Noncommutative geometry and lower dimensional volumes in Riemannian geometry

In this paper we explain how to define "lower dimensional'' volumes of any compact Riemannian manifold as the integrals of local Riemannian invariants. For instance we give sense to the area and the length of such a manifold in any dimension. Our reasoning is motivated by an idea of Connes and involves in an essential way noncommutative geometry and the analysis of Dirac operators on spin manifolds. However, the ultimate definitions of the lower dimensional volumes don't involve noncommutative geometry or spin structures at all.Comment: 12 page

Classical and quantum ergodicity on orbifolds

We extend to orbifolds classical results on quantum ergodicity due to Shnirelman, Colin de Verdi\`ere and Zelditch, proving that, for any positive, first-order self-adjoint elliptic pseudodifferential operator P on a compact orbifold X with positive principal symbol p, ergodicity of the Hamiltonian flow of p implies quantum ergodicity for the operator P. We also prove ergodicity of the geodesic flow on a compact Riemannian orbifold of negative sectional curvature.Comment: 14 page

Curvature in Noncommutative Geometry

Our understanding of the notion of curvature in a noncommutative setting has progressed substantially in the past ten years. This new episode in noncommutative geometry started when a Gauss-Bonnet theorem was proved by Connes and Tretkoff for a curved noncommutative two torus. Ideas from spectral geometry and heat kernel asymptotic expansions suggest a general way of defining local curvature invariants for noncommutative Riemannian type spaces where the metric structure is encoded by a Dirac type operator. To carry explicit computations however one needs quite intriguing new ideas. We give an account of the most recent developments on the notion of curvature in noncommutative geometry in this paper.Comment: 76 pages, 8 figures, final version, one section on open problems added, and references expanded. Appears in "Advances in Noncommutative Geometry - on the occasion of Alain Connes' 70th birthday

Noncommutative Induced Gauge Theory

We consider an external gauge potential minimally coupled to a renormalisable scalar theory on 4-dimensional Moyal space and compute in position space the one-loop Yang-Mills-type effective theory generated from the integration over the scalar field. We find that the gauge invariant effective action involves, beyond the expected noncommutative version of the pure Yang-Mills action, additional terms that may be interpreted as the gauge theory counterpart of the harmonic oscillator term, which for the noncommutative $\phi^4$-theory on Moyal space ensures renormalisability. The expression of a possible candidate for a renormalisable action for a gauge theory defined on Moyal space is conjectured and discussed.Comment: 20 pages, 6 figure

Conifers in cold environments synchronize maximum growth rate of tree-ring formation with day length

Intra-annual radial growth rates and durations in trees are reported to differ greatly in relation to species, site and environmental conditions. However, very similar dynamics of cambial activity and wood formation are observed in temperate and boreal zones. Here, we compared weekly xylem cell production and variation in stem circumference in the main northern hemisphere conifer species (genera Picea, Pinus, Abies and Larix) from 1996 to 2003. Dynamics of radial growth were modeled with a Gompertz function, defining the upper asymptote (A), x-axis placement (β) and rate of change (κ). A strong linear relationship was found between the constants β and κ for both types of analysis. The slope of the linear regression, which corresponds to the time at which maximum growth rate occurred, appeared to converge towards the summer solstice. The maximum growth rate occurred around the time of maximum day length, and not during the warmest period of the year as previously suggested. The achievements of photoperiod could act as a growth constraint or a limit after which the rate of tree-ring formation tends to decrease, thus allowing plants to safely complete secondary cell wall lignification before winter

A systematic review of rodent pest research in Afro-Malagasy small-holder farming systems: Are we asking the right questions?

Rodent pests are especially problematic in terms of agriculture and public health since they can inflict considerable economic damage associated with their abundance, diversity, generalist feeding habits and high reproductive rates. To quantify rodent pest impacts and identify trends in rodent pest research impacting on small-holder agriculture in the Afro-Malagasy region we did a systematic review of research outputs from 1910 to 2015, by developing an a priori defined set of criteria to allow for replication of the review process. We followed the Preferred Reporting Items for Systematic Reviews and Meta-Analyses guidelines. We reviewed 162 publications, and while rodent pest research was spatially distributed across Africa (32 countries, including Madagascar), there was a disparity in number of studies per country with research biased towards four countries (Tanzania [25%], Nigeria [9%], Ethiopia [9%], Kenya [8%]) accounting for 51% of all rodent pest research in the Afro-Malagasy region. There was a disparity in the research themes addressed by Tanzanian publications compared to publications from the rest of the Afro-Malagasy region where research in Tanzania had a much more applied focus (50%) compared to a more basic research approach (92%) in the rest of the Afro-Malagasy region. We found that pest rodents have a significant negative effect on the Afro-Malagasy small-holder farming communities. Crop losses varied between cropping stages, storage and crops and the highest losses occurred during early cropping stages (46% median loss during seedling stage) and the mature stage (15% median loss). There was a scarcity of studies investigating the effectiveness of various management actions on rodent pest damage and population abundance. Our analysis highlights that there are inadequate empirical studies focused on developing sustainable control methods for rodent pests and rodent pests in the Africa-Malagasy context is generally ignored as a research topic

Spectral action for Bianchi type-IX cosmological models

In this paper we prove a rationality phenomena for the coefficients of the heat kernel expansion of the Dirac-Laplacian of Bianchi IX cosmological models. Due the complexities arising from the anisotropic nature of the model, we present a novel method of writing the heat coefficients as Wodzicki resiudes of certain Laplacians and then provide an elegant proof of the rationality result. That is, we show that each coefficient is described by a several variable polynomial with rational coefficients of the cosmic expansion factors and their higher derivatives of a certain order. This result confirms the arithmetic nature of the complicated terms in the expansion