1,150,193 research outputs found
On Density-Critical Matroids
For a matroid having rank-one flats, the density is
unless , in which case . A matroid is
density-critical if all of its proper minors of non-zero rank have lower
density. By a 1965 theorem of Edmonds, a matroid that is minor-minimal among
simple matroids that cannot be covered by independent sets is
density-critical. It is straightforward to show that is the only
minor-minimal loopless matroid with no covering by independent sets. We
prove that there are exactly ten minor-minimal simple obstructions to a matroid
being able to be covered by two independent sets. These ten matroids are
precisely the density-critical matroids such that but for all proper minors of . All density-critical matroids of density
less than are series-parallel networks. For , although finding all
density-critical matroids of density at most does not seem straightforward,
we do solve this problem for .Comment: 16 page
Critical sets of the total variance of state detect all SLOCC entanglement classes
We present a general algorithm for finding all classes of pure multiparticle
states equivalent under Stochastic Local Operations and Classsical
Communication (SLOCC). We parametrize all SLOCC classes by the critical sets of
the total variance function. Our method works for arbitrary systems of
distinguishable and indistinguishable particles. We also discuss the Morse
indices of critical points which have the interpretation of the number of
independent non-local perturbations increasing the variance and hence
entanglement of a state. We illustrate our method by two examples.Comment: 4 page
Apparent horizons in simplicial Brill wave initial data
We construct initial data for a particular class of Brill wave metrics using
Regge calculus, and compare the results to a corresponding continuum solution,
finding excellent agreement. We then search for trapped surfaces in both sets
of initial data, and provide an independent verification of the existence of an
apparent horizon once a critical gravitational wave amplitude is passed. Our
estimate of this critical value, using both the Regge and continuum solutions,
supports other recent findings.Comment: 7 pages, 6 EPS figures, LaTeX 2e. Submitted to Class. Quant. Gra
Check-hybrid GLDPC Codes: Systematic Elimination of Trapping Sets and Guaranteed Error Correction Capability
In this paper, we propose a new approach to construct a class of check-hybrid
generalized low-density parity-check (CH-GLDPC) codes which are free of small
trapping sets. The approach is based on converting some selected check nodes
involving a trapping set into super checks corresponding to a 2-error
correcting component code. Specifically, we follow two main purposes to
construct the check-hybrid codes; first, based on the knowledge of the trapping
sets of the global LDPC code, single parity checks are replaced by super checks
to disable the trapping sets. We show that by converting specified single check
nodes, denoted as critical checks, to super checks in a trapping set, the
parallel bit flipping (PBF) decoder corrects the errors on a trapping set and
hence eliminates the trapping set. The second purpose is to minimize the rate
loss caused by replacing the super checks through finding the minimum number of
such critical checks. We also present an algorithm to find critical checks in a
trapping set of column-weight 3 LDPC code and then provide upper bounds on the
minimum number of such critical checks such that the decoder corrects all error
patterns on elementary trapping sets. Moreover, we provide a fixed set for a
class of constructed check-hybrid codes. The guaranteed error correction
capability of the CH-GLDPC codes is also studied. We show that a CH-GLDPC code
in which each variable node is connected to 2 super checks corresponding to a
2-error correcting component code corrects up to 5 errors. The results are also
extended to column-weight 4 LDPC codes. Finally, we investigate the eliminating
of trapping sets of a column-weight 3 LDPC code using the Gallager B decoding
algorithm and generalize the results obtained for the PBF for the Gallager B
decoding algorithm
On Identifying Critical Nuggets Of Information During Classification Task
In large databases, there may exist critical nuggets - small collections of records or instances that contain domain-specific important information. This information can be used for future decision making such as labeling of critical, unlabeled data records and improving classification results by reducing false positive and false negative errors. In recent years, data mining efforts have focussed on pattern and outlier detection methods. However, not much effort has been dedicated to finding critical nuggets within a data set. This work introduces the idea of critical nuggets, proposes an innovative domain-independent method to measure criticality, suggests a heuristic to reduce the search space for finding critical nuggets, and isolates and validates critical nuggets from some real world data sets. It seems that only a few subsets may qualify to be critical nuggets, underlying the importance of finding them. The proposed methodology can detect them. This work also identifies certain properties of critical nuggets and provides experimental validation of the properties. Critical nuggets were then applied to 2 important classification task related performance metrics - classification accuracy and misclassification costs. Experimental results helped validate that critical nuggets can assist in improving classification accuracies in real world data sets when compared with other standalone classification algorithms. The improvements in accuracy using the critical nuggets were statistically significant. Extensive studies were also undertaken on real world data sets that utilized critical nuggets to help minimize misclassification costs. In this case as well the critical nuggets based approach yielded statistically significant, lower misclassification costs than than standalone classification methods
Real root finding for equivariant semi-algebraic systems
Let be a real closed field. We consider basic semi-algebraic sets defined
by -variate equations/inequalities of symmetric polynomials and an
equivariant family of polynomials, all of them of degree bounded by .
Such a semi-algebraic set is invariant by the action of the symmetric group. We
show that such a set is either empty or it contains a point with at most
distinct coordinates. Combining this geometric result with efficient algorithms
for real root finding (based on the critical point method), one can decide the
emptiness of basic semi-algebraic sets defined by polynomials of degree
in time . This improves the state-of-the-art which is exponential
in . When the variables are quantified and the
coefficients of the input system depend on parameters , one
also demonstrates that the corresponding one-block quantifier elimination
problem can be solved in time
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