A generalisation of the Hopf Construction and harmonic morphisms into \s^2

In this paper we construct a new family of harmonic morphisms \varphi:V^5\to\s^2, where $V^5$ is a 5-dimensional open manifold contained in an ellipsoidal hypersurface of \c^4=\r^8. These harmonic morphisms admit a continuous extension to the completion ${V^{\ast}}^5$, which turns out to be an explicit real algebraic variety. We work in the context of a generalization of the Hopf construction and equivariant theory.Comment: 10 page

Gutierrezia mendocina (Asteraceae, Astereae), una nueva especie sudamericana

A taxonomic revision of the South American species of Gutierrezia has revealed a new species from Argentina, characterised by its stems with the base prostrate and rooting, turbinate heads and white ray florets. This species, found in the Department of Tunuyán of Mendoza Province (Argentina), is described and illustrated and a key to differentiate it from allied species is given.En el marco de la revisión taxonómica del género Gutierrezia para Sudamérica fue hallada una nueva especie de Argentina, caracterizada por sus tallos con la porción basal postrada, enraizante en los nudos, capítulos con involucro turbinado y flores liguladas blancas. Se describe e ilustra la especie encontrada en el Departamento de Tunuyán de la Provincia de Mendoza (Argentina), y se incluye una clave para diferenciarla de las especies afines

On the Stability of the Equator Map for Higher Order Energy Functionals

Let Bn ⊂ ℝn and Sn ⊂ Rn+1 denote the Euclidean n-dimensional unit ball and sphere, respectively. The extrinsic k-energy functional is defined on the Sobolev space Wk,2 (Bn, Sn) as follows: Ekext(u) = ∫Bn |Δs u|2 dx when k = 2s, and Ekext(u) = ∫Bn|∇ Δs u|2 dx when k = 2s + 1. These energy functionals are a natural higher order version of the classical extrinsic bienergy, also called Hessian energy. The equator map u∗: Bn → Sn, defined by u∗(x) = (x/|x|,0), is a critical point of Ekext(u) provided that n ≥ 2k + 1. The main aim of this paper is to establish necessary and sufficient conditions on k and n under which u∗: Bn → Sn is minimizing or unstable for the extrinsic k-energy

On the second variation of the biharmonic Clifford torus in S-4

The flat torus T = S-1 (1/2) x S-1 (1/2) admits a proper biharmonic isometric immersion into the unit 4-dimensional sphere S-4 given by Phi = i o phi, where phi : T -> S-3 (1/root 2) is the minimal Clifford torus and i : S-3 (1 root 2) -> S-4 is the biharmonic small hypersphere. The first goal of this paper is to compute the biharmonic index and nullity of the proper biharmonic immersion Phi. After, we shall study in the detail the kernel of the generalised Jacobi operator I-2 Phi. We shall prove that it contains a direction which admits a natural variation with vanishing first, second and third derivatives, and such that the fourth derivative is negative. In the second part of the paper, we shall analyse the specific contribution of phi to the biharmonic index and nullity of Phi. In this context, we shall study a more general composition (Phi) over tilde = i o (phi) over tilde, where (phi) over tilde : M-m -> Sn-1 (1/root 2), m >= 1, n >= 3, is a minimal immersion and i : Sn-1 (1/root 2) -> S-n is the biharmonic small hypersphere. First, we shall determine a general sufficient condition which ensures that the second variation of (Phi) over tilde is nonnegatively defined on C((phi) over tilde -1TSn-1(1/root 2)). Then, we complete this type of analysis on our Clifford torus and, as a complementary result, we obtain the p-harmonic index and nullity of phi. In the final section, we compare our general results with those which can be deduced from the study of the equivariant second variation

Local log-law of the wall: numerical evidences and reasons

Numerical studies performed with a primitive equation model on two-dimensional sinusoidal hills show that the local velocity profiles behave logarithmically to a very good approximation, from a distance from the surface of the order of the maximum hill height almost up to the top of the boundary layer. This behavior is well known for flows above homogeneous and flat topographies (``law-of-the-wall'') and, more recently, investigated with respect to the large-scale (``asymptotic'') averaged flows above complex topography. Furthermore, this new-found local generalized law-of-the-wall involves effective parameters showing a smooth dependence on the position along the underlying topography. This dependence is similar to the topography itself, while this property does not absolutely hold for the underlying flow, nearest to the hill surface.Comment: 9 pages, Latex, 2 figure