Soliton representations and Sobolev diffeomorphism symmetry in CFT

We show that any positive energy representation of the group of (orientation preserving) smooth diffeomorphisms of the circle Diff(S^1) can be extended to a strongly continuous unitary projective representation of the group of (orientation preserving) fractional Sobolev diffeomorphisms D^s(S^1) with real Sobolev exponent s>3. For some positive energy representations, i.e for the positive energy vacuum representations of Diff(S^1) with positive integer central charge, we can improve the implementation to the group D^s(S^1) with s>2. We show that a conformal net of von Neumann algebras on the circle is always D^s(S^1)-covariant, s>3. Furthermore, we show that a given positive energy representation U of Diff(S^1) cannot be extended to some less-smooth diffeomorphisms, and from this fact we obtain an uncountable family of proper soliton representations. From these soliton representations we construct irreducible unitary projective positive energy representations of the group ΛSU(N) consisting of loops with support not containing the point -1 (resp. B_0, the stabilizer subgroup of -1) which do not extend to LSU(N) (resp. Diff(S^1))

Solitons and nonsmooth diffeomorphisms in conformal nets

We show that any solitonic representation of a conformal (diffeomorphism covariant) net on S^1 has positive energy and construct an uncountable family of mutually inequivalent solitonic representations of any conformal net, using nonsmooth diffeomorphisms. On the loop group nets, we show that these representations induce representations of the subgroup of loops compactly supported in S^1 \ {-1} which do not extend to the whole loop group. In the case of the U(1)-current net, we extend the diffeomorphism covariance to the Sobolev diffeomorphisms D^s(S^1), s > 2, and show that the positive-energy vacuum representations of Diff_+(S^1) with integer central charges extend to D^s(S^1). The solitonic representations constructed above for the U(1)-current net and for Virasoro nets with integral central charge are continuously covariant with respect to the stabilizer subgroup of Diff_+(S^1) of -1 of the circle.Comment: 33 pages, 3 TikZ figure

A computational study of the number of connected components of positive Thompson links

Almost a decade ago Vaughan Jones introduced a method to produce knots from elements of the Thompson groups $F$, which was later extended to the Brown-Thompson group $F_3$. In this article we define a way to produce permutations out of elements of the $F$ and $F_3$ that we call Thompson permutations. The number of orbits of each Thompson permutation coincides with the number of connected components of the link. We explore the positive elements of $F_3$ of fixed \emph{width} and \emph{height} and make some conjectures based on numerical experiments. In order to define the Thompson permutations we need to assign an orientation to each link produced from elements of $F$ and $F_3$. We prove that all oriented links can be produced in this way

Positive energy representations of Sobolev diffeomorphism groups of the circle

We show that any positive energy projective representation of Diff(S^1) extends to a strongly continuous projective unitary representation of the fractional Sobolev diffeomorphisms D^s(S^1) with s>3, and in particular to C^k-diffeomorphisms Diff^k(S^1) with k >= 4. A similar result holds for the universal covering groups provided that the representation is assumed to be a direct sum of irreducibles. As an application we show that a conformal net of von Neumann algebras on S^1 is covariant with respect to D^s(S^1), s > 3. Moreover every direct sum of irreducible representations of a conformal net is also D^s(S^1)-covariant.Comment: 30 pages, 1 TikZ figur

Accelerated Corneal Collagen Cross-Linking Using Topography-Guided UV-A Energy Emission: Preliminary Clinical and Morphological Outcomes

Purpose. To assess the clinical and morphological outcomes of topography-guided accelerated corneal cross-linking. Design. Retrospective case series. Methods. 21 eyes of 20 patients with progressive keratoconus were enrolled. All patients underwent accelerated cross-linking using an ultraviolet-A (UVA) exposure with an energy release varying from 7.2 J/cm2 up to 15 J/cm2, according to the topographic corneal curvature. Uncorrected (UDVA) and corrected (CDVA) distance visual acuity, topography, in vivo confocal microscopy (IVCM), and anterior segment optic coherence tomography (AS-OCT) were evaluated preoperatively and at the 1, 3, 6, and 12 months postoperatively. Results. 12 months after surgery UDVA and CDVA did not significantly vary from preoperative values. The average topographic astigmatism decreased from -4.61±0.74 diopters (D) to -3.20±0.81 D and coma aberration improved from 0.95 ± 0.03 μm to 0.88 ± 0.04 μm after surgery. AS-OCT and IVCM documented differential effects on the treated areas using different energies doses. The depths of demarcation line and keratocyte apoptosis were assessed. Conclusions. Preliminary results show correspondence between the energy dose applied and the microstructural stromal changes induced by the cross-linking at various depths in different areas of treated cornea. One year after surgery a significant reduction in the topographic astigmatism and comatic aberration was detected. None of the patients developed significant complications

Therapeutic effect of topical 5-fluorouracil in conjunctival squamous carcinoma is associated with changes in matrix metalloproteinases and tissue inhibitor of metalloproteinases expression

To evaluate matrix metalloproteinases (MMP)-2, MMP-9, and tissue inhibitor of metalloproteinase (TIMP)-1 expression in a case of conjunctival intraepithelial squamous cell carcinoma (SCC) treated with topical 5-fluorouracil (5-FU) chemotherapy

Expression of Toll-like receptors in healthy and allergic conjunctiva

To investigate the expression of toll-like receptors (TLRs) in healthy and active vernal keratoconjunctivitis (VKC) conjunctiva

Identification and functional inference on the MLO-family in viridiplantae

Powdery mildew (PM) is a widespread plant disease of temperate climates caused by ascomycete fungi of the order Erysiphales. PM is an important agricultural issue since it can cause significant economic losses. Specific members of the MLO gene family act as susceptibility factors towards the PM disease. A step towards the stability of crop productions would be thus the characterization of MLO genes at the genomic level. We carried out a genome-wide characterization of the MLO gene family in twenty-three plant and two algal genomes providing manual curated MLO protein catalogues. In total,180 novel proteins containing the MLO domain were identified. Evolutionary history and phylogenetic relationships were studied through maximum likelihood analysis. This highlighted eight different clades,including a new monocot-specific clade (VIII) identified for the first time. In addition,15 and 67 putative PM susceptibility genes,clustering in clade IV and V,respectively,were identified. Results of this work may help to address further biological questions concerning MLOs involved in PM susceptibility. In follow-up studies,it could be investigated whether the silencing or loss-of-function mutations in one or more of these candidate genes may lead to PM resistance