http://popups.lib.uliege.be/1373-5411/index.php?id=2805 Index terms fr 0 http://popups.lib.uliege.be/1373-5411/index.php?id=2424 This paper deals with the Euler and Incursive algorithms of the nonlinear pendulum. The Euler algorithm is unstable. The incursive algorithms show a stable solution as an orbital stabilify for small values of the time step. For larger values of the time step, the incursive algorithms show an orbital stability for small values of the initial conditions and a chaotic sea for larger initial conditions. Tue, 20 Aug 2024 11:27:28 +0200 Tue, 08 Oct 2024 14:44:03 +0200 http://popups.lib.uliege.be/1373-5411/index.php?id=2424 http://popups.lib.uliege.be/1373-5411/index.php?id=3867 Algorithmic topology is the spanning of an algorithm on a topological structure. The common calculus with paper and pen shows that all the recursive functions can be spanned on Euclidean planes. It is known that two topological structures are identical if and only if cut-pasting operations don't need to transform one in the other. Dubois' third stage (identification of incursive algorithm last row and column respectively with its first row and column) gives to incursive algorithms a spanning only on a torus that can be transformed in Euclidean plane only by cut-pasting operations. Thus incursive algorithms couldn't reduce to recursive algorithms and Church's hypothesis couldn't be true. Now, observe the affinity between topologic cut-pasting operations, Dubois' third stage and quantum entanglement. This last one can be considered either two "entanglements" in incursive algorithms or a cut-pasting operation on Euclidean plane on which such an algorithm is spanned to transform such a plane in torus. Is quantum entanglement simply the inadequacy of algorithms that can be spanned only on Euclidean plains to represent quantum mechanics? The same question could have value for some complex biological systems. Mon, 30 Sep 2024 14:20:43 +0200 Tue, 08 Oct 2024 14:43:38 +0200 http://popups.lib.uliege.be/1373-5411/index.php?id=3867 http://popups.lib.uliege.be/1373-5411/index.php?id=2804 This paper firstly recalls the mathematical model of the Newtonian gravitation law applied to the solar system dealing with the Sun and the Mercury planet. The orbits of all the planets are given by closed ellipses in the Newton paradigm. This paper continues with the introduction of the relativistic correction to this Newton gravitation law, for exhibiting the precession of orbits and, more particularly, the advance of the perihelion of the Mercury planet. The following section gives the Euler and the First Incursive algorithms for the classical Newton gravitation law and the relativistic correction of this Newtonian law. The last section of this paper gives the result of the numerical simulations of the Mercury orbit around the Sun. It is shown that the simulation with the Euler algorithm is not stable and does not give a closed ellipse to the Mercury orbit with the Newtonian law. The simulation with the First Incursive algorithm gives a perfect simulation of the Newtonian orbit in 88 days. Then, the simulation with this First Incursive algorithm of the relativistic Newtonian gravitation shows the correct Mercury precession angle that is equal to 80 = 180 degrees, after 15,000 centuries, in agreement with the experimental data and Einstein relativity. Tue, 03 Sep 2024 14:55:18 +0200 Tue, 03 Sep 2024 14:55:27 +0200 http://popups.lib.uliege.be/1373-5411/index.php?id=2804