Relations Between Adjacency and Modularity Graph Partitioning

In this paper the exact linear relation between the leading eigenvector of the unnormalized modularity matrix and the eigenvectors of the adjacency matrix is developed. Based on this analysis a method to approximate the leading eigenvector of the modularity matrix is given, and the relative error of the approximation is derived. A complete proof of the equivalence between normalized modularity clustering and normalized adjacency clustering is also given. Some applications and experiments are given to illustrate and corroborate the points that are made in the theoretical development.Comment: 11 page

Stochastic Data Clustering

In 1961 Herbert Simon and Albert Ando published the theory behind the long-term behavior of a dynamical system that can be described by a nearly uncoupled matrix. Over the past fifty years this theory has been used in a variety of contexts, including queueing theory, brain organization, and ecology. In all these applications, the structure of the system is known and the point of interest is the various stages the system passes through on its way to some long-term equilibrium. This paper looks at this problem from the other direction. That is, we develop a technique for using the evolution of the system to tell us about its initial structure, and we use this technique to develop a new algorithm for data clustering.Comment: 23 page

The Evaluation Of Barrier Option Prices Under Stochastic Volatility

This paperc onsiders the problem o fnumerically evaluating barrier option prices when the dynamics of the underlying are driven by stochastic volatility following the square root process of Heston (1993). We develop a method of lines approach to evaluate the price as well as the delta and gamma of the option. The method is able to effciently handle bothc ontinuously monitored and discretely monitored barrier options and can also handle barrier options with early exercise features. In the latter case, we can calculate the early exercise boundary of an American barrier option in both the continuously and discretely monitored cases.barrier option; stochastic volatility; continuously monitored; discretely monitored; free boundary problem; method of lines; Monte Carlo simulation

Book Review. - Literatur

Book Review. - Literatu

Cranmer\u27s Legacy

At Oxford in the nineteenth century at the height of the Tractarian Movement, the Evangelicals believed that the Tractarians had shown themselves opposed to the principles of the Reformation. Partially to counteract this movement a proposal was made to erect a memorial to the Reformation martyrs. Cranmer, Latimer, and Ridley- the three men who had been burnt opposite Balliol College -were to be honored especially. Latimer and Ridley had died together there on October 16, 1555. Five months later, on the 21st of March, 1556, in the same place where Ridley and Latimer had suffered, Thomas Cranmer was burnt. The proposed memorial to these men was oversubscribed. Designed by Sir Gilbert Scot, it still stands today near St. Mary Magdalene\u27s Church as a testimony of the faith of these martyrs. Thus, in the minds of some, Oxford University reaffirmed the historic Protestantism of the Church of England and of the University