The exact form of a gradient-following learning algorithm for completely recurrent networks running in continually sampled time is derived and used as the basis for practical algorithms for temporal supervised learning tasks. These algorithms have: (1) the advantage that they do not require a

In this paper we derive a precise estimate on the growth of potential functions of complete noncompact shrinking solitons. Based on this, we prove that a complete noncompact gradient shrinking Ricci soliton has at most Euclidean volume growth. The latter result can be viewed as an analog

ABSTRACT. In this paper, we first apply an integral identity on Ricci solitons to prove that closed locally conformally flat gradient Ricci solitons are of constant sectional curvature. We then generalize this integral identity to complete noncompact gradient shrinking Ricci solitons, under

Abstract. In this paper we show how techniques coming from stochastic analysis, such as stochastic completeness (in the form of the weak maximum principle at infinity), parabolicity and L p-Liouville type results for the weighted Laplacian associated to the potential may be used to obtain

Abstract. In this paper we prove that any complete conformal gradient soliton with non-negative Ricci tensor is either isometric to a direct product R×Nn−1, or globally conformally equivalent to the Euclidean space Rn or to the round sphere Sn. In particular, we show that any complete, noncompact

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curvature, the scalar curvature R has at least one maximum point on X, which is the only minimum point of the potential function f. Furthermore, R> 0 on X unless (X, g) is Ricci flat. We also show that there is exponentially decay for scalar curvature for ǫ-pinched complete non-compact expanding solitons

Abstract. In this paper, we study the volume growth property of a non-compact complete Riemannian manifold X. We improve the volume growth theorem of Calabi (1975) and Yau (1976), Cheeger, Gromov and Taylor (1982). Then we use our new result to study gradient Ricci solitons. We also show that on X

Abstract. A new type of gradient estimate is established for diffusion semigroups on non-compact complete Riemannian manifolds. As applications, a global Harnack inequality with power and a heat kernel estimate are derived for diffusion semigroups on arbitrary complete Riemannian manifolds. 1