Integral formulas for wave functions of quantum many-body problems and representations of gl(n)

We derive explicit integral formulas for eigenfunctions of quantum integrals of the Calogero-Sutherland-Moser operator with trigonometric interaction potential. In particular, we derive explicit formulas for Jack's symmetric functions. To obtain such formulas, we use the representation of these eigenfunctions by means of traces of intertwining operators between certain modules over the Lie algebra $\frak gl_n$, and the realization of these modules on functions of many variables.Comment: 6 pages. One reference ([FF]) has been corrected. New references and an introduction have been adde

Quantum integrable systems and representations of Lie algebras

In this paper the quantum integrals of the Hamiltonian of the quantum many-body problem with the interaction potential K/sinh^2(x) (Sutherland operator) are constructed as images of higher Casimirs of the Lie algebra gl(N) under a certain homomorphism from the center of U(gl(N)) to the algebra of differential operators in N variables. A similar construction applied to the affine gl(N) at the critical level k=-N defines a correspondence between higher Sugawara operators and quantum integrals of the Hamiltonian of the quantum many-body problem with the potential equal to constant times the Weierstrass function. This allows one to give a new proof of the Olshanetsky-Perelomov theorem stating that this Hamiltonian defines a completely integrable quantum system. We also give a new expression for eigenfunctions of the quantum integrals of the Sutherland operator as traces of intertwining operators between certain representations of gl(N).Comment: 17 pages, no figure

Automated Inference of Past Action Instances in Digital Investigations

As the amount of digital devices suspected of containing digital evidence increases, case backlogs for digital investigations are also increasing in many organizations. To ensure timely investigation of requests, this work proposes the use of signature-based methods for automated action instance approximation to automatically reconstruct past user activities within a compromised or suspect system. This work specifically explores how multiple instances of a user action may be detected using signature-based methods during a post-mortem digital forensic analysis. A system is formally defined as a set of objects, where a subset of objects may be altered on the occurrence of an action. A novel action-trace update time threshold is proposed that enables objects to be categorized by their respective update patterns over time. By integrating time into event reconstruction, the most recent action instance approximation as well as limited past instances of the action may be differentiated and their time values approximated. After the formal theory if signature-based event reconstruction is defined, a case study is given to evaluate the practicality of the proposed method.Comment: International Journal of Information Securit