Non-Local Matrix Generalizations of W-Algebras

There is a standard way to define two symplectic (hamiltonian) structures, the first and second Gelfand-Dikii brackets, on the space of ordinary linear differential operators of order $m$, $L = -d^m + U_1 d^{m-1} + U_2 d^{m-2} + \ldots + U_m$. In this paper, I consider in detail the case where the $U_k$ are $n\times n$-matrix-valued functions, with particular emphasis on the (more interesting) second Gelfand-Dikii bracket. Of particular interest is the reduction to the symplectic submanifold $U_1=0$. This reduction gives rise to matrix generalizations of (the classical version of) the {\it non-linear} $W_m$-algebras, called $V_{m,n}$-algebras. The non-commutativity of the matrices leads to {\it non-local} terms in these $V_{m,n}$-algebras. I show that these algebras contain a conformal Virasoro subalgebra and that combinations $W_k$ of the $U_k$ can be formed that are $n\times n$-matrices of conformally primary fields of spin $k$, in analogy with the scalar case $n=1$. In general however, the $V_{m,n}$-algebras have a much richer structure than the $W_m$-algebras as can be seen on the examples of the {\it non-linear} and {\it non-local} Poisson brackets of any two matrix elements of $U_2$ or $W_3$ which I work out explicitly for all $m$ and $n$. A matrix Miura transformation is derived, mapping these complicated second Gelfand-Dikii brackets of the $U_k$ to a set of much simpler Poisson brackets, providing the analogue of the free-field realization of the $W_m$-algebras.Comment: 43 pages, a reference and a remark on the conformal properties for $U_1\ne 0$ adde

KCNQ channels regulate age-related memory impairment:KCNQ regulates age-related memory

In humans KCNQ2/3 heteromeric channels form an M-current that acts as a brake on neuronal excitability, with mutations causing a form of epilepsy. The M-current has been shown to be a key regulator of neuronal plasticity underlying associative memory and ethanol response in mammals. Previous work has shown that many of the molecules and plasticity mechanisms underlying changes in alcohol behaviour and addiction are shared with those of memory. We show that the single KCNQ channel in Drosophila (dKCNQ) when mutated show decrements in associative short- and long-term memory, with KCNQ function in the mushroom body α/βneurons being required for short-term memory. Ethanol disrupts memory in wildtype flies, but not in a KCNQ null mutant background suggesting KCNQ maybe a direct target of ethanol, the blockade of which interferes with the plasticity machinery required for memory formation. We show that as in humans, Drosophila display age-related memory impairment with the KCNQ mutant memory defect mimicking the effect of age on memory. Expression of KCNQ normally decreases in aging brains and KCNQ overexpression in the mushroom body neurons of KCNQ mutants restores age-related memory impairment. Therefore KCNQ is a central plasticity molecule that regulates age dependent memory impairment

Multi-Component KdV Hierarchy, V-Algebra and Non-Abelian Toda Theory

I prove the recently conjectured relation between the $2\times 2$-matrix differential operator $L=\partial^2-U$, and a certain non-linear and non-local Poisson bracket algebra ($V$-algebra), containing a Virasoro subalgebra, which appeared in the study of a non-abelian Toda field theory. Here, I show that this $V$-algebra is precisely given by the second Gelfand-Dikii bracket associated with $L$. The Miura transformation is given which relates the second to the first Gelfand-Dikii bracket. The two Gelfand-Dikii brackets are also obtained from the associated (integro-) differential equation satisfied by fermion bilinears. The asymptotic expansion of the resolvent of $(L-\xi)\Psi=0$ is studied and its coefficients $R_l$ yield an infinite sequence of hamiltonians with mutually vanishing Poisson brackets. I recall how this leads to a matrix KdV hierarchy which are flow equations for the three component fields $T, V^+, V^-$ of $U$. For $V^\pm=0$ they reduce to the ordinary KdV hierarchy. The corresponding matrix mKdV equations are also given, as well as the relation to the pseudo- differential operator approach. Most of the results continue to hold if $U$ is a hermitian $n\times n$-matrix. Conjectures are made about $n\times n$-matrix $m^{\rm th}$-order differential operators $L$ and associated $V_{(n,m)}$-algebras.Comment: 20 pages, revised: several references to earlier papers on multi-component KdV equations are adde

Classical A_n--W-Geometry

This is a detailed development for the $A_n$ case, of our previous article entitled "W-Geometries" to be published in Phys. Lett. It is shown that the $A_n$--W-geometry corresponds to chiral surfaces in $CP^n$. This is comes out by discussing 1) the extrinsic geometries of chiral surfaces (Frenet-Serret and Gauss-Codazzi equations) 2) the KP coordinates (W-parametrizations) of the target-manifold, and their fermionic (tau-function) description, 3) the intrinsic geometries of the associated chiral surfaces in the Grassmannians, and the associated higher instanton- numbers of W-surfaces. For regular points, the Frenet-Serret equations for $CP^n$--W-surfaces are shown to give the geometrical meaning of the $A_n$-Toda Lax pair, and of the conformally-reduced WZNW models, and Drinfeld-Sokolov equations. KP coordinates are used to show that W-transformations may be extended as particular diffeomorphisms of the target-space. This leads to higher-dimensional generalizations of the WZNW and DS equations. These are related with the Zakharov- Shabat equations. For singular points, global Pl\"ucker formulae are derived by combining the $A_n$-Toda equations with the Gauss-Bonnet theorem written for each of the associated surfaces.Comment: (60 pages

Rational vs Polynomial Character of Wnl_n^l-Algebras

The constraints proposed recently by Bershadsky to produce $W^l_n$ algebras are a mixture of first and second class constraints and are degenerate. We show that they admit a first-class subsystem from which they can be recovered by gauge-fixing, and that the non-degenerate constraints can be handled by previous methods. The degenerate constraints present a new situation in which the natural primary field basis for the gauge-invariants is rational rather than polynomial. We give an algorithm for constructing the rational basis and converting the base elements to polynomials.Comment: 18 page

Enhancing pharmaceutical packaging through a technology ecosystem to facilitate the reuse of medicines and reduce medicinal waste

The idea of reusing dispensed medicines is appealing to the general public provided its benefits are illustrated, its risks minimized, and the logistics resolved. For example, medicine reuse could help reduce medicinal waste, protect the environment and improve public health. However, the associated technologies and legislation facilitating medicine reuse are generally not available. The availability of suitable technologies could arguably help shape stakeholders’ beliefs and in turn, uptake of a future medicine reuse scheme by tackling the risks and facilitating the practicalities. A literature survey is undertaken to lay down the groundwork for implementing technologies on and around pharmaceutical packaging in order to meet stakeholders’ previously expressed misgivings about medicine reuse (’stakeholder requirements’), and propose a novel ecosystem for, in effect, reusing returned medicines. Methods: A structured literature search examining the application of existing technologies on pharmaceutical packaging to enable medicine reuse was conducted and presented as a narrative review. Results: Reviewed technologies are classified according to different stakeholders’ requirements, and a novel ecosystem from a technology perspective is suggested as a solution to reusing medicines. Conclusion: Active sensing technologies applying to pharmaceutical packaging using printed electronics enlist medicines to be part of the Internet of Things network. Validating the quality and safety of returned medicines through this network seems to be the most effective way for reusing medicines and the correct application of technologies may be the key enabler

Islam in Turkish Cinema

This essay describes the history of Turkish cinema from its origins in 1896 until the present, focusing upon how Islam and religious Muslims are portrayed in the movies. For historical, political, and cultural reasons, Islam and religious Muslims have often been portrayed in a negative light, even though Turkey is a predominantly Muslim country. Present day Turkish cinema offers a more eclectic array of movies, including movies that show Islamic practices in daily life and movies that portray Islam in a favorable light