135,413 research outputs found
Representation and generation of plans using graph spectra
Numerical comparison of spaces with one another is often achieved with set scalar
measures such as global and local integration, connectivity, etc., which capture a
particular quality of the space but therefore lose much of the detail of its overall
structure. More detailed methods such as graph edit distance are difficult to calculate,
particularly for large plans. This paper proposes the use of the graph spectrum, or the
ordered eigenvalues of a graph adjacency matrix, as a means to characterise the space
as a whole. The result is a vector of high dimensionality that can be easily measured
against others for detailed comparison.
Several graph types are investigated, including boundary and axial representations, as
are several methods for deriving the spectral vector. The effectiveness of these is
evaluated using a genetic algorithm optimisation to generate plans to match a given
spectrum, and evolution is seen to produce plans similar to the initial targets, even in
very large search spaces. Results indicate that boundary graphs alone can capture the
gross topological qualities of a space, but axial graphs are needed to indicate local
relationships. Methods of scaling the spectra are investigated in relation to both global
local changes to plan arrangement. For all graph types, the spectra were seen to
capture local patterns of spatial arrangement even as global size is varied
Representation and generation of plans using graph spectra
Numerical comparison of spaces with one another is often achieved with set scalar measures such as global and local integration, connectivity, etc., which capture a particular quality of the space but therefore lose much of the detail of its overall structure. More detailed methods such as graph edit distance are difficult to calculate, particularly for large plans. This paper proposes the use of the graph spectrum, or the ordered eigenvalues of a graph adjacency matrix, as a means to characterise the space as a whole. The result is a vector of high dimensionality that can be easily measured
against others for detailed comparison.
Several graph types are investigated, including boundary and axial representations, as are several methods for deriving the spectral vector. The effectiveness of these is evaluated using a genetic algorithm optimisation to generate plans to match a given spectrum, and evolution is seen to produce plans similar to the initial targets, even in very large search spaces. Results indicate that boundary graphs alone can capture the
gross topological qualities of a space, but axial graphs are needed to indicate local relationships. Methods of scaling the spectra are investigated in relation to both global local changes to plan arrangement. For all graph types, the spectra were seen to capture local patterns of spatial arrangement even as global size is varied
Pion-Nucleon Scattering in a Large-N Sigma Model
We review the large-N_c approach to meson-baryon scattering, including recent
interesting developments. We then study pion-nucleon scattering in a particular
variant of the linear sigma-model, in which the couplings of the sigma and pi
mesons to the nucleon are echoed by couplings to the entire tower of I=J
baryons (including the Delta) as dictated by large-N_c group theory. We sum the
complete set of multi-loop meson-exchange
\pi N --> \pi N and \pi N --> \sigma N Feynman diagrams, to leading order in
1/N_c. The key idea, reviewed in detail, is that large-N_c allows the
approximation of LOOP graphs by TREE graphs, so long as the loops contain at
least one baryon leg; trees, in turn, can be summed by solving classical
equations of motion. We exhibit the resulting partial-wave S-matrix and the
rich nucleon and Delta resonance spectrum of this simple model, comparing not
only to experiment but also to pion-nucleon scattering in the Skyrme model. The
moral is that much of the detailed structure of the meson-baryon S-matrix which
hitherto has been uncovered only with skyrmion methods, can also be described
by models with explicit baryon fields, thanks to the 1/N_c expansion.Comment: This LaTeX file inputs the ReVTeX macropackage; figures accompany i
Spectra of Monadic Second-Order Formulas with One Unary Function
We establish the eventual periodicity of the spectrum of any monadic
second-order formula where:
(i) all relation symbols, except equality, are unary, and
(ii) there is only one function symbol and that symbol is unary
Spectral dimension of quantum geometries
The spectral dimension is an indicator of geometry and topology of spacetime
and a tool to compare the description of quantum geometry in various approaches
to quantum gravity. This is possible because it can be defined not only on
smooth geometries but also on discrete (e.g., simplicial) ones. In this paper,
we consider the spectral dimension of quantum states of spatial geometry
defined on combinatorial complexes endowed with additional algebraic data: the
kinematical quantum states of loop quantum gravity (LQG). Preliminarily, the
effects of topology and discreteness of classical discrete geometries are
studied in a systematic manner. We look for states reproducing the spectral
dimension of a classical space in the appropriate regime. We also test the
hypothesis that in LQG, as in other approaches, there is a scale dependence of
the spectral dimension, which runs from the topological dimension at large
scales to a smaller one at short distances. While our results do not give any
strong support to this hypothesis, we can however pinpoint when the topological
dimension is reproduced by LQG quantum states. Overall, by exploring the
interplay of combinatorial, topological and geometrical effects, and by
considering various kinds of quantum states such as coherent states and their
superpositions, we find that the spectral dimension of discrete quantum
geometries is more sensitive to the underlying combinatorial structures than to
the details of the additional data associated with them.Comment: 39 pages, 18 multiple figures. v2: discussion improved, minor typos
correcte
Dynamics of Flux Tubes in Large N Gauge Theories
The gluonic field created by a static quark anti-quark pair is described via
the AdS/CFT correspondence by a string connecting the pair which is located on
the boundary of AdS. Thus the gluonic field in a strongly coupled large N CFT
has a stringy spectrum of excitations. We trace the stability of these
excitations to a combination of large N suppressions and energy conservation.
Comparison of the physics of the N=infinity flux tube in the {\cal N}=4 SYM
theory at weak and strong coupling shows that the excitations are present only
above a certain critical coupling. The density of states of a highly excited
string with a fold reaching towards the horizon of AdS is in exact agreement at
strong coupling with that of the near-threshold states found in a ladder
diagram model of the weak-strong coupling transition. We also study large
distance correlations of local operators with a Wilson loop, and show that the
fall off at weak coupling and N=infinity (i.e. strictly planar diagrams)
matches the strong coupling predictions given by the AdS/CFT correspondence,
rather than those of a weakly coupled U(1) gauge theory.Comment: 22 pages, 4 figures; v2: clarifications in section 5, 1 reference
added; v3: the final version (minor changes, 1 more reference added
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