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.

The possibility to apply the Frenet trihedron and formulas for the complex movement of a point on a plane with the predefined plane displacement

Material particles interact with the working moving surfaces of machines in various technological processes. Mechanics considers a technique to describe the movement of a point and decompose the speed and acceleration into single unit vectors of the accompanying trajectory trihedron for simple movement. The shape of the spatial curve uniquely sets the movement of the accompanying Frenet trihedral as a solid body. This paper has considered the relative movement of a material particle in the static plane of the accompanying Frenet trihedron, which moves along a flat curve with variable curvature. Frenet formulas were used to build a system of differential equations of relative particle movement. In contrast to the conventional approach, the chosen independent variable was not the time but the length of the arc of the guide curve along which the trihedron moves. The system of equations has been built in the projections onto the unit vectors of the moving trihedron; it has been solved by numerical methods. The use of the accompanying curve trihedron as a moving coordinate system makes it possible to solve the problems of the complex movement of a point. The shape of the curve guide assigned by parametric equations in its length function determines the portable movement of the trihedron and makes it possible to use Frenet formulas to describe the relative movement of a point in the trihedron system. This approach enables setting the portable movement of the trihedron osculating plane along a curve with variable curvature, thereby revealing additional possibilities for solving problems on a complex movement of a point at which rotational motion around a fixed axis is a partial case. The proposed approach has been considered using an example of the relative movement of cargo in the body of a truck moving along the road with a curvilinear axis of variable curvature. The charts of the relative trajectory of cargo slip and the relative speed for the predefined speed of the truck have been constructed

Classification of time series of temperature variations from climatically homogeneous regions using Hurst Space Analysis

<p>We used the Hurst Space Analysis (HSA), a technique that we recently developed to cluster or differentiate records from an arbitrary complex system based on the presence and influence of cycles in their statistical functions, to classify climatic data from climatically homogeneous regions according to their long-term persistent (LTP) character. For our analysis we selected four types of HadCRUT4 cells of temperature records over regions homogeneous in both climate and topography, which are sufficiently populated with ground observational stations. These cells bound: Pannonian and West Siberian plains, Rocky Mountains and Himalayas mountainous regions, Arctic and sub-Arctic climates of Island and Alaska, and Gobi and Sahara deserts.</p><p>It was shown for LTP records across different complex systems that their statistical functions are rarely, as in theory, and due to their power-law dynamics, ideal linear functions on log-log graphs of time scale dependence. Instead, they frequently exhibit existence of transient crossovers in behavior, signs of trends that arise as effects of periodic or aperiodic cycles. HSA was developed so to use methods of scaling analysis – the time dependent Detrended Moving Average (tdDMA) algorithm and Wavelet Transform spectral analysis (WTS) – to analyse these cycles in data. In HSA we defined a space of <em>p</em>-vectors <em>h<sup>ts</sup></em> (that we dubbed the Hurst space) that represent record <em>ts</em> in any dataset, which are populated by tdDMA scaling exponents α calculated on subsets of time scale windows of time series <em>ts</em> that bound cyclic peaks in their WTS. In order to be able to quantify any such time series <em>ts</em> with a single number, we projected their relative unit vectors <em>s<sup>ts</sup> = (h<sup>ts</sup> – m) / (∑<sub>i=1</sub><sup>n</sup> (h<sub>i</sub><sup>ts</sup> - m<sub>i</sub>)<sup>2</sup>)<sup>1/2</sup></em>  (with <em>m<sub>i</sub> = 1/n ∑<sub>ts=1</sub><sup>n</sup> h<sub>i</sub><sup>ts</sup></em>) onto a unit vector <em>e</em> of an assigned preferred direction in the Hurst space. The definition of the ’preferred’ direction depends on the characteristic behavior one wants to investigate with HSA - projection of unit vectors <em>s<sup>ts</sup></em> of any record  with a ’preferred’ behavior onto the unit vector <em>e</em> is then always positive.</p><p>By using HSA we were able to cluster records from our selected climatically homogeneous regions according to the 'preferred' characteristic that those do not 'belong to the ocean'. We further extended HSA constructed from our dataset to group teleconnection indices that may influence their climate dynamics. In this way our results suggested that there probably exists a necessity to examine cycles in climate records as important elements of natural variability.</p>

Prime gradient noise

Procedural noise functions are fundamental tools in computer graphics used for synthesizing virtual geometry and texture patterns. Ideally, a procedural noise function should be compact, aperiodic, parameterized, and randomly accessible. Traditional lattice noise functions such as Perlin noise, however, exhibit periodicity due to the axial correlation induced while hashing the lattice vertices to the gradients. In this paper, we introduce a parameterized lattice noise called prime gradient noise (PGN) that minimizes discernible periodicity in the noise while enhancing the algorithmic efficiency. PGN utilizes prime gradients, a set of random unit vectors constructed from subsets of prime numbers plotted in polar coordinate system. To map axial indices of lattice vertices to prime gradients, PGN employs Szudzik pairing, a bijection F: ℕ2 → ℕ. Compositions of Szudzik pairing functions are used in higher dimensions. At the core of PGN is the ability to parameterize noise generation though prime sequence offsetting which facilitates the creation of fractal noise with varying levels of heterogeneity ranging from homogeneous to hybrid multifractals. A comparative spectral analysis of the proposed noise with other noises including lattice noises show that PGN significantly reduces axial correlation and hence, periodicity in the noise texture. We demonstrate the utility of the proposed noise function with several examples in procedural modeling, parameterized pattern synthesis, and solid texturing.

Raman Spectroscopy of Twisted Bilayer Graphene

When two periodic two-dimensional structures are superposed, any mismatch rotation angle between the layers generates a Moiré pattern superlattice, whose size depends on the twisting angle θ. If the layers are composed by different materials, this effect is also dependent on the lattice parameters of each layer. Moiré superlattices are commonly observed in bilayer graphene, where the mismatch angle between layers can be produced by growing twisted bilayer graphene (TBG) samples by CVD or folding the monolayer back upon itself. In TBG, it was shown that the coupling between the Dirac cones of the two layers gives rise to van Hove singularities (vHs) in the density of electronic states, whose energies vary with θ. The understanding of the behavior of electrons and their interactions with phonons in atomically thin heterostructures is crucial for the engineering of novel 2D devices. Raman spectroscopy has been often used to characterize twisted bilayer graphene and graphene heterostructures. Here, we review the main important effects in the Raman spectra of TBG discussing firstly the appearance of new peaks in the spectra associated with phonons with wavevectors within the interior of the Brillouin zone of graphene corresponding to the reciprocal unit vectors of the Moiré superlattice, and that are folded to the center of the reduced Brillouin Zone (BZ) becoming Raman active. Another important effect is the giant enhancement of G band intensity of TBG that occurs only in a narrow range of laser excitation energies and for a given twisting angle. Results show that the vHs in the density of states is not only related to the folding of the commensurate BZ, but mainly associated with the Moiré pattern that does not necessarily have a translational symmetry. Finally, we show that there are two different resonance mechanisms that activate the appearance of the extra peaks: the intralayer and interlayer electron–phonon processes, involving electrons of the same layer or from different layers, respectively. Both effects are observed for twisted bilayer graphene, but Raman spectroscopy can also be used to probe the intralayer process in any kind of graphene-based heterostructure, like in the graphene/h-BN junctions.

Control scheme based on capacitor voltage second-order complex-vector feedforward for LCL-filtered inverter

For an inverter with LCL filter, voltage at the point of common coupling (PCC) is equivalent to the filter capacitor voltage in practical applications. However, the PCC voltage background harmonics will cause the distortion of grid current. Some problems exist in the traditional methods for harmonics suppression, such as insufficient phase margin, increasing design complexity and computation burden. Thus, a scheme of second-order complex-vector feedforward (SOCVF) of capacitor voltage is proposed in this article. First, based on the vectorial principle, the active unit vectors of capacitor voltage were calculated. Then, its fundamental component was obtained by the SOCVF-function. Furthermore, the references of the grid current were obtained with the instantaneous power theory. Hence, the proposed scheme helps effectively dampen the LCL resonance to make the system stable, and works well at suppressing the injected grid current distortion simultaneously. Simulation and experimental results validated the effectiveness of the proposed scheme.

USuRPER: Unit-sphere representation periodogram for full spectra

We introduce an extension of the periodogram concept to time-resolved spectroscopy. USuRPER, the unit-sphere representation periodogram, is a novel technique that opens new horizons in the analysis of astronomical spectra. It can be used to detect a wide range of periodic variability of the spectrum shape. Essentially, the technique is based on representing spectra as unit vectors in a multidimensional hyperspace, hence its name. It is an extension of the phase-distance correlation periodogram we had introduced in previous papers, to very high-dimensional data such as spectra. USuRPER takes the overall shape of the spectrum into account, which means that it does not need to be reduced into a single quantity such as radial velocity or temperature. Through simulations, we demonstrate its performance in various types of spectroscopic variability: single-lined and double-lined spectroscopic binary stars, and pulsating stars. We also show its performance on actual data of a rapidly oscillating Ap star. USuRPER is a new tool to explore large time-resolved spectroscopic databases such as APOGEE, LAMOST, and the RVS spectra of Gaia. We have made a public GitHub repository with a Python implementation of USuRPER available to the community, to experiment with it and apply it to a wide range of spectroscopic time series.