Sectional curvature and Weitzenb\"ock formulae

We establish a new algebraic characterization of sectional curvature bounds $\sec\geq k$ and $\sec\leq k$ using only curvature terms in the Weitzenb\"ock formulae for symmetric $p$-tensors. By introducing a symmetric analogue of the Kulkarni-Nomizu product, we provide a simple formula for such curvature terms. We also give an application of the Bochner technique to closed $4$-manifolds with indefinite intersection form and $\sec>0$ or $\sec\geq0$, obtaining new insights into the Hopf Conjecture, without any symmetry assumptions.Comment: LaTeX2e, 25 pages, final version. To appear in Indiana Univ. Math.

Strongly positive curvature

We begin a systematic study of a curvature condition (strongly positive curvature) which lies strictly between positive curvature operator and positive sectional curvature, and stems from the work of Thorpe in the 1970s. We prove that this condition is preserved under Riemannian submersions and Cheeger deformations, and that most compact homogeneous spaces with positive sectional curvature satisfy it.Comment: LaTeX2e, 26 page

Symbolic Sequences and Tsallis Entropy

We address this work to investigate symbolic sequences with long-range correlations by using computational simulation. We analyze sequences with two, three and four symbols that could be repeated $l$ times, with the probability distribution $p(l)\propto 1/ l^{\mu}$. For these sequences, we verified that the usual entropy increases more slowly when the symbols are correlated and the Tsallis entropy exhibits, for a suitable choice of $q$, a linear behavior. We also study the chain as a random walk-like process and observe a nonusual diffusive behavior depending on the values of the parameter $\mu$.Comment: Published in the Brazilian Journal of Physic

When Ellipsis Can Save Defectiveness and When It Can’t

We discuss cases of salvation and non-salvation by deletion in the domain of lexical gaps, and distinguish two types of defectiveness: (a) defectiveness that can be saved by PF deletion, which we take to signal the lack of an eligible allomorph for certain environments within a language, and (b) defectiveness that cannot be saved by PF deletion, which we take to signal the lack of a proper alloseme for a given environment. With ellipsis modeled as an instruction for nonpronunci-ation on the PF branch of the grammar, only gaps on the Exponent List can be saved by it