Polarisation of Graded Bundles

We construct the full linearisation functor which takes a graded bundle of degree $k$ (a particular kind of graded manifold) and produces a $k$-fold vector bundle. We fully characterise the image of the full linearisation functor and show that we obtain a subcategory of $k$-fold vector bundles consisting of symmetric $k$-fold vector bundles equipped with a family of morphisms indexed by the symmetric group ${\mathbb S}_k$. Interestingly, for the degree 2 case this additional structure gives rise to the notion of a symplectical double vector bundle, which is the skew-symmetric analogue of a metric double vector bundle. We also discuss the related case of fully linearising $N$-manifolds, and how one can use the full linearisation functor to "superise" a graded bundle

PDBFlex: exploring flexibility in protein structures.

The PDBFlex database, available freely and with no login requirements at http://pdbflex.org, provides information on flexibility of protein structures as revealed by the analysis of variations between depositions of different structural models of the same protein in the Protein Data Bank (PDB). PDBFlex collects information on all instances of such depositions, identifying them by a 95% sequence identity threshold, performs analysis of their structural differences and clusters them according to their structural similarities for easy analysis. The PDBFlex contains tools and viewers enabling in-depth examination of structural variability including: 2D-scaling visualization of RMSD distances between structures of the same protein, graphs of average local RMSD in the aligned structures of protein chains, graphical presentation of differences in secondary structure and observed structural disorder (unresolved residues), difference distance maps between all sets of coordinates and 3D views of individual structures and simulated transitions between different conformations, the latter displayed using JSMol visualization software

Graded bundles and homogeneity structures

We introduce the concept of a graded bundle which is a natural generalization of the concept of a vector bundle and whose standard examples are higher tangent bundles T^nQ playing a fundamental role in higher order Lagrangian formalisms. Graded bundles are graded manifolds in the sense that we can choose an atlas whose local coordinates are homogeneous functions of degrees 0,1,...,n. We prove that graded bundles have a convenient equivalent description as homogeneity structures, i.e. manifolds with a smooth action of the multiplicative monoid of non-negative reals. The main result states that each homogeneity structure admits an atlas whose local coordinates are homogeneous. Considering a natural compatibility condition of homogeneity structures we formulate, in turn, the concept of a double (r-tuple, in general) graded bundle - a broad generalization of the concept of a double (r-tuple) vector bundle. Double graded bundles are proven to be locally trivial in the sense that we can find local coordinates which are simultaneously homogeneous with respect to both homogeneity structures.Comment: 19 pages, the revised version to be published in J. Geom. Phy