On fermionic representation of the framed topological vertex

The Gromov-Witten invariants of \mathbb{C}^3 with branes is encoded in the topological vertex which has a very complicated combinatorial expression. A simple formula for the topological vertex was proposed by Aganagic et al in the fermionic picture. We will propose a similar formula for the framed topological vertex and prove it in the case when there are one or two branes.Comment: An acknowledgement adde

Structural reliability analysis for implicit performance function using radial basis function network

AbstractThis is the second paper of our work on structural reliability analysis for implicit performance function. The first paper proposed structural reliability analysis methods using multilayer perceptron artificial neural network [Deng, J., Gu, D.S., Li, X.B., Yue, Z.Q., 2005. Structural reliability analysis for implicit performance function using artificial neural network. Structural Safety 25 (1), 25–48]. This paper presents three radial basis function network (RBF) based reliability analysis methods, i.e. RBF based MCS, RBF based FORM, and RBF based SORM. In these methods, radial basis function network technique is adopted to model and approximate the implicit performance functions or partial derivatives. The RBF technique uses a small set of the actual data of the implicit performance functions, which are obtained via physical experiments or normal numerical analysis such as finite element methods for the complicated structural system, and are used to develop a trained RBF generalization algorithm. Then a large number of the function values and partial derivatives of implicit performance functions can be readily obtained by simply extracting information from the established and successfully trained RBF network. These function values and derivatives are used in conventional MCS, FORM or SORM to constitute RBF based reliability analysis algorithms. Examples are presented in the paper to illustrate how the proposed RBF based methods are used in structural reliability analysis. The results are well compared with those obtained by the conventional reliability methods such as the Monte-Carlo simulation, multilayer perceptrons networks, the response surface method, the FORM method 2, and so on. The examples showed the proposed approach is applicable to structural reliability analysis involving implicit performance functions