Tight frames of k-plane ridgelets and the problem of representing objects that are smooth away from d-dimensional singularities in Rn

For each pair (n, k) with 1 ≤ k < n, we construct a tight frame (ρλ : λ ∈ Λ) for L2 (Rn), which we call a frame of k-plane ridgelets. The intent is to efficiently represent functions that are smooth away from singularities along k-planes in Rn. We also develop tools to help decide whether k-plane ridgelets provide the desired efficient representation. We first construct a wavelet-like tight frame on the X-ray bundle χn,k—the fiber bundle having the Grassman manifold Gn,k of k-planes in Rn for base space, and for fibers the orthocomplements of those planes. This wavelet-like tight frame is the pushout to χn,k, via the smooth local coordinates of Gn,k, of an orthonormal basis of tensor Meyer wavelets on Euclidean space Rk(n−k) × Rn−k. We then use the X-ray isometry [Solmon, D. C. (1976) J. Math. Anal. Appl. 56, 61–83] to map this tight frame isometrically to a tight frame for L2(Rn)—the k-plane ridgelets. This construction makes analysis of a function f ∈ L2(Rn) by k-plane ridgelets identical to the analysis of the k-plane X-ray transform of f by an appropriate wavelet-like system for χn,k. As wavelets are typically effective at representing point singularities, it may be expected that these new systems will be effective at representing objects whose k-plane X-ray transform has a point singularity. Objects with discontinuities across hyperplanes are of this form, for k = n − 1.

A few autoreactive cells in an autoimmune infiltrate control a vast population of nonspecific cells: a tale of smart bombs and the infantry.

Inflammatory infiltrates in tissue-specific autoimmune disease comprise a collection of T cells with specificity for an antigen in the target organ. These specific cells recruit a population of nonspecific T cells and macrophages. The rare tissue-specific T cells in the infiltrate have the capacity to regulate both the influx and the efflux of cells from the tissue. Administration of an altered peptide ligand for the specific T cell which triggers autoimmunity can lead to the regression of the entire inflammatory ensemble in a few hours. Interleukin 4 is a critical cytokine involved in the regression of the inflammatory infiltrate.

Phototaxis away from blue light by an Escherichia coli mutant accumulating protoporphyrin IX.

The hemH gene of Escherichia coli encodes ferrochelatase (EC 4.99.1.1), the enzyme that catalyzes the last step in the production of heme, namely the synthesis of heme from protoporphyrin IX plus Fe2+. The behavioral responses to light were studied in E. coli carrying a hemH mutation. It was shown that the hemH mutant displayed a tumbling response upon illumination and a running response upon removal of the light. The most effect light to induce a tumbling response in the hemH mutant was blue light (396-450 nm). The chemotaxis machinery was needed for the light-induced tumbling response in the hemH mutant. The bacterial defect is an analog of the human inherited disease erythropoietic protoporphyria.