The apolar bilinear form in geometric modeling

Some recent methods of Computer Aided Geometric Design are related to the apolar bilinear form, an inner product on the space of homogeneous multivariate polynomials of a fixed degree, already known in 19th century invariant theory. Using a generalized version of this inner product, we derive in a straightforward way some of the recent results in CAGD, like Marsden's identity, the expression for the de Boor-Fix functionals, and recursion schemes for the computation of B-patches and their derivatives.

     

Equivalence of functional limit theorems for stationary point processes and their Palm distributions

Summary Let P be the distribution of a stationary point process on the real line and let P ⁰ be its Palm distribution. In this paper we consider two types of functional limit theorems, those in terms of the number of points of the point process in (0, t] and those in terms of the location of the nth point right of the origin. The former are most easily obtained under P and the latter under P ⁰. General conditions are presented that guarantee equivalence of either type of functional limit theorem under both probability measures, and under a third, P ₁, which plays a role in the proofs and is obtained from P by shifting the origin to the first point of the process on the right.In a brief final section the obtained results for either type of functional limit theorem are extended to equivalences between the two types by applying well-known results about processes drifting to infinity and the corresponding inverse processes.     

Epistemic irrelevance in credal nets: The case of imprecise Markov trees

We focus on credal nets, which are graphical models that generalise Bayesian nets to imprecise probability. We replace the notion of strong independence commonly used in credal nets with the weaker notion of epistemic irrelevance, which is arguably more suited for a behavioural theory of probability. Focusing on directed trees, we show how to combine the given local uncertainty models in the nodes of the graph into a global model, and we use this to construct and justify an exact message-passing algorithm that computes updated beliefs for a variable in the tree. The algorithm, which is linear in the number of nodes, is formulated entirely in terms of coherent lower previsions, and is shown to satisfy a number of rationality requirements. We supply examples of the algorithm’s operation, and report an application to on-line character recognition that illustrates the advantages of our approach for prediction. We comment on the perspectives, opened by the availability, for the first time, of a truly efficient algorithm based on epistemic irrelevance.     

Employing different loss functions for the classification of images via supervised learning

Supervised learning methods are powerful techniques to learn a function from a given set of labeled data, the so-called training data. In this paper the support vector machines approach is applied to an image classification task. Starting with the corresponding Tikhonov regularization problem, reformulated as a convex optimization problem, we introduce a conjugate dual problem to it and prove that, whenever strong duality holds, the function to be learned can be expressed via the dual optimal solutions. Corresponding dual problems are then derived for different loss functions. The theoretical results are applied by numerically solving a classification task using high dimensional real-world data in order to obtain optimal classifiers. The results demonstrate the excellent performance of support vector classification for this particular problem.     

Scroll codes

We study algebraic geometric codes obtained from rational normal scrolls. We determine the complete weight hierarchy and spectrum of these codes.      

Farkas-type results for fractional programming problems

Considering a constrained fractional programming problem, within the present paper we present some necessary and sufficient conditions, which ensure that the optimal objective value of the considered problem is greater than or equal to a given real constant. The desired results are obtained using the Fenchel–Lagrange duality approach applied to an optimization problem with convex or difference of convex (DC) objective functions and finitely many convex constraints. These are obtained from the initial fractional programming problem using an idea due to Dinkelbach. We also show that our general results encompass as special cases some recently obtained Farkas-type results.     

Balinski—Tucker simplex tableaus: Dimensions, degeneracy degrees, and interior points of optimal faces

This paper shows the relationship between degeneracy degrees and multiple solutions in linear programming (LP) models. The usual definition of degeneracy is restricted to vertices of a polyhedron. We introduce degeneracy for nonempty subsets of polyhedra and show that for LP-models for which the feasible region contains at least one vertex it holds that the dimension of the optimal face is equal to the degeneracy degree of the optimal face of the corresponding dual model. This result is obtained by means of the so-called Balinski—Tucker (B—T) Simplex Tableaus. Furthermore, we give a strong polynomial algorithm for constructing such a B—T Simplex Tableau when a solution in the relative interior of the optimal face is known. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.     

The purity of laboratory chemicals with regard to measurement uncertainty

The purity P of laboratory chemicals is often declared in the form P > or = xy% (e.g., P > or = 97%). With a randomly chosen set of 40 compounds we found that their purity is generally closer to 100% than to the lower limit. The distribution of the purity data as found in the laboratory depends on the analytical technique used. Whereas purities determined by chromatography do not exceed 100% (because the sum of all observed peak areas is set to 100%), the purities obtained by titration can exceed 100% (because the functionality of the compound is measured). Therefore, the data for these two groups need to be dealt with in different ways. For purities based on titration we propose to use a rectangular distribution with a range from Pmin to 101%, an expected purity value which is the mean and a standard uncertainty of the purity u(P) of 29% of the range. Purities determined by chromatography can be described with a triangular distribution (ramp function). One leg of the triangle represents the range from Pmin to 100% and the right-angle is located at 100%. The expected value is the median and the uncertainty u(P) is 24% of the range. These proposals match the experimental data well.     

Semiclassical theory for atom—surface scattering: Calculations on Ar + W(110)

By using a semiclassical approach we derive an effective potential which governs the motion of the colliding atom/molecule. The effective potential depends on the surface temperature and the phonon excitation during the collision. Numerical results on Ar + W(110) are discussed and compared with experimental data.