Certifying the Existence of Epipolar Matrices

Given a set of point correspondences in two images, the existence of a fundamental matrix is a necessary condition for the points to be the images of a 3-dimensional scene imaged with two pinhole cameras. If the camera calibration is known then one requires the existence of an essential matrix. We present an efficient algorithm, using exact linear algebra, for testing the existence of a fundamental matrix. The input is any number of point correspondences. For essential matrices, we characterize the solvability of the Demazure polynomials. In both scenarios, we determine which linear subspaces intersect a fixed set defined by non-linear polynomials. The conditions we derive are polynomials stated purely in terms of image coordinates. They represent a new class of two-view invariants, free of fundamental (resp.~essential)~matrices

The diacylglycerol kinase α/Atypical PKC/β1 integrin pathway in SDF-1α mammary carcinoma invasiveness

Diacylglycerol kinase α (DGKα), by phosphorylating diacylglycerol into phosphatidic acid, provides a key signal driving cell migration and matrix invasion. We previously demonstrated that in epithelial cells activation of DGKα activity promotes cytoskeletal remodeling and matrix invasion by recruiting atypical PKC at ruffling sites and by promoting RCP-mediated recycling of α5β1 integrin to the tip of pseudopods. In here we investigate the signaling pathway by which DGKα mediates SDF-1α-induced matrix invasion of MDA-MB-231 invasive breast carcinoma cells. Indeed we showed that, following SDF-1α stimulation, DGKα is activated and localized at cell protrusion, thus promoting their elongation and mediating SDF-1α induced MMP-9 metalloproteinase secretion and matrix invasion. Phosphatidic acid generated by DGKα promotes localization at cell protrusions of atypical PKCs which play an essential role downstream of DGKα by promoting Rac-mediated protrusion elongation and localized recruitment of β1 integrin and MMP-9. We finally demonstrate that activation of DGKα, atypical PKCs signaling and β1 integrin are all essential for MDA-MB-231 invasiveness. These data indicates the existence of a SDF-1α induced DGKα - atypical PKC - β1 integrin signaling pathway, which is essential for matrix invasion of carcinoma cells

The adjacency matrix and the discrete Laplacian acting on forms

We study the relationship between the adjacency matrix and the discrete Laplacian acting on 1-forms. We also prove that if the adjacency matrix is bounded from below it is not necessarily essentially self-adjoint. We discuss the question of essential self-adjointness and the notion of completeness

Direct derivation of the Veneziano-Yankielowicz superpotential from matrix model

We derive the Veneziano-Yankielowicz superpotential directly from the matrix model by fixing the measure precisely. The essential requirement here is that the effective superpotential of the matrix model corresponding to the ${\cal N}=4$ supersymmetric Yang-Mills theory vanishes except for the tree gauge kinetic term. Thus we clarify the reason why the matrix model reproduces the Veneziano-Yankielowicz superpotential correctly in the Dijkgraaf-Vafa theory.Comment: 13 pages, to appear in Phys. Lett.