1,721,081 research outputs found
Commuting holomorphic maps and linear fractional models
No full textIn this paper we give a contribution to the investigation of the connection between iteration theory and the study of sets of commuting holomorphic maps in the open unit disk of C
Boundary Constructions of Petals at the Wolff Point in the Parabolic Case.
No full textIn the same spirit of the classical Leau-Fatou flower theorem, we prove the existence of a petal, with vertex at the Wolff point, for a holomorphic self-map of the open unit disk of parabolic type. The result is obtained in the framework of two interesting dynamical situations which require different kinds of regularity of at the Wolff point of non-automorphism type and or injective of automorphism type, and $Re(f''(\tau))=0.
The Mittag-Leffler Theorem for regular functions of a quaternionic variable
Full text linkWe prove a version of the classical Mittag-Leffler Theorem for regular functions over quaternions. Our result relies upon an appropriate notion of principal part, that is inspired by the recent definition of spherical analyticity
On Fixed Points of Regular Mobius Transformations over Quaternions
No full textIn this paper we give a complete description of the fixed-point set
for regular Möbius transformations of a quaternionic variable; furthermore we
apply these results for the proof of a rigidity property for commuting hyperbolic
regular Möbius transformations
A family of Cauchy-Riemann type operators
No full textA natural question is whether and in which sense the denition of a holomorphic function depends on the choice of the two vectors 1, i that form a basis of C over R. In fact these two vectors determine both the form of the Cauchy-Riemann operator, and the splitting of a holomorphic function in its harmonic real and imaginary components.
In this paper we consider the basis 1, exp(i heta) of C over R, and define as heta-holomorphic the functions that belong to the kernel of a Cauchy-Riemann type operator determined by this basis. We study properties of these functions, and discuss the relation between them and classical holomorphic functions. This analysis will lead us to discover the special role that heta= pi/2 plays, that renders the theory of holomorphic functions special among this family of theories
Landau-Toeplitz theorems for slice regular functions
No full textThe theory of slice regular functions of a quaternionic variable extends the notion of holomorphic function to the quaternionic setting. This theory, already rich of results, is sometimes surprisingly different from the theory of holomorphic functions of a complex variable. However, several fundamental results in the two environments are similar, even if their proofs for the case of quaternions need new technical tools. In this paper we prove the Landau-Toeplitz Theorem for slice regular functions, in a formulation that involves an appropriate notion of regular 2-diameter. We then show that the Landau-Toeplitz inequalities hold in the case of the regular n-diameter, for all n ≥ 2. Finally, a 3-diameter version of the Landau-Toeplitz Theorem is proved using the notion of slice 3-diameter
Slice conformality and Riemann manifolds on quaternions and octonions
Full text linkIn this paper we establish quaternionic and octonionic analogs of the
classical Riemann surfaces. The construction of these manifolds has nice
peculiarities and the scrutiny of Bernhard Riemann approach to Riemann
surfaces, mainly based on conformality, leads to the definition of slice
conformal or slice isothermal parameterization of quaternionic or octonionic
Riemann manifolds. These new classes of manifolds include slice regular
quaternionic and octonionic curves, graphs of slice regular functions, the
and dimensional spheres, the helicoidal and catenoidal and
dimensional manifolds. Using appropriate Riemann manifolds, we also give a
unified definition of the quaternionic and octonionic logarithm and -th root
function.Comment: Published online in: Math. Z. (2022) - (open access
On a definition of logarithm of quaternionic functions
Full text linkFor a slice-regular quaternionic function , the classical exponential function is not slice-regular in general. An alternative definition of an exponential function, the -exponential , was given in the work by Altavilla and de Fabritiis (2019): if is a slice-regular function, then is a slice-regular function as well. The study of a -logarithm of a slice-regular function becomes of great interest for basic reasons, and is performed in this paper. The main result shows that the existence of such a depends only on the structure of the zero set of the vectorial part of the slice-regular function , besides the topology of its domain of definition. We also show that, locally, every slice-regular nonvanishing function has a -logarithm and, at the end, we present an example of a nonvanishing slice-regular function on a ball which does not admit a -logarithm on that ball
Ideals of regular functions of a quaternionic variable
Full text linkIn this paper we prove that, for any natural number n, the ideal generated by n slice regular functions f_1 , . . . , 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
On a continuation of quaternionic and octonionic logarithm along curves and the winding number
Full text linkThis paper focuses on the problem of finding a continuous extension of the
hypercomplex logarithm along a path. While a branch of the complex logarithm
can be defined in a small open neighbourhood of a strictly negative real point,
no continuous branch of the hypercomplex logarithm can be defined in any open
set which contains a strictly negative real
point (here represents the algebra of quaternions or
octonions).
To overcome these difficulties, we introduced the logarithmic manifold
and then showed that if then
is an
immersion and a diffeomorphism between and .
In this paper, we consider lifts of paths in to the
logarithmic manifold ; even though is simply connected, in general, given a path in , the existence of a lift of this path to
is not guaranteed. There is an obvious equivalence
between the problem of lifting a path in and the
one of finding a continuation of the hypercomplex logarithm
along this path.Comment: 30 pages, 4 figure
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