Harmonic Functions Research Articles

Bohr Inequalities for Certain Classes of Harmonic Mappings

Bohr Inequalities for Certain Classes of Harmonic Mappings

Wind Load Model and Response Spectra of Overbridges Subjected to Train-Induced Wind Loads

The vibration of pedestrian overbridges due to train-induced wind loads (TIWLs) during the passage of a high-speed train threatens the overbridge safety and causes pedestrian discomfort. To incorporate the structural dynamic magnification effect on overbridges under TIWLs in the design process, a framework to calculate the structural dynamic response, equivalent static TIWLs and response spectra of overbridges is established. A new simplified TIWL model with a “three-segment” wind pressure time history described by harmonic functions is proposed. This model facilitates the acquisition of analytical solutions for the structural dynamic response, streamlining the analysis process. The dynamic response and equivalent static TIWLs based on dynamic magnification factors (DMFs) of overbridges are obtained from the analytical method. The DMFs are primarily affected by the structural natural frequency and the characteristic frequency of TIWLs. Furthermore, based on the analytical solution of the overbridge dynamic response, parametric analysis is performed, and the response spectra representing the dynamic displacement and acceleration of overbridges are developed. The response spectra can be used to quickly calculate the displacement, internal force, and acceleration of the overbridges during engineering applications, which is more convenient but slightly less accurate than the analytical method.

The quinta huapanguera in the huapango arribeño and in the son huasteco: an ethnomusicological vision of this musical instrument in two traditional expressions

The quinta huapanguera is a type of Mexican guitar that is part of the instrumental ensemble that performs music in the Huasteca, which constitutes the main musical instrument of the son huasteco and the huapango arribeño. These two musical expressions are currently recreated in the Huasteca area and in the Sierra Gorda (San Luis Potosí, Querétaro, Hidalgo, Guanajuato, Tamaulipas, Veracruz, Puebla). The son huasteco and the huapango arribeño have common and different sociomusical features and functions. Therefore, the role of the quinta huapanguera varies, since it fulfills different rhythmic, harmonic and technical-musical functions, so comparing these roles will allow an analysis and systematization from a musicological perspective. Certain actions of individuals within sociocultural, historical and geopolitical contexts propitiate and define the existence of this sonorous instrument and the musical genres it performs. Identifying such anthropological issues will contribute to an understanding of the phenomena and codes that cause and make possible the genesis and development of musical instruments with their sonorous expressions. The ethnographic analysis that allows to see music as a cultural practice, the musical analysis that makes possible the definition of the structures and sonorous qualities of the quinta huapanguera and the two musical genres it plays, will allow to collect the necessary data and information to obtain a holistic vision of this sonorous instrument. The comparative analysis of the quinta huapanguera in both sonorous expressions, from the Ethnomusicology or also called Musical Anthropology, will contribute to the promotion of the cultural identity, since it will allow the understanding of these sociomusical phenomena.

Weighted composition operators on harmonic Fock spaces

ABSTRACT In this paper, we study weighted composition operators on Fock spaces of harmonic functions (briefly as harmonic Fock spaces). The basic properties of such operators (boundedness, compactness, invertibility and symmetries) are characterized completely in simple algebraic terms, involving their symbols.

An analogue of Hartogs lemma for separately harmonic functions with variable radius of harmonicity

An analogue of Hartogs lemma for separately harmonic functions with variable radius of harmonicity

Multi-harmonic sources identification and evaluation method based on cloud-edge-end collaboration

With the increase of power electronic devices such as distributed renewable energies, energy storage, and flexible loads, the disturbance sources are rapidly raised and distributed in multiple voltage levels in the distributed network. The harmonics they produce are random and uncertain, and the massive amounts of data propose high demands on server computing capability, which restrict harmonic sources evaluation and significantly influence the judgments of the power quality in the distribution network. In order to improve the power quality and the harmonic source evaluation function of the distribution network, in this paper, a framework for multi-harmonic source identification and evaluation based on cloud-edge-end collaboration is proposed. Firstly, based on the analysis of the functional requirements harmonic sources evaluation in multiple voltage levels, a novel framework of cloud-edge-end collaboration is proposed. Secondly, the cloud-edge-end collaborative harmonic sources evaluation method which executes different service strategies according to voltage levels and grid operation based on the multi-level interaction of edge computing is proposed. Further, the working principle of the proposed novel framework with multiple evaluation methods including the dominant harmonic source identification, concentrated multiple harmonic sources evaluation (CMHSE) and hierarchical harmonic sources evaluation (HHSE) is described in detail in order to show the completed work processes of the proposed framework. Finally, the performance of the proposed method is verified based on the real data measured from different realistic urban power grids in China. The proposed method can effectively evaluate harmonic contributions in distribution systems and in turn provide a basis for harmonic mitigation measures.

Enhancement of intra-cardiac flow-field data using adaptive Kernel filtering

A method of determining the optimal kernel size for filtering noise in vortex dominated flow-fields, as found in the cardiac chambers is presented in this paper. Using synthetic flow fields generated using harmonic functions and perturbed using Gaussian noises of different amplitudes and spreads, the effect of kernel size on noise removal using the Median filter is tested systematically. It is shown that there exists an optimal kernel size at which the Median filter works best. The size of the optimal kernel is shown to be related to the vortex size. When applied to MRI generated cardiac flow-fields, the approach is seen to reveal underlying vortex patterns thereby aiding as an effective tool in the diagnosis and prognosis of cardiac diseases based on vortices as clinical biomarkers. The behavior of the restored cardiac flow fields which are filtered with the optimal kernel size and also with some values preceding and succeeding it are similar to that observed in studies with synthetic flow fields. This confirms that the optimal size of the kernel is related to the cardiac vortex size as is observed in the case of synthetic flow fields.

Four-Dimensional Differential Operator

The hyper-exponential function is a great family of special functions. When we configure a complex plane with a vector of an arbitrary direction in three-dimensional space as the real axis and time as the imaginary axis, by using the variables on this complex plane the hyper-exponential function of second-order proves to be a harmonic function. Based on this, I use quaternion to define my own four-dimensional differential operator and show its characteristics

Min-max harmonic maps and a new characterization of conformal eigenvalues

Given a surface M and a fixed conformal class c one defines \Lambda_{k}(M,c) to be the supremum of the k -th nontrivial Laplacian eigenvalue over all metrics g\in c of unit volume. It has been observed by Nadirashvili that the metrics achieving \Lambda_{k}(M,c) are closely related to harmonic maps to spheres. In the present paper, we identify \Lambda_{1}(M,c) and \Lambda_{2}(M,c) with min-max quantities associated to the energy functional for sphere-valued maps. As an application, we obtain several new eigenvalue bounds, including a sharp isoperimetric inequality for the first two Steklov eigenvalues. This characterization also yields an alternative proof of the existence of maximal metrics realizing \Lambda_{1}(M,c) , \Lambda_{2}(M,c) , and moreover allows us to obtain a regularity theorem for maximal Radon measures satisfying a natural compactness condition.

Harmonic maps between 2-dimensional simplicial complexes: conformal and singular metrics

Harmonic maps between 2-dimensional simplicial complexes: conformal and singular metrics

Harmonic maps on locally conformal almost cosymplectic manifolds

We investigate harmonic maps on almost contact metric manifolds which are locally conformal to almost cosymplectic manifolds. We obtain the necessary and sufficient conditions for the holomorphy to imply harmonicity and then we find obstructions to the existence of non-constant pluriharmonic maps. We also establish some results on the stability of the identity map on a locally conformal almost cosymplectic manifold of pointwise constant [Formula: see text]-holomorphic sectional curvature.

Criteria for close-to-convexity, convex in one direction of planar harmonic mappings

Criteria for close-to-convexity, convex in one direction of planar harmonic mappings

Sharp Estimates Involving a Generalized Symmetric Sălăgean q-Differential Operator for Harmonic Functions via Quantum Calculus

In this study, we apply q-symmetric calculus operator theory and investigate a generalized symmetric Sălăgean q-differential operator for harmonic functions in an open unit disk. We consider a newly defined operator and establish new subclasses of harmonic functions in complex order. We determine the sharp results, such as the sufficient necessary coefficient bounds, the extreme of closed convex hulls, and the distortion theorems for a new family of harmonic functions. Further, we illustrate how we connect the findings of previous studies and the results of this article.

On Bohr's inequality for special subclasses of stable starlike harmonic mappings

Abstract The focus of this article is to explore the Bohr inequality for a specific subset of harmonic starlike mappings introduced by Ghosh and Vasudevarao (Some basic properties of certain subclass of harmonic univalent functions, Complex Var. Elliptic Equ. 63 (2018), no. 12, 1687–1703.). This set is denoted as ℬ H 0 ( M ) ≔ { f = h + g ¯ ∈ ℋ 0 : ∣ z h ″ ( z ) ∣ ≤ M − ∣ z g ″ ( z ) ∣ } {{\mathcal{ {\mathcal B} }}}_{H}^{0}\left(M):= \{f=h+\overline{g}\in {{\mathcal{ {\mathcal H} }}}_{0}:| z{h}^{^{\prime\prime} }\left(z)| \le M-| z{g}^{^{\prime\prime} }\left(z)| \} for z ∈ D z\in {\mathbb{D}} , where 0 < M ≤ 1 0\lt M\le 1 . It is worth mentioning that the functions belonging to the class ℬ H 0 ( M ) {{\mathcal{ {\mathcal B} }}}_{H}^{0}\left(M) are recognized for their stability as starlike harmonic mappings. With this in mind, this research has a twofold goal: first, to determine the optimal Bohr radius for this specific subclass of harmonic mappings, and second, to extend the Bohr-Rogosinski phenomenon to the same subclass.

Improved modal analyses using the novel quadrilateral overlapping elements

In this work, the novel quadrilateral overlapping element incorporated with the harmonic trigonometric functions is presented to reduce the efforts in mesh division for modal analysis. The new partition of unity functions in the overlapping element are cubic polynomials. Therefore, low-order polynomials can be utilized to augment local approximation space without the linear dependence problem. Moreover, the original polynomial approximation spaces are further enriched by the harmonic trigonometric functions without introducing additional rank deficiencies in the system matrices. Thus, the presented enriched overlapping finite element method (EOFEM) provides very precise eigenfrequencies and eigenmodes by using a very coarse mesh. However, the shape functions in the presented EOFEM do not possess the Kronecker delta function property. Hence, a conforming method that modifies the shape functions rather than the variational principle is proposed to impose the essential boundary conditions. Several numerical examples in two dimensions are then investigated to elaborate on the performance of the presented EOFEM in modal analyses. Besides, a block-diagonal lumped mass matrix is presented and further extended to the structure vibration analysis by incorporating the explicit time integration method. It is proven that the enriched overlapping element is computationally more efficient than the quadratic finite element and the original overlapping element.

Analysis of Defects and Harmonic Grid Generation in Domains with Angles and Cutouts

A survey of works concerning difficulties associated with harmonic grid generation in plane domains with angles and cutouts is given, and some new results are presented. It is well known that harmonic grids produced by standard methods in domains with cutouts or reentrant angles (i.e., interior angles greater than π) may contain defects, such as self-overlappings or exit beyond the domain boundary. It is established that, near the vertex of a reentrant angle, these defects follow from the asymptotics constructed for the underlying harmonic mapping, according to which the grid line leaving the angle vertex is tangent to one of the angle sides at the vertex (an effect referred to as “adhesion”), except for a special case. A survey of results is given for domains z of three types with angles or cutouts (L-shaped, horseshoe, and a domain with a rectangular cutout), for which standard methods for harmonic grid generation encounter difficulties. Applying the multipole method to such domains yields a harmonic mapping for them with high accuracy: the a posteriori error estimate of the mapping in the C(z) norm is 10–7 in the case of using 120 approximative functions.

Windows into Beethoven’s Lessons in Bonn: Kirnberger’s Die wahren Grundsätze zum Gebrauch der Harmonie (1773) and Vogler’s Gründe der Kuhrpfälzischen Tonschule in Beyspielen (1776/1778)

Beethoven’s lessons in Vienna with Haydn, Albrechtsberger, and Salieri are well known, but considerably less has been written about his earlier studies in Bonn. This article examines what Beethoven may have learned from two treatises that Gustav Nottebohm (Nottebohn 1873) connected to Beethoven’s Bonn manuscripts: Johann Philipp Kirnberger’s Die wahren Grundsätze zum Gebrauch der Harmonie (Kirnberger 1773) and Georg Joseph Vogler’s Gründe der Kuhrpfälzischen Tonschule in Beyspielen (1776, revised in 1778). I corroborate the evidence that links these treatises to Beethoven, analyze and categorize their contents, and suggest some parallels between materials in these treatises and Beethoven’s Bonn works including his “Elector” piano sonatas (WoO 47; 1783) and his two unusual preludes for piano or organ (op. 39; 1789, published in 1803). From what we know of Beethoven’s studies in Vienna, several pillars of standard eighteenth-century musical education are missing: the study of solfeggio, thoroughbass, and harmony. This article makes the case that Beethoven encountered this training in Bonn. From Kirnberger’s Grundsätze, he would have learned about the fundamental bass, harmonic function and progression, and the principles of prolongation. In Vogler’s book, he would have encountered solfeggio exercises, common thoroughbass patterns including the Rule of the Octave, invertible sequences, diminution patterns, modulations schemes to every key, the fundamental bass, and more. Although these two treatises were not the only books Beethoven likely studied in Bonn, they offer probable windows into his formative lessons in music theory, improvisation, and composition.

Study of the Symmetry Energy and the Nuclear Equation of State for 13O -13B and 13N-13C Mirror Nuclei

In parallel with the shell model using the harmonic oscillator's single-particle wave functions, the Hartree-Fock approximation was also used to calculate the neutron skin thickness, the mirror charge radii, and the differences in proton radii for 13O-13B and 13N-13C mirror nuclei. The calculations were done for both mirror nuclei in the psdpn model space. Depending on the type of potential used, the calculated values of skin thickness are affected. The symmetry energy and the symmetry energy's slope at nuclear saturation density were also determined, and the ratio of the density to the saturation density of nuclear matter and the symmetry energy has a nearly linear correlation. The mirror energy displacement was calculated, and the findings corresponded well with the available experimental data for the binding energies of the studied mirror nuclei. The measured values of the symmetry energy coefficient for the pair of mirror nuclei agreed with the computed ones, and this coefficient's value rises exponentially as the difference in charge radius increases.

Magnetotelluric Forward Modeling in Spherical Coordinate System Under Non-planar Wave Field Source

In the traditional magnetotelluric forward modeling, the spherical earth model is typically stretched into the rectangular coordinate system using geographical projection, inevitably producing modeling errors especially for cases with large-scale data coverage. In these cases, the earth can not be considered as be flat, and the external source is not laterally uniform. In this paper, we developed a MT modeling algorithm in the spherical coordinate system based on the use of the spherical harmonic function P10 for the source. A model with public solutions is used to verify our algorithm. A more realistic model is designed to show the application of our algorithm.

Estimate on the Bloch constant for certain harmonic mappings under a differential operator

Estimate on the Bloch constant for certain harmonic mappings under a differential operator