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This book is about neural networks in general and recurrent neural network in particular. Neural networks are networks of biological neurons. This definition is the traditional one, where a neural network was considered as a simple calculator. However, modern usage of the term refers to they are networks of Artificial neurons, which are simple to handle. On the other hand, recurrent neural network consists of the graph G = (V, E) and a family of formal neurons (Xi, Yi, Sigmai, Si), each associated to one of the vertices i V . The general interest to study neural networks in general and recurrent neural network in particular is that their remarkable ability to derive meaning from complicated or imprecise data, can be used to extract patterns and detect trends that are too complex to be noticed by either humans or other computer techniques.