Harmonic Degree Research Articles

UNVEILING THE RELATIONSHIP BETWEEN M-POLYNOMIAL BASED TOPOLOGICAL INDICES AND INVERSE GRAPHS OF FINITE CYCLIC GROUPS: A COMPREHENSIVE STUDY

In the discipline of graph theory, topological indices are extremely important. The M-polynomial is a powerful tool for determining a graph's topological indices. The use of M-polynomials to describe macro-molecules and biochemical networking is a novel concept. Also, the M-polynomial of various micro-structural allows us to calculate a variety of topological indices. The chemical substances and biochemical networks are correlated with their chemical characteristics and bio-active compounds using these findings. In this research, we use the M-polynomial to create special essential topological indices of inverse graphs on finite cyclic groups, such as Randic, Zagreb, Augmented Zagreb, Harmonic, Inverse sum, and Symmetric division degree indices.

Realizing the Calculation of a Fully Normalized Associated Legendre Function Based on an FPGA.

A large number of fully normalized associated Legendre function (fnALF) calculations are required to compute Earth's gravity field elements using ultra high-order gravity field coefficient models. In the surveying and mapping industry, researchers typically rely on CPU-based systems for these calculations, which leads to limitations in execution speed and power efficiency. Although modern CPUs improve instruction execution efficiency through instruction-level parallelism, the constraints of a shared memory architecture impose further limitations on the execution speed and power efficiency. This results in exponential increases in computation time as demand rises alongside high power consumption. In this article, we present a new computational implementation of an fnALF based on the ZYNQ platform. We design a task-parallel "pipeline" architecture which converts the original serial logic into a more efficient hardware implementation, and we utilize a redundant calculation layer to handle repetitive coefficient computations separately. The experimental results demonstrate that our system achieved accurate and rapid calculations. Under the only one geocentric residual latitude condition, we measured the computation times for spherical harmonic coefficient degrees of 360, 720, and 1080 to be 0.155922 s, 0.520950 s, and 1.401609 s, respectively. In the case of the multiple geocentric residual latitudes condition, our design generally yielded efficiency gains of over three times those of MATLAB R2020b implementation. Additionally, our calculated results were used to determine the geoid height in the field with an error of less than ±0.1m, confirming the reliability of our computations.

Linking deep-time subduction history to modern day expressions of dynamic topography

Dynamic topography refers to vertical deflections of Earth’s surface from viscous flow within the mantle. Here we investigate how past subduction history affects present dynamic topography. We assimilate two plate reconstructions into TERRA forward mantle convection models to calculate past mantle states and predict Earth’s present dynamic topography; a comparison is made with a database of observed oceanic residual topography. The two assimilated plate reconstructions ‘Earthbyte’ and ‘Tomopac’ show divergent subduction histories across an extensive deep-time interval within Pacific-Panthalassa. We find that introducing an alternative subduction history perturbs our modelled present-day dynamic topography on the same order as the choice of radial viscosity. Additional circum-Pacific intra-oceanic subduction in Tomopac consistently produces higher correlations to the geoid (more than 20% improvement). At spherical harmonic degrees 1–40, dynamic topography models with intra-oceanic subduction produce universally higher correlations with observations and improve fit by up to 37%. In northeast Asia, Tomopac models show higher correlations (0.46 versus 0.18) to observed residual topography and more accurately predict approximately 1 km of dynamic subsidence within the Philippine Sea plate. We demonstrate that regional deep-time changes in subduction history have widespread impacts on the spatial distribution and magnitude of present-day dynamic topography. Specifically, we find that local changes to plate motion histories can induce dynamic topography changes in faraway regions located thousands of kilometres away. Our results affirm that present-day residual topography observations provide a powerful, additional constraint for reconstructing ancient subduction histories.

Variations of Heat Flux and Elastic Thickness of Mercury From 3‐D Thermal Evolution Modeling

AbstractMercury's low obliquity and 3:2 spin‐orbit resonance create a surface temperature distribution with large latitudinal and longitudinal variations. These propagate via thermal conduction through the thin silicate shell influencing the interior temperature distribution. We use 3‐D thermal evolution models to investigate the effects of lateral variations of surface temperature and crustal thickness on the surface and core‐mantle boundary (CMB) heat fluxes, and elastic thickness distribution. Surface temperature variations cause a long‐wavelength perturbation of the heat fluxes, as well as of the elastic lithosphere thickness, while variations in crustal thickness affect these quantities at small spatial scales. Like the surface temperature, the present‐day CMB heat flux pattern is characterized primarily by spherical harmonics of degrees two and four, which could affect dynamo generation. Placing robust constraints on the thermal evolution based on the available estimates of the age and thickness of the elastic lithosphere remains challenging due to their large uncertainties.

FSH3D: 3D Representation via Fibonacci Spherical Harmonics

AbstractSpherical harmonics are a favorable technique for 3D representation, employing a frequency‐based approach through the spherical harmonic transform (SHT). Typically, SHT is performed using equiangular sampling grids. However, these grids are non‐uniform on spherical surfaces and exhibit local anisotropy, a common limitation in existing spherical harmonic decomposition methods. This paper proposes a 3D representation method using Fibonacci Spherical Harmonics (FSH3D). We introduce a spherical Fibonacci grid (SFG), which is more uniform than equiangular grids for SHT in the frequency domain. Our method employs analytical weights for SHT on SFG, effectively assigning sampling errors to spherical harmonic degrees higher than the recovered band‐limited function. This provides a novel solution for spherical harmonic transformation on non‐equiangular grids. The key advantages of our FSH3D method include: 1) With the same number of sampling points, SFG captures more features without bias compared to equiangular grids; 2) The root mean square error of 32‐degree spherical harmonic coefficients is reduced by approximately 34.6% for SFG compared to equiangular grids; and 3) FSH3D offers more stable frequency domain representations, especially for rotating functions. FSH3D enhances the stability of frequency domain representations under rotational transformations. Its application in 3D shape reconstruction and 3D shape classification results in more accurate and robust representations. Our code is publicly available at https://github.com/Miraclelzk/Fibonacci-Spherical-Harmonics.

Density Waves of Vortex Fluids on a Sphere

Abstract We aim to find one highly nontrivial example of the solutions to the vortex fluid dynamical equation on the unit sphere (S^2) and compare it with the numerical simulation. Since the rigid rotating steady solution for vortex fluids on S^2 is
already known to us, we consider the perturbations above it. After decomposing the perturbation of the vortex number density and vortex charge density into spherical harmonics, we find that the perturbations are propagating waves. To
be precise, the velocities for different single-mode vortex number density waves are all the same, while the velocities for single-mode vortex charge density waves depend on the degree of the spherical harmonics l, which is a signal of the existence of dispersion. Meanwhile, we find that there is a beat phenomenon for the positive (or negative) vortex density wave. Numerical simulation based on the canonical equations for point vortex model agrees perfectly with our theoretical calculations.

The HVDC commutation failure mechanism impacted by harmonics and its suppression method

Harmonics are an important cause of voltage distortion, and voltage reduction and voltage distortion are the main causes of commutation failure. But most of the current research studies on commutation failure are based on the drop in fundamental voltage after a fault, and there are few reports on the impact of voltage distortion on commutation failure. Aiming to solve the above problem, this article first analyzes the generation and transmission characteristics of harmonics in AC and DC systems, as well as the coupling degree of harmonics in DC systems. Then, starting from the mechanism of the commutation process, a commutation failure analysis method based on the harmonic voltage time area is proposed. This method can quantitatively analyze the impact of different harmonics on commutation failure. On this basis, the harmonic criteria for causing commutation failure are established, and the harmonic characteristics of AC voltage when harmonics cause subsequent commutation failure are quantitatively analyzed. Finally, a preventive controller is proposed to suppress commutation failure caused by different harmonics, where the above analysis is verified by the simulations.

A RANDOMIZED CARRIER-BASED DISCONTINUOUS PULSE WIDTH MODULATION STRATEGY FOR NEUTRAL POINT CLAMPED THREE-LEVEL INVERTER

This article introduces a novel approach called randomized carrier-based discontinuous pulse width modulation (RCDPWM) to address the challenge of mitigating both low-order and high-order harmonics concentrated at the switching frequency and its integer multiples in the output voltage and current of a three-level inverter. The proposed strategy reduces switching losses and a wide dispersion of voltage and current harmonics with significantly smaller amplitudes by integrating features from discontinuous and randomized PWM methods. This characteristic proves beneficial for enhancing efficiency and electromagnetic compatibility. The RCDPWM strategy involves three schemes: two simple schemes (one random parameter), randomized pulse position modulation (RPP-DPWM) and randomized carrier frequency modulation (RCF-DPWM), and a dual scheme combining the two previous ones (RPPRCF-DPWM). Power spectral density (PSD) is used to assess the effectiveness of the proposed strategy in quantitatively analyzing the harmonic dispersion degree. The results demonstrate that the proposed RCDPWM strategy significantly expands harmonic clusters around the switching frequency, reducing its intensity. The spectrum analysis revealed that the RPPRCF-DPWM scheme is the most effective in spreading the output voltage and current spectrum compared to simple schemes (RPP-DPWM and RCF-DPWM).

C-band 15w High PAE Second Harmonic Suppression Power Amplifier MMIC For Radar Networks

Abstract This paper introduces the design process of a high-efficiency power amplifier, which adopts the 0.25µm GaN HEMT technology and a two-stage amplifier structure. In order to control harmonics for the second time, the SHS topology was added to the final matching network. After obtaining the physical object, the chip was tested using a 10% duty cycle pulse, and the test results showed that the second harmonic suppression degree of MMIC in the 5-6GHz band exceeded -45dBc, the pout was higher than 15W, the gain was greater than 22dB, and the PAE ranged from 52% to 60%. In addition, the PA MMIC had a size of 3 × 1.3 mm2.

Scaling of the geomagnetic secular variation timescale

SUMMARY The ratio of the magnetic power spectrum and the secular variation spectrum measured at the Earth’s surface provides a timescale $\tau _{\rm sv}(l)$ as a function of spherical harmonic degree l. $\tau _{\rm sv}$ is often assumed to be representative of timescales related to the dynamo inside the outer core and its scaling with l is debated. To assess the validity of this surmise and to study the time variation of the geomagnetic field $\dot{\boldsymbol {B}}$ inside the outer core, we introduce a magnetic timescale spectrum $\tau (l,r)$ that is valid for all radius r above the inner core and reduces to the usual $\tau _{\rm sv}$ at and above the core–mantle boundary (CMB). We study $\tau$ in a numerical geodynamo model. At the CMB, we find that $\tau \sim l^{-1}$ is valid at both the large and small scales, in agreement with previous numerical studies on $\tau _{\rm sv}$. Just below the CMB, the scaling undergoes a sharp transition at small l. Consequently, in the interior of the outer core, $\tau$ exhibits different scaling at the large and small scales, specifically, the scaling of $\tau$ becomes shallower than $l^{-1}$ at small l. We find that this transition at the large scales stems from the fact that the horizontal components of the magnetic field evolve faster than the radial component in the interior. In contrast, the magnetic field at the CMB must match onto a potential field, hence the dynamics of the radial and horizontal magnetic fields are tied together. The upshot is $\tau _{\rm sv}$ becomes unreliable in estimating timescales inside the outer core. Another question concerning $\tau$ is whether an argument based on the frozen-flux hypothesis can be used to explain its scaling. To investigate this, we analyse the induction equation in the spectral space. We find that away from both boundaries, the magnetic diffusion term is negligible in the power spectrum of $\dot{\boldsymbol {B}}$. However, $\dot{\boldsymbol {B}}$ is controlled by the radial derivative in the induction term, thus invalidating the frozen-flux argument. Near the CMB, magnetic diffusion starts to affect $\dot{\boldsymbol {B}}$ rendering the frozen-flux hypothesis inapplicable. We also examine the effects of different velocity boundary conditions and find that the above results apply for both no-slip and stress-free conditions at the CMB.

Onset of thermal convection in a solid spherical shell with melting at either or both boundaries

SUMMARY Thermal convection in planetary solid (rocky or icy) mantles sometimes occurs adjacent to liquid layers with a phase equilibrium at the boundary. The possibility of a solid–liquid phase change at the boundary has been shown to greatly help convection in the solid layer in spheres and plane layers and a similar study is performed here for a spherical shell with a radius-independent central gravity subject to a destabilizing temperature difference. The solid–liquid phase change is considered as a mechanical boundary condition and applies at either or both horizontal boundaries. The boundary condition is controlled by a phase change number, Φ, that compares the timescale for latent heat exchange in the liquid side to that necessary to build a topography at the boundary. We introduce a numerical tool, available at https://github.com/amorison/stablinrb, to carry out the linear stability analysis of the studied setup as well as other similar situations (Cartesian geometry, arbitrary temperature and viscosity depth-dependent profiles). Decreasing Φ makes the phase change more efficient, which reduces the importance of viscous resistance associated to the boundary and makes the critical Rayleigh number for the onset of convection smaller and the wavelength of the critical mode larger, for all values of the radii ratio, γ. In particular, for a phase change boundary condition at the top or at both boundaries, the mode with a spherical harmonics degree of 1 is always favoured for Φ ≲ 10−1. Such a mode is also favoured for a phase change at the bottom boundary for small (γ ≲ 0.45) or large (γ ≳ 0.75) radii ratio. Such dynamics could help explaining the hemispherical dichotomy observed in the structure of many planetary objects.

Degree based hybrid topological indices and entropies of hydrogen bonded benzo-trisimidazole frameworks

Hydrogen bonded organic frameworks, especially those comprising of benzo-trisimidazole are porous molecular materials facilitated by hydrogen-bonds. As these materials find several applications in catalysis, hydrogen storage, and environmental remediation through sequestration, we have investigated the topological and entropy properties of these novel covalent organic frameworks. We have obtained the analytical expressions for vertex degree based topological indices in order to characterize their topological complexities. Our computations reveal that there are isentropic hydrogen bonded networks. Furthermore we demonstrate the utility of variations of hybrid arithmetic, geometric, harmonic and Zagreb degree based topological and entropy indices of these networks.

Differentiable and accelerated spherical harmonic and Wigner transforms

Many areas of science and engineering encounter data defined on spherical manifolds. Modelling and analysis of spherical data often necessitates spherical harmonic transforms, at high degrees, and increasingly requires efficient computation of gradients for machine learning or other differentiable programming tasks. We develop novel algorithmic structures for accelerated and differentiable computation of generalised Fourier transforms on the sphere S2 and rotation group SO(3), i.e. spherical harmonic and Wigner transforms, respectively. We present a recursive algorithm for the calculation of Wigner d-functions that is both stable to high harmonic degrees and extremely parallelisable. By tightly coupling this with separable spherical transforms, we obtain algorithms that exhibit an extremely parallelisable structure that is well-suited for the high throughput computing of modern hardware accelerators (e.g. GPUs). We also develop a hybrid automatic and manual differentiation approach so that gradients can be computed efficiently. Our algorithms are implemented within the JAX differentiable programming framework in the S2FFT▪ software code. Numerous samplings of the sphere are supported, including equiangular and HEALPix sampling. Computational errors are at the order of machine precision for spherical samplings that admit a sampling theorem. When benchmarked against alternative C codes we observe up to a 400-fold acceleration. Furthermore, when distributing over multiple GPUs we achieve very close to optimal linear scaling with increasing number of GPUs due to the highly parallelised and balanced nature of our algorithms. Provided access to sufficiently many GPUs our transforms thus exhibit an unprecedented effective linear time complexity.

Spherical‐Harmonic Distribution Analysis of Coronae in Relation to Volcanic Features on Venus

AbstractVenus boasts an abundance of volcanoes and volcano‐like structures. Synthetic aperture radar images of the surface have revealed extensive evidence of volcanism, including lava flows and edifices. Volcanic activity is further supported by crater statistics, and analysis of topography and gravity data. Unique to Venus, coronae are quasi‐circular volcano‐tectonic features exhibiting diverse volcanic characteristics. Despite this, volcanism is often under‐represented in formation models. We identify a new subset of coronae that display topographic changes subsequent to the emplacement of lava flows within their fracture annuli, pointing to the critical role of volcanic and magmatic processes in the formation of these coronae. Through spherical‐harmonic distribution analysis, we find that this new subset is spatially related to the full coronae database, pointing to an intrinsic process of coronae formation. Furthermore, coronae exhibit strong correlations and similar spectral shapes at low spherical harmonic degrees with large volcanoes, suggesting a shared geodynamic origin. Our findings underscore the pivotal role of volcanism in coronae formation and highlight the need for future research to integrate magmatic and volcanic processes more comprehensively into geophysical models. Such models would better capture the complex interactions between volcanic emplacement, magmatic activity, and lithospheric dynamics on Venus.

Higher-order composition of short- and long-period effects for satellite analytical ephemeris computation

The construction of an analytic orbit theory in closed form of the eccentricity that takes into account the main effects of the Geopotential is notably simplified when splitting the removal of periodic effects in several stages. Conversely, this splitting of the closed-form analytical solution into several transformations reduces the evaluation efficiency for dense ephemeris output. However, the advantage is twofold when the different parts of the mean-to-osculating transformation are composed into a single transformation. To show that, Brouwer’s solution is extended to the second order of the zonal harmonic of the second degree by the sequential elimination of short and long period terms. Then, the generating functions of the different transformations are composed into a single one, from which a single mean-to-osculating transformation is derived. The new, unique transformation notably speeds up the evaluation process, commonly improving evaluation efficiency by at least one third with respect to the customary decomposition of the analytical solution into three different parts.

On Difference Pattern Synthesis for Spherical Sensor Arrays.

An innovative method for synthesizing optimum difference patterns of the spherical sensor array is introduced, along with a sidelobe tapering technique. Firstly, we suggest employing the spherical harmonics of degree ±1 to synthesize the spherical array difference pattern; secondly, we study the mapping relationship between the difference pattern of the spherical sensor array and the difference pattern of the uniformly spaced linear array (ULA) with odd-numbered elements; finally, we enhance the Zolotarev difference pattern, which is a counterpart to the Dolph-Chebyshev sum pattern that traditionally allows synthesis only for ULA with even-numbered elements. Our modification extends its applicability to synthesize difference patterns for ULA with odd-numbered elements. Leveraging the optimal difference pattern, a generalized Bayliss difference pattern synthesis method designed for the ULA with odd-numbered elements is further proposed. To illustrate the effectiveness of our approach, we present several design examples through experimental simulation.

Attribute reductions based on δ-fusion condition entropy and harmonic similarity degree in interval-valued decision systems

This paper defines an improved similarity degree based on inclusion degree as well as advanced information system based on interval coverage and credibility, and thus an attribute reduction framework embodying 4×2 = 8 reduct algorithms is systematically constructed for application and optimization in interval-valued decision systems. Firstly, a harmonic similarity degree is constructed by introducing interval inclusion degree and harmonic average mechanism, which has better semantic interpretation and robustness. Secondly, interval credibility degree and coverage degree are defined for information fusion, and they are combined to propose a δ-fusion condition entropy. The improved condition entropy achieves the information reinforcement and integrity by dual quantization fusion of credibility and coverage, and it obtains measure development from granularity monotonicity to non-monotonicity. In addition, information and joint entropies are also constructed to obtain system equations. Furthermore, 8 reduct algorithms are designed by using attribute significance for heuristic searches. Finally, data experiments show that our five novel reduct algorithms are superior to the three contrast algorithms on classification performance, which also further verify the effectiveness of proposed similarity degree, information measures and attribute reductions.

Reversal of drift direction during the Laschamp geomagnetic excursion

Earth's magnetic field changes in both space and time: the temporal changes are called geomagnetic and paleomagnetic secular variations. Westward drift has been noted as a feature of secular variation for several centuries, but eastward drift has received less attention. We use three global geomagnetic field models covering the past 100 kyr to extend temporal coverage for tracking the zonal (azimuthal) motion of the radial magnetic field. The models we use are GGF100k (100–0 ka), GGFSS70 (70–15 ka), LSMOD.2 (50–30 ka); the extent of the models enables the inclusion of the extreme secular variations found during excursions, particularly the Laschamp excursion (42–40 ka). GGFSS70 and LSMOD.2 have higher temporal resolution than GGF100k, but their underlying data have poorer spatial coverage. Spatial structure is greatly diminished in all models for spherical harmonic degrees l>4.We use two types of time-longitude plots, one of the full radial field to expose reverse and intense flux patches at the core-mantle boundary. The second time-longitude plot is processed to enhance zonal motion signatures and allows us to use Radon drift analyses to uncover characteristic time scales of both westward and eastward drift at mid to high latitudes in both the northern and southern hemispheres. Our results differ across the three models, which we attribute to varying degrees of resolution, accuracy, and data distribution. Nevertheless, recurrent episodes of both eastward and westward drift ranging from ±0.05o/yr to ±0.18o/yr occur in both the northern and southern hemispheres. Westward drift dominates. We also observe 8–20 kyr intervals between occurrences of high-latitude reverse flux patches correlated with strong drift signals. Focusing on the period 50–30 ka, we observe dominant eastward drift preceding the Laschamp excursion and westward drift subsequently. In a period not associated with an excursion, 90–80 ka, we see strong mid to high latitude drift signatures in the northern hemisphere.

Co-estimation of core and lithospheric magnetic fields by a maximum entropy method

SUMMARY Satellite observations of the geomagnetic field contain signals generated in Earth’s interior by electrical currents in the core and by magnetized rocks in the lithosphere. At short wavelengths the lithospheric signal dominates, obscuring the signal from the core. Here we present details of a method to co-estimate separate models for the core and lithospheric fields, which are allowed to overlap in spherical harmonic degree, that makes use of prior information to aid the separation. Using a maximum entropy method we estimate probabilistic models for the time-dependent core field and the static lithospheric field that satisfy constraints provided by satellite observations while being consistent with prior knowledge of the spatial covariance and expected magnitude of each field at its source surface. For the core field, we find that between spherical harmonic degree 13 and 22 power adds coherently to the established structures, and present a synthetic test that illustrates the aspects of the small scale core field that can reliably be retrieved. For the large scale lithospheric field we also find encouraging results, with the strongest signatures below spherical harmonic degree 13 occurring at locations of known prominent lithospheric field anomalies in the northern part of Eastern Europe, Australia and eastern North America. Although the amplitudes of the small scale core field and large scale lithospheric field are likely underestimated we find no evidence that obvious artefacts are introduced. Compared with conventional maps of the core–mantle boundary field our results suggest more localized normal flux concentrations close to the tangent cylinder, and that low latitude flux concentrations occur in pairs with opposite polarities. Future improvements in the recovery of the small scale core field and large scale lithospheric field will depend on whether more detailed prior information can be reliably extracted from core dynamo and lithospheric magnetization simulations.

Mio ben ricordati by Franz Schubert and Mikhail Glinka: Concerning the Problem of “the Personal” and “the Alien”

Mio ben ricordati is one of Pietro Metastasio’s popular operatic texts that was set to music by many composers, including Franz Schubert and Mikhail Glinka. The reason for comparison was not only the closeness of their compositions during the time of their creation (respectively, 1820 and 1828), but also the similarity of the situations of their lives: both composers, when achieving mastery of the basics of music, studied with Italian teachers, both composed masterpieces in the field of vocal miniature works in their native languages. The article attempts to place Schubert’s canzona and Glinka’s arietta in the context of the tradition of the musical manifestation of Mio ben ricordati that had developed by their time, revealing their common and specialized features, as well as to compare the works of the two composers with each other, to determine the degree of their closeness to their Italian prototypes and the peculiarities of the style of the original author of the text. Thus, from the perspective of form, Glinka’s arietta turns out to be more traditional than Schubert’s canzona, but at the same time, in some peculiarities of its melody and tonal-harmonic plan, it forms a prototype of the composer’s later, more famous romance I Recall a Wonderful Moment. In Schubert’s musical interpretation of Mio ben ricordati, individual moments of the poetic text are especially emphasized that were clearly important for Metastasio, and techniques characteristic of the art of Italian opera are applied – namely, the parallel minor/major keys (chiaroscuro) and the harmonic color of the lowered harmonic second degree. On the other hand, both composers use the triple meter, bring in repetitions in the text, traditional for the arias of Mio ben ricordati, and also rely on the periodicity intrinsic to Rossini’s works.