Monogenic Functions Research Articles

On the Fueter–Sce theorem for generalized partial-slice monogenic functions

AbstractIn a recent paper, we introduced the concept of generalized partial-slice monogenic functions. The class of these functions includes both monogenic functions and slice monogenic functions with values in a Clifford algebra. In this paper, we establish a version of the Fueter–Sce theorem in this new setting, which allows to construct monogenic functions in higher dimensions starting from generalized partial-slice monogenic functions. We also prove that an alternative construction can be obtained by using the dual Radon transform. It turns out that these two constructions are closely related to the generalized CK-extension.

A Cauchy integral formula for (p, q)-monogenic functions with α-weight

A Cauchy integral formula for (p, q)-monogenic functions with α-weight

Approximating special monogenic functions in Clifford analysis

Approximating special monogenic functions in Clifford analysis

Generalized Lower Order and Generalized Lower Type of Entire Monogenic Functions of Slow Growth

Generalized Lower Order and Generalized Lower Type of Entire Monogenic Functions of Slow Growth

Invariance of Iterated Global Differential Operator for Slice Monogenic Functions

Invariance of Iterated Global Differential Operator for Slice Monogenic Functions

Harmonic and Monogenic Functions on Toroidal Domains

Harmonic and Monogenic Functions on Toroidal Domains

Characterization of continuous homomorphisms on entire slice monogenic functions

Abstract This paper is inspired by a class of infinite order differential operators arising in quantum mechanics. They turned out to be an important tool in the investigation of evolution of superoscillations with respect to quantum fields equations. Infinite order differential operators act naturally on spaces of holomorphic functions or on hyperfunctions. Recently, infinite order differential operators have been considered and characterized on the spaces of entire monogenic functions, i.e. functions that are in the kernel of the Dirac operators. The focus of this paper is the characterization of infinite order differential operators that act continuously on a different class of hyperholomorphic functions, called slice hyperholomorphic functions with values in a Clifford algebra or also slice monogenic functions. This function theory has a very reach associated spectral theory and both the function theory and the operator theory in this setting are subjected to intensive investigations. Here we introduce the concept of proximate order and establish some fundamental properties of entire slice monogenic functions that are crucial for the characterization of infinite order differential operators acting on entire slice monogenic functions.

A unified notion of regularity in one hypercomplex variable

We define a very general notion of regularity for functions taking values in an alternative real ⁎-algebra. Over Clifford numbers, this notion subsumes the well-established notions of monogenic function and slice-monogenic function. Over quaternions, in addition to subsuming the notions of Fueter-regular function and of slice-regular function, it gives rise to an entirely new theory, which we develop in some detail.

Generalized Partial-Slice Monogenic Functions: A Synthesis of Two Function Theories

In this paper, we review the notion of generalized partial-slice monogenic functions that was introduced by the authors in Xu and Sabadini (Generalized partial-slice monogenic functions, arXiv:2309.03698, 2023). The class of these functions includes both the theory of monogenic functions and of slice monogenic functions over Clifford algebras and it is obtained via a synthesis operator which combines a generalized Cauchy–Riemann operator with an operator acting on slices. Besides recalling the fundamental features, we provide a notion of ∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$*$$\\end{document}-product based on the CK-extension and discuss the smoothness of generalized partial-slice functions.

Entire Monogenic Functions of Given Proximate Order and Continuous Homomorphisms

Infinite order differential operators appear in different fields of mathematics and physics. In the past decade they turned out to play a crucial role in the theory of superoscillations and provided new insight in the study of the evolution as initial data for the Schrödinger equation. Inspired by the infinite order differential operators arising in quantum mechanics, in this paper we investigate the continuity of a class of infinite order differential operators acting on spaces of entire hyperholomorphic functions. Precisely, we consider homomorphisms acting on functions in the kernel of the Dirac operator. For this class of functions, often called monogenic functions, we introduce the proximate order and prove some fundamental properties. As an important application, we are able to characterize infinite order differential operators that act continuously on spaces of monogenic entire functions.

P-21 Maturity onset diabetes of the young 7: Case report

Abstract Introduction Maturity Onset Diabetes of the Young (MODY) is a rare form of diabetes which is different from both type 1 and type 2 diabetes, and runs strongly in families. MODY is autosomal dominant inherited type of diabetes, caused by a mutation (or change) in a single gene. If a parent has this gene mutation, any child they have, has a 50% chance of inheriting it from them. Recently, 14 gene mutations have been found to be associated with MODY. The Krüppel-like factor 11 (KLF11) gene mutation is the pathogenic gene of MODY7. Clinical Case A 17-year-old female patient with a previous history of obesity was admitted at outpatient office for regular control of weight and body mass index (BMI). Two years ago the patient was diagnosed with obesity, hyperinsulinemia and insulin resistance, with body weight (BW) 90 kg, body height (BH) 167 cm and BMI 32,3. The patient is on therapy with Metformin 2×500 mg. Results of the moment of examination: BW 100 kg, BH 168 cm, BMI 35,4; HbA1c 5,96%; Fasting plasma glucose (FPG) 6.8 mmol/l; Insulinemia 13,17; C-peptide 1,00; oral glucose tolerant test (OGTT): 0’ 6,5 mmol/l 120’ 11,0 mmol/l; insulinemia during OGTT (μU/mL): 0’ 14,05; 120’ 98,75; C-peptide during OGTT (ng/mL): 0’ 1,00; 120’ 6.95; Fundus examination and Doppler ultrasonography of both lower limbs showed no abnormalities. Family history was positive for Diabetes mellitus type 2 (mother, grandmother). The Immunological tests for Diabetes mellitus type 1 were negative for antibodies: GAD65, IAA, ICA and IA-2; Because there was doubt for MODY, genetic tests were performed with targeted resequencing of 4800 clinically relevant genes. The presence in the KLF11 gene associated with MODY7 was determined, c.911>G, p.(Lys304Arg). Тhe change is found in heterozygous form and is inherited from the mother. The patient continued treatment with metformin and lifestyle changes and this resulted in weight loss and good glycoregulation. Control one year later showed weight loss and good glycoregulation, FPG 5.1 mmol/l; HbA1c 5.5%; BW 89 kg, BH 168 cm, BMI 31.5; The level of blood glucose was controlled and stable. The patient's mother and grandmother were also treated with metformin. Conclusion Young patients with signs of DM2 should be tested for genetic types of diabetes. The number of patients with MODY is small, especially regarding MODY7, which rarely occurs. MODY is the most common type of monogenic β-cell function defect. It accounts for 2% to 15% of young diabetic patients. Studies have found that oral hypoglycemic drugs and dietary interventions are beneficial for MODY7 patients, and the control of the intake of staple food carbohydrates helps to control blood glucose which was also shown in our patient (1). 1. Chen L, Hou X. Clinical and Functional Characteristics of a Novel KLF11 Cys354Phe Variant Involved in Maturity-Onset Diabetes of the Young. J Diabetes Res. 2021;2021:7136869

Partial slice regularity and Fueter's theorem in several quaternionic variables

Abstract We extend some definitions and give new results about the theory of slice analysis in several quaternionic variables. The sets of slice functions that are slice, slice regular, and circular with respect to given variables are characterized. We introduce new notions of partial spherical value and derivative for functions of several variables that extend those of one variable. We recover some of their properties as circularity, harmonicity, some relations with differential operators, and a Leibniz rule with respect to the slice product as well as studying their behavior in the context of several variables. Then, we prove our main result, which is a generalization of Fueter’s theorem for slice regular functions in several variables. This extends the link between slice regular and axially monogenic functions well known in the one variable context.

The Cauchy integral formula for (p, q)-monogenic functions with α-weight in superspace

First, some properties for -monogenic functions with α-weight in superspace are obtained. Then the Cauchy–Pompeiu formula in superspace is given. Finally, the Cauchy integral formula and the Cauchy integral theorem for -monogenic functions with α-weight in superspace are proved.

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This paper aims to study the jump problem for monogenic functions in fractal hypersurfaces of Euclidean spaces. The notion of the Marcinkiewicz exponent has been taken into consideration. A new solvability condition is obtained, basing the work on specific properties of the Teodorescu transform in Clifford analysis. It is shown that this condition improves those involving the Minkowski dimension.

Modified Quaternionic Analysis on the Ball

A natural generalization of the classical complex analysis to higher dimensions is the theory of monogenic functions (see, e.g. Brackx et al. in Clifford analysis, Pitman, Boston, 1982). Unfortunately the powers of the underlying variable are not in this system. But they are in the modified system introduced by the first author in Leutwiler (Complex Var Theory Appl 20:19–51, 1992). Since this modified system lives in the upper half space {mathbb {R}}_{+}^{3} it is natural to transplant it—with the so-called Cayley mapping—to the unit ball B(0, 1) in {mathbb {R}}^{3}. It is this transplanted system which we are going to investigate in this article. It represents the counterpart to the theory of (H)-solutions studied in Leutwiler (Complex Var Theory Appl 20:19–51, 1992).

The Laurent expansion and residue theorem of weighted monogenic functions

ABSTRACT Firstly, the definition of p order homogeneous weighted right monogenic polynomials is given, and the hypercomplex variables are introduced in order to construct a basis of all homogeneous weighted right monogenic polynomials of degree p, then the second Taylor expansion of the weighted right monogenic functions is obtained. Secondly, the weighted left monogenic functions are constructed from continuous functions in different regions, and corresponding Taylor expansions are given. Finally, on the basis of the previous conclusions, the Laurent expansion and residue theorem of the weighted left monogenic functions are proved.

Properties of a polyanalytic functional calculus on the S‐spectrum

AbstractThe Fueter mapping theorem gives a constructive way to extend holomorphic functions of one complex variable to monogenic functions, that is, null solutions of the generalized Cauchy–Riemann operator in , denoted by . This theorem is divided in two steps. In the first step, a holomorphic function is extended to a slice hyperholomorphic function. The Cauchy formula for this type of functions is the starting point of the S‐functional calculus. In the second step, a monogenic function is obtained by applying the Laplace operator in four real variables, namely, Δ, to a slice hyperholomorphic function. The polyanalytic functional calculus, that we study in this paper, is based on the factorization of . Instead of applying directly the Laplace operator to a slice hyperholomorphic function, we apply first the operator and we get a polyanalytic function of order 2, that is, a function that belongs to the kernel of . We can represent this type of functions in an integral form and then we can define the polyanalytic functional calculus on S‐spectrum. The main goal of this paper is to show the principal properties of this functional calculus. In particular, we study a resolvent equation suitable for proving a product rule and generate the Riesz projectors.

Integral theorems in finite-dimensional commutative algebra

For monogenic (continuous and differentiable in the sense of G\^ateaux) functions given in special real subspaces of an arbitrary finite-dimensional commutative associative algebra over the complex field and taking values in this algebra, we establish basic properties analogous to properties of holomorphic functions of a complex variable. Methods for proving results are based on a representation of monogenic functions via holomorphic functions of complex variables that allows to establish analogues of Cauchy-Riemann conditions and the continuity of G\^ateaux derivatives of all orders for monogenic functions. In such a way, analogues of a number of classical theorems of complex analysis (the Cauchy integral theorem for a curvilinear integral, the Cauchy integral formula, the Morera theorem, the Taylor theorem) are proved and different equivalent definitions for the mentioned monogenic functions are established. An analogue of the Cauchy theorem for an integral over non piecewise smooth surfaces is proved.

The Fine Structure of the Spectral Theory on the S-Spectrum in Dimension Five

Holomorphic functions play a crucial role in operator theory and the Cauchy formula is a very important tool to define the functions of operators. The Fueter–Sce–Qian extension theorem is a two-step procedure to extend holomorphic functions to the hyperholomorphic setting. The first step gives the class of slice hyperholomorphic functions; their Cauchy formula allows to define the so-called S-functional calculus for noncommuting operators based on the S-spectrum. In the second step this extension procedure generates monogenic functions; the related monogenic functional calculus, based on the monogenic spectrum, contains the Weyl functional calculus as a particular case. In this paper we show that the extension operator from slice hyperholomorphic functions to monogenic functions admits various possible factorizations that induce different function spaces. The integral representations in such spaces allow to define the associated functional calculi based on the S-spectrum. The function spaces and the associated functional calculi define the so-called fine structure of the spectral theories on the S-spectrum. Among the possible fine structures there are the harmonic and polyharmonic functions and the associated harmonic and polyharmonic functional calculi. The study of the fine structures depends on the dimension considered and in this paper we study in detail the case of dimension five, and we describe all of them. The five-dimensional case is of crucial importance because it allows to determine almost all the function spaces will also appear in dimension greater than five, but with different orders.

Orthogonal Harmonic and Quaternionic Monogenic Functions in the Exterior of a Spheroid

Orthogonal Harmonic and Quaternionic Monogenic Functions in the Exterior of a Spheroid