10,198 research outputs found
A Short Survey on Data Clustering Algorithms
With rapidly increasing data, clustering algorithms are important tools for
data analytics in modern research. They have been successfully applied to a
wide range of domains; for instance, bioinformatics, speech recognition, and
financial analysis. Formally speaking, given a set of data instances, a
clustering algorithm is expected to divide the set of data instances into the
subsets which maximize the intra-subset similarity and inter-subset
dissimilarity, where a similarity measure is defined beforehand. In this work,
the state-of-the-arts clustering algorithms are reviewed from design concept to
methodology; Different clustering paradigms are discussed. Advanced clustering
algorithms are also discussed. After that, the existing clustering evaluation
metrics are reviewed. A summary with future insights is provided at the end
Asymptotic Behavior of Colored Jones polynomial and Turaev-Viro Invariant of figure eight knot
In this paper we investigate the asymptotic behavior of the colored Jones
polynomials and the Turaev-Viro invariants for the figure eight knot. More
precisely, we consider the -th colored Jones polynomials evaluated at
-th root of unity with a fixed limiting ratio, , of and
. We find out the asymptotic expansion formula (AEF) of the colored
Jones polynomials of the figure eight knot with close to . Nonetheless,
we show that the exponential growth rate of the colored Jones polynomials of
the figure eight knot with close to is strictly less than those with
close to . It is known that the Turaev Viro invariant of the figure
eight knot can be expressed in terms of a sum of its colored Jones polynomials.
Our results show that this sum is asymptotically equal to the sum of the terms
with close to 1. As an application of the asymptotic behavior of the
colored Jones polynomials, we obtain the asymptotic expansion formula for the
Turaev-Viro invariants of the figure eight knot. Finally, we suggest a possible
generalization of our approach so as to relate the AEF for the colored Jones
polynomials and the AEF for the Turaev-Viro invariants for general hyperbolic
knots.Comment: 40 pages, 0 figure
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