The action of the symmetric group S4 on the Tetrahedron algebra, introduced by Hartwig and Terwilliger [HT05], is studied. This action gives a grading of the algebra which is related to its decomposition in [HT05] into a direct sum of three subalgebras isomorphic to the Onsager algebra. The ideals of both the Tetrahedron algebra and the Onsager algebra are determined.

1. Davey Smith G, Ebrahim S. ‘Mendelian randomization’: can genetic epidemiology contribute to understanding environmental determinants of disease? Int J Epidemiol 2003;32:1–22. 2. Burgess S, Timpson NJ, Ebrahim S, Davey Smith G. Mendelian randomization: where are we now and where are we going? Int J Epidemiol 2015;44:379–88. 3. Haycock PC, Burgess S, Wade KH, Bowden J, Relton C, Davey Smith G....

Toeplitz matrices have applications to different problems of statistical mechanics. Recently it was used for calculation of entanglement entropy in spin chains. In the paper we review these recent developments. We use Fisher-Hartwig formula, as well as the recent results concerning the asymp-totics of the block Toeplitz determinants, to calculate entanglement entropy of large block of spins in ...

In this paper a representation is given for the Drazin inverse of a 2 × 2 operator matrix, extending to Banach spaces results of Hartwig, Li and Wei [SIAM J. Matrix Anal. Appl., 27 (2006) pp. 757–771]. Also, formulae are derived for the Drazin inverse of an operator matrix M under some new conditions.

We describe the asymptotics of the spectral norm of finite Toeplitz matrices generated by functions with Fisher-Hartwig singularities as the matrix dimension goes to infinity. In the case of positive generating functions, our result provides the asymptotics of the largest eigenvalue, which is of interest in time series with long-range memory.

The introduction of a fluorinated moiety into organic molecules can result in profound changes of the biological activities of these molecules, and therefore, this strategy has been widely used in drug design. Among various fluorinated moieties, the trifluoromethyl group (CF3) has been the focus of increasing attention owing to its capacity to act as a lipophilic electron-withdrawing group. In ...

Ann Thorac Surg Sievers, Uwe Schönbeck and Claus Bartels Jan Felix Christiansen, Dirk Hartwig, J.F. Matthias Bechtel, Harald Klüter, HansH. atherosclerotic coronary arteries Diseased vein grafts express elevated inflammatory cytokine levels compared with http://ats.ctsnetjournals.org/cgi/content/full/77/5/1575 located on the World Wide Web at: The online version of this article, along with upda...