Real Hypersurfaces of Type A in Complex Two-Plane Grassmannians Related to The Reeb Vector Field

Y. J. Suh and H. Lee (Bull. Korean. Math. Soc. 47, 551-561 (2010)) characterized real hypersurfaces $M$ of type $B$ by the invariance of vector bundle $JTM^\perp$ under the shape operator and the orthogonality of $JTM^\perp$ and $\mathcal {J}TM^\perp$, where $TM^\perp$, $J$ and $\mathcal J$ are the normal bundle of $M$, K\"ahler structure and Quaternionic K\"ahler structure of $G_2({\mathbb{C}}^{m+2})$ respectively. In this paper, we characterize real hypersurfaces $M$ of type A by the invariance of the vector bundle $JTM^\perp$ under the shape operator with the Reeb vector field in $\mathcal {J}TM^\perp$.Comment: 8 page

Torsion and its interaction with other internal forces in EN 1993-1-1: a new approach

This is the accepted version of the following article: [Real, E., Bordallo, J. and Real, E. (2016), Torsion and its interaction with other internal forces in EN 1993-1-1 – a new approach. Steel Construction, 9: 240–248. doi:10.1002/stco.201500005], which has been published in final form at http://onlinelibrary.wiley.com/doi/10.1002/stco.201500005/abstractThis paper presents and clarifies the treatments included in EN 1993-1-1, relating to checking the resistance of the steel cross-sections under torsion and its interaction with other internal forces. Specifically, the origin of the formulations for shear-torsion interaction, which was not found in the literature, is presented. Furthermore, a very simple formulation based on the expressions used for shear-torsion interaction is developed, in order to take into account bending-warping torsion interaction for symmetrical double T cross-sections (IP and HE steel profiles). Such formulation overcomes the overly conservative approach stated in EN 1993-1-1, for Class 1 and Class 2 cross-sections (plastic and compact cross-sections). Finally, a rigorous method for the determination of the bending resistance of cross-sections is proposed, considering the interaction with shear and torsion. The proposal is well suited to the concept of cross-sectional class and it is perfectly consistent with the approaches set out to consider the bending-shear and bending-warping torsion interactions.Peer ReviewedPostprint (author's final draft

Real Time in a Real Operating System

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