A New Approach to Slice Regularity on Real Algebras

Ghiloni, Riccardo; Perotti, Alessandro

doi:10.1007/978-3-0346-0246-4_8

Cited by 33 publications

(39 citation statements)

References 35 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Let us recall some basic material concerning slice functions over H, see [20] and also [19,21,22] for a full account presented via the stem function approach, including generalizations.…”

Section: Preliminariesmentioning

confidence: 99%

On a Class of Orientation-Preserving Maps of $$\pmb {\mathbb {R}}^4$$

Ghiloni¹,

Perotti²

2020

J Geom Anal

Self Cite

View full text Add to dashboard Cite

The purpose of this paper is to present several new, sometimes surprising, results concerning a class of hyperholomorphic functions over quaternions, the so-called slice regular functions. The concept of slice regular function is a generalization of the one of holomorphic function in one complex variable. The results we present here show that such a generalization is multifaceted and highly non-trivial. We study the behavior of the Jacobian matrix J f of a slice regular function f proving in particular that det(J f ) ≥ 0, i.e. f is orientationpreserving. We give a complete characterization of the fibers of f making use of a new notion we introduce here, the one of wing of f . We investigate the singular set N f of f , i.e. the set in which J f is singular. The singular set N f turns out to be equal to the branch set of f , i.e. the set of points y such that f is not a homeomorphism locally at y. We establish the quasi-openness properties of f . As a consequence we deduce the validity of the Maximum Modulus Principle for f in its full generality. Our results are sharp as we show by explicit examples.2010 MSC: Primary 30G35; Secondary 30C15, 32A30, 57R45.Keywords: Quaternionic hyperholomorphic functions, Orientation-preserving maps, Singular and branch sets of differentiable maps, Quasi-openness, Maximum Modulus Principle. than two. During the last century several generalizations were introduced mainly over quaternions H, octonions O and Clifford algebras R m , see [5,27]. These generalized theories share many analytic and differential behaviors with the theory of holomorphic functions of a complex variable. However they do not include the classical theory of polynomials with noncommutative coefficients on one side, see [29].In 2006 Gentili and Struppa [13,14] remedied to this 'algebraic' lack introducing a new theory, the one of slice regular functions over quaternions. Such a theory was extended to octonions and Clifford algebras in [8,15,16]. In paper [20] we gave a unified and generalized approach valid over all real alternative * -algebras, based on the concept of stem function. The theory has developed rapidly, see e.g. [12,23] and references therein. It has proven also its effectiveness in applications to quaternionic functional calculus and mathematical foundation of quaternionic quantum mechanics (see e.g. [9,17,18]), classification of orthogonal complex structures in R 4 (see e.g. [3,4,10]), and operator semigroup theory in noncommutative setting (see e.g. [7,24,25]).The stem function approach over quaternions reads as follows, see [20]. Let H be the real division algebra of quaternions and let S H = {J ∈ H | J 2 = −1} be the 2-sphere of its imaginary units. For each J ∈ S H , we denote by C J = Span(1, J) ≃ C the subalgebra of H generated by J. Then we have the 'slice' decompositionGiven a (non-empty) subset D of C, we define the circularization Ω D of D as follows:

show abstract

“…Let us recall some basic material concerning slice functions over H, see [20] and also [19,21,22] for a full account presented via the stem function approach, including generalizations.…”

Section: Preliminariesmentioning

confidence: 99%

On a Class of Orientation-Preserving Maps of $$\pmb {\mathbb {R}}^4$$

Ghiloni¹,

Perotti²

2020

J Geom Anal

Self Cite

View full text Add to dashboard Cite

show abstract

“…The quadratic cone of R p,n is different from Q n if p = 0 (cf. [8]). However, since the trace t(x) does not depend on the signature, we still have x K ≡ 0 on the quadratic cone for every K = ∅ such that |K | ≡ 3, 4 (mod 4).…”

Section: Examples: Quaternions Octonions and Clifford Algebrasmentioning

confidence: 99%

“…We recall some definitions from and : Definition We call quadratic cone of A the set

{scriptQ}_{A} : = R \cup \{{x \in A | t (x) \in double-struckR, n (x) \in double-struckR, 4 n (x) > t (x)}^{2}\} .

We also set

{double-struckS}_{A} : = \{J \in Q_{A} | J^{2} = - 1\}

. Elements of

S_{A}

are called square roots of −1 in the algebra A .…”

Section: Introductionmentioning

confidence: 99%

Global differential equations for slice regular functions

Ghiloni

Perotti

2013

Mathematische Nachrichten

Self Cite

View full text Add to dashboard Cite

Key words Slice regular functions, functions of a hypercomplex variable, global differential operators MSC (2010) 30C15, 30G35, 32A30, 13J30In the present paper, we give a system of global differential equations which are satisfied by slice regular functions on a real alternative algebra. By means of the concepts of stem function and slice function, we are able to improve some results obtained recently in the quaternionic and slice monogenic case and to extend them to this general setting. In particular, we describe the precise relation existing between the global differential equations and the condition of slice regularity.

show abstract

“…Further studies showed that on suitable open sets called axially symmetric slice domains, this class of function coincides with the class of functions of the form f (q) = f (x + Iy) = α(x, y) + Iβ(x, y) when the quaternion q is written in the form x + Iy (I being a suitable quaternion such that I 2 = −1) and the pair (α, β) satisfies the Cauchy-Riemann system and the conditions α(x, −y) = α(x, y), β(x, −y) = −β(x, y). This class of functions when α and β are real quaternionic or, more in general, Clifford algebra valued is well known: they are the so-called holomorphic functions of a paravector variable, see [126,149], which were later studied in the setting of real alternative algebras in [116].…”

Section: Introductionmentioning

confidence: 99%

“…The case of several variables was treated in [2,79], but a lot of work has still to be done in this direction since the theory is at the beginning. The generalization of slice regularity to real alternative algebra was developed in the papers [26,116,117,118,119] and, finally, while some results associated with the global operator associated to slice hyperholomorphicity are in the papers [53,70,80,120].…”

Section: Introductionmentioning

confidence: 99%

Entire Slice Regular Functions

Colombo

Sabadini

Struppa

2016

SpringerBriefs in Mathematics

111

View full text Add to dashboard Cite

The notion of supershift (in itself a generalization of the notion of superoscillation arising in quantum mechanics) expresses the fact that the sampling of a function in an interval allows to compute the values of the function far from the interval. In this paper we study the relation between supershift and real analyticity. We use a classical result due to Serge Bernstein to show that real analyticity for a complex valued function implies a strong form of supershift. On the other hand, we use a parametric version of a result by Leonid Kantorovitch to show that the converse is not true. We additionally study the stability of functions that satisfy suitable supershift requirements under multiplication by complex numbers or primitivization.

show abstract

A New Approach to Slice Regularity on Real Algebras

Abstract: A new approach to slice regularity on real algebras

Cited by 33 publications

References 35 publications

On a Class of Orientation-Preserving Maps of $$\pmb {\mathbb {R}}^4$$

On a Class of Orientation-Preserving Maps of $$\pmb {\mathbb {R}}^4$$

Global differential equations for slice regular functions

Entire Slice Regular Functions

Contact Info

Product

Resources

About