On the Conformal Mappings and the Global Operator G

After Gentili and Struppa introduced in 2006 the theory of quaternionic slice regular functions, the theory has focused on functions on the so-called slice domains. The present work defines the class of speared domains, which is a rather broad extension of the class of slice domains, and proves that the theory is extremely interesting on speared domains. A Semi-global Extension Theorem and a Semi-global Representation Formula are proven for slice regular functions on speared domains: they generalize and strengthen some known local properties of slice regular functions on slice domains. A proper subclass of speared domains, called hinged domains, is defined and studied in detail. For slice regular functions on a hinged domain, a Global Extension Theorem and a Global Representation Formula are proven. The new results are based on a novel approach: one can associate to each slice regular function f:Ω→H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f:\\Omega \\rightarrow {\\mathbb {H}}$$\\end{document} a family of holomorphic stem functions and a family of induced slice regular functions. As we tighten the hypotheses on Ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega $$\\end{document} (from an arbitrary quaternionic domain to a speared domain, to a hinged domain), these families represent f better and better and allow to prove increasingly stronger results.

In a recent paper [Trans. Amer. Math. Soc. 378 (2025), pp. 851–883], the concept of generalized partial-slice monogenic (or regular) function was introduced over Clifford algebras. The present paper shall extend the study of generalized partial-slice monogenic functions from the associative case of Clifford algebras to non-associative alternative algebras, such as octonions. The new class of functions encompasses the regular functions [Rend. Sem. Mat. Univ. Padova 50 (1973), pp. 251–267] and slice regular functions [Rocky Mountain J. Math. 40 (2010), no. 1, 225–241] over octonions, indeed both appear in the theory as special cases. In the non-associative setting of octonions, we shall develop some fundamental properties such as identity theorem, Representation Formula, Cauchy (and Cauchy-Pompeiu) integral formula, maximum modulus principle, Fueter polynomials, Taylor series expansion. As a complement, the paper also introduces and discusses the notion of generalized partial-slice (and regular) functions. Although the study is limited to the case of octonions, it is clear from the statements and the arguments in the proofs that the results hold more in general in real alternative algebras equipped with a notion of conjugation.

In the present paper we introduce the class of slice-polynomial functions: slice regular functions {defined over the quaternions, outside the real axis,} whose restriction to any complex half-plane is a polynomial. These functions naturally emerge in the twistor interpretation of slice regularity introduced in \cite{gensalsto} and developed in \cite{AAtwistor}. To any slice-polynomial function $P$ we associate its {\em companion} $P^\vee$ and its {\em extension} to the real axis $P_\mathbb{R}$, that are quaternionic functions naturally related to $P$. Then, using the theory of twistor spaces, we are able to show that for any quaternion $q$ the {cardinality of simultaneous} pre-images of $q$ via $P$, $P^\vee$ and $P_\mathbb{R}$ is generically constant, giving a notion of degree. With the brand new tool of slice-polynomial functions, we {compute} the twistor discriminant locus of a cubic scroll $\mathcal{C}$ in $\mathbb{CP}^3$ and we conclude by giving some qualitative results on the complex structures induced by $\mathcal{C}$ via the twistor projection.

In a recent paper, we showed that the classical Bergman theory admits two possible formulations for the class of slice regular functions with quaternionic values. In the so called formulation of the first kind, we provide a Bergman kernel which is defined on and is a reproducing kernel. In the so called formulation of the second kind, we use the Representation Formula for slice regular functions to define a second Bergman kernel; this time the kernel is still defined on U, but the integral representation of f is based on an integral computed only on and the integral does not depend on , (here denotes the sphere unit of purely imaginary quaternions, and represents the complex plane with imaginary unit I). In this paper, we extend the second formulation of the Bergman theory to the case of slice monogenic functions and we focus our attention on the so-called Bergman–Sce transform. This integral transform is defined by using the Bergman kernel and the Sce mapping theorem and associates to every slice monogenic function f, an axially monogenic function . Copyright © 2011 John Wiley & Sons, Ltd.

Geometric function theory over quaternionic slice domains

The Green's function formalism in Condensed Matter Physics is reviewed within the equation of motion approach. Composite operators and their Green's functions naturally appear as building blocks of generalized perturbative approaches and require fully self-consistent treatments in order to be properly handled. It is shown how to unambiguously set the representation of the Hilbert space by fixing both the unknown parameters, which appear in the linearized equations of motion and in the spectral weights of non-canonical operators, and the zero-frequency components of Green's functions in a way that algebra and symmetries are preserved. To illustrate this procedure some examples are given: the complete solution of the two-site Hubbard model, the evaluation of spin and charge correlators for a narrow-band Bloch system, the complete solution of the three-site Heisenberg model, and a study of the spin dynamics in the Double-Exchange model.

This works introduces several notions of subharmonicity for real-valued functions of one quaternionic variable. These notions are related to the theory of slice regular quaternionic functions introduced by Gentili and Struppa in 2006. The interesting properties of these new classes of functions are studied and applied to construct the analogs of Green's functions.

In this paper we show a version of the Fueter mapping theorem that can be stated in integral form based on the Cauchy formulas for slice monogenic (or slice regular) functions. More precisely, given a holomorphic function f of a paravector variable, we generate a monogenic function by an integral transform whose kernel is particularly simple. This procedure allows us to define a functional calculus for n-tuples of commuting operators (called ℱ-functional calculus) based on a new notion of spectrum, called ℱ-spectrum, for the n-tuples of operators. Analogous results are shown for the quaternionic version of the theory and for the related ℱ-functional calculus. Copyright © 2010 John Wiley & Sons, Ltd.

Quaternionic exponentially dichotomous operators through S-spectral splitting and applications to Cauchy problem

The study of asymptotic representations of solutions of differential equations, in particular, of the second order differential equations, which contain nonlinearities of various types in the right-hand side, play an important role in the development of the qualitative theory of differential equations. This paper considers the type of differential equations of the second order, which contain in the right part the product of a regularly varying function from an unknown function and a rapidly varying function from the derivative of an unknown function when the corresponding arguments tend to zero or infinity. Necessary and sufficient conditions for the existence of slowly varying Pω(Y0, Y1, ±∞)-solutions of such equations have been obtained. Asymptotic representations of such solutions and their first-order derivatives are also obtained. Note that when additional conditions are imposed on the coefficients of the characteristic equation, such Pω(Y0, Y1, ±∞)-solutions of the equation exist as a one-parameter family. Similar results were obtained earlier when considering second-order equations, which contain in the right-hand side the product of a rapidly varying function from an unknown function and a regularly varying function from the derivative of an unknown function when the arguments tend to zero or infinity. For the equations considered in this paper, similar results are new.

The mathematical concept of bicomplex numbers (quaternions) is introduced in electromagnetics, and is directly applied to the derivation of analytical solutions of Maxwell's equations. It is demonstrated that, with the assistance of a bicomplex vector field, a novel entity combining both the electric and the magnetic fields, the number of unknown quantities is practically reduced by half, whereas the Helmholtz equation is no longer necessary in the development of the final solution. The most important advantage of the technique is revealed in the analysis of electromagnetic propagation through inhomogeneous media, where the coefficients of the (second order) Helmholtz equation are variable, causing severe complications to the solution procedure. Unlike conventional methods, bicomplex algebra invokes merely first order differential equations, solvable even when their coefficients vary, and hence enables the extraction of several closed form solutions, not easily derivable via standard analytical techniques.

This paper applied a new concept of a nonlinear model reference PID controller for a planar robot arm. The controller consists of two second order differential equations, each in terms of the angle error of a single joint from the designed trajectory. The coefficients of each differential equation are specifiable giving the ability to specify the behavior of the robot. Simulations are presented that show the ability of the robot arm with the nonlinear model reference PID controller to follow the precise designed trajectory.

We obtain new algebraic analytic presentations for solutions and coefficients of nonlinear second order differential equations and systems of such equations. Bibliography: 18 titles.

We are going to present a suitable bases to treat the space- and timefractional diffusion equation with the Galerkin method to obtain spectral convergence in both, time and space. Furthermore, by carefully choosing a Fourier ansatz in space, we can guarantee the resulting matrices to be sparse, even though fractional order differential equations are global operator. This is due to the fact that the chosen basis consists of eigenfunctions of the given fractional differential operator. Numerical experiments validate the theoretically predicted spectral convergence for smooth problems. Additionally, we show that this method is also capable of computing approximation of the solution of the wave equation by letting the order of the spatial and temporal derivative approach two arbitrarily close.

The conformal transformation properties of the recently discovered pseudoparticle solution to a pure Yang-Mills theory are studied. It is shown that the solution is invariant under an O(5) subgroup of conformal transformations. A formalism is developed which renders this invariance explicit and which allows a very compact group-theoretical analysis of the propagation of fermions in the field of the pseudoparticle.