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A discussion is given of early literature pertaining to the theory of vibration from the time of Pythagoras up through 1750. The paper attempts to give an analytical interpretation to early anecdotal works concerning Pythagoras and to publications of Galileo, Huygens, Hooke, Taylor, John Bernoulli, Leibniz and Euler. To bridge the “culture gap,” mathematical developments by the latter cited authors are, whenever appropriate, rephrased in modern notation, using, for the most part, only those techniques that should have been well known to the authors at the time. The emphasis is on what might be loosely called the physics (or the mathematical physics) of vibration.

Standard model has to be generalized to a “New Physics” beyond the Standard Model. Main problem is the lack of consistency SM with gravity. We analyse Kerr-Newman spinning particle which is consistent with gravity by nature and, contrary to opinion that gravity conflicts with quantum theory, we obtain that spinning Kerr’s gravity collaborates with quantum theory in the process of formation of spinning particle. The most dramatic is the shift of the fundamental scale from Planck to Compton distances.

In the QCD axion dark matter scenario with post-inflationary Peccei-Quinn symmetry breaking, the number density of axions, and hence the dark matter density, depends on the length of string per unit volume at cosmic time t, by convention written ζ/t². The expectation has been that the dimensionless parameter ζ tends to a constant ζ₀, a feature of a string network known as scaling. It has recently been claimed that in larger numerical simulations ζ shows a logarithmic increase with time. This case would result in a large enhancement of the string density at the QCD transition, and a substantial revision to the axion mass required for the axion to constitute all of the dark matter. With a set of new simulations of global strings we compare the standard scaling (constant-ζ) model to the logarithmic growth. We also study the approach to scaling, through measuring the root-mean-square velocity v as well as the scaled mean string separation x. We find good evidence for a fixed point in the phase-space analysis in the variables (x, v), providing a strong indication that standard scaling is taking place. We show that the approach to scaling can be well described by a two parameter velocity-one-scale (VOS) model, and show that the values of the parameters are insensitive to the initial state of the network. We conclude that the apparent corrections to ζ are artifacts of the initial conditions, rather than a property of the scaling network.