The bar-radical of a class of almost alternative baric algebras

In this article we prove that, if (U, ) is a finite dimensional baric algebra of (gamma, delta) type over a field F of characteristic not equal 2,3,5 such that gamma(2) - delta(2) + delta = 1 and 0,1, then rad(U) = R(U)boolean AND(bar(U))(2), where R(U) is the nilradical (maximal nil ideal) of U.

On the Multiplicity of Orthogonal Geodesics in Riemannian Manifold With Concave Boundary. Applications to Brake Orbits and Homoclinics

Let (M, g) be a complete Riemannian Manifold, Omega subset of M an open subset whose closure is diffeomorphic to an annulus. If partial derivative Omega is smooth and it satisfies a strong concavity assumption, then it is possible to prove that there are at least two geometrically distinct geodesics in (Omega) over bar = Omega boolean OR partial derivative Omega starting orthogonally to one connected component of partial derivative Omega and arriving orthogonally onto the other one. The results given in [6] allow to obtain a proof of the existence of two distinct homoclinic orbits for an autonomous Lagrangian system emanating from a nondegenerate maximum point of the potential energy, and a proof of the existence of two distinct brake orbits for a. class of Hamiltonian systems. Under a further symmetry assumption, it is possible to show the existence of at least dim(M) pairs of geometrically distinct geodesics as above, brake orbits and homoclinics.

A three-state model for the photophysics of guanine

The nonadiabatic photochemistry of the guanine molecule (2-amino-6-oxopurine) and some of its tautomers has been studied by means of the high-level theoretical ab initio quantum chemistry methods CASSCF and CASPT2. Accurate computations, based by the first time on minimum energy reaction paths, states minima, transition states, reaction barriers, and conical intersections on the potential energy hypersurfaces of the molecules lead to interpret the photochemistry of guanine and derivatives within a three-state model. As in the other purine DNA nucleobase, adenine, the ultrafast subpicosecond fluorescence decay measured in guanine is attributed to the barrierless character of the path leading from the initially populated (1)(pi pi* L-a) spectroscopic state of the molecule toward the low-lying methanamine-like conical intersection (gs/pi pi* L-a)(CI). On the contrary, other tautomers are shown to have a reaction energy barrier along the main relaxation profile. A second, slower decay is attributed to a path involving switches toward two other states, (1)(pi pi* L-b) and, in particular, (1)(n(o)pi*), ultimately leading to conical intersections with the ground state. A common framework for the ultrafast relaxation of the natural nucleobases is obtained in which the predominant role of a pi pi*-type state is confirmed.