Homogeneity and projective equivalence of differential equation fields

We propose definitions of homogeneity and projective equivalence for systems of ordinary differential equations of order greater than two, which allow us to generalize the concept of a spray (for systems of order two). We show that the Euler-Lagrange fields of parametric Lagrangians of order greater than one which are regular (in a natural sense that we define) form a projective equivalence class of homogeneous systems. We show further that the geodesics, or base integral curves, of projectively equivalent homogeneous differential equation fields are the same apart from orientation-preserving reparametrization; that is, homogeneous differential equation fields determine systems of paths

A new multi-spectral imaging system for examining paintings

A new multispectral system developed at the National Gallery is presented. The system is capable of measuring the spectral reflectance per pixel of a painting. These spectra are found to be almost as accurate as those recorded with a spectrophotometer; there is no need for any spectral reconstruction apart from a simple cubic interpolation between measured points. The procedure for recording spectra is described and the accuracy of the system is quantified. An example is presented of the use of the system to scan a painting of St. Mary Magdalene by Crivelli. The multispectral data are used in an attempt to identify some of the pigments found in the painting by comparison with a library of spectra obtained from reference pigments using the same system. In addition, it is shown that the multispectral data can be used to render a color image of the original under a chosen illuminant and that interband comparison can help to elucidate features of the painting, such as retouchings and underdrawing, that are not visible in trichromatic images

Results from computational analysis of a mixed compression supersonic inlet

A numerical study was performed to simulate the critical flow through a supersonic inlet. This flow field has many phenomena such as shock waves, strong viscous effects, turbulent boundary layer development, boundary layer separations, and mass flow suction through the walls, (bleed). The computational tools used were two full Navier-Stokes (FNS) codes. The supersonic inlet that was analyzed is the Variable Diameter Centerbody, (VDC), inlet. This inlet is a candidate concept for the next generation supersonic involved effort in generating an efficient grid geometry and specifying boundary conditions, particularly in the bleed region and at the outflow boundary. Results for a critical inlet operation compare favorably to Method of Characteristics predictions and experimental data

Shape maps for second order partial differential equations

We analyse the singularity formation of congruences of solutions of systems of second order PDEs via the construction of \emph{shape maps}. The trace of such maps represents a congruence volume whose collapse we study through an appropriate evolution equation, akin to Raychaudhuri's equation. We develop the necessary geometric framework on a suitable jet space in which the shape maps appear naturally associated with certain linear connections. Explicit computations are given, along with a nontrivial example

Double structures and jets

We show how the double vector bundle structure of the manifold of double velocities, with its submanifolds of holonomic and semiholonomic double velocities, is mirrored by a structure of holonomic and semiholonomic subgroups in the principal prolongation of the first jet group. We use the actions of these groups to construct holonomic and semiholonomic submanifolds in the manifold of double contact elements, and show that these give rise to affine bundles where a semiholonomic element has well-defined holonomic and curvature components.Comment: Based on a talk given at a meeting in Krakow for the eightieth birthday of W. M. Tulczyje

Lepage equivalents and the Variational Bicomplex

We show how to construct, for a Lagrangian of arbitrary order, a Lepage equivalent satisfying the closure property: that the Lepage equivalent vanishes precisely when the Lagrangian is null. The construction uses a homotopy operator for the horizontal differential of the variational bicomplex. A choice of symmetric linear connection on the manifold of independent variables, and a global homotopy operator constructed using that connection, may then be used to extend any global Lepage equivalent to one satisfying the closure property. In the second part of the paper we investigate the role of vertical endomorphisms in constructing such Lepage equivalents. These endomorphisms may be used directly to construct local homotopy operators. Together with a symmetric linear connection they may also be used to construct global vertical tensors, and these define infinitesimal nonholonomic projections which in turn may be used to construct Lepage equivalents. We conjecture that these global vertical tensors may also be used to define global homotopy operators