A Cauchy integral formula for(p,q)-monogenic functions withα-weight

Firstly,some properties for(p,q)-monogenic functions withα-weight in Clifford analysis are given.Then,the Cauchy-Pompeiu formula is proved.Finally,the Cauchy integral formula and the Cauchy integral theorem for(p,q)-monogenic functions withα-weight are given.

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.

CAUCHY TYPE INTEGRALS AND A BOUNDARY VALUE PROBLEM IN A COMPLEX CLIFFORD ANALYSIS

Clifford analysis is an important branch of modern analysis;it has a very important theoretical significance and application value,and its conclusions can be applied to the Maxwell equation,Yang-Mill field theory,quantum mechanics and value problems.In this paper,we first give the definition of a quasi-Cauchy type integral in complex Clifford analysis,and get the Plemelj formula for it.Second,we discuss the H?lder continuity for the Cauchy-type integral operators with values in a complex Clifford algebra.Finally,we prove the existence of solutions for a class of linear boundary value problems and give the integral representation for the solution.

KIRCHHOFF FORMULA WITH LINE INTEGRAL FORM AND ITS APPLICATIONS

In this paper, Kirchhoff formula has been transformed from surface integral form into a line integral form. The new form of the formula can be applied to separate geometrical optical field from diffraction field, and reduce the time of numerical computation greatly. Based on the new form, an analytical formula of diffraction field in the far zone has been presented for the polygonal aperture illuminated by a uniform plane wave.

Computing Streamfunction and Velocity Potential in a Limited Domain of Arbitrary Shape.Part I:Theory and Integral Formulae

The non-uniqueness of solution and compatibility between the coupled boundary conditions in computing velocity potential and streamfunction from horizontal velocity in a limited domain of arbitrary shape are revisited theoretically with rigorous mathematic treatments.Classic integral formulas and their variants are used to formulate solutions for the coupled problems.In the absence of data holes,the total solution is the sum of two integral solutions.One is the internally induced solution produced purely and uniquely by the domain internal divergence and vorticity,and its two components（velocity potential and streamfunction） can be constructed by applying Green＇s function for Poisson equation in unbounded domain to the divergence and vorticity inside the domain.The other is the externally induced solution produced purely but non-uniquely by the domain external divergence and vorticity,and the non-uniqueness is caused by the harmonic nature of the solution and the unknown divergence and vorticity distributions outside the domain.By setting either the velocity potential（or streamfunction） component to zero,the other component of the externally induced solution can be expressed by the imaginary（or real） part of the Cauchy integral constructed using the coupled boundary conditions and solvability conditions that exclude the internally induced solution.The streamfunction（or velocity potential） for the externally induced solution can also be expressed by the boundary integral of a double-layer（or singlelayer） density function.In the presence of data holes,the total solution includes a data-hole-induced solution in addition to the above internally and externally induced solutions.

ON QUADRATURE FORMULAE FOR SINGULAR INTEGRALS OF ARBITRARY ORDER

Some quadrature formulae for the numerical evaluation of singular integrals of arbitrary order are established and both the estimate of remainder and the convergence of each quadrature formula derived here are also given.

Plemelj Formula for Cauchy Type Integral on Certain Distinguished Boundary in Universal Clifford Analysis

Under the foundation of Cauchy integral formula on certain distinguished boundary for functions with values in universal Clifford algebra, we define the Cauchy type integral with values in a universal Clifford algebra, obtain its Cauchy principal value and Plemelj formula on certain distinguished boundary.

General Integral Formulasof （0, q）（q 〉 0）Differential Forms on the Analytic Varieties

In this paper, firstly using different method and technique we derive the cor-responding integral representation formulas of （0, q）（q 〉 0） differential forms for the twotypes of the bounded domains in complex submanifolds with codimension-m. Secondly weobtain the unified integral representation formulas of （0, q）（q 〉 0） differential forms for thegeneral bounded domain in complex submanifold with codimension-m, which include Hatzi-afratis formula, i.e. Koppelman type integral formula for the bounded domain with smoothboundary in analytic varieties. In particular, when m -- 0, we obtain the unified integralrepresentation formulas of （0, q）（q 〉 0） differential forms for general bounded domain in Cn,which are the generalization and the embodiment of Koppelman-Leray formula.

ORTHOGONAL POLYNOMIALS AND DETERMINANT FORMULAS OF FUNCTION-VALUED PADE-TYPE APPROXIMATION USING FOR SOLUTION OF INTEGRAL EQUATIONS

To solve Fredholm integral equations of the second kind, a generalized linear functional is introduced and a new function-valued Padé-type approximation is defined. By means of the power series expansion of the solution, this method can construct an approximate solution to solve the given integral equation. On the basis of the orthogonal polynomials, two useful determinant expressions of the numerator polynomial and the denominator polynomial for Padé-type approximation are explicitly given.

THE PERMUTATION FORMULA OF SINGULAR INTEGRALS WITH BOCHNER-MARTINELLI KERNEL ON STEIN MANIFOLDS

Using the method of localization, the authors obtain the permutation formula of singular integrals with Bochner-Martinelli kernel for a relative compact domain with C^（1） smooth boundary on a Stein manifold. As an application the authors discuss the regularization problem for linear singular integral equations with Bochner-Martinelli kernel and variable coefficients; using permutation formula, the singular integral equation can be reduced to a fredholm equation.

AN INTEGRATION BY PARTS FORMULA FOR STOCHASTIC HEAT EQUATIONS WITH FRACTIONAL NOISE

In this paper,we establish the integration by parts formula for the solution of fractional noise driven stochastic heat equations using the method of coupling.As an application,we also obtain the shift Harnack inequalities.

The Leray-Stokes Type Integral Representation Formulas on the Analytic Varieties

The closure of the bounded domains D in Cnconsists of a chain of the slit spaces,and may be divided into two types. Based on the two types of bounded domains in C^n, firstly using different method and technique we derive the corresponding integral representation formulas of differentiable functions for complex n-m(0 ≤ m < n) dimensional analytic varieties in the two types of the bounded domains. Secondly we obtain the unified integral representation formulas of differentiable functions for complex n-m(0 ≤ m < n) dimensional analytic varieties in the general bounded domains. When functions are holomorphic, the integral formulas in this paper include formulas of Stout^([1]), Hatziafratis^([2]) and the author^([3]),and are the extension of all the integral representations for holomorphic functions in the existing papers to analytic varieties. In particular, when m = 0, firstly we gave the integral representation formulas of differentiable functions for the two types of bounded domains in C^n. Therefore they can make the concretion of Leray-Stokes formula. Secondly we obtain the unified integral representation formulas of differentiable functions for general bounded domains in C^n. So they can make the Leray-Stokes formula generalizations.

HURWITZ-HODGE INTEGRAL IDENTITIES FROM THE ORBIFOLD MARIO-VAFA FORMULA

In this paper, we present some polynomial identities of Hurwitz-Hodge integral. Subsequently, we present how to obtain some Hurwitz-Hodge integral identities from the polynomial identity. Lastly, we give a recursion formula for Hurwitz-Hodge integral （TbL λgλ1）ag.

Numerical integral formula of two-dimension based on optimal approximation

There are such problems as convergence and stability of numerical calculations during multivariate interpolation. Moreover, it is very difficult to construct a overall multivariate numerical interpolation formula to ensure convergence for a set of irregular nodes. In this paper by means of an optimal binary interpolation formula given in a reproducing kernel space, a high precision overall two dimension numerical integral formula is established and its advantage is that it ensures the convergence for arbitrary irregular node set in the integral domain.

COMPUTATION FORMULAS OF GENERALIZED INVERSE PAD APPROXIMANTS USING FOR SOLUTION OF INTEGRAL EQUATIONS

For the generalized inverse function-valued Pade approximants, its intact computation formulas are given. The explicit determinantal formulas for the denominator scalar polynomials and the numerator function-valued polynomials are first established. A useful existence condition is given by means of determinant form.

Lamb's integral formulas of two-phase saturated medium for soil dynamic with drainage

When dynamic force is applied to a saturated porous soil, drainage is common. In this paper, the saturated porous soil with a two-phase saturated medium is simulated, and Lamb＇s integral formulas with drainage and stress formulas for a two-phase saturated medium are given based on Biot＇s equation and Betti＇s theorem （the reciprocal theorem）. According to the basic solution to Biot＇s equation, Green＇s function Gij and three terms of Green＇s function G4i, Gi4, and G44 of a two-phase saturated medium subject to a concentrated force on a spherical coordinate are presented. The displacement field with drainage, the magnitude of drainage, and the pore pressure of the center explosion source are obtained in computation. The results of the classical Sharpe＇s solutions and the solutions of the two-phase saturated medium that decays to a single-phase medium are compared. Good agreement is observed.

PLEMELJ FORMULA OF CAUCHY TYPE INTEGRAL ON STEIN MANIFOLDS

Let D be a bounded domain with piecewise C^（1） smooth orientable boundary on Stein manifolds, and let Φ（z） be a Cauchy type integral with Bochner-Martinelli kernel Ω（φ^V, S^-, S） and Holder continuous density function φ（ζ）, the authors define a solid angular coefficient α（t） at the point t∈δD, prove that there exist the interior and outer limit values Φ^±（t） under the meaning of the Cauchy principal value, and obtain the more general Plemelj formula and jump formula.

A Pfaffian Formula of the Generalized Inverse Function-Valued Padé Approximant Used in Solving Integral Equations

The generalized inverse function-valued Padé approximant was defined to solve the integral equations. However, it is difficult to compute the approximants by some high-order determinant formulas. In this paper, to simplify computation of the function-valued Padé approximants, an efficient Pfaffian formula for the determinants was extended from the matrix form to the function-valued form. As an important application, a Pfaffian formula of [4/4] type Padé approximant was established.

BOUNDARY INTEGRAL FORMULA OF ELASTICPROBLEMS IN CIRCLE PLANE

By bianalytic functions, the boundary integral formula of the stress function for the elastic problem in a circle plane is developed. But this integral formula includes a strongly singular integral and can not be directly calculated. After the stress function is expounded to Fourier series, making use of some formulas in generalized functions to the convolutions, the boundary integral formula which does not include strongly singular integral is derived further. Then the stress function can be got simply by the integration of the values of the stress function and its derivative on the boundary. Some examples are given. It shows that the boundary integral formula of the stress function for the elastic problem is convenient.

BOUNDARY INTEGRAL FORMULAS FOR ELASTIC PLANE PROBLEM OF EXTERIOR CIRCULAR DOMAIN

After the stress function and the normal derivative on the boundary for the plane problem of exterior circular domain are expanded into Laurent series, comparing them with the Laurent series of the complex stress function and making use of some formulas in Fourier series and the convolutions, the boundary integral formula of the stress function is derived further. Then the stress function can be obtained directly by the integration of the stress function and its normal derivative on the boundary. Some examples are given. It shows that the boundary integral formula of the stress function is convenient to be used for solving the elastic plane problem of exterior circular domain.