1,900 research outputs found
Nominal Logic Programming
Nominal logic is an extension of first-order logic which provides a simple
foundation for formalizing and reasoning about abstract syntax modulo
consistent renaming of bound names (that is, alpha-equivalence). This article
investigates logic programming based on nominal logic. We describe some typical
nominal logic programs, and develop the model-theoretic, proof-theoretic, and
operational semantics of such programs. Besides being of interest for ensuring
the correct behavior of implementations, these results provide a rigorous
foundation for techniques for analysis and reasoning about nominal logic
programs, as we illustrate via examples.Comment: 46 pages; 19 page appendix; 13 figures. Revised journal submission as
of July 23, 200
Super-resolution, Extremal Functions and the Condition Number of Vandermonde Matrices
Super-resolution is a fundamental task in imaging, where the goal is to
extract fine-grained structure from coarse-grained measurements. Here we are
interested in a popular mathematical abstraction of this problem that has been
widely studied in the statistics, signal processing and machine learning
communities. We exactly resolve the threshold at which noisy super-resolution
is possible. In particular, we establish a sharp phase transition for the
relationship between the cutoff frequency () and the separation ().
If , our estimator converges to the true values at an inverse
polynomial rate in terms of the magnitude of the noise. And when no estimator can distinguish between a particular pair of
-separated signals even if the magnitude of the noise is exponentially
small.
Our results involve making novel connections between {\em extremal functions}
and the spectral properties of Vandermonde matrices. We establish a sharp phase
transition for their condition number which in turn allows us to give the first
noise tolerance bounds for the matrix pencil method. Moreover we show that our
methods can be interpreted as giving preconditioners for Vandermonde matrices,
and we use this observation to design faster algorithms for super-resolution.
We believe that these ideas may have other applications in designing faster
algorithms for other basic tasks in signal processing.Comment: 19 page
Negative thermal expansion of MgB in the superconducting state and anomalous behavior of the bulk Gr\"uneisen function
The thermal expansion coefficient of MgB is revealed to change
from positive to negative on cooling through the superconducting transition
temperature . The Gr\"uneisen function also becomes negative at
followed by a dramatic increase to large positive values at low temperature.
The results suggest anomalous coupling between superconducting electrons and
low-energy phonons.Comment: 5 figures. submitted to Phys. Rev. Let
A dependent nominal type theory
Nominal abstract syntax is an approach to representing names and binding
pioneered by Gabbay and Pitts. So far nominal techniques have mostly been
studied using classical logic or model theory, not type theory. Nominal
extensions to simple, dependent and ML-like polymorphic languages have been
studied, but decidability and normalization results have only been established
for simple nominal type theories. We present a LF-style dependent type theory
extended with name-abstraction types, prove soundness and decidability of
beta-eta-equivalence checking, discuss adequacy and canonical forms via an
example, and discuss extensions such as dependently-typed recursion and
induction principles
First-Order Provenance Games
We propose a new model of provenance, based on a game-theoretic approach to
query evaluation. First, we study games G in their own right, and ask how to
explain that a position x in G is won, lost, or drawn. The resulting notion of
game provenance is closely related to winning strategies, and excludes from
provenance all "bad moves", i.e., those which unnecessarily allow the opponent
to improve the outcome of a play. In this way, the value of a position is
determined by its game provenance. We then define provenance games by viewing
the evaluation of a first-order query as a game between two players who argue
whether a tuple is in the query answer. For RA+ queries, we show that game
provenance is equivalent to the most general semiring of provenance polynomials
N[X]. Variants of our game yield other known semirings. However, unlike
semiring provenance, game provenance also provides a "built-in" way to handle
negation and thus to answer why-not questions: In (provenance) games, the
reason why x is not won, is the same as why x is lost or drawn (the latter is
possible for games with draws). Since first-order provenance games are
draw-free, they yield a new provenance model that combines how- and why-not
provenance
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Improving the condition number of estimated covariance matrices
High dimensional error covariance matrices and their inverses are used to weight the
contribution of observation and background information in data assimilation procedures. As
observation error covariance matrices are often obtained by sampling methods, estimates are
often degenerate or ill-conditioned, making it impossible to invert an observation error
covariance matrix without the use of techniques to reduce its condition number. In this paper
we present new theory for two existing methods that can be used to ‘recondition’ any covariance
matrix: ridge regression, and the minimum eigenvalue method. We compare these methods
with multiplicative variance inflation, which cannot alter the condition number of a matrix, but
is often used to account for neglected correlation information. We investigate the impact of
reconditioning on variances and correlations of a general covariance matrix in both a theoretical
and practical setting. Improved theoretical understanding provides guidance to users regarding
method selection, and choice of target condition number. The new theory shows that, for the
same target condition number, both methods increase variances compared to the original
matrix, with larger increases for ridge regression than the minimum eigenvalue method. We
prove that the ridge regression method strictly decreases the absolute value of off-diagonal
correlations. Theoretical comparison of the impact of reconditioning and multiplicative
variance inflation on the data assimilation objective function shows that variance inflation alters
information across all scales uniformly, whereas reconditioning has a larger effect on scales
corresponding to smaller eigenvalues. We then consider two examples: a general correlation
function, and an observation error covariance matrix arising from interchannel correlations. The
minimum eigenvalue method results in smaller overall changes to the correlation matrix than
ridge regression, but can increase off-diagonal correlations. Data assimilation experiments reveal
that reconditioning corrects spurious noise in the analysis but underestimates the true signal
compared to multiplicative variance inflation
Positive approximations of the inverse of fractional powers of SPD M-matrices
This study is motivated by the recent development in the fractional calculus
and its applications. During last few years, several different techniques are
proposed to localize the nonlocal fractional diffusion operator. They are based
on transformation of the original problem to a local elliptic or
pseudoparabolic problem, or to an integral representation of the solution, thus
increasing the dimension of the computational domain. More recently, an
alternative approach aimed at reducing the computational complexity was
developed. The linear algebraic system , is considered, where is a properly normalized (scalded) symmetric
and positive definite matrix obtained from finite element or finite difference
approximation of second order elliptic problems in ,
. The method is based on best uniform rational approximations (BURA)
of the function for and natural .
The maximum principles are among the major qualitative properties of linear
elliptic operators/PDEs. In many studies and applications, it is important that
such properties are preserved by the selected numerical solution method. In
this paper we present and analyze the properties of positive approximations of
obtained by the BURA technique. Sufficient conditions for
positiveness are proven, complemented by sharp error estimates. The theoretical
results are supported by representative numerical tests
A mobile element based phylogeny of Old World monkeys
SINEs (Short INterspersed Elements) are a class of non-autonomous mobile elements that are states, making them useful genetic systems for phylogenetic studies. Alu elements are the most successful SINE in primate genomes and have been utilized for resolving primate phylogenetic relationships and human population genetics. However, no Alu based phylogenetic analysis has yet been performed to resolve relationships among Old World monkeys. Using both a computational approach and polymerase chain reaction display methodology, we identified 285 new Alu insertions from sixteen Old World monkey taxa that were informative at various levels of catarrhine phylogeny. We have utilized these elements along with 12 previously reported loci to construct a phylogenetic tree of the selected taxa. Relationships among all major clades are in general agreement with other molecular and morphological data sets but have stronger statistical support. © 2005 Elsevier Inc. All rights reserved
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