Darboux curves in three dimensional Heisenberg group

In this paper, a new characterization for darboux curves in Heis₃ is completely given. Then, a new classification for translation surface , which is generated by darboux curve in Heis₃ is obtained.

Periods of Abelian Differentials With Prescribed Singularities

We give a necessary and sufficient condition for a representation of the fundamental group of a closed surface of genus at least $2$ to ${\mathbb{C}}$ to be the holonomy of a translation surface with a prescribed list of conical singularities. Equivalently, we determine the period maps of abelian differentials with prescribed list of multiplicities of zeros. Our main result was also obtained, independently, by Bainbridge, Johnson, Judge, and Park.

Affine Equivalence and Saddle Connection Graphs of Half-Translation Surfaces

To every half-translation surface, we associate a saddle connection graph, which is a subgraph of the arc graph. We prove that every isomorphism between two saddle connection graphs is induced by an affine homeomorphism between the underlying half-translation surfaces. We also investigate the automorphism group of the saddle connection graph and the corresponding quotient graph.

Estimation of Transition Frequency during Continuous Translation Surface Perturbation

Depending on task requirements, a human is able to select distinct strategies such as the use of an ankle strategy and hip strategy to maintain their balance. Postural control actions often co-occur with other movements, and such movements may bring about a change from one type of postural coordination to another. The selection of a postural control strategy has typically been investigated by the transition of the center of mass (COM), center of pressure (COP), and in between angle joint motion along with their characteristics. In this paper, we proposed a method using the logistic function of the sigmoid model based on cross-correlation coefficient (CCF) data for investigating and observing the transition of postural control strategies of COM–COP and ankle-hip angles towards anterior–posterior (AP) continuous translation perturbation. Subjects were required to stand on the motion base platform where perturbations with an increasing frequency (0.2 Hz to 0.8 Hz) and decreasing frequency (0.8 Hz to 0.2 Hz) in steps of 0.02 Hz, were induced. As the frequency increased, the COM and COP displacements were decreased, with the opposite trend observable with decreasing frequency. This pattern was also observed at the head peak-to-peak amplitude. Meanwhile, ankle and hip angular displacements were increased during increasing frequency and decreased during decreasing frequency. In this paper, the proposed sigmoid model could identify the transition frequency of COM–COP and ankle–hip transition. The mean transition frequency of COM–COP during increasing frequency was 0.44 Hz, and the ankle–hip transition frequency was 0.42 Hz. Meanwhile, for decreasing frequency, the COM–COP transition frequency was 0.55 Hz, and for the ankle–hip transition the frequency was 0.56 Hz. With frequencies, both increasing and decreasing, the COM–COP and ankle–hip transition frequencies occurred almost at the same frequency. Furthermore, the transition occurred at a lower time scale during increasing frequency compared to decreasing frequency. In conclusion, the continuous translation surface perturbation provided information on the behavior of postural control strategies. A sudden change in ‘phase angle’ was observed, where either an ankle or hip strategy was implemented to maintain balance. Besides, the transition frequency of postural control strategies could be determined to occur between 0.4 Hz and 0.6 Hz, based on the average value, for healthy young subjects in the AP plane. Furthermore, the proposed sigmoid model was believed to be able to be used in the determination of transition frequency in postural control strategies.

Classification of j-maximal spacelike affine translation surfaces in the Minkovski space i31 with density

An affine translation surface is a graph of a function introduced by Liu and Yu in 2013. The article considers the spacelike affine translation surfaces in the Minkowski space with density establishing the Lagrange’s equation type for -maximal surface, classifying -maximal spacelike affine translation surfaces. The result obtains two parameters and . From that, the Calabi – Bernstein theorem in this space is not true because two function and are defined on

Semi-parallel and harmonic surfaces in semi-Euclidean 4-space with index 2

The present paper mainly deals with construction and investigation further properties of semi-parallel and harmonic surfaces. The first part of the study shall be devoted to investigate and present necessary and sufficient conditions of being semi-parallel surfaces by considering semi-parallelity condition R(X,Y).h = 0. In the light of the condition, the fact of a part of semi-parallel surfaces can be created by translation surfaces is captured. As a last result, we present that M must be a translation surface in the case of it is a harmonic surface.

A Non-varying Phenomenon with an Application to the Wind-Tree Model

We exhibit a non-varying phenomenon for the counting problem of cylinders, weighted by their area, passing through two marked (regular) Weierstrass points of a translation surface in a hyperelliptic connected component $\mathcal{H}^{hyp}(2g-2)$ or $\mathcal{H}^{hyp}(g-1,g-1)$, $g> 1$. As an application, we obtain the non-varying phenomenon for the counting problem of (weighted) periodic trajectories on the Ehrenfest wind-tree model, a billiard in the plane endowed with $\mathbb{Z}^2$-periodically located identical rectangular obstacles.

Dual translation surfaces in the three dimensional simply isotropic space $\mathbb{I}_{3}^{1}$

In this paper, we study the dual translation surfaces in three dimensional simply isotropic space. We give classification of dual translation surface with constant dual isotropic mean curvature or constant dual isotropic Guassian curvature.