The Weil algebra and the Van Est isomorphism

This paper belongs to a series devoted to the study of the cohomology of classifying spaces. Generalizing the Weil algebra of a Lie algebra and Kalkman's BRST model, here we introduce the Weil algebra $W(A)$ associated to any Lie algebroid $A$. We then show that this Weil algebra is related to the Bott-Shulman-Stasheff complex (computing the cohomology of the classifying space) via a Van Est map and we prove a Van Est isomorphism theorem. As application, we generalize and find a simpler more conceptual proof of the main result of Bursztyn et.al. on the reconstructions of multiplicative forms and of a result of Weinstein-Xu and Crainic on the reconstruction of connection 1-forms. This reveals the relevance of the Weil algebra and Van Est maps to the integration and the pre-quantization of Poisson (and Dirac) manifolds.Comment: 28 pages. Final version, to appear in "Annales de l'Institut Fourier

A Sensitivity Matrix Methodology for Inverse Problem Formulation

We propose an algorithm to select parameter subset combinations that can be estimated using an ordinary least-squares (OLS) inverse problem formulation with a given data set. First, the algorithm selects the parameter combinations that correspond to sensitivity matrices with full rank. Second, the algorithm involves uncertainty quantification by using the inverse of the Fisher Information Matrix. Nominal values of parameters are used to construct synthetic data sets, and explore the effects of removing certain parameters from those to be estimated using OLS procedures. We quantify these effects in a score for a vector parameter defined using the norm of the vector of standard errors for components of estimates divided by the estimates. In some cases the method leads to reduction of the standard error for a parameter to less than 1% of the estimate

Peptides encoded by short ORFs control development and define a new eukaryotic gene family

Despite recent advances in developmental biology and in genomics, key questions remain regarding the organisation of cells into embryos. One possibility is that novel types of genes might await discovery and could provide some of the answers. Genome annotation depends strongly on comparison with previously known gene sequences, and so genes having previously uncharacterised structure and function can be missed. Here we present the characterisation of tarsal-less, a new such type of gene. Tarsal-less has two unusual features: first, it contains more than one coding unit, a structure more similar to some bacterial genes. Second, it codes for small peptides rather than proteins, and in fact these peptides represent the smallest gene products known to date. Functional analysis of this gene in the fruitfly Drosophila shows that it has important functions throughout development, including tissue morphogenesis and pattern formation. We identify genes similar to tarsal-less in other species, and thus define a tarsal-less-related gene family. We expect that a combination of bioinformatic and functional methods, such as the ones we use in this study, will identify and characterize more genes of this type. Potentially, thousands of such new genes may exist

Deformations of Lie brackets and representations up to homotopy

We show that representations up to homotopy can be differentiated in a functorial way. A van Est type isomorphism theorem is established and used to prove a conjecture of Crainic and Moerdijk on deformations of Lie brackets.Comment: 28 page

Modular Classes of Lie Groupoid Representations up to Homotopy

We describe a construction of the modular class associated to a representation up to homotopy of a Lie groupoid. In the case of the adjoint representation up to homotopy, this class is the obstruction to the existence of a volume form, in the sense of Weinstein's "The volume of a differentiable stack"

Implicit self-consistent electrolyte model in plane-wave density-functional theory

The ab-initio computational treatment of electrochemical systems requires an appropriate treatment of the solid/liquid interfaces. A fully quantum mechanical treatment of the interface is computationally demanding due to the large number of degrees of freedom involved. In this work, we describe a computationally efficient model where the electrode part of the interface is described at the density-functional theory (DFT) level, and the electrolyte part is represented through an implicit solvation model based on the Poisson-Boltzmann equation. We describe the implementation of the linearized Poisson-Boltzmann equation into the Vienna Ab-initio Simulation Package (VASP), a widely used DFT code, followed by validation and benchmarking of the method. To demonstrate the utility of the implicit electrolyte model, we apply it to study the surface energy of Cu crystal facets in an aqueous electrolyte as a function of applied electric potential. We show that the applied potential enables the control of the shape of nanocrystals from an octahedral to a truncated octahedral morphology with increasing potential