The inverse spectral problem for the discrete cubic string

Given a measure $m$ on the real line or a finite interval, the "cubic string" is the third order ODE $-\phi'''=zm\phi$ where $z$ is a spectral parameter. If equipped with Dirichlet-like boundary conditions this is a nonselfadjoint boundary value problem which has recently been shown to have a connection to the Degasperis-Procesi nonlinear water wave equation. In this paper we study the spectral and inverse spectral problem for the case of Neumann-like boundary conditions which appear in a high-frequency limit of the Degasperis--Procesi equation. We solve the spectral and inverse spectral problem for the case of $m$ being a finite positive discrete measure. In particular, explicit determinantal formulas for the measure $m$ are given. These formulas generalize Stieltjes' formulas used by Krein in his study of the corresponding second order ODE $-\phi''=zm\phi$.Comment: 24 pages. LaTeX + iopart, xypic, amsthm. To appear in Inverse Problems (http://www.iop.org/EJ/journal/IP

Testing the Hubble Law with the IRAS 1.2 Jy Redshift Survey

We test and reject the claim of Segal et al. (1993) that the correlation of redshifts and flux densities in a complete sample of IRAS galaxies favors a quadratic redshift-distance relation over the linear Hubble law. This is done, in effect, by treating the entire galaxy luminosity function as derived from the 60 micron 1.2 Jy IRAS redshift survey of Fisher et al. (1995) as a distance indicator; equivalently, we compare the flux density distribution of galaxies as a function of redshift with predictions under different redshift-distance cosmologies, under the assumption of a universal luminosity function. This method does not assume a uniform distribution of galaxies in space. We find that this test has rather weak discriminatory power, as argued by Petrosian (1993), and the differences between models are not as stark as one might expect a priori. Even so, we find that the Hubble law is indeed more strongly supported by the analysis than is the quadratic redshift-distance relation. We identify a bias in the the Segal et al. determination of the luminosity function, which could lead one to mistakenly favor the quadratic redshift-distance law. We also present several complementary analyses of the density field of the sample; the galaxy density field is found to be close to homogeneous on large scales if the Hubble law is assumed, while this is not the case with the quadratic redshift-distance relation.Comment: 27 pages Latex (w/figures), ApJ, in press. Uses AAS macros, postscript also available at http://www.astro.princeton.edu/~library/preprints/pop682.ps.g

Inverse problems associated with integrable equations of Camassa-Holm type; explicit formulas on the real axis, I

The inverse problem which arises in the Camassa--Holm equation is revisited for the class of discrete densities. The method of solution relies on the use of orthogonal polynomials. The explicit formulas are obtained directly from the analysis on the real axis without any additional transformation to a "string" type boundary value problem known from prior works

On the tau-functions of the Degasperis-Procesi equation

The DP equation is investigated from the point of view of determinant-pfaffian identities. The reciprocal link between the Degasperis-Procesi (DP) equation and the pseudo 3-reduction of the $C_{\infty}$ two-dimensional Toda system is used to construct the N-soliton solution of the DP equation. The N-soliton solution of the DP equation is presented in the form of pfaffian through a hodograph (reciprocal) transformation. The bilinear equations, the identities between determinants and pfaffians, and the $\tau$-functions of the DP equation are obtained from the pseudo 3-reduction of the $C_{\infty}$ two-dimensional Toda system.Comment: 27 pages, 4 figures, Journal of Physics A: Mathematical and Theoretical, to be publishe

Multi loop soliton solutions and their interactions in the Degasperis-Procesi equation

In this article, we construct loop soliton solutions and mixed soliton - loop soliton solution for the Degasperis-Procesi equation. To explore these solutions we adopt the procedure given by Matsuno. By appropriately modifying the $\tau$-function given in the above paper we derive these solutions. We present the explicit form of one and two loop soliton solutions and mixed soliton - loop soliton solutions and investigate the interaction between (i) two loop soliton solutions in different parametric regimes and (ii) a loop soliton with a conventional soliton in detail.Comment: Published in Physica Scripta (2012

A class of Poisson-Nijenhuis structures on a tangent bundle

Equipping the tangent bundle TQ of a manifold with a symplectic form coming from a regular Lagrangian L, we explore how to obtain a Poisson-Nijenhuis structure from a given type (1,1) tensor field J on Q. It is argued that the complete lift of J is not the natural candidate for a Nijenhuis tensor on TQ, but plays a crucial role in the construction of a different tensor R, which appears to be the pullback under the Legendre transform of the lift of J to co-tangent manifold of Q. We show how this tangent bundle view brings new insights and is capable also of producing all important results which are known from previous studies on the cotangent bundle, in the case that Q is equipped with a Riemannian metric. The present approach further paves the way for future generalizations.Comment: 22 page

Endochondral bone formation in toothless (osteopetrotic) rats: failures of chondrocyte patterning and type X collagen expression

The pacemaker of endochondral bone growth is cell division and hypertrophy of chondrocytes. The developmental stages of chondrocytes, characterized by the expression of collagen types II and X, are arranged in arrays across the growth zone. Mutations in collagen II and X genes as well as the absence of their gene products lead to different, altered patterns of chondrocyte stages which remain aligned across the growth plate (GP). Here we analyze GP of rats bearing the mutation toothless (tl) which, apart from bone defects, develop a progressive, severe chondrodystrophy during postnatal weeks 3 to 6. Mutant GP exhibited disorganized, non-aligned chondrocytes and mineralized metaphyseal bone but without cartilage mineralization or cartilaginous extensions into the metaphysis. Expression of mRNA coding for collagen types II (Col II) and X (Col X) was examined in the tibial GP by in situ hybridization. Mutant rats at 2 weeks exhibited Col II RNA expression and some hypertrophied chondrocytes (HC) but no Col X RNA was detected. By 3rd week, HC had largely disappeared from the central part of the mutant GP and Col II RNA expression was present but weak and in 2 separate bands. Peripherally the GP contained HC but without Col X RNA expression. This abnormal pattern was exacerbated by the fourth week. Bone mineralized but cartilage in the GP did not. These data suggest that the tl mutation involves a regulatory function for chondrocyte maturation, including Col X RNA synthesis and mineralization, and that the GP abnormalities are related to the Col X deficiency. The differences in patterning in the tl rat GP compared to direct Col X mutations may be explained by compensatory effects

Projective dynamics and classical gravitation

Given a real vector space V of finite dimension, together with a particular homogeneous field of bivectors that we call a "field of projective forces", we define a law of dynamics such that the position of the particle is a "ray" i.e. a half-line drawn from the origin of V. The impulsion is a bivector whose support is a 2-plane containing the ray. Throwing the particle with a given initial impulsion defines a projective trajectory. It is a curve in the space of rays S(V), together with an impulsion attached to each ray. In the simplest example where the force is identically zero, the curve is a straight line and the impulsion a constant bivector. A striking feature of projective dynamics appears: the trajectories are not parameterized. Among the projective force fields corresponding to a central force, the one defining the Kepler problem is simpler than those corresponding to other homogeneities. Here the thrown ray describes a quadratic cone whose section by a hyperplane corresponds to a Keplerian conic. An original point of view on the hidden symmetries of the Kepler problem emerges, and clarifies some remarks due to Halphen and Appell. We also get the unexpected conclusion that there exists a notion of divergence-free field of projective forces if and only if dim V=4. No metric is involved in the axioms of projective dynamics.Comment: 20 pages, 4 figure

Control of actin polymerization via the coincidence of phosphoinositides and high membrane curvature

The conditional use of actin during clathrin-mediated endocytosis in mammalian cells suggests that the cell controls whether and how actin is used. Using a combination of biochemical reconstitution and mammalian cell culture, we elucidate a mechanism by which the coincidence of PI(4,5)P2 and PI(3)P in a curved vesicle triggers actin polymerization. At clathrin-coated pits, PI(3)P is produced by the INPP4A hydrolysis of PI(3,4)P2, and this is necessary for actin-driven endocytosis. Both Cdc42⋅guanosine triphosphate and SNX9 activate N-WASP–WIP- and Arp2/3-mediated actin nucleation. Membrane curvature, PI(4,5)P2, and PI(3)P signals are needed for SNX9 assembly via its PX–BAR domain, whereas signaling through Cdc42 is activated by PI(4,5)P2 alone. INPP4A activity is stimulated by high membrane curvature and synergizes with SNX9 BAR domain binding in a process we call curvature cascade amplification. We show that the SNX9-driven actin comets that arise on human disease–associated oculocerebrorenal syndrome of Lowe (OCRL) deficiencies are reduced by inhibiting PI(3)P production, suggesting PI(3)P kinase inhibitors as a therapeutic strategy in Lowe syndrome.J.L. Gallop is supported by a Wellcome Trust Research Career Development Fellowship (grant WT095829AIA). F.  Daste, A.  Walrant, J.R. Gadsby, and J. Mason are supported by an H2020 European Research Council Starting Grant (281971) awarded to J.L. Gallop. Gurdon Institute funding is provided by the Wellcome Trust (grant 092096) and Cancer Research UK (grant C6946/A14492). The Swedish Medical Research Council and the Swedish Foundation for Strategic Research supported the work of M.R. Holst and R. Lundmark. S.F. Lee is funded by a Royal Society University Research Fellowship (grant UF120277). M. Mettlen is funded by grant MH73125 to Sandra L. Schmid (University of Texas Southwestern Medical Center)

Variants of ST8SIA1 Are Associated with Risk of Developing Multiple Sclerosis

Multiple sclerosis (MS) is an inflammatory demyelinating disease of the central nervous system of unknown etiology with both genetic and environmental factors playing a role in susceptibility. To date, the HLA DR15/DQ6 haplotype within the major histocompatibility complex on chromosome 6p, is the strongest genetic risk factor associated with MS susceptibility. Additional alleles of IL7 and IL2 have been identified as risk factors for MS with small effect. Here we present two independent studies supporting an allelic association of MS with polymorphisms in the ST8SIA1 gene, located on chromosome 12p12 and encoding ST8 alpha-N-acetyl-neuraminide alpha-2,8-sialyltransferase 1. The initial association was made in a single three-generation family where a single-nucleotide polymorphism (SNP) rs4762896, was segregating together with HLA DR15/DQ6 in MS patients. A study of 274 family trios ( affected child and both unaffected parents) from Australia validated the association of ST8SIA1 in individuals with MS, showing transmission disequilibrium of the paternal alleles for three additional SNPs, namely rs704219, rs2041906, and rs1558793, with p = 0.001, p = 0.01 and p = 0.01 respectively. These findings implicate ST8SIA1 as a possible novel susceptibility gene for MS