Minimal translation surfaces in the Heisenberg group Nil3

A translation surface in the Heisenberg group $\mathrm{Nil}_3$ is a surface constructed by multiplying (using the group operation) two curves. We completely classify minimal translation surfaces in the Heisenberg group $\mathrm{Nil}_3$.Comment: 10 page

Discrete Integrable Systems and Hodograph Transformations Arising from Motions of Discrete Plane Curves

We consider integrable discretizations of some soliton equations associated with the motions of plane curves: the Wadati-Konno-Ichikawa elastic beam equation, the complex Dym equation, and the short pulse equation. They are related to the modified KdV or the sine-Gordon equations by the hodograph transformations. Based on the observation that the hodograph transformations are regarded as the Euler-Lagrange transformations of the curve motions, we construct the discrete analogues of the hodograph transformations, which yield integrable discretizations of those soliton equations.Comment: 19 page

Hamiltonian flows on null curves

The local motion of a null curve in Minkowski 3-space induces an evolution equation for its Lorentz invariant curvature. Special motions are constructed whose induced evolution equations are the members of the KdV hierarchy. The null curves which move under the KdV flow without changing shape are proven to be the trajectories of a certain particle model on null curves described by a Lagrangian linear in the curvature. In addition, it is shown that the curvature of a null curve which evolves by similarities can be computed in terms of the solutions of the second Painlev\'e equation.Comment: 14 pages, v2: final version; minor changes in the expositio

Explicit solutions to the semi-discrete modified KdV equation and motion of discrete plane curves

We construct explicit solutions to continuous motion of discrete plane curves described by a semi-discrete potential modified KdV equation. Explicit formulas in terms the $\tau$ function are presented. B\"acklund transformations of the discrete curves are also discussed. We finally consider the continuous limit of discrete motion of discrete plane curves described by the discrete potential modified KdV equation to motion of smooth plane curves characterized by the potential modified KdV equation

Closed trajectories of a particle model on null curves in anti-de Sitter 3-space

We study the existence of closed trajectories of a particle model on null curves in anti-de Sitter 3-space defined by a functional which is linear in the curvature of the particle path. Explicit expressions for the trajectories are found and the existence of infinitely many closed trajectories is proved.Comment: 12 pages, 1 figur

Timelike surfaces of constant mean curvature 1 in anti-de Sitter 3-space

It is shown that timelike surfaces of constant mean curvature 1 in anti-de Sitter 3-space can be constructed from a pair of Lorentz holomorphic and Lorentz antiholomorphic null curves in PSL(2,R) via Bryant type representation formulae. These formulae are used to investigate an explicit one-to-one correspondence, the so-called Lawson correspondence, between timelike surfaces of constant mean curvature 1 in anti-de Sitter 3-space and timelike minimal surfaces in Minkowski 3-space. The hyperbolic Gauss map of timelike surfaces in anti-de Sitter 3-space, which is a close analogue of the classical Gauss map is considered. It is discussed that the hyperbolic Gauss map plays an important role in the study of timelike surfaces of constant mean curvature 1 in anti-de Sitter 3-space. In particular, the relationship between the Lorentz holomorphicity of the hyperbolic Gauss map and timelike surfaces of constant mean curvature 1 in anti-de Sitter 3-space is studied.Comment: 47 pages, 24 figures, references revised, Annals of Global Analysis and Geometr

Improvements in vascular health by a low-fat diet, but not a high-fat diet, are mediated by changes in adipocyte biology

<p>Abstract</p> <p>Background</p> <p>Low-fat (LF) and high-fat (HF) weight loss diets improve brachial artery flow-mediated dilation (FMD) in obese individuals, although results are conflicting. Moreover, the role that adipose tissue plays in mediating these diet-related effects are unknown.</p> <p>Objective</p> <p>This study examined how modulations in FMD by HF and LF diets relate to changes in adipocyte parameters.</p> <p>Design</p> <p>Obese subjects (n = 17) were randomized to a HF diet (60% kcal as fat) or a LF diet (25% kcal as fat) for 6 weeks. Both groups were restricted by 25% of energy needs.</p> <p>Results</p> <p>Body weight decreased (<it>P <</it>0.05) in both groups (HF: -6.6 ± 0.5 kg, LF: -4.7 ± 0.6 kg). Fat mass and waist circumference were reduced (<it>P <</it>0.05) in the LF group only (-4.4 ± 0.3 kg; -3.6 ± 0.8 cm, respectively). FMD improved (<it>P <</it>0.05) in the LF group (7.4 ± 0.8% to 9.8 ± 0.8; 32% increase) and was impaired in the HF group (8.5 ± 0.6% to 6.9 ± 0.7; 19% reduction). Increases in plasma adiponectin (<it>P <</it>0.05, 16 ± 5%), and decreases in resistin (<it>P <</it>0.05, -26 ± 11%), were shown by the LF diet only. Greater decreases in leptin were observed with LF (-48 ± 9%) versus HF (-28 ± 12%) (<it>P <</it>0.05, diet × time). Increased FMD by the LF diet was associated with increased adiponectin, and decreased fat mass, waist circumference, leptin, and resistin.</p> <p>Conclusion</p> <p>Beneficial modulations in vascular health by LF diets may be mediated by improvements in adipocyte parameters.</p