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A proof of the product integral using derivatives and and the limit of a difference quotient to certify its identity as an integral of the natural logarithm.

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On the basis of the conclusions selectively gleaned from the history of ontological thought, it was assumed that the real geometry of the space of the Universe is the geometry of topologically fixed 3-dimensional hyper sphere. It claims that the understanding of metaphysical principles of space formation is important, resulting in the simplest mathematical approach to the calculation of actual astronomical distances. The comparisons of calculated lengths of trajectories of photons with the actual intergalactic distances reported by observers for different \(z\)-values are carried out using the selected assumptions. Photon mileage \(L_f = f (L_z, R_D)\) don't equal to the astronomical distance \(L_z\) detected by \(z\) value if \(R_D\) is the discrete quantum length of one of 4 space dimensions, associated with the curvature of space as a whole. The research concludes there is a good agreement between data obtained this way for star brightness with experimental data, when we speak about the hypothesis presupposing an accelerating growth of the Universe and abnormal brightness of supernovae; besides conclusions imply that adjustments to this hypothesis forward in the search of the values of brightness attenuation undetectable by today's tools are needed, otherwise, it could be talk about a possible expansion of our Universe in our era without accelerating.

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An essay