Harmonic Series Research Articles

Dual-heterodyne Kelvin probe force microscopy.

We present a new open-loop implementation of Kelvin probe force microscopy (KPFM) that provides access to the Fourier spectrum of the time-periodic surface electrostatic potential generated under optical (or electrical) pumping with an atomic force microscope. The modulus and phase coefficients are probed by exploiting a double heterodyne frequency mixing effect between the mechanical oscillation of the cantilever, modulated components of the time-periodic electrostatic potential at harmonic frequencies of the pump, and an ac bias modulation signal. Each harmonic can be selectively transferred to the second cantilever eigenmode. We show how phase coherent sideband generation and signal demodulation at the second eigenmode can be achieved by using two numerical lock-in amplifiers configured in cascade. Dual-heterodyne KPFM (DHe-KPFM) can be used to map any harmonic (amplitude/phase) of the time-periodic surface potential at a standard scanning speed. The Fourier spectrum (series of harmonics) can also be recorded in spectroscopic mode (DHe-KPFM spectroscopy), and 2D dynamic images can be acquired in data cube mode. The capabilities of DHe-KPFM in terms of time-resolved measurements, surface photovoltage (SPV) imaging, and detection of weak SPV signals are demonstrated through a series of experiments on difference surfaces: a reference substrate, a bulk organic photovoltaic heterojunction thin film, and an optoelectronic interface obtained by depositing caesium lead bromide perovskite nanosheets on a graphite surface. The conclusion provides perspectives for future improvements and applications.

A New Generalization of the Alternating Harmonic Series

Kilmer and Zheng (2021) recently introduced a generalized version of the alternating harmonic series. In this paper, we introduce a new generalization of the alternating harmonic series. A special case of our generalization converges to the Kilmer-Zheng series. Then we investigate several interesting and useful properties of this generalized, such as a summation formula related to the Hurwitz -Lerch Zeta function, a duplication formula, an integral representation, derivatives, and the recurrence relationship. Some important special cases of the main results are also discussed.

PROCESSING OF VIBRO - ACOUSTIC SIGNALS OF PRECESSIONAL PLANETARY TRANSMISSIONS

The operation of working machines is accompanied by the constant occurrence of vibrations and shocks, generated by various constructive, kinematic and dynamic factors. The study of the vibro-acoustic aspect of mechanical transmissions and, in particular, of precessional planetary ones is a primary concern for engineers in the field of machine building technologies. The paper presents the research of precessional planetary transmissions under the vibro-acoustic aspect in the anechoic chamber, which is a room, insulated with sound-absorbing material against external noises and isolated anti-vibration. To process the vibro-acoustic signals received as a result of the measurements, the Fourier harmonic series will be applied, a mathematical function that can be used to find the basic frequencies that make up a wave. The final result is the detection of vibration and noise generating sources, the decoding of signals by means of Fourier series and the advancement of constructive-technological solutions to minimize the vibro-acoustic level.

Derivative and partial integral methods and Gauss integral's indefinite integral solution and its use in wave function in quantum mechanics and exact solutions of the wave function depending on position and time

This study, , solution of Gaussian Integral with differential equation [18,19]; By using the partial integral method, the indefinite integral solution by taking x under the differential [29,30,31] and the indefinite integral solution by taking the Gaussian integral under the differential [33,38,39] both harmonic series and function solutions are found. Indefinite integral solution of Gaussian Integral [38] In Quantum Physics, the wave function solution f(x,α) in terms of α variable in x position and k space by substituting it in wave function [44,45] and in one-dimensional time dependent Schrödinger equation, wave function equation [66, 67] is found. When the wave function in direct space is differentiated by by the partial integral method without using the approximate value of w(k)[50] in space k, the general wave equation in k space by position [74] and approximately the wave function f(x,α) [80] is found. min in space k depending on location and time the exact solution of the wave equations [83,89] and the Taylor Series solution give the wave equation [87,90] in k space according to position and time.

A New Modified Helmholtz Equation for the Expression of the Gravity Gradient and the Intensity of an Electrostatic Field in Spherical Harmonics

In this work, it is shown that the geometry of a gravity field generated by a spheroid with low eccentricity can be described with the help of a newly modified Helmholtz equation. To distinguish this equation from the modified Helmholtz equation, we refer to it as the G-modified Helmholtz equation. The use of this new equation to study the spheroid’s gravity field is advantageous in expressing the gravity vector as a vector series of spherical harmonics. The solution of the G-modified Helmholtz equation involves both the gravity intensity g (or simply gravity g) and the intensity E of an electrostatic field as shown in sequel. An electrostatic field generated by an oblate spheroid charged with l electrons (uniform ellipsoidal charge distribution) is demonstrated to be a special case. Both gravity intensity g and intensity E are governed by the same law and can be expressed as a series of spherical harmonics, and thus the G-modified Helmholtz equation is useful for describing the gravity and electrostatic fields.

Improvement of methods for analyzing the technical condition of electromechanical equipment of nuclear power plants

The article analyzes the problem of diagnostics of NPP electric drive valves. Failure of this type of equipment can become the initial event for design or beyond design accident. Therefore, during the period of preventive maintenance at NPPs diagnostics of up to 2000 units of valves is carried out mainly by registration and analysis of the motor current signal during opening and closing. However, the applied statistical and frequency analysis methods do not always provide the necessary quality of diagnostics. The article points out the disadvantage of the applied methods – insensitivity to the dynamics of electromechanical equipment in terms of nonlinear manifestations of defects. As a tool for analyzing these manifestations the phase-plane method is proposed and its theoretical basis is outlined. The diagnostic signal is described as a vector diagram of a harmonic series subjected to amplitude and phase modulation. The most characteristic forms of phase trajectories - cardioids, nephroids, epicycloids – are shown. The main advantages of the phase-plane method over traditional frequency methods – sensitivity, possibility to identify the type of modulation, phase shift, dynamic and chaotic manifestations – are demonstrated by examples of processing diagnostic signals of NPP equipment. An important advantage of signal representation in the form of phase portrait is the visualization of dynamic features of the signal. The result of the work is a new method that allows to extract additional information about the object state by analyzing the non-stationary component of the initial data. It is proposed to introduce the phase-plane method into the practice of processing diagnostic signals of electric drive valves and other electromechanical equipment operated at NPPs.

Vlasov-Fokker-Planck-Maxwell simulations for plasmas in inertial confinement fusion

This work presents a code to simulate collisions and laser-plasma interactions of multiple-species plasmas by solving 1D3V Vlasov-Fokker-Planck-Maxwell equations. The distribution functions of all species are expanded into spherical harmonic series of arbitrary order. Each electron or ion species possesses its distinct and suitable velocity mesh and the number of harmonics. Collisions between different ion species are considered and treated with the Fokker-Planck collision term. The code employs the total variation diminishing (TVD) scheme to avoid numerical instabilities in the region of steep density, rarefied plasma-vacuum interface, and non-periodic boundary conditions, which makes it possible to simulate cases that are close to actual physical scenes. The number of particles is globally conserved for each species, the energy for each species is conserved for collisions between the same species, and total energy is conserved for collisions between different species.

Stable identification of sources located on the cerebral cortex from EEG over the scalp

In this work, we present a stable algorithm for the inverse problem of identifying cortical sources from electroencephalographic measurements on the scalp. This inverse problem is ill-posed due to the numerical instability it presents, i.e., small changes in the measurements can produce large variations in the location of the sources. A boundary value problem is used to find correlations between the sources and measurements. In the case in which the head is modeled using two concentric spheres. We use spherical harmonics to find the solution to the inverse source cortical problem. To handle the numerical instability of this problem, we use the Tikhonov regularization method and a cut-off of the harmonic expansion series. From numerical tests, we found these parameters with which we get good approximations. Finally, we illustrate the algorithm proposed with synthetic examples

A different kind of meantone temperament and a novel mapping from the harmonic series to Pythagorean pitch classes

The logarithm mapping of natural numbers is a sum of products of coefficients. If these coefficients are arbitrary parameters, a new mapping of natural numbers to some subset of real numbers appears. This mapping preserves some crucial logarithm properties and constructs a new musical sound with a spectrum of inharmonic overtones. The simplest one-parameter mapping to a subset of polynomials with integer coefficients is constructed. The parameter defines a new “perfect fifth” similarly to the meantone temperament. It is an interesting case when the mapping parameter is defined from the linear relation between the new “major third” and the new “perfect fifth.” The most interesting case of a linear relation is the condition of zeroing the syntonic comma. Here, the new meantone temperament, based on the new “perfect fifth,” simultaneously coincides with Pythagorean- and five-limit-like tunings.

Thermal analysis of photon-like particles in rainbow gravity

This work is devoted to study the thermodynamic behavior of photon-like particles within the rainbow gravity formalism. To to do this, we chose two particular ansatzs to accomplish our calculations. First, we consider a dispersion relation which avoids UV divergences, getting a positive effective cosmological constant. We provide numerical analysis for the thermodynamic functions of the system and bounds are estimated. Furthermore, a phase transition is also expected for this model. Second, we consider a dispersion relation employed in the context of Gamma Ray Bursts. Remarkably, for this latter case, the thermodynamic properties are calculated in an analytical manner and they turn out to depend on the harmonic series Hn, gamma Γ(z), polygamma ψn(z) and zeta Riemann functions ζ(z).

Comparison between the geological features of Venus and Earth based on gravity aspects

We probe the gravitational properties of two neighboring planets, Earth and Venus. To justify a comparison between gravity models of the two planets, spherical harmonic series were considered up to a degree and order of 100. The topography and gravity aspects, including {Gamma }_{zz} (vertical derivative of the vertical component of the gravity field), strike alignment (SA), comb factor (CF), and I2 invariant derived from the Marussi tensor, were calculated for the two planets at specifically selected zones that provided sufficient resolution. From Γzz we discovered that the N-NW edge of Lakshmi Planum does not show any subduction-like features. Its Γzz signature resembles passive continental margins on Earth, like those surrounding the Indian Peninsula. Moreover, according to SA and CF, the Pacific and Philippine-North American Contact Zone on Earth indicates significantly higher level of deformation due to convergent motion of the plates, whereas the deformation level on Venus is significantly smaller and local, when considering an equatorial rifting zone (ERZ) of Venus (between Atla-Beta Regios) as diverging boundaries. The strain mode on the East African Rift system is smaller in comparison with ERZ as its Venusian analog. The topography-I2 analysis suggests a complicated nature of the topographic rise on Beta Regio. We show that specific regions in this volcanic rise are in incipient stages of upward motion, with denser mantle material approaching the surface and thinning the crust, whereas some risen districts show molten and less dense underlying crustal materials. Other elevated districts appear to be due to mantle plumes and local volcanic activities with large density of underlying material.

The Lower Limit of Pitch Perception for Pure Tones and Low-Frequency Complex Sounds

The lower limit of pitch (LLP) perception was explored for pure tones, sinusoidally amplitude-modulated (SAM) tones with a carrier frequency of 125 Hz, and trains of 125-Hz tone pips, using an adaptive procedure to estimate the lowest repetition rate for which a tonal/humming quality was heard. The LLP was similar for the three stimulus types, averaging 19 Hz. There were marked individual differences, which were correlated to some extent across stimulus types. The pure-tone stimuli contained a single resolved harmonic, whereas the SAM tones and tone-pip trains contained only unresolved components, whose frequencies did not necessarily form a harmonic series. The similarity of the LLP across stimulus types suggests that the LLP is determined by the repetition period of the sound for pure tones, and the envelope repetition period for complex stimuli. The results are consistent with the idea that the LLP is determined by a periodicity analysis in the auditory system, and that the longest time interval between waveform or envelope peaks for which this analysis can be performed is approximately 53 ms.

Approximation of Aperiodic Signals Using Non-Integer Harmonic Series: The Generalized NAFASS Approach

In this paper, the non-orthogonal amplitude-frequency analysis of smoothed signals (NAFASS) method) is used to approximate discrete aperiodic signals from various complex systems with the non-integer harmonic series (NIHS). When approximating by the NIHS, there is a problem in determining the dispersion law for harmonic frequencies. In the original version of the NAFASS approach, the frequency dispersion law was determined from a linear-difference equation. However, many complex systems in nature have frequency distributions that differ from the linear law, which is used in the conventional Fourier analysis of periodic signals. This paper proposes a generalization of the NAFASS method for describing aperiodic signals by the NIHS with a frequency distribution that satisfies a recursive formula, which coincides with the local generalized geometric mean (GGM). The methodology of the generalized NAFASS method is demonstrated using descriptions of financial data (prices for metals) and sound data (sounds of insects) as examples. The results show the effectiveness of the generalized NAFASS approach for describing real-world time data. This discovery allows us to propose a new classification scheme for smoothed and aperiodic signals captured as responses and envelopes from various complex systems.

Spherical Harmonics for the 1D Radiative Transfer Equation. I. Reflected Light

A significant challenge in radiative transfer theory for atmospheres of exoplanets and brown dwarfs is the derivation of computationally efficient methods that have adequate fidelity to more precise, numerically demanding solutions. In this work, we extend the capability of the first open-source radiative transfer model for computing the reflected light of exoplanets at any phase geometry, PICASO (Planetary Intensity Code for Atmospheric Spectroscopy Observations). Until now, PICASO has implemented two-stream approaches to the solving the radiative transfer equation for reflected light, in particular following the derivations of Toon et al. In order to improve the model accuracy, we have considered higher-order approximations of the phase functions; namely, we have increased the order of approximation from two to four, using spherical harmonics. The spherical harmonics approximation decouples spatial and directional dependencies by expanding the intensity and phase function into a series of spherical harmonics, or Legendre polynomials, allowing for analytical solutions for low-order approximations to optimize computational efficiency. We rigorously derive the spherical harmonics method for reflected light and benchmark the four-term method (SH4) against Toon et al. and two independent and higher-fidelity methods (CDISORT and doubling method). On average, the SH4 method provides an order-of-magnitude increase in accuracy, compared to Toon et al. Finally, we implement SH4 within PICASO and observe only a modest increase in computational time, compared to two-stream methods (20% increase).

ID: 216730 Comparing Tractography Methods for Preoperative Deep Brain Stimulation Planning in Essential Tremor Patients

Accurate lead placement is essential to successful treatment with deep brain stimulation (DBS). The Ventralis Intermediate Nucleus (VIM) of the thalamus has been identified as a surgical target for DBS neuromodulation of motor input with successful outcomes [1-3]. To aid in this, prospective pre-operative tractography can be implemented for targeting the VIM in ET patients [4,5]. For this, FSL’s ball-and-stick model based probabilistic tracking” (Probtrackx) has been validated to be an effective tractography method (TM) [5]. However, with many other tractography widely available and embedded in commercial surgical platforms, there is benefit in evaluating the relative performance of tractography methods (TMs). Here, we assessed the relative performance of five TMs from the MRTRIX toolbox: three of these techniques were based on fiber orientation distribution functions (ODFs) represented by spherical harmonic series: iFOD1, iFOD2, iFOD-rk4, and two tensor-based models: Tensor-Deterministic, and Tensor-Probabilistic.

Movement patterns in tuba playing: comparison of an embouchure dystonia case with healthy professional tuba players using real-time MRI imaging.

Musculoskeletal problems in professional brass musicians are very common and often involve the muscles of the embouchure. In rare cases, embouchure dystonia (EmD), a task-specific movement disorder with a wide symptomatic and phenotypic variability, occurs. Following trumpeters and horn players, professional tuba players with and without EmD have now been studied using the latest real-time MRI technology to better understand the underlying pathophysiology. In the present study, the tongue movement patterns of 11 healthy professional artists and one subject suffering from EmD were compared. The tongue position in the anterior, intermediary and posterior oral cavity were converted into pixel positions based on seven previously generated profile lines, using the established software MATLAB. These data allow a structured comparison of tongue movement patterns between the patient and the healthy subjects, as well as between individual exercises. The main focus of the analysis was on an ascending 7-note harmonic series performed in different playing techniques (slurred, tongued, tenuto and staccato). Playing the ascending harmonics, a noticeable ascending tongue movement could be observed in the anterior part of the oral cavity in healthy tubists. In the posterior region, there was a slight decrease in oral cavity space. In the EmD patient, hardly any movement was observed at the tongue apex, but in the middle and posterior regions of the oral cavity there was an increase in size the higher the tone became. These distinct differences are relevant for the characterization and a better understanding of the clinical presentation of EmD. Concerning different playing techniques, it was apparent, that notes played slurred or staccato resulted in a larger oral cavity when compared to notes played tongued or tenuto, respectively. By using real-time MRI videos, the tongue movements of tuba players can be clearly observed and analyzed. The differences between healthy and diseased tuba players demonstrate the great effects of movement disorders in a small area of the tongue. In order to better understand the compensation of this motor control dysfunction, further studies should investigate further parameters of tone production in all brass players with a larger number of EmD patients additional to the observed movement patterns.

Mapping understory plant communities in deciduous forests from Sentinel-2 time series

Understory plant communities are an integral component of deciduous forests, playing a vital role in the overall health of the ecosystem. However, remote sensing of understory plant communities is challenging due to the obstruction by the forest canopy. In this study, we proposed an automated dense Sentinel-2 time series-based approach for understory plant communities and created maps of four understory classes (i.e., native shrubs of greenbrier and mountain laurel, and invasive shrubs of barberry and the assemblage of mixed invasive) at 10 m resolution in Connecticut's deciduous forests in 2020. A harmonic time series model and three years of Sentinel-2 time series from 2019 to 2021 were used to classify understory species based on their unique, intra-annual phenology characteristics. The time series model coefficients captured the subtle phenology differences and created synthetic cloud-free images within a short temporal window in the spring prior to canopy leaf-on (hereafter called “observation window”). During the observation window, Sentinel-2 data penetrated the deciduous overstory canopy and observed the unique trajectories of different understory species due to their phenology differences. We also calculated spatial texture features (i.e., mean, second moment, and contrast from gray level co-occurrence matrix) based on the synthetic images created within the observation window to capture the different conditions of leaf growth and distinct spatial patterns within deciduous forests. By using the spectral, temporal, and spatial features as input variables from dense Sentinel-2 data, auxiliary data (i.e., LiDAR and soil drainage layer), a random forest classifier, and a new strategy to iteratively select representative sample (namely ISRS), understory species maps were created with an overall accuracy of approximately 93%, and the user's and producer's accuracies varied from 39% to 99% for the three mapped understory species and one assemblage of species. The proposed method created an accurate binary map of understory presence with an overall accuracy of 95%, a producer's accuracy of 84%, and a user's accuracy of 68%. Additionally, we separated the invasive (i.e. barberry and mixed invasive of multi-flora rose, oriental bittersweet, honeysuckle, winged euonymus, and autumn olive) and native (greenbrier and mountain laurel) species with an overall accuracy of 94%. We estimated that the invasive species cover an area of 649.33 ± 140.59 km2, which occupied a large proportion (∼53%) of the shrub understory in Connecticut's deciduous forests.

Tidal Forcing on the Sun and the 11-Year Solar-Activity Cycle

The hypothesis that tidal forces on the Sun are related to the modulations of the solar-activity cycle has gained increasing attention. The works proposing physical mechanisms of planetary action via tidal forcing have in common that quasi-alignments between Venus, Earth, and Jupiter (V-E-J configurations) would provide a basic periodicity of $\approx 11.0$ years able to synchronize the operation of solar dynamo with these planetary configurations. Nevertheless, the evidence behind this particular tidal forcing is still controversial. In this context we develop, for the first time, the complete Sun's tide-generating potential (STGP) in terms of a harmonic series, where the effects of different planets on the STGP are clearly separated and identified. We use a modification of the spectral analysis method devised by Kudryavtsev (J. Geodesy. 77, 829, 2004; Astron. Astrophys. 471, 1069, 2007b) that permits to expand any function of planetary coordinates to a harmonic series over long time intervals. We build a catalog of 713 harmonic terms able to represent the STGP with a high degree of precision. We look for tidal forcings related to V-E-J configurations and specifically the existence of periodicities around $11.0$ years. Although the obtained tidal periods range from $\approx$ 1000 years to 1 week, we do not find any $\approx$ 11.0 years period. The V-E-J configurations do not produce any significant tidal term at this or other periods. The Venus tidal interaction is absent in the 11-year spectral band, which is dominated by Jupiter's orbital motion. The planet that contributes the most to the STGP in three planets configurations, along with Venus and Earth, is Saturn. An $\approx 11.0$ years tidal period with a direct physical relevance on the 11-year-like solar-activity cycle is highly improbable.

ПРИБЛИЖЕННЫЕ АСИМПТОТИЧЕСКИЕ ВЫРАЖЕНИЯ ЭЛЕКТРОМАГНИТНОГО ПОЛЯ ИСТОЧНИКОВ ВБЛИЗИ ГЛАДКОЙ ВЫПУКЛОЙ ПРОВОДЯЩЕЙ ПОВЕРХНОСТИ ВРАЩЕНИЯ

Approximate asymptotic solution is developed for the electromagnetic field due of a surface magnetic and electric currents located near an arbitrary smooth convex conducting surface of rotation of large size. The expressions of field are valid in a near-surface layer with a thickness of the order of the wavelength and have the form of a sum of a series of azimuthal harmonics and an integral over a continuous spectrum of eigenfunctions of the field. The coefficients of the series are expressed by integrals from the product of a given distribution of sources with eigenfunctions. For eigenfunctions, both general integral representations and closed expressions in terms of Airy functions are obtained, which are uniformly valid along surface of a rotation for the case of one pole by the parabolic equation method. The field expressions take into account the inseparability of the field by the sum in term of E- and H-type fields, are uniformly valid in the near-surface layer, excluding the vicinity of the poles of the rotation surface, and have no discontinuities on the caustics of surface rays. The expressions obtained were used to calculate the mutual admittances between two annular slots on a semi-infinite and smoothly truncated conical surface. Numerical results obtained by the proposed and rigorous methods for a semi-infinite cone were compared. It is shown that in the case of in-phase annular slots in a semi-infinite cone, the obtained asymptotic expression for mutual admittance coincides with the main term of the asymptotic of a strict solution based on the method of eigenfunctions.

The Gravity Force Generated by a Non-Rotating Level Ellipsoid of Revolution with Low Eccentricity as a Series of Spherical Harmonics

The gravity force of a gravity field generated by a non-rotating level ellipsoid of revolution enclosing mass M is given as a solution of a partial differential equation along with a boundary condition of Dirichlet type. The partial differential equation is formulated herein on the basis of the behavior of spherical gravity fields. A classical solution to this equation is represented on the basis of spherical harmonics. The series representation of the solution is exploited in order to conduct a rigorous asymptotic analysis with respect to eccentricity. Finally, the Dirichlet boundary problem is solved for the case of an ellipsoid of revolution (spheroid) with low eccentricity. This has been accomplished on the basis of asymptotic analysis, which resulted in the determination of the coefficients participating in the spherical harmonics expansion. The limiting case of this series expresses the gravity force of a non-rotating sphere.