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In the classical complex analysis of several variables Lars Hoermander had been proven for Hartog's extension theorem. His proof is based on solving the inhomogeneous Cauchy–Riemann system. However...

In this paper we study a generalization of Clifford algebra depending on parameters introduced by Tutschke and Vanegas in 2008. We also introduce some related notions, a Cauchy–Pompeiu integral formula, and two boundary value problems for monogenic functions with values in this Cliford algebra.

The Cauchy–Riemann type systems in ℝ n+1 are systems of first-order partial differential equations whose solutions satisfy the Laplace equation. The systems can be written in the form , where E 0 = I m is the identity matrix, E i (i ≥ 1) are m × m constant matrices satisfying the condition E i E j + E j E i = − 2δ ij I m . We introduce a more general condition for E i , they are variable coefficients and satisfy the condition E i (x) · E j (x) + E j (x) · E i (x) = − 2a ij (x)I m , where [a ij ] is a symmetric and positive matrix. Each 𝒞2-solution of the system now becomes a solution of a second-order elliptic system with the leading part . In this article, we introduce the Levi functions and construct fundamental solutions of the systems ‘in the large’.

A multi-monogenic function u is separately monogenic in several variables x (j), j = 1, … , n with n ≥ 2, if runs in the Euclidean space and u is monogenic with respect to x (j). Using an algebraic structure of Clifford type depending on parameters, this article discusses the properties of u in case the dimensions m j are different from each other.

In the present article, we consider the initial value problems (IVPs) of type where t is the time variable and L is a linear partial differential operator of the first order acting with respect to space like variables and taking value in the quaternion algebra. We formulate conditions on the coefficients of the operator L under which L is associated with the Cauchy–Riemann operator in quaternionic analysis. Hence we can construct all linear operators for which the IVP with an arbitrary regular function ϕ is always solvable.

By using the higher-order Cauchy–Pompeiu formula for functions taking values in a Clifford algebra, the class of generalized regular functions is defined. Some properties, such as the Cauchy integral representation, the Cauchy theorem, Momera's theorem, total analyticity and extension theorem of Hartog's type from theory of regular functions in Clifford analysis, are generalized for those functions. †Dedicated to Professor H. Begehr on his 65th Birthday.

Joint Research on Environmental Science and Technology for the Earth

From the very beginning, Complex Analysis is closely connected with partial differential equations. The article surveys present generalizations of these connexions to higher dimensions.

This article deals with the Runge type approximations problem, the solution of inhomogeneous system and the additive Cousin problem for Clifford-algebra-valued functions ƒ(x, y) which are regular with respect to x and real-analytic in y. Some main results concerning these three fundamental problems from complex analysis are proved for the class of functions mentioned above.

The article deals with initial value problems of type δw/δt = Fw, w(0, ·) = φ where t is the time and F is a linear first order operator acting in the z = x ☎ iy-plane. In view of the classical Cauchy-Kovalevkaya Theorem, the initial value problem is solvable provided F has holomorphic coefficients and the initial function is holomorphic. On the other hand, the Lewy example [H. Lewy (1957). An example of a smooth linear partial differential equation without solution. Ann. of Math., 66, 155–158.] shows that there are equations of the above form with infinitely differentiable coefficients not having any solutions. The article in hand constructs, conversely, all linear operators F for which the initial value problem with an arbitrary holomorphic initial function is always solvable. In particular, we shall see that there are equations of that type whose coefficients are only continuous.

The Cousin Problem is important in Theory of holomorphic functions of several complex variables. We study in this paper the additive Cousin problem for biregular functions with values in a Clifford algebra, which are a generalization of analytic functions. It is shown that this problem is solvable for polydomains.

This dissertation addresses the challenges of giving computers the ability of doing face recognition, i.e. discriminate between different faces. Face recognition systems are commonly trained with a database of face images, becoming “familiar” with the given faces. Many reported methods rely heavily on training database size and representativenes. But collecting training images covering, for instance, a wide range of viewpoints, different expressions and illumination conditions is difficult and costly. Moreover, there may be only one face image per person at low image resolution or quality. In these situations, face recognition techniques usually suffer serious performance drop. Here we present effective algorithms that deal with single image per person database, despite issues with illumination, face expression and pose variation. Illumination changes the appearance of a face in images. Thus, we use a new pyramid based fusion method for face recognition under arbitrary unknown lighting. This extended approach with logarithmic transform works efficiently with a single image. The produced image has better contrast at both low and high ranges, i.e. has more visible details than the original one. An improved method works with high dynamic range images, useful for outdoor face images. Face expressions also modify the images’ appearance. An extended Hidden Markov Models (HMM) with a flexible encoding scheme treats images as an ensemble of horizontal and vertical strips. Each person is modeled by Joint Multiple Hidden Markov Models (JM-HMMs). This approach offers computational advantages and the good learning ability from just a single sample per class. A fast method simulated JM-HMM functionality is then derived. The new method with abstract observations and a simplified similarity measurement does not require retraining HMMs for new images or subjects. Pose invariant recognition from a single sample image per person was overcome by using the wire frame Candide face model for the synthesis of virtual views. This is one of the support functions of our face recognition system, WAWO. The extensive experiments clearly show that WAWO outperforms the state-of-the-art systems in FERET tests.

This paper deals with the initial value problem of the type $$ \frac{{\partial w}} {{\partial t}} = L\left( {t,x,w,\frac{{\partial w}} {{\partial x_i }}} \right) $$ (1) $$ w(0,x) = \phi (x) $$ (2) where t is the time, L is a linear differential operator of first order in Quaternionic Analysis and ϕ is a regular function taking values in the Quaternionic Algebra. The necessary and sufficient conditions on the coefficients of the operator L under which L is associated to the generalized Cauchy-Riemann operator of the Quaternionic Analysis are proved.

In this paper we prove an Extension Theorem for solutions of a general linear system of partial differential equations in several variables and give an example to apply this Theorem.