Counting and Generating Terms in the Binary Lambda Calculus (Extended version)

In a paper entitled Binary lambda calculus and combinatory logic, John Tromp presents a simple way of encoding lambda calculus terms as binary sequences. In what follows, we study the numbers of binary strings of a given size that represent lambda terms and derive results from their generating functions, especially that the number of terms of size n grows roughly like 1.963447954. .. n. In a second part we use this approach to generate random lambda terms using Boltzmann samplers.Comment: extended version of arXiv:1401.037

B-splines in EMD and Graph Theory in Pattern Recognition

With the development of science and technology, a large amount of data is waiting for further scientific exploration. We can always build up some good mathematical models based on the given data to analyze and solve the real life problems. In this work, we propose three types of mathematical models for different applications.;In chapter 1, we use Bspline based EMD to analysis nonlinear and no-stationary signal data. A new idea about the boundary extension is introduced and applied to the Empirical Mode Decomposition(EMD) algorithm. Instead of the traditional mirror extension on the boundary, we propose a ratio extension on the boundary.;In chapter 2 we propose a weighted directed multigraph for text pattern recognition. We set up a weighted directed multigraph model using the distances between the keywords as the weights of arcs. We then developed a keyword-frequency-distance-based algorithm which not only utilizes the frequency information of keywords but also their ordering information.;In chapter 3, we propose a centrality guided clustering method. Different from traditional methods which choose a center of a cluster randomly, we start clustering from a LEADER - a vertex with highest centrality score, and a new member is added into an existing community if the new vertex meet some criteria and the new community with the new vertex maintain a certain density.;In chapter 4, we define a new graph optimization problem which is called postman tour with minimum route-pair cost. And we model the DNA sequence assembly problem as the postman tour with minimum route-pair cost problem