Helical phase in two-dimensional magnets due to four-spin interactions

We demonstrate that in ferromagnets with the D3h point group of symmetry a possible origin of phase transition from a collinear ferromagnetic state to a non-collinear state can be the fourth order contributions to the free energy density that are allowed by this point group of symmetry. At the same time, Dzyaloshinskii-Moria interaction vanishes in such materials. Via symmetry analysis we derive seven possible fourth order contributions to the free energy density with respect to the unit vector of the local magnetization direction but only two of them can be considered as independent. Moreover, for two-dimensional systems only one survives. Considered symmetry class is essential because a large group of two-dimensional intrinsic ferromagnets belongs to it, for example a monolayer Fe3GeTe2. The four-spin chiral exchange does also manifest itself in peculiar magnon spectra and favors spin waves.

A generalized meshing analysis method and its application to toroidal surface enveloping conical worm drive

A generalized method for the meshing analysis of conical worm drive is proposed, whose mathematical model is more general and whose application scope is expanded. A universal mathematical model, which can be conveniently applied to left-handed and right-handed conical worm pairs and their tooth flanks on different sides, is established by introducing the helical spin coefficient and tooth side coefficient of the conical worm. The pressure angle at the reference point, which is a key parameter for calculating the curvature parameters and lubrication angle, is determined based on the unit normal vector of the worm helical surface and is no longer determined by the tooth profile angle in the worm shaft section. The above improvement breaks away from the limitation of the classic meshing analysis method based on the reference-point-based meshing theory and thus expands its application scope. The toroidal surface enveloping conical worm drive is taken as an instance to illustrate the proposed method and the numerical example studies are conducted. The approaches to determine the reference point, the normal unit vector, and the curvature parameters at the reference point are all demonstrated in detail. The numerical results all manifest that the method presented in the current work is correct and practicable.

On the spectrum of a multidimensional periodic magnetic Shrödinger operator with a singular electric potential

We prove absolute continuity of the spectrum of a periodic $n$-dimensional Schrödinger operator for $n\geqslant 4$. Certain conditions on the magnetic potential $A$ and the electric potential $V+\sum f_j\delta_{S_j}$ are supposed to be fulfilled. In particular, we can assume that the following conditions are satisfied. (1) The magnetic potential $A\colon{\mathbb{R}}^n\to{\mathbb{R}}^n$ either has an absolutely convergent Fourier series or belongs to the space $H^q_{\mathrm{loc}}({\mathbb{R}}^n;{\mathbb{R}}^n)$, $2q>n-1$, or to the space $C({\mathbb{R}}^n;{\mathbb{R}}^n)\cap H^q_{\mathrm{loc}}({\mathbb{R}}^n;{\mathbb{R}}^n)$, $2q>n-2$. (2) The function $V\colon{\mathbb{R}}^n\to\mathbb{R}$ belongs to Morrey space ${\mathfrak{L}}^{2,p}$, $p\in \big(\frac{n-1}{2},\frac{n}{2}\big]$, of periodic functions (with a given period lattice), and $$\lim\limits_{\tau\to+0}\sup\limits_{0<r\leqslant\tau}\sup\limits_{x\in{\mathbb{R}}^n}r^2\bigg(\big(v(B^n_r)\big)^{-1}\int_{B^n_r(x)}|{\mathcal{V}}(y)|^pdy\bigg)^{1/p}\leqslant C,$$ where $B^n_r(x)$ is a closed ball of radius $r>0$ centered at a point $x\in{\mathbb{R}}^n$, $B^n_r=B^n_r(0)$, $v(B^n_r)$ is volume of the ball $B^n_r$, $C=C(n,p;A)>0$. (3) $\delta_{S_j}$ are $\delta$-functions concentrated on (piecewise) $C^1$-smooth periodic hypersurfaces $S_j$, $f_j\in L^p_{\mathrm{loc}}(S_j)$, $j=1,\ldots,m$. Some additional geometric conditions are imposed on the hypersurfaces $S_j$, and these conditions determine the choice of numbers $p\geqslant n-1$. In particular, let hypersurfaces $S_j$ be $C^2$-smooth, the unit vector $e$ be arbitrarily taken from some dense set of the unit sphere $S^{n-1}$ dependent on the magnetic potential $A$, and the normal curvature of the hypersurfaces $S_j$ in the direction of the unit vector $e$ be nonzero at all points of tangency of the hypersurfaces $S_j$ and the lines $\{x_0+te\colon t\in\mathbb{R}\}$, $x_0\in{\mathbb{R}}^n$. Then we can choose the number $p>\frac{3n}{2}-3$, $n\geqslant 4$.

Identification of students’ difficulties in understanding of vector concepts using test of understanding of vector

This study is aimed to identify students’ difficulties in understanding vector concepts in physics because many students think that vector concept is very difficult to understand. This research used an embedded approach research design with quantitative descriptive methods and the sampling used a random sampling technique. Total sample of 142 students from two different schools in Central Lombok district. Test of understanding of vector (TUV) used to test the understanding of students consist of 20 item questions, then followed by interview session with several students. Kruskal-Wallis non-parametric descriptive and inferential statistic was used to performed data analysis. The results of this study indicate that (i) students’ ability to understand vector concepts is still lacking and tends to be very lacking; (ii) the most difficult items for students are the unit vector graphic representation and the graphical representation of vector multiplication. The concept of vector is still considered very difficult for students, especially if the item questions use graphical representations. For further researchers, it is better to conduct a study related to what kind of learning system can support and reduce the difficulties faced by students in learning vector concepts especially on graphical representation.

Cooperative Coevolution with Two-Stage Decomposition for Large-Scale Global Optimization Problems

Cooperative coevolution (CC) is an effective framework for solving large-scale global optimization (LSGO) problems. However, CC with static decomposition method is ineffective for fully nonseparable problems, and CC with dynamic decomposition method to decompose problems is computationally costly. Therefore, a two-stage decomposition (TSD) method is proposed in this paper to decompose LSGO problems using as few computational resources as possible. In the first stage, to decompose problems using low computational resources, a hybrid-pool differential grouping (HPDG) method is proposed, which contains a hybrid-pool-based detection structure (HPDS) and a unit vector-based perturbation (UVP) strategy. In the second stage, to decompose the fully nonseparable problems, a known information-based dynamic decomposition (KIDD) method is proposed. Analytical methods are used to demonstrate that HPDG has lower decomposition complexity compared to state-of-the-art static decomposition methods. Experiments show that CC with TSD is a competitive algorithm for solving LSGO problems.

Voltage Sag, Voltage Swell and Harmonics Reduction Using Unified Power Quality Conditioner (UPQC) Under Nonlinear Loads

In light of the widespread usage of power electronics devices, power quality (PQ) has become an increasingly essential factor. Due to nonlinear characteristics, the power electronic devices produce harmonics and consume lag current from the utility. The UPQC is a device that compensates for harmonics and reactive power while also reducing problems related to voltage and current. In this work, a three-phase, three-wire UPQC is suggested to reduce voltage-sag, voltage-swell, voltage and current harmonics. The UPQC is composed of shunt and series Active Power Filters (APFs) that are controlled utilizing the Unit Vector Template Generation (UVTG) technique. Under nonlinear loads, the suggested UPQC system can be improved PQ at the point of common coupling (PCC) in power distribution networks. The simulation results show that UPQC reduces the effect of supply voltage changes and harmonic currents on the power line under nonlinear loads, where the Total Harmonic Distortion (THD) of load voltages and source currents obtained are less than 5%, according to the IEEE-519 standard.

The Multipole Structure and Symmetry Classification of Even-Type Deviators Decomposed from the Material Tensor

The number of distinct components of a high-order material/physical tensor might be remarkably reduced if it has certain symmetry types due to the crystal structure of materials. An nth-order tensor could be decomposed into a direct sum of deviators where the order is not higher than n, then the symmetry classification of even-type deviators is the basis of the symmetry problem for arbitrary even-order physical tensors. Clearly, an nth-order deviator can be expressed as the traceless symmetric part of tensor product of n unit vectors multiplied by a positive scalar from Maxwell’s multipole representation. The set of these unit vectors shows the multipole structure of the deviator. Based on two steps of exclusion, the symmetry classifications of all even-type deviators are obtained by analyzing the geometric symmetry of the unit vector sets, and the general results are provided. Moreover, corresponding to each symmetry type of the even-type deviators up to sixth-order, the specific multipole structure of the unit vector set is given. This could help to identify the symmetry types of an unknown physical tensor and possible back-calculation of the involved physical coefficients.