Amplitude equations for systems with long-range interactions

We derive amplitude equations for interface dynamics in pattern forming systems with long-range interactions. The basic condition for the applicability of the method developed here is that the bulk equations are linear and solvable by integral transforms. We arrive at the interface equation via long-wave asymptotics. As an example, we treat the Grinfeld instability, and we also give a result for the Saffman-Taylor instability. It turns out that the long-range interaction survives the long-wave limit and shows up in the final equation as a nonlocal and nonlinear term, a feature that to our knowledge is not shared by any other known long-wave equation. The form of this particular equation will then allow us to draw conclusions regarding the universal dynamics of systems in which nonlocal effects persist at the level of the amplitude description.Comment: LaTeX source, 12 pages, 4 figures, accepted for Physical Review

Pattern formation in directional solidification under shear flow. I: Linear stability analysis and basic patterns

An asymptotic interface equation for directional solidification near the absolute stabiliy limit is extended by a nonlocal term describing a shear flow parallel to the interface. In the long-wave limit considered, the flow acts destabilizing on a planar interface. Moreover, linear stability analysis suggests that the morphology diagram is modified by the flow near the onset of the Mullins-Sekerka instability. Via numerical analysis, the bifurcation structure of the system is shown to change. Besides the known hexagonal cells, structures consisting of stripes arise. Due to its symmetry-breaking properties, the flow term induces a lateral drift of the whole pattern, once the instability has become active. The drift velocity is measured numerically and described analytically in the framework of a linear analysis. At large flow strength, the linear description breaks down, which is accompanied by a transition to flow-dominated morphologies, described in a companion paper. Small and intermediate flows lead to increased order in the lattice structure of the pattern, facilitating the elimination of defects. Locally oscillating structures appear closer to the instability threshold with flow than without.Comment: 20 pages, Latex, accepted for Physical Review

Root polytopes and abelian ideals

We study the root polytope $\mathcal P_\Phi$ of a finite irreducible crystallographic root system $\Phi$ using its relation with the abelian ideals of a Borel subalgebra of a simple Lie algebra with root system $\Phi$. We determine the hyperplane arrangement corresponding to the faces of codimension 2 of $\mathcal P_\Phi$ and analyze its relation with the facets of $\mathcal P_\Phi$. For $\Phi$ of type $A_n$ or $C_n$, we show that the orbits of some special subsets of abelian ideals under the action of the Weyl group parametrize a triangulation of $\mathcal P_\Phi$. We show that this triangulation restricts to a triangulation of the positive root polytope $\mathcal P_\Phi^+$.Comment: 41 pages, revised version, accepted for publication in Journal of Algebraic Combinatoric

Patient satisfaction with calcipotriol/betamethasone dipropionate cutaneous foam for the treatment of plaque psoriasis: The LION real-life multicenter prospective observational cohort study

Topical treatment is the mainstay for mild or moderate psoriasis, but patients are generally little satisfied. Calcipotriol/betamethasone dipropionate (Cal/BD) cutaneous foam has shown to improve signs and symptoms in plaque psoriasis patients. This study assessed patient's satisfaction with Cal/BD foam in a real-life Italian dermatological clinical practice. A multicenter, 4-week observational prospective cohort study enrolled, in 17 Italian dermatology clinics, adult patients with plaque psoriasis on the body and/or scalp. Treatment satisfaction was assessed by 9-item Treatment Satisfaction Questionnaire for Medication (TSQM-9), preference over previous treatments by Patient Preference Questionnaire (PPQ), and change in disease state by Psoriasis Area Severity Index (PASI). Overall 256 patients were eligible, with a mean (SD) age of 55.6 (15.4) years, 59.4% were males. Psoriasis severity was mild in 52.0% of patients, moderate in 43.3%, and severe in 4.7%. Scalp involvement was present in 36.7% of patients. Previous antipsoriatic treatments had been received by 80.5% of patients. TSQM-9 median (25th–75th percentile) scores were 83.3 (66.7–88.9) for effectiveness, 77.8 (66.7–88.9) for convenience, and 78.6 (64.3–92.9) for global satisfaction. Mean (SD) PASI value decreased from 7.3 (4.8) to 2.1 (2.7) after 4 weeks. More than 90% of patients previously treated for psoriasis evaluated the Cal/BD foam more effective, easier to use and better tolerated compared to previous topical treatments at PPQ. This observational study provides real-life evidence of a high level of satisfaction with effectiveness and convenience of the Cal/BD foam in a cohort of plaque psoriasis patients, with an objective improvement in PASI