Riemann System Research Articles

The Static Maxwell System in Three Dimensional Axially Symmetric Inhomogeneous Media and Axially Symmetric Generalization of the Cauchy–Riemann System

In this paper we discuss different generalizations of the Cauchy–Riemann system and their connection with the static Maxwell system. In particular, this allows us to present relations between slice-monogenic functions and hypermonogenic functions, as well as to provide a physical interpretation of slice-monogenic functions. Furthermore, we present an explicit and complete set of basic solutions of a new class of axial-hypermonogenic functions in \(\mathbb R^3\). In the end we determine the symmetry operators for the class of axial-hypermonogenic functions.

Matrix approach to hypercomplex Appell polynomials

Recently the authors presented a matrix representation approach to real Appell polynomials essentially determined by a nilpotent matrix with natural number entries. It allows to consider a set of real Appell polynomials as solution of a suitable first order initial value problem. The paper aims to confirm that the unifying character of this approach can also be applied to the construction of homogeneous Appell polynomials that are solutions of a generalized Cauchy–Riemann system in Euclidean spaces of arbitrary dimension. The result contributes to the development of techniques for polynomial approximation and interpolation in non-commutative Hypercomplex Function Theories with Clifford algebras.

RELATIONS BETWEEN QUATERNIONIC DIFFERENTIAL AND THE CORRESPONDING CAUCHY RIEMANN SYSTEM

Abstract. In this paper, we investigate several properties of quaternionicfunctions. We research some di erentials of quaternionic functions, andrelations between the di erentials and the corresponding Cauchy Riemannsystem in Cli ord analysis. 1. IntroductionThe quaternion eld Tis a non-commutative four dimensional skew eld ofreal numbers. The eld Tis identi ed with C 2 and R 4 , where R denotes the eld of real numbers and C denotes the eld of complex numbers.Naser [9] have studied some properties of hyperholomorphic functions overthe eld Tand quaternionic conjugate harmonic functions in 1971. In 1995,N^ono [10] has shown several properties of regular hypercomplex functions ontwo approaches. And N^ono [10] has searched properties of hyperholomorphicfunctions by using partial di erential equation.In 2011, Luna-Elizarraras and Shapiro [1] have shown that some propertiesfor regular functions in one complex analysis are feasible in Cli ord analysis.Also Luna-Elizarraras and Shapiro [1] have explained the derivatives of func-tions in one complex variable theory and the quaternionic analysis. In 2011,Koriyama et al. [8] have given regularities on quaternionic functions for severaldi erential operators in Cli ord analysis. Also Koriyama et al. [8] have shownthe corresponding Cauchy Riemann system for each operator D

Boundary value problem for a generalized Cauchy–Riemann equation with singular coefficients

For a generalized Cauchy–Riemann system whose coefficients admit higher-order singularities on a segment, we obtain an integral representation of the general solution and study a boundary value problem combining the properties of the linear conjugation problem and the Riemann–Hilbert problem in function theory.

Integral Representations for a Generalized Cauchy–Riemann System with Singular Coefficients

We consider the Cauchy–Riemann system with supersingular coefficients and find integral representations of solutions which allows us to study the behavior of the solution in neighborhoods of singularities.

CONIC REGULAR FUNCTIONS OF CONIC QUATERNION VARIABLES IN THE SENSE OF CLIFFORD ANALYSIS

Abstract. The aim of this paper is to research certain properties ofconic regular functions of conic quaternion variables in C 2 . We generalizethe properties of conic regular functions and the Cauchy theorem of conicregular functions in conic quaternion analysis. 1. IntroductionWe introduce the four dimensional commutative conic quaternions, not quater-nions, and its associated function theory and analysis. Conic quaternions havethe following advantages: It is a classical four dimensional function theory andhas something that is impossible with quaternions and other non-commutativeor non-associative systems. Muses [11, 12] discussed speci c examples andtheorems, specially, the relation of hypernumbers to time, developed in termsof hypernumber computation. Davenport [1] worked with numbers that havefour distinct components and constructed a formal algebra formed upon a ba-sis commutative ring and a consistent de nition of multiplication and someoperators. Kajiwara etal: [2, 3] obtained mathematical results of quaternionalgebra, properties of several operators in quaternions and regenerations forthe inhomogeneous Cauchy Riemann system of quaternion and Cli ord anal-ysis. Koriyama etal: [8] gave some de nitions and properties of regularities ofquaternionic functions with regular mappings in a domain in C

О решениях полиномиального роста многомерной обобщенной системы Коши - Римана

О решениях полиномиального роста многомерной обобщенной системы Коши - Римана

Generalized Growth Orders for Polymonogenic Functions and Related Inequalities

In this paper we introduce generalized growth orders and growth types in the sense of Shah and Seremeta in the context of polymonogenic functions. They provide a refinement of previously introduced growth orders in the study of the asymptotic growth behavior of functions satisfying higher dimensional Cauchy–Riemann type equations. We prove a number of insight giving inequalities and manage to extend some basic results that were recently proved for the context of the usual first order Cauchy–Riemann system in Clifford analysis.

On the solutions of generalized Cauchy–Riemann system

It is well known that the first order elliptic system with two unknown functions of two real variables can be represented in the form of a single equation with one unknown function of the complex variable. The last equation is known as the generalized Cauchy–Riemann system. In the past extensive research has been done regarding the last system, having created a comprehensive theory for such systems. But in the majority of research the last system was studied for bounded domains of the complex plane. As for unbounded domains, the theory for such systems is far from being complete. Therefore, it seems reasonable to study such systems in unbounded domains. In this regard, this paper is an attempt to study some properties of the solutions of such systems on the whole complex plane. The paper, mainly deals with the question of existence of the solutions of the generalized Cauchy–Riemann system in function spaces, in particular, in the space of bounded analytic functions. Without summability assumption on the coefficients of the equation, existence of the solutions for the whole complex plane is proved.

Regularity at the boundary and tangential regularity of solutions of the Cauchy–Riemann system

Regularity at the boundary and tangential regularity of solutions of the Cauchy–Riemann system

Review of Edward Gollin and Alexander Rehding eds.,The Oxford Handbook of Neo-Riemannian Theories(Oxford University Press, 2011)

[1] In the Preface the impressive compilation of twenty contributed chapters making up Oxford Handbook of Neo-Riemannian Theories, editors Ed Gollin and Alexander Rehding begin with Riemann himself:The theoretical work of Hugo Riemann (1849-1919) has attracted much attention in recent years.... [T]he renewal of Riemann's ideas has reinvigorated the discipline of music theory, offering the prospect of establishing a new music-theoretical paradigm, complement and stand alongside the two well-established systems of Heinrich Schenker and Allen Forte. (ix)After a quick overview of the development of the field of neo-Riemannian theory, the editors list the purpose of the book as twofold: to provide contemporary perspectives on Riemann's scholarship and illustrate the way the Riemannian perspective shapes and informs contemporary analytical and theoretical scholarship. In these endeavors, this beautifully produced volume succeeds superbly. twenty chapters by leading researchers in the field are consistently illuminating and thought-provoking, and the overall read presents a thorough, multi-dimensional depiction of the German theorist, his evolving thoughts on the nature and organizing principles of tonality, and modern-day appropriations of his ideas in the context of North American academic music theory.Example 1. Chapter titles and authors.[2] 600-page text is organized into six parts: Intellectual Contexts (Holtmeier, Steege, Hyer, Gelbart/Rehding), Dualism (Bent, Klumpenhouwer, Rehding, Tymoczko), Tone Space (Gollin, Clark, Cohn), Harmonic Space (Engebretsen, Gollin, Kopp), Temporal Space (Caplin, Burnham, Berry), and Transformation, Analysis, Criticism (Rings, R. Cook, Harrison). Each of the six sections is preceded by a one- two-page introduction, including a brief synopsis of each chapter, and the entire volume ends with a short glossary of mostly German terms that Riemann used in his writings. list of the chapter titles is given for reference in Example 1. organization here has been carefully planned by the editors in order capitalize on strong points of contact between chapters that have been placed consecutively. Although this strategy leads a couple of oddly categorized chapters (Clark's fabulous contribution, in particular), it generally achieves its goal brilliantly, creating a compelling momentum in what might be called the middleground, chapter-by-chapter progression. logical sequence the overall content, however, does not preclude the reader from jumping in at any point in the volume; the material of any given chapter is self-contained, and yet there is commendably minimal direct repetition of foundational material over the course of the volume.[3] editors' choice concentrate on Riemann in a compilation whose title suggests a neo-Riemannian survey jars initially, but it reflects a deep and, of late, growing concern in our discipline situate our lines of theoretical and analytical inquiry in the rich historical continuum of the field. This approach represents a dramatic shift from earlier perspectives and deserves a brief digression here. We can trace this epistemological swivel through the recent history of neo-Riemannian theory.[4] Accounts detailing the origins of the modern field of neo-Riemannian theory will often cite David Lewin's A Formal Theory of Generalized Tonal Functions (1982) as the initial entry into this line of inquiry. Elements of Riemann's own theory only apply in passing there however, as Lewin makes clear: The formalistic approach of the discussion...to matters such as the interrelations of triads and systems can be particularly associated with the spirit and work of Hugo Riemann. For that reason, I shall call the basic abstract structure we will study a 'Riemann System' (Lewin 1982, 25). startling aspect of the article at the time was not its appeal Riemann, but the ease with which Lewin carried over the tools of formalism generally associated with atonal theory the tonal realm; the nod the past was honorific. …

Erratum to: The range of the tangential Cauchy–Riemann system to a CR embedded manifold

We prove that every compact, pseudoconvex, orientable, CR manifold of $\C^n$, bounds a complex manifold in the $C^\infty$ sense. In particular, the tangential Cauchy-Riemann system has closed range.

The range of the tangential Cauchy–Riemann system to a CR embedded manifold

We prove that every compact, pseudoconvex, orientable, CR manifold of \({\Bbb{C}}^{n}\), bounds a complex manifold in the C∞ sense. In particular, \(\bar{\partial}_{b}\) has closed range.

On some constructive aspects of monogenic function theory in ℝ4

As is well known, a possible generalization to ℝ4 of the classical Cauchy–Riemann system leads to the so-called Riesz system. The main goal of this paper is to construct explicitly a complete orthonormal system of polynomial solutions of this system with respect to a certain inner product. This will be done in the spaces of square integrable functions on the unit ball over ℝ. Copyright © 2011 John Wiley & Sons, Ltd.

A boundary value problem for monogenic functions in parameter-depending Clifford algebras

In this article a Dirichlet problem in R 3 is solved for functions which are monogenic with respect to a Clifford algebra depending on parameters. If the parameters of the Clifford algebra depend on the variable x ∈ R 3, then the coefficients of the Cauchy–Riemann system for monogenic functions depend on x.

The Cauchy problem for Chaplygin's system

We discuss the Cauchy problem for the so-called Chaplygin system which often appears in gas, aero- and hydrodynamics. This system can be thought of as a nonlinear analogue of the Cauchy–Riemann system in the plane. We pose Cauchy data on a part of the boundary and apply variational approach to construct a solution to this ill-posed problem. The problem actually gives insight to fundamental questions related to instable problems for nonlinear equations.

Solutions of Robin Problems for Overdetermined Inhomogeneous Cauchy–Riemann Systems on the Unit Polydisc

The inhomogeneous Robin condition with general coefficient for the overdetermined inhomogeneous Cauchy–Riemann system of equations on the polydisc is studied using Fourier analysis. It is shown that this problem for the case of nonholomorphic general coefficient, is actually a problem with essential singularity in the domain, but still well-posed under certain compatibility conditions. Under proper assumptions, the unique solution is given explicitly.

On monogenic primitives of monogenic functions

The theory of functions with values in Clifford algebras shows a lot of analogies to the complex function theory in the complex one-dimensional case. The class of holomorphic functions is now the set of null solutions of a generalized Cauchy–Riemann system, the class of monogenic functions. Analogously to the complex case one can define a derivative of monogenic functions by applying the adjoint Cauchy–Riemann operator. The main goal of this article is to study whether the derivative can be “inverted” and how monogenic primitives of monogenic functions can be constructed.

Non-subelliptic estimates for the tangential Cauchy–Riemann system

We prove non-subelliptic estimates for the tangential Cauchy-Riemann system over a weakly “q-pseudoconvex” higher codimensional submanifold M of \(\mathbb{C}^{n}\). Let us point out that our hypotheses do not suffice to guarantee subelliptic estimates, in general. Even more: hypoellipticity of the tangential C-R system is not in question (as shows the example by Kohn of (Trans AMS 181:273–292,1973) in case of a Levi-flat hypersurface). However our estimates suffice for existence of smooth solutions to the inhomogeneous C-R equations in certain degree. The main ingredients in our proofs are the weighted L2 estimates by Hormander (Acta Math 113:89–152,1965) and Kohn (Trans AMS 181:273–292,1973) of Sect. 2 and the tangential \(\bar\partial\)-Neumann operator by Kohn of Sect 4; for this latter we also refer to the book (Adv Math AMS Int Press 19,2001). As for the notion of q pseudoconvexity we follow closely Zampieri (Compositio Math 121:155–162,2000). The main technical result, Theorem 2.1, is a version for “perturbed” q-pseudoconvex domains of a similar result by Ahn (Global boundary regularity of the \(\bar\partial\)-equation on q-pseudoconvex domains, Preprint, 2003) who generalizes in turn Chen-Shaw (Adv Math AMS Int Press 19, 2001).

On characteristics of solutions to a model generalized Cauchy–Riemann system degenerating on a part of the boundary†

The equation , given in the domain G={z: r0<|z|< R}, is considered. We formally construct a full solution and prove an analogy of the Liouville theorem when R=∞ . Dedicated to Professor Guochun Wen on the occasion of his 75th birthday.