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 $M$-th colored Jones polynomials evaluated at $(N+1/2)$-th root of unity with a fixed limiting ratio, $s$, of $M$ and $(N+1/2)$. We find out the asymptotic expansion formula (AEF) of the colored Jones polynomials of the figure eight knot with $s$ close to $1$. Nonetheless, we show that the exponential growth rate of the colored Jones polynomials of the figure eight knot with $s$ close to $1/2$ is strictly less than those with $s$ close to $1$. 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 $s$ 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

"Heterotic" Discrete Flavor Model

We present an extended 331 model with $T'$ discrete flavor symmetry that simultaneously explains the need to have exactly three generations and provides acceptable quark and lepton masses and mixings. New fermionic states and gauge bosons are predicted within the reach of the LHC. We discuss the relevance to the 126 GeV scalar discovered at the LHC.Comment: 13 pages, v3: version to appear in PR

Time lagged ordinal partition networks for capturing dynamics of continuous dynamical systems

We investigate a generalised version of the recently proposed ordinal partition time series to network transformation algorithm. Firstly we introduce a fixed time lag for the elements of each partition that is selected using techniques from traditional time delay embedding. The resulting partitions define regions in the embedding phase space that are mapped to nodes in the network space. Edges are allocated between nodes based on temporal succession thus creating a Markov chain representation of the time series. We then apply this new transformation algorithm to time series generated by the R\"ossler system and find that periodic dynamics translate to ring structures whereas chaotic time series translate to band or tube-like structures -- thereby indicating that our algorithm generates networks whose structure is sensitive to system dynamics. Furthermore we demonstrate that simple network measures including the mean out degree and variance of out degrees can track changes in the dynamical behaviour in a manner comparable to the largest Lyapunov exponent. We also apply the same analysis to experimental time series generated by a diode resonator circuit and show that the network size, mean shortest path length and network diameter are highly sensitive to the interior crisis captured in this particular data set

Human computer interaction for international development: past present and future

Recent years have seen a burgeoning interest in research into the use of information and communication technologies (ICTs) in the context of developing regions, particularly into how such ICTs might be appropriately designed to meet the unique user and infrastructural requirements that we encounter in these cross-cultural environments. This emerging field, known to some as HCI4D, is the product of a diverse set of origins. As such, it can often be difficult to navigate prior work, and/or to piece together a broad picture of what the field looks like as a whole. In this paper, we aim to contextualize HCI4D—to give it some historical background, to review its existing literature spanning a number of research traditions, to discuss some of its key issues arising from the work done so far, and to suggest some major research objectives for the future