Harmonic Functions Research Articles

Defect pairs in nematic cell alignment on doubly connected domains

We consider the orientational order that arises from the alignment of anisotropic biological cells confined in bounded domains. Especially, it is known that topological defects, where the orientation angles are not well-defined, play an important role in the global pattern formation of cell alignment. Hence, it is important to investigate the relationship between the shape of the region and the location of the defects. The orientational order of cells is theoretically modelled as a harmonic function achieving the minimum elastic energy state of the nematic liquid crystal, and an analytical formula in simply connected domains is given by Miyazako & Nara (Miyazako, Nara 2022 R. Soc. Open Sci . 9, 211663 ( doi:10.1098/rsos.211663 )). In this paper, we extend this study to construct an analytic formula of cell angle in the doubly connected domains, where topological defects and boundary shapes are represented as the positions of logarithmic singularities and as conformal maps from a given region to a standard annular region, respectively. In addition, using this analytical formula, we propose a numerical algorithm to minimize elastic energy, from which we investigate cell alignment in doubly connected domains with anisotropic quadrilateral boundaries. Our numerical simulations suggest that pairs of topological defects tend to be located around corners of the boundaries, which will be a design principle of the boundary geometry for controlling the defect positions and cell alignment.

Harmonic and Monogenic Functions on Toroidal Domains

Harmonic and Monogenic Functions on Toroidal Domains

Almost sharp lower bound for the nodal volume of harmonic functions

AbstractThis paper focuses on a relation between the growth of harmonic functions and the Hausdorff measure of their zero sets. Let be a real‐valued harmonic function in with and . We prove where the doubling index is a notion of growth defined by This gives an almost sharp lower bound for the Hausdorff measure of the zero set of , which is conjectured to be linear in . The new ingredients of the article are the notion of stable growth, and a multiscale induction technique for a lower bound for the distribution of the doubling index of harmonic functions. It gives a significant imuprovement over the previous best‐known bound , which implied Nadirashvili's conjecture.

Модели составных гармонических полуволн и связь дискретизации времени с энтропией временных параметров сигналов

The problem of finding the absolute error of stepwise and linear interpolation of the control signal from uniform samples from it using models of composite harmonic half-waves is solved. Previously, during the inspection of the control object, the maximum values of the signal parameters and half-waves are determined: speed, acceleration and sharpness, there are no spectrum parameters. To determine the values of the intervals of uniform sampling of time, two groups of models of "harmonic half-waves" are considered. The first group of models is described by harmonic time functions whose parameters are consistent. The second group of models is described by composite harmonic functions of time, thereby the time parameters of the signals are consistent. It is proved that with an increase in the entropy of the maximum values of the signal parameters, the value of the time sampling interval increases without increasing the interpolation error. Thus, the entropy value of the signal parameters serves as an indicator of their inconsistency. The results of modeling and graphs obtained in the environment of the mathematical package are presented. The results are intended to optimize the loading of input tasks and primary information processing of processors in robust real-time automation systems, for example, used to control high-speed trains when braking in sliding or skidding mode.

Energy‐principle‐based analysis of elastic instability in polymer cylinders subject to surface tension

AbstractAn elastic instability model of polymer cylinder subject to surface tension is studied by energy functional analysis. The no‐classical boundary conditions and equilibrium equation are derived by introducing stress double harmonic function. The criteria of instability with surface tension contribution are deduced by substituting stress solution into boundary conditions. The results show that the criterion equation of elastic instability is relevant to the radius and intrinsic scale and instable wavelength or wavenumber of cylinder. Therefore, the relation between shear modulus and surface tension has the same trend comparing with the result of Barrierès' et al. This study provides insights into the instability of polymer and has important implications for the development of nanoscale cylindrical element in micro electro mechanical systems (MEMS) or nano electro mechanical systems (NEMS).

Stochastic response and stability analysis of nonlinear vehicle energy-regenerative suspension system with time delay

With the rapid development of new energy vehicles, a novel vehicle energy-regenerative suspension for simultaneous vibration suppression and energy harvesting has become one of the key development directions of the next generation of vehicle design. In order to improve the performance of suspension system, the time-delayed feedback control is applied to the novel suspension system subjected to different external excitation. At the same time, there are few studies related to the stochastic dynamics of nonlinear vehicle energy-regenerative suspension system with time delay. This paper studies stochastic response and stability of nonlinear vehicle energy-regenerative suspension with time-delayed feedback control under the excitation of Gaussian white noise. First of all, a stochastic dynamic model of the nonlinear vehicle energy-regenerative suspension with time delay system is established. Secondly, using the stochastic averaging method based on generalized harmonic function, the steady-state probability density function (PDF) of amplitude is obtained. The accuracy of the presented approach is verified through the comparison of analytical and numerical solutions. In addition, through combining the stochastic averaging method with the Khasminskii method, the top Lyapunov exponent (TLE) of the system is obtained, and then the influence of time delay and system parameters on the asymptotic Lyapunov stability is discussed. Finally, the effects of time delay and system parameters on system performance are also studied.

Linear measure of non-iterative dynamic system of harmonic functions

Linear measure of non-iterative dynamic system of harmonic functions

Harmonic functions with traces in Q type spaces related to weights

Harmonic functions with traces in Q type spaces related to weights

On the use of elliptic PDEs for the parameterisation of planar multipatch domains

AbstractThis paper presents a parameterisation framework based on (inverted) elliptic PDEs for addressing the planar parameterisation problem of finding a valid description of the domain’s interior given no more than a spline-based description of its boundary contours. The framework is geared towards isogeometric analysis (IGA) applications wherein the physical domain is comprised of more than four sides, hence requiring more than one patch. We adopt the concept of harmonic maps and propose several PDE-based problem formulations capable of finding a valid map between a convex parametric multipatch domain and the piecewise-smooth physical domain with an equal number of sides. In line with the isoparametric paradigm of IGA, we treat the parameterisation problem using techniques that are characteristic for the analysis step. As such, this study proposes several IGA-based numerical algorithms for the problem’s governing equations that can be effortlessly integrated into a well-developed IGA software suite. We augment the framework with mechanisms that enable controlling the parametric properties of the outcome. Parametric control is accomplished by, among other techniques, the introduction of a curvilinear coordinate system in the convex parametric domain, for which more general elliptic PDEs are adopted. Depending on the application, parametric control allows for building desired features into the computed map, such as homogeneous cell sizes or boundary layers.

Electro-mechanical analysis of piezoelectric sandwich structure

Piezoelectric sandwich structures are a fundamental construction in most piezoelectric devices. Accurately modeling their electro-mechanical response remains challenging due to complex coupling and interface phenomena. The Green’s function solution for electro-mechanical field of piezoelectric sandwich structures under a normal concentrated force, a basic case for common loadings, is derived using a new type of harmonic function and the Mirror Image Method. We investigate the distribution of the electro-mechanical field and the impact of material parameters and force direction on electromechanical components at the interfaces. This solution provides essential theoretical guidance for material selection and loading design in piezoelectric devices.

A new subclass of harmonic univalent functions associated with q-calculus

The purpose of the present paper is to introduce a new subclass of harmonic univalent functions by applying q-calculus. Coefficient inequalities, extreme points, distortion bounds, covering results, convolution condition and convex combination are determined for this class. Finally, we discuss a class preserving integral operator for this class.

Interior solution of azimuthally symmetric case of Laplace equation in orthogonal similar oblate spheroidal coordinates

Curvilinear coordinate systems distinct from the rectangular Cartesian coordinate system are particularly valuable in the field calculations as they facilitate the expression of boundary conditions of differential equations in a reasonably simple way when the coordinate surfaces fit the physical boundaries of the problem. The recently finalized orthogonal similar oblate spheroidal (SOS) coordinate system can be particularly useful for a physical processes description inside or in the vicinity of the bodies or particles with the geometry of an oblate spheroid. The solution of the azimuthally symmetric case of the Laplace equation was found for the interior space in the orthogonal SOS coordinates. In the frame of the derivation of the harmonic functions, the Laplace equation was separated by a special separation procedure. A generalized Legendre equation was introduced as the equation for the angular part of the separated Laplace equation. The harmonic functions were determined as relations involving generalized Legendre functions of the first and of the second kind. Several lower-degree functions are reported. Recursion formula facilitating determination of the higher-degree harmonic functions was found. The general solution of the azimuthally symmetric Laplace equation for the interior space in the SOS coordinates is reported.

Finite-dimensional perturbation of the Dirichlet boundary value problem for the biharmonic equation

Abstract The biharmonic equation is one of the important equations of mathematical physics, describing the behaviour of harmonic functions in higher-dimensional spaces. The main purpose of this study was to construct a finite-dimensional perturbation for the Dirichlet boundary value problem associated with the biharmonic equation. The methodological basis for this study was an integrated approach that includes mathematical analysis, algebraic methods, operator theory, and the theorem on the existence and uniqueness of a solution for a boundary value. The main tool is a finite-dimensional perturbation, which allows for examining the properties and behaviour of boundary value problems in as much detail as possible. In the study, descriptions of correctly solvable internal boundary value problems for a biharmonic equation in non-simply connected domains were considered in detail. The study is also devoted to the search for solutions and the analytical representation of resolvents of boundary value problems for a biharmonic equation in multi-connected domains. Within the framework of the study, theorems and their consequences were proved, and a finite-dimensional perturbation was constructed for the Dirichlet boundary value problem. Analytical representations of resolvents of boundary value problems for a biharmonic equation in multi-connected domains were also obtained. The examination of a finite-dimensional perturbation of the Dirichlet boundary value problem for a biharmonic equation has expanded the understanding of the properties of this equation in various contexts.

Vibration characteristics of a cylindrical sandwich shell with an FG auxetic honeycomb core and nanocomposite face layers subjected to supersonic fluid flow

In this paper, the flutter analysis is studied for a cylindrical sandwich shell subjected to external supersonic fluid flow. The sandwich shell consists of a re-entrant auxetic honeycomb (AH) core made of functionally graded materials (FGMs) which is covered with two polymeric face layers enriched with either carbon nanotubes (CNTs), graphene nanoplatelets (GNPs), or graphene oxide powders (GOPs). The volume fraction (percentage) of the ceramic phase in the functionally graded auxetic honeycomb (FGAH) core varies from zero at the inner surface to one at the outer one based on either power-law function (P-FGM), exponential function (E-FGM), or sigmoid function (S-FGM). It is assumed that the nanofillers are distributed uniformly inside the face layers. The first-order shear deformation theory (FSDT) and the piston theory are employed to provide the mathematical models of the shell and the aerodynamic pressure, respectively. The governing equations and boundary conditions are derived via Hamilton’s principle. An exact solution is performed in the circumferential direction via harmonic trigonometric functions (sine and cosine) and an approximate solution is performed in the axial direction utilizing the differential quadrature method (DQM). The natural frequencies and corresponding damping ratios are attained through the presented semi-analytical solution and the impacts of several factors on the natural frequencies and the critical aerodynamic pressure of the shell are examined. These factors the type of the FGM, the material index, and inclined angle of the cells in the FGAH core, the thickness of the FGAH core, the type and mass fraction of the nanofillers in the nanocomposite face layers, and boundary conditions.

The Liouville theorem for a class of Fourier multipliers and its connection to coupling

AbstractThe classical Liouville property says that all bounded harmonic functions in , that is, all bounded functions satisfying , are constant. In this paper, we obtain necessary and sufficient conditions on the symbol of a Fourier multiplier operator , such that the solutions to are Lebesgue a.e. constant (if is bounded) or coincide Lebesgue a.e. with a polynomial (if is polynomially bounded). The class of Fourier multipliers includes the (in general non‐local) generators of Lévy processes. For generators of Lévy processes, we obtain necessary and sufficient conditions for a strong Liouville theorem where is positive and grows at most exponentially fast. As an application of our results above, we prove a coupling result for space‐time Lévy processes.

Schwarz-Pick lemma for (α,β)-harmonic functions in the unit disc

We obtain Schwarz–Pick lemma for (α,β)-harmonic functions u in the disc, where α and β are complex parameters satisfying ℜα+ℜβ>−1. We prove sharp estimate of ‖Du(0)‖ for such functions in terms of Lp norm of the boundary function. Also, we give asymptotically sharp estimate of ‖Du(z)‖. This estimate is generalized to higher order derivatives. Our results extend earlier results for α-harmonic and Tα-harmonic functions.

SDUST2020MGCR: a global marine gravity change rate model determined from multi-satellite altimeter data

Abstract. Investigating the global time-varying gravity field mainly depends on GRACE/GRACE-FO gravity data. However, satellite gravity data exhibit low spatial resolution and signal distortion. Satellite altimetry is an important technique for observing the global ocean and provides many consecutive years of data, which enables the study of high-resolution marine gravity variations. This study aims to construct a high-resolution marine gravity change rate (MGCR) model using multi-satellite altimetry data. Initially, multi-satellite altimetry data and ocean temperature–salinity data from 1993 to 2019 are utilized to estimate the altimetry sea level change rate (SLCR) and steric SLCR, respectively. Subsequently, the mass-term SLCR is calculated. Finally, based on the mass-term SLCR, the global MGCR model on 5′ × 5′ grids (SDUST2020MGCR) is constructed by applying the spherical harmonic function method and mass load theory. Comparisons and analyses are conducted between SDUST2020MGCR and GRACE2020MGCR resolved from GRACE/GRACE-FO gravity data. The spatial distribution characteristics of SDUST2020MGCR and GRACE2020MGCR are similar in the sea areas where gravity changes significantly, such as the eastern seas of Japan, the western seas of the Nicobar Islands, and the southern seas of Greenland. The statistical mean values of SDUST2020MGCR and GRACE2020MGCR in global and local oceans are all positive, indicating that MGCR is rising. Nonetheless, differences in spatial distribution and statistical results exist between SDUST2020MGCR and GRACE2020MGCR, primarily attributable to spatial resolution disparities among altimetry data, ocean temperature–salinity data, and GRACE/GRACE-FO data. Compared with GRACE2020MGCR, SDUST2020MGCR has higher spatial resolution and excludes stripe noise and leakage errors. The high-resolution MGCR model constructed using altimetry data can reflect the long-term marine gravity change in more detail, which is helpful in studying seawater mass migration and its associated geophysical processes. The SDUST2020MGCR model data are available at https://doi.org/10.5281/zenodo.10701641 (Zhu et al., 2024).

An energy model for harmonic functions with junctions

We consider an energy model for harmonic graphs with junctions and study the regularity properties of minimizers and their free boundaries.

Recurrent and (strongly) resolvable graphs

We develop a new approach to recurrence and the existence of non-constant harmonic functions on infinite weighted graphs. The approach is based on the capacity of subsets of metric boundaries with respect to intrinsic metrics. The main tool is a connection between polar sets in such boundaries and null sets of paths. This connection relies on suitably diverging functions of finite energy.

Fourier-Hankel-Abel Nyquist-limited tomography: A spherical harmonic basis function approach to tomographic velocity-map image reconstruction.

A new method for the fully generalized reconstruction of three-dimensional (3D) photoproduct distributions from velocity-map imaging (VMI) projection data is presented. This approach, dubbed Fourier-Hankel-Abel Nyquist-limited TOMography (FHANTOM), builds on recent previous work in tomographic image reconstruction [C. Sparling and D. Townsend, J. Chem. Phys. 157, 114201 (2022)] and takes advantage of the fact that the distributions produced in typical VMI experiments can be simply described as a sum over a small number of spherical harmonic functions. Knowing the solution is constrained in this way dramatically simplifies the reconstruction process and leads to a considerable reduction in the number of projections required for robust tomographic analysis. Our new method significantly extends basis set expansion approaches previously developed for the reconstruction of photoproduct distributions possessing an axis of cylindrical symmetry. FHANTOM, however, can be applied generally to any distribution-cylindrically symmetric or otherwise-that can be suitably described by an expansion in spherical harmonics. Using both simulated and real experimental data, this new approach is tested and benchmarked against other tomographic reconstruction strategies. In particular, the reconstruction of photoelectron angular distributions recorded in a strong-field ionization regime-marked by their extensive expansion in terms of spherical harmonics-serves as a key test of the FHANTOM methodology. With the increasing use of exotic optical polarization geometries in photoionization experiments, it is anticipated that FHANTOM and related reconstruction techniques will provide an easily accessible and relatively low-cost alternative to more advanced 3D-VMI spectrometers.