Research Reports

This paper considers invariant almost Hermitian structures on a flag manifold $G/P=U/K$ where $G$ is a complex semi-simple Lie group, $P$ is a parabolic subgroup of $G$, $U$ is a compact real form of $G$ and $K=U\cap P$ is the centralizer of a torus. The main result shows that there are nearly-K\"{a}hler structures in $G/P$ which are not K\"{a}hler if and only if $G/P$ has height three. This proves for the flag manifolds a conjecture by J.A. Wolf and A. Gray.

rp-2003-46.pdf

Using an integral representation for the first kind Hankel (Hankel-Bessel) function we obtain the so-called Basset formula, an integral representation for the second kind modified Bessel function. Using the Sonine-Bessel integral representation we obtain the Fourier cosine integral transform of the zero order Bessel function. As an application we present the calculation of the Green's function associated with a second order partial differential equation , particularly a wave equation for a lossy two-dimensional medium. This application is associated to the transient electromagnetic field radiated by a pulsed source in the presence of dispersive media, which is of great importance in the theory of geophysical prospecting, lightning studies

rp-2003-44.pdf

In this paper we establish the existence of positive and multiple solutions for the quasilinear elliptic problem\begin{eqnarray*}\begin{array}{ccl}-\Delta_p u = g(x,u) & {\rm in} & \Omega\\ u = 0 & {\rm on} & \partial \Omega,\end{array}\end{eqnarray*}where $\Omega \subset \mathbb{R}^N$ is an open bounded domain with smooth boundary $\partial \Omega$, $g:\Omega\times\mathbb{R}\to \mathbb{R}$ is a Carath\'eodory function such that $g(x,0)=0$ and which is asymptotically linear. We suppose that $g(x,t)/t$ tends to an $L^r$-function, $r>N/p$ if $1N$, which can change sign. We consider both cases, resonant and nonresonant.

rp-2003-42.pdf

In this paper we show how to describe the general theory of linear metric compatible connection with the theory of Clifford valued differential forms. This is done by realizing that for each spacetime point the algebra of Clifford bivectors is isomorphic to the algebra of $Sl(2,\mathbb{C)}$. In that way the pullback of the linear connection under a trivialization of the bundleis represented by a Clifford valued 1-form. That observation makes it possible to realize Einstein's gravitational theory can be formulated in a way which is similar to a $Sl(2,\mathbb{C)}$ gauge theory. Some aspects of such approach is discussed. Also, the theory of covariant spinor derivatives of spinor fields is introduced in a novel way, allowing for a physical interpretation of some rules postulated for that covariant spinor derivative in the standard theory of these objects. We use our methods to investigate some polemical issues in gravitational theories and in particular we scrutinize a supposedly "unified" field theory of gravitation and electromagnetism proposed by M. Sachs and recently used in a series of papers. Our results show that Sachs did not attain his objective and that recent papers based on that theory are ill conceived and completely invalid both as Mathematics and Physics.

rp-2003-40.pdf