5,728 research outputs found
Kripke Models for Classical Logic
We introduce a notion of Kripke model for classical logic for which we
constructively prove soundness and cut-free completeness. We discuss the
novelty of the notion and its potential applications
Step-Indexed Normalization for a Language with General Recursion
The Trellys project has produced several designs for practical dependently
typed languages. These languages are broken into two
fragments-a_logical_fragment where every term normalizes and which is
consistent when interpreted as a logic, and a_programmatic_fragment with
general recursion and other convenient but unsound features. In this paper, we
present a small example language in this style. Our design allows the
programmer to explicitly mention and pass information between the two
fragments. We show that this feature substantially complicates the metatheory
and present a new technique, combining the traditional Girard-Tait method with
step-indexed logical relations, which we use to show normalization for the
logical fragment.Comment: In Proceedings MSFP 2012, arXiv:1202.240
Testing the Universality of the Stellar IMF with Chandra and HST
The stellar initial mass function (IMF), which is often assumed to be
universal across unresolved stellar populations, has recently been suggested to
be "bottom-heavy" for massive ellipticals. In these galaxies, the prevalence of
gravity-sensitive absorption lines (e.g. Na I and Ca II) in their near-IR
spectra implies an excess of low-mass ( ) stars over that
expected from a canonical IMF observed in low-mass ellipticals. A direct
extrapolation of such a bottom-heavy IMF to high stellar masses (
) would lead to a corresponding deficit of neutron stars and black
holes, and therefore of low-mass X-ray binaries (LMXBs), per unit near-IR
luminosity in these galaxies. Peacock et al. (2014) searched for evidence of
this trend and found that the observed number of LMXBs per unit -band
luminosity () was nearly constant. We extend this work using new and
archival Chandra X-ray Observatory (Chandra) and Hubble Space Telescope (HST)
observations of seven low-mass ellipticals where is expected to be the
largest and compare these data with a variety of IMF models to test which are
consistent with the observed . We reproduce the result of Peacock et al.
(2014), strengthening the constraint that the slope of the IMF at
must be consistent with a Kroupa-like IMF. We construct an IMF model
that is a linear combination of a Milky Way-like IMF and a broken power-law
IMF, with a steep slope ( ) for stars < 0.5 (as
suggested by near-IR indices), and that flattens out ( ) for
stars > 0.5 , and discuss its wider ramifications and limitations.Comment: Accepted for publication in ApJ; 7 pages, 2 figures, 1 tabl
Proof-graphs for Minimal Implicational Logic
It is well-known that the size of propositional classical proofs can be huge.
Proof theoretical studies discovered exponential gaps between normal or cut
free proofs and their respective non-normal proofs. The aim of this work is to
study how to reduce the weight of propositional deductions. We present the
formalism of proof-graphs for purely implicational logic, which are graphs of a
specific shape that are intended to capture the logical structure of a
deduction. The advantage of this formalism is that formulas can be shared in
the reduced proof.
In the present paper we give a precise definition of proof-graphs for the
minimal implicational logic, together with a normalization procedure for these
proof-graphs. In contrast to standard tree-like formalisms, our normalization
does not increase the number of nodes, when applied to the corresponding
minimal proof-graph representations.Comment: In Proceedings DCM 2013, arXiv:1403.768
Statistical properties of the combined emission of a population of discrete sources: astrophysical implications
We study the statistical properties of the combined emission of a population
of discrete sources (e.g. X-ray emission of a galaxy due to its X-ray binaries
population). Namely, we consider the dependence of their total luminosity
L_tot=SUM(L_k) and of fractional rms_tot of their variability on the number of
sources N or, equivalently, on the normalization of the luminosity function. We
show that due to small number statistics a regime exists, in which L_tot grows
non-linearly with N, in an apparent contradiction with the seemingly obvious
prediction =integral(dN/dL*L*dL) ~ N. In this non-linear regime, the
rms_tot decreases with N significantly more slowly than expected from the rms ~
1/sqrt(N) averaging law. For example, for a power law luminosity function with
a slope of a=3/2, in the non-linear regime, L_tot ~ N^2 and the rms_tot does
not depend at all on the number of sources N. Only in the limit of N>>1 do
these quantities behave as intuitively expected, L_tot ~ N and rms_tot ~
1/sqrt(N). We give exact solutions and derive convenient analytical
approximations for L_tot and rms_tot.
Using the total X-ray luminosity of a galaxy due to its X-ray binary
population as an example, we show that the Lx-SFR and Lx-M* relations predicted
from the respective ``universal'' luminosity functions of high and low mass
X-ray binaries are in a good agreement with observations. Although caused by
small number statistics the non-linear regime in these examples extends as far
as SFR<4-5 Msun/yr and log(M*/Msun)<10.0-10.5, respectively.Comment: MNRAS, accepted for publicatio
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