This paper, we discuss the solutions＇ characterize of Cauchy-Riemann equation and the extension phenomenon of Hartogs in C^n and, a series of new extended results of the solutions for Cauchy-Riemann equations is obta...This paper, we discuss the solutions＇ characterize of Cauchy-Riemann equation and the extension phenomenon of Hartogs in C^n and, a series of new extended results of the solutions for Cauchy-Riemann equations is obtained by using the latest developments of the solutions＇ extension. Furthermore, the case of the extension＇s limitation for the solutions is also given.展开更多
The purpose of the research is to assign a formally exact elliptic complex of length two to the Cauchy-Riemann Operator. The Neumann problem for this complex in a bounded domain with smooth boundary in R<sup>2&l...The purpose of the research is to assign a formally exact elliptic complex of length two to the Cauchy-Riemann Operator. The Neumann problem for this complex in a bounded domain with smooth boundary in R<sup>2</sup> will be studied, helping therefore to solve a usual boundary value problem for the Cauchy-Riemann operator.展开更多
In this note,we mainly make use of a method devised by Shaw[15]for studying Sobolev Dolbeault cohomologies of a pseudoconcave domain of the type Ω=Ω\∪_(j=1^(m))Ω_(j),where Ω and {Ω_(j)}_(j=1^(m)■Ω are bounded ...In this note,we mainly make use of a method devised by Shaw[15]for studying Sobolev Dolbeault cohomologies of a pseudoconcave domain of the type Ω=Ω\∪_(j=1^(m))Ω_(j),where Ω and {Ω_(j)}_(j=1^(m)■Ω are bounded pseudoconvex domains in ℂ^(n) with smooth boundaries,and Ω_(1),…,Ω_(m) are mutually disjoint.The main results can also be quickly obtained by virtue of[5].展开更多
The Riemann-Hilbert boundary value problem for the inhomogeneous Cauchy-Riemann equation in the polydomain is considered. The sufficient and necessary solvable conditions and integral expressions of solution for the a...The Riemann-Hilbert boundary value problem for the inhomogeneous Cauchy-Riemann equation in the polydomain is considered. The sufficient and necessary solvable conditions and integral expressions of solution for the above problem are given.展开更多
In this paper we study the function , for z∈C. We derive a functional equation that relates G(z) and G(1−z) for all z∈C, and we prove: 1) that G and the Riemann zeta function ζ have exactly the same zeros in the cr...In this paper we study the function , for z∈C. We derive a functional equation that relates G(z) and G(1−z) for all z∈C, and we prove: 1) that G and the Riemann zeta function ζ have exactly the same zeros in the critical region D:= {z∈C:ℜz∈(0,1)};2) the Riemann hypothesis, i.e., that all of the zeros of G in D are located on the critical line := {z∈D:ℜz =1/2};and that 3) all the zeros of the Riemann zeta function located on the critical line are simple.展开更多
基金Supported by the EDSF of Shandong Province(J04A11)
文摘This paper, we discuss the solutions＇ characterize of Cauchy-Riemann equation and the extension phenomenon of Hartogs in C^n and, a series of new extended results of the solutions for Cauchy-Riemann equations is obtained by using the latest developments of the solutions＇ extension. Furthermore, the case of the extension＇s limitation for the solutions is also given.
文摘The purpose of the research is to assign a formally exact elliptic complex of length two to the Cauchy-Riemann Operator. The Neumann problem for this complex in a bounded domain with smooth boundary in R<sup>2</sup> will be studied, helping therefore to solve a usual boundary value problem for the Cauchy-Riemann operator.
文摘In this note,we mainly make use of a method devised by Shaw[15]for studying Sobolev Dolbeault cohomologies of a pseudoconcave domain of the type Ω=Ω\∪_(j=1^(m))Ω_(j),where Ω and {Ω_(j)}_(j=1^(m)■Ω are bounded pseudoconvex domains in ℂ^(n) with smooth boundaries,and Ω_(1),…,Ω_(m) are mutually disjoint.The main results can also be quickly obtained by virtue of[5].
文摘The Riemann-Hilbert boundary value problem for the inhomogeneous Cauchy-Riemann equation in the polydomain is considered. The sufficient and necessary solvable conditions and integral expressions of solution for the above problem are given.
文摘In this paper we study the function , for z∈C. We derive a functional equation that relates G(z) and G(1−z) for all z∈C, and we prove: 1) that G and the Riemann zeta function ζ have exactly the same zeros in the critical region D:= {z∈C:ℜz∈(0,1)};2) the Riemann hypothesis, i.e., that all of the zeros of G in D are located on the critical line := {z∈D:ℜz =1/2};and that 3) all the zeros of the Riemann zeta function located on the critical line are simple.
