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.

Self-organized criticality: sandpiles, singularities, and scaling.

We present an overview of the statistical mechanics of self-organized criticality. We focus on the successes and failures of hydrodynamic description of transport, which consists of singular diffusion equations. When this description applies, it can predict the scaling features associated with these systems. We also identify a hard driving regime where singular diffusion hydrodynamics fails due to fluctuations and give an explicit criterion for when this failure occurs.

Genetic instability of cancer cells is proportional to their degree of aneuploidy

Genetic and phenotypic instability are hallmarks of cancer cells, but their cause is not clear. The leading hypothesis suggests that a poorly defined gene mutation generates genetic instability and that some of many subsequent mutations then cause cancer. Here we investigate the hypothesis that genetic instability of cancer cells is caused by aneuploidy, an abnormal balance of chromosomes. Because symmetrical segregation of chromosomes depends on exactly two copies of mitosis genes, aneuploidy involving chromosomes with mitosis genes will destabilize the karyotype. The hypothesis predicts that the degree of genetic instability should be proportional to the degree of aneuploidy. Thus it should be difficult, if not impossible, to maintain the particular karyotype of a highly aneuploid cancer cell on clonal propagation. This prediction was confirmed with clonal cultures of chemically transformed, aneuploid Chinese hamster embryo cells. It was found that the higher the ploidy factor of a clone, the more unstable was its karyotype. The ploidy factor is the quotient of the modal chromosome number divided by the normal number of the species. Transformed Chinese hamster embryo cells with a ploidy factor of 1.7 were estimated to change their karyotype at a rate of about 3% per generation, compared with 1.8% for cells with a ploidy factor of 0.95. Because the background noise of karyotyping is relatively high, the cells with low ploidy factor may be more stable than our method suggests. The karyotype instability of human colon cancer cell lines, recently analyzed by Lengnauer et al. [Lengnauer, C., Kinzler, K. W. & Vogelstein, B. (1997) Nature (London) 386, 623–627], also corresponds exactly to their degree of aneuploidy. We conclude that aneuploidy is sufficient to explain genetic instability and the resulting karyotypic and phenotypic heterogeneity of cancer cells, independent of gene mutation. Because aneuploidy has also been proposed to cause cancer, our hypothesis offers a common, unique mechanism of altering and simultaneously destabilizing normal cellular phenotypes.

Coexistence of linear zones and pinwheels within orientation maps in cat visual cortex

Revealing the layout of cortical maps is important both for understanding the processes involved in their development and for uncovering the mechanisms underlying neural computation. The typical organization of orientation maps in the cat visual cortex is radial; complete orientation cycles are mapped around orientation singularities. In contrast, long linear zones of orientation representation have been detected in the primary visual cortex of the tree shrew. In this study, we searched for the existence of long linear sequences and wide linear zones within orientation preference maps of the cat visual cortex. Optical imaging based on intrinsic signals was used. Long linear sequences and wide linear zones of preferred orientation were occasionally detected along the border between areas 17 and 18, as well as within area 18. Adjacent zones of distinct radial and linear organizations were observed across area 18 of a single hemisphere. However, radial and linear organizations were not necessarily segregated; long (7.5 mm) linear sequences of preferred orientation were found embedded within a typical pinwheel-like organization of orientation. We conclude that, although the radial organization is dominant, perfectly linear organization may develop and perform the processing related to orientation in the cat visual cortex.

Wheat leaves emit nitrous oxide during nitrate assimilation

Nitrous oxide (N2O) is a key atmospheric greenhouse gas that contributes to global climatic change through radiative warming and depletion of stratospheric ozone. In this report, N2O flux was monitored simultaneously with photosynthetic CO2 and O2 exchanges from intact canopies of 12 wheat seedlings. The rates of N2O-N emitted ranged from <2 pmol⋅m−2⋅s−1 when NH\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{equation*}{\mathrm{_{4}^{+}}}\end{equation*}\end{document} was the N source, to 25.6 ± 1.7 pmol⋅m−2⋅s−1 (mean ± SE, n = 13) when the N source was shifted to NO\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{equation*}{\mathrm{_{3}^{-}}}\end{equation*}\end{document}. Such fluxes are among the smallest reported for any trace gas emitted by a higher plant. Leaf N2O emissions were correlated with leaf nitrate assimilation activity, as measured by using the assimilation quotient, the ratio of CO2 assimilated to O2 evolved. 15N isotopic signatures on N2O emitted from leaves supported direct N2O production by plant NO\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{equation*}{\mathrm{_{3}^{-}}}\end{equation*}\end{document} assimilation and not N2O produced by microorganisms on root surfaces and emitted in the transpiration stream. In vitro production of N2O by both intact chloroplasts and nitrite reductase, but not by nitrate reductase, indicated that N2O produced by leaves occurred during photoassimilation of NO\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{equation*}{\mathrm{_{2}^{-}}}\end{equation*}\end{document} in the chloroplast. Given the large quantities of NO\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{equation*}{\mathrm{_{3}^{-}}}\end{equation*}\end{document} assimilated by plants in the terrestrial biosphere, these observations suggest that formation of N2O during NO\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{equation*}{\mathrm{_{2}^{-}}}\end{equation*}\end{document} photoassimilation could be an important global biogenic N2O source.

The cranium of Parapithecus grangeri, an Egyptian Oligocene anthropoidean primate

A nearly complete skull of Parapithecus grangeri from the early Oligocene of Egypt is described. The specimen is relatively undistorted and is undoubtedly the most complete higher primate skull yet found in the African Oligocene, which also makes it the most complete Oligocene primate cranium worldwide. Belonging in superfamily Parapithecoidea, a group regarded by some as the sister group to all other Anthropoidea, this skull reveals important information about the radiation of stem anthropoideans. This cranium is about 15% larger than size estimates based on a fragmentary cranium of its contemporary and close relative Apidium phiomense. It is about the same size as that of the gray gentle lemur, Hapalemur griseus, or of platyrrhines such as the owl monkey, Aotus trivirgatus, or the titi monkey, Callicebus torquatus. Comparatively small orbits and size differences in jaws and teeth show it was both diurnal and dimorphic. This is the only specimen of the species that shows (from sockets) that there were four small upper incisors. Several mandibular specimens of the species establish that there were no permanent lower incisors and that the symphysis was fused. Like other early anthropoideans this species possessed a lower encephalization quotient and less-developed orbital frontality than later anthropoideans. There is full postorbital closure and fusion of the metopic suture, and the ectotympanic forms a rim to the auditory aperture. A probable frontal/alisphenoid contact is a potentially derived resemblance to Catarrhini. A proposed separate genus for the species P. grangeri is not sustained.