243 research outputs found
Fourier bases and Fourier frames on self-affine measures
This paper gives a review of the recent progress in the study of Fourier
bases and Fourier frames on self-affine measures. In particular, we emphasize
the new matrix analysis approach for checking the completeness of a mutually
orthogonal set. This method helps us settle down a long-standing conjecture
that Hadamard triples generates self-affine spectral measures. It also gives us
non-trivial examples of fractal measures with Fourier frames. Furthermore, a
new avenue is open to investigate whether the Middle Third Cantor measure
admits Fourier frames
Wavelets and graph -algebras
Here we give an overview on the connection between wavelet theory and
representation theory for graph -algebras, including the higher-rank
graph -algebras of A. Kumjian and D. Pask. Many authors have studied
different aspects of this connection over the last 20 years, and we begin this
paper with a survey of the known results. We then discuss several new ways to
generalize these results and obtain wavelets associated to representations of
higher-rank graphs. In \cite{FGKP}, we introduced the "cubical wavelets"
associated to a higher-rank graph. Here, we generalize this construction to
build wavelets of arbitrary shapes. We also present a different but related
construction of wavelets associated to a higher-rank graph, which we anticipate
will have applications to traffic analysis on networks. Finally, we generalize
the spectral graph wavelets of \cite{hammond} to higher-rank graphs, giving a
third family of wavelets associated to higher-rank graphs
A Review of Recent Developments in Atomic Processes for Divertors and Edge Plasmas
The most promising concepts for power and particle control in tokamaks and
other fusion experiments rely upon atomic processes to transfer the power and
momentum from the edge plasma to the plasma chamber walls. This places a new
emphasis on processes at low temperatures (1-200 eV) and high densities
(10^20-10^22 m^-3). The most important atomic processes are impurity and
hydrogen radiation, ionization, excitation, recombination, charge exchange,
radiation transport, molecular collisions, and elastic scattering of atoms,
molecules and ions. Important new developments have occurred in each of these
areas. The best available data for these processes and an assessment of their
role in plasma wall interactions are summarized, and the major areas where
improved data are needed are reviewed.Comment: Preprint for the 11th PSI meeting, postscript with 22 figures, 40
page
Natural Orbitals and BEC in traps, a diffusion Monte Carlo analysis
We investigate the properties of hard core Bosons in harmonic traps over a
wide range of densities. Bose-Einstein condensation is formulated using the
one-body Density Matrix (OBDM) which is equally valid at low and high
densities. The OBDM is calculated using diffusion Monte Carlo methods and it is
diagonalized to obtain the "natural" single particle orbitals and their
occupation, including the condensate fraction. At low Boson density, , where and is the hard core diameter, the condensate is
localized at the center of the trap. As increases, the condensate moves
to the edges of the trap. At high density it is localized at the edges of the
trap. At the Gross-Pitaevskii theory of the condensate
describes the whole system within 1%. At corrections are
3% to the GP energy but 30% to the Bogoliubov prediction of the condensate
depletion. At , mean field theory fails. At , the Bosons behave more like a liquid He droplet than a trapped Boson
gas.Comment: 13 pages, 14 figures, submitted Phys. Rev.
The Complexity of the Empire Colouring Problem
We investigate the computational complexity of the empire colouring problem
(as defined by Percy Heawood in 1890) for maps containing empires formed by
exactly countries each. We prove that the problem can be solved in
polynomial time using colours on maps whose underlying adjacency graph has
no induced subgraph of average degree larger than . However, if , the problem is NP-hard even if the graph is a forest of paths of arbitrary
lengths (for any , provided .
Furthermore we obtain a complete characterization of the problem's complexity
for the case when the input graph is a tree, whereas our result for arbitrary
planar graphs fall just short of a similar dichotomy. Specifically, we prove
that the empire colouring problem is NP-hard for trees, for any , if
(and polynomial time solvable otherwise). For arbitrary
planar graphs we prove NP-hardness if for , and , for . The result for planar graphs also proves the NP-hardness of colouring
with less than 7 colours graphs of thickness two and less than colours
graphs of thickness .Comment: 23 pages, 12 figure
Eco-evolutionary dynamics on deformable fitness landscapes
Conventional approaches to modelling ecological dynamics often do not include evolutionary changes in the genetic makeup of component species and, conversely, conventional approaches to modelling evolutionary changes in the genetic makeup of a population often do not include ecological dynamics. But recently there has been considerable interest in understanding the interaction of evolutionary and ecological dynamics as coupled processes. However, in the context of complex multi-species ecosytems, especially where ecological and evolutionary timescales are similar, it is difficult to identify general organising principles that help us understand the structure and behaviour of complex ecosystems. Here we introduce a simple abstraction of coevolutionary interactions in a multi-species ecosystem. We model non-trophic ecological interactions based on a continuous but low-dimensional trait/niche space, where the location of each species in trait space affects the overlap of its resource utilisation with that of other species. The local depletion of available resources creates, in effect, a deformable fitness landscape that governs how the evolution of one species affects the selective pressures on other species. This enables us to study the coevolution of ecological interactions in an intuitive and easily visualisable manner. We observe that this model can exhibit either of the two behavioural modes discussed in the literature; namely, evolutionary stasis or Red Queen dynamics, i.e., continued evolutionary change. We find that which of these modes is observed depends on the lag or latency between the movement of a species in trait space and its effect on available resources. Specifically, if ecological change is nearly instantaneous compared to evolutionary change, stasis results; but conversely, if evolutionary timescales are closer to ecological timescales, such that resource depletion is not instantaneous on evolutionary timescales, then Red Queen dynamics result. We also observe that in the stasis mode, the overall utilisation of resources by the ecosystem is relatively efficient, with diverse species utilising different niches, whereas in the Red Queen mode the organisation of the ecosystem is such that species tend to clump together competing for overlapping resources. These models thereby suggest some basic conditions that influence the organisation of inter-species interactions and the balance of individual and collective adaptation in ecosystems, and likewise they also suggest factors that might be useful in engineering artificial coevolution
Velocity-space sensitivity of the time-of-flight neutron spectrometer at JET
The velocity-space sensitivities of fast-ion diagnostics are often described by so-called weight functions. Recently, we formulated weight functions showing the velocity-space sensitivity of the often dominant beam-target part of neutron energy spectra. These weight functions for neutron emission spectrometry (NES) are independent of the particular NES diagnostic. Here we apply these NES weight functions to the time-of-flight spectrometer TOFOR at JET. By taking the instrumental response function of TOFOR into account, we calculate time-of-flight NES weight functions that enable us to directly determine the velocity-space sensitivity of a given part of a measured time-of-flight spectrum from TOFOR
Relationship of edge localized mode burst times with divertor flux loop signal phase in JET
A phase relationship is identified between sequential edge localized modes (ELMs) occurrence times in a set of H-mode tokamak plasmas to the voltage measured in full flux azimuthal loops in the divertor region. We focus on plasmas in the Joint European Torus where a steady H-mode is sustained over several seconds, during which ELMs are observed in the Be II emission at the divertor. The ELMs analysed arise from intrinsic ELMing, in that there is no deliberate intent to control the ELMing process by external means. We use ELM timings derived from the Be II signal to perform direct time domain analysis of the full flux loop VLD2 and VLD3 signals, which provide a high cadence global measurement proportional to the voltage induced by changes in poloidal magnetic flux. Specifically, we examine how the time interval between pairs of successive ELMs is linked to the time-evolving phase of the full flux loop signals. Each ELM produces a clear early pulse in the full flux loop signals, whose peak time is used to condition our analysis. The arrival time of the following ELM, relative to this pulse, is found to fall into one of two categories: (i) prompt ELMs, which are directly paced by the initial response seen in the flux loop signals; and (ii) all other ELMs, which occur after the initial response of the full flux loop signals has decayed in amplitude. The times at which ELMs in category (ii) occur, relative to the first ELM of the pair, are clustered at times when the instantaneous phase of the full flux loop signal is close to its value at the time of the first ELM
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