Utilizing scale-free networks to support the search for scientific publications

When searching for scientific publications, users today often rely on search engines such as Yahoo.com. Whereas searching for publications whose titles are known is considered to be an easy task, users who are looking for important publications in research fields they are unfamiliar with face greater diffiulties since few or no indications of a publication’s importance to the respective fields are given. In this paper we investigate the application of the theory of scale-free networks to derive importance indicators for a collection of publications. A tool was developed to support the user in his publication search by visualizing the publications’ importance indicators derived from the number of citations received and the publication’s age as well as visualizing part of the citation network structure. A preliminary user study indicates the utility of our approach and warrants further research in that direction

A Word Sense-Oriented User Interface for Interactive Multilingual Text Retrieval

In this paper we present an interface for supporting a user in an interactive cross-language search process using semantic classes. In order to enable users to access multilingual information, different problems have to be solved: disambiguating and translating the query words, as well as categorizing and presenting the results appropriately. Therefore, we first give a brief introduction to word sense disambiguation, cross-language text retrieval and document categorization and finally describe recent achievements of our research towards an interactive multilingual retrieval system. We focus especially on the problem of browsing and navigation of the different word senses in one source and possibly several target languages. In the last part of the paper, we discuss the developed user interface and its functionalities in more detail

Bivariate spline interpolation with optimal approximation order

Let be a triangulation of some polygonal domain f c R2 and let S9 (A) denote the space of all bivariate polynomial splines of smoothness r and degree q with respect to A. We develop the first Hermite-type interpolation scheme for S9 (A), q >_ 3r + 2, whose approximation error is bounded above by Kh4+i, where h is the maximal diameter of the triangles in A, and the constant K only depends on the smallest angle of the triangulation and is independent of near-degenerate edges and nearsingular vertices. Moreover, the fundamental functions of our scheme are minimally supported and form a locally linearly independent basis for a superspline subspace of Sr, (A). This shows that the optimal approximation order can be achieved by using minimally supported splines. Our method of proof is completely different from the quasi-interpolation techniques for the study of the approximation power of bivariate splines developed in [71 and [181

Strong unicity in nonlinear approximation and free knot splines

We give a necessary alternation condition for unique local best approximation from Sm,k, the set of splines of degree m with k free knots. This result is related to a conjecture of L.L. Schumaker. Moreover, we give a characterization of functions from the interior of the strong unicity set for S1m,k, the set of splines of degree m with k free simple knots, and show that this set is dense in the unicity set. Then we give a general characterization of suns for strong unicity and show that S1m,k is a set of this type, although it is not a sun

Bivariate Interpolation by Splines and Approximation Order

We construct Hermite interpolation sets for bivariate spline spaces of arbitrary degree and smoothness one on non-rectangular domains with uniform type triangulations. This is done by applying a general method for constructing Lagrange interpolation sets for bivariate spline spaecs of arbitrary degree and smoothness. It is shown that Hermite interpolation yields (nearly) optimal approximation order. Applications to data fitting problems and numerical examples are given

Strong unicity of best approximations : a numerical aspect

The set of functions in C(T) which have a strongly unique best approximation from a given finite-dimensional subspace is denoted by SU(G). Since strong unicity plays an important role in numerical computations and since there the functions are only known up to some error, it is natural to ask what are the functions from the interior of SU(G). A complete characterization of those functions is given and the result is applied to weak Chebyshev and spline subspaces

Bivariate Segment Approximation

In this note we state some problems on approximation by univariate splines with free knots, bivariate segment approximation and tensor product splines with variable knot lines. There is a vast literature on approximation and interpolation by univariate splines with fixed knots (see e.g. the books of de Boor [1], Braess [2], DeVore & Lorentz [4], Powell [20], Schumaker [21], Nürnberger [13] and the book of Chui [3] on multivariate splines). On the other hand, numerical examples show that in general, the error is much smaller if variable knots are used for the approximation of functions instead of fixed knots. This is true for univariate splines as well as for bivariate splines. But approximation by splines with free knots leads to rather difficult nonlinear problems.[...

The metric projection for free knot splines

Only few results are known on continuity properties of the set-valued metric projection in nonlinear uniform approximation. In this paper we investigate this mapping in the case of best uniform approximation by splines of degree m with k free knots. A characterization of those functions at which the metric projection is upper semicontinuous is given. It follows that the metric projection is upper semicontinuous if and only if k ≤ m, and that it is upper semicontinuous at all "normal" functions. On the other hand, it is shown that the metric projection is never lower semicontinuous

Interpolation by spline spaces on classes of triangulations

We describe a general method for constructing triangulations Δ which are suitable for interpolation by Srq(Δ),