71,782 research outputs found
Pattern formation for the Swift-Hohenberg equation on the hyperbolic plane
We present an overview of pattern formation analysis for an analogue of the
Swift-Hohenberg equation posed on the real hyperbolic space of dimension two,
which we identify with the Poincar\'e disc D. Different types of patterns are
considered: spatially periodic stationary solutions, radial solutions and
traveling waves, however there are significant differences in the results with
the Euclidean case. We apply equivariant bifurcation theory to the study of
spatially periodic solutions on a given lattice of D also called H-planforms in
reference with the "planforms" introduced for pattern formation in Euclidean
space. We consider in details the case of the regular octagonal lattice and
give a complete descriptions of all H-planforms bifurcating in this case. For
radial solutions (in geodesic polar coordinates), we present a result of
existence for stationary localized radial solutions, which we have adapted from
techniques on the Euclidean plane. Finally, we show that unlike the Euclidean
case, the Swift-Hohenberg equation in the hyperbolic plane undergoes a Hopf
bifurcation to traveling waves which are invariant along horocycles of D and
periodic in the "transverse" direction. We highlight our theoretical results
with a selection of numerical simulations.Comment: Dedicated to Klaus Kirchg\"assne
Bifurcation of hyperbolic planforms
Motivated by a model for the perception of textures by the visual cortex in
primates, we analyse the bifurcation of periodic patterns for nonlinear
equations describing the state of a system defined on the space of structure
tensors, when these equations are further invariant with respect to the
isometries of this space. We show that the problem reduces to a bifurcation
problem in the hyperbolic plane D (Poincar\'e disc). We make use of the concept
of periodic lattice in D to further reduce the problem to one on a compact
Riemann surface D/T, where T is a cocompact, torsion-free Fuchsian group. The
knowledge of the symmetry group of this surface allows to carry out the
machinery of equivariant bifurcation theory. Solutions which generically
bifurcate are called "H-planforms", by analogy with the "planforms" introduced
for pattern formation in Euclidean space. This concept is applied to the case
of an octagonal periodic pattern, where we are able to classify all possible
H-planforms satisfying the hypotheses of the Equivariant Branching Lemma. These
patterns are however not straightforward to compute, even numerically, and in
the last section we describe a method for computation illustrated with a
selection of images of octagonal H-planforms.Comment: 26 pages, 11 figure
Emerging Consciousness as a Result of Complex-Dynamical Interaction Process
A quite general interaction process within a multi-component system is analysed by the extended effective potential method, liberated from usual limitations of perturbation theory or integrable model. The obtained causally complete solution of the many-body problem reveals the phenomenon of dynamic multivaluedness, or redundance, of emerging, incompatible system realisations and dynamic entanglement of system components within each realisation. The ensuing concept of dynamic complexity (and related intrinsic chaoticity) is absolutely universal and can be applied to the problem of consciousness that emerges now as a high enough, properly specified level of unreduced complexity of a suitable interaction process. This complexity level can be identified with the appearance of bound, permanently localised states in the multivalued brain dynamics from strongly chaotic states of unconscious intelligence, by analogy with classical behaviour emergence from quantum states at much lower levels of world dynamics. We show that the main properties of this dynamically emerging consciousness (and intelligence, at the preceding complexity level) correspond to empirically derived properties of natural versions and obtain causally substantiated conclusions about their artificial realisation, including the fundamentally justified paradigm of genuine machine consciousness. This rigorously defined machine consciousness is different from both natural consciousness and any mechanistic, dynamically single-valued imitation of the latter. We use then the same, truly universal concept of complexity to derive equally rigorous conclusions about mental and social implications of the machine consciousness paradigm, demonstrating its indispensable role in the next stage of civilisation development
Superlattice Patterns in Surface Waves
We report novel superlattice wave patterns at the interface of a fluid layer
driven vertically. These patterns are described most naturally in terms of two
interacting hexagonal sublattices. Two frequency forcing at very large aspect
ratio is utilized in this work. A superlattice pattern ("superlattice-I")
consisting of two hexagonal lattices oriented at a relative angle of 22^o is
obtained with a 6:7 ratio of forcing frequencies. Several theoretical
approaches that may be useful in understanding this pattern have been proposed.
In another example, the waves are fully described by two superimposed hexagonal
lattices with a wavelength ratio of sqrt(3), oriented at a relative angle of
30^o. The time dependence of this "superlattice-II" wave pattern is unusual.
The instantaneous patterns reveal a time-periodic stripe modulation that breaks
the 6-fold symmetry at any instant, but the stripes are absent in the time
average. The instantaneous patterns are not simply amplitude modulations of the
primary standing wave. A transition from the superlattice-II state to a 12-fold
quasi-crystalline pattern is observed by changing the relative phase of the two
forcing frequencies. Phase diagrams of the observed patterns (including
superlattices, quasicrystalline patterns, ordinary hexagons, and squares) are
obtained as a function of the amplitudes and relative phases of the driving
accelerations.Comment: 15 pages, 14 figures (gif), to appear in Physica
Orientational Harmonic Model for Illusory Boundary Formation in Biological Vision
An extension to the Boundary Contour System model is proposed to account for boundary completion through vertices with arbitrary numbers of orientations, in a manner consistent with psychophysical observartions, by way of harmonic resonance in a neural architecture
Proliferation of anomalous symmetries in colloidal monolayers subjected to quasiperiodic light fields
Quasicrystals provide a fascinating class of materials with intriguing
properties. Despite a strong potential for numerous technical applications, the
conditions under which quasicrystals form are still poorly understood.
Currently, it is not clear why most quasicrystals hold 5- or 10-fold symmetry
but no single example with 7 or 9-fold symmetry has ever been observed. Here we
report on geometrical constraints which impede the formation of quasicrystals
with certain symmetries in a colloidal model system. Experimentally, colloidal
quasicrystals are created by subjecting micron-sized particles to
two-dimensional quasiperiodic potential landscapes created by n=5 or seven
laser beams. Our results clearly demonstrate that quasicrystalline order is
much easier established for n = 5 compared to n = 7. With increasing laser
intensity we observe that the colloids first adopt quasiperiodic order at local
areas which then laterally grow until an extended quasicrystalline layer forms.
As nucleation sites where quasiperiodicity originates, we identify highly
symmetric motifs in the laser pattern. We find that their density strongly
varies with n and surprisingly is smallest exactly for those quasicrystalline
symmetries which have never been observed in atomic systems. Since such high
symmetry motifs also exist in atomic quasicrystals where they act as
preferential adsorption sites, this suggests that it is indeed the deficiency
of such motifs which accounts for the absence of materials with e.g. 7-fold
symmetry
Image Sampling with Quasicrystals
We investigate the use of quasicrystals in image sampling. Quasicrystals
produce space-filling, non-periodic point sets that are uniformly discrete and
relatively dense, thereby ensuring the sample sites are evenly spread out
throughout the sampled image. Their self-similar structure can be attractive
for creating sampling patterns endowed with a decorative symmetry. We present a
brief general overview of the algebraic theory of cut-and-project quasicrystals
based on the geometry of the golden ratio. To assess the practical utility of
quasicrystal sampling, we evaluate the visual effects of a variety of
non-adaptive image sampling strategies on photorealistic image reconstruction
and non-photorealistic image rendering used in multiresolution image
representations. For computer visualization of point sets used in image
sampling, we introduce a mosaic rendering technique.Comment: For a full resolution version of this paper, along with supplementary
materials, please visit at
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