7,521 research outputs found
An L-Point Characterization of Normality and Normalizer of an L-Subgroup of an L-Group
AbstractIn this paper, we study the notion of normal L-subgroup of an L-group and provide its characterization by an L-point. We also provide a construction of the normalizer of an L-subgroup of a given L-group by using L-points. Moreover, we also discuss the product, homomorphic images and homomorphic preimages of normalizers
Diversity of knot solitons in liquid crystals manifested by linking of preimages in torons and hopfions
Topological solitons are knots in continuous physical fields classified by
non-zero Hopf index values. Despite arising in theories that span many branches
of physics, from elementary particles to condensed matter and cosmology, they
remain experimentally elusive and poorly understood. We introduce a method of
experimental and numerical analysis of such localized structures in liquid
crystals that, similar to the mathematical Hopf maps, relates all points of the
medium's order parameter space to their closed-loop preimages within the
three-dimensional solitons. We uncover a surprisingly large diversity of
naturally occurring and laser-generated topologically nontrivial solitons with
differently knotted nematic fields, which previously have not been realized in
theories and experiments alike. We discuss the implications of the liquid
crystal's non-polar nature on the knot soliton topology and how the medium's
chirality, confinement and elastic anisotropy help to overcome the constrains
of the Hobart-Derrick theorem, yielding static three-dimensional solitons
without or with additional defects. Our findings will establish chiral nematics
as a model system for experimental exploration of topological solitons and may
impinge on understanding of such nonsingular field configurations in other
branches of physics, as well as may lead to technological application
Complex patterns on the plane: different types of basin fractalization in a two-dimensional mapping
Basins generated by a noninvertible mapping formed by two symmetrically
coupled logistic maps are studied when the only parameter \lambda of the system
is modified. Complex patterns on the plane are visualised as a consequence of
basins' bifurcations. According to the already established nomenclature in the
literature, we present the relevant phenomenology organised in different
scenarios: fractal islands disaggregation, finite disaggregation, infinitely
disconnected basin, infinitely many converging sequences of lakes, countable
self-similar disaggregation and sharp fractal boundary. By use of critical
curves, we determine the influence of zones with different number of first rank
preimages in the mechanisms of basin fractalization.Comment: 19 pages, 11 figure
On the number of solutions of a transcendental equation arising in the theory of gravitational lensing
The equation in the title describes the number of bright images of a point
source under lensing by an elliptic object with isothermal density. We prove
that this equation has at most 6 solutions. Any number of solutions from 1 to 6
can actually occur.Comment: 26 pages, 12 figure
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