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2024

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Vol 1, No 1 (2023)

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Why the big bang never happened

If general relativity is correct, then the origin of the universe is a simple mathematical problem. The Friedmann equation in cosmology is a well-structured ordinary diﬀerential equation, and the global properties of its solutions can be qualitatively analyzed by the phase-trajectory method. In this paper we show that the total energy density of matter in the universe is positive, and the total pressure near the Big Bang is negative. By analyzing the global properties of the solutions to the Friedmann equation according to these two conditions of state functions, we ﬁnd that the Big Bang is impossible, and the space must be a closed 3-dimensional sphere, the cosmological constant is likely to be zero, and the evolution of the universe should be cyclic. The analysis and the proof are simple and straight forward, therefore these conclusions should be reliable.

Fermat surfaces and hypercubes

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On types of stability in Hamiltonian systems

Alexander Dmitrievich Bruno, Alexander Borisovich Batkhin

Article ID: 2269
Vol 1, Issue 1, 2023

DOI: https://doi.org/10.54517/mss.v1i1.2269

Received: 3 August 2023; Accepted: 21 October 2023; Available online: 3 November 2023; Issue release: 15 November 2023

VIEWS - 749 (Abstract)

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Abstract

We consider conditions of three types of stability: Lyapunov, formal and weak of a stationary solution, and of a periodic solution in a Hamiltonian system with a finite number of degrees of freedom. The conditions contain restrictions on the order of resonances and some inequalities for initial coefficients of the normal forms of the Hamiltonian functions. We show that the number-theoretical analysis of frequencies can help in proof of stability. We also estimate the orders of solutions’ divergence from the stationary or the periodic ones under lack of formal stability.

Keywords

stationary solution; periodic solution; normal form; formal stability; weak stability

References

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17. Batkhin AB, Khaydarov ZK. Calculation of a strong resonance condition in a Hamiltonian system. Computational Mathematics and Mathematical Physics 2023; 63(5): 687–703. doi: 10.1134/S0965542523050068

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21. Markeev AP. Stability of planar rotations of a satellite in a circular orbit. Mechanics of Solids 2006; 41(4): 46–63.

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Copyright (c) 2023 Alexander Dmitrievich Bruno, Alexander Borisovich Batkhin

License URL: https://creativecommons.org/licenses/by/4.0

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Cairo University, Egypt

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