## Publications (2)0 Total impact

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**ABSTRACT:**Let $M^4$ be a PL-manifold of nonnegative curvature that is homeomorphic to a product of two spheres, $S^2 \times S^2$. We prove that $M$ is a direct metric product of two spheres endowed with some polyhedral metrics. In other words, $M$ is a direct metric product of the surfaces of two convex polyhedra in $\mathbb{R}^3$. The background for the question is the following. The classical H.Hopf's hypothesis states: for any Riemannian metric on $S^2 \times S^2$ of nonnegative sectional curvature the curvature cannot be strictly positive at all points. There is no quick answer to this question: it is known that a Riemannian metric on $S^2 \times S^2$ of nonnegative sectional curvature need not be a product metric. However, M.Gromov has pointed out that the condition of nonnegative curvature in the PL-case appears to be stronger than nonnegative sectional curvature of Riemannian manifolds and analogous to some condition on the curvature operator. So the motivation for the question addressed in this text is to settle the PL-version of the Hopf's hypothesis. -
##### Conference Paper: Co-ranking Authors and Documents in a Heterogeneous Network

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**ABSTRACT:**Recent graph-theoretic approaches have demonstrated remarkable successes for ranking networked entities, but most of their applications are limited to homogeneous networks such as the network of citations between publications. This paper proposes a novel method for co-ranking authors and their publications using several networks: the social network connecting the authors, the citation network connecting the publications, as well as the authorship network that ties the previous two together. The new co-ranking framework is based on coupling two random walks, that separately rank authors and documents following the PageRankparadigm. As a result, improved rankings of documents and their authors depend on each other in a mutually reinforcing way, thus taking advantage of the additional information implicit in the heterogeneous network of authors and documents.Data Mining, 2007. ICDM 2007. Seventh IEEE International Conference on; 11/2007

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87 | Citations | |