Relative tensor products for modules over von Neumann algebras

We give an overview of relative tensor products (RTPs) for von Neumann algebra modules. For background, we start with the categorical definition and go on to examine its algebraic formulation, which is applied to Morita equivalence and index. Then we consider the analytic construction, with particular emphasis on explaining why the RTP is not generally defined for every pair of vectors. We also look at recent work justifying a representation of RTPs as composition of unbounded operators, noting that these ideas work equally well for L^p modules. Finally, we prove some new results characterizing preclosedness of the map (\xi, \eta) \mapsto \xi \otimes_\phi \eta.Comment: 17 pages; to appear in Contemporary Mathematic

Two adaptation processes in auditory hair cells together can provide an active amplifier

The hair cells of the vertebrate inner ear convert mechanical stimuli to electrical signals. Two adaptation mechanisms are known to modify the ionic current flowing through the transduction channels of the hair bundles: a rapid process involves calcium ions binding to the channels; and a slower adaptation is associated with the movement of myosin motors. We present a mathematical model of the hair cell which demonstrates that the combination of these two mechanisms can produce `self-tuned critical oscillations', i.e. maintain the hair bundle at the threshold of an oscillatory instability. The characteristic frequency depends on the geometry of the bundle and on the calcium dynamics, but is independent of channel kinetics. Poised on the verge of vibrating, the hair bundle acts as an active amplifier. However, if the hair cell is sufficiently perturbed, other dynamical regimes can occur. These include slow relaxation oscillations which resemble the hair bundle motion observed in some experimental preparations.Comment: 13 pages, 6 figures,REVTeX 4, To appear in Biophysical Journa

Rapid, multiplexed microfluidic phage display

The development of a method for high-throughput, automated proteomic screening could impact areas ranging from fundamental molecular interactions to the discovery of novel disease markers and therapeutic targets. Surface display techniques allow for efficient handling of large molecular libraries in small volumes. In particular, phage display has emerged as a powerful technology for selecting peptides and proteins with enhanced, target-specific binding affinities. Yet, the process becomes cumbersome and time-consuming when multiple targets are involved.Here we demonstrate for the first time a microfluidic chip capable of identifying high affinity phage displayed peptides for multiple targets in just a single round and without the need for bacterial infection. The chip is shown to be able to yield well-established control consensus sequences while simultaneously identifying new sequences for clinically important targets. Indeed, the confined parameters of the device allow not only for highly controlled assay conditions but also introduce a significant time-reduction to the phage display process. We anticipate that this easily-fabricated, disposable device has the potential to impact areas ranging from fundamental studies of protein, peptide, and molecular interactions, to applications such as fully automated proteomic screening

Braiding Interactions in Anyonic Quantum Walks

The anyonic quantum walk is a dynamical model describing a single anyon propagating along a chain of stationary anyons and interacting via mutual braiding statistics. We review the recent results on the effects of braiding statistics in anyonic quantum walks in quasi-one dimensional ladder geometries. For anyons which correspond to spin-1/2 irreps of the quantum groups $SU(2)_k$, the non-Abelian species $(1<k<\infty)$ gives rise to entanglement between the walker and topological degrees of freedom which is quantified by quantum link invariants over the trajectories of the walk. The decoherence is strong enough to reduce the walk on the infinite ladder to classical like behaviour. We also present numerical results on mixing times of $SU(2)_2$ or Ising model anyon walks on cyclic graphs. Finally, the possible experimental simulation of the anyonic quantum walk in Fractional Quantum Hall systems is discussed.Comment: 13 pages, submitted to Proceedings of the 2nd International Conference on Theoretical Physics (ICTP 2012