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Preprint
Negative Größen bei Diophant?
(Universität Kassel, FB 17, Mathematik/Informatik, 2005)
In this paper we champion Diophantus of Alexandria and Isabella Basmakova against Norbert Schappacher. In two publications ([46] and [47]) he puts forward inter alia two propositions: Questioning Diophantus' originality he considers affirmatively the possibility, that the Arithmetica are the joint work of a team of authors like Bourbaki. And he calls Basmakova's claim (in [5]), that Diophantus uses negative numbers, a "nonsense", reproaching her for her "thoughtlessness". First, we disprove Schappacher's Bourbaki ...
Preprint
A generic polynomial solution for the differential equation of hypergeometric type and six sequences of orthogonal polynomials related to it
(2005)
In this work, we present a generic formula for the polynomial solution families of the well-known differential equation of hypergeometric type s(x)y"n(x) + t(x)y'n(x) - lnyn(x) = 0 and show that all the three classical orthogonal polynomial families as well as three finite orthogonal polynomial families, extracted from this equation, can be identified as special cases of this derived polynomial sequence. Some general properties of this sequence are also given.
Preprint
Modellierung und numerische Simulation der Thermoregulation von Früh- und Neugeborenen
(Universität Kassel, FB 17, Mathematik/Informatik, 2005)
Bei frühgeborenen Säuglingen spielt die Thermoregulation zur Aufrechterhaltung einer überlebenswichtigen Körpertemperatur durch Wärmeproduktion, -abgabe bzw. -aufnahme eine entscheidende Rolle. Der Einsatz moderner Inkubatoren soll die körpereigenen Thermoregulatoren unterstützen, und es ist im Hinblick auf verschiedene medizinische Fragestellungen wünschenswert, diesen Prozess modellieren zu können. Wir stellen ein einfaches Modell auf der Basis von partiellen Differentialgleichungen vor und beschreiben detailliert ...
Preprint
Asymptotics of number fields and the Cohen-Lenstra heuristics
(Universität Kassel, FB 17, Mathematik/Informatik, 2005)
We study the asymptotics conjecture of Malle for dihedral groups Dl of order 2l, where l is an odd prime. We prove the expected lower bound for those groups. For the upper bounds we show that there is a connection to class groups of quadratic number fields. The asymptotic behavior of those class groups is predicted by the Cohen-Lenstra heuristics. Under the assumption of this heuristic we are able to prove the expected upper bounds.
Preprint
Bieberbach's Conjecture, the de Branges and Weinstein Functions and the Askey-Gasper Inequality
(2005)
The Bieberbach conjecture about the coefficients of univalent functions of the unit disk was formulated by Ludwig Bieberbach in 1916 [Bieberbach1916]. The conjecture states that the coefficients of univalent functions are majorized by those of the Koebe function which maps the unit disk onto a radially slit plane. The Bieberbach conjecture was quite a difficult problem, and it was surprisingly proved by Louis de Branges in 1984 [deBranges1985] when some experts were rather trying to disprove it. It turned out that ...
Preprint
A crack on the interface of piezo-electric bodies
(Universität Kassel, FB 17, Mathematik/Informatik, 2005)
Singularities of elastic and electric fields are investigated at the tip of a crack on the interface of two anisotropic piezo-electric media under various boundary conditions on the crack surfaces. The Griffith formulae are obtained for increments of energy functionals due to growth of the crack and the notion of the energy release matrix is introduced. Normalization conditions for bases of singular solution are proposed to adapt them to the energy, stress, and deformation fracture criteria. Connections between these ...
Preprint
Degrees of Free Word-Order and Freely Rewriting Restarting Automata
(Universität Kassel, FB 17, Mathematik/Informatik, 2005)
In natural languages with a high degree of word-order freedom syntactic phenomena like dependencies (subordinations) or valencies do not depend on the word-order (or on the individual positions of the individual words). This means that some permutations of sentences of these languages are in some (important) sense syntactically equivalent. Here we study this phenomenon in a formal way. Various types of j-monotonicity for restarting automata can serve as parameters for the degree of word-order freedom and for the ...
Preprint
On the Descriptional Complexity of Simple RL-Automata
(2006)
Analysis by reduction is a method used in linguistics for checking the correctness of sentences of natural languages.
This method is modelled by restarting automata. Here we study a new type of restarting automaton, the so-called t-sRL-automaton, which is an RL-automaton that is rather restricted in that it has a window of size 1 only, and that it works under a minimal acceptance condition.
On the other hand, it is allowed to perform up to t rewrite (that is, delete) steps per cycle.
We focus on the descriptional ...
Preprint
A generalization of Student’s t-distribution from the viewpoint of special functions
(2005)
Student’s t-distribution has found various applications in mathematical statistics. One of the main properties of the t-distribution is to converge to the normal distribution as the number of samples tends to infinity. In this paper, by using a Cauchy integral we introduce a generalization of the t-distribution function with four free parameters and show that it converges to the normal distribution again. We provide a comprehensive treatment of mathematical properties of this new distribution. Moreover, since the ...
Preprint
Artificial boundary conditions for the Stokes and Navier-Stokes equations in domains that are layer-like at infinity
(Universität Kassel, FB 17, Mathematik/Informatik, 2005)
Artificial boundary conditions are presented to approximate solutions to Stokes- and Navier-Stokes problems in domains that are layer-like at infinity. Based on results about existence and asymptotics of the solutions v^infinity, p^infinity to the problems in the unbounded domain Omega the error v^infinity - v^R, p^infinity - p^R is estimated in H^1(Omega_R) and L^2(Omega_R), respectively. Here v^R, p^R are the approximating solutions on the truncated domain Omega_R, the parameter R controls the exhausting of Omega. ...