389 research outputs found
Five-Torsion in the Homology of the Matching Complex on 14 Vertices
J. L. Andersen proved that there is 5-torsion in the bottom nonvanishing
homology group of the simplicial complex of graphs of degree at most two on
seven vertices. We use this result to demonstrate that there is 5-torsion also
in the bottom nonvanishing homology group of the matching complex on
14 vertices. Combining our observation with results due to Bouc and to
Shareshian and Wachs, we conclude that the case is exceptional; for all
other , the torsion subgroup of the bottom nonvanishing homology group has
exponent three or is zero. The possibility remains that there is other torsion
than 3-torsion in higher-degree homology groups of when and .Comment: 11 page
Cosmic Neutrinos and the Energy Budget of Galactic and Extragalactic Cosmic Rays
Although kilometer-scale neutrino detectors such as IceCube are discovery
instruments, their conceptual design is very much anchored to the observational
fact that Nature produces protons and photons with energies in excess of
10^{20} eV and 10^{13} eV, respectively. The puzzle of where and how Nature
accelerates the highest energy cosmic particles is unresolved almost a century
after their discovery. We will discuss how the cosmic ray connection sets the
scale of the anticipated cosmic neutrino fluxes. In this context, we discuss
the first results of the completed AMANDA detector and the science reach of its
extension, IceCube.Comment: 13 pages, Latex2e, 3 postscript figures included. Talk presented at
the International Workshop on Energy Budget in the High Energy Universe,
Kashiwa, Japan, February 200
Random geometric complexes
We study the expected topological properties of Cech and Vietoris-Rips
complexes built on i.i.d. random points in R^d. We find higher dimensional
analogues of known results for connectivity and component counts for random
geometric graphs. However, higher homology H_k is not monotone when k > 0. In
particular for every k > 0 we exhibit two thresholds, one where homology passes
from vanishing to nonvanishing, and another where it passes back to vanishing.
We give asymptotic formulas for the expectation of the Betti numbers in the
sparser regimes, and bounds in the denser regimes. The main technical
contribution of the article is in the application of discrete Morse theory in
geometric probability.Comment: 26 pages, 3 figures, final revisions, to appear in Discrete &
Computational Geometr
High Energy Neutrino Astronomy: Towards Kilometer-Scale Detectors
Of all high-energy particles, only neutrinos can directly convey astronomical
information from the edge of the universe---and from deep inside the most
cataclysmic high-energy processes. Copiously produced in high-energy
collisions, travelling at the velocity of light, and not deflected by magnetic
fields, neutrinos meet the basic requirements for astronomy. Their unique
advantage arises from a fundamental property: they are affected only by the
weakest of nature's forces (but for gravity) and are therefore essentially
unabsorbed as they travel cosmological distances between their origin and us.
Many of the outstanding mysteries of astrophysics may be hidden from our sight
at all wavelengths of the electromagnetic spectrum because of absorption by
matter and radiation between us and the source. For example, the hot dense
regions that form the central engines of stars and galaxies are opaque to
photons. In other cases, such as supernova remnants, gamma ray bursters, and
active galaxies, all of which may involve compact objects or black holes at
their cores, the precise origin of the high-energy photons emerging from their
surface regions is uncertain. Therefore, data obtained through a variety of
observational windows---and especially through direct observations with
neutrinos---may be of cardinal importance. In this talk, the scientific goals
of high energy neutrino astronomy and the technical aspects of water and ice
Cherenkov detectors are examined, and future experimental possibilities,
including a kilometer-square deep ice neutrino telescope, are explored.Comment: 13 pages, Latex, 6 postscript figures, uses aipproc.sty and epsf.sty.
Talk presented at the International Symposium on High Energy Gamma Ray
Astronomy, Heidelberg, June 200
Phase Transitions on Nonamenable Graphs
We survey known results about phase transitions in various models of
statistical physics when the underlying space is a nonamenable graph. Most
attention is devoted to transitive graphs and trees
Atmospheric Muon Flux at Sea Level, Underground, and Underwater
The vertical sea-level muon spectrum at energies above 1 GeV and the
underground/underwater muon intensities at depths up to 18 km w.e. are
calculated. The results are particularly collated with a great body of the
ground-level, underground, and underwater muon data. In the hadron-cascade
calculations, the growth with energy of inelastic cross sections and pion,
kaon, and nucleon generation in pion-nucleus collisions are taken into account.
For evaluating the prompt muon contribution to the muon flux, we apply two
phenomenological approaches to the charm production problem: the recombination
quark-parton model and the quark-gluon string model. To solve the muon
transport equation at large depths of homogeneous medium, a semi-analytical
method is used. The simple fitting formulas describing our numerical results
are given. Our analysis shows that, at depths up to 6-7 km w. e., essentially
all underground data on the muon intensity correlate with each other and with
predicted depth-intensity relation for conventional muons to within 10%.
However, the high-energy sea-level data as well as the data at large depths are
contradictory and cannot be quantitatively decribed by a single nuclear-cascade
model.Comment: 47 pages, REVTeX, 15 EPS figures included; recent experimental data
and references added, typos correcte
HIV Medication Adherence and HIV Symptom Severity: The Roles of Sleep Quality and Memory
Cosmic Neutrinos from the Sources of Galactic and Extragalactic Cosmic Rays
Although kilometer-scale neutrino detectors such as IceCube are discovery
instruments, their conceptual design is very much anchored to the observational
fact that Nature produces protons and photons with energies in excess of 10^20
eV and 10^13 eV, respectively. The puzzle of where and how Nature accelerates
the highest energy cosmic particles is unresolved almost a century after their
discovery. From energetics considerations we anticipate order 10~100 neutrino
events per kilometer squared per year pointing back at the source(s) of both
galactic and extragalactic cosmic rays. In this context, we discuss the results
of the AMANDA and IceCube neutrino telescopes which will deliver a
kilometer-square-year of data over the next 3 years.Comment: 8 pages, 4 figure
Center of mass, spin supplementary conditions, and the momentum of spinning particles
We discuss the problem of defining the center of mass in general relativity
and the so-called spin supplementary condition. The different spin conditions
in the literature, their physical significance, and the momentum-velocity
relation for each of them are analyzed in depth. The reason for the
non-parallelism between the velocity and the momentum, and the concept of
"hidden momentum", are dissected. It is argued that the different solutions
allowed by the different spin conditions are equally valid descriptions for the
motion of a given test body, and their equivalence is shown to dipole order in
curved spacetime. These different descriptions are compared in simple examples.Comment: 45 pages, 7 figures. Some minor improvements, typos fixed, signs in
some expressions corrected. Matches the published version. Published as part
of the book "Equations of Motion in Relativistic Gravity", D. Puetzfeld et
al. (eds.), Fundamental Theories of Physics 179, Springer, 201
On Eigenvalues of Random Complexes
We consider higher-dimensional generalizations of the normalized Laplacian
and the adjacency matrix of graphs and study their eigenvalues for the
Linial-Meshulam model of random -dimensional simplicial complexes
on vertices. We show that for , the eigenvalues of
these matrices are a.a.s. concentrated around two values. The main tool, which
goes back to the work of Garland, are arguments that relate the eigenvalues of
these matrices to those of graphs that arise as links of -dimensional
faces. Garland's result concerns the Laplacian; we develop an analogous result
for the adjacency matrix. The same arguments apply to other models of random
complexes which allow for dependencies between the choices of -dimensional
simplices. In the second part of the paper, we apply this to the question of
possible higher-dimensional analogues of the discrete Cheeger inequality, which
in the classical case of graphs relates the eigenvalues of a graph and its edge
expansion. It is very natural to ask whether this generalizes to higher
dimensions and, in particular, whether the higher-dimensional Laplacian spectra
capture the notion of coboundary expansion - a generalization of edge expansion
that arose in recent work of Linial and Meshulam and of Gromov. We show that
this most straightforward version of a higher-dimensional discrete Cheeger
inequality fails, in quite a strong way: For every and , there is a -dimensional complex on vertices that
has strong spectral expansion properties (all nontrivial eigenvalues of the
normalised -dimensional Laplacian lie in the interval
) but whose coboundary expansion is bounded
from above by and so tends to zero as ;
moreover, can be taken to have vanishing integer homology in dimension
less than .Comment: Extended full version of an extended abstract that appeared at SoCG
2012, to appear in Israel Journal of Mathematic
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