Let M be an n-dimensional projective algebraic manifold in certain projective space CP . The hyperplane line bundle of CP restricts to an ample line bundle L on M , which is called a polarization of M . A Kähler metric g is called a polarized metric, if the corresponding Kähler form represents the first Chern class c1(L) of L in H (M,Z). Given any polarized Kähler metric g, there is a Hermitian...

We prove a number of new results on the large-scale geometry \(L^p\)-metrics group area-preserving diffeomorphisms each orientable surface. Our proofs use in key way Fulton-MacPherson type compactification configuration space n points surface due to Axelrod-Singer and Kontsevich. This allows us apply Schwarz-Milnor lemma spaces, natural approach which we carry out successfully for first time. A...

We study a one-parameter family of billiard maps Tp given by the convex tables j.rIP + IyIP = 1 for p E [l, m]. We note that the topological entropy is positive if p $ {l, 2, a}. We study the linear stability of some periodic orbits and observe that the stability of these orbits changes for p = 2. For p $ { 1, 2, m}, there exist elliptic periodic orbits suggesting that Tp is not ergodic for all...

Continuing the study of the relationship between TC 0 , AC 0 and arithmetic circuits, started by Agrawal et al. 1], we answer a few questions left open in this paper. Our main result is that the classes DiiAC 0 and GapAC 0 coincide, under poly-time, log-space, and log-time uniformity. From that we can derive that under logspace uniformity, the following equalities hold:

We introduce the notion of cotype of a metric space, and prove that for Banach spaces it coincides with the classical notion of Rademacher cotype. This yields a concrete version of Ribe's theorem, settling a long standing open problem in the non-linear theory of Banach spaces. We apply our results to several problems in metric geometry. Namely, we use metric cotype in the study of uniform and c...

In trying to improve Weinstock's results on approximation by holomorphic functions on certain product domains, we are led to estimates in Sobolev spaces for the ∂-operator on polycylinders for (γ, q)-forms. This generalizes our results for the same operator on poly-cylinders previously obtained, and can be applied to a number of other problems such as the Corona problem. 1. Introduction. Had we...

Let G = (V, L) denote a network with nodes V and links L. Each link l ∈ L is associated with a performance metric that varies stochastically according to a distribution with unknown parameter θl. Let P be a given set of candidate probing paths in G. Each path py ∈ P consists of one or more pairwise adjacent links in G. We assume that the monitoring system can inject probes on any selected paths...