The tangential k-Cauchy-Fueter type operator and Penrose type integral formula on the generalized complex Heisenberg group

The tangential k-Cauchy-Fueter operator and k-CF functions are counterparts of the tangential Cauchy–Riemann operator and CR functions on the Heisenberg group in the theory of several complex variables,respectively.In this paper,we introduce a Lie group that the Heisenberg group can be imbedded into and call it generalized complex Heisenberg.We investigate quaternionic analysis on the generalized complex Heisenberg.We also give the Penrose integral formula for k-CF functions and construct the tangential k-Cauchy-Fueter complex.

On the Method of Solution for the Non-Homogeneous Generalized Riemann-Hilbert Boundary Value Problems

This paper studies the non-homogeneous generalized Riemann-Hilbert(RH)problems involving two unknown functions.Using the uniformization theorem,such problems are transformed into the case of homogeneous type.By the theory of classical boundary value problems,we adopt a novel method to obtain the sectionally analytic solutions of problems in strip domains,and analyze the conditions of solvability and properties of solutions in various domains.

LEIBNIZ′FORMULA OF GENERALIZED DIFFERENCE WITH RESPECT TO A CLASS OF DIFFERENTIAL OPERATORS AND RECURRENCE FORMULA OF THEIR GREEN′S FUNCTION

In this paper, Leibniz' formula of generalized divided difference with respect to a class of differential operators whose basic sets of solutions have power form, is considered. The recurrence formula of Green function about the operators is also given.

A GENERALIZED GAUSS-TYPE QUADRATURE FORMULA AND ITS APPLICATIONS TO PSEUDOSPECTRAL METHOD

A generalized Gauss-type quadrature formula is introduced, which assists in selection of collocation points in pseudospectral method for differential equations with two-point derivative boundary conditions. Some results on the related Jacobi interpolation are established. A pseudospectral scheme is proposed for the Kuramoto-Sivashisky equation. A skew symmetric decomposition is used for dealing with the nonlinear convection term. The stability and convergence of the proposed scheme are proved. The error estimates are obtained. Numerical results show the efficiency of this approach.

A generalized Collins formula derived by virtue of the displacement-squeezing related squeezed coherent state representation

Based on the displacement-squeezing related squeezed coherent state representation ｜z〉g and using the technique of integration within an ordered product of operators, this paper finds a generalized Fresnel operator, whose matrix element in the coordinate representation leads to a generalized Collins formula （Huygens-Fresnel integration transformation describing optical diffraction）. The generalized Fresnel operator is derived by a quantum mechanical mapping from z to sz -- rz^＊ in the ｜Z〉g representation, while ｜z〉g in phase space is graphically denoted by an ellipse.

Operator Formulas Involving Generalized Stirling Number Derived by Virtue of Normal Ordering of Vacuum Projector

By virtue of the normal ordering of vacuum projector we directly derive some new complicated operatoridentities, regarding to the generalized Stirling number.

Derivation of the Gutenberg-Richter empirical formula from the solution of the generalized logistic equation

We have written a new equation to study the statistics of earthquake distributions. We call this equation “the generalized logistic equation”. The Gutenberg-Richter frequency-magnitude formula was derived from the solution of the generalized logistic equation as an asymptotic case for the approximation of large magnitudes. To illustrate how the solution of the generalized logistic equation works, it was used to approximate the observed cumulative distribution of earthquakes in four different geological provinces: the Central Atlantic (40N - 25N, 5W - 35W), Canary Islands, Magellan Mountains (20N - 9S, 148E - 170E), and the Sea of Japan. This approximation showed an excellent correlation between the theoretical curves and observed data for earthquakes of magnitudes 1 < m < 9.

The SMW Formula for Bounded Homogeneous Generalized Inverses with Applications

In this paper, we present an extension of the so-called classical Sherman-Morrison-Woodbury (for short SMW) formula for bounded homogeneous generalized inverse in Banach spaces. Some particular cases and applications will be also considered. Our results generalize the results of many authors for finite dimensional matrices and Hilbert space operators in the literature.

Construction of Recursion Formulas for Lax Pairs of (2+1)-Dimensional Equation: Modified Generalized Dispersive Long Wave Equation

In this article, we study the Lax pairs of -dimensional equation: the modified generalized dispersive long wave (MGDLW) equation. Based on the well-known binary Darboux transformation, we dig out the recursion formulas of the first part of the Lax pairs. Then by further discussion and doing some revisional work, we make the recursion formulas fit for the second part of Lax pairs. At last, some solutions to the MGDLW equation are worked out by using the recursion formula.

Green Function of Generalized Time Fractional Diffusion Equation Using Addition Formula of Mittag-Leffler Function

In this paper, we use the Mittag-Leffler addition formula to solve the Green function of generalized time fractional diffusion equation in the whole plane and prove the convergence of the Green function.

A new generalized fractional Dirac soliton hierarchy and its fractional Hamiltonian structure

Based on the differential forms and exterior derivatives of fractional orders, Wu first presented the generalized Tu formula to construct the generalized Hamiltonian structure of the fractional soliton equation. We apply the generalized Tu formula to calculate the fractional Dirac soliton equation hierarchy and its Hamiltonian structure. The method can be generalized to the other fractional soliton hierarchy.

On the generalized Cauchy function and new Conjecture on its exterior singularities

This article studies on Cauchy’s function f （z） and its integral, （2πi）J[ f （z）] ≡ ■C f （t）dt/（t z） taken along a closed simple contour C, in regard to their comprehensive properties over the entire z = x ＋ iy plane consisted of the simply connected open domain D ＋ bounded by C and the open domain D outside C. （1） With f （z） assumed to be C n （n ∞-times continuously differentiable） z ∈ D ＋ and in a neighborhood of C, f （z） and its derivatives f （n） （z） are proved uniformly continuous in the closed domain D ＋ = [D ＋ ＋ C]. （2） Cauchy’s integral formulas and their derivatives z ∈ D ＋ （or z ∈ D ） are proved to converge uniformly in D ＋ （or in D = [D ＋C]）, respectively, thereby rendering the integral formulas valid over the entire z-plane. （3） The same claims （as for f （z） and J[ f （z）]） are shown extended to hold for the complement function F（z）, defined to be C n z ∈ D and about C. （4） The uniform convergence theorems for f （z） and F（z） shown for arbitrary contour C are adapted to find special domains in the upper or lower half z-planes and those inside and outside the unit circle ｜z｜ = 1 such that the four general- ized Hilbert-type integral transforms are proved. （5） Further, the singularity distribution of f （z） in D is elucidated by considering the direct problem exemplified with several typ- ical singularities prescribed in D . （6） A comparative study is made between generalized integral formulas and Plemelj’s formulas on their differing basic properties. （7） Physical sig- nificances of these formulas are illustrated with applicationsto nonlinear airfoil theory. （8） Finally, an unsolved inverse problem to determine all the singularities of Cauchy function f （z） in domain D , based on the continuous numerical value of f （z） z ∈ D ＋ = [D ＋ ＋ C], is presented for resolution as a conjecture.

DUAL FUNCTIONALS OF SAID-BZIER TYPE GENERALIZED BALL BASES OVER TRIANGLE DOMAIN AND THEIR APPLICATION

A family of Said-Bézier type generalized Ball （SBGB） bases and surfaces with a parameter H over triangular domain is introduced,which unifies Bézier surface and Said-Ball surface and includes several intermediate surfaces. To convert different bases and surfaces,the dual functionals of bases are presented. As an application of dual functionals,the subdivision formulas for surfaces are established.

Generalized Normal Derivatives and Their Applications in DDMs with Nonmatching Grids and DG Methods

A class of normal-like derivatives for functions with low regularity defined on Lipschitz domains are introduced and studied.It is shown that the new normal-like derivatives,which are called the generalized normal derivatives,preserve the major prop- erties of the existing standard normal derivatives.The generalized normal derivatives are then applied to analyze the convergence of domain decomposition methods (DDMs) with nonmatching grids and discontinuous Galerkin (DG) methods for second-order el- liptic problems.The approximate solutions generated by these methods still possess the optimal energy-norm error estimates,even if the exact solutions to the underlying elliptic problems admit very low regularities.

A New Class of Generalized Landau Linear Positive Operator Sequence and Its Properties of Approximation

In this paper two sequences of generalized Landau linear positive operators are introduced. They can be applied in approximating continuous functions with arbitrary growth order, defined on a finite interval or the whole real axis. The properties of approximation are studied and their asymptotic formulae are presented. These results show that their degrees of approximation are the best among existing operator sequences of Landau type, for example, their degrees of approximation for C 2[0, 1] are O(1/n 2) but corresponding degree of ordinary Landau operators are only O(1/n).

General calculation formulas and recurrence relations of radial matrix elements for relativistic n-dimensional hydrogen atom of spin S=0

In this paper, the general calculation formulas of radial matrix elements for relativistic n-dimensional hydrogen atom of spin S=0 are obtained, and the recurrence relation of different power order radial matrix elements are also derived.

Lucas Symbolic Formulae and Generating Functions for Chebyshev Polynomials

This work shows that each kind of Chebyshev polynomials may be calculated from a symbolic formula similar to the Lucas formula for Bernoulli polynomials. It exposes also a new approach for obtaining generating functions of them by operator calculus built from the derivative and the positional operators.

A Formula of General Solution for a Class of Homogeneous Trinomial Recurrence of Variable Coefficients with Two Indices

In practice, it is very difficult to find the solution of recurrence relation by using the characteristic roots. By applying iteration and induction we present an explicit formula of general solution for a class of homogeneous trinomial recurrence of variable coefficients with two indices. It provides a concrete and applicable model to solve the relevant problem with computer.

General Interpolation Formulae for Barycentric Blending Interpolation

General interpolation formulae for barycentric interpolation and barycen- tric rational Hermite interpolation are established by introducing multiple parameters, which include many kinds of barycentric interpolation and barycentric rational Her- mite interpolation. We discussed the interpolation theorem, dual interpolation and special cases. Numerical example is given to show the effectiveness of the method.

General formula for phonon-assisted n-photon absorption in solids

A general formula for phonon-assisted n-photon absorption in solids is obtained by （n ＋ 1）-th order perturbation technique. The complicated calculation process for transition element of n-photon absorption is simply demonstrated by a diagram approach that is proposed in this work. We find that the transition element for the n-photon absorption has a simple form, i.e., it is just the first term of the n-th order fist kind Bessel function.