2,089 research outputs found
Dynamical Systems, Topology and Conductivity in Normal Metals
New observable integer-valued numbers of the topological origin were revealed
by the present authors studying the conductivity theory of single crystal 3D
normal metals in the reasonably strong magnetic field (). Our
investigation is based on the study of dynamical systems on Fermi surfaces for
the motion of semi-classical electron in magnetic field. All possible
asymptotic regimes are also found for based on the topological
classification of trajectories.Comment: Latex, 51 pages, 14 eps figure
Areas of Same Cardinal Direction
Cardinal directions, such as North, East, South, and West, are the foundation for qualitative spatial reasoning, a common field of GIS, Artificial Intelligence, and cognitive science. Such cardinal directions capture the relative spatial direction relation between a reference object and a target object, therefore, they are important search criteria in spatial databases. The projection-based model for such direction relations has been well investigated for point-like objects, yielding a relation algebra with strong inference power. The Direction Relation Matrix defines the simple region-to-region direction relations by approximating the reference object to a minimum bounding rectangle. Models that capture the direction between extended objects fall short when the two objects are close to each other. For instance, the forty-eight contiguous states of the US are colloquially considered to be South of Canada, yet they include regions that are to the North of some parts of Canada. This research considers the cardinal direction as a field that is distributed through space and may take on varying values depending on the location within a reference object. Therefore, the fundamental unit of space, the point, is used as a reference to form a point-based cardinal direction model. The model applies to capture the direction relation between point-to-region and region-to-region configurations. As such, the reference object is portioned into areas of same cardinal direction with respect to the target object. This thesis demonstrates there is a set of 106 cardinal point-to-region relations, which can be normalized by considering mirroring and 90° rotations, to a subset of 22 relations. The differentiating factor of the model is that a set of base relations defines the direction relation anywhere in the field, and the conceptual neighborhood graph of the base relations offers the opportunity to exploit the strong inference of point-based direction reasoning for simple regions of arbitrary shape. Considers the tiles and pockets of same cardinal direction, while a coarse model provides a union of all possible qualitative direction values between a reference region and a target region
Generalized modular transformations in 3+1D topologically ordered phases and triple linking invariant of loop braiding
In topologically ordered quantum states of matter in 2+1D (space-time
dimensions), the braiding statistics of anyonic quasiparticle excitations is a
fundamental characterizing property which is directly related to global
transformations of the ground-state wavefunctions on a torus (the modular
transformations). On the other hand, there are theoretical descriptions of
various topologically ordered states in 3+1D, which exhibit both point-like and
loop-like excitations, but systematic understanding of the fundamental physical
distinctions between phases, and how these distinctions are connected to
quantum statistics of excitations, is still lacking. One main result of this
work is that the three-dimensional generalization of modular transformations,
when applied to topologically ordered ground states, is directly related to a
certain braiding process of loop-like excitations. This specific braiding
surprisingly involves three loops simultaneously, and can distinguish different
topologically ordered states. Our second main result is the identification of
the three-loop braiding as a process in which the worldsheets of the three
loops have a non-trivial triple linking number, which is a topological
invariant characterizing closed two-dimensional surfaces in four dimensions. In
this work we consider realizations of topological order in 3+1D using
cohomological gauge theory in which the loops have Abelian statistics, and
explicitly demonstrate our results on examples with topological
order
Uniform Semiclassical Approximation for the Wigner Symbol in Terms of Rotation Matrices
A new uniform asymptotic approximation for the Wigner symbol is given in
terms of Wigner rotation matrices (-matrices). The approximation is uniform
in the sense that it applies for all values of the quantum numbers, even those
near caustics. The derivation of the new approximation is not given, but the
geometrical ideas supporting it are discussed and numerical tests are
presented, including comparisons with the exact -symbol and with the
Ponzano-Regge approximation.Comment: 44 pages plus 20 figure
Thirty Years of Turnstiles and Transport
To characterize transport in a deterministic dynamical system is to compute
exit time distributions from regions or transition time distributions between
regions in phase space. This paper surveys the considerable progress on this
problem over the past thirty years. Primary measures of transport for
volume-preserving maps include the exiting and incoming fluxes to a region. For
area-preserving maps, transport is impeded by curves formed from invariant
manifolds that form partial barriers, e.g., stable and unstable manifolds
bounding a resonance zone or cantori, the remnants of destroyed invariant tori.
When the map is exact volume preserving, a Lagrangian differential form can be
used to reduce the computation of fluxes to finding a difference between the
action of certain key orbits, such as homoclinic orbits to a saddle or to a
cantorus. Given a partition of phase space into regions bounded by partial
barriers, a Markov tree model of transport explains key observations, such as
the algebraic decay of exit and recurrence distributions.Comment: Updated and corrected versio
The State of the Art in Cartograms
Cartograms combine statistical and geographical information in thematic maps,
where areas of geographical regions (e.g., countries, states) are scaled in
proportion to some statistic (e.g., population, income). Cartograms make it
possible to gain insight into patterns and trends in the world around us and
have been very popular visualizations for geo-referenced data for over a
century. This work surveys cartogram research in visualization, cartography and
geometry, covering a broad spectrum of different cartogram types: from the
traditional rectangular and table cartograms, to Dorling and diffusion
cartograms. A particular focus is the study of the major cartogram dimensions:
statistical accuracy, geographical accuracy, and topological accuracy. We
review the history of cartograms, describe the algorithms for generating them,
and consider task taxonomies. We also review quantitative and qualitative
evaluations, and we use these to arrive at design guidelines and research
challenges
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