multi stage solvers optimized for damping and propagation explicit multi stage solvers are routinely used to solve the semi discretized equations that arise in computational fluid dynamics cfd problems often they are used in combination with multi grid methods in that case the role of the multi stage solver is to efficiently reduce the high frequency modes on the current grid and is called a smoother in the past when optimizing the coefficients of the scheme only the damping characteristics of the smoother were taken into account and the interaction with the remainder of the multi grid cycle was neglected recently it had been found that coefficients that result in less damping but allow for a higher courant friedrichs lewy cfl number are often superior to schemes that try to optimize damping alone while this is certainly true for multi stage schemes used as a stand alone solver we investigate in this paper if using higher cfl numbers also yields better results in a multi grid setting we compare the results with a previous study we conducted and where a more accurate model of the multi grid cycle was used to optimize the various parameters of the solver we show that the use of the more accurate model results in better coefficients and that in a multi grid setting propagation is of little importance we also look into the gains to be made when we allow the parameters to be different for the pre and post smoother and show that even better coefficients can be found in this way

non finite specifiability of projections of functional dependency families abstract a functional dependency fd implicational dependency id family is a family of relational database instances consisting of all instances which satisfy a given set of fds ids an id family is finitely specifiable if it can be defined using a finite set of ids it is known that all fd families are finitely specifiable id families and also that each projection of each fd family while not necessarily an fd family is an id family it is shown here that if the projection of an fd family is not an fd family then it is not a finitely specifiable id family

solving cauchy integral equations of the first kind by the adomian decomposition method a new approach is presented to resolve cauchy integral equations of the first kind in the general case by first considering a regularized integral equation and then transforming it into a canonical form suitable for applying the adomian decomposition method adm we obtain a decomposition solution f e of the regularized integral equation and prove the convergence of our new combined method as the regularization parameter e 1 the obtained solution is shown to be a sufficiently good approximate solution for the particular cauchy integral equation the proposed method has been tested for a variety of cauchy integral equations which are particularly important in engineering applications e g airfoil design etc

multi staging of jacobi relaxation in multigrid methods for steady euler equations ii multi stage versions of jacobi relaxation are studied for use in multigrid methods for steady euler equations in particular versions adapted to the nonlinear tvd discretization are analysed it is shown that multi stage relaxation versions lead to more efficient multigrid methods than possible with classic relaxation methods

axiomatisation of functional dependencies in incomplete relations abstract incomplete relations are relations which contain null values whose meaning is value is at present unknown such relations give rise to two types of functional dependency fd the first type called the strong fd sfd is satisfied in an incomplete relation if for all possible worlds of this relation the fd is satisfied in the standard way the second type called the weak fd wfd is satisfied in an incomplete relation if there exists a possible world of this relation in which the fd is satisfied in the standard way we exhibit a sound and complete axiom system for both strong and weak fds which takes into account the interaction between sfds and wfds an interesting feature of the combined axiom system is that it is not k ary for any natural number k 0 we show that the combined implication problem for sfds and wfds can be solved in time polynomial in the size of the input set of fds finally we show that armstrong relations exist for sfds and wfds

distribution results on the algebra generated by toeplitz sequences a finite dimensional approach abstract let a be an l 1 symbol defined on q d q with d 1 and let us consider the multi indexed sequence of toeplitz matrices t n a with n n d it is well known that t n a is spectrally distributed as a in the sense of the singular values in this paper we prove that a sequence as t n a is spectrally distributed in the sense of the singular values as the measurable function a for a l 1 and and ranging in any finite set of values

optimal design of an optical lens we wish to construct an optimal optical lens in the sense that it focuses all rays from a given source onto a given focal point the construction can be reduced mathematically to a global study of a system of two nonlinear functional differential equations we prove that this system has precisely two analytic solutions

a generalization of the runge kutta iteration iterative solvers in combination with multi grid have been used extensively to solve large algebraic systems one of the best known is the runge kutta iteration we show that a generally used formulation a jameson numerical solution of the euler equations for compressible inviscid fluids in f angrand a dervieux j a desideri r glowinski eds numerical methods for the euler equations of fluid dynamics siam philadelphia 1985 pp 199 245 does not allow to form all possible polynomial transmittance functions and we propose a new formulation to remedy this without using an excessive number of coefficients after having converted the optimal parameters found in previous studies e g b van leer c h tai k g powell design of optimally smoothing multi stage schemes for the euler equations aiaa paper 89 1923 1989 we compare them with those that we obtain when we optimize for an integrated 2 grid v cycle and show that this results in superior performance using a low number of stages we also propose a variant of our new formulation that roughly follows the idea of the martinelli jameson scheme a jameson analysis and design of numerical schemes for gas dynamics 1 artificial diffusion upwind biasing limiter and their effect on multigrid convergence int j comput fluid dyn 4 1995 171 218 j v lassaline optimal multistage relaxation coefficients for multigrid flow solvers http www ryerson ca jvl papers cfd2005 pdf used on the advection diffusion equation which that can be extended to other types gains in the order of 30 50 have been shown with respect to classical iterative schemes on the advection equation better results were also obtained on the advection diffusion equation than with the martinelli jameson coefficients but with less than half the number of matrix vector multiplications

functional dependencies in horn theories abstract this paper studies functional dependencies in horn theories both when the theory is represented by its clausal form and when it is defined as the horn envelope of a set of models we provide polynomial algorithms for the recognition of whether a given functional dependency holds in a given horn theory as well as polynomial algorithms for the generation of some representative sets of functional dependencies we show that some problems of inferring functional dependencies e g constructing an irredundant fd cover are computationally difficult we also study the structure of functional dependencies that hold in a horn theory showing that every such functional dependency is in fact a single positive term boolean function and prove that for any horn theory the set of its minimal functional dependencies is quasi acyclic finally we consider the problem of condensing a horn theory prove that any horn theory has a unique condensation and develop an efficient polynomial algorithm for condensing horn theories

closed sets and translations of relation schemes abstract the concept of translation of relation schemes is introduced some characterizations of various closed sets in relation schemes such as generators coatoms or antikeys are investigated from different aspects the connections between these objects in a given relation scheme alone and in the prime and translated relation schemes are presented it is shown that translating a relation scheme can be done in o f r time and testing whether a subset of attributes is an antikey can be done in o f r 2 time where f is the number of functional dependencies and r is the number of attributes it is shown that computing the intersection of all antikeys is np complete but computing their union can be done in polynomial time a connection between three well known results of beeri et al demetrovics and thi and ginsburg and hull about relations representing a given relation scheme or sperner system is presented

algorithms for generating an armstrong relation and inferring functional dependencies in the relational datamodel abstract the main purpose of this paper is to give some new combinatorial algorithms for generating an armstrong relation from a given relation scheme s and inferring functional dependencies fds which hold in a relation r we estimate the time complexities of them it is known that worst case time complexities of generating an armstrong relation and inferring the fds are exponential however we show that our algorithms are effective in many cases i e in these cases they require polynomial time in the size of s in a case generating an armstrong relation in the size of r in a case inferring the fds for bcnf class of relations and relation schemes we also present two effective algorithms constructing an armstrong relation and finding a set of fds we give a class of relations and relation schemes for which generating an armstrong relation and inferring the fds are solved in polynomial time in this paper we prove that if relations and relation schemes satisfy certain additional properties then the fd relation equivalence problem is solved in polynomial time the keys play important roles in logical and structural investigations for fd in the relational datamodel this paper gives some results concerning keys of relation and relation scheme

optimization of the runge kutta iteration with residual smoothing iterative solvers in combination with multi grid have been used extensively to solve large algebraic systems one of the best known is the runge kutta iteration previously haelterman et al 2009 3 we reformulated the runge kutta scheme and established a model of a complete v cycle which was used to optimize the coefficients of the multi stage scheme and resulted in a better overall performance we now look into aspects of central and upwind residual smoothing within the same optimization framework we consider explicit and implicit residual smoothing and either apply it within the runge kutta time steps as a filter for restriction or as a preconditioner for the discretized equations we also shed a different light on the very high cfl numbers obtained by upwind residual smoothing and point out that damping the high frequencies by residual smoothing is not necessarily a good idea

reasoning with models we develop a model based approach to reasoning in which the knowledge base is represented as a set of models satisfying assignments rather than a logical formula and the set of queries is restricted we show that for every propositional knowledge base kb there exists a set of characteristic models with the property that a query is true in kb if and only if it is satisfied by the models in this set we characterize a set of functions for which the model based representation is compact and provides efficient reasoning these include cases where the formula based representation does not support efficient reasoning in addition we consider the model based approach to abductive reasoning and show that for any propositional kb reasoning with its model based representation yields an abductive explanation in time that is polynomial in its size some of our technical results make use of the monotone theory a new characterization of boolean functions recently introduced the notion of restricted queries is inherent in our approach this is a wide class of queries for which reasoning is efficient and exact even when the model based representation kb provides only an approximate representation of the domain in question moreover we show that the theory developed here generalizes the model based approach to reasoning with horn expressions and captures even the notion of reasoning with horn approximations

optimizing a multigrid runge kutta smoother for variable coefficient convection diffusion equations abstract the theory of generally locally toeplitz or glt for short sequences of matrices is proposed in the analysis of a multigrid solver for the linear systems generated by finite volume finite difference approximations of variable coefficients linear convection diffusion equations in 1d proposed by birken in 2012 and extended here to 2d problems the multigrid solver is used with a runge kutta smoother optimal coefficients for the smoother are found by considering the unsteady linear advection equation and using optimization algorithms in particular in order to reduce the issues of having multiple local minima the sequential quadratic programming sqp mixed with genetic and particle swarm optimization algorithms are proposed numerical results show that our proposals are competitive with respect to other multigrid implementations

solution of the euler equations for two dimensional transonic flow by a multigrid method the assumption of potential o w implies that the o w is irrotational this is not strictly correct when shock waves are present an exact description of transonic inviscid o w requires the solution of the euler equations the numerical solution of the euler equations for steady transonic o ws is therefore a problem of great interest to the aeronautical community it also presents a testing challenge to applied mathematicians and numerical analysts

algorithms for k meet semidistributive lattices in this paper we consider k meet semidistributive lattices and we are interested in the computation of the set colored poset associated to an implicational base the parameter k is of interest since for any finite lattice l there exists an integer k for which l is k meet semidistributive when k 1 they are known as meet semidistributive lattices we first give a polynomial time algorithm to compute an implicational base of a k meet semidistributive lattice from its associated colored poset in other words for a fixed k finding a minimal implicational base of a k meet semidistributive lattice l from a context fca literature of l can be done not just in output polynomial time which is open in the general case but in polynomial time in the size of the input this result generalizes that in 26 second we derive an algorithm to compute a set colored poset from an implicational base which is based on the enumeration of minimal transversals of a hypergraph and turns out to be in polynomial time for k meet semidistributive lattices 13 20 finally we show that checking whether a given implicational base describes a k meet semidistributive lattice can be done in polynomial time

an optical lens for focusing two pairs of points we construct an optical lens in the x y plane which focuses two pairs of points i e all the rays from a given point x i are focused by the lens at a given point y i for i 1 2 the points x 1 x 2 y 1 y 2 lie on the x axis and the lens has the form left gamma text 1 text text f text 1 text y text underline leqslant text x text underline leqslant text gamma text 2 text text f text 2 text y text text left y right text underline leqslant text y text 0 right where 1 2 are given and f i 0 0 f i y f i y we then let x 2 x 1 y 2 y 1 and investigate the limiting lens we show that this limit is generally not a symmetric lens i e f 1 f 2 0

locally toeplitz sequences spectral properties and applications abstract the notion of locally toeplitz sequence of matrices is introduced which extends the notion of toeplitz sequence of matrices the singular value distribution and in the hermitian case the eigenvalue distribution is completely characterized for locally toeplitz sequences and their sums obtaining weighted szeg formulas which extend well known results due to tyrtyshnikov concerning toeplitz matrices indeed any toeplitz sequence t n f where f is bounded is proved to be a locally toeplitz sequence moreover sufficient conditions are given for the product of two locally toeplitz sequences to be also locally toeplitz by combining these theoretic results we are able to explicitly compute the asymptotic spectral distribution of a large class of matrices arising in the applications including the algebra generated by toeplitz sequences and virtually all matrices resulting from the discretization of a unidimensional differential operator with non constant coefficients finally a large number of examples is discussed

equivalence of relational database schemes we investigate the question of when two database schemes embody the same information we argue that this question reduces to the equivalence of the sets of fixed points of the project join mappings associated with the two database schemes in question when data dependencies are given we need only consider those fixed points that satisfy the dependencies a polynomial algorithm to test the equivalence of database schemes when there are no dependencies is given we also provide an exponential algorithm to handle the case where there are functional and or multivalued dependencies furthermore we give a polynomial time test to determine whether a project join mapping preserves a set of functional dependencies and a polynomial time algorithm for equivalence of database schemes whose project join mappings do preserve the given set of functional dependencies lastly we introduce the update sets approach to database design as an application of these results

a lens focusing light at two different wavelengths abstract a single element lens capable of focusing light from a source at two different wavelengths to two distinct foci in the image space is studied in this paper results are obtained concerning the existence and uniqueness of the lens the investigation is centred around a system of functional differential equations derived from fermat s principle and an asymptotic method is developed for the special case when the image space foci are close and the source nearly monochromatic

dynamic functional dependencies and database aging a simple extension of the relational model is introduced to study the effects of dynamic constraints on database evolution both static and dynamic constraints are used in conjunction with the model the static constraints considered here are functional dependencies fds the dynamic constraints involve global updates and are restricted to certain analogs of fds called dynamic fds the results concern the effect of the dynamic constraints on the static constraints satisfied by the database in the course of time the effect of the past history of the database on the static constraints is investigated using the notions of age and age closure the connection between the static constraints and the potential future evolution of the database is briefly discussed using the notions of survivability and survivability closure

on the asymptotic spectrum of finite element matrix sequences we derive a new formula for the asymptotic eigenvalue distribution of stiffness matrices obtained by applying p 1 finite elements with standard mesh refinement to the semielliptic pde of second order in divergence form nabla k nabla t u f on o u g on partial omega here omega subset r 2 and k is supposed to be piecewise continuous and pointwise symmetric semipositive definite the symbol describing this asymptotic eigenvalue distribution depends on the pde but also both on the numerical scheme for approaching the underlying bilinear form and on the geometry of triangulation of the domain our work is motivated by recent results on the superlinear convergence behavior of the conjugate gradient method which requires the knowledge of such asymptotic eigenvalue distributions for sequences of matrices depending on a discretization parameter h when h to 0 we compare our findings with similar results for the finite difference method which were published in recent years in particular we observe that our sequence of stiffness matrices is part of the class of generalized locally toeplitz sequences for which many theoretical tools are available this enables us to derive some results on the conditioning and preconditioning of such stiffness matrices

an equivalence between relational database dependencies and a fragment of propositional logic it is known that there is an eqmvalence between functional dependencies m a relatmonal database and a certain fragment of proposmonal logic thins eqmvalence is extended to include both functional and multivalued dependencmes thus for each dependency there is a corresponding statement m proposmonal logic it ms then shown that a dependency funcuonal or multivalued is a consequence of a set of dependencies ff and only ff the corresponding proposiuonal statement s a consequence of the corresponding set of proposmonal statements examples are given to show that these techniques are valuable mn provmdmg much shorter proofs of theorems about dependencies than have been obtained by more tradmonal means it is shown that this eqmvalence cannot be extended to include either join dependencies or embedded multmvalued dependencies

optimal runge kutta smoothers for the p multigrid discontinuous galerkin solution of the 1d euler equations this work presents a family of original runge kutta methods specifically designed to be effective relaxation schemes in the numerical solution of the steady state solution of purely advective problems with a high order accurate discontinuous galerkin space discretization and a p multigrid solution algorithm the design criterion for the construction of the runge kutta methods here developed is different form the one traditionally used to derive optimal runge kutta smoothers for the h multigrid algorithm which are designed in order to provide a uniform damping of the error modes in the high frequency range only the method here proposed is instead designed in order to provide a variable amount of damping of the error modes over the entire frequency spectrum the performance of the proposed schemes is assessed in the solution of the steady state quasi one dimensional euler equations for two test cases of increasing difficulty some preliminary results showing the performance for multidimensional applications are also presented

generalized locally toeplitz sequences spectral analysis and applications to discretized partial differential equations abstract starting from the finite difference discretization of an elliptic second order pde as i j 1 d x i a i j x x j u x b x over a bounded domain we introduce the notion of generalized locally toeplitz sequence of matrices the singular value distribution the eigenvalue distribution in the hermitian case is studied and characterized for generalized locally toeplitz sequences in terms of weighted multidimensional szeg formulas this extends preceding results attributed to tilli which concern the unilevel case the application of this theoretic analysis to the numerical solution of pdes is finally discussed

a note on solving cauchy integral equations of the first kind by iterations abstract the successive approximation method was applied for the first time by n i ioakimidis to solve practical cases of a cauchy singular integral equation the airfoil one in this paper we study a more general case we prove the convergence of the method in this general case the proposed method has been tested for two kernels which are particularly important in practice finally some numerical examples illustrate the accuracy of the method

the glt class as a generalized fourier analysis and applications recently the class of generalized locally toeplitz glt sequences has been introduced as a generalization both of classical toeplitz sequences and of variable coefficient differential operators and for every sequence of the class it has been demonstrated that it is possible to give a rigorous description of the asymptotic spectrum in terms of a function the symbol that can be easily identified this generalizes the notion of a symbol for differential operators discrete and continuous or for toeplitz sequences for which it is identified through the fourier coefficients and is related to the classical fourier analysis the glt class has nice algebraic properties and indeed it has been proven that it is stable under linear combinations and products in this paper we prove that the considered class is closed under inversion as well when the sequence which is inverted shows a sparsely vanishing symbol sparsely vanishing symbol a symbol which vanishes at most in a set of zero lebesgue measure furthermore we show that the glt class virtually includes any finite difference or finite element discretization of pdes and based on this we demonstrate that our results on glt sequences can be used in a pde setting in various directions 1 as a generalized fourier analysis for the study of iterative and semi iterative methods when dealing with variable coefficients non rectangular domains non uniform gridding or triangulations 2 in order to provide a tool for the stability analysis of pde numerical schemes e g a necessary von neumann criterium for variable coefficient systems of pdes is obtained uniformly with respect to the boundary conditions 3 for a multigrid analysis of convergence and for providing spectral information on large preconditioned systems in the variable coefficient case etc the final part of the paper deals indeed with problems 1 3 and other possible directions in which the glt analysis can be conveniently employed

finitely specifiable implicational dependency families une famille de dependance d implication id est une famille d exemples de base de donnees relationnelles composee de tous les cas qui satisfont un ensemble donne d ids dans cet article on montre que la collection de familles de dependances d implication specifiable de maniere finie n est pas bloquee sous les dependances de jointure ou de projection

fuzzy functional dependencies and independencies in extended fuzzy relational database models abstract in this paper we have considered data dependencies more specifically restricted fuzzy functional dependencies in the framework of extended fuzzy relational database model a formal system is presented for the derivation of fuzzy functional dependencies which is both sound and complete the problem of lossless join decomposition is discussed in the framework of extended fuzzy relational model then we introduce a formal statement of fuzzy functional independency which expresses the fact that the corresponding restricted fuzzy functional dependency is not a legitimate constraint of the real world a formal system is presented for the derivation of restricted fuzzy functional dependencies and independencies the various problems such as the membership problem the contradiction problem and the problem of canonical represents are discussed a cover called minimal cover of restricted fuzzy functional independencies is shown to be unique

optimal multistage schemes for euler equations with residual smoothing a numerical technique composed of the van leer tai powell optimization and a modified procedure is applied to design multistage schemes that give optimal damping of high frequencies for given upwind biased spatial differencing with implicit and explicit residual smoothing the analysis is done for a scalar convection equation in one space dimension the object of this technique is to make the schemes suited for multigrid acceleration the optimal multistage schemes and their damping properties are presented in this paper by keeping the multistage coefficients from the one dimensional analysis and simply redefining the courant number the schemes can be applied to multidimensional problems a fast euler code is used to evaluate the suitability of the schemes for multidimensional multigrid computations numerical results show that the modified schemes effectively enhance the convergence performance on both single and multiple grids

aggregation based algebraic multigrid for convection diffusion equations we consider the iterative solution of large sparse linear systems arising from the upwind finite difference discretization of convection diffusion equations the system matrix is then an m matrix with nonnegative row sum and further when the convective flow has zero divergence the column sum is also nonnegative possibly up to a small correction term we investigate aggregation based algebraic multigrid methods for this class of matrices a theoretical analysis is developed for a simplified two grid scheme with one damped jacobi postsmoothing step an uncommon feature of this analysis is that it applies directly to problems with variable coefficients e g to problems with recirculating convective flow on the basis of this theory we develop an approach in which a guarantee is given on the convergence rate thanks to an aggregation algorithm that allows an explicit control of the location of the eigenvalues of the preconditioned matrix some issues that remain beyond the analysis are discussed in the

high accuracy multigrid solution of the 3d convection diffusion equation we present an explicit fourth order compact finite difference scheme for approximating the three dimensional 3d convection diffusion equation with variable coefficients this 19 point formula is defined on a uniform cubic grid fourier smoothing analysis is performed to show that the smoothing factor of certain relaxation techniques used with the scheme is smaller than 1 we design a parallelization oriented multigrid method for fast solution of the resulting linear system using a four color gauss seidel relaxation technique for robustness and efficiency and a scaled residual injection operator to reduce the cost of multigrid inter grid transfer operator numerical experiments on a 16 processor vector computer are used to test the high accuracy of the discretization scheme as well as the fast convergence and the parallelization or vectorization efficiency of the solution method several test problems are solved and highly accurate solutions of the 3d convection diffusion equations are obtained for small to medium values of the grid reynolds number effects of using different residual projection operators are compared on both vector and serial computers

order dependency in the relational model abstract the relational model is formally extended to include fixed orderings on attribute domains a new constraint called order dependency is then introduced to incorporate semantic information involving these orderings it is shown that this constraint can be applied to enhance the efficiency of an implemented database the thrust of the paper is to study logical implication for order dependency the main theoretical results consist in i introducing a formalism analogous to propositional calculus for analyzing order dependency ii exhibiting a sound and complete set of inference rules for order dependency and iii demonstrating that determining logical implication for order dependency is co np complete it is also shown that there are sets of order dependencies for which no armstrong relations exist

identifying the minimal transversals of a hypergraph and related problems the paper considers two decision problems on hypergraphs hypergraph saturation and recognition of the transversal hypergraph and discusses their significance for several search problems in applied computer science hypergraph saturation i e given a hypergraph cal h decide if every subset of vertices is contained in or contains some edge of cal h is shown to be co np complete a certain subproblem of hypergraph saturation the saturation of simple hypergraphs i e sperner families is shown to be under polynomial transformation equivalent to transversal hypergraph recognition i e given two hypergraphs cal h 1 cal h 2 decide if the sets in cal h 2 are all the minimal transversals of cal h 1 the complexity of the search problem related to the recognition of the transversal hypergraph the computation of the transversal hypergraph is an open problem this task needs time exponential in the input size it is unknown whether an output polynomial algorithm exists for several important subcases for instance if an upper or lower bound is imposed on the edge size or for acyclic hypergraphs output polynomial algorithms are presented computing or recognizing the minimal transversals of a hypergraph is a frequent problem in practice which is pointed out by identifying important applications in database theory boolean switching theory logic and artificial intelligence ai particularly in model based diagnosis

on completing tables to satisfy functional dependencies abstract several sufficiency conditions on a given set of functional dependencies are presented to ensure that a partially specified table can be extended to a completely specified table satisfying the dependencies in particular each table over a minimal key can be extended to such a table over the entire attribute set

representing lattices using many valued relations this paper provides a representation theorem of lattices using many valued relations we show that any many valued relation can be associated to a unique lattice which is a meet sublattice of a product of chains conversely to any lattice we can associate a many valued relation such that its associated lattice is isomorphic to the initial one thereby we obtain a representation theorem of lattices using many valued relations moreover since several many valued relations might have the same associated lattice we give a characterization of the minimal many valued relation that can be associated to a lattice we then sketch a polynomial time algorithm which computes such a minimal relation from either a lattice or an arbitrary relation this representation presents several advantages it is smaller than the usual binary representation all known reconstruction algorithms working on binary relation can be used without loss of efficiency it can be used by existing data mining processes

abstract functional dependency structures abstract the functional dependency fd structures known mainly from database relationships theory appear to include other kinds of dependencies among objects like functions boolean or random variables etc we examine the representations of fd relations in the matroid theory manner a galois connection of the fd relations with closure system is elaborated

horn approximations of empirical data abstract formal ai systems traditionally represent knowledge using logical formulas sometimes however a model based representation is more compact and enables faster reasoning than the corresponding formula based representation the central idea behind our work is to represent a large set of models by a subset of characteristic models more specifically we examine model based representations of horn theories and show that there are large horn theories that can be exactly represented by an exponentially smaller set of characteristic models we show that deduction based on a set of characteristic models requires only polynomial time as it does using horn theories more surprisingly abduction can be performed in polynomial time using a set of characteristic models whereas abduction using horn theories is np complete finally we discuss algorithms for generating efficient representations of the horn theory that best approximates a general set of models

characterizations for functional dependency and boyce codd normal form families abstract a functional dependency fd family was recently defined 20 as the set of all instances satisfying some set of functional dependencies a boyce codd normal form abbreviated bcnf family is defined here as an fd family specified by some bcnf set of functional dependencies the purpose of this paper is to present set theoretic algebraic characterizations relating to both types of families two characterizations of f i the smallest fd family containing the family i of instances are established the first involves the notion of agreement a concept related to that of a closed set of attributes the second describes f i as the smallest family of instances containing i and closed under four specific operations on instances companion results are also given for bcnf families the remaining results concern characterizations involving the well known operations of projection join and union two characterizations for when the projection of an fd family is again an fd family are given several corollaries are obtained including the effective decidability of whether a projection of an fd family is an fd family the problem for bcnf families disappears since it is shown that the projection of a bcnf family is always a bcnf family analogous to results for fd families presented in 20 characterizations of when the join and union of bcnf families are bcnf families are given finally the collections of all fd families and all bcnf families are characterized in terms of inverse projection operations and intersection

on a numerical minimization problem for a new class of nonlinear partial differential equations arising in nonspherical geometrical optics and beamforming lenticular antennas 11 abstract an improved mixed numerical and analytical procedure for solving the problem of determining the surfaces i and i of a three dimensional aperture or volume extremized and ellipticized bifocal nonspherical singlet optical radar or sonar lens antenna defined by a system of four nonlinear partial differential equations with specified symmetry and boundary conditions subject to a minimizing side condition on the antenna focal length of f number is presented the index of refraction n0 the maximum lens diameter 2b0 and the bifocal off axis angle 0 are assumed given the results obtained are of approximately twice the accuracy of those computed in part i

on the structure of armstrong relations for functional dependencies an armstrong relation for a set of functional dependencies fds is a relation that satisfies each fd implied by the set but no fd that is not implied by it the structure and size number of tuples of armstrong relatsons are investigated upper and lower bounds on the size of minimal sized armstrong relations are derived and upper and lower bounds on the number of distinct entries that must appear m an armstrong relation are given it is shown that the time complexity of finding an armstrong relation gwen a set of functional dependencies is precisely exponential in the number of attributes also shown s the falsity of a natural conjecture which says that almost all relations obeying a given set of fds are armstrong relations for that set of fds finally armstrong relations are used to generahze a result obtained by demetrovics using quite complicated methods about the possible sets of keys for a relauon