142 research outputs found
Hidden structure in the randomness of the prime number sequence?
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
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
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
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
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 . 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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