On boundary values for rectifiable curves of a generalization of the Cauchy-type integral related to the Helmholtz operator in $R^2$

Cauchy积分、分数阶Sobolev空间与弦弧曲线

Cauchy Integral, Fractional Sobolev Spaces and Chord-Arc Curves

本文研究可求长Jordan曲线上调和延拓型与Gagliardo型分数阶Sobolev空间的等价条件，证明Lipschitz曲线上两空间相等，并探讨Plemelj-Calderón问题。

Multipliers of integrals of Cauchy - Stieltjes type

Let ${\rm {\mathbb G}}$ be a domain with closed rectifiable Jordan curve $\ell $ . Let $K({\rm {\mathbb G}})$ be the space of all analytic functions in ${\rm {\mathbb G}} $ representable by a Cauchy - Stieltjes integral. Let ${\rm {\mathfrak M}}(K)$ be the class of all multipliers of the space $K({\rm {\mathbb G}}).$ In this paper we prove that if $f$ is bounded analytic function on ${\rm {\mathbb G}}$ and $${\kern 1pt} {\kern 1pt} {\kern 1pt} \mathop{ess\sup}\limits_{η\in \ell } \int_{\ell} \frac{|f(ζ)-f(η)|}{|ζ-η|} |dζ|{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {…

The Kerzman-Stein operator for piecewise continuously differentiable regions

The Kerzman-Stein operator is the skew-hermitian part of the Cauchy operator defined with respect to an unweighted hermitian inner product on a rectifiable curve. If the curve is continuously differentiable, the Kerzman-Stein operator is compact on the Hilbert space of square integrable functions; when there is a corner, the operator is noncompact. Here we give a complete description of the spectrum for a finite symmetric wedge and we show how this reveals the essential spectrum for curves that are piecewise continuously differentiable. We also give an explicit construction for a smooth curve …

通用Beta入射角：柯西刚性与无限排列

Universal Beta Incidence Angles: Cauchy Rigidity and Infinite Arrangements

该研究证明了通用Beta入射角的平方余弦律，通过两种方法证明并推导其刚性、可扩展性及相关准则，给出正交坐标下的缩放F律。

Applications of Differential Chains to Complex Analysis and Dynamics

This thesis is divided into three parts. In the first part, we give an introduction to J. Harrison's theory of differential chains. In the second part, we apply these tools to generalize the Cauchy theorems in complex analysis. Instead of requiring a piecewise smooth path over which to integrate, we can now do so over non- rectifiable curves and divergence-free vector fields supported away from the singularities of the holomorphic function in question. In the third part, we focus on applications to dynamics, in particular, flows on compact Riemannian manifolds. We prove that the asymptotic cyc…

切片函数的特征值问题

Eigenvalue problems for slice functions

本文研究四元数切片正则函数和轴向单演函数中的两类特征值问题，通过四维Laplace算子与Fueter定理关联，并应用于Helmholtz和Klein-Gordon方程的轴向单演解表示。