Conjugation Orbits of Loxodromic Pairs in SU(n,1)

Let ${\bf H}_{\mathbb C}^n$ be the $n$-dimensional complex hyperbolic space and ${\rm SU}(n,1)$ be the (holomorphic) isometry group. An element $g$ in ${\rm SU}(n,1)$ is called loxodromic or hyperbolic if it has exactly two fixed points on the boundary $\partial {\bf H}_{\mathbb C}^n$. We classify ${\rm SU}(n,1)$ conjugation orbits of pairs of loxodromic elements in ${\rm SU}(n,1)$.Comment: final version. to appear in Bull. Sci. Mat

Validation of the Vitiligo Noticeability Scale: a patient-reported outcome measure of Vitiligo treatment success

Background: Patient-reported outcome measures are rarely used in vitiligo trials. The Vitiligo Noticeability Scale (VNS) is a new patient-reported outcome measure assessing how ‘noticeable’ the vitiligo patches are after treatment. The noticeability of vitiligo after treatment is an important indicator of treatment success from the patient’s perspective. Objectives: This study aimed to evaluate the construct validity, acceptability and interpretability of the VNS. Our main hypothesis was that the VNS would be a better and more consistent indicator of treatment success than percentage repigmentation. Methods: Clinicians (n=33) and patients with vitiligo (n=101) examined 39 image pairs, each depicting a vitiligo lesion pre- and post-treatment. Using an online questionnaire, respondents gave a global assessment of treatment success and a VNS score for treatment response. Clinicians also estimated percentage repigmentation of lesions (75). Treatment success was defined as ‘Yes’ on global assessment, a VNS score of 4 or 5, and more than 75% repigmentation. Agreement between respondents and the different scales was assessed using kappa statistics. Results: VNS scores were associated with both patient- and clinician-reported global treatment success (κ = 0.54 and κ = 0.47, respectively). Percentage repigmentation showed a weaker association with patient- and clinician-reported global treatment success (κ = 0.39 and κ = 0.29, respectively). VNS scores of 4 or 5 can be interpreted as representing treatment success. Images depicting post-treatment hyperpigmentation were less likely to be rated as successful. Conclusions: The VNS is a valid patient-reported measure of vitiligo treatment success. Further validation of the VNS is required, using larger sets of clinical pre- and post-treatment images, affecting a wider range of anatomical sites

Determination of melanin types and relative concentrations: an observational study using a non-invasive inverse skin reflectance analysis

Objective: Melanin is a major skin colour pigment that made up of eumelanin (the dark brown–black colour) and pheomelanin (the light red–yellow colour) pigments. Skin-whitening products typically contain depigmentation agents that reduce the level of pigmentation by changing the pheomelanin–eumelanin production. Similarly, in skin pigment treatment of skin disorders, the melanin production is managed accordingly. To assess and improve treatment efficacy, it is important to have a measurement tool that is capable of determining the melanin types objectively. So far, the efficacy assessment is subjective. In this study, an inverse skin reflectance pigmentation analysis system that determines eumelanin and pheomelanin content is developed and evaluated in an observational study involving 36 participants with skin photo type IV. Methods: The reflectance spectra of the left forearms of participants were analysed by the pigmentation analysis system to determine their skin parameters – pheomelanin and eumelanin concentrations, melanosome volume fraction, and epidermal thickness. The determined skin parameters are then inputted into the realistic skin model (RSM) of the Advanced Systems Analyses Program (asap®) to generate the ground truth reflectance spectra for the given skin parameters to validate the system. Results: The developed pigmentation analysis system is found to be accurate with a spectral error of 0.0163 ± 0.009 between measured reflectance and the reflectance output of the analysis system and RSM. The regression analysis shows a strong linear relationship (R2 = 0.994) indicating good precision. The relative concentrations of pheomelanin (38.23 ± 15.04) and eumelanin (1.68 ± 0.91) analysed by the system gives a ratio of pheomelanin to eumelanin of 0.048 ± 0.029; this value is consistent with previously reported figure of 0.049. Conclusion: The proposed pigmentation analysis system is able to determine melanin types and their relative concentrations. It has the potential to assess the efficacy of the skin-whitening and pigmentation treatments objectively in a non-invasive manner