95,725 research outputs found
Semiclassical wave functions and energy spectra in polygon billiards
A consistent scheme of semiclassical quantization in polygon billiards by
wave function formalism is presented. It is argued that it is in the spirit of
the semiclassical wave function formalism to make necessary rationalization of
respective quantities accompanied the procedure of the semiclassical
quantization in polygon billiards. Unfolding rational polygon billiards (RPB)
into corresponding Riemann surfaces (RS) periodic structures of the latter are
demonstrated with 2g independent periods on the respective multitori with g as
their genuses. However it is the two dimensional real space of the real linear
combinations of these periods which is used for quantizing RPB. A class of
doubly rational polygon billiards (DRPB) is distinguished for which these real
linear relations are rational and their semiclassical quantization by wave
function formalism is presented. It is shown that semiclassical quantization of
both the classical momenta and the energy spectra are determined completely by
periodic structure of the corresponding RS. Each RS is then reduced to
elementary polygon patterns (EPP) as its basic periodic elements. Each such EPP
can be glued to a torus of genus g. Semiclassical wave functions (SWF) are then
constructed on EPP. The SWF for DRPB appear to be exact. They satisfy the
Dirichlet, the Neumannn or the mixed boundary conditions. Not every mixing is
allowed however and a respective incompleteness of SWF is discussed. Dens
families of DRPB are used for approximate semiclassical quantization of RPB.
General rational polygons are quantized by approximating them by DRPB. An
extension of the formalism to irrational polygons is described as well. The
semiclassical approximations constructed in the paper are controlled by general
criteria of the eigenvalue theory. A relation between the superscar solutions
and SWF constructed in the paper is also discussed.Comment: 34 pages, 5 figure
Computational Universality and 1/f Noise in Elementary Cellular Automata
It is speculated that there is a relationship between 1/f noise and
computational universality in cellular automata. We use genetic algorithms to
search for one-dimensional and two-state, five-neighbor cellular automata which
have 1/f-type spectrum. A power spectrum is calculated from the evolution
starting from a random initial configuration. The fitness is estimated from the
power spectrum in consideration of the similarity to 1/f-type spectrum. The
result shows that the rule with the highest average fitness has a propagating
structure like other computationally universal cellular automata, although
computational universality of the rule has not been proved yet
Periodic behaviour of coronal mass ejections, eruptive events, and solar activity proxies during solar cycles 23 and 24
We report on the parallel analysis of the periodic behaviour of coronal mass
ejections (CMEs) based on 21 years [1996 -- 2016] of observations with the
SOHO/LASCO--C2 coronagraph, solar flares, prominences, and several proxies of
solar activity. We consider values of the rates globally and whenever possible,
distinguish solar hemispheres and solar cycles 23 and 24. Periodicities are
investigated using both frequency (periodogram) and time-frequency (wavelet)
analysis. We find that these different processes, in addition to following the
11-year Solar Cycle, exhibit diverse statistically significant
oscillations with properties common to all solar, coronal, and heliospheric
processes: variable periodicity, intermittence, asymmetric development in the
northern and southern solar hemispheres, and largest amplitudes during the
maximum phase of solar cycles, being more pronounced during solar cycle 23 than
the weaker cycle 24. However, our analysis reveals an extremely complex and
diverse situation. For instance, there exists very limited commonality for
periods of less than one year. The few exceptions are the periods of 3.1--3.2
months found in the global occurrence rates of CMEs and in the sunspot area
(SSA) and those of 5.9--6.1 months found in the northern hemisphere. Mid-range
periods of 1 and 2 years are more wide spread among the
studied processes, but exhibit a very distinct behaviour with the first one
being present only in the northern hemisphere and the second one only in the
southern hemisphere. These periodic behaviours likely results from the
complexity of the underlying physical processes, prominently the emergence of
magnetic flux.Comment: 33 pages, 15 figures, 2 table
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