Extended Bernoulli and Stirling matrices and related combinatorial identities

In this paper we establish plenty of number theoretic and combinatoric identities involving generalized Bernoulli and Stirling numbers of both kinds. These formulas are deduced from Pascal type matrix representations of Bernoulli and Stirling numbers. For this we define and factorize a modified Pascal matrix corresponding to Bernoulli and Stirling cases.Comment: Accepted for publication in Linear Algebra and its Application

Collective Field Theory for Quantum Hall States

We develop a collective field theory for fractional quantum Hall (FQH) states. We show that in the leading approximation for a large number of particles, the properties of Laughlin states are captured by a Gaussian free field theory with a (filling fraction dependent) background charge. Gradient corrections to the Gaussian field theory arise from ultraviolet regularization. They are the origin of the gravitational anomaly and are described by the Liouville theory of quantum gravity. The field theory simplifies the computation of correlation functions in FQH states and makes manifest the effect of quantum anomalies.Comment: v1: 20 pages; v2: 6 pages, considerably revised and rewritten for the sake of clarity and brevity, v3: 7 pages, updated to reflect the published version which includes a discussion of the effects spi

Optimization via Low-rank Approximation for Community Detection in Networks

Community detection is one of the fundamental problems of network analysis, for which a number of methods have been proposed. Most model-based or criteria-based methods have to solve an optimization problem over a discrete set of labels to find communities, which is computationally infeasible. Some fast spectral algorithms have been proposed for specific methods or models, but only on a case-by-case basis. Here we propose a general approach for maximizing a function of a network adjacency matrix over discrete labels by projecting the set of labels onto a subspace approximating the leading eigenvectors of the expected adjacency matrix. This projection onto a low-dimensional space makes the feasible set of labels much smaller and the optimization problem much easier. We prove a general result about this method and show how to apply it to several previously proposed community detection criteria, establishing its consistency for label estimation in each case and demonstrating the fundamental connection between spectral properties of the network and various model-based approaches to community detection. Simulations and applications to real-world data are included to demonstrate our method performs well for multiple problems over a wide range of parameters.Comment: 45 pages, 7 figures; added discussions about computational complexity and extension to more than two communitie

Translating Khan on Singer: Global Solvent Versus Local Interpretation

This work focuses on Peter Singer’s book, One World: The Ethics of Globalisation, and a reading of it recently presented by M. Ali Khan. Khan’s response to Singer is acutely critical, but ultimately fails to situate Singer’s offering in its proper historical context. In this sense, Khan’s response is not sufficient. We demonstrate that Singer’s offering is permeated by a universalising discourse marked by asymmetric power relations clearly described by Edward Said in Orientalism and, more surprisingly, by Fyodor Dostoyevsky in The Possessed. We illustrate how Singer’s narrative and the counter-narrative of Khan represent a continuation of a longer historical disputation between the West and the East.Orientalism, Globalisation, Economy, Language, Translation, Communication, Domination, Dialogue, Local, Global, Community