Uniformization and an Index Theorem for Elliptic Operators Associated with Diffeomorphisms of a Manifold

We consider the index problem for a wide class of nonlocal elliptic operators on a smooth closed manifold, namely differential operators with shifts induced by the action of an isometric diffeomorphism. The key to the solution is the method of uniformization: We assign to the nonlocal problem a pseudodifferential operator with the same index, acting in sections of an infinite-dimensional vector bundle on a compact manifold. We then determine the index in terms of topological invariants of the symbol, using the Atiyah-Singer index theorem.Comment: 16 pages, no figure

Lieb-Thirring inequalities for geometrically induced bound states

We prove new inequalities of the Lieb-Thirring type on the eigenvalues of Schr\"odinger operators in wave guides with local perturbations. The estimates are optimal in the weak-coupling case. To illustrate their applications, we consider, in particular, a straight strip and a straight circular tube with either mixed boundary conditions or boundary deformations.Comment: LaTeX2e, 14 page

Upper and lower limits for the number of S-wave bound states in an attractive potential

New upper and lower limits are given for the number of S-wave bound states yielded by an attractive (monotonic) potential in the context of the Schrodinger or Klein-Gordon equation.Comment: 21 pages, 4 figures, to appear in J. Math. Phy

Identification of genetic alterations in pancreatic cancer by the combined use of tissue microdissection and array-based comparative genomic hybridisation

Pancreatic ductal adenocarcinoma (PDAC) is characterised pathologically by a marked desmoplastic stromal reaction that significantly reduces the sensitivity and specificity of cytogenetic analysis. To identify genetic alterations that reflect the characteristics of the tumour in vivo, we screened a total of 23 microdissected PDAC tissue samples using array-based comparative genomic hybridisation (array CGH) with 1 Mb resolution. Highly stringent statistical analysis enabled us to define the regions of nonrandom genomic changes. We detected a total of 41 contiguous regions (>3.0 Mb) of copy number changes, such as a genetic gain at 7p22.2–p15.1 (26.0 Mb) and losses at 17p13.3–p11.2 (13.6 Mb), 18q21.2–q22.1 (12.0 Mb), 18q22.3–q23 (7.1 Mb) and 18q12.3–q21.2 (6.9 Mb). To validate our array CGH results, fluorescence in situ hybridisation was performed using four probes from those regions, showing that these genetic alterations were observed in 37–68% of a separate sample set of 19 PDAC cases. In particular, deletion of the SEC11L3 gene (18q21.32) was detected at a very high frequency (13 out of 19 cases; 68%) and in situ RNA hybridisation for this gene demonstrated a significant correlation between deletion and expression levels. It was further confirmed by reverse transcription–PCR that SEC11L3 mRNA was downregulated in 16 out of 16 PDAC tissues (100%). In conclusion, the combination of tissue microdissection and array CGH provided a valid data set that represents in vivo genetic changes in PDAC. Our results raise the possibility that the SEC11L3 gene may play a role as a tumour suppressor in this disease