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}$

Grb2 depletion under non-stimulated conditions inhibits PTEN, promotes Akt-induced tumor formation and contributes to poor prognosis in ovarian cancer

In the absence of extracellular stimulation the adaptor protein growth factor receptor-bound protein (Grb2) and the phospholipase Plcγ1 compete for the same binding site on fibroblast growth factor receptor 2 (FGFR2). Reducing cellular Grb2 results in upregulation of Plcγ1 and depletion of the phospholipid PI(4,5)P2. The functional consequences of this event on signaling pathways are unknown. We show that the decrease in PI(4,5)P2 level under non-stimulated conditions inhibits PTEN activity leading to the aberrant activation of the oncoprotein Akt. This results in excessive cell proliferation and tumor progression in a xenograft mouse model. As well as defining a novel mechanism of Akt phosphorylation with important therapeutic consequences, we also demonstrate that differential expression levels of FGFR2, Plcγ1 and Grb2 correlate with patient survival. Oncogenesis through fluctuation in the expression levels of these proteins negates extracellular stimulation or mutation and defines them as novel prognostic markers in ovarian cancer

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

Ein Beitrag aus formal-logischer Sicht zur Diskussion des Abstimmungsproblems bei Initiative und Gegenvorschlag

None