DSpace Collection:https://opendata.uni-halle.de//handle/1981185920/135332024-08-08T10:40:55Z2024-08-08T10:40:55ZOn the downward Löwenheim-Skolem Theorem for elementary submodelsKunik, Matthiashttps://opendata.uni-halle.de//handle/1981185920/1173222024-03-19T09:44:30Z2024-03-01T00:00:00ZTitle: 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.2024-03-01T00:00:00ZRadially symmetric solutions of the ultra-relativistic Euler equations in several space dimensionsKunik, MatthiasKolb, AdrainMüller, SiegfriedThein, Ferdinandhttps://opendata.uni-halle.de//handle/1981185920/1171962024-03-18T09:15:04Z2024-02-22T00:00:00ZTitle: 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.2024-02-22T00:00:00ZOn the formulas for Pi(x) and Psi(x) of Riemann and von MangoldtKunik, Matthiashttps://opendata.uni-halle.de//handle/1981185920/1108702023-07-11T01:20:40Z2023-01-01T00:00:00ZTitle: 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.2023-01-01T00:00:00ZReduced set theoryKunik, Matthiashttps://opendata.uni-halle.de//handle/1981185920/1076992023-06-24T02:42:48Z2023-06-22T00:00:00ZTitle: 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