Harmonic Functions Research Articles

Harmonic maps between pseudo-Riemannian surfaces

We study locally harmonic maps between a Riemann surface or a Lorentz surface M and a Riemann or Lorentz surface N. All four cases are written using a unified formalism. Therefore properties and solutions to the harmonic map problem can be studied in a unified way.It is known that harmonic maps between Riemannian surfaces are classified by the classification of the solutions of a sinh-Gordon equation. We extend this result to all the four cases of harmonic maps between Riemannian or pseudo-Riemannian surfaces. The calculation of the corresponding harmonic map can be calculated by the solutions of the corresponding Beltrami equations in all the cases.We study the one-soliton solutions of this equation and we find the corresponding harmonic maps in a unified way.Next, we discuss a Bäcklund transformation of the harmonic map equations that provides a connection between the solutions of two sine or sinh-Gordon type equations. Finally, we give an example of a harmonic map that is constructed by the use of a Bäcklund transformation.

A proof of Guo-Wang’s conjecture on the uniqueness of positive harmonic functions in the unit ball

A proof of Guo-Wang’s conjecture on the uniqueness of positive harmonic functions in the unit ball

Application of B-spline basis functions as harmonic functions for the concurrent shape and mesh morphing of airfoils

The automatic deformation of the computational mesh along with the deformed geometry in design optimization cycles is a valuable procedure, as it reduces the required time for the construction of new meshes. The introduction of harmonic coordinates for the deformation of objects included within a closed mesh (cage) has been introduced in computer graphics. Harmonic coordinates result from solutions to the Laplace’s equation (harmonic functions) using a numerical solver. In this work, a modification to the classical harmonic coordinates’ concept is introduced for the deformation of 2D geometries (and the corresponding computational mesh) which are defined as B-spline curves. The B-spline basis functions are used as harmonic functions along the mesh boundary, being also the geometry to be deformed. Thus, any deformation of the B-spline boundary, through the movement of the curve’s control points, can be successfully propagated to the interior of the computational domain, resulting in the concurrent and conformable modification of the B-spline boundary and the entire computational mesh. For the computation of harmonic coordinates a node-centered Finite-Volume based Laplace solver for unstructured meshes is used, enhanced with an agglomeration multigrid scheme. The proposed method is applied and assessed for the shape and mesh morphing of airfoils.

Exploring new fuzzy fractional integral operators with applications over fuzzy number convex and harmonic convex mappings

Exploring new fuzzy fractional integral operators with applications over fuzzy number convex and harmonic convex mappings

Dirac Eigenvalue Optimisation and Harmonic Maps to Complex Projective Spaces

Abstract Consider a Dirac operator on an oriented compact surface endowed with a Riemannian metric and spin structure. Provided the area and the conformal class are fixed, how small can the $k$-th positive Dirac eigenvalue be? This problem mirrors the maximization problem for the eigenvalues of the Laplacian, which is related to the study of harmonic maps into spheres. We uncover the connection between the critical metrics for Dirac eigenvalues and harmonic maps into complex projective spaces. Using this approach we show that for many conformal classes on a torus the first nonzero Dirac eigenvalue is minimised by the flat metric. We also present a new geometric proof of Bär’s theorem stating that the first nonzero Dirac eigenvalue on the sphere is minimised by the standard round metric.

On prescribing the number of singular points in a Cosserat-elastic solid

Abstract In a geometrically non-linear Cosserat model for micro-polar elastic solids, we prove that critical points of the Cosserat energy functional with an arbitrary large (finite) number of singularities do exist, whereas Cosserat energy minimizers are known to be locally Hölder continuous. To reach that goal, we first develop a technique to insert dipole pairs of singularities into smooth maps while controlling the amount of Cosserat energy needed to do so. We then use this method to force an arbitrary number of singular points into (weak) Cosserat-elastic solids by prescribing smooth boundary data. The boundary data themselves are given in such a way, that they contain no topological obstruction to regularity. Throughout this paper, we often exploit connections between harmonic maps and Cosserat-elastic solids, so that we are able to adapt and incorporate ideas of R. Hardt and F.-H. Lin for harmonic maps with singularities, as well as of F. Béthuel for dipole pairs of singularities.

On manifolds with nonnegative Ricci curvature and the infimum of volume growth order < 2

Abstract We study open (complete and noncompact) n-manifolds M with nonnegative Ricci curvature, under the condition that any asymptotic cone of M splits off an ℝ k {\mathbb{R}^{k}} factor. In particular, we obtain two rigidity results for open n-manifolds M with Ric M ≥ 0 {\mathrm{Ric}_{M}\geq 0} and the infimum of volume growth order < 2 {<2} . The first result asserts that there exists a nonconstant linear growth harmonic function on M if and only if M is isometric to the metric product ℝ × N {\mathbb{R}\times N} for some compact manifold N. The second asserts that the Riemannian universal cover of M has Euclidean volume growth if and only if M is flat with an n - 1 {n-1} dimensional soul.

A Survey on Schwarz Lemma and Kobayashi Metrics for Harmonic and Holomorphic Functions

M. Mateljević [34] mainly consider various version of Schwarz lemma and its relatives related to harmonic and holomorphic functions including several variables. His methods (results) unify very recent approaches. This considerations include domains on which he can compute Kobayashi-Finsler norm. We intend do to a light application on the methods of [34].

Simulation of Motion Nonlinear Error Compensation of CNC Machine Tools With Multi-Axis Linkage

In order to solve the problem of nonlinear error for a dual rotary table five-axis CNC machine tool due to the linkage of rotary and translational axes, the simulation of motion nonlinear error compensation for a multi-axis linkage CNC machine tool is proposed. The adjacent points in the tool position file are selected as the tool position points for building the model, and then the nonlinear error model resolved by the harmonic function is established according to the error distribution in the classical post-processing. The nonlinear error between the two tool position points is quickly predicted by the analytical expression of this model, and the real-time error compensation of the intermediate interpolation points is realized. Finally, MALTLAB simulation analysis is performed on the tool position file of an impeller part machining to verify the effectiveness of the proposed algorithm. The experimental results show that it can be seen from the distribution curve of the nonlinear error that it is about 10% after compensation as before compensation, thus verifying the effectiveness of the nonlinear error compensation mechanism. The correctness of the nonlinear error analysis and compensation method and the effectiveness of post-processing are verified.

FHLight: A novel method of indoor scene illumination estimation using improved loss function

In augmented reality tasks, especially in indoor scenes, achieving illumination consistency between virtual objects and real environments is a critical challenge. Currently, mainstream methods are illumination parameters regression and illumination map generation. Among these two categories of methods, few works can effectively recover both high-frequency and low-frequency illumination information within indoor scenes. In this work, we argue that effective restoration of low-frequency illumination information forms the foundation for capturing high-frequency illumination details. In this way, we propose a novel illumination estimation method called FHLight. Technically, we use a low-frequency spherical harmonic irradiance map (LFSHIM) restored by the low-frequency illumination regression network (LFIRN) as prior information to guide the high-frequency illumination generator (HFIG) to restore the illumination map. Furthermore, we suggest an improved loss function to optimize the network training procedure, ensuring that the model accurately restores both low-frequency and high-frequency illumination information within the scene. We compare FHLight with several competitive methods, and the results demonstrate significant improvements in metrics such as RMSE, si-RMSE, and Angular error. In addition, visual experiments further confirm that FHLight is capable of generating scene illumination maps with genuine frequencies, effectively resolving the illumination consistency issue between virtual objects and real scenes. The code is available at https://github.com/WA-tyro/FHLight.git.

Harmonic extension through cylindrical and conical surfaces

This article describes recent progress concerning the extension of harmonic functions which vanish on a cylindrical or conical surface. The results, which are based on detailed analyses of various expansions of Green functions, can be viewed as partial generalizations of the well-known Schwarz reflection principle to surfaces other than hyperplanes and spheres. Some unexpected features emerge which suggest avenues for further exploration.

Cross-sectional zero-dimension temperature model for thin-walled circular tubes in space environment

A sufficiently accurate, yet computationally efficient prediction of temperature field is essential for design of spacecraft structures. This paper presents a model dimension reduction method for thermal analysis of thin-walled circular tubes in space environment, which takes into account radiation heat transfer among the internal surfaces besides heat conduction along the circumferential direction and radiative emission from the external surface. Temperature distribution on the tube cross section is approximated by a series of harmonic functions, so that a one-dimensional problem is reduced to that of zero dimension. The relation between average and perturbation temperatures that depend only on time is broadened to fully coupled one. By comparison to the previous model and the plane finite element model, the accuracy and economy of the new model are illustrated. The results show that internal radiation exchange plays an important role in thermal analysis of thin-walled circular tubes used extensively in spacecraft appendages. Furthermore, differences between present and previous models are analyzed and a two-way analysis of variance is performed to determine the effect of various physical and geometric parameters on the temperature distribution and response of the tubes. The work can be further developed to analyze thermally induced deformation and vibration of spacecraft structures.

New subclass of meromorphic harmonic functions defined by symmetric q-calculus and domain of Janowski functions

In this article, by applying the convolution principle and symmetric q-calculus, we develop a new generalized symmetric q-difference operator of convolution type, which is applicable in the domain E⁎={τ:τ∈C and 0<|τ|<∞}. Utilizing this operator, we construct, analyze, and evaluate two new sets of meromorphically harmonic functions in the Janowski domain. Furthermore, we investigate the convolution properties and necessary conditions for a function F to belong to the class MSHP,R(q,q−1), examining the sufficiency conditions for F to satisfy these properties. Moreover, we examine key geometric properties of the function F in the class MSH‾P,R(q,q−1), including the distortion bound, convex combinations, the extreme point theorem, and weighted mean estimates.

Certain Subclass of Harmonic Multivalent Functions Defined by New Linear Operator

The main goal of the present paper is to introduce a new class of harmonic multivalent functions defined by a new linear operator in the open unit disc . Thus, some geometric properties have examined, including coefficient inequality, extreme points, convolution conditions, convex linear combinations, and integral transforms for the class .

A Study on Harmonic Functions

The class of functions that have bounded boundary rotation and bounded radius rotation are the generalization of the convex and the starlike functions, respectively. The concept of such functions was introduced by Löwner [1]. But he did not use the present terminology. It was Paatero [2, 3] who systematically developed their properties and made an exhaustive study of the class of functions that have bounded boundary rotation. We will examine in this paper, the subclass of $${\mathcal S}_{H}$$ that is related to the class of functions that have bounded radius rotation.

Hermite–Hadamard-type inequalities for multiplicative harmonic s -convex functions

UDC 517.5 We study the concept of multiplicative harmonic s -convex functions and establish Hermite–Hadamard integral inequalities for this class of functions. Furthermore, we derive a set of Hermite–Hadamard-type inequalities applicable to the product and quotient of multiplicative harmonic s -convex functions. In addition, we deduce new inequalities involving multiplicative integrals for the product and quotient of harmonic convex and multiplicative harmonic s -convex functions. Some results for the class of multiplicative harmonic convex functions are obtained as special cases of our results. The obtained results are verified by providing examples with included graphs.

On Cauchy Problem Solution for a Harmonic Function in a Simply Connected Domain with Multi-Component Boundary

Here, we present an investigation of the Cauchy problem solvability for the Laplace equation in a simply connected plane domain with a multi-component boundary. We reduce our investigation to solution of two singular integral equations. If the problem is resolvable, then we restore its solution via the integral Cauchy formula. We present the examples of the solvable problems. The construction involves an auxiliary approximate conformal mapping or the solution of certain singular integral equations.

On Removable Singularities of Harmonic Functions on a Stratified Set

On Removable Singularities of Harmonic Functions on a Stratified Set

Interior noise characteristics, source identification and its quantification contribution of 400 km/h high-speed train in different operating environments

Noise level is one of the core technical indicators of high-speed trains, especially for a speed of 400 km/h. Noise source identification and its contribution quantification are key for interior noise control. This article takes a certain high-speed train running up to 400 km/h as the research object. Firstly, based on the interior noise spectrum and experimental results of source identification based on spherical harmonic function, the main source locations and energy distribution inside the train are determined. Secondly, through polynomial fitting, the correlation between vibration and noise in various areas inside and outside the train and speed is determined. Subsequently, by calculating the energy contribution through area integration, the noise contribution of each interior region is determined, and then the variation law of the noise contribution of each region with the speed is determined, and the quantitative contributions of various areas inside the train are given when the high-speed train is running at 400 km/h. Finally, the vibration and noise inside the train, aerodynamic noise on the train body surface, and noise in the bogie area in open lines and tunnel operating environments are compared and analyzed, as well as their variation laws with speed. The main noise sources inside the train under two types of operating environments are identified, and the contribution rates of noise sources in different areas inside the train are analyzed, further studying the interior noise and vibration transmission characteristics. The results show that, when the high-speed train is running at 400 km/h in open line operating environment, the significant frequency range of interior noise is 40 Hz ∼ 2000 Hz, and dominant interior noise sources are located in the roof and floor. In the tunnel operating environment, the significant frequency range of interior noise is 160 Hz ∼ 1000 Hz, and the primary sources of interior noise are predominantly located in the left window and floor. For the sidewall area, the interior noise comes mainly from the vibration of the inner sidewall when in open line operating environment, while in tunnel operating environment, the interior noise comes mainly from the aerodynamic excitation of the body surface.

A nonvariational form of the Neumann problem for the Poisson equation

We present a nonvariational setting for the Neumann problem for the Poisson equation for solutions that are Hölder continuous and that may have infinite Dirichlet integral. We introduce a distributional normal derivative on the boundary for the solutions that extends that for harmonic functions which has been introduced in a previous paper and we solve the nonvariational Neumann problem for data in the interior with a negative Schauder exponent and for data on the boundary that belong to a certain space of distributions on the boundary.