Casimirs and Lax operators from the structure of Lie algebras

This paper uses the structure of the Lie algebras to identify the Casimir invariant functions and Lax operators for matrix Lie groups. A novel mapping is found from the cotangent space to the dual Lie algebra which enables Lax operators to be found. The coordinate equations of motion are given in terms of the structure constants and the Hamiltonian.

Centrality and spectral radius in dynamic communication networks

We explore the influence of the choice of attenuation factor on Katz centrality indices for evolving communication networks. For given snapshots of a network observed over a period of time, recently developed communicability indices aim to identify best broadcasters and listeners in the network. In this article, we looked into the sensitivity of communicability indices on the attenuation factor constraint, in relation to spectral radius (the largest eigenvalue) of the network at any point in time and its computation in the case of large networks. We proposed relaxed communicability measures where the spectral radius bound on attenuation factor is relaxed and the adjacency matrix is normalised in order to maintain the convergence of the measure. Using a vitality based measure of both standard and relaxed communicability indices we looked at the ways of establishing the most important individuals for broadcasting and receiving of messages related to community bridging roles. We illustrated our findings with two examples of real-life networks, MIT reality mining data set of daily communications between 106 individuals during one year and UK Twitter mentions network, direct messages on Twitter between 12.4k individuals during one week.

Solvability and spectral properties of integral equations on the real line: I. Weighted spaces of continuous functions

We consider in this paper the solvability of linear integral equations on the real line, in operator form (λ−K)φ=ψ, where and K is an integral operator. We impose conditions on the kernel, k, of K which ensure that K is bounded as an operator on . Let Xa denote the weighted space as |s|→∞}. Our first result is that if, additionally, |k(s,t)|⩽κ(s−t), with and κ(s)=O(|s|−b) as |s|→∞, for some b>1, then the spectrum of K is the same on Xa as on X, for 01. As an example where kernels of this latter form occur we discuss a boundary integral equation formulation of an impedance boundary value problem for the Helmholtz equation in a half-plane.