Relatórios de Pesquisa

We show some analytical properties of SDDE including a closed formula for a strong solution of $\dot{x}(t) = x(t-1) \circ dB_t$ with an initial adapted process $\varphi_t$ in the delay interval $[-1,0]$. SDDE's on a manifold $M$ depend intrinsically on a connection $\nabla$. The main geometric result in this article concerns the horizontal lift of solutions of SDDE on a manifold $M$ to an SDDE in the frame bundle $BM$, hence the lifted equation should come together with the prolonged horizontal connection $\nabla^H$ on $BM$.

rp-2002-42.pdf

Lyapunov graphs carry dynamical information of gradient-like fows as well as topological information of its phase space which is taken to be a closed orientable n-manifold.In this article we will show that an abstract Lyapunov graph L(h0,..., hn,k) in dimension n greater than two, with cycle number k, satisfies the Poincaré-Hopf inequalities if and only if it satisfies the Morse inequalities and the first Betti number g 1³ k. We also show a continuation theorem for abstract Lyapunov graphs with the presence of cycles. Finally, a family of Lyapunov graphs L(h0,..., hn,k) with fixed pre-assigned data (h0,..., hn,k) is associated with the Morse polytope, Pk(h0,..., hn), determined by the Morse inequalities for the given data.

rp-2002-40.pdf

Let K be an infinite field and denote UTn(K) the algebra of n x n upper triangular matrices over K. We describe all elementary gradings on this algebra. Further we produce linear bases of the respective relatively free graded algebras, and prove that one can distinguish the elementary gradings by their graded identities. We describe bases of the graded polynomial identities in several "typical" cases. Although we consider elementary gradings by the cyclic group of order two the same methods serve for elementary gradings by any finite group.An extended version of this report will be published elsewhere.

rp-2002-38.pdf

Let $X$ be an homogeneous space and let $E$ be an UMD Banach space with a normalized unconditional basis $(e_j)_{j\geq 1}$. Given an operator $T$ from $L^{\infty}_c(X)$ in $L^1(X)$, we consider the vector-valued extension ${\widetilde T}$ of $T$ given by ${\widetilde T}(\sum_jf_je_j)=\sum_jT(f_j)e_j$. We prove a weighted integral inequality for the vector-valued extension of the Hardy-Littlewood maximal operator and a weighted Fefferman-Stein inequality between the vector-valued extensions of the Hardy-Littlewood and the sharp maximal operators, in the context of Orlicz spaces. We give sufficient conditions on the kernel of a singular integral operator to have the boundedness of the vector-valued extension of this operator on $L^p(X,Wd\mu;E)$ for $1

rp-2002-36.pdf