589 research outputs found
Finding Cycles and Trees in Sublinear Time
We present sublinear-time (randomized) algorithms for finding simple cycles
of length at least and tree-minors in bounded-degree graphs. The
complexity of these algorithms is related to the distance of the graph from
being -minor-free (resp., free from having the corresponding tree-minor).
In particular, if the graph is far (i.e., -far) {from} being
cycle-free, i.e. if one has to delete a constant fraction of edges to make it
cycle-free, then the algorithm finds a cycle of polylogarithmic length in time
\tildeO(\sqrt{N}), where denotes the number of vertices. This time
complexity is optimal up to polylogarithmic factors.
The foregoing results are the outcome of our study of the complexity of {\em
one-sided error} property testing algorithms in the bounded-degree graphs
model. For example, we show that cycle-freeness of -vertex graphs can be
tested with one-sided error within time complexity
\tildeO(\poly(1/\e)\cdot\sqrt{N}). This matches the known
query lower bound, and contrasts with the fact that any minor-free property
admits a {\em two-sided error} tester of query complexity that only depends on
the proximity parameter \e. For any constant , we extend this result
to testing whether the input graph has a simple cycle of length at least .
On the other hand, for any fixed tree , we show that -minor-freeness has
a one-sided error tester of query complexity that only depends on the proximity
parameter \e.
Our algorithm for finding cycles in bounded-degree graphs extends to general
graphs, where distances are measured with respect to the actual number of
edges. Such an extension is not possible with respect to finding tree-minors in
complexity.Comment: Keywords: Sublinear-Time Algorithms, Property Testing, Bounded-Degree
Graphs, One-Sided vs Two-Sided Error Probability Updated versio
Molecular and fossil evidence place the origin of cichlid fishes long after Gondwanan rifting.
Cichlid fishes are a key model system in the study of adaptive radiation, speciation and evolutionary developmental biology. More than 1600 cichlid species inhabit freshwater and marginal marine environments across several southern landmasses. This distributional pattern, combined with parallels between cichlid phylogeny and sequences of Mesozoic continental rifting, has led to the widely accepted hypothesis that cichlids are an ancient group whose major biogeographic patterns arose from Gondwanan vicariance. Although the Early Cretaceous (ca 135 Ma) divergence of living cichlids demanded by the vicariance model now represents a key calibration for teleost molecular clocks, this putative split pre-dates the oldest cichlid fossils by nearly 90 Myr. Here, we provide independent palaeontological and relaxed-molecular-clock estimates for the time of cichlid origin that collectively reject the antiquity of the group required by the Gondwanan vicariance scenario. The distribution of cichlid fossil horizons, the age of stratigraphically consistent outgroup lineages to cichlids and relaxed-clock analysis of a DNA sequence dataset consisting of 10 nuclear genes all deliver overlapping estimates for crown cichlid origin centred on the Palaeocene (ca 65-57 Ma), substantially post-dating the tectonic fragmentation of Gondwana. Our results provide a revised macroevolutionary time scale for cichlids, imply a role for dispersal in generating the observed geographical distribution of this important model clade and add to a growing debate that questions the dominance of the vicariance paradigm of historical biogeography
An SU(5) Heterotic Standard Model
We introduce a new heterotic Standard Model which has precisely the spectrum
of the Minimal Supersymmetric Standard Model (MSSM), with no exotic matter. The
observable sector has gauge group SU(3) x SU(2) x U(1). Our model is obtained
from a compactification of heterotic strings on a Calabi-Yau threefold with Z_2
fundamental group, coupled with an invariant SU(5) bundle. Depending on the
region of moduli space in which the model lies, we obtain a spectrum consisting
of the three generations of the Standard Model, augmented by 0, 1 or 2 Higgs
doublet conjugate pairs. In particular, we get the first compactification
involving a heterotic string vacuum (i.e. a {\it stable} bundle) yielding
precisely the MSSM with a single pair of Higgs.Comment: 15 page
Moduli in N=1 heterotic/F-theory duality
The moduli in a 4D N=1 heterotic compactification on an elliptic CY, as well
as in the dual F-theoretic compactification, break into "base" parameters which
are even (under the natural involution of the elliptic curves), and "fiber" or
twisting parameters; the latter include a continuous part which is odd, as well
as a discrete part. We interpret all the heterotic moduli in terms of
cohomology groups of the spectral covers, and identify them with the
corresponding F-theoretic moduli in a certain stable degeneration. The argument
is based on the comparison of three geometric objects: the spectral and cameral
covers and the ADE del Pezzo fibrations. For the continuous part of the
twisting moduli, this amounts to an isomorphism between certain abelian
varieties: the connected component of the heterotic Prym variety (a modified
Jacobian) and the F-theoretic intermediate Jacobian. The comparison of the
discrete part generalizes the matching of heterotic 5brane / F-theoretic 3brane
impurities.Comment: Latex, 26 pages. Acknowledgements adde
The Optimisation of Bayesian Classifier in Predictive Spatial Modelling for Secondary Mineral Deposits
This paper discusses the general concept of Bayesian Network classifier and the optimisation of a predictive spatial model using Naive Bayes (NB) on secondary mineral deposit data. A different NB modelling approaches to mineral distribution data was used to predict the occurrence of a particular mineral deposit in a given area, which include; predictive attributes sub-selection, normalised attributes selection, NB dependent attributes and the strictness to NB model assumptions of attributes independence selection. The performance of the model was determined by selecting a model with the best predictive accuracy. The NB classifier that violates assumptions of attributes independence was used to compare with other forms of NB. The aim is to improve the general performance of the model through the best selection of predictive attribute data. The paper elaborates the workings of a Bayesian Network learning model, the concept of NB and its application to predicting mineral deposit potentials. The result of the optimised NB model based on predictive accuracies and the Receivr Operating Characteristics (ROC) value is also determined
Higgs Doublets, Split Multiplets and Heterotic SU(3)_C x SU(2)_L x U(1)_Y Spectra
A methodology for computing the massless spectrum of heterotic vacua with
Wilson lines is presented. This is applied to a specific class of vacua with
holomorphic SU(5)-bundles over torus-fibered Calabi-Yau threefolds with
fundamental group Z_2. These vacua lead to low energy theories with the
standard model gauge group SU(3)_C x SU(2)_L x U(1)_Yand three families of
quark/leptons. The massless spectrum is computed, including the multiplicity of
Higgs doublets.Comment: 11+1 p
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