142 research outputs found

    Hidden structure in the randomness of the prime number sequence?

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    We report a rigorous theory to show the origin of the unexpected periodic behavior seen in the consecutive differences between prime numbers. We also check numerically our findings to ensure that they hold for finite sequences of primes, that would eventually appear in applications. Finally, our theory allows us to link with three different but important topics: the Hardy-Littlewood conjecture, the statistical mechanics of spin systems, and the celebrated Sierpinski fractal.Comment: 13 pages, 5 figures. New section establishing connection with the Hardy-Littlewood theory. Published in the journal where the solved problem was first describe

    The mathematical basis for deterministic quantum mechanics

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    If there exists a classical, i.e. deterministic theory underlying quantum mechanics, an explanation must be found of the fact that the Hamiltonian, which is defined to be the operator that generates evolution in time, is bounded from below. The mechanism that can produce exactly such a constraint is identified in this paper. It is the fact that not all classical data are registered in the quantum description. Large sets of values of these data are assumed to be indistinguishable, forming equivalence classes. It is argued that this should be attributed to information loss, such as what one might suspect to happen during the formation and annihilation of virtual black holes. The nature of the equivalence classes is further elucidated, as it follows from the positivity of the Hamiltonian. Our world is assumed to consist of a very large number of subsystems that may be regarded as approximately independent, or weakly interacting with one another. As long as two (or more) sectors of our world are treated as being independent, they all must be demanded to be restricted to positive energy states only. What follows from these considerations is a unique definition of energy in the quantum system in terms of the periodicity of the limit cycles of the deterministic model.Comment: 17 pages, 3 figures. Minor corrections, comments and explanations adde

    BiEntropy, TriEntropy and Primality

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    The order and disorder of binary representations of the natural numbers < 2^8 is measured using the BiEntropy function. Significant differences are detected between the primes and the non primes. The BiEntropic prime density is shown to be quadratic with a very small Gaussian distributed error. The work is repeated in binary using a monte carlo simulation for a sample of the natural numbers < 2^32 and in trinary for all natural numbers < 3^9 with similar but cubic results. We find a significant relationship between BiEntropy and TriEntropy such that we can discriminate between the primes and numbers divisible by six. We discuss the theoretical underpinnings of these results and show how they generalise to give a tight bound on the variance of Pi(x) - Li(x) for all x. This bound is much tighter than the bound given by Von Koch in 1901 as an equivalence for proof of the Riemann Hypothesis. Since the primes are Gaussian due to a simple induction on the binary derivative, this implies that the twin primes conjecture is true. We also provide absolutely convergent asymptotes for the numbers of Fermat and Mersenne primes in the appendices.Comment: 18 Pages, 12 Colour Figures, 10 Tables. Minor updates & typos. Supplementary materials now available at https://doi.org/10.6084/m9.figshare.11743749. Empirical results first presented at ANPA 40, University of Liverpool, 11th August 2019. Theoretical Basis first presented at PANPA 2019 meeting, Anstruther, Fife, Scotland, 21st August 201

    Physics of Interpulse Emission in Radio Pulsars

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    The magnetized induced Compton scattering off the particles of the ultrarelativistic electron-positron plasma of pulsar is considered. The main attention is paid to the transverse regime of the scattering, which holds in a moderately strong magnetic field. We specifically examine the problem on induced transverse scattering of the radio beam into the background, which takes place in the open field line tube of a pulsar. In this case, the radiation is predominantly scattered backwards and the scattered component may grow considerably. Based on this effect, we for the first time suggest a physical explanation of the interpulse emission observed in the profiles of some pulsars. Our model can naturally account for the peculiar spectral and polarization properties of the interpulses. Furthermore, it implies a specific connection of the interpulse to the main pulse, which may reveal itself in the consistent intensity fluctuations of the components at different timescales. Diverse observational manifestations of this connection, including the moding behavior of PSR B1822-09, the peculiar temporal and frequency structure of the giant interpulses in the Crab pulsar, and the intrinsic phase correspondence of the subpulse patterns in the main pulse and the interpulse of PSR B1702-19, are discussed in detail. It is also argued that the pulse-to-pulse fluctuations of the scattering efficiency may lead to strong variability of the interpulse, which is yet to be studied observationally. In particular, some pulsars may exhibit transient interpulses, i.e. the scattered component may be detectable only occasionally.Comment: 28 pages, 2 figures. Accepted for publication in Ap

    Crystal properties of eigenstates for quantum cat maps

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    Using the Bargmann-Husimi representation of quantum mechanics on a torus phase space, we study analytically eigenstates of quantized cat maps. The linearity of these maps implies a close relationship between classically invariant sublattices on the one hand, and the patterns (or `constellations') of Husimi zeros of certain quantum eigenstates on the other hand. For these states, the zero patterns are crystals on the torus. As a consequence, we can compute explicit families of eigenstates for which the zero patterns become uniformly distributed on the torus phase space in the limit 0\hbar\to 0. This result constitutes a first rigorous example of semi-classical equidistribution for Husimi zeros of eigenstates in quantized one-dimensional chaotic systems.Comment: 43 pages, LaTeX, including 7 eps figures Some amendments were made in order to clarify the text, mainly in the 4 first sections. Figures are unchanged. To be published in: Nonlinearit
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