpler clauses this rule can be used for general first order theorem proving with any resolution based calculus compatible with the standard notion of redundancy

on the complexity of qualitative spatial reasoning a maximal tractable fragment of the region connection calculus the computational properties of qualitative spatial reasoning have been investigated to some degree however the question for the boundary between polynomial and np hard reasoning problems has not been addressed yet in this paper we explore this boundary in the region connection calculus rcc 8 we extend bennett s encoding of rcc 8 in modal logic based on this encoding we prove that reasoning is np complete in general and identify a maximal tractable subset of the relations in rcc 8 that contains all base relations further we show that for this subset path consistency is sufficient for deciding consistency

rayleigh wave scattering in a wedge with mixed boundary conditions we consider a problem of rayleigh wave scattering in a single elastic wedge with one face tractionfree and the other face displacementfree the problem is treated by the method previously applied t

haar wavelet method for some nonlinear volterra integral equations of the first kind we present here a simple efficient haar wavelet method for numerical solution of a class of nonlinear volterra integral equations of the first kind the present method is based on converting nonlinear volterra integral equations of the first kind into linear volterra integral equations of the second kind numerical examples are given to illustrate efficiency and accuracy of the present method comparison with numerical methods in the past has also been done solving some nonlinear first kind volterra integral equations using haar wavelet haar wavelet method is simple and fast solved examples demonstrate the accuracy and efficiency of the present method present method gives better results than the numerical methods described in past necessary accuracy may be obtained by increasing the number of calculation points

generalized satisfiability for the description logic alc the standard reasoning problem concept satisfiability in the basic description logic alc is pspace complete and it is exptime complete in the presence of general concept inclusions several fragments of alc notably logics in the fl el and dl lite families have an easier satisfiability problem for some of these logics satisfiability can be decided in polynomial time we classify the complexity of the standard variants of the satisfiability problem for all possible boolean and quantifier fragments of alc with and without general concept inclusions

exp time tableaux for alc abstract the last years have seen two major advances in knowledge representation and reasoning first many interesting problems ranging from semi structured data to linguistics were shown to be expressible in logics whose main deductive problems are exptime complete second experiments in automated reasoning have substantially broadened the meaning of practical tractability instances of realistic size for pspace complete problems are now within reach for implemented systems still there is a gap between the reasoning services needed by the expressive logics mentioned above and those provided by the current systems indeed the algorithms based on tree automata which are used to prove exptime completeness require exponential time and space even in simple cases on the other hand current algorithms based on tableau methods can take advantage of such cases but require double exponential time in the worst case we propose a tableau calculus for the description logic alc for checking the satisfiability of a concept with respect to a tbox with general axioms and transform it into the first simple tableau based decision procedure working in single exponential time to guarantee the ease of implementation we also discuss the effects that optimizations propositional backjumping simplification semantic branching etc might have on our complexity result and introduce a few optimizations ourselves

resolvents for weakly singular kernels and fractional differential equations the solution r t s of the resolvent equation 1 r t s b t s stb t u r u s du where b t s denotes a given weakly singular matrix is obtained by means of fixed point mappings the result is a series that begins with some singular terms after which the remainder of the terms defines a continuous function in particular the resolvent is calculated for the kernel b t s t s q 1 of the scalar abel integral equation of the second kind 2 x t a t 0t1 t s 1 qx s ds where 0 q 1 it is then used to derive closed form formulas for the resolvents corresponding to q 1 2 and 1 3 furthermore a closed form formula for the resolvent for b t s s t2 s2 q 1 is derived thereby demonstrating that the results of this paper apply not only to kernels of convolution type but also to those of non convolution type as well finally the resolvent for the kernel of 2 is used to find a general solution of a linear fractional differential equation of caputo type

the complexity of concept languages abstract a basic feature of terminological knowledge representation systems is to represent knowledge by means of taxonomies here called terminologies and to provide a specialized reasoning engine to do inferences on these structures the taxonomy is built through a representation language called a concept language or description logic which is given a well defined set theoretic semantics the efficiency of reasoning has often been advocated as a primary motivation for the use of such systems the main contributions of the paper are 1 a complexity analysis of concept satisfiability and subsumption for a wide class of concept languages 2 algorithms for these inferences that comply with the worst case complexity of the reasoning task they perform

numerical solution of a nonuniquely solvable volterra integral equation using extrapolation methods in this work the numerical solution of a volterra integral equation with a certain weakly singular kernel depending on a real parameter is considered although for certain values of this equation possesses an infinite set of solutions we have been able to prove that euler s method converges to a particular solution it is also shown that the error allows an asymptotic expansion in fractional powers of the stepsize so that general extrapolation algorithms like the e algorithm can be applied to improve the numerical results this is illustrated by means of some examples

a proper hierarchy of propositional sequent calculi abstract it is known that lk a system of propositional sequent calculus without a cut rule written in tree form does not p simulate lk with a full cut rule written in tree form 1 6 it is also known that lk with a full cut rule in tree form p simulates lk in dag form in this paper we show that lk in tree form with a cut rule where the complexity of cut formulas is bounded by k 1 has an exponential speed up over the one bounded by k we also show that lk in tree form with a bounded complexity cut rule does not polynomially simulate cut free lk in dag form

attributive concept descriptions with complements abstract we investigate the consequences of adding unions and complements to attributive concept descriptions employed in terminological knowledge representation languages it is shown that deciding coherence and subsumption of such descriptions are pspace complete problems that can be decided with linear space

two theses of knowledge representation language restrictions taxonomic classification and the utility of representation services abstract levesque and brachman argue that in order to provide timely and correct responses in the most critical applications general purpose knowledge representation systems should restrict their languages by omitting constructs which require nonpolynomial worst case response times for sound and complete classification they also separate terminological and assertional knowledge and restrict classification to purely terminological information we demonstrate that restricting the terminological language and classifier in these ways limits these general purpose facilities so severely that they are no longer generally applicable we argue that logical soundness completeness and worst case complexity are inadequate measures for evaluating the utility of representation services and that this evaluation should employ the broader notions of utility and rationality found in decision theory we suggest that general purpose representation services should provide fully expressive languages classification over relevant contingent information approximate forms of classification involving defaults and rational management of inference tools

product integration methods for second kind abel integral equations the construction and convergence of high order product integration methods for the second kind abel equation are discussed and the results of de hoog and weiss are generalised backward difference methods are introduced and numerical results are presented which verify the theoretical rates of convergence