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The paper describes an algorithm to compute the canonical basis of an irreducible module over a quantized enveloping algebra of a finite-dimensional semisimple Lie algebra. The algorithm works for any module that is constructed as a submodule of a tensor product of modules with known canonical bases.
Abstract:We give an algorithm for finding the module of linear dependencies of the roots of a monic integral polynomial. Using this, we describe an algorithm for constructing the algebraic hull of a given matrix Lie algebra in characteristic zero.
Let G be a simple algebraic group over an algebraically closed field with Lie algebra g. Then the orbits of nilpotent elements of g under the adjoint action of G have been classified. We describe a simple algorithm for finding a representative of a nilpotent orbit. We use this to compute lists of representatives of these orbits for the Lie algebras of exceptional type. Then we give two applications. The first one concerns settling a conjecture by Elashvili on the index of centralizers of nilpotent orbits, for the case where the Lie algebra is of exceptional type. The second deals with minimal dimensions of centralizers in centralizers.
Data on the duration of major depressive episodes (MDE) in the general population are sparse.
To assess the duration of MDE and its clinical and socio-demographic determinants in a study group drawn from the general population with newly originated episodes of major depression.
The Netherlands Mental Health Survey and Incidence Study is a prospective epidemiological survey in the adult population (n=7076), using the Composite International Diagnostic Interview. Duration of MDE over 2 years was assessed with a Life Chart Interview.
The median duration of MDE was 3.0 months; 50% of participants recovered within 3 months, 63% within 6 months, 76% within 12 months and nearly 20% had not recovered at 24 months. Determinants of persistence were severity of depression and comorbid dysthymia. A recurrent episode predicted shorter duration.
Although half of those affected with MDE recovered rapidly, the risk of chronicity (duration 24months or more) was considerable. This underlines the necessity of diagnosing and treating those at risk.
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