On the Calculation of Monogenic Primitives

The theory of functions with values in the algebra of quaternions shows a lot of analogies to the function theory in the complex one-dimensional case. The class of holomorphic functions is replaced by the set of null solutions of a generalized Cauchy-Riemann system, the class of monogenic functions. R. Delanghe described in “On Primitives of Monogenic Functions”, Complex Variables and Elliptic Equations, 51, No. 8-11 (2006) the basic ideas of an algorithm to construct monogenic primitives of the Fueter polynomials. Main goal of this paper is to complete the proposed algorithm for the anti-derivation and to calculate the primitives explicitly.     

Duality in complex Clifford analysis

The aim of this paper is to characterize the dual and bidual of complex Clifford modules of holomorphic functions which are defined over domains in $\text{C}$^{n + 1} and satisfy generalized Cauchy-Riemann equations. In one instance the generalized Cauchy-Riemann equation reduces to a holomorphic extension of Maxwell's equations in vacuo.     

An Introduction to Commutative Quaternions

A Scheffers theorem states that for commutative hypercomplex numbers the differential calculus does exist and the functions can be introduced in the same way as they are for the complex variable. This property could open new applications of commutative quaternions in comparison with non-commutative Hamilton quaternions. In this article we introduce some quaternionic systems, their algebraic properties and the differential conditions (Generalized Cauchy-Riemann conditions) that their functions must satisfy. Then we show that the functional mapping, studied in the geometry associated with the quaternions, does have the same properties of the conformal mapping performed by the functions of complex variable. We also summarize the expressions of the elementary functions.     

A Note on the Closed-Form Determination of Zeros and Poles of Generalized Analytic Functions

The generalized method of Burniston and Siewert for the derivation of closed-form formulae for the zeros (and/or poles) of analytic functions inside a closed contour in the complex plane is further extended to the case of generalized analytic functions with real and imaginary parts satisfying homogeneous generalized Cauchy-Riemann equations. Two special cases and one generalization of this approach are also considered in brief.     

On a new generalization of Fibonacci quaternions

In this paper, we present a new generalization of the Fibonacci quaternions that are emerged as a generalization of the best known quaternions in the literature, such as classical Fibonacci quaternions, Pell quaternions, k -Fibonacci quaternions. We give the generating function and the Binet formula for these quaternions. By using the Binet formula, we obtain some well-known results. Also, we correct some results in [3] and [4] which have been overlooked that the quaternion multiplication is non commutative.     

Operators Associated to the Cauchy-Riemann Operator in Elliptic Complex Numbers

In this article we provide a generalized version of the result of L.H. Son and W. Tutschke [2] on the solvability of first order systems on the plane whose initial functions are arbitrary holomorphic functions. This is achieved by considering the more general concept of holomorphicity with respect to the structure polynomial X ²+?? X+??. It is shown that the Son-Tutschke lemma on the construction of complex linear operators associated to the Cauchy-Riemann operator remains valid when interpreted for a large class of real parameters ?? and ?? including the elliptic case but also cases that are not elliptic. For the elliptic case, first order interior estimates are obtained via the generalized version of the Cauchy representation theorem for elliptic numbers and thus the method of associated operators is applied to solve initial value problems with initial functions that are holomorphic in elliptic complex numbers.     

On isometries of matrix spaces

Let A _kbe the group of isometries of the space of n-by-n matrices over reals (resp. complexes, quaternions) with respect to the Ky Fan k-norm (see the Introduction for the definitions). Let Γ⁰ be the group of transformations of this space consisting of all products of left and right multiplications by the elements of SO(n)(resp. U(n), Sp(n)). It is shown that, except for three particular casesA_k coincides with the normalizer of Γ in Δ group of isometries of the above matrix space with respect to the standard inner product. We also give an alternative treatment of the case D = R n = 4k = 2 which was studied in detail by Johnson, Laffey, and Li [4].     

Weingarten surfaces and nonlinear partial differential equations

The sine-Gordon equation has been known for a long time as the equation satisfied by the angle between the two asymptotic lines on a surface inR ³ with constant Gauss curvature –1. In this paper, we consider the following question: Does any other soliton equation have a similar geometric interpretation? A method for finding all the equations that have such an interpretation using Weingarten surfaces inR ³ is given. It is proved that the sine-Gordon equation is the only partial differential equation describing a class of Weingarten surfaces inR ³ and having a geometricso(3)-scattering system. Moreover, it is shown that the elliptic Liouville equation and the elliptic sinh-Gordon equation are the only partial differential equations describing classes of Weingarten surfaces inR ³ and having geometricso(3,C)-scattering systems.     

On the Variety of Solutions of Generalized Cauchy-Riemann Systems with a Singular Line

In the article it is proved that a specific generalized Cauchy-Riemann system with singularity on a circle has only the trivial solution in the class of infinitely differentiable functions.     

The Virtual Waiting Time for the Batch Arrival Queueing Systems

Abstract

In the article the G ^η I/G/1-type batch arrival system with infinite waiting-room is considered. The explicit formulae for the distribution of the virtual waiting time at any fixed moment t and as t → ∞ are obtained. The study is based on generalization of Korolyuk's method for semi-markov random walks.     

Existence of Generalized Cauchy-Riemann Side-Systems in Several Complex Variables

In the present paper we have deduced the necessary and sufficient conditions on which an initial value problem $\fraca {\partial w}{\partial z_j} = a_j(z,\overline {z})\overline {w}+b_j(z,\overline {z})w+c_j(z,\overline {z}), \, j = 1,\ldots , n,\, w(z_0,\overline {z_0}) = w_0$ is locally solvable in the class of generalized analytic functions of several complex variables, which are functions fulfilling generalized Cauchy-Riemann System, $\fraca {\partial w}{\partial \overline {z_k}} = \overline {\alpha _k(z,\overline {z})}\, \overline {w}+ \overline {\beta _k(z,\overline {z})}w+ \overline {\gamma _k(z,\overline {z})},\, k = 1,\ldots , n$ .     

Determinants of Cauchy-Riemann operators as τ-functions

The -functions introduced by Sato, Miwa, and Jimbo for the deformation theory associated with the Riemann-Hilbert problem on P ¹ is shown to be a determinant for a singular Cauchy-Riemann operator whose domain incorporates functions with prescribed branching behavior. The analysis relies heavily on previous work of Malgrange on monodromy preserving deformation theory.     

复Banach空间中C-R方程的全纯解

二重复数是复数的一种推广，在其上的全纯映照族对应于Ｃ２上满足复Ｃａｕｃｈｙ－Ｒｉｅｍａｎｎ方程的全纯映照族．可以证明，这样的映照族本质上是由二个单复变数的全纯函数的直乘积所组成的族．本文证明：即使在Ｂａｎａｃｈ空间中，Ｃａｕｃｈｙ－Ｒｉｅｍａｎｎ方程的全纯解，具有同样的性质．     

Riemann-Hilbert and Poincaré problems with discontinuous boundary conditions for some model systems of partial differential equations

We study the solvability of the Riemann-Hilbert and Poincaré problems for systems of Cauchy-Riemann and Bitsadze equations in Sobolev spaces. For a generalized system of Cauchy-Riemann equations, we pose a boundary value problem and prove its unique solvability in the Sobolev space W ₂¹ (D). By supplementing the Riemann-Hilbert boundary conditions with some new conditions, we obtain a statement of the Poincaré problem with discontinuous boundary conditions for a system of second-order Bitsadze equations; we also prove the unique solvability of this problem in Sobolev spaces.     

On the growth type of entire monogenic functions

In this paper we establish an explicit relation between the growth type of general entire solutions to the generalized Cauchy-Riemann system in \mathbbRⁿ⁺¹{\mathbb{R}^{n+1}} and their Taylor coefficients. This formula then enables us to compute the growth type of some higher dimensional generalizations of the trigonometric and special functions that are null-solutions to this system.     

INDECOMPOSABLE INJECTIVE MODULES AND A THEOREM OF KAPLANSKY

Abstract

Every tripotent e of a generalized Jordan triple system of second order uniquely defines a decomposition of the space of the triple into a direct sum of eight components. This decomposition is a generalization of the Peirce decomposition for the Jordan triple system. The relations between components are studied in the case when e is a left unit.     

POINTWISE CONVERGENCE FOR EXPANSIONS IN SPHERICAL MONOGENICS

We offer a new approach to deal with the pointwise convergence of FourierLaplace series on the unit sphere of even-dimensional Euclidean spaces. By using spherical monogenics defined through the generalized Cauchy-Riemann operator, we obtain the spherical monogenic expansions of square integrable functions on the unit sphere. Based on the generalization of Fueter＇s theorem inducing monogenic functions from holomorphic functions in the complex plane and the classical Carleson＇s theorem, a pointwise convergence theorem on the new expansion is proved. The result is a generalization of Carleson＇s theorem to the higher dimensional Euclidean spaces. The approach is simpler than those by using special functions, which may have the advantage to induce the singular integral approach for pointwise convergence problems on the spheres.     

On the Bennequin Invariant and the Geometry of Wave Fronts

The theory of Arnold's invariants of plane curves and wave fronts is applied to the study of the geometry of wave fronts in the standard 2-sphere, in the Euclidean plane and in the hyperbolic plane. Some enumerative formulae similar to the Plücker formulae in algebraic geometry are given in order to compute the generalized Bennequin invariant J ⁺ in terms of the geometry of the front. It is shown that in fact every coefficient of the polynomial invariant of Aicardi can be computed in this way. In the case of affine wave fronts, some formulae previously announced by S.L. Tabachnikov are proved. This geometric point of view leads to a generalization to generic wave fronts of a result shown by Viro for smooth plane curves. As another application, the Fabricius-Bjerre and Weiner formulae for smooth plane and spherical curves are generalized to wave fronts.     

Basics of a generalized Wiman–Valiron theory for monogenic Taylor series of finite convergence radius

In this paper, we develop the basic concepts for a generalized Wiman–Valiron theory for Clifford algebra valued functions that satisfy inside an n + 1-dimensional ball the higher dimensional Cauchy-Riemann system ${\frac{\partial f}{\partial x_0} + \sum_{i=1}^n e_i\frac{\partial f}{\partial x_i}=0}In this paper, we develop the basic concepts for a generalized Wiman–Valiron theory for Clifford algebra valued functions that satisfy inside an n + 1-dimensional ball the higher dimensional Cauchy-Riemann system \frac?f?x₀ + ?_i=1ⁿ e_i\frac?f?x_i=0{\frac{\partial f}{\partial x_0} + \sum_{i=1}^n e_i\frac{\partial f}{\partial x_i}=0} . These functions are called monogenic or Clifford holomorphic inside the ball. We introduce growth orders, the maximum term and a generalization of the central index for monogenic Taylor series of finite convergence radius. Our goal is to establish explicit relations between these entities in order to estimate the asymptotic growth behavior of a monogenic function in a ball in terms of its Taylor coefficients. Furthermore, we exhibit a relation between the growth order of such a function f and the growth order of its partial derivatives.