Jesús A. Álvarez López

D665/D665a INDEX VS. FREQUENCIES AS INDICATORS OF BRYOPHYTERESPONSE TO PHYSICOCHEMICAL GRADIENTS

The chlorophyll-to-phaeophytin (D665/D665a) ratio is an index of physiological stress in aquatic bryophytes. In the present study, the usefulness of this index for evaluating water contamination was investigated. Samples of bryophytes and water were collected from 188 stretches of river in northwest Spain. Water quality was characterized by standard procedures. Ecological profile methods were used to investigate (for each species and each water quality variable) whether stress index or frequency of occurrence varied significantly along environmental gradients. Most stress profiles showed significant departures from uniformity, whereas many frequency profiles did not, indicating that the stress index approach is more sensitive to differences in water quality than are approaches based simply on presence/absence data. To further investigate relationships between water quality variables and D665/D665a ratio, canonical correspondence analysis (CCA) was used. The first axis extracted by CCA was clearly closely related to degree of organic pollution. The second axis was most strongly correlated with pH. Taken together, these results suggest that D665/D665a ratio in aquatic bryophytes may be of value as an indicator of river water pollution.

Long Time Behaviour of Leafwise Heat Flow for Riemannian Foliations

For any Riemannian foliation F on a closed manifold M with an arbitrary bundle-like metric, leafwise heat flow of differential forms is proved to preserve smoothness on M at infinite time. This result and its proof have consequences about the space of bundle-like metrics on M, about the dimension of the space of leafwise harmonic forms, and mainly about the second term of the differentiable spectral sequence of F.

Hodge decomposition along the leaves of a Riemannian foliation

Abstract A Hodge decomposition theorem is proved for the leafwise Laplacian of a Riemannian foliation on a closed Riemannian manifold.

Generic coarse geometry of leaves

In this chapter we recall basic concepts and preliminaries about foliated spaces, as well as fix the notation, so that the foliated space versions of our main theorems follow directly from their pseudogroup versions.

Lefschetz Distribution of Lie Foliations

Let \( \mathcal{F} \) be a Lie foliation on a closed manifold M with structural Lie group G. Its transverse Lie structure can be considered as a transverse action Φ of G on (M,\( \mathcal{F} \)); i.e., an “action” which is defined up to leafwise homotopies. This Φ induces an action Φ* of G on the reduced leafwise cohomology \( \bar H\left( \mathcal{F} \right) \). By using leafwise Hodge theory, the supertrace of Φ* can be defined as a distribution L dis(\( \mathcal{F} \)) on G called the Lefschetz distribution of \( \mathcal{F} \). A distributional version of the Gauss-Bonett theorem is proved, which describes L dis(\( \mathcal{F} \) ) around the identity element. On any small enough open subset of G, L dis(\( \mathcal{F} \)) is described by a distributional version of the Lefschetz trace formula.

On turbulent relations

This paper extends the theory of turbulence of Hjorth to certain classes of equivalence relations that cannot be induced by Polish actions. It applies this theory to analyze the quasi-isometry relation and finite Gromov-Hausdorff distance relation in the space of isometry classes of pointed proper metric spaces, called the Gromov space.

Some Examples of First Exit Times

The purpose of this article is to compute the expected first exit times of Brownian motion from a variety of domains in the Euclidean plane and in the hyperbolic plane.

Nonreduction of Relations in the Gromov Space to Polish Actions

It is shown that, in the Gromov space of isometry classes of pointed proper metric spaces, the equivalence relations defined by existence of coarse quasi-isometries or being at finite Gromov-Hausdorff distance, cannot be reduced to the equivalence relation defined by any Polish action.

Examples and Open Problems

This final chapter contains a variety of examples that serve to illustrate our main theorems, as well as a list of open problems on the topics that we have studied in this book.

Coarse Quasi-Isometries

This chapter is devoted to the study of coarse quasi-isometries. Of particular interest is the coarse version of the Arzela-Ascoli theorem.