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

On the Lieb-Thirring constants L_gamma,1 for gamma geq 1/2

Let $E_i(H)$ denote the negative eigenvalues of the one-dimensional Schr\"odinger operator $Hu:=-u^{\prime\prime}-Vu,\ V\geq 0,$ on $L_2({\Bbb R})$. We prove the inequality \sum_i|E_i(H)|^\gamma\leq L_{\gamma,1}\int_{\Bbb R} V^{\gamma+1/2}(x)dx, (1) for the "limit" case $\gamma=1/2.$ This will imply improved estimates for the best constants $L_{\gamma,1}$ in (1), as $1/2<\gamma<3/2.Comment: AMS-LATEX, 15 page

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

Exploiting inflammation for therapeutic gain in pancreatic cancer

Pancreatic ductal adenocarcinoma (PDAC) is an aggressive malignancy associated with &#60;5% 5-year survival, in which standard chemotherapeutics have limited benefit. The disease is associated with significant intra- and peritumoral inflammation and failure of protective immunosurveillance. Indeed, inflammatory signals are implicated in both tumour initiation and tumour progression. The major pathways regulating PDAC-associated inflammation are now being explored. Activation of leukocytes, and upregulation of cytokine and chemokine signalling pathways, both have been shown to modulate PDAC progression. Therefore, targeting inflammatory pathways may be of benefit as part of a multi-target approach to PDAC therapy. This review explores the pathways known to modulate inflammation at different stages of tumour development, drawing conclusions on their potential as therapeutic targets in PDAC

Alterations of tumor suppressor gene p16(INK4a )in pancreatic ductal carcinoma

BACKGROUND: Cell cycle inhibitor and tumor suppressor gene p16 / MTS-1 has been reported to be altered in a variety of human tumors. The purpose of the study was to evaluate primary pancreatic ductal adenocarcinomas for potentially inactivating p16 alterations. METHODS: We investigated the status of p16 gene by polymerase chain reaction (PCR), nonradioisotopic single strand conformation polymorphism (SSCP), DNA sequencing and hypermethylation analysis in 25 primary resected ductal adenocarcinomas. In addition, we investigated p16 protein expression in these cases by immunohistochemistry (IHC) using a monoclonal antibody clone (MS-887-PO). RESULTS: Out of the 25 samples analyzed and compared to normal pancreatic control tissues, the overall frequency of p16 alterations was 80% (20/25). Aberrant promoter methylation was the most common mechanism of gene inactivation present in 52% (13/25) cases, followed by coding sequence mutations in 16% (4/25) cases and presumably homozygous deletion in 12% (3/25) cases. These genetic alterations correlated well with p16 protein expression as complete loss of p16 protein was found in 18 of 25 tumors (72%). CONCLUSION: These findings confirm that loss of p16 function could be involved in pancreatic cancer and may explain at least in part the aggressive behaviour of this tumor type

Topics in Mathematical Analysis

This volume contains most of the lectures of the "Minicorsi of Mathematical Analysis" held at the University of Padova in the years 2000-2003

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

Plk1 regulates mitotic Aurora A function through βTrCP-dependent degradation of hBora

Polo-like kinase 1 (Plk1) and Aurora A play key roles in centrosome maturation, spindle assembly, and chromosome segregation during cell division. Here we show that the functions of these kinases during early mitosis are coordinated through Bora, a partner of Aurora A first identified in Drosophila. Depletion of human Bora (hBora) results in spindle defects, accompanied by increased spindle recruitment of Aurora A and its partner TPX2. Conversely, hBora overexpression induces mislocalization of Aurora A and monopolar spindle formation, reminiscent of the phenotype seen in Plk1-depleted cells. Indeed, Plk1 regulates hBora. Following Cdk1-dependent recruitment, Plk1 triggers hBora destruction by phosphorylating a recognition site for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text{SCF}}^{{\text{ $ \beta $ - TrCP}}}$$\end{document}. Plk1 depletion or inhibition results in a massive accumulation of hBora, concomitant with displacement of Aurora A from spindle poles and impaired centrosome maturation, but remarkably, co-depletion of hBora partially restores Aurora A localization and bipolar spindle formation. This suggests that Plk1 controls Aurora A localization and function by regulating cellular levels of hBora

MICALs in control of the cytoskeleton, exocytosis, and cell death

MICALs form an evolutionary conserved family of multidomain signal transduction proteins characterized by a flavoprotein monooxygenase domain. MICALs are being implicated in the regulation of an increasing number of molecular and cellular processes including cytoskeletal dynamics and intracellular trafficking. Intriguingly, some of these effects are dependent on the MICAL monooxygenase enzyme and redox signaling, while other functions rely on other parts of the MICAL protein. Recent breakthroughs in our understanding of MICAL signaling identify the ability of MICALs to bind and directly modify the actin cytoskeleton, link MICALs to the docking and fusion of exocytotic vesicles, and uncover MICALs as anti-apoptotic proteins. These discoveries could lead to therapeutic advances in neural regeneration, cancer, and other diseases