38 research outputs found

    Multi-Harnack smoothings of real plane branches

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    We introduce a new method for the construction of smoothings of a real plane branch (C,0)(C, 0) by using Viro Patchworking method. Since real plane branches are Newton degenerated in general, we cannot apply Viro Patchworking method directly. Instead we apply the Patchworking method for certain Newton non degenerate curve singularities with several branches. These singularities appear as a result of iterating deformations of the strict transforms of the branch at certain infinitely near points of the toric embedded resolution of singularities of (C,0)(C,0). We characterize the MM-smoothings obtained by this method by the local data. In particular, we analyze the class of multi-Harnack smoothings, those smoothings arising in a sequence MM-smoothings of the strict transforms of (C,0) which are in maximal position with respect to the coordinate lines. We prove that there is a unique the topological type of multi-Harnack smoothings, which is determined by the complex equisingularity type of the branch. This result is a local version of a recent Theorem of Mikhalkin

    Non-existence of torically maximal hypersurfaces

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    Torically maximal curves (known also as simple Harnack curves) are real algebraic curves in the projective plane such that their logarithmic Gau{\ss} map is totally real. In this paper we show that hyperplanes in projective spaces are the only torically maximal hypersurfaces of higher dimensions.Comment: 10 pages. V2 merges the first version of this paper with the first version of arXiv:1510.0026

    On the curvature of the real amoeba

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    For a real smooth algebraic curve A \subset (\mathhbb{C}^*)^2, the amoeba A⊂R2\mathcal{A} \subset \mathbb{R}^2 is the image of AA under the map Log : (x,y)↦(log⁡∣x∣,log⁡∣y∣)(x,y) \mapsto (\log |x|, \log | y |). We describe an universal bound for the total curvature of the real amoeba ARA\mathcal{A}_{\mathbb{R} A} and we prove that this bound is reached if and only if the curve AA is a simple Harnack curve in the sense of Mikhalkin

    Stress Clamp Experiments on Multicellular Tumor Spheroids

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    The precise role of the microenvironment on tumor growth is poorly understood. Whereas the tumor is in constant competition with the surrounding tissue, little is known about the mechanics of this interaction. Using a novel experimental procedure, we study quantitatively the effect of an applied mechanical stress on the long-term growth of a spheroid cell aggregate. We observe that a stress of 10 kPa is sufficient to drastically reduce growth by inhibition of cell proliferation mainly in the core of the spheroid. We compare the results to a simple numerical model developed to describe the role of mechanics in cancer progression.Comment: 5 pages, 4 figure

    Undulation Instability of Epithelial Tissues

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    Treating the epithelium as an incompressible fluid adjacent to a viscoelastic stroma, we find a novel hydrodynamic instability that leads to the formation of protrusions of the epithelium into the stroma. This instability is a candidate for epithelial fingering observed in vivo. It occurs for sufficiently large viscosity, cell-division rate and thickness of the dividing region in the epithelium. Our work provides physical insight into a potential mechanism by which interfaces between epithelia and stromas undulate, and potentially by which tissue dysplasia leads to cancerous invasion.Comment: 4 pages, 3 figure

    On the Bezout theorem in the real case

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    Depto. de Álgebra, Geometría y TopologíaFac. de Ciencias MatemáticasTRUEpu