The unit ball of the complex P(3H)

Let H be a two-dimensional complex Hilbert space and P(3H) the space of 3-homogeneous polynomials on H. We give a characterization of the extreme points of its unit ball, P(3H), from which we deduce that the unit sphere of P(3H) is the disjoint union of the sets of its extreme and smooth points. We also show that an extreme point of P(3H) remains extreme as considered as an element of L(3H). Finally we make a few remarks about the geometry of the unit ball of the predual of P(3H) and give a characterization of its smooth points.

Geometry in preduals of spaces of 2-homogeneous polynomials on Hilbert spaces

Let H be a (real or complex) Hilbert space. Using spectral theory and properties of the Schatten–Von Neumann operators, we prove that every symmetric tensor of unit norm in HoH is an infinite absolute convex combination of points of the form xox with x in the unit sphere of the Hilbert space. We use this to obtain explicit characterizations of the smooth points of the unit ball of HoH .

Judging where a ball will go: the case of curved free kicks in football

This study examined whether adding spin to a ball in the free kick situation in football affects a professional footballer's perception of the ball's future arrival position. Using a virtual reality set-up, participants observed the flight paths of aerodynamically realistic free kicks with (+/- 600 rpm) and without sidespin. With the viewpoint being fixed in the centre of the goal, participants had to judge whether the ball would have ended up in the goal or not. Results show that trajectories influenced by the Magnus force caused by sidespin gave rise to a significant shift in the percentage of goal responses. The resulting acceleration that causes the ball to continually change its heading direction as the trajectory unfolds does not seem to be taken into account by the participants when making goal judgments. We conclude that the visual system is not attuned to such accelerated motion, which may explain why goalkeepers appear to misjudge the future arrival point of such curved free kicks.

Support vector machine classification for large data sets via minimum enclosing ball clustering

Support vector machine (SVM) is a powerful technique for data classification. Despite of its good theoretic foundations and high classification accuracy, normal SVM is not suitable for classification of large data sets, because the training complexity of SVM is highly dependent on the size of data set. This paper presents a novel SVM classification approach for large data sets by using minimum enclosing ball clustering. After the training data are partitioned by the proposed clustering method, the centers of the clusters are used for the first time SVM classification. Then we use the clusters whose centers are support vectors or those clusters which have different classes to perform the second time SVM classification. In this stage most data are removed. Several experimental results show that the approach proposed in this paper has good classification accuracy compared with classic SVM while the training is significantly faster than several other SVM classifiers.

Cinderella comes to the Ball: Article 14 and the right to non-discrimination in the ECHR

Article 14 ECHR has often been derided as a Cinderella provision, but during the last few years, this has started to change. This article examines how Article 14 has developed, and may live up to its potential as a powerful non-discrimination principle. The case law developments in relation to the “ambit” requirement in Article 14, the development of indirect discrimination case law, and the approval of positive action, all point to a more substantive conception of equality, which offers protection to disadvantaged and vulnerable groups.

Upper and lower bounds of 180 degrees unit element of a ridge waveguide: Calculations and measurements

The upper and lower bounds on the actual solution of any microwave structure is of general interest. The purpose of this letter is to compare some calculations using the mode-matching and finite-element methods, with some measurements on a 180 degrees ridge waveguide insert between standard WR62 rectangular waveguides. The work suggests that the MMM produces an upper bound, while the FEM places a lower bound on the measurement. (C) 2001 John Wiley & Sons, Inc.