Harmonic Functions Research Articles

Harmonic functions associated with Pascal distribution series

The primary objective of this paper is to explore the application of a specific convolution operator, which incorporates the Pascal distribution series. Through this investigation, we establish essential inclusion relations between the harmonic class HΥ(ϱ,ξ) and other classes of harmonic functions, namely, harmonic starlike functions and harmonic convex functions, all defined within the confines of the open unit disk. Moreover, we derive consequential outcomes from our main findings. The study delves into the intriguing interplay among these harmonic function classes, shedding light on their underlying properties and mutual connections. The presented results contribute to a deeper understanding of harmonic functions and provide a valuable basis for further research in this area.

Frequency Scanning of Weakly Damped Single-Phase VSCs With Chirp Error Control

The chirp sweep is a go-to wide-band impedance measurement technique when speed and simplicity are of main concern. Additionally, the time-frequency trait of chirp scans provide unique benefits for systems exhibiting frequency couplings - a phenomenon often encountered in single-phase VSCs. Unfortunately, time domain interpretation of chirp responses are inaccurate if the VSC-based system exhibits weak damping, as the transient response induced by the chirp cannot be safely neglected. We quantify this error for linear(ised) time-invariant and time-periodic systems, with impedance representations of scalar transfer functions and harmonic transfer functions, respectively. An observer termed the Multiple Chirp Reference Frame Filter is proposed which enables real-time estimation of the harmonic transfer functions and their corresponding chirp errors. By controlling the relative errors to negligible values, the chirp exhibits low rate around resonances and high rate elsewhere. An experimental admittance sweep of a single-phase STATCOM operated by dispatchable Virtual Oscillator Control, consolidates the proposed approach as a simple frequency scan technique for weakly damped and frequency coupled systems.

Harmonic maps from $S^3$ into $S^2$ with low Morse index

Harmonic maps from $S^3$ into $S^2$ with low Morse index

Klangfarbenmelodie, Chromophony, and Timbral Function in Arnold Schoenberg’s “Farben”

Arnold Schoenberg’s concept of Klangfarbenmelodie (melody of timbres) is one of the most important yet least understood compositional innovations of the twentieth century. By examining significant factors in Klangfarbenmelodie’s theoretical formulation, proposing functional roles that timbre can fulfill, and locating examples of timbre realizing those musical functions in “Farben,” the third of Schoenberg’s Five Orchestral Pieces, Op. 16, this article demonstrates some of the ways timbre can shape music and our musical experience. While musical logic based on timbre operates according to laws of its own, not those of pitch, parallels can nonetheless be drawn between harmonic functions and timbral functions. Timbral developments are shown to articulate the formal process in “Farben” and create coherent progressions, modulations, and cadences that illustrate some possibilities of how timbre can function in music.

3D Hybrid Segmented Layer Modeling of On-Load Magnetic Fields and Torques for Coreless Axial-Flux Permanent Magnet Synchronous Motor

This paper presents a novel 3-dimensional hybrid segmented layer model (3D HSLM) for coreless axial-flux permanent magnet synchronous motors (AFPMSMs). In the proposed 3D HSLM, the magnetic field sources are divided into the PM array and current-driven electro-magnets, which are captured by the combination of the magnetic scalar potential and vector potential models. This paper proposes a geometry function and cylindrical harmonic index modulation functions in order to correctly capture the curvature effect by the PMs and coils with arbitrary shapes and also the radial edge effect from the finite lengths of the rotor and the stator. The proposed modeling method also enables accurate estimation of the motor characteristics on the on-load conditions using the magnetic vector potential model. The fidelity of the proposed 3D HSLM is validated against the equivalent FEM models for various cases with different PM and coil shapes, in terms of the airgap magnetic fields, motor torque, and flux linkage. The comparison results show that the proposed 3D HSLM yields a modeling accuracy of more than 98% while significantly reducing the calculation time by two orders of magnitude as compared to the FEM models.

Two-level distributed fully-predictive frequency control scheme for inverter-based AC Microgrid considering communication delay

Amid the current development of decentralized renewable energy sources, the research of multi-agent-based control for the next generation of the intelligent power system has become one of the most trending and worth-noticing topics in the smart grid research community. Serving as a contribution to the research orientation, this paper proposes a novel two-layer distributed fully-predictive frequency control scheme for an inverter-based three-phase three-wire AC microgrid. The key feature of the suggested methodology is the incorporation of objective functions and dynamic models as constraints in the predictive formulation of both primary and secondary layers in the microgrid control hierarchy. In our study, the controller is designed based on constrained cost function minimization using Quadratic Programming (QP) and the utilization of local measurements and information from surrounding distributed generators to achieve the desired control action. Along with the integration of virtual impedance, second-order generalized integrator phase-lock loop, and harmonic suppression constraint functions, the proposed control scheme shows great response against all variations of load changes, communication delay, communication network failure, and Plug-n-Play capability test. The performance of the proposed controller is verified and evaluated on the 4-DGs microgrid model using MATLAB/Simulink environment.

Harnack’s Inequality for Harmonic Functions on Stratified Sets

Harnack’s Inequality for Harmonic Functions on Stratified Sets

A Fast and Super-Resolution Method of Vortex-Based Imaging

With characteristics of particular wavefronts and orthogonal modes, the sparse imaging method based on vortex waves could obtain high-resolution reconstruction. However, building a large-scale observation matrix is time-consuming and heavily burdens the whole imaging procedure. Inspired by the spherical harmonic expansion of plane waves, this paper provides a fast super-resolution imaging method based on vortex waves. Featuring spherical harmonic functions and a transformation, an orthogonal matching pursuit algorithm for high-resolution imaging exploits a low-rank matrix decomposition of the observation matrix. Simulation results show that the proposed method exhibits significant efficiency advantages while retaining high resolution, achieving an efficiency of up to 51.4% when applied to a large-scale observation matrix.

Uniqueness theorems for weighted harmonic functions in the upper half-plane

AbstractWe consider a class of weighted harmonic functions in the open upper half-plane known as α-harmonic functions. Of particular interest is the uniqueness problem for such functions subject to a vanishing Dirichlet boundary value on the real line and an appropriate vanishing condition at infinity. We find that the non-classical case (α ≠ 0) allows for a considerably more relaxed vanishing condition at infinity compared to the classical case (α = 0) of usual harmonic functions in the upper half-plane. The reason behind this dichotomy is different geometry of zero sets of certain polynomials naturally derived from the classical binomial series. These findings shed new light on the theory of harmonic functions, for which we provide sharp uniqueness results under vanishing conditions at infinity along geodesics or along rays emanating from the origin.

$n$-harmonicity, minimality, conformality and cohomology


 
 
 By studying cohomology classes that are related with n-harmonic morphisms and F-harmonic maps, we augment and extend several results on F-harmonic maps, harmonic maps in [1, 3, 14], p-harmonic morphisms in [17], and also revisit our previous results in [9, 10, 21] on Riemannian submersions and n-harmonic morphisms which are submersions. The results, for example Theorem 3.2 obtaine by utilizing the n-conservation law (2.6), are sharp.
 
 

Some New Classes of Harmonic Hemivariational Inequalities

Some new classes of harmonic hemivariational inequalities are introduced and investigated in this paper. It has been shown that the optimality conditions of the sum of two harmonic convex functions can be characterized by the harmonic hemivariational inequalities. Several special cases such as harmonic complementarity problems and related harmonic problems are discussed. The auxiliary principle technique is applied to suggest and analyze some iterative schemes for harmonic hemivariational inequalities. We prove the convergence of these iterative methods under some weak conditions. Our method of proof of the convergence criteria is simple compared to other techniques. Results obtained in this paper continue to hold for new and known classes of harmonic variational inequalities and related optimization problems. The ideas and techniques of this paper may inspire further research in various branches of pure and applied sciences.

Estimation of the Bounds of Some Classes of Harmonic Functions with Symmetric Conjugate Points

In this paper, we introduce some classes of univalent harmonic functions with respect to the symmetric conjugate points by means of subordination, the analytic parts of which are reciprocal starlike (or convex) functions. Further, by combining with the graph of the function, we discuss the bound of the Bloch constant and the norm of the pre-Schwarzian derivative for the classes.

On Some Classes of Harmonic Functions Associated with the Janowski Function

In this paper, we introduce some classes of univalent harmonic functions with respect to the symmetric conjugate points by means of subordination, the analytic parts of which are reciprocal starlike (or convex) functions. Further, we discuss the geometric properties of the classes, such as the integral expression, coefficient estimation, distortion theorem, Jacobian estimation, growth estimates, and covering theorem.

4D potential energy surface of NCCN-H2 collision: Rotational dynamics by p-H2 and o-H2 at interstellar temperatures.

The rotational excitation rates of NCCN species are studied for its collision with hydrogen (H2) in temperatures ranging from 1 to 100K. Such collisions can occur in the interstellar medium with H2 in either para (p-) or ortho (o-) state, of which the p-H2 state can be approximated via its collision with He (using a scaling factor) or with a reduced rigid rotor-H2 surface (by averaging over various orientations of H2). In the current work, a four-dimensional (4D) abinitio potential energy surface (PES) is considered to study the collision dynamics of H2 in both p- and o-states and the results are compared with previous approximations. The 4D surface is constructed using the explicitly correlated coupled-cluster method CCSD(T)-F12b with the augmented triple zeta basis AVTZ and then fitted into an artificial neural networks (NN) model to augment the surface and account for missing data points. The radial coefficients are obtained from this NN fitted 4D PES via a least square fit over two spherical harmonics functions. The cross sections (σ) are computed using the close-coupling (CC) method (until 230 cm-1) for both p- and o-H2 collisions, and the rates are obtained by Boltzmann distribution over the translational energy of H2 until 100K. The o-H2 rates are found to be higher by 25%-30% and 10%-20% compared to the p-H2 rates for Δj = 2 and higher order transitions, respectively. The coupled-state/centrifugal sudden approximated rates are also computed and found to have deviations as large as 40% when compared to CC rates, thus making quantitative descriptions unreliable.

A multiplicity estimate for the Jacobi operator of a nonflat Yang–Mills field over mathbb {S}^m

Here we provide refinements of the stability results of Simons and Xin, concerning the stability of Yang–Mills fields and harmonic maps respectively. The result also implies the earlier Morse index estimates for both cases.

Refined Bohr inequality for functions in and in complex Banach spaces

In this paper, we first obtain a refined version of the Bohr inequality of norm-type for holomorphic mappings with lacunary series on the polydisk in under some restricted conditions. Next, we determine the refined version of the Bohr inequality for holomorphic functions defined on a balanced domain G of a complex Banach space X and take values in the unit disk . Furthermore, as a consequence of one of these results, we obtain a refined version of the Bohr-type inequality for harmonic functions defined on a balanced domain . All the results are proved to be sharp.

Deciphering the Influence of Ground-State Distributions on the Calculation of Photolysis Observables.

Nonadiabatic molecular dynamics offers a powerful tool for studying the photochemistry of molecular systems. Key to any nonadiabatic molecular dynamics simulation is the definition of its initial conditions (ICs), ideally representing the initial molecular quantum state of the system of interest. In this work, we provide a detailed analysis of how ICs may influence the calculation of experimental observables by focusing on the photochemistry of methylhydroperoxide (MHP), the simplest and most abundant organic peroxide in our atmosphere. We investigate the outcome of trajectory surface hopping simulations for distinct sets of ICs sampled from different approximate quantum distributions, namely harmonic Wigner functions and ab initio molecular dynamics using a quantum thermostat (QT). Calculating photoabsorption cross-sections, quantum yields, and translational kinetic energy maps from the results of these simulations reveals the significant effect of the ICs, in particular when low-frequency (∼ a few hundred cm-1) normal modes are connected to the photophysics of the molecule. Overall, our results indicate that sampling ICs from ab initio molecular dynamics using a QT is preferable for flexible molecules with photoactive low-frequency modes. From a photochemical perspective, our nonadiabatic dynamics simulations offer an explanation for a low-energy tail observed at high excitation energy in the translational kinetic energy map of MHP.

Infinite Energy Harmonic Maps from Riemann Surfaces to Cat(0) Spaces

Infinite Energy Harmonic Maps from Riemann Surfaces to Cat(0) Spaces

Uniform W1,p estimates and large-scale regularity for Dirichlet problems in perforated domains

In this paper we study the Dirichlet problem for Laplace's equation in a domain ωε,η perforated periodically with small holes in Rd, where ε represents the scale of the minimal distances between holes and η the ratio between the scale of sizes of holes and ε. We establish W1,p estimates for solutions with bounding constants depending explicitly on ε and η. The proof relies on a large-scale Lipschitz estimate for harmonic functions in perforated domains. The results are optimal for d≥2.

Dynamic modeling and analysis of discontinuous wave propagation in a rod

The wave propagation in an impacting rod is instant and nearly discontinuous, which remains a challenging problem for both dynamic modeling and experimental test. This paper presents an Enriched Finite Element Method (Enriched FEM) with controllable dispersion error to accurately and efficiently model the discontinuous wave propagation. The Enriched FEM approximates the displacement solutions based on a linear interpolation enriched with a series of harmonic functions. The paper focuses on the dispersion error of the enriched finite element from spatial-temporal discretization and the sampling theorem of the enriched finite element with different cutoff numbers, and then gives a simple quantitative criterion on the interdependent relationship between the size of mesh and time step so as to reduce the numerical dispersion error. In addition, the paper presents the experimental study on the wave propagation of an elastic rod colliding with a rigid wall. The experimental results explicitly show the incidence and reflection of the stress wave in the rod in the contact duration. They also indicate that the amplitude of the strain almost does not decay with the wave propagation along the rod. The numerical results based on the Hertzian contact model are in good agreement with the experimental observations, especially at wave interfaces.