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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’s last theorem appears not as a unique property of natural numbers but as the bottom line of extended possible issues involving larger dimensions and powers when observed from a natural vector space viewpoint. The fabric of this general Fermat’s theorem structure consists of a well-defined set of vectors associated with dimensional vector spaces and the Minkowski norms one can define there. Here, a special vector set is studied and named a Fermat surface. Besides, a connection between Fermat surfaces and hypercubes is unveiled.
Some applications of partial differential equations in medical image processing
Article ID: 3148
Vol 3, Issue 2, 2025
DOI: https://doi.org/10.54517/mss3148
Vol 3, Issue 2, 2025
Received: 9 December 2024; Accepted: 6 February 2025; Available online: 8 April 2025; Issue release: 30 June 2025
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Abstract
The second-order nonlinear diffusion parabolic partial differential equations models have been widely applied in image restoration. However, the numerical results in the literature treat only the case without source term. In this paper, we have developed a general calculation code which is based in a consistent explicit approximation finite difference method scheme. Furthermore, the paper provides satisfying answers with a nonlinear source term relying on the image solution and its gradient. Numerical experiments are presented to show the robustness of the cases with source term to obtain better results in image denoising restoration using measures as Peak Signal-to-Noise Ratio (PSNR) and SNR of filtering and noisy image.
Keywords
nonlinear parabolic; reaction-diffusion; medical image processing
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Cairo University, Egypt
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