Minimal length of two intersecting simple closed geodesics

On a hyperbolic Riemann surface, given two simple closed geodesics that intersect n times, we address the question of a sharp lower bound L n on the length attained by the longest of the two geodesics. We show the existence of a surface S n on which there exists two simple closed geodesics of length L n intersecting n times and explicitly find L n for $${n\leq 3}$

Minimal length of two intersecting simple closed geodesics

On a hyperbolic Riemann surface, given two simple closed geodesics that intersect n times, we address the question of a sharp lower bound Ln on the length attained by the longest of the two geodesics. We show the existence of a surface Sn on which there exists two simple closed geodesics of length Ln intersecting n times and explicitly find Ln for n ≤ 3

Trace coordinates of Teichmüller space of Riemann surfaces of signature (0,4)(0,4)

Summary: We explicitly give $calT$, the Teichmüller space of four-holed spheres (which we call $X$ pieces) in trace coordinates, as well as its modular group and a fundamental domain for the action of this group on $calT$ which is its moduli space. As a consequence, we see that on any hyperbolic Riemann surface, two closed geodesics of lengths smaller than $2operatornamearccosh(2)$ intersect at most once; two closed geodesics of lengths smaller than $2operatornamearccosh(3)$ are both non-dividing or intersect at most once

Fibroblast growth factor receptor 4: a putative key driver for the aggressive phenotype of hepatocellular carcinoma

Recently, we found upregulation of fibroblast growth factor receptor 4 (FGFR4) in a subset of hepatocellular carcinoma (HCC). Here, we provide mechanistic insight into the role of FGFR4-mediated signalling for the aggressive behaviour of HCC cells. To overexpress FGFR4, hepatoma/hepatocarcinoma cells were transfected with a construct coding for FGFR4. For downmodulation of endogenous FGFR4, we used small interfering RNA or adenoviral infection with dominant-negative FGFR4 constructs being either kinase dead (kdFGFR4) or coding for the autoinhibitory soluble domain (solFGFR4). FGFR4 overexpression in non-tumourigenic hepatocarcinoma cells significantly reduced cell-matrix adhesion, enabled cells to grow anchorage-independently in soft agar, to disintegrate the lymph-/blood-endothelial barrier for intra-/extravasation of tumour cells and to form tumours in SCID mice. Transcriptome analysis revealed altered expression of genes involved in cell-matrix interactions. Conversely, in highly tumourigenic cell lines, kdFGFR4 or solFGFR4 lowered the proportion of cells in S phase of the cell cycle, enhanced the G0/G1 and G2/M-phase proportions, reduced anchorage-independent growth in vitro and attenuated disintegration of the lymph-/blood-endothelium and tumour formation in vivo. These findings were confirmed by altered expression profiles of genes being important for late stages of cell division. Deregulated FGFR4 expression appears to be one of the key drivers of the malignant phenotype of HCC cells. Accordingly, blockade of FGFR4-mediated signalling by soluble dominant-negative constructs, like solFGFR4, may be a feasible and promising therapeutic approach to antagonize aggressive behaviour of hepatoma/hepatocarcinoma cells. © The Author 2014. Published by Oxford University Press. All rights reserved. For Permissions, please email: [email protected]

Randomized, open-label phase 2 study comparing frontline dovitinib versus sorafenib in patients with advanced hepatocellular carcinoma

Randomized, open-label phase 2 study comparing frontline dovitinib versus sorafenib in patients with advanced hepatocellular carcinom