150 research outputs found
Optimal solutions to matrix-valued Nehari problems and related limit theorems
In a 1990 paper Helton and Young showed that under certain conditions the
optimal solution of the Nehari problem corresponding to a finite rank Hankel
operator with scalar entries can be efficiently approximated by certain
functions defined in terms of finite dimensional restrictions of the Hankel
operator. In this paper it is shown that these approximants appear as optimal
solutions to restricted Nehari problems. The latter problems can be solved
using relaxed commutant lifting theory. This observation is used to extent the
Helton and Young approximation result to a matrix-valued setting. As in the
Helton and Young paper the rate of convergence depends on the choice of the
initial space in the approximation scheme.Comment: 22 page
Computing Linear Matrix Representations of Helton-Vinnikov Curves
Helton and Vinnikov showed that every rigidly convex curve in the real plane
bounds a spectrahedron. This leads to the computational problem of explicitly
producing a symmetric (positive definite) linear determinantal representation
for a given curve. We study three approaches to this problem: an algebraic
approach via solving polynomial equations, a geometric approach via contact
curves, and an analytic approach via theta functions. These are explained,
compared, and tested experimentally for low degree instances.Comment: 19 pages, 3 figures, minor revisions; Mathematical Methods in
Systems, Optimization and Control, Birkhauser, Base
Tropically convex constraint satisfaction
A semilinear relation S is max-closed if it is preserved by taking the
componentwise maximum. The constraint satisfaction problem for max-closed
semilinear constraints is at least as hard as determining the winner in Mean
Payoff Games, a notorious problem of open computational complexity. Mean Payoff
Games are known to be in the intersection of NP and co-NP, which is not known
for max-closed semilinear constraints. Semilinear relations that are max-closed
and additionally closed under translations have been called tropically convex
in the literature. One of our main results is a new duality for open tropically
convex relations, which puts the CSP for tropically convex semilinaer
constraints in general into NP intersected co-NP. This extends the
corresponding complexity result for scheduling under and-or precedence
constraints, or equivalently the max-atoms problem. To this end, we present a
characterization of max-closed semilinear relations in terms of syntactically
restricted first-order logic, and another characterization in terms of a finite
set of relations L that allow primitive positive definitions of all other
relations in the class. We also present a subclass of max-closed constraints
where the CSP is in P; this class generalizes the class of max-closed
constraints over finite domains, and the feasibility problem for max-closed
linear inequalities. Finally, we show that the class of max-closed semilinear
constraints is maximal in the sense that as soon as a single relation that is
not max-closed is added to L, the CSP becomes NP-hard.Comment: 29 pages, 2 figure
Polynomial-sized Semidefinite Representations of Derivative Relaxations of Spectrahedral Cones
We give explicit polynomial-sized (in and ) semidefinite
representations of the hyperbolicity cones associated with the elementary
symmetric polynomials of degree in variables. These convex cones form a
family of non-polyhedral outer approximations of the non-negative orthant that
preserve low-dimensional faces while successively discarding high-dimensional
faces. More generally we construct explicit semidefinite representations
(polynomial-sized in , and ) of the hyperbolicity cones associated with
th directional derivatives of polynomials of the form where the are symmetric
matrices. These convex cones form an analogous family of outer approximations
to any spectrahedral cone. Our representations allow us to use semidefinite
programming to solve the linear cone programs associated with these convex
cones as well as their (less well understood) dual cones.Comment: 20 pages, 1 figure. Minor changes, expanded proof of Lemma
An exact duality theory for semidefinite programming based on sums of squares
Farkas' lemma is a fundamental result from linear programming providing
linear certificates for infeasibility of systems of linear inequalities. In
semidefinite programming, such linear certificates only exist for strongly
infeasible linear matrix inequalities. We provide nonlinear algebraic
certificates for all infeasible linear matrix inequalities in the spirit of
real algebraic geometry: A linear matrix inequality is infeasible if and only
if -1 lies in the quadratic module associated to it. We also present a new
exact duality theory for semidefinite programming, motivated by the real
radical and sums of squares certificates from real algebraic geometry.Comment: arXiv admin note: substantial text overlap with arXiv:1108.593
Probe-caught spontaneous and deliberate mind wandering in relation to self-reported inattentive, hyperactive and impulsive traits in adults.
Research has revealed a positive relationship between types of mind wandering and ADHD at clinical and subclinical levels. However, this work did not consider the relationship between mind wandering and the core symptoms of ADHD: inattention, hyperactivity and impulsivity. Given that the DMS-V attributes mind wandering to inattention only, and that only inattention is thought to result from impairment to the executive function linked to mind wandering, the present research sought to examine this relationship in 80 undiagnosed adults. Using both standard and easy versions of the Sustained Attention to Response Task (SART) we measured both spontaneous and deliberate mind wandering. We found that spontaneous mind wandering was related to self-reported inattentive traits when the task was cognitively more challenging (standard SART). However, hyperactive and impulsive traits were related to spontaneous mind wandering independent of task difficulty. The results suggest inattentive traits are not uniquely related to mind wandering; indeed, adults with hyperactive/impulsive traits were more likely to experience mind wandering, suggesting that mind wandering might not be useful diagnostic criteria for inattention
Stream denitrification across biomes and its response to anthropogenic nitrate loading
Author Posting. © The Author(s), 2008. This is the author's version of the work. It is posted here by permission of Nature Publishing Group for personal use, not for redistribution. The definitive version was published in Nature 452 (2008): 202-205, doi:10.1038/nature06686.Worldwide, anthropogenic addition of bioavailable nitrogen (N) to the
biosphere is increasing and terrestrial ecosystems are becoming increasingly N
saturated, causing more bioavailable N to enter groundwater and surface waters.
Large-scale N budgets show that an average of about 20-25% of the N added to the
biosphere is exported from rivers to the ocean or inland basins, indicating
substantial sinks for N must exist in the landscape. Streams and rivers may be
important sinks for bioavailable N owing to their hydrologic connections with
terrestrial systems, high rates of biological activity, and streambed sediment
environments that favor microbial denitrification. Here, using data from 15N
tracer experiments replicated across 72 streams and 8 regions representing several
biomes, we show that total biotic uptake and denitrification of nitrate increase with
stream nitrate concentration, but that the efficiency of biotic uptake and
denitrification declines as concentration increases, reducing the proportion of instream
nitrate that is removed from transport. Total uptake of nitrate was related
to ecosystem photosynthesis and denitrification was related to ecosystem
respiration. Additionally, we use a stream network model to demonstrate that
excess nitrate in streams elicits a disproportionate increase in the fraction of nitrate
that is exported to receiving waters and reduces the relative role of small versus
large streams as nitrate sinks.Funding for this research was provided by the National Science
Foundation
It Takes Two–Skilled Recognition of Objects Engages Lateral Areas in Both Hemispheres
Our object recognition abilities, a direct product of our experience with objects, are fine-tuned to perfection. Left temporal and lateral areas along the dorsal, action related stream, as well as left infero-temporal areas along the ventral, object related stream are engaged in object recognition. Here we show that expertise modulates the activity of dorsal areas in the recognition of man-made objects with clearly specified functions. Expert chess players were faster than chess novices in identifying chess objects and their functional relations. Experts' advantage was domain-specific as there were no differences between groups in a control task featuring geometrical shapes. The pattern of eye movements supported the notion that experts' extensive knowledge about domain objects and their functions enabled superior recognition even when experts were not directly fixating the objects of interest. Functional magnetic resonance imaging (fMRI) related exclusively the areas along the dorsal stream to chess specific object recognition. Besides the commonly involved left temporal and parietal lateral brain areas, we found that only in experts homologous areas on the right hemisphere were also engaged in chess specific object recognition. Based on these results, we discuss whether skilled object recognition does not only involve a more efficient version of the processes found in non-skilled recognition, but also qualitatively different cognitive processes which engage additional brain areas
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