E Martinez

Derivations of differential forms along the tangent bundle projection II

We study derivations of the algebra of differential forms along the tangent bundle projection τ : TM → M and of the module of vector-valued forms along τ . It is shown that a satisfactory classification and characterization of such derivations requires the extra availability of a connection on TM . The present theory completely explains and generalizes the calculus of forms associated to a given second-order vector field, which was previously introduced by one of us.

Derivations of forms along a map: the framework for time-dependent second-order equations

Abstract A comprehensive theory is presented concerning derivations of scalar and vector-valued forms along the projection π : R × TM → R × M . It is the continuation of previous work on derivations of forms along the tangent bundle projection and is prompted by the need for a scheme which is adapted to the study of time-dependent second-order equations. The overall structure of the theory closely follows the pattern of this preceding work, but there are many features which are certainly not trivial transcripts of the time-independent situation. As before, a crucial ingredient in the classification of derivations is a non-linear connection on the bundle π. In the presence of a given second-order system, such a connection is canonically defined and gives rise to two important operations: the dynamical covariant derivative, which is a derivation of degree 0, and the Jacobi endomorphism, which is a type (1, 1) tensor field along π. The theory is developed in such a way that all results readily apply to the more general situation of a bundle π : J 1 E → E , where E is fibred over R, but need not be the trivial fibration R × M → R .

A global synthesis of the effects of diversified farming systems on arthropod diversity within fields and across agricultural landscapes

Agricultural intensification is a leading cause of global biodiversity loss, which can reduce the provisioning of ecosystem services in managed ecosystems. Organic farming and plant diversification are farm management schemes that may mitigate potential ecological harm by increasing species richness and boosting related ecosystem services to agroecosystems. What remains unclear is the extent to which farm management schemes affect biodiversity components other than species richness, and whether impacts differ across spatial scales and landscape contexts. Using a global metadataset, we quantified the effects of organic farming and plant diversification on abundance, local diversity (communities within fields), and regional diversity (communities across fields) of arthropod pollinators, predators, herbivores, and detritivores. Both organic farming and higher in-field plant diversity enhanced arthropod abundance, particularly for rare taxa. This resulted in increased richness but decreased evenness. While these responses were stronger at local relative to regional scales, richness and abundance increased at both scales, and richness on farms embedded in complex relative to simple landscapes. Overall, both organic farming and in-field plant diversification exerted the strongest effects on pollinators and predators, suggesting these management schemes can facilitate ecosystem service providers without augmenting herbivore (pest) populations. Our results suggest that organic farming and plant diversification promote diverse arthropod metacommunities that may provide temporal and spatial stability of ecosystem service provisioning. Conserving diverse plant and arthropod communities in farming systems therefore requires sustainable practices that operate both within fields and across landscapes.

Lie algebroid structures on a class of affine bundles

We introduce the notion of a Lie algebroid structure on an affine bundle whose base manifold is fibered over R. It is argued that this is the framework which one needs for coming to a time-dependent generalization of the theory of Lagrangian systems on Lie algebroids. An extensive discussion is given of a way one can think of forms acting on sections of the affine bundle. It is further shown that the affine Lie algebroid structure gives rise to a coboundary operator on such forms. The concept of admissible curves and dynamical systems whose integral curves are admissible brings an associated affine bundle into the picture, on which one can define in a natural way a prolongation of the original affine Lie algebroid structure.

Symmetry theory and Lagrangian inverse problem for time-dependent second-order differential equations

A set XGamma of vector fields in the evolution space E playing the role of Newtonian vector fields, with respect to a second-order equation field Gamma , is introduced and endowed with a Cinfinity (E)-module structure. A dual set M*Gamma is used for giving an answer to the Lagrangian inverse problem. The symmetry theory is also developed in this framework and, in particular, the characterisation of symmetries of Gamma in terms of the transformation properties of the Lagrangian L is also given.

Complete separability of time-dependent second-order ordinary differential equations.

Extending previous work on the geometric characterization of separability in the autonomous case, necessary and sufficient conditions are established for the complete separability of a system of time-dependent second-order ordinary differential equations. In deriving this result, extensive use is made of the theory of derivations of scalar and vector-valued forms along the projection π:J1E→E of the first jet bundle of a fibre bundleE → ℝ. Two illustrative examples are discussed, which fully demonstrate all aspects of the constructive nature of the theory.

Note on generalized connections and affine bundles

We develop an alternative view on the concept of connections over a vector bundle map, which consists of a horizontal lift procedure to a prolonged bundle. We further focus on prolongations to an affine bundle and introduce the concept of affineness of a generalized connection.

Habitat Heterogeneity Affects Plant and Arthropod Species Diversity and Turnover in Traditional Cornfields.

The expansion of the agricultural frontier by the clearing of remnant forests has led to human-dominated landscape mosaics. Previous studies have evaluated the effect of these landscape mosaics on arthropod diversity at local spatial scales in temperate and tropical regions, but little is known about fragmentation effects in crop systems, such as the complex tropical traditional crop systems that maintain a high diversity of weeds and arthropods in low-Andean regions. To understand the factors that influence patterns of diversity in human-dominated landscapes, we investigate the effect of land use types on plant and arthropod diversity in traditionally managed cornfields, via surveys of plants and arthropods in twelve traditional cornfields in the Colombian Andes. We estimated alpha and beta diversity to analyze changes in diversity related to land uses within a radius of 100 m to 1 km around each cornfield. We observed that forests influenced alpha diversity of plants, but not of arthropods. Agricultural lands had a positive relationship with plants and herbivores, but a negative relationship with predators. Pastures positively influenced the diversity of plants and arthropods. In addition, forest cover seemed to influence changes in plant species composition and species turnover of herbivore communities among cornfields. The dominant plant species varied among fields, resulting in high differentiation of plant communities. Predator communities also exhibited high turnover among cornfields, but differences in composition arose mainly among rare species. The crop system evaluated in this study represents a widespread situation in the tropics, therefore, our results can be of broad significance. Our findings suggest that traditional agriculture may not homogenize biological communities, but instead could maintain the regional pool of species through high beta diversity.

A geometric characterisation of Lagrangian second-order differential equations

The authors introduce a set of 1-forms M*Gamma =( alpha epsilon Lambda 1(TM):S*(LGamma alpha )=0), related to a second-order differential equation field Gamma , in order to find geometric conditions for this vector field to be a solution of the dynamics defined by a Lagrangian function. The results they get are compared with those of other previous approaches.

Symmetries of second-order differential equations and decoupling

Necessary and sufficient conditions are discussed, which characterize complete separability of second-order ordinary differential equations via symmetry properties of the system.