Rigidity of asymptotically conical shrinking gradient Ricci solitons

We show that if two gradient Ricci solitons are asymptotic along some end of each to the same regular cone, then the soliton metrics must be isometric on some neighborhoods of infinity of these ends. Our theorem imposes no restrictions on the behavior of the metrics off of the ends in question and in particular does not require their geodesic completeness. As an application, we prove that the only complete connected gradient shrinking Ricci soliton asymptotic to a rotationally symmetric cone is the Gaussian soliton on R^n.Comment: 44 page

The gradient of a graph

In this article I introduce a dynamic interpretation of the gradient of a graph which leads naturally into the notion of differentiation

New modification of the hestenes-stiefel with strong wolfe line search

. The method of the nonlinear conjugate gradient is widely used in solving large-scale unconstrained optimization since been proven in solving optimization problems without using large memory storage. In this paper, we proposed a new modification of the Hestenes-Stiefel conjugate gradient parameter that fulfils the condition of sufficient descent using a strong Wolfe-Powell line search. Besides, the conjugate gradient method with the proposed conjugate gradient also guarantees low computation of iteration and CPU time by comparing with other classical conjugate gradient parameters. Numerical results have shown that the conjugate gradient method with the proposed conjugate gradient parameter performed better than the conjugate gradient method with other classical conjugate gradient parameters