arXiv subjects

Guangbin Ren

Publications and source records attributed to Guangbin Ren.

At least 37 records · Page 2Linked to original sources

arXiv2021

Structure of octonionic Hilbert spaces with applications in the Parseval equality and Cayley-Dickson algebras

Contrary to the simple structure of the tensor product of the quaternionic Hilbert space, the octonionic situation becomes more involved. It turns out that an octonionic Hilbert space can be decomposed as an orthogonal direct sum of two subspaces, each of them isomorphic to a tensor product of an irreducible octonionic Hilbert space with a real Hilbert space. As an application, we find that for a given orthogonal basis the octonionic Parseval equality holds if and only if the basis is weak associative. Fortunately, there always exists a weak associative orthogonal basis in an octonionic Hilbert space. This completely removes the obstacles caused by the failure of the octonionic Parseval equality. As another application, we provide a new approach to studythe Cayley-Dickson algebras, which turn out to be specific examples of octonionic Hilbert spaces. An explicit weak associative orthonormal basis is constructed in each Cayley-Dickson algebra.

math.FA↗

arXiv2021

Adaptive Fourier decomposition of slice regular functions

In the slice Hardy space over the unit ball of quaternions, we introduce the slice hyperbolic backward shift operator $\mathcal S_a$ with the decomposition process $$f=e_a\langle f, e_a\rangle+B_{a}*\mathcal S_a f,$$ where $e_a$ denotes the slice normalized Szegö kernel and $ B_a $ the slice Blaschke factor. Iterating the above decomposition process, a corresponding maximal selection principle gives rise to the slice adaptive Fourier decomposition. This leads to a adaptive slice Takenaka-Malmquist orthonormal system.

math.CV↗

arXiv2021

Para-linearity as the nonassociative counterpart of linearity

In an octonionic Hilbert space $H$, the octonionic linearity is taken to fail for the maps induced by the octonionic inner products, and it should be replaced with the octonionic para-linearity. However, to introduce the notion of the octonionic para-linearity we encounter an insurmountable obstacle. That is, the axiom $$\left\langle pu ,u\right\rangle=p\left\langle u ,u\right\rangle$$ for any octonion $p$ and element $u\in H$ introduced by Goldstine and Horwitz in 1964 can not be interpreted as a property to be obeyed by the octonionic para-linear maps. In this article, we solve this critical problem by showing that this axiom is in fact non-independent from others. This enables us to initiate the study of octonionic para-linear maps. We can thus establish the octonionic Riesz representation theorem which, up to isomorphism, identifies two octonionic Hilbert spaces with one being the dual of the other. The dual space consists of continuous left \almost linear functionals and it becomes a right $Ø$-module under the multiplication defined in terms of the second associators which measures the failure of $Ø$-linearity. This right multiplication has an alternative expression $${(f\odot p)(x)}=pf(p^{-1}x)p,$$ which is a generalized Moufang identity. Remarkably, the multiplication is compatible with the canonical norm, i.e., $$\fsh{f\odot p}=\fsh{f}\abs{p}.$$ Our final conclusion is that para-linearity is the nonassociative counterpart of linearity.

math.FA↗

arXiv2021

Evaluation of blood coagulation by optical vortex tracking

We investigated a new optical approach for the detection of the coagulation dynamic process by means of the information on the optical vortex. In our study, laser speckle was captured using a high-speed CMOS camera, and the statistical information of the optical vortex characterized the change in coagulation properties with time. Similar to the scattering particles, the motion of the optical vortex is restricted during coagulation, as a result, the whole process of coagulation can be detected by calculating the mean square displacement(MSD) of the optical vortex. The results demonstrate a close correlation between coagulation parameters measured using the optical vortex method and thrombelastography(TEG), creating a powerful opportunity for self-testing and real-time detection of coagulation.

physics.med-ph↗

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↗

arXiv2020

Octonionic bimodule

The structure of octonionic bimodules is formulated in this paper. It turns out that every octonionic bimodule is a tensor product, the category of octonionic bimodules is isomorphic to the category of real vector spaces. We show that there is also a real part structure on octonionic bimodules similar to the quaternion case. Different from the quaternion setting , the octonionic bimodule sturcture is uniquely determined by its left module structure and hence the real part can be obtained only by left multiplication. The structure of octonionic submodules generated by one element is more involved, which leads to many obstacles to further development of the octonionic functional analysis. We introduce a notion of cyclic decomposition to deal with this difficulty. Using this concept, we give a complete description of the submodule generated by one element in octonionic bimodules. This paper clears the barrier of the structure of $Ø$-modules for the later study of octonionic functional analysis.

math.RA↗

arXiv2020

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 $\mathbb B\subset \mathbb H$ and of the right half-space $\mathbb H_+$. Especially, we find a new phenomenon that the classical Hopf lemma about $f'(ξ)>1$ at the boundary point may fail in general in quaternions, and its quaternionic variant should involve the Lie bracket reflecting the non-commutative feature of quaternions.

math.CV↗

arXiv2019

Classification of left octonion modules

It is natural to study octonion Hilbert spaces as the recently swift development of the theory of quaternion Hilbert spaces. In order to do this, it is important to study first its algebraic structure, namely, octonion modules. In this article, we provide complete classification of left octonion modules. In contrast to the quaternionic setting, we encounter some new phenomena. That is, a submodule generated by one element $m$ may be the whole module and may be not in the form $Øm$. This motivates us to introduce some new notions such as associative elements, conjugate associative elements, cyclic elements. We can characterize octonion modules in terms of these notions. It turns out that octonions admit two distinct structures of octonion modules, and moreover, the direct sum of their several copies exhaust all octonion modules with finite dimensions.

math.RA↗

arXiv2019

Slice Dirac operator over octonions

The slice Dirac operator over octonions is a slice counterpart of the Dirac operator over quaternions. It involves a new theory of stem functions, which is the extension from the commutative $ O(1) $ case to the non-commutative $ O(3) $ case. For functions in the kernel of the slice Dirac operator over octonions, we establish the representation formula, the Cauchy integral formula (and, more in general, the Cauchy-Pompeiu formula), and the Taylor as well as the Laurent series expansion formulas.

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

Slice regular functions of several octonionic variables

Octonionic analysis is becoming eminent due to the role of octonions in the theory of G2 manifold. In this article, a new slice theory is introduced as a generalization of the holomorphic theory of several complex variables to the noncommutative or nonassociative realm. The Bochner-Martinelli formula is established for slice functions of several octonionic variables as well as several quaternionic variables. In this setting, we find the Hartogs phenomena for slice regular functions

math.CV↗

arXiv2018

Growth theorems in slice analysis of several variables

In this paper, we define a class of slice mappings of several Clifford variables, and the corresponding slice regular mappings. Furthermore, we establish the growth theorem for slice regular starlike or convex mappings on the unit ball of several slice Clifford variables, as well as on the bounded slice domain which is slice starlike and slice circular.

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↗

arXiv2018

Slice starlike functions over quaternions

In this paper, we initiate the study of the geometric function theory for slice starlike functions over quaternions and its subclasses. This allows us to answer negatively some questions about the Bieberbach conjecture, the growth, distortion, and covering theorems for slice regular functions. Precisely, we find that the Bieberbach conjecture holds true for slice starlike functions in contrast to the fact that the Bieberbach conjecture fails for biholomorphic starlike mappings in higher dimensions. We also establish some sharp versions of the growth, distortion, and covering theorems for quaternions.

math.CV↗

arXiv2017

The Growth and Distortion Theorems for Slice Monogenic Functions

The sharp growth and distortion theorems are established for slice monogenic extensions of univalent functions on the unit disc $\mathbb D\subset \mathbb C$ in the setting of Clifford algebras, based on a new convex combination identity. The analogous results are also valid in the quaternionic setting for slice regular functions and we can even prove the Koebe type one-quarter theorem in this case. Our growth and distortion theorems for slice regular (slice monogenic) extensions to higher dimensions of univalent holomorphic functions hold without extra geometric assumptions, in contrast to the setting of several complex variables in which the growth and distortion theorems fail in general and hold only for some subclasses with the starlike or convex assumption.

math.CV↗

arXiv2016

Scaling limits of discrete holomorphic functions

One of the most natural and challenging issues in discrete complex analysis is to prove the convergence of discrete holomorphic functions to their continuous counterparts. This article is to solve the open problem in the general setting. To this end we introduce new concepts of discrete surface measure and discrete outer normal vector and establish the discrete Cauchy-Pompeiu integral formula, \begin{eqnarray*} f(ζ)=\displaystyle{\int_{\partial B^h}} \mathcal{K}^h(z,ζ) f(z)dS^h(z)+\displaystyle{\int_{B^h}} E^h(ζ-z) \partial_{\bar z}^h f (z)dV^h(z),\end{eqnarray*} which results in the uniform convergence of the scaling limits of discrete holomorphic functions up to second order derivatives in the standard square lattices.

math.CV↗

arXiv2016

Julia theory for slice regular functions

Slice regular functions have been extensively studied over the past decade, but much less is known about their boundary behavior. In this paper, we initiate the study of Julia theory for slice regular functions. More specifically, we establish the quaternionic versions of the Julia lemma, the Julia-Carathéodory theorem, the boundary Schwarz lemma, and the Burns-Krantz rigidity theorem for slice regular self-mappings of the open unit ball $\mathbb B$ and of the right half-space $\mathbb H^+$. Our quaternionic boundary Schwarz lemma involves a Lie bracket reflecting the non-commutativity of quaternions. Together with some explicit examples, it shows that the slice derivative of a slice regular self-mapping of $\mathbb B$ at a boundary fixed point is not necessarily a positive real number, in contrast to that in the complex case, meaning that its commonly believed version turns out to be totally wrong.

math.CV↗