The eta invariant and equivariant index of transversally elliptic operators

We prove a formula for the multiplicities of the index of an equivariant transversally elliptic operator on a G-manifold. The formula is a sum of integrals over blowups of the strata of the group action and also involves eta invariants of associated elliptic operators. Among the applications, we obtain an index formula for basic Dirac operators on Riemannian foliations, a problem that was open for many years.

Constraint algorithm for k-presymplectic Hamiltonian systems. Application to singular field theories

The k-symplectic formulation of field theories is especially simple, since only tangent and cotangent bundles are needed in its description. Its defining elements show a close relationship with those in the symplectic formulation of mechanics. It will be shown that this relationship also stands in the presymplectic case. In a natural way,one can mimick the presymplectic constraint algorithm to obtain a constraint algorithmthat can be applied to k-presymplectic field theory, and more particularly to the Lagrangian and Hamiltonian formulations offield theories defined by a singular Lagrangian, as well as to the unified Lagrangian-Hamiltonian formalism (Skinner--Rusk formalism) for k-presymplectic field theory. Two examples of application of the algorithm are also analyzed.

Harmonic active contours.

We propose a segmentation method based on the geometric representation of images as 2-D manifolds embedded in a higher dimensional space. The segmentation is formulated as a minimization problem, where the contours are described by a level set function and the objective functional corresponds to the surface of the image manifold. In this geometric framework, both data-fidelity and regularity terms of the segmentation are represented by a single functional that intrinsically aligns the gradients of the level set function with the gradients of the image and results in a segmentation criterion that exploits the directional information of image gradients to overcome image inhomogeneities and fragmented contours. The proposed formulation combines this robust alignment of gradients with attractive properties of previous methods developed in the same geometric framework: 1) the natural coupling of image channels proposed for anisotropic diffusion and 2) the ability of subjective surfaces to detect weak edges and close fragmented boundaries. The potential of such a geometric approach lies in the general definition of Riemannian manifolds, which naturally generalizes existing segmentation methods (the geodesic active contours, the active contours without edges, and the robust edge integrator) to higher dimensional spaces, non-flat images, and feature spaces. Our experiments show that the proposed technique improves the segmentation of multi-channel images, images subject to inhomogeneities, and images characterized by geometric structures like ridges or valleys.

Contribution of recombination to the evolutionary history of HIV.

PURPOSE OF REVIEW: An improved understanding of how recombination affects the evolutionary history of HIV is crucial to understand its current and future evolution. The present review aims to disentangle the manifold effects of recombination on HIV by discussing its effects on the evolutionary history and the adaptive potential of HIV in the context of concepts from evolutionary genetics and genomics. RECENT FINDINGS: The increasing occurrence of secondary contacts between divergent subtype populations (during coinfection) results in increased observations of recombinants worldwide. Recombination is heterogeneous along the HIV genome. Consequences of recombination of HIV evolution are, in combination with other demographic processes, expected to either homogenize the genetic composition of HIV populations (homogenization) or provide the potential for novel adaptations (diversification). New methods in population genomics allow deep characterization of recombinant genome (the segment composition and origin) and their evolutionary trajectories. SUMMARY: HIV recombinants increase worldwide and invade geographical regions where pure subtypes were previously predominant. This trend is expected to continue in the future, as ease to travel worldwide increases opportunities for recombination between divergent HIV strains. While the effects of recombination in HIV are much researched, more effort is required to characterize current HIV recombinant composition and dynamics. This can be achieved with new population genetic and genomic methods.

Quantifying rock fabrics: a test of autocorrelation of the spatial distribution of cristals

A novel test of spatial independence of the distribution of crystals or phases in rocksbased on compositional statistics is introduced. It improves and generalizes the commonjoins-count statistics known from map analysis in geographic information systems.Assigning phases independently to objects in RD is modelled by a single-trial multinomialrandom function Z(x), where the probabilities of phases add to one and areexplicitly modelled as compositions in the K-part simplex SK. Thus, apparent inconsistenciesof the tests based on the conventional joins{count statistics and their possiblycontradictory interpretations are avoided. In practical applications we assume that theprobabilities of phases do not depend on the location but are identical everywhere inthe domain of de nition. Thus, the model involves the sum of r independent identicalmultinomial distributed 1-trial random variables which is an r-trial multinomialdistributed random variable. The probabilities of the distribution of the r counts canbe considered as a composition in the Q-part simplex SQ. They span the so calledHardy-Weinberg manifold H that is proved to be a K-1-affine subspace of SQ. This isa generalisation of the well-known Hardy-Weinberg law of genetics. If the assignmentof phases accounts for some kind of spatial dependence, then the r-trial probabilitiesdo not remain on H. This suggests the use of the Aitchison distance between observedprobabilities to H to test dependence. Moreover, when there is a spatial uctuation ofthe multinomial probabilities, the observed r-trial probabilities move on H. This shiftcan be used as to check for these uctuations. A practical procedure and an algorithmto perform the test have been developed. Some cases applied to simulated and realdata are presented.Key words: Spatial distribution of crystals in rocks, spatial distribution of phases,joins-count statistics, multinomial distribution, Hardy-Weinberg law, Hardy-Weinbergmanifold, Aitchison geometry

Kuranishi type Moduli Spaces for proper CR submersions fibering over the circle

Kuranishi's fundamental result (1962) associates to any compact complex manifold X&sub&0&/sub& a finite-dimensional analytic space which has to be thought of as a local moduli space of complex structures close to X&sub&0&/sub&. In this paper, we give an analogous statement for Levi-flat CR manifolds fibering properly over the circle by describing explicitely an infinite-dimensional Kuranishi type local moduli space of Levi-flat CR structures. We interpret this result in terms of Kodaira-Spencer deformation theory making clear the likenesses as well as the differences with the classical case. The article ends with applications and examples.

Equidistribution estimates for Fekete points on complex manifolds

We study the equidistribution of Fekete points in a compact complex manifold. These are extremal point configurations defined through sections of powers of a positive line bundle. Their equidistribution is a known result. The novelty of our approach is that we relate them to the problem of sampling and interpolation on line bundles, which allows us to estimate the equidistribution of the Fekete points quantitatively. In particular we estimate the Kantorovich-Wasserstein distance of the Fekete points to its limiting measure. The sampling and interpolation arrays on line bundles are a subject of independent interest, and we provide necessary density conditions through the classical approach of Landau, that in this context measures the local dimension of the space of sections of the line bundle. We obtain a complete geometric characterization of sampling and interpolation arrays in the case of compact manifolds of dimension one, and we prove that there are no arrays of both sampling and interpolation in the more general setting of semipositive line bundles.

Tenseness of Riemannian flows

We show that any transversally complete Riemannian foliation &em&F&/em& of dimension one on any possibly non-compact manifold M is tense; namely, (M,&em&F&/em&) admits a Riemannian metric such that the mean curvature form of &em&F&/em& is basic. This is a partial generalization of a result of Domínguez, which says that any Riemannian foliation on any compact manifold is tense. Our proof is based on some results of Molino and Sergiescu, and it is simpler than the original proof by Domínguez. As an application, we generalize some well known results including Masa's characterization of tautness.

Multisensor change detection with nonlinear canonical correlation

The analysis of multi-modal and multi-sensor images is nowadays of paramount importance for Earth Observation (EO) applications. There exist a variety of methods that aim at fusing the different sources of information to obtain a compact representation of such datasets. However, for change detection existing methods are often unable to deal with heterogeneous image sources and very few consider possible nonlinearities in the data. Additionally, the availability of labeled information is very limited in change detection applications. For these reasons, we present the use of a semi-supervised kernel-based feature extraction technique. It incorporates a manifold regularization accounting for the geometric distribution and jointly addressing the small sample problem. An exhaustive example using Landsat 5 data illustrates the potential of the method for multi-sensor change detection.

GeoKernels: Modeling of Spatial Data on GeoManifolds

This paper presents a review of methodology for semi-supervised modeling with kernel methods, when the manifold assumption is guaranteed to be satisfied. It concerns environmental data modeling on natural manifolds, such as complex topographies of the mountainous regions, where environmental processes are highly influenced by the relief. These relations, possibly regionalized and nonlinear, can be modeled from data with machine learning using the digital elevation models in semi-supervised kernel methods. The range of the tools and methodological issues discussed in the study includes feature selection and semisupervised Support Vector algorithms. The real case study devoted to data-driven modeling of meteorological fields illustrates the discussed approach.

Rapid affinity purification of erythropoietin from biological samples using disposable monoliths.

Identification of post-translational modifications of proteins in biological samples often requires access to preanalytical purification and concentration methods. In the purification step high or low molecular weight substances can be removed by size exclusion filters, and high abundant proteins can be removed, or low abundant proteins can be enriched, by specific capturing tools. In this paper is described the experience and results obtained with a recently emerged and easy-to-use affinity purification kit for enrichment of the low amounts of EPO found in urine and plasma specimens. The kit can be used as a pre-step in the EPO doping control procedure, as an alternative to the commonly used ultrafiltration, for detecting aberrantly glycosylated isoforms. The commercially available affinity purification kit contains small disposable anti-EPO monolith columns (6 ?L volume, Ø7 mm, length 0.15 mm) together with all required buffers. A 24-channel vacuum manifold was used for simultaneous processing of samples. The column concentrated EPO from 20 mL urine down to 55 ?L eluate with a concentration factor of 240 times, while roughly 99.7% of non-relevant urine proteins were removed. The recoveries of Neorecormon (epoetin beta), and the EPO analogues Aranesp and Mircera applied to buffer were high, 76%, 67% and 57%, respectively. The recovery of endogenous EPO from human urine was 65%. High recoveries were also obtained when purifying human, mouse and equine EPO from serum, and human EPO from cerebrospinal fluid. Evaluation with the accredited EPO doping control method based on isoelectric focusing (IEF) showed that the affinity purification procedure did not change the isoform distribution for rhEPO, Aranesp, Mircera or endogenous EPO. The kit should be particularly useful for applications in which it is essential to avoid carry-over effects, a problem commonly encountered with conventional particle-based affinity columns. The encouraging results with EPO propose that similar affinity monoliths, with the appropriate antibodies, should constitute useful tools for general applications in sample preparation, not only for doping control of EPO and other hormones such as growth hormone and insulin but also for the study of post-translational modifications of other low abundance proteins in biological and clinical research, and for sample preparation prior to in vitro diagnostics.

Aitchison Geometry for Probability and Likelihood as a new approach to mathematical statistics

The Aitchison vector space structure for the simplex is generalized to a Hilbert space structure A2(P) for distributions and likelihoods on arbitrary spaces. Centralnotations of statistics, such as Information or Likelihood, can be identified in the algebraical structure of A2(P) and their corresponding notions in compositional data analysis, such as Aitchison distance or centered log ratio transform.In this way very elaborated aspects of mathematical statistics can be understoodeasily in the light of a simple vector space structure and of compositional data analysis. E.g. combination of statistical information such as Bayesian updating,combination of likelihood and robust M-estimation functions are simple additions/perturbations in A2(Pprior). Weighting observations corresponds to a weightedaddition of the corresponding evidence.Likelihood based statistics for general exponential families turns out to have aparticularly easy interpretation in terms of A2(P). Regular exponential families formfinite dimensional linear subspaces of A2(P) and they correspond to finite dimensionalsubspaces formed by their posterior in the dual information space A2(Pprior).The Aitchison norm can identified with mean Fisher information. The closing constant itself is identified with a generalization of the cummulant function and shown to be Kullback Leiblers directed information. Fisher information is the local geometry of the manifold induced by the A2(P) derivative of the Kullback Leibler information and the space A2(P) can therefore be seen as the tangential geometry of statistical inference at the distribution P.The discussion of A2(P) valued random variables, such as estimation functionsor likelihoods, give a further interpretation of Fisher information as the expected squared norm of evidence and a scale free understanding of unbiased reasoning

Quantum metric spaces as a model for pregeometry

A new arena for the dynamics of spacetime is proposed, in which the basic quantum variable is the two-point distance on a metric space. The scaling dimension (that is, the Kolmogorov capacity) in the neighborhood of each point then defines in a natural way a local concept of dimension. We study our model in the region of parameter space in which the resulting spacetime is not too different from a smooth manifold.

One billion years of bZIP transcription factor evolution: conservation and change in dimerization and DNA-binding site specificity.

The genomic era has revealed that the large repertoire of observed animal phenotypes is dependent on changes in the expression patterns of a finite number of genes, which are mediated by a plethora of transcription factors (TFs) with distinct specificities. The dimerization of TFs can also increase the complexity of a genetic regulatory network manifold, by combining a small number of monomers into dimers with distinct functions. Therefore, studying the evolution of these dimerizing TFs is vital for understanding how complexity increased during animal evolution. We focus on the second largest family of dimerizing TFs, the basic-region leucine zipper (bZIP), and infer when it expanded and how bZIP DNA-binding and dimerization functions evolved during the major phases of animal evolution. Specifically, we classify the metazoan bZIPs into 19 families and confirm the ancient nature of at least 13 of these families, predating the split of the cnidaria. We observe fixation of a core dimerization network in the last common ancestor of protostomes-deuterostomes. This was followed by an expansion of the number of proteins in the network, but no major dimerization changes in interaction partners, during the emergence of vertebrates. In conclusion, the bZIPs are an excellent model with which to understand how DNA binding and protein interactions of TFs evolved during animal evolution.

Quantum metric spaces as a model for pregeometry

A new arena for the dynamics of spacetime is proposed, in which the basic quantum variable is the two-point distance on a metric space. The scaling dimension (that is, the Kolmogorov capacity) in the neighborhood of each point then defines in a natural way a local concept of dimension. We study our model in the region of parameter space in which the resulting spacetime is not too different from a smooth manifold.