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A Study of Extensions of Classical Summation Theorems for the Series ₃F₂ and ₄F₃ with Applications
(2021-03-16)
Very recently, Masjed-Jamei & Koepf [Some summation theorems for generalized hypergeometric functions, Axioms, 2018, 7, 38, 10.3390/axioms 7020038] established some summation theorems for the generalized hypergeometric functions. The aim of this paper is to establish extensions of some of their summation theorems in the most general form. As an application, several Eulerian-type and Laplace-type integrals have also been given. Results earlier obtained by Jun et al. and Koepf et al. follow special cases of our main findings.
Aufsatz
On Multivariate Bernstein Polynomials
(2020-08-21)
In this paper, we first revisit and bring together as a sort of survey, properties of Bernstein polynomials of one variable. Secondly, we extend the results from one variable to several ones, namely—uniform convergence, uniform convergence of the derivatives, order of convergence, monotonicity, fixed sign for the p-th derivative, and deduction of the upper and lower bounds of Bernstein polynomials from those of the corresponding functions.
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A Logic Based Approach to Finding Real Singularities of Implicit Ordinary Differential Equations
(2020-06-17)
We discuss the effective computation of geometric singularities of implicit ordinary differential equations over the real numbers using methods from logic. Via the Vessiot theory of differential equations, geometric singularities can be characterised as points where the behaviour of a certain linear system of equations changes. These points can be discovered using a specifically adapted parametric generalisation of Gaussian elimination combined with heuristic simplification techniques and real quantifier elimination ...
Aufsatz
Efficient Solution of Distributed MIP in Control of Networked Systems
(2021-01-25)
Certain classes of optimization‐based control problems stated for networked systems involving hybrid dynamics and logical constraints can be cast into Mixed‐Integer Programming (MIP) problems. Since these belong to the complexity class NP‐hard, the motivation arises to find approximations of the optimal solution by distributed solution efficiently. For the cases that the cost functional is linear or quadratic and the constraints are linear, this paper proposes an alternative to the standard centralized schemes, by ...
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The Impact of Visualizing Nested Sets. An Empirical Study on Tree Diagrams and Unit Squares
(2017-01-06)
It is an ongoing debate, what properties of visualizations increase people’s performance when solving Bayesian reasoning tasks. In the discussion of the properties of two visualizations, i.e., the tree diagram and the unit square, we emphasize how both visualizations make relevant subset relations transparent. Actually, the unit square with natural frequencies reveals the subset relation that is essential for the Bayes’ rule in a numerical and geometrical way whereas the tree diagram with natural frequencies does it ...
Aufsatz
Lie algebra for rotational subsystems of a driven asymmetric top
(2022-05-11)
We present an analytical approach to construct the Lie algebra of finite-dimensional subsystems of the driven asymmetric top rotor. Each rotational level is degenerate due to the isotropy of space, and the degeneracy increases with rotational excitation. For a given rotational excitation, we determine the nested commutators between drift and drive Hamiltonians using a graph representation. We then generate the Lie algebra for subsystems with arbitrary rotational excitation using an inductive argument.
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Numerical Simulation of Viscoelastic Fluid‐Structure Interaction Problems and Drug Therapy in the Eye
(2021-01-25)
The vitreous is a fluid‐like viscoelastic transparent medium located in the center of the human eye and is surrounded by hyperelastic structures like the sclera, lens and iris. This naturally gives rise to a fluid‐structure interaction (FSI) problem. While the healthy vitreous is viscoelastic and described by a viscoelastic Burgers‐type equation, the aging vitreous liquefies and is therefore modeled by the Navier‐Stokes equations. We derive a monolithic variational formulation employing the arbitrary Lagrangian ...
Aufsatz
Derivation of Analytical, Closed‐form Formulas for the Calculations of Instantaneous Screw Axes of Arbitrary Rigid 3D Multi‐Body Systems
(2021-01-25)
A method to calculate the parameters for each instantaneous and relative instantaneous screw axis in an arbitrary rigid multi‐body system is presented. At first a kinematic analysis is performed which calculates the displacements of all nodes and the rotation of each body. In the next step the displacements of the nodes and the rotation of the bodies are used to calculate the instantaneous screw axis of each body. With the information on the instantaneous screw axes the instantaneous relative screw axes can be computed ...
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Fourier Analysis of DG Schemes for Advection‐Diffusion
(2021-01-25)
This work compares the wave propagation properties of discontinuous Galerkin (DG) schemes for advection‐diffusion problems in particular with respect to the discretization of diffusion terms. Extending previous investigations, the advection discretization now additionally varies between the choices of central or upwind fluxes. The results show that a previously recognized better performance of central schemes for well‐resolved problems only hold for even polynomial degrees and that upwind‐type discretizations also ...
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On the non-global linear stability and spurious fixed points of MPRK schemes with negative RK parameters
(2024-02-13)
Recently, a stability theory has been developed to study the linear stability of modified Patankar–Runge–Kutta (MPRK) schemes. This stability theory provides sufficient conditions for a fixed point of an MPRK scheme to be stable as well as for the convergence of an MPRK scheme towards the steady state of the corresponding initial value problem, whereas the main assumption is that the initial value is sufficiently close to the steady state. Initially, numerical experiments in several publications indicated that these ...