ON THE LOWER BOUND FOR A CLASS OF HARMONIC FUNCTIONS IN THE HALF SPACE

The main objective is to derive a lower bound from an upper one for harmonic functions in the half space, which extends a result of B. Y. Levin from dimension 2 to dimension n 〉 2. To this end, we first generalize the Carleman＇s formula for harmonic functions in the half plane to higher dimensional half space, and then establish a Nevanlinna＇s representation for harmonic functions in the half sphere by using HSrmander＇s theorem.

Establishment of European Regional Ionosphere Model Based on Spherical Harmonic Functions

In order to study the temporal and spatial variation characteristics of the regional ionosphere and the modeling accuracy,the experiment is based on the spherical harmonic function model,using the GPS,Glonass,and Galileo dual-frequency observation data from the 305th-334th day of the European CORS network in 2019 to establish a global ionospheric model.By analyzing and evaluating the accuracy of the global ionospheric puncture points,VTEC,and comparing code products,the test results showed that the GPS system has the most dense puncture electricity distribution,the Glonass system is the second,and the Galileo system is the weakest.The values of ionospheric VTEC calculated by GPS,Glonass and Galileo are slightly different,but in terms of trends,they are the same as those of ESA,JPL and UPC.GPS data has the highest accuracy in global ionospheric modeling.GPS,Glonass and Galileo have the same trend,but Glonass data is unstable and fluctuates greatly.

NUMERICAL COMPARISON OF TWO BOUNDARY ELEMENT METHODS FOR PLANE HARMONIC FUNCTIONS

A direct boundary element method (BEM) has been studied in the paper based on a set of sufficient and necessary boundary integral equations (BIE) for the plane harmonic functions. The new sufficient and necessary BEM leads to accurate results while the conventional insufficient BEM will lead to inaccurate results when the conventional BIE has multiple solutions. Theoretical and numerical analyses show that it is beneficial to use the sufficient and necessary BEM, to avoid hidden dangers due to non-unique solution of the conventional BIE.

CONSTRUCTION SPACE HARMONIC FUNCTIONS IN POLYN0MIAL FORM AND SPHERICAL FUNCTIONS BY COMPLEX-FUNCTIONAL

In this paper, applying the theory of complex-functional, not only the spaceharmonic functions in polynomial form. but aIso the spherical functions are obtained.

Hlder Estimate of Harmonic Functions on a Class of p.c.f. Self-Similar Sets

In this paper we establish sharp H/51der estimates of harmonic functions on a class of connected post critically finite （p.c.f.） self-similar sets, and show that functions in the domain of Laplacian enjoy the same property. Some weU-known examples, such as the Sierpinski gasket, the unit interval, the level 3 Sierpinski gasket, the hexagasket, the 3-dimensional Sierpinski gasket, and the Vicsek set are also considered.

The Oscillation Inequality of Harmonic Functions on Post Critically Finite Self-Similar Sets

In this paper we establish the oscillation inequality of harmonic functions and HOlder estimate of the functions in the domain of the Laplacian on connected post critically finite （p.c.f.） self-similar sets.

Nonconstant Harmonic Functions on the Level 3 Sierpinski Gasket

We give a detailed description of nonconstant harmonic functions on the level 3 Sierpinski gasket. Then we extend the method onβ-set with 1/3〈β〈1/2.

A New Class of Harmonic Univalent Functions by the Generalized Salagean Operator

A complex-valued harmonic functions that are univalent and sense preserving in the unit disk U can be written in the form f = h ＋ g^-, where h and g are analytic in U. We define and investigate a new class SHPλ（α,β）by generalized Salagean operator of harmonic univalent functions. We give sufficient coefficient conditions for normalized harmonic functions in the class SHPλ（α,β） These conditions are also shown to be necessary when the coefficients are negative. This leads to distortion bounds and extreme points.

SINGULAR BOUNDARY PROPERTIES OF HARMONIC FUNCTIONS AND FRACTAL ANALYSIS

This paper shows an important relation between the fractal analysis and the boundary properties of harmonic functions.It is proved that the multifractal analysis of a finite measureμonIR^(d)determines the(non-tangential)boundary increasing properties ofPμ,the Poisson integral ofμwhich is harmonic onIR^(d+1)_(+).Some examples are given.

Unique Continuation on Quadratic Curves for Harmonic Functions

The unique continuation on quadratic curves for harmonic functions is dis-cussed in this paper.By using complex extension method,the conditional stability of unique continuation along quadratic curves for harmonic functions is illustrated.The nu-merical algorithm is provided based on collocation method and Tikhonov regularization.The stability estimates on parabolic and hyperbolic curves for harmonic functions are demonstrated by numerical examples respectively.

A New Class of Harmonic Univalent Functions

A complex-valued harmonic function that is univalent and sense preserving in the unit disk U can be written in the form of f = h + g,where h and g are analytic in U.We define and investigate a new class LH_λ(α,β) by generalized Salagean operator of harmonic univalent functions.We give sufficient coefficient conditions for normalized harmonic functions in the class LH_λ(α,β).These conditions are also shown to be necessary when the coefficients are negative.This leads to distortion bounds and extreme points.

Boundary behavior of harmonic functions on metric measure spaces with non-negative Ricci curvature

Let(X,d,μ)be a metric measure space with non-negative Ricci curvature.This paper is concerned with the boundary behavior of harmonic function on the(open)upper half-space X×R_(+).We derive that a function f of bounded mean oscillation(BMO)is the trace of harmonic function u(x,t)on X×R_(+),u(x,0)=f(x),whenever u satisfies the following Carleson measure condition supx_(B),r_(B) ∫_(0)^(r_(B) ) f_(B(x_(B),r_(B))) |t■u(x,t)|^(2)dμ(x)dt/t≤C<∞,where ■=(■_(x),■_(t))denotes the total gradient and B(x_(B),r_(B)) denotes the(open)ball centered at x_(B) with radius r_(B).Conversely,the above condition characterizes all the harmonic functions whose traces are in BMO space.

BoundaryValues of Generalized Harmonic Functions Associated with the Rank-One Dunkl Operator

We consider the local boundary values of generalized harmonic functions associated with the rank-one Dunkl operator D in the upper half-plane R2+=R×(0,∞),where(Df)(x)=f′0(x)+(λ/x)[f(x)-f(-x)]for givenλ≥0.A C2 function u in R2+is said to beλ-harmonic if(D2x+■2y)u=0.For aλ-harmonic function u in R2+and for a subset E of■R2+=R symmetric about y-axis,we prove that the following three assertions are equivalent:(i)u has a finite non-tangential limit at(x,0)for a.e.x∈E;(ii)u is non-tangentially bounded for a.e.x∈E;(iii)(Su)(x)<∞for a.e.x∈E,where S is a Lusin-type area integral associated with the Dunkl operator D.

Growth of certain harmonic functions in an n-dimensional cone

We give the growth properties of harmonic functions at infinity in a cone, which generalize the results obtained by Siegel-Talvila.

On the accuracy of the potential harmonic functions

The matrix elements of the correlation function between symmetric potential harmonics am simplified into the analytical summations of the grand angular momenta by dy using the recurrence and coupling relations of the potential harmonics. 'The correlation-function potential-harmonic and generalized-Laguerre-function method (CFPHGLF), recently developed by us, is applied to the S states of the helium-like systems for Z = 2 to 9. The results exhibit good convergence with the bases in tern of both the angular and radial directions. The final eigen-energies agree excellently with the best s-limits of the variational configuration interaction (CI) method for the involved low-lying S states. The accuracy of the potential harmonic (PH) expansion scheme is discussed relative to the exact Hylleraas CI results (HCI), and Hartree-Fock results. Moreover, suggestion is given for the future improvement of the PH scheme.

Several Hermite-Hadamard Type Inequalities for Harmonically Convex Functions in the Second Sense with Applications

In this paper, we first introduce the concept ＂harmonically convex functions＂ in the second sense and establish several Hermite-Hadamard type inequalities for harmonically convex functions in the second sense. Finally, some applications to special mean are shown.

INTEGRATION OF A ZONAL FUNCTION RELATED TO THE LOCAL HARMONIC ANALYSIS ON SPHERES

In studying Local harmonic analysis on the sphere S',R.S.Strichartz introduced certain zonab functionsid(x,y))which satisfy the equation

RIEMANN BOUNDARY VALUE PROBLEMS FOR SOME K-REGULAR FUNCTIONS IN CLIFFORD ANALYSIS

In this paper,we study the R m（m〉0） Riemann boundary value problems for regular functions,harmonic functions and bi-harmonic functions with values in a universal clifford algebra C（Vn,n）.By using Plemelj formula,we get the solutions of R m（m〉0） Riemann boundary value problems for regular functions.Then transforming the Riemann boundary value problems for harmonic functions and bi-harmonic functions into the Riemann boundary value problems for regular functions,we obtain the solutions of R m（m〉0） Riemann boundary value problems for harmonic functions and bi-harmonic functions.

The Jaffa Transform for Hessian Matrix Systems and the Laplace Equation

Hessian matrices are square matrices consisting of all possible combinations of second partial derivatives of a scalar-valued initial function. As such, Hessian matrices may be treated as elementary matrix systems of linear second-order partial differential equations. This paper discusses the Hessian and its applications in optimization, and then proceeds to introduce and derive the notion of the Jaffa Transform, a new linear operator that directly maps a Hessian square matrix space to the initial corresponding scalar field in nth dimensional Euclidean space. The Jaffa Transform is examined, including the properties of the operator, the transform of notable matrices, and the existence of an inverse Jaffa Transform, which is, by definition, the Hessian matrix operator. The Laplace equation is then noted and investigated, particularly, the relation of the Laplace equation to Poisson’s equation, and the theoretical applications and correlations of harmonic functions to Hessian matrices. The paper concludes by introducing and explicating the Jaffa Theorem, a principle that declares the existence of harmonic Jaffa Transforms, which are, essentially, Jaffa Transform solutions to the Laplace partial differential equation.

Accurate estimation of Li/Ni mixing degree of lithium nickel oxide cathode materials

Li/Ni mixing negatively influences the discharge capacity of lithium nickel oxide and high-nickel ternary cathode materials.However,accurately measuring the Li/Ni mixing degree is difficult due to the preferred orientation of labbased XRD measurements using Bragg–Brentano geometry.Here,we find that employing spherical harmonics in Rietveld refinement to eliminate the preferred orientation can significantly decrease the measurement error of the Li/Ni mixing ratio.The Li/Ni mixing ratio obtained from Rietveld refinement with spherical harmonics shows a strong correlation with discharge capacity,which means the electrochemical capacity of lithium nickel oxide and high-nickel ternary cathode can be estimated by the Li/Ni mixing degree.Our findings provide a simple and accurate method to estimate the Li/Ni mixing degree,which is valuable to the structural analysis and screening of the synthesis conditions of lithium nickel oxide and high-nickel ternary cathode materials.