Sub-Riemannian calculus and monotonicity of the perimeter for graphical strips

We consider the class of minimal surfaces given by the graphical strips ${{\mathcal S}}We consider the class of minimal surfaces given by the graphical strips S{{\mathcal S}} in the Heisenberg group \mathbb H¹{{\mathbb {H}}^1} and we prove that for points p along the center of \mathbb H¹{{\mathbb {H}}^1} the quantity \fracs_H(S?B(p,r))r^Q-1{\frac{\sigma_H(\mathcal S\cap B(p,r))}{r^{Q-1}}} is monotone increasing. Here, Q is the homogeneous dimension of \mathbb H¹{{\mathbb {H}}^1} . We also prove that these minimal surfaces have maximum volume growth at infinity.     

Local Behavior of <Emphasis Type="Italic">p</Emphasis>-harmonic Green’s Functions in Metric Spaces

We describe the behavior of p-harmonic Green’s functions near a singularity in metric measure spaces equipped with a doubling measure and supporting a Poincaré inequality.     

Urinary Metabolomics Study of Patients with Bicuspid Aortic Valve Disease

Bicuspid aortic valve (BAV) is the most common congenital heart defect responsible for valvular and aortic complications in affected patients. Causes and mechanisms of this pathology are still elusive and thus the lack of early detection biomarkers leads to challenges in its diagnosis and prevention of associated cardiovascular anomalies. The aim of this study was to explore the potential use of urine Nuclear Magnetic Resonance (NMR) metabolomics to evaluate a molecular fingerprint of BAV. Both multivariate and univariate statistical analyses were performed to compare the urinary metabolome of 20 patients with BAV with that of 24 matched controls. Orthogonal partial least squared discriminant analysis (OPLS-DA) showed statistically significant discrimination between cases and controls, suggesting seven metabolites (3-hydroxybutyrate, alanine, betaine, creatine, glycine, hippurate, and taurine) as potential biomarkers. Among these, glycine, hippurate and taurine individually displayed medium sensitivity and specificity by receiver operating characteristic (ROC) analysis. Pathway analysis indicated two metabolic pathways likely perturbed in BAV subjects. Possible contributions of gut microbiota activity and energy imbalance are also discussed. These results constitute encouraging preliminary findings in favor of the use of urine-based metabolomics for early diagnosis of BAV.     

Magnetic-field-induced quantum criticality in a spin-<Emphasis Type="Italic">S</Emphasis> planar ferromagnet with single-ion anisotropy

The effects of single-ion anisotropy on quantum criticality in a d-dimensional spin-S planar ferromagnet is explored by means of the two-time Green’s function method. We work at the Tyablikov decoupling level for exchange interactions and the Anderson-Callen decoupling level for single-ion anisotropy. In our analysis a longitudinal external magnetic field is used as the non-thermal control parameter and the phase diagram and the quantum critical properties are established for suitable values of the single-ion anisotropy parameter D. We find that the single-ion anisotropy has sensible effects on the structure of the phase diagram close to the quantum critical point. However, for values of the uniaxial crystal-field parameter below a positive threshold, the conventional magnetic-field-induced quantum critical scenario remains unchanged.     

On the Cauchy Problem for Hyperbolic Operators with Double Characteristics

We show that for a class of hyperbolic second order operators with double characteristics the Ivrii–Petkov condition is also sufficient for the well-posedness of the Cauchy problem in C ^∞.     

Geometric gradient-flow dynamics with singular solutions

The gradient-flow dynamics of an arbitrary geometric quantity is derived using a generalization of Darcy’s Law. We consider flows in both Lagrangian and Eulerian formulations. The Lagrangian formulation includes a dissipative modification of fluid mechanics. Eulerian equations for self-organization of scalars, 1-forms and 2-forms are shown to reduce to nonlocal characteristic equations. We identify singular solutions of these equations corresponding to collapsed (clumped) states and discuss their evolution.     

Vlasov moments, integrable systems and singular solutions

The Vlasov equation governs the evolution of the single-particle probability distribution function (PDF) for a system of particles interacting without dissipation. Its singular solutions correspond to the individual particle motions. The operation of taking the moments of the Vlasov equation is a Poisson map. The resulting Lie-Poisson Hamiltonian dynamics of the Vlasov moments is found to be integrable is several cases. For example, the dynamics for coasting beams in particle accelerators is associated by a hodograph transformation to the known integrable Benney shallow-water equation. After setting the context, the Letter focuses on geodesic Vlasov moment equations. Continuum closures of these equations at two different orders are found to be integrable systems whose singular solutions characterize the geodesic motion of the individual particles.     

Geometry of Vlasov kinetic moments: A bosonic Fock space for the symmetric Schouten bracket

The dynamics of Vlasov kinetic moments is shown to be Lie-Poisson on the dual Lie algebra of symmetric contravariant tensor fields. The corresponding Lie bracket is identified with the symmetric Schouten bracket and the moment Lie algebra is related with a bundle of bosonic Fock spaces, where creation and annihilation operators are used to construct the cold plasma closure. Kinetic moments are also shown to define a momentum map, which is infinitesimally equivariant. This momentum map is the dual of a Lie algebra homomorphism, defined through the Schouten bracket. Finally the moment Lie-Poisson bracket is extended to anisotropic interactions.     

Geometric second derivative estimates in Carnot groups and convexity

We prove some new a priori estimates for H ₂-convex functions which are zero on the boundary of a bounded smooth domain Ω in a Carnot group . Such estimates are global and are geometric in nature as they involve the horizontal mean curvature of ∂Ω. As a consequence of our bounds we show that if has step two, then for any smooth H ₂-convex function in vanishing on ∂Ω one has

Efficient and simple synthesis of 6-aryl-1,4-dimethyl-9H-carbazoles

A synthetic method for the preparation of 6-aryl-1,4-dimethyl-9H-carbazoles involving a palladium catalyzed coupling reaction of 1,4-dimethyl-9H-carbazole-6-boronic acids and (hetero)aryl halides is described.