The Higher Derivatives of Bianalytic Functions

By using the Cauchy integral formula of bianalytic functions, the formula of higher derivatives of bianalytic functions and Weierstrass Theorem are obtained.

HIGHER RADIAL DERIVATIVE OF BLOCH TYPE FUNCTIONS

In this paper, The integral characterizations of alpha-Bloch (little alpha-Bloch) axe given in terms of higher radial derivative, and their characterizations of Caxleson type measure are obtained.

On the Variational Problems of the Functionals with Derivatives of Higher Orders and Undetermined Boundary

According to the necessary condition of the functional taking the extremum, that is its first variation is equal to zero, the variational problems of the functionals for the undetermined boundary in the calculus of variations are researched, the functionals depend on single argument, arbitrary unknown functions and their derivatives of higher orders. A new view point is posed and demonstrated, i.e. when the first variation of the functional is equal to zero, all the variational terms are not independent to each other, and at least one of them is equal to zero. Some theorems and corollaries of the variational problems of the functionals are obtained.

Nonlinear Jordan Higher Derivations of Triangular Algebras

In this paper, we prove that any nonlinear Jordan higher derivation on triangular algebras is an additive higher derivation. As a byproduct, we obtain that any nonlinear Jordan derivation on nest algebras over infinite dimensional Hilbert suaces is inner.

Lie Higher Derivations on Nest Algebras

Let N be a nest on a Banach space X, and Alg N be the associated nest algebra. It is shown that if there exists a non-trivial element in N which is complemented in X, then D = （Ln）n∈N is a Lie higher derivation of AlgAl if and only if each Ln has the form Ln（A） ： Tn（A） ＋ hn（A）I for all A ∈ AlgN, where （Tn）n∈N is a higher derivation and （hn）n∈N is a sequence of additive functionals satisfying hn（[A,B]） = 0 for all A,B ∈ AlgN and all n ∈ N.

A REFINEMENT OF THE SCHWARZ-PICK ESTIMATES AND THE CARATHéODORY METRIC IN SEVERAL COMPLEX VARIABLES

In this article,we first establish an asymptotically sharp result on the higher order Fréchet derivatives for bounded holomorphic mappings f(x)=f(0)+∞∑s=1Dskf(0)(x^(sk))/(sk)!:B_(X)→B_(Y),where B_X is the unit ball of X.We next give a sharp result on the first order Fréchet derivative for bounded holomorphic mappings F(X)=F(0+)∞∑s=KD^(s)f(0)(x^(8)/s!):B_(X)→B_(Y),where B_(X)is the unit ball of X.The results that we derive include some results in several complex variables,and extend the classical result in one complex variable to several complex variables.

A Kind of Identities Involving Complete Bell Polynomials

The purpose of this paper is to establish a formula of higher derivative by Faà di Bruno formula, and apply it to some known results to get some identities involving complete Bell polynomials.

Rayleigh reciprocity relations： Applications

Classical reciprocity relations have wide applications in acoustics, from field representation to generalized optical theorem. In this paper we introduce our recent results on the applications and generalization of classical Rayleigh reciprocity relation： higher derivative reciprocity relations as a generalization of the classical one and a theoretical proof on the Green＇s function retrieval from volume noises.

A High Order Spectral Volume Formulation for Solving Equations Containing Higher Spatial Derivative Terms II:Improving the Third Derivative Spatial Discretization Using the LDG2 Method

In this paper,the second in a series,we improve the discretization of the higher spatial derivative terms in a spectral volume(SV)context.The motivation for the above comes from[J.Sci.Comput.,46(2),314–328],wherein the authors developed a variant of the LDG(Local Discontinuous Galerkin)flux discretization method.This variant(aptly named LDG2),not only displayed higher accuracy than the LDG approach,but also vastly reduced its unsymmetrical nature.In this paper,we adapt the LDG2 formulation for discretizing third derivative terms.A linear Fourier analysis was performed to compare the dispersion and the dissipation properties of the LDG2 and the LDG formulations.The results of the analysis showed that the LDG2 scheme(i)is stable for 2nd and 3rd orders and(ii)generates smaller dissipation and dispersion errors than the LDG formulation for all the orders.The 4th order LDG2 scheme is howevermildly unstable:as the real component of the principal eigen value briefly becomes positive.In order to circumvent the above,a weighted average of the LDG and the LDG2 fluxes was used as the final numerical flux.Even a weight of 1.5%for the LDG(i.e.,98.5%for the LDG2)was sufficient tomake the scheme stable.Thisweighted scheme is still predominantly LDG2 and hence generated smaller dissipation and dispersion errors than the LDG formulation.Numerical experiments are performed to validate the analysis.In general,the numerical results are very promising and indicate that the approach has a great potential for higher dimension Korteweg-de Vries(KdV)type problems.

A High Order Spectral Volume Formulation for Solving Equations Containing Higher Spatial Derivative Terms:Formulation and Analysis for Third Derivative Spatial Terms Using the LDG Discretization Procedure

In this paper,we develop a formulation for solving equations containing higher spatial derivative terms in a spectral volume(SV)context;more specifically the emphasis is on handling equations containing third derivative terms.This formulation is based on the LDG(Local Discontinuous Galerkin)flux discretization method,originally employed for viscous equations containing second derivatives.A linear Fourier analysis was performed to study the dispersion and the dissipation properties of the new formulation.The Fourier analysis was utilized for two purposes:firstly to eliminate all the unstable SV partitions,secondly to obtain the optimal SV partition.Numerical experiments are performed to illustrate the capability of this formulation.Since this formulation is extremely local,it can be easily parallelized and a h-p adaptation is relatively straightforward to implement.In general,the numerical results are very promising and indicate that the approach has a great potential for higher dimension Korteweg-de Vries(KdV)type problems.

Local Well-posedness of the Derivative Schrödinger Equation in Higher Dimension for Any Large Data

In this paper,the authors consider the local well-posedness for the derivative Schrödinger equation in higher dimension ut-iΔu+|u|^(2)(→γ·▽u)+u^(2)(→λ·▽-u)=0,(x,t)∈R^(n)×R,→γ,→λ∈R^(n);n≥2 It is shown that the Cauchy problem of the derivative Schrödinger equation in higher dimension is locally well-posed in H^(s)(R^(n))(s>n/2)for any large initial data.Thus this result can compare with that in one dimension except for the endpoint space H^(n/2).

Jordan Higher Derivable Maps on Triangular Algebras by Commutative Zero Products

In this paper, the structure of Jordan higher derivable maps on triangular algebras by commutative zero products is given. As an application, the form of Jordan higher derivable maps of nest algebras by commutative zero products is obtained.

Generalized*-Lie Higher Derivable Mappings on*-Rings

Let R be a*-ring with the center Z(R)and N be the set of nonnegative integers.In this paper,it is shown that if R contains a nontrivial self-adjoint idempotent which admits a generalized Lie higher derivable mapping△={G_(n)}_(n∈N)associated with a*-Lie higher derivable mapping L={L_(n)}_(n∈N),then for any X,Y in R and for each n in N there exists an element Z_(X,Y)(depending on X and Y)in the center Z(R)such that G_(n)(X+Y)=G_(n)(X)+G_(n)(Y)+Z_(X,Y).

Characterization of Lie Higher Derivations on Triangular Algebras

Let A and B be unital rings, and M be an （A, B）-bimodule, which is faithful as a left A-module and also as a right B-module. Let U = Tri（A,M, B） be the triangular algebra. In this paper, we give some different characterizations of Lie higher derivations on U.

A Note on the Stability of Jordan Triple Higher Ring Derivations

In the present paper, we establish the stability and the superstability of a functional inequality corresponding to the functional equation fn（xyx） = ∑i＋j＋k=n fi（x）fj （y）fk（x）. In addition, we take account of the problem of Jacobson radical ranges for such functional inequality.

The Refined Schwarz-Pick Estimates for Positive Real Part Holomorphic Functions in Several Complex Variables

In this article,the refined Schwarz-Pick estimates for positive real part holomorphic functions p(x)=p(0)+Σ_(m=k)^(∞)D^(M)p(0)(x^(m))/m!:G→Care given,where k is a positive integer,and G is a balanced domain in complex Banach spaces.In particular,the results of first order Fréchet derivative for the above functions and higher order Frechet derivatives for positive real part holomorphic functions p(x)=p(0)+Σ_(s=1)^(∞)D^(sk)p(0)(x^(sk))/(sk)!:G→Care sharp for G=B,where B is the unit ball of complex Banach spaces or the unit ball of complex Hilbert spaces.Their results reduce to the classical result in one complex variable,and generalize some known results in several complex variables.