814 research outputs found
A lower bound for nodal count on discrete and metric graphs
According to a well-know theorem by Sturm, a vibrating string is divided into
exactly N nodal intervals by zeros of its N-th eigenfunction. Courant showed
that one half of Sturm's theorem for the strings applies to the theory of
membranes: N-th eigenfunction cannot have more than N domains. He also gave an
example of a eigenfunction high in the spectrum with a minimal number of nodal
domains, thus excluding the existence of a non-trivial lower bound. An analogue
of Sturm's result for discretizations of the interval was discussed by
Gantmacher and Krein. The discretization of an interval is a graph of a simple
form, a chain-graph. But what can be said about more complicated graphs? It has
been known since the early 90s that the nodal count for a generic eigenfunction
of the Schrodinger operator on quantum trees (where each edge is identified
with an interval of the real line and some matching conditions are enforced on
the vertices) is exact too: zeros of the N-th eigenfunction divide the tree
into exactly N subtrees. We discuss two extensions of this result in two
directions. One deals with the same continuous Schrodinger operator but on
general graphs (i.e. non-trees) and another deals with discrete Schrodinger
operator on combinatorial graphs (both trees and non-trees). The result that we
derive applies to both types of graphs: the number of nodal domains of the N-th
eigenfunction is bounded below by N-L, where L is the number of links that
distinguish the graph from a tree (defined as the dimension of the cycle space
or the rank of the fundamental group of the graph). We also show that if it the
genericity condition is dropped, the nodal count can fall arbitrarily far below
the number of the corresponding eigenfunction.Comment: 15 pages, 4 figures; Minor corrections: added 2 important reference
Tomonaga-Luttinger features in the resonant Raman spectra of quantum wires
The differential cross section for resonant Raman scattering from the
collective modes in a one dimensional system of interacting electrons is
calculated non-perturbatively using the bosonization method. The results
indicate that resonant Raman spectroscopy is a powerful tool for studying
Tomonaga-Luttinger liquid behaviour in quasi-one dimensional electron systems.Comment: 4 pages, no figur
Fractional Exclusion Statistics and Anyons
Do anyons, dynamically realized by the field theoretic Chern-Simons
construction, obey fractional exclusion statistics? We find that they do if the
statistical interaction between anyons and anti-anyons is taken into account.
For this anyon model, we show perturbatively that the exchange statistical
parameter of anyons is equal to the exclusion statistical parameter. We obtain
the same result by applying the relation between the exclusion statistical
parameter and the second virial coefficient in the non-relativistic limit.Comment: 9 pages, latex, IFT-498-UN
N=1 Type IIA brane configurations, Chirality and T-duality
We consider four-dimensional N=1 field theories realized by type IIA brane
configurations of NS-branes and D4-branes, in the presence of orientifold
six-planes and D6-branes. These configurations are known to present interesting
effects associated to the appearance of chiral symmetries and chiral matter in
the four-dimensional field theory. We center on models with one compact
direction (elliptic models) and show that, under T-duality, the configurations
are mapped to a set of type IIB D3-branes probing N=1 orientifolds of C^2/Z_N
singularities. We explicitly construct these orientifolds, and show the field
theories on the D3-brane probes indeed reproduces the field theories
constructed using the IIA brane configurations. This T-duality map allows to
understand the type IIB realization of several exotic brane dynamics effects on
the type IIA side: Flavour doubling, the splitting of D6-branes and O6-planes
in crossing a NS-brane and the effect of a non-zero type IIA cosmological
constant turn out to have surprisingly standard type IIB counterparts.Comment: 39 pages, Latex, 7 eps figures. References adde
A surface containing a line and a circle through each point is a quadric
We prove that a surface in real 3-space containing a line and a circle
through each point is a quadric. We also give some particular results on the
classification of surfaces containing several circles through each point.Comment: Improved exposition, 4 figures adde
Solar Wakes of Dark Matter Flows
We analyze the effect of the Sun's gravitational field on a flow of cold dark
matter (CDM) through the solar system in the limit where the velocity
dispersion of the flow vanishes. The exact density and velocity distributions
are derived in the case where the Sun is a point mass. The results are extended
to the more realistic case where the Sun has a finite size spherically
symmetric mass distribution. We find that regions of infinite density, called
caustics, appear. One such region is a line caustic on the axis of symmetry,
downstream from the Sun, where the flow trajectories cross. Another is a
cone-shaped caustic surface near the trajectories of maximum scattering angle.
The trajectories forming the conical caustic pass through the Sun's interior
and probe the solar mass distribution, raising the possibility that the solar
mass distribution may some day be measured by a dark matter detector on Earth.
We generalize our results to the case of flows with continuous velocity
distributions, such as that predicted by the isothermal model of the Milky Way
halo.Comment: 30 pages, 8 figure
Measuring Black Hole Spin using X-ray Reflection Spectroscopy
I review the current status of X-ray reflection (a.k.a. broad iron line)
based black hole spin measurements. This is a powerful technique that allows us
to measure robust black hole spins across the mass range, from the stellar-mass
black holes in X-ray binaries to the supermassive black holes in active
galactic nuclei. After describing the basic assumptions of this approach, I lay
out the detailed methodology focusing on "best practices" that have been found
necessary to obtain robust results. Reflecting my own biases, this review is
slanted towards a discussion of supermassive black hole (SMBH) spin in active
galactic nuclei (AGN). Pulling together all of the available XMM-Newton and
Suzaku results from the literature that satisfy objective quality control
criteria, it is clear that a large fraction of SMBHs are rapidly-spinning,
although there are tentative hints of a more slowly spinning population at high
(M>5*10^7Msun) and low (M<2*10^6Msun) mass. I also engage in a brief review of
the spins of stellar-mass black holes in X-ray binaries. In general,
reflection-based and continuum-fitting based spin measures are in agreement,
although there remain two objects (GROJ1655-40 and 4U1543-475) for which that
is not true. I end this review by discussing the exciting frontier of
relativistic reverberation, particularly the discovery of broad iron line
reverberation in XMM-Newton data for the Seyfert galaxies NGC4151, NGC7314 and
MCG-5-23-16. As well as confirming the basic paradigm of relativistic disk
reflection, this detection of reverberation demonstrates that future large-area
X-ray observatories such as LOFT will make tremendous progress in studies of
strong gravity using relativistic reverberation in AGN.Comment: 19 pages. To appear in proceedings of the ISSI-Bern workshop on "The
Physics of Accretion onto Black Holes" (8-12 Oct 2012). Revised version adds
a missing source to Table 1 and Fig.6 (IRAS13224-3809) and corrects the
referencing of the discovery of soft lags in 1H0707-495 (which were in fact
first reported in Fabian et al. 2009
Orientifolds of K3 and Calabi-Yau Manifolds with Intersecting D-branes
We investigate orientifolds of type II string theory on K3 and Calabi-Yau
3-folds with intersecting D-branes wrapping special Lagrangian cycles. We
determine quite generically the chiral massless spectrum in terms of
topological invariants and discuss both orbifold examples and algebraic
realizations in detail. Intriguingly, the developed techniques provide an
elegant way to figure out the chiral sector of orientifold models without
computing any explicit string partition function. As a new example we derive a
non-supersymmetric Standard-like Model from an orientifold of type IIA on the
quintic Calabi-Yau 3-fold with wrapped D6-branes. In the case of supersymmetric
intersecting brane models on Calabi-Yau manifolds we discuss the D-term and
F-term potentials, the effective gauge couplings and the Green-Schwarz
mechanism. The mirror symmetric formulation of this construction is provided
within type IIB theory. We finally include a short discussion about the lift of
these models from type IIB on K3 to F-theory and from type IIA on Calabi-Yau
3-folds to M-theory on G_2 manifolds.Comment: 82 pages, harvmac, 5 figures. v2: references added. v3: T^6
orientifold corrected, JHEP versio
The general purpose analog computer and computable analysis are two equivalent paradigms of analog computation
In this paper we revisit one of the rst models of analog
computation, Shannon's General Purpose Analog Computer (GPAC).
The GPAC has often been argued to be weaker than computable analysis.
As main contribution, we show that if we change the notion of GPACcomputability
in a natural way, we compute exactly all real computable
functions (in the sense of computable analysis). Moreover, since GPACs
are equivalent to systems of polynomial di erential equations then we
show that all real computable functions can be de ned by such models
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