The Kazhdan-Lusztig conjecture for finite W-algebras

We study the representation theory of finite W-algebras. After introducing parabolic subalgebras to describe the structure of W-algebras, we define the Verma modules and give a conjecture for the Kac determinant. This allows us to find the completely degenerate representations of the finite W-algebras. To extract the irreducible representations we analyse the structure of singular and subsingular vectors, and find that for W-algebras, in general the maximal submodule of a Verma module is not generated by singular vectors only. Surprisingly, the role of the (sub)singular vectors can be encapsulated in terms of a `dual' analogue of the Kazhdan-Lusztig theorem for simple Lie algebras. These involve dual relative Kazhdan-Lusztig polynomials. We support our conjectures with some examples, and briefly discuss applications and the generalisation to infinite W-algebras.Comment: 11 page

Plasma exchange induces vitamin D deficiency

BACKGROUND: Plasma exchange is used in the treatment of diseases mediated by pathogenic circulating proteins, or for transplant desensitization. Its non-targeted nature results in the depletion of physiologically important molecules, and it is often complicated by hypocalcaemia.  AIM: To determine the effects of plasma exchange on vitamin D binding protein (DBP) and associated vitamin D metabolites.  DESIGN: Single-centre prospective cohort study of 11 patients.  METHODS: DBP and vitamin D metabolites were measured before and immediately after five plasma exchanges, and 7 and 28 days after discontinuation of plasma exchange.  RESULTS: Plasma exchange reduced plasma DBP concentration from 196.9 ± 53.2 to 98.5 ± 34 μg/ml (P = 0.0001), 1,25-dihydroxyvitamin D from 103 ± 52 to 42 ± 4 pmol/l (P = 0.003) and 25-hydroxyvitamin D from 49.7 ± 29 to 22 ± 9.4 nmol/l (P = 0.0017), through their removal in effluent. After 7 days, DBP and 1,25-dihydroxyvitamin D were not significantly different from baseline, but 25-hydroxyvitamin D remained significantly lower after 7 days (26.4 ± 9.8 nmol/l, P = 0.02) and 28 days (30.8 ± 15.5 nmol/l, P = 0.048). Corrected calcium decreased from 2.23 ± 0.11 to 1.98 ± 0.08 mmol/l (P = 0.0007) immediately after five treatments. Plasma calcium was significantly associated with 1,25-dihydroxyvitamin D (r(2) = 0.79, P < 0.0001).  CONCLUSION: Plasma exchange induced an acute reversible decrease in plasma 1,25-dihydroxyvitamin D, DBP, calcium and a sustained decrease in plasma 25-hydroxyvitamin D

Definition and validation of the Design Space for co-milled nasal powder containing nanosized lamotrigine

<p><b>Objective:</b> Design of Experiment (DoE), that is a tool of Quality by Design (QbD) paradigm, with which experiments can be planned more effectively and provide more information, while after Design Space (DS) can be set up, which assure the quality of the desired product. The aim of this study was to find the optimal drug-excipient ratio and the optimal process parameters (milling time, milling speed) of our previously used dry co-milling method and validate the DS.</p> <p><b>Materials and methods:</b> Lamotrigine (LAM), an antiepileptic drug was used as a model API. Poly-vinyl alcohol (PVA) was chosen according to our previous study as a hydrophylic matrix polymer. Milling time, speed, and the API:additive ratio was varied to find out their effect on the product. The optimization was performed on particle size of LAM, its standard deviation and the <i>in vitro</i> dissolution of the samples. Response surface modeling completed the statistical analysis that assessed the effects of independent variables on the responses.</p> <p><b>Results:</b> Due to the DS estimation, a more economical sample preparation method was set up. Finally, the sample that was prepared according to the optimized parameters (1.5 h, 400 rpm, 0.8 PVA:LAM ratio) showed around 100 nm drug particles and 97% drug release in five minutes.</p> <p><b>Conclusion:</b> From the DS generated by the software, an optimal formulation was obtained and the results validated the experimental design. The QbD approach was a useful and effective tool of understanding the parameters that affect the quality of the desired product.</p