H. De Meyer

Exponentially-fitted explicit Runge–Kutta methods

An exponentially-fitted explicit Runge–Kutta method is constructed, which exactly integrates differential initial-value problems whose solutions are linear combinations of functions of the form exp(ωx) and exp(−ωx) (ω∈R or iR); this method is compared to a previously constructed method of Simos. Numerical experiments show the efficiency of the new method.

Exponentially fitted Runge-Kutta methods

Abstract Exponentially fitted Runge–Kutta methods with s stages are constructed, which exactly integrate differential initial-value problems whose solutions are linear combinations of functions of the form {x j exp (ωx),x j exp (−ωx)} , ( ω∈ R or i R , j=0,1,…,j max ), where 0⩽j max ⩽⌊s/2−1⌋ , the lower bound being related to explicit methods, the upper bound applicable for collocation methods. Explicit methods with s∈{2,3,4} belonging to that class are constructed. For these methods, a study of the local truncation error is made, out of which follows a simple heuristic to estimate the ω-value. Error and step length control is introduced based on Richardson extrapolation ideas. Some numerical experiments show the efficiency of the introduced methods. It is shown that the same techniques can be applied to construct implicit exponentially fitted Runge–Kutta methods.

On the existence and construction of T -transitive closures

In this paper, it is shown that any fuzzy relation R on an arbitrary universe X has a T - transitive closure. The triangular norm T involved is not subject to any conditions. This existential result can be turned into an explicit expression in the case of a left-continuous triangular norm T . Furthermore, this continuity restriction vanishes in the case of a finite universe X . In that case, the explicit expression can also be operationalized.

Frequency evaluation in exponential fitting multistep algorithms for ODEs

We consider the linear multistep algorithms for first order ODEs and examine the problem of how the λ-frequencies should be tuned in order to obtain the maximal benefit from the exponential fitting versions of such algorithms. We find out that the key of the answer consists in analysing the behaviour of the error. On further investigating the simple case of two-step bdf algorithms we produce formulae for the optimal λs and show that, if the optimal λs are used, the order of the method is increased by one unit. The reported numerical illustrations suggest that further investigations along these lines deserve a real attention.

On the cycle-transitivity of the dice model

We introduce the notion of a dice model as a framework for describing a class of probabilistic relations. We investigate the transitivity of the probabilistic relation generated by a dice model and prove that it is a special type of cycle-transitivity that is situated between moderate stochastic transitivity or product-transitivity on the one side, and Łukasiewicz-transitivity on the other side. Finally, it is shown that any probabilistic relation with rational elements on a three-dimensional space of alternatives which possesses this particular type of cycle-transitivity, can be represented by a dice model. The same does not hold in higher dimensions.

Knowledge accumulation and resolution of data inconsistencies during the integration of microbial information sources

The Internet has emerged as an ever-increasing environment of multiple heterogeneous and autonomous data sources that contain relevant but overlapping information on microorganisms. Microbiologists might therefore seriously benefit from the design of intelligent software agents that assist in the navigation through this information-rich environment, together with the development of data mining tools that can aid in the discovery of new information. These applications heavily depend upon well-conditioned data samples that are correlated with multiple information sources, hence, accurate database merging operations are desirable. Information systems designed for joining the related knowledge provided by different microbial data sources are hampered by the labeling mechanism for referencing microbial strains and cultures that suffers from syntactical variation in the practical usage of the labels, whereas, additionally, synonymy and homonymy are also known to exist amongst the labels. This situation is even complicated by the observation that the label equivalence knowledge is itself fragmentarily recorded over several data sources which can be suspected of providing information that might be both incomplete and incorrect. This paper presents how extraction and integration of label equivalence information from several distributed data sources has led to the construction of a so-called integrated strain database, which helps to resolve most of the above problems. Given the fact that information retrieved from autonomous resources might be overlapping, incomplete, and incorrect, much energy was spent into the completion of missing information, the discovery of new associations between information objects, and the development and application of tools for error detection and correction. Through a thorough evaluation of the different levels of incompleteness and incorrectness encountered within the incorporated data sources, we have finally given proof of the added value of the integrated strain database as a necessary service provider for the seamless integration of microbial information sources.

Algorithms for the computation of T-transitive closures

We present two weight-driven algorithms for the computation of the T-transitive closure of a symmetric binary fuzzy relation on a finite universe X with cardinality n (or, equivalently, of a symmetric (n/spl times/n)-matrix with elements in [0, 1]), with T a triangular norm. The first algorithm is proven to be valid for any triangular norm T, whereas the second algorithm is shown to be valid when T is either the minimum operator or an Archimedean triangular norm. Furthermore, we investigate how these algorithms can be appropriately adapted to generate the T-transitive closure of nonsymmetric binary fuzzy relations (or matrices) as well.

A class of rational cardinality-based similarity measures

A systematic way of generating similarity measures for ordinary sets is presented in the form of a rational expression solely based on cardinalities of the sets involved. Twenty-eight measures are examined carefully and completely classified on the basis of their boundary behaviour and properties of reflexivity and monotonicity. Two types of reflexivity (reflexivity and local reflexivity) and three types of monotonicity (involving, respectively, two, three and four sets) are considered. In addition, 17 of these measures are shown to be T-transitive, with the t-norm T ranging from the drastic product Z to the minimum operator M. The given class of rational cardinality-based measures covers some well-known similarity measures.

On a new type of mixed interpolation

Abstract We approximate every function f by a function f n ( x ) of the form a cos kx + b sin kx + Σ n −2 i =0 c i x i so that f ( jh ) = f n ( jh ) for the n + 1 equidistant points jh , j = 0,…, n . That interpolation function f n ( x ) is proved to be unique and can be written as the sum of the n th-degree interpolation polynomial based on the same points and two correction terms. The error term is also discussed. The results for this mixed type of interpolation reduce to the known results of the polynomial case as the parameter k is tending to 0. This new interpolation theory will be used in the future for the construction of quadrature rules and multistep methods for ordinary differential equations.

Algorithms for computing the min-transitive closure and associated partition tree of a symmetric fuzzy relation

Abstract Based on a previously derived weight-driven algorithm for the computation of the T -transitive closure of an arbitrary binary fuzzy relation on a finite universe, with T a triangular norm, we establish three algorithms for the computation of the min-transitive closure of a symmetric matrix with elements in [0,1]. As a by-product, these algorithms enable to generate the partition tree associated with the min-transitive closure of the given matrix in descending order of the cutting parameter.