Yuying Qiao

Decomposition of k-monogenic functions in superspace

We give the decomposition of k-monogenic functions in superspace, which not only induces Almansi theorem, but also contacts with Fischer decomposition. Besides, the Painlevé theorem and the uniqueness theorem for monogenic functions in superspace are shown. Furthermore, we deduce the two theorems for k-monogenic functions from the decomposition correspondingly.

Cauchy integral formula and Plemelj formula of bihypermonogenic functions in real Clifford analysis

In the first part of this article, we give the definition of bihypermonogenic functions in Clifford analysis. Using the idea of quasi-permutation, introduced by Sha Huang [Quasi-permutations and generalized regular functions in real Clifford analysis, J. Sys. Sci. and Math. Sci 18 (1998), pp. 380–384], we state an equivalent condition for bihypermonogenicity. In the second part, we discuss the Cauchy integral formula and Plemelj formula for the bihypermonogenic functions in real Clifford analysis.

Complex k-hypermonogenic functions in complex Clifford analysis

First, we consider the relationships between complex k-hypermonogenic functions and complex monogenic functions. Then we discuss necessary and sufficient conditions for complex k-hypermonogenic functions which is similar to the Cauchy–Riemann equation. Finally, we establish three theorems about complex partial differential equations.

Solutions of the Dirac and related equations in superspace

In this paper, we establish the connection between supermonogenic functions (i.e. solutions to the Dirac equation in superspace) and -supermonogenic functions (i.e. solutions to the higher order Dirac equations in superspace) by an expansion of Almansi type for -supermonogenic functions. The expansion generalizes the classical Almansi expansion for polyharmonic functions as well as the Fischer decomposition of polynomials. In order to obtain the expansion, we construct the 0-normalized system of functions with respect to the Dirac operator in superspace. Moreover, using the system we get a non-trivial solution to the modified Dirac equation in superspace which is closely related to Helmholtz equation.

Poincaré–Bertrand transformation formula of Cauchy-type singular integrals in Clifford analysis

This article studies the Poincaré–Bertrand transformation formula of Cauchy-type singular integrals of double multi-variables Clifford functions. First, it discusses some properties for several singular integrals about Clifford functions, and then proves the existence, for Cauchy principal value, of some Cauchy-type singular integrals with a parameter. Finally, it confirms the Poincaré–Bertrand transformation formula.

Clifford analysis with higher order kernel over unbounded domains

In this article we study Clifford analysis with higher order kernel over unbounded domains. First we derive a higher order Cauchy-Pompeiu formula for the functions with rth order continuous differentiability over an unbounded domain whose complementary set contains non-empty open set. Then we obtain higher order Cauchy integral formula for k-regular functions and prove Cauchy inequality. Based on the higher order Cauchy integral formula, we define higher order Cauchy-type integrals and discuss boundary behaviours of the higher order Cauchy-type integrals.

Some properties of T-operator with bihypermonogenic kernel in Clifford analysis

Abstract In this paper, we give the definition of T-operator with bihypermonogenic kernel in Clifford analysis and discuss a series of properties of this operator, such as uniform boundness, Hölder continuity and - integrability. T-operator is a singular integral operator which is defined in the n-dimensional Euclidean space valued in the noncommutative Clifford algebra. The properties of T-operator play an important role in solving differential equations.

Cauchy-type integral formula and Plemelj formula of hypermonogenic functions on unbounded domains

In this paper, we discuss the Cauchy-type integral formula of hypermonogenic functions on unbounded domains in real Clifford analysis, then we extend the Plemelj formula and Cauchy–Pompeiu formula of hypermonogenic functions on bounded domains to unbounded domains. We also deal with the Green-type formula on unbounded domains and get several important corollaries.

Some properties of quasi-Cauchy-type singular integral operator on unbounded domains

Firstly, this paper gives the Cauchy-type integral formula and the Plemelj formula of hypermonogenic functions on unbounded domains and discusses the Hölder continuity of the quasi-Cauchy-type singular integral operator for hypermonogenic functions on unbounded domains. Secondly, this paper studies the relation between and and introduces the modified quasi-Cauchy-type singular integral operator . Finally, this paper proves the existence and uniqueness of the fixed point of the operator by the Banach’s Contraction Mapping Principle and gives a Mann iterative sequence which strongly converges to the fixed point of the operator .