arXiv subjects

Xinyuan Dou

Publications and source records attributed to Xinyuan Dou.

13 recordsLinked to original sources

arXiv2026

Slice Fueter-regular functions on arbitrary domains in octonions

This paper is concerned with a class of generalized slice Fueter-regular functions on arbitrary domains in O with local stem functions. Some classical theorems such as the maximum modulus principle will be generalized to our setting. Some new phenomena such as the conditional uniqueness of stem vectors will be discovered by means of new technical tools, e.g., the CCL equivalence relation and the Bers-Vekua continuation. And a natural connection between the theory of slice Fueter-regular functions and that of Riemann domains will be revealed via the quotient space under the CCL equivalence relation.

math.CV↗

arXiv2026

On the Enestrom-Kakeya Theorem for polynomials of an octonionic variable

To study the zeros of octonionic polynomials, we generalize the well-known Enestrom-Kakeya Theorem to the case of octonions. In this paper, we first deal with octonionic polynomials with nonnegative and monotonic coefficients, and prove that its zero set is contained within the closed sphere of octonion space. Then, we also consider the octonionic polynomials which the coefficients muduli is monotonic and the real parts of the coefficients is monotonic respectively, and get some results.

math.AG↗

arXiv2026

Growth theorems for slice Dirac-regular mappings over Clifford algebras

In this paper, we define a class of slice Dirac-regular mappings of several variables over Clifford algebras, based on the concept of O(3)-stem mappings. We prove that the slice mappings vanish under the slice Dirac operator, which is equivalent to its O(3)-stem mappings satisfy the generalized version of the Cauchy-Riemann equation. Moreover, we establish the growth theorem for slice Dirac-regular starlike mappings in the slice cones of Clifford algebras, as well as for slice Dirac-regular k-fold symmetric mappings.

math.CV↗

arXiv2025

Convergence domains of sedenionic star-power series

The theory of slice regular (also called hyperholomorphic) functions is a generalization of complex analysis originally given in the quaternionic framework, and then further extended to Clifford algebras, octonions, and to real alternative algebras. Recently, we have extended this theory to the case of (real) even-dimensional Euclidean space. We provided several tools for studying slice regular functions in this context, such as a path-representation formula, slice-solutions, hyper-solutions and hyper-sigma balls. When adding an algebra structure to Euclidean space we have more properties related to the possibility of multiplying elements in the algebra. In this paper, we focus on exploring the properties specific to the context of the algebra of sedenions which is rather peculiar, being the algebra non associative and non alternative. The consideration of a non alternative algebra is a novelty in the context of slice regularity which opens the way to a number of further progresses. We offer an explicit characterization for slice-solutions and hyper-solutions, and we determine the convergence domain of the power series in the form $q\mapsto\sum_{n\in\mathbb{N}}(q-p)^{*n}a_n$ with sedenionic coefficients. Unlike the case of complex numbers and quaternions, the convergence domain in this context is determined by two convergence radii instead of one. These results are closely tied to the presence of zero divisors in the algebra of sedenions.

math.CV↗

arXiv2024

Weak slice regular functions on the $n$-dimensional quadratic cone of octonions

In the literature on slice analysis in the hypercomplex setting, there are two main approaches to define slice regular functions in one variable: one consists in requiring that the restriction to any complex plane is holomorphic (with the same complex structure of the complex plane), the second one makes use of stem and slice functions. So far, in the setting of several hypercomplex variables, only the second approach has been considered, i.e. the one based on stem functions. In this paper, we use instead the first definition on the so-called $n$-dimensional quadratic cone of octonions. These two approaches yield the same class of slice regular functions on axially symmetric slice-domains, however, they are different on other types of domains. We call this new class of functions weak slice regular. We show that there exist weak slice regular functions which are not slice regular in the second approach. Moreover, we study various properties of these functions, including a Taylor expansion.

math.CV↗

arXiv2024

Domains of existence of slice regular functions in one quaternionic variable

Recently, we introduced domains of slice regularity in the space $\mathbb{H}$ of quaternions and also proved that domains of slice regularity satisfy a symmetry with respect to paths, called $2$-path-symmetry. In this paper, we give a full characterization by showing that all $2$-path-symmetric slice-open sets are domains of slice regularity. In fact, we will prove a counterpart of the Cartan-Thullen theorem for slice regular functions, namely that a slice-open set is a domain of existence for some slice regular function if and only if it is a domain of slice regularity, if and only if it is slice-regularly convex, if and only if it is $2$-path-symmetric. As a tool, we also prove an interpolation theorem of independent interest.

math.CV↗

arXiv2024

Algebra of slice regular functions on non-symmetric domains in several quaternionic variables

The primary objective of this paper is to establish an algebraic framework for the space of weakly slice regular functions over several quaternionic variables. We recently introduced a $*$-product that maintains the path-slice property within the class of path-slice functions. It is noteworthy that this $*$-product is directly applicable to weakly slice regular functions, as every slice regular function defined on a slice-open set inherently possesses path-slice properties. Building on this foundation, we propose a precise definition of an open neighborhood for a path $\gamma$ in the path space $\mathscr{P}(\mathbb{C}^n)$. This definition is pivotal in establishing the holomorphism of stem functions. Consequently, we demonstrate that the $*$-product of two weakly slice regular functions retains its weakly slice regular nature. This retention is facilitated by holomorphy of stem functions and their relationship with weakly slice regular functions, providing a comprehensive algebraic structure for this class of functions.

math.CV↗

arXiv2024

Zeroes of weakly slice regular functions of several quaternionic variables on non-axially symmetric domains

In this research, we study zeroes of weakly slice regular functions within the framework of several quaternionic variables, specifically focusing on non-axially symmetric domains. Our recent work introduces path-slice stem functions, along with a novel $*$-product, tailored for weakly slice regular functions. This innovation allows us to explore new techniques for conjugating and symmetrizing path-slice functions. A key finding of our study is the discovery that the zeroes of a path-slice function are comprehensively encapsulated within the zeroes of its symmetrized counterpart. This insight is particularly significant in the context of path-slice stem functions. We establish that for weakly slice regular functions, the processes of conjugation and symmetrization gain prominence once the function's slice regularity is affirmed. Furthermore, our investigation sheds light on the intricate nature of the zeroes of a slice regular function. We ascertain that these zeroes constitute a path-slice analytic set. This conclusion is drawn from the observed phenomenon that the zeroes of the symmetrization of a slice regular function also form a path-slice analytic set. This finding marks an advancement in understanding the complex structure and properties of weakly slice regular functions in quaternionic analysis.

math.CV↗

arXiv2024

Path-slice star-product on non-axially symmetric domains in several quaternionic variables

This paper extends the $*$-product from slice analysis to weakly slice analysis in several quaternionic variables, focusing on non-axially symmetric domains. It diverges from traditional applications in axially symmetric domains to address slice regularity in more complicated cases. The approach involves redefining the $*$-product for path-slice functions, borrowing techniques from strongly slice analysis. Key to this work is the introduction of relative stem-preserving set pairs and real-path-connected sets, which help establish a direct link between path-slice functions and their stem functions. The study culminates in conditions under which weakly slice regular functions form an algebra in specific slice domains, broadening the scope of slice analysis.

math.CV↗

arXiv2021

Extension theorem and representation formula in non-axially symmetric domains for slice regular functions

Slice analysis is a generalization of the theory of holomorphic functions of one complex variable to quaternions. Among the new phenomena which appear in this context, there is the fact that the convergence domain of $f(q)=Σ_{n\in\mathbb{N}}(q-p)^{*n} a_n$, given by a $σ$-ball $Σ(p,r)$, is not open in $\mathbb{H}$ unless $p\in\mathbb{R}$. This motivates us to investigate, in this article, what is a natural topology for slice regular functions. It turns out that the natural topology is the so-called slice topology, which is different from the Euclidean topology and nicely adapts to the slice structure of quaternions. We extend the function theory of slice regular functions to any domains in the slice topology. Many fundamental results in the classical slice analysis for axially symmetric domains fail in our general setting. We can even construct a counterexample to show that a slice regular function in a domain cannot be extended to an axially symmetric domain. In order to provide positive results we need to consider so-called path-slice functions instead of slice functions. Along this line, we can establish an extension theorem and a representation formula in a slice-domain.

math.CV↗

arXiv2020

A representation formula for slice regular functions over slice-cones in several variables

The aim of this paper is to extend the so called slice analysis to a general case in which the codomain is a real vector space of even dimension, i.e. is of the form $\mathbb{R}^{2n}$. We define a cone $\mathcal{W}_\mathcal{C}^d$ in $[End(\mathbb{R}^{2n})]^d$ and we extend the slice-topology $τ_s$ to this cone. Slice regular functions can be defined on open sets in $\left(τ_s,\mathcal{W}_\mathcal{C}^d\right)$ and a number of results can be proved in this framework, among which a representation formula. This theory can be applied to some real algebras, called left slice complex structure algebras. These algebras include quaternions, octonions, Clifford algebras and real alternative $*$-algebras but also left-alternative algebras and sedenions, thus providing brand new settings in slice analysis.

math.CV↗

arXiv2019

Riemann slice-domains over quaternions I

We construct a counterexample to a well-known extension theorem for slice regular functions, which motivates us to develop a theory of Riemann slice-domains by introducing a new topology on quaternions. By some paths describing axial symmetry in Riemann slice-domains, we rectify the classical extension formula in the theory of slice regular functions and prove a representation formula over slice-domains of regularity. This proof involves an intertwining relation between imaginary units of quaternions and a fixed matrix corresponding to a complex structure.

math.CV↗

arXiv2018

Riemann slice-domains over quaternions II

We generalize the representation formula from slice-domains of regularity to general Riemann slice-domains. This result allows us to extend the $*$-product of slice regular functions on axially symmetric domains to certain Riemann slice-domains by introducing holomorphic stem systems and tensor holomorphic functions. In particular, we construct a power series expansions of slice regular functions on certain Riemann slice-domains, which implies a relation between stem holomorphic, tensor holomorphic and slice regular functions.

math.CV↗