https://opendata.uni-halle.de//handle/1981185920/13533 2026-04-15T12:05:53Z https://opendata.uni-halle.de//handle/1981185920/117322 Title: On the downward Löwenheim-Skolem Theorem for elementary submodels Author(s): Kunik, Matthias Abstract: We introduce a new definition of a model for a formal mathematical system. The definition is based upon the substitution in the formal systems, which allows a purely algebraic approach to model theory. This is very suitable for applications due to a general syntax used in the formal systems. For our models we present a new proof of the downward Löwenheim-Skolem Theorem for elementary submodels. 2024-03-01T00:00:00Z https://opendata.uni-halle.de//handle/1981185920/117196 Title: Radially symmetric solutions of the ultra-relativistic Euler equations in several space dimensions Author(s): Kunik, Matthias; Kolb, Adrain; Müller, Siegfried; Thein, Ferdinand Abstract: The ultra-relativistic Euler equations for an ideal gas are described in terms of the pressure, the spatial part of the dimension- less four-velocity and the particle density. Radially symmetric solutions of these equations are studied in two and three space dimensions. Of particular interest in the solutions are the formation of shock waves and a pressure blow up. For the investigation of these phenomena we develop a one-dimensional scheme using radial symmetry and integral conservation laws. We compare the numerical results with solutions of multi-dimensional high-order numerical schemes for general initial data in two space dimensions. The presented test cases and results may serve as interesting benchmark tests for multi-dimensional solvers. 2024-02-22T00:00:00Z https://opendata.uni-halle.de//handle/1981185920/110870 Title: On the formulas for Pi(​x) and Psi(x) of Riemann and von Mangoldt Author(s): Kunik, Matthias Abstract: Using the Mellin transform and the complex exponential integral we derive various representation formulas for the factors of the entire functions in Hadamards product theorem. The application of these results on Riemann’s zeta function leads to a new derivation of Rie- mann’s prime number formula for Pi(x). We will thereby present a correct version of this formula, which is given in a wrong way in the literature. Using the nontrivial zeros of the Zeta function we also obtain explicit formulas for regularizations of von Mangoldt’s function Psi(x). These regularizations are based on cardinal B-splines and Gaussian integration kernels, which are related by the Central Limit Theorem. Our results will then be generalized to a windowed Mellin or Fourier transform with a Gaussian window function. 2023-01-01T00:00:00Z https://opendata.uni-halle.de//handle/1981185920/107699 Title: Reduced set theory Author(s): Kunik, Matthias Abstract: We present a new fragment of axiomatic set theory for pure sets and for the iteration of power sets within given transitive sets. It turns out that this formal system admits an interesting hierarchy of models with true membership relation and with only finite or countably infinite ordinals. Still a considerable part of mathematics can be formalized within this system. 2023-06-22T00:00:00Z