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An introduction is given to Clifford Analysis over pseudo-Euclidean space of arbitrary signature, called for short Ultrahyperbolic Clifford Analysis (UCA). UCA is regarded as a function theory of Clifford-valued functions, satisfying a first order partial differential equation involving a vector-valued differential operator, called a Dirac operator. The formulation of UCA presented here pays special attention to its geometrical setting. This permits to identify tensors which qualify as geometrically invariant Dirac operators and to take a position on the naturalness of contravariant and covariant versions of such a theory. In addition, a formal method is described to construct the general solution to the aforementioned equation in the context of covariant UCA.
SIMULATION AND MATHEMATICAL OPTIMIZATION OF THE HYDRATION OF CONCRETE FOR AVOIDING THERMAL CRACKS
(2010)
After mixing of concrete, the hardening starts by an exothermic chemical reaction known as hydration. As the reaction rate depends on the temperature the time in the description of the hydration is replaced by the maturity which is defined as an integral over a certain function depending on the temperature. The temperature distribution is governed by the heat equation with a right hand side depending on the maturity and the temperature itself. We compare of the performance of different time integration schemes of higher order with an automatic time step control. The simulation of the heat distribution is of importance as the development of mechanical properties is driven by the hydration. During this process it is possible that the tensile stresses exceed the tensile strength and cracks occur. The goal is to produce cheap concrete without cracks. Simple crack-criterions use only temperature differences, more involved ones are based on thermal stresses. If the criterion predicts cracks some changes in the input data are needed. This can be interpreted as optimization. The final goal will be to adopt model based optimization (in contrast to simulation based optimization) to the problem of the hydration of young concrete and the avoidance of cracks. The first step is the simulation of the hydration, which we focus in this paper.
NONZONAL WAVELETS ON S^N
(2010)
In the present article we will construct wavelets on an arbitrary dimensional sphere S^n due the approach of approximate Identities. There are two equivalently approaches to wavelets. The group theoretical approach formulates a square integrability condition for a group acting via unitary, irreducible representation on the sphere. The connection to the group theoretical approach will be sketched. The concept of approximate identities uses the same constructions in the background, here we select an appropriate section of dilations and translations in the group acting on the sphere in two steps. At First we will formulate dilations in terms of approximate identities and than we call in translations on the sphere as rotations. This leads to the construction of an orthogonal polynomial system in L²(SO(n+1)). That approach is convenient to construct concrete wavelets, since the appropriate kernels can be constructed form the heat kernel leading to the approximate Identity of Gauss-Weierstra\ss. We will work out conditions to functions forming a family of wavelets, subsequently we formulate how we can construct zonal wavelets from a approximate Identity and the relation to admissibility of nonzonal wavelets. Eventually we will give an example of a nonzonal Wavelet on $S^n$, which we obtain from the approximate identity of Gauss-Weierstraß.
This paper describes the application of interval calculus to calculation of plate deflection, taking in account inevitable and acceptable tolerance of input data (input parameters). The simply supported reinforced concrete plate was taken as an example. The plate was loaded by uniformly distributed loads. Several parameters that influence the plate deflection are given as certain closed intervals. Accordingly, the results are obtained as intervals so it was possible to follow the direct influence of a change of one or more input parameters on output (in our example, deflection) values by using one model and one computing procedure. The described procedure could be applied to any FEM calculation in order to keep calculation tolerances, ISO-tolerances, and production tolerances in close limits (admissible limits). The Wolfram Mathematica has been used as tool for interval calculation.
THE FOURIER-BESSEL TRANSFORM
(2010)
In this paper we devise a new multi-dimensional integral transform within the Clifford analysis setting, the so-called Fourier-Bessel transform. It appears that in the two-dimensional case, it coincides with the Clifford-Fourier and cylindrical Fourier transforms introduced earlier. We show that this new integral transform satisfies operational formulae which are similar to those of the classical tensorial Fourier transform. Moreover the L2-basis elements consisting of generalized Clifford-Hermite functions appear to be eigenfunctions of the Fourier-Bessel transform.
In the past, several types of Fourier transforms in Clifford analysis have been studied. In this paper, first an overview of these different transforms is given. Next, a new equation in a Clifford algebra is proposed, the solutions of which will act as kernels of a new class of generalized Fourier transforms. Two solutions of this equation are studied in more detail, namely a vector-valued solution and a bivector-valued solution, as well as the associated integral transforms.
MICROPLANE MODEL WITH INITIAL AND DAMAGE-INDUCED ANISOTROPY APPLIED TO TEXTILE-REINFORCED CONCRETE
(2010)
The presented material model reproduces the anisotropic characteristics of textile reinforced concrete in a smeared manner. This includes both the initial anisotropy introduced by the textile reinforcement, as well as the anisotropic damage evolution reflecting fine patterns of crack bridges. The model is based on the microplane approach. The direction-dependent representation of the material structure into oriented microplanes provides a flexible way to introduce the initial anisotropy. The microplanes oriented in a yarn direction are associated with modified damage laws that reflect the tension-stiffening effect due to the multiple cracking of the matrix along the yarn.
The application of a recent method using formal power series is proposed. It is based on a new representation for solutions of Sturm-Liouville equations. This method is used to calculate the transmittance and reflectance coefficients of finite inhomogeneous layers with high accuracy and efficiency. Tailoring the refraction index profile defining the inhomogeneous media it is possible to develop very important applications such as optical filters. A number of profiles were evaluated and then some of them selected in order to perform an improvement of their characteristics via the modification of their profiles.
A UNIFIED APPROACH FOR THE TREATMENT OF SOME HIGHER DIMENSIONAL DIRAC TYPE EQUATIONS ON SPHERES
(2010)
Using Clifford analysis methods, we provide a unified approach to obtain explicit solutions of some partial differential equations combining the n-dimensional Dirac and Euler operators, including generalizations of the classical time-harmonic Maxwell equations. The obtained regular solutions show strong connections between hypergeometric functions and homogeneous polynomials in the kernel of the Dirac operator.
Euclidean Clifford analysis is a higher dimensional function theory offering a refinement of classical harmonic analysis. The theory is centered around the concept of monogenic functions, i.e. null solutions of a first order vector valued rotation invariant differential operator called the Dirac operator, which factorizes the Laplacian. More recently, Hermitean Clifford analysis has emerged as a new and successful branch of Clifford analysis, offering yet a refinement of the Euclidean case; it focusses on the simultaneous null solutions, called Hermitean (or h-) monogenic functions, of two Hermitean Dirac operators which are invariant under the action of the unitary group. In Euclidean Clifford analysis, the Clifford-Cauchy integral formula has proven to be a corner stone of the function theory, as is the case for the traditional Cauchy formula for holomorphic functions in the complex plane. Previously, a Hermitean Clifford-Cauchy integral formula has been established by means of a matrix approach. This formula reduces to the traditional Martinelli-Bochner formula for holomorphic functions of several complex variables when taking functions with values in an appropriate part of complex spinor space. This means that the theory of Hermitean monogenic functions should encompass also other results of several variable complex analysis as special cases. At present we will elaborate further on the obtained results and refine them, considering fundamental solutions, Borel-Pompeiu representations and the Teoderescu inversion, each of them being developed at different levels, including the global level, handling vector variables, vector differential operators and the Clifford geometric product as well as the blade level were variables and differential operators act by means of the dot and wedge products. A rich world of results reveals itself, indeed including well-known formulae from the theory of several complex variables.