Bifurcation along curves for the p-Laplacian with radial symmetry

We study the global structure of the set of radial solutions of a nonlinear Dirichlet problem involving the p-Laplacian with p>2, in the unit ball of $R^N$, N \ges 1. We show that all non-trivial radial solutions lie on smooth curves of respectively positive and negative solutions and bifurcating from the line of trivial solutions. This involves a local bifurcation result of Crandall-Rabinowitz type, and global continuation arguments relying on monotonicity properties of the equation. An important part of the analysis is dedicated to the delicate issue of differentiability of the inverse p-Laplacian. We thus obtain a complete description of the global continua of positive/negative solutions bifurcating from the first eigenvalue of a weighted, radial, p-Laplacian problem, by using purely analytical arguments, whereas previous related results were proved by topological arguments or a mixture of analytical and topological arguments. Our approach requires stronger hypotheses but yields much stronger results, bifurcation occuring along smooth curves of solutions, and not only connected sets.Comment: Minor changes to the statement and proof of Theorem 1

Geometric lifting of the canonical basis and semitoric degenerations of Richardson varieties

In the sl\_n case, A. Berenstein and A. Zelevinsky studied the Sch\"{u}tzenberger involution in terms of Lusztig's canonical basis, [3]. We generalize their construction and formulas for any semisimple Lie algebra. We use for this the geometric lifting of the canonical basis, on which an analogue of the Sch\"{u}tzenberger involution can be given. As an application, we construct semitoric degenerations of Richardson varieties, following a method of P. Caldero, [6]Comment: 22 pages, 3 figure

Well, Papa, can you multiply triplets?

We show that the classical algebra of quaternions is a commutative $\Z_2\times\Z_2\times\Z_2$-graded algebra. A similar interpretation of the algebra of octonions is impossible.Comment: 3 page