Riemann–Hilbert problems, Toeplitz operators and Q-classes

We generalize the notion of Q-classes C(Q1,Q2) , which was introduced in the context of Wiener–Hopf factorization, by considering very general 2 × 2 matrix functions Q1, Q2. This allows us to use a mainly algebraic approach to obtain several equivalent representations for each class, to study the intersections of Q-classes and to explore their close connection with certain non-linear scalar equations. The results are applied to various factorization problems and to the study of Toeplitz operators with symbol in a Q-class. We conclude with a group theoretic interpretation of some of the main results.Fundação para a Ciência e a Tecnologia (FCT/Portugal), through Project PTDC/MAT/121837/2010 and Project Est- C/MAT/UI0013/2011. The first author was also supported by the Center for Mathematical Analysis, Geometry, and Dynamical Systems and the second author was also supported by the Centre of Mathematics of the University of Minho through the FEDER Funds Programa Operacional Factores de Competitividade COMPET

A comprehensive analysis of pattern recognition receptors in normal and inflamed human epidermis: upregulation of dectin-1 in psoriasis.

Contains fulltext : 89335.pdf (publisher's version ) (Closed access)Human epidermis plays an important role in host defense by acting as a physical barrier and signaling interface between the environment and the immune system. Pattern recognition receptors (PRRs) are crucial to maintain homeostasis and provide protection during infection, but are also causally involved in monogenic auto-inflammatory diseases. This study aimed to investigate the epidermal expression of PRRs and several associated host defense molecules in healthy human skin, psoriasis, and atopic dermatitis (AD). Using microarray analysis and real-time quantitative PCR, we found that many of these genes are transcribed in normal human epidermis. Only a few genes were differentially induced in psoriasis (CLEC7A (dectin-1), Toll-like receptor (TLR) 4, and mannose receptor C type 1 (MRC1)) or AD (MRC1, IL1RN, and IL1beta) compared with normal epidermis. A remarkably high expression of dectin-1 mRNA was observed in psoriatic epidermis and this was corroborated by immunohistochemistry. In cultured primary human keratinocytes, dectin-1 expression was induced by IFN-gamma, IFN-alpha, and Th17 cytokines. Keratinocytes were unresponsive, however, to dectin-1 ligands such as beta-glucan or heat-killed Candida albicans, nor did we observe synergy with TLR2/TLR5 ligands. In conclusion, upregulation of dectin-1 in psoriatic lesions seems to be under control of psoriasis-associated cytokines. Its role in the biology of skin inflammation and infection remains to be explored.1 november 201