We introduce a natural symmetry condition for a pseudodifferential operator on a manifold with cylindrical ends ensuring that the operator admits a doubling across the boundary. For such operators we prove an explicit index formula containing, apart from the Atiyah-Singer integral, a finite number of residues of the logarithmic derivative of the conormal symbol.

In this case study I explore why the conflict of the Basque country which involves Euskadi Ta Askatasuna (ETA) persists. The purpose is to gain an understanding of the existence of ETA. Working somewhat  as an ethnographer the method has been to observe Basque society and through semi- structured interviews gain an understanding of ETAs existence. As exemplified with the hermeneutic circle, this case study starts out with a preunderstanding of the reasons for the existence of ETA, a preunderstanding which alters many times during my visit to the Basque country. Keys to understanding the continued existence of ETA is that historical “injustices” are perceived to still exist in the Basque country. Perceptions which question the legitimacy of the Spanish state in the Basque country. Further the social base of ETA, as exemplified in election support for Herri Batasuna (HB), still remains rather strong despite the party‟s illegalization and finally, the perceived lack of a functioning democracy in Spain and the lack of legitimacy of the Spanish state in the Basque country are among the reasons for the continued existence of ETA.

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Many sports biomechanics research studies follow a traditional task analysis concept that there is only one best possible movement pattern and thus focus on the examination of kinematics and kinetics of movement without considering the influence of constraints that are imposed on it. This study developed an interdisciplinary approach by utilizing the principles of ecological task analysis and movement coordination from areas of motor leaning and biomechanics to examine the skill of placement hitting in slo-pitch softball. The choice of evaluating this slo-pitch batting skill to assess movement patterns is pragmatic because of its popularity of the sport and uniqueness of the batting movement. Therefore, the purpose of this study was to examine the influence of two task constraints (stride technique and designated field location) and an environmental constraint (pitched ball location) on the participants batting performances, kinematics, and movement patterns. A three-way ANOVA of 2 fields (same and opposite) x 2 locations of pitch (inside and outside) x 3 strides (open, parallel and closed) repeated measure study was conducted in this study. The results showed that participants were more successful in placing the ball to the same field instead of the opposite field. The pitched ball location and stride techniques did not have a consistent impact on the results across the different hitting conditions. To achieve these batting performance results, participants demonstrated different joint movements and different coordination patterns. Hence, this study supports the rationale of ecological task analysis but not traditional task analysis. Further, to understand the generalizability of the findings, a Euclidean distance analysis was conducted to evaluate the degree of dissimilarity between the individual and group mean results. The results indicated that participants generally showed a low degree of dissimilarity, so they were quite homogeneous as a group. Hence, the results from this study not only enable us to evaluate a human movement skill under the influence of different constraints but educators may apply the findings to other players. A similar interdisciplinary approach is warranted for future research studies in order to better understand the mechanics of human motion.

Zugl.: Trier, Univ., Diss., 1999/2000

Universiẗat, Diss., 2000--Wuppertal.

Universiẗat, Diss., 2003--Bonn.

Universiẗat, Diss., 2004--Münster (Westfalen).

University, Diss., 2004--Bonn.

University, Diss., 2005--Heidelberg.