非交换线性代数与四元数变量的多重次调和函数

The Monge-Ampère equation for $(n-1)$-quaternionic PSH functions on a hyperKähler manifold

We prove the existence of unique smooth solutions to the quaternionic Monge-Ampère equation for $(n-1)$-quaternionic plurisubharmonic functions on a hyperKähler manifold and thus obtain solutions for the quaternionic form type equation. We derive $C^0$ estimate by establishing a Cherrier-type inequality as in Tosatti and Weinkove [22]. By adopting the approach of Dinew and Sroka [9] to our context, we obtain $C^1$ and $C^2$ estimates without assuming the flatness of underlying hyperKähler metric comparing to previous results [14].

Cegrell classes and a variational approach for the quaternionic Monge-Ampere equation

In this paper, we introduce finite energy classes of quaternionic plurisubharmonic functions of Cegrell type and study the quaternionic Monge-Ampere operator on these classes on quaternionic hyperconvex domains of Hn. We extend the domain of definition of quaternionic Monge-Ampere operator to some Cegrell classes, the functions of which are not necessarily bounded. We show that integration by parts and comparison principle are valid on some classes. Moreover, we use the variational method to solve the quaternionic Monge-Ampere equations when the right hand side is a positive mea- sure of finit…

Pluripotential theory on quaternionic manifolds

On any quaternionic manifold of dimension greater than 4 a class of plurisubharmonic functions (or, rather, sections of an appropriate line bundle) is introduced. Then a Monge-Ampère operator is defined. It is shown that it satisfies a version of theorems of A. D. Alexandrov and Chern-Levine-Nirenberg. These notions and results were previously known in the special case of hypercomplex manifolds. One of the new technical aspects of the present paper is the systematic use of the Baston differential operators, for which we prove a new multiplicativity property.

Valuations on convex sets, non-commutative determinants, and pluripotential theory

A new method of constructing translation invariant continuous valuations on convex subsets of the quaternionic space $\HH^n$ is presented. In particular new examples of $Sp(n)Sp(1)$-invariant translation invariant continuous valuations are constructed. This method is based on the theory of plurisubharmonic functions of quaternionic variables.

On quaternionic functions: I. Local theory

Several sets of quaternionic functions are described and studied with respect to hyperholomorphy, addition and (non commutative) multiplication, on open sets of $\mathbb H$. The aim is to get a local function theory.

多重次调和函数与复空间的奇点不变量

Singularity invariants of plurisubharmonic functions and complex spaces

本文结合多重位势理论与交换代数，研究多重次调和函数的奇点不变量，建立其与全纯函数不变量的关系，给出对数规范阈值的尖锐下界，并推广至复空间。

两个四元数变量切片正则多项式的结式

Resultants of slice regular polynomials in two quaternionic variables

本文为两个四元数变量的切片正则多项式定义了基于Dieudonné行列式的非交换结式，并借此研究多项式公共零点的存在性。

四元数马勒测度

Quaternionic Mahler measure

引入非交换多项式的四元数马勒测度，扩展经典复马勒测度。建立单变量和双变量切片正则多项式的四元数马勒测度的存在性，研究实及切片正则多项式的四元数马勒测度并考虑相关莱默问题，证明多种公式。

Ideals of regular functions of a quaternionic variable

In this paper we prove that, for any $n\in \mathbb N$, the ideal generated by $n$ slice regular functions $f_1,\ldots,f_n$ having no common zeros concides with the entire ring of slice regular functions. The proof required the study of the non-commutative syzygies of a vector of regular functions, that manifest a different character when compared with their complex counterparts.

关于四元数全纯函数概念的研究

On the notion of a quaternionic holomorphic function

本文提出一种物理上更恰当的四元数全纯函数定义及Cauchy-Riemann方程的四元数推广，证明其与复全纯函数在代数性质和微分规则上完全相似，并通过实例验证理论有效性。

Quaternionic Analysis and the Algebrodynamics

We present the ``algebrodynamical'' approach to field-particle theory based on a nonlinear generalization of the Cauchy-Riemann conditions to non-commutative algebras of quaternion-like type. For complex quaternions the theory is Lorentz invariant and naturally carries some gauge and twistor structures. Point- and string-like singularities are considered as particle-like formations; their electric charge is ``self-quantized''. A novel ``causal Minkowski geometry with additional phase'' is presented that is induced by the structure of primordial biquaternion algebra. On this geometrical backgro…

On a noncommutative algebraic geometry

Several sets of quaternionic functions are described and studied with respect to hy-perholomorphy, addition and (non commutative) multiplication, on open sets of H, then Hamil-ton 4-manifolds analogous to Riemann surfaces, for H instead of C, are defined, and so begin to describe a class of four dimensional manifolds.

Boundary Julia theory for slice regular functions

The theory of slice regular functions is nowadays widely studied and has found its elegant applications to a functional calculus for quaternionic linear operators and Schur analysis. However, much less is known about their boundary behaviors. In this paper, we initiate the study of the boundary Julia theory for quaternions. More precisely, we establish the quaternionic versions of the Julia lemma, the Julia-Carathéodory theorem, the boundary Schwarz lemma, the Hopf lemma, and the Burns-Krantz rigidity theorem for slice regular self-mappings of the open unit ball $\mathbb B\subset \mathbb H$ an…

A Cauchy kernel for slice regular functions

In this paper we show how to construct a regular, non commutative Cauchy kernel for slice regular quaternionic functions. We prove an (algebraic) representation formula for such functions, which leads to a new Cauchy formula. We find the expression of the derivatives of a regular function in terms of the powers of the Cauchy kernel, and we present several other consequent results.

Entire slice regular functions

Entire functions in one complex variable are extremely relevant in several areas ranging from the study of convolution equations to special functions. An analog of entire functions in the quaternionic setting can be defined in the slice regular setting, a framework which includes polynomials and power series of the quaternionic variable. In the first chapters of this work we introduce and discuss the algebra and the analysis of slice regular functions. In addition to offering a self-contained introduction to the theory of slice-regular functions, these chapters also contain a few new results (…

Douglas-Ahlfors 恒等式的高维版本

Higher Dimensional Versions of the Douglas-Ahlfors Identities

本文研究 Douglas-Ahlfors 恒等式在高维球体中的推广，探讨了欧氏空间调和函数、四元正则函数及 Clifford 单演函数设定下的等价关系，发现 Clifford 代数设定下需用不同规则替代原关系式。