This is a series of studies on Wu's conjecture and on its resolution to be presented herein. Both are devoted to expound all the comprehensive properties of Cauchy's function f(z) (z = x + iy) and its integral ...This is a series of studies on Wu's conjecture and on its resolution to be presented herein. Both are devoted to expound all the comprehensive properties of Cauchy's function f(z) (z = x + iy) and its integral J[f(z)]≡(2πi)-∮cf(t)(t-z)-1dt taken along the unit circle as contour C,inside which(the open domain D+) f(z) is regular but has singularities distributed in open domain Doutside C. Resolution is given to the inverse problem that the singularities of f(z) can be determined in analytical form in terms of the values f(t) of f(z) numerically prescribed on C(|t| = 1) ,as so enunciated by Wu's conjecture. The case of a single singularity is solved using complex algebra and analysis to acquire the solution structure for a standard reference. Multiple singularities are resolved by reducing them to a single one by elimination in principle,for which purpose a general asymptotic method is developed here for resolution to the conjecture by induction,and essential singularities are treated with employing the generalized Hilbert transforms. These new methods are applicable to relevant problems in mathematics,engineering and technology in analogy with resolving the inverse problem presented here.展开更多
In this article, we establish the Gauss Green type theorems for Clifford-valued functions in Clifford analysis. The boundary conditions in theorems obtained are very general by using the geometric measure theoretic me...In this article, we establish the Gauss Green type theorems for Clifford-valued functions in Clifford analysis. The boundary conditions in theorems obtained are very general by using the geometric measure theoretic method. The Cauchy-Pompeiu formula for Clifford-valued functions under the weak condition will be derived as their simple application. Furthermore, Cauchy formula for monogenic functions under the weak condition is derived directly from the Cauchy-Pompeiu formula.展开更多
When concentrated forces are applied at any points of the outer region of an ellipse in an infinite plate, the complex potentials are determined using the conformal mapping method and Cauchy's integral formula. And t...When concentrated forces are applied at any points of the outer region of an ellipse in an infinite plate, the complex potentials are determined using the conformal mapping method and Cauchy's integral formula. And then, based on the superposition principle, the analyt- ical solutions for stress around an elliptical hole in an infinite plate subjected to a uniform far-field stress and concentrated forces, are obtained. Tangential stress concentration will occur on the hole boundary when only far-field uniform loads are applied. When concen- trated forces are applied in the reversed directions of the uniform loads, tangential stress concentration on the hole boundary can be released significantly. In order to minimize the tangential stress concentration, we need to determine the optimum positions and values of the concentrated forces. Three different optimization methods are applied to achieve this aim. The results show that the tangential stress can be released significantly when the op- timized concentrated forces are applied.展开更多
In this paper we prove that, with at most O(N^5/12+ε) exceptions, all positive odd integers n ≤ N with n ≡ 0 or 1(mod 3) can be written as a sum of a prime and two squares of primes.
By virtue of Cauchy’s integral formula in the theory of complex functions,the authors establish an integral representation for the weighted geometric mean,apply this newly established integral representation to show ...By virtue of Cauchy’s integral formula in the theory of complex functions,the authors establish an integral representation for the weighted geometric mean,apply this newly established integral representation to show that the weighted geometric mean is a complete Bernstein function,and find a new proof of the well-known weighted arithmetic-geometric mean inequality.展开更多
文摘This is a series of studies on Wu's conjecture and on its resolution to be presented herein. Both are devoted to expound all the comprehensive properties of Cauchy's function f(z) (z = x + iy) and its integral J[f(z)]≡(2πi)-∮cf(t)(t-z)-1dt taken along the unit circle as contour C,inside which(the open domain D+) f(z) is regular but has singularities distributed in open domain Doutside C. Resolution is given to the inverse problem that the singularities of f(z) can be determined in analytical form in terms of the values f(t) of f(z) numerically prescribed on C(|t| = 1) ,as so enunciated by Wu's conjecture. The case of a single singularity is solved using complex algebra and analysis to acquire the solution structure for a standard reference. Multiple singularities are resolved by reducing them to a single one by elimination in principle,for which purpose a general asymptotic method is developed here for resolution to the conjecture by induction,and essential singularities are treated with employing the generalized Hilbert transforms. These new methods are applicable to relevant problems in mathematics,engineering and technology in analogy with resolving the inverse problem presented here.
基金supported by NNSF of China(11171260)RFDP of Higher Education of China(20100141110054)
文摘In this article, we establish the Gauss Green type theorems for Clifford-valued functions in Clifford analysis. The boundary conditions in theorems obtained are very general by using the geometric measure theoretic method. The Cauchy-Pompeiu formula for Clifford-valued functions under the weak condition will be derived as their simple application. Furthermore, Cauchy formula for monogenic functions under the weak condition is derived directly from the Cauchy-Pompeiu formula.
基金supported by the National Natural Science Foundation of China [grant numbers 11172101, 11572126]
文摘When concentrated forces are applied at any points of the outer region of an ellipse in an infinite plate, the complex potentials are determined using the conformal mapping method and Cauchy's integral formula. And then, based on the superposition principle, the analyt- ical solutions for stress around an elliptical hole in an infinite plate subjected to a uniform far-field stress and concentrated forces, are obtained. Tangential stress concentration will occur on the hole boundary when only far-field uniform loads are applied. When concen- trated forces are applied in the reversed directions of the uniform loads, tangential stress concentration on the hole boundary can be released significantly. In order to minimize the tangential stress concentration, we need to determine the optimum positions and values of the concentrated forces. Three different optimization methods are applied to achieve this aim. The results show that the tangential stress can be released significantly when the op- timized concentrated forces are applied.
基金Project supported by National Natural Science Foundation(No. 90304009)Foundation of Qufu Normal University for Ph.D.
文摘In this paper we prove that, with at most O(N^5/12+ε) exceptions, all positive odd integers n ≤ N with n ≡ 0 or 1(mod 3) can be written as a sum of a prime and two squares of primes.
文摘By virtue of Cauchy’s integral formula in the theory of complex functions,the authors establish an integral representation for the weighted geometric mean,apply this newly established integral representation to show that the weighted geometric mean is a complete Bernstein function,and find a new proof of the well-known weighted arithmetic-geometric mean inequality.