School of Mathematics and Statistics

586.30
3.05
257

Member activityView all

• Article: The K\
[hide abstract]
ABSTRACT: We consider complexes $(\X, d)$ of nuclear Fr\'echet spaces and continuous boundary maps $d_n$ with closed ranges and prove that, up to topological isomorphism, $(H_{n}(\X, d))^*$ $\iso$ $H^{n}(\X^*,d^*),$ where $(H_{n}(\X,d))^*$ is the strong dual space of the homology group of $(\X,d)$ and $H^{n}(\X^*,d^*)$ is the cohomology group of the strong dual complex $(\X^*,d^*)$. We use this result to establish the existence of topological isomorphisms in the K\"{u}nneth formula for the cohomology of complete nuclear $DF$-complexes and in the K\"{u}nneth formula for continuous Hochschild cohomology of nuclear $\hat{\otimes}$-algebras which are Fr\'echet spaces or $DF$-spaces for which all boundary maps of the standard homology complexes have closed ranges. We describe explicitly continuous Hochschild and cyclic cohomology groups of certain tensor products of $\hat{\otimes}$-algebras which are Fr\'echet spaces or nuclear $DF$-spaces.
• Source
Article: Dark soliton dynamics in confined Bose-Einstein condensates
[hide abstract]
ABSTRACT: Dilute atomic Bose-Einstein condensates are inherently nonlinear systems and support solitary wave solutions. An important distinction from optical systems is the inhomogeneous background density, which results from the traps used to confine the atoms. As in optical systems, dark solitary waves in three- dimensional geometries are unstable to transverse excitations, which lead to a bending of the dark soliton plane and decay into vortex rings. Highly elon- gated geometries can now be achieved experimentally, in which the condensate dynamics are eectively one-dimensional, and the motion of the dark soliton is governed by the inhomogeneous longitudinal density. We show that a dark soliton is fundamentally unstable to such a changing background density, by means of numerical simulations of the soliton under various potentials (e.g. steps, ramps, harmonic traps, and optical lattices). This leads to the emission of radiation in the form of sound waves. The power emitted is found to be proportional to the square of the soliton acceleration. The latter quantity is shown to be proportional to the deformation of the apparent soliton profile, arising from the sound field in the region of the soliton. We demonstrate that the ensuing interactions between the soliton and sound field, and therefore the dynamics of the soliton, can be controlled experimentally via manipulation of the emitted sound, achieved by modifying the trap geometry. In this manner, it is possible to induce a rapid decay of the soliton, stabilise the soliton, or even pump energy into the soliton by means of parametric driving.
• Article: A case of mu-synthesis as a quadratic semidefinite program
[hide abstract]
ABSTRACT: We analyse a special case of the robust stabilization problem under structured uncertainty. We obtain a new criterion for the solvability of the spectral Nevanlinna-Pick problem, which is a special case of the $\mu$-synthesis problem of $H^\infty$ control in which $\mu$ is the spectral radius. Given $n$ distinct points $\la_1,\dots,\la_n$ in the unit disc and $2\times 2$ nonscalar complex matrices $W_1,\dots,W_n$, the problem is to determine whether there is an analytic $2\times 2$ matrix function $F$ on the disc such that $F(\la_j)=W_j$ for each $j$ and the supremum of the spectral radius of $F(\la)$ is less than 1 for $\la$ in the disc. The condition is that the minimum of a quadratic function of pairs of positive $3n$-square matrices subject to certain linear matrix inequalities in the data be attained and be zero.
03/2013;