R ring (always commutative and Noetherian) (R,m,k) local ring with maximal ideal m and k = R/m L,M,N, . . . R-modules (always finitely generated) M HomR(M,R), the dual of M D(M) the Auslander dual of M (Definition 2) σM : M wM∗∗ the natural evaluation map; KM = Ker(σM ), CM = Coker(σM ) G-dimR(M),G-dim(M) Gorenstein dimension of M (Definition 16) G-dim(M) <loc ∞ M has locally finite Gorenstein ...

We introduce an algorithm for causally estimating three-dimensional shape and motion of an object represented locally as a rigid collection of planes supporting projectively deforming texture patches. Due to occlusions and the local nature of any causal algorithm, a drift in the estimates accumulates over time. We describe a method to perform global registration of local estimates of motion and...

Taking full advantage of two independent projectively equivalent metrics on the ellipsoid leading to Liouville integrability of the geodesic flow via the well-known Jacobi-Moser system, we disclose a novel integrable system on the sphere S, namely the dual Moser system. The latter falls, along with the Jacobi-Moser and Neumann-Uhlenbeck systems, into the category of (locally) Stäckel systems. M...

We show that there exist non-trivial piecewise-linear (PL) knots with isolated singularities Sn−2 ⊂ Sn, n ≥ 5, whose complements have the homotopy type of a circle. This is in contrast to the case of smooth, PL locally-flat, and topological locally-flat knots, for which it is known that if the complement has the homotopy type of a circle, then the knot is trivial. It is well-known that if the c...

We will review work with Tatiana Toro yielding a characterization of those domains for which the harmonic measure has a density whose logarithm has vanishing mean oscillation. 2000 Mathematics Subject Classification: 31B25, 35R35, 42B35, 51M25.

1. A principal bundle 33(X, G) with base space X and group G is called "locally flat" if the structural group G can be reduced to a totally disconnected subgroup GiCG, or in other words: if there exists a totally disconnected subgroup GiQG, a principal bundle 33i(X, Gi) and an injection j: Soi—>33 [l ]. For instance, if a differentiable bundle 33 admits an infinitesimal connection whose curvatu...

By the SYZ construction, a mirror pair (X,Xˇ) of complex torus X and partner Xˇ is described as special Lagrangian fibrations X→B Xˇ→B on same base space B. Then, by transform, we can construct simple projectively flat bundle from each affine multisection with unitary local system along it. However, there are ambiguities choices transition functions it, this causes difficulties when try to func...