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.     

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.     

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.     

Geometric Roots of –1 in Clifford Algebras Cℓ p,q with p + q ≤ 4

It is known that Clifford (geometric) algebra offers a geometric interpretation for square roots of –1 in the form of blades that square to –1. This extends to a geometric interpretation of quaternions as the side face bivectors of a unit cube. Research has been done [1] on the biquaternion roots of –1, abandoning the restriction to blades. Biquaternions are isomorphic to the Clifford (geometric) algebra Cℓ ₃ of \mathbb R³{{\mathbb R^3}} . All these roots of –1 find immediate applications in the construction of new types of geometric Clifford Fourier transformations.     

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.     

An application of <Emphasis Type="Italic">H</Emphasis>-differentiability to nonnegative and unrestricted generalized complementarity problems

This paper deals with nonnegative nonsmooth generalized complementarity problem, denoted by GCP(f,g). Starting with H-differentiable functions f and g, we describe H-differentials of some GCP functions and their merit functions. We show how, under appropriate conditions on H-differentials of f and g, minimizing a merit function corresponding to f and g leads to a solution of the generalized complementarity problem. Moreover, we generalize the concepts of monotonicity, P ₀-property and their variants for functions and use them to establish some conditions to get a solution for generalized complementarity problem. Our results are generalizations of such results for nonlinear complementarity problem when the underlying functions are C ¹, semismooth, and locally Lipschitzian.     

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, $\fraca {\partial w}{\partial \overline {z_k}} = \overline {\alpha _k(z,\overline {z})}\, \overline {w}+ \overline {\beta _k(z,\overline {z})}w+ \overline {\gamma _k(z,\overline {z})},\, k = 1,\ldots , n$ .     

On latently real matrices and block quaternions

Let a complex n × n matrix A be unitarily similar to its entrywise conjugate matrix [`(A)] \bar{A} . If in the relation [`(A)] = P^*AP \bar{A} = {P^*}AP the unitary matrix P can be chosen symmetric (skew-symmetric), then A is called a latently real matrix (respectively, a generalized block quaternion). The differences in the systems of elementary divisors of these two matrix classes are found that explain why latently real matrices can be made real via unitary similarities, whereas, in general, block quaternions cannot. Bibliography: 5 titles.     

M-P invertible matrices and unitary groups over Fq2

The Moor-Penrose generalized inverses (M-P inverses for short) of matrices over a finite field F_q ² which is a generalization of the Moor-Penrose generalized inverses over the complex field, are studied in the present paper. Some necessary and sufficient conditions for anm xn matrixA over F_q ² having an M-P inverse are obtained, which make clear the set ofm xn matrices over F_q ² having M-P inverses and reduce the problem of constructing and enumerating the M-P invertible matrices to that of constructing and enumerating the non-isotropic subspaces with respect to the unitary group. Based on this reduction, both the construction problem and the enumeration problem are solved by borrowing the results in geometry of unitary groups over finite fields.     

Properties of normalr-potent matrices

The main goal of this paper is to present several new results concerning r-potent matrices which are also assumed to be normal. Included will be a theorem on the Moore-Penrose generalized inverse of a normal r-potent matrix, an equivalent characterization of a normal r-potent matrix, and some other basic properties. Finally, a generalization of the algebraic formulation of Cochran's theorem will be developed for complex normalr-potent matrices.     

REMARKS ON UNIVERSAL COEFFICIENT THEOREMS FOR GENERALIZED HOMOLOGY THEORIES

Abstract

New proofs of universal coefficient theorems for generalized homology theories (cf. ∮ 2, ∮ 3) including L. G. Brown's result, relating Brown-Douglas-Fillmore's Ext (X) with complex K-theory are presented. They are all based on a theorem asserting the existence of a chain functor for a generalized homology theory (cf. ∮ 1), which was originally designed for the construction of strong homology theories on strong shape categories.     

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.