Lie algebroid structures on a class of affine bundles

We introduce the notion of a Lie algebroid structure on an affine bundle whose base manifold is fibred over the real numbers. It is argued that this is the framework which one needs for coming to a time-dependent generalization of the theory of Lagrangian systems on Lie algebroids. An extensive discussion is given of a way one can think of forms acting on sections of the affine bundle. It is further shown that the affine Lie algebroid structure gives rise to a coboundary operator on such forms. The concept of admissible curves and dynamical systems whose integral curves are admissible, brings an associated affine bundle into the picture, on which one can define in a natural way a prolongation of the original affine Lie algebroid structure.Comment: 28 page

Egorov's theorem for transversally elliptic operators on foliated manifolds and noncommutative geodesic flow

The main result of the paper is Egorov's theorem for transversally elliptic operators on compact foliated manifolds. This theorem is applied to describe the noncommutative geodesic flow in noncommutative geometry of Riemannian foliations.Comment: 23 pages, no figures. Completely revised and improved version of dg-ga/970301

Nambu-Poisson manifolds and associated n-ary Lie algebroids

We introduce an n-ary Lie algebroid canonically associated with a Nambu-Poisson manifold. We also prove that every Nambu-Poisson bracket defined on functions is induced by some differential operator on the exterior algebra, and characterize such operators. Some physical examples are presented

Detection of chromosome aberrations in metaphase and interphase tumor cells by in situ hybridization using chromosome-specific library probes

Chromosome aberrations in two glioma cell lines were analyzed using biotinylated DNA library probes that specifically decorate chromosomes 1, 4, 7, 18 and 22 from pter to qter. Numerical changes, deletions and rearrangements of these chromosomes were radily visualized in metaphase spreads, as well as in early prophase and interphase nuclei. Complete chromosomes, deleted chromosomes and segments of translocated chromosomes were rapidly delineated in very complex karyotypes. Simultaneous hybridizations with additional subregional probes were used to further define aberrant chromosomes. Digital image analysis was used to quantitate the total complement of specific chromosomal DNAs in individual metaphase and interphase cells of each cell line. In spite of the fact that both glioma lines have been passaged in vitro for many years, an under-representation of chromosome 22 and an over-representation of chromosome 7 (specifically 7p) were observed. These observations agree with previous studies on gliomas. In addition, sequences of chromosome 4 were also found to be under-represented, especially in TC 593. These analyses indicate the power of these methods for pinpointing chromosome segments that are altered in specific types of tumors

A spinor approach to Walker geometry

A four-dimensional Walker geometry is a four-dimensional manifold M with a neutral metric g and a parallel distribution of totally null two-planes. This distribution has a natural characterization as a projective spinor field subject to a certain constraint. Spinors therefore provide a natural tool for studying Walker geometry, which we exploit to draw together several themes in recent explicit studies of Walker geometry and in other work of Dunajski (2002) and Plebanski (1975) in which Walker geometry is implicit. In addition to studying local Walker geometry, we address a global question raised by the use of spinors.Comment: 41 pages. Typos which persisted into published version corrected, notably at (2.15

Preliminary Biomarkers for Identification of Human Ascending Thoracic Aortic Aneurysm

Background: Human ascending thoracic aortic aneurysms (ATAAs) are life threatening and constitute a leading cause of mortality in the United States. Previously, we demonstrated that collagens α2(V) and α1(XI) mRNA and protein expression levels are significantly increased in ATAAs. Methods and Results: In this report, the authors extended these preliminary studies using high‐throughput proteomic analysis to identify additional biomarkers for use in whole blood real‐time RT‐PCR analysis to allow for the identification of ATAAs before dissection or rupture. Human ATAA samples were obtained from male and female patients aged 65±14 years. Both bicuspid and tricuspid aortic valve patients were included and compared with nonaneurysmal aortas (mean diameter 2.3 cm). Five biomarkers were identified as being suitable for detection and identification of ATAAs using qRT‐PCR analysis of whole blood. Analysis of 41 samples (19 small, 13 medium‐sized, and 9 large ATAAs) demonstrated the overexpression of 3 of these transcript biomarkers correctly identified 79.4% of patients with ATAA of ≥4.0 cm (P<0.001, sensitivity 0.79, CI=0.62 to 0.91; specificity 1.00, 95% CI=0.42 to 1.00). Conclusion: A preliminary transcript biomarker panel for the identification of ATAAs using whole blood qRT‐PCR analysis in men and women is presented

The "Symplectic Camel Principle" and Semiclassical Mechanics

Gromov's nonsqueezing theorem, aka the property of the symplectic camel, leads to a very simple semiclassical quantiuzation scheme by imposing that the only "physically admissible" semiclassical phase space states are those whose symplectic capacity (in a sense to be precised) is nh + (1/2)h where h is Planck's constant. We the construct semiclassical waveforms on Lagrangian submanifolds using the properties of the Leray-Maslov index, which allows us to define the argument of the square root of a de Rham form.Comment: no figures. to appear in J. Phys. Math A. (2002

On paraquaternionic submersions between paraquaternionic K\"ahler manifolds

In this paper we deal with some properties of a class of semi-Riemannian submersions between manifolds endowed with paraquaternionic structures, proving a result of non-existence of paraquaternionic submersions between paraquaternionic K\"ahler non locally hyper paraK\"ahler manifolds. Then we examine, as an example, the canonical projection of the tangent bundle, endowed with the Sasaki metric, of an almost paraquaternionic Hermitian manifold.Comment: 13 pages, no figure

Poisson-Jacobi reduction of homogeneous tensors

The notion of homogeneous tensors is discussed. We show that there is a one-to-one correspondence between multivector fields on a manifold $M$, homogeneous with respect to a vector field $\Delta$ on $M$, and first-order polydifferential operators on a closed submanifold $N$ of codimension 1 such that $\Delta$ is transversal to $N$. This correspondence relates the Schouten-Nijenhuis bracket of multivector fields on $M$ to the Schouten-Jacobi bracket of first-order polydifferential operators on $N$ and generalizes the Poissonization of Jacobi manifolds. Actually, it can be viewed as a super-Poissonization. This procedure of passing from a homogeneous multivector field to a first-order polydifferential operator can be also understood as a sort of reduction; in the standard case -- a half of a Poisson reduction. A dual version of the above correspondence yields in particular the correspondence between $\Delta$-homogeneous symplectic structures on $M$ and contact structures on $N$.Comment: 19 pages, minor corrections, final version to appear in J. Phys. A: Math. Ge

Truncated and Helix-Constrained Peptides with High Affinity and Specificity for the cFos Coiled-Coil of AP-1

Protein-based therapeutics feature large interacting surfaces. Protein folding endows structural stability to localised surface epitopes, imparting high affinity and target specificity upon interactions with binding partners. However, short synthetic peptides with sequences corresponding to such protein epitopes are unstructured in water and promiscuously bind to proteins with low affinity and specificity. Here we combine structural stability and target specificity of proteins, with low cost and rapid synthesis of small molecules, towards meeting the significant challenge of binding coiled coil proteins in transcriptional regulation. By iteratively truncating a Jun-based peptide from 37 to 22 residues, strategically incorporating i-->i+4 helix-inducing constraints, and positioning unnatural amino acids, we have produced short, water-stable, alpha-helical peptides that bind cFos. A three-dimensional NMR-derived structure for one peptide (24) confirmed a highly stable alpha-helix which was resistant to proteolytic degradation in serum. These short structured peptides are entropically pre-organized for binding with high affinity and specificity to cFos, a key component of the oncogenic transcriptional regulator Activator Protein-1 (AP-1). They competitively antagonized the cJun–cFos coiled-coil interaction. Truncating a Jun-based peptide from 37 to 22 residues decreased the binding enthalpy for cJun by ~9 kcal/mol, but this was compensated by increased conformational entropy (TDS ≤ 7.5 kcal/mol). This study demonstrates that rational design of short peptides constrained by alpha-helical cyclic pentapeptide modules is able to retain parental high helicity, as well as high affinity and specificity for cFos. These are important steps towards small antagonists of the cJun-cFos interaction that mediates gene transcription in cancer and inflammatory diseases