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

Accurate angle-of-arrival measurement using particle swarm optimization

As one of the major methods for location positioning, angle-of-arrival (AOA) estimation is a significant technology in radar, sonar, radio astronomy, and mobile communications. AOA measurements can be exploited to locate mobile units, enhance communication efficiency and network capacity, and support location-aided routing, dynamic network management, and many location-based services. In this paper, we propose an algorithm for AOA estimation in colored noise fields and harsh application scenarios. By modeling the unknown noise covariance as a linear combination of known weighting matrices, a maximum likelihood (ML) criterion is established, and a particle swarm optimization (PSO) paradigm is designed to optimize the cost function. Simulation results demonstrate that the paired estimator PSO-ML significantly outperforms other popular techniques and produces superior AOA estimates

DNSS: Dual-Normal-Space Sampling for 3-D ICP Registration

Rigid registration is a fundamental process in many applications that require alignment of different datasets. Iterative closest point (ICP) is a widely used algorithm that iteratively finds point correspondences and updates the rigid transformation. One of the key variants of ICP to its success is the selection of points, which is directly related to the convergence and robustness of the ICP algorithm. Besides uniform sampling, there are a number of normal-based and feature-based approaches that consider normal, curvature, and/or other signals in the point selection. Among them, normal-space sampling (NSS) is one of the most popular techniques due to its simplicity and low computational cost. The rationale of NSS is to sample enough constraints to determine all the components of transformation, but this paper finds that NSS actually can constrain the translational normal space only. This paper extends the fundamental idea of NSS and proposes Dual NSS (DNSS) to sample points in both translational and rotational normal spaces. Compared with NSS, this approach has similar simplicity and efficiency without any need of additional information, but has a much better effectiveness. Experimental results show that DNSS can outperform the normal-based and feature-based methods in terms of convergence and robustness. For example, DNSS can achieve convergence from an orthogonal initial position while no other methods can achieve. Note to Practitioners-ICP is commonly used to align different data to a same coordination system. While NSS is often used to speed up the alignment process by down-sampling the data uniformly in the normal space. The implementation of NSS only has three steps: 1) construct a set of buckets in the normal-space; 2) put all points of the data into buckets based on their normal direction; and 3) uniformly pick points from all the buckets until the desired number of points is selected. The algorithm is simple and fast, so that it is still the common practice. However, the weakness of NSS comes from the reason that it cannot handle rotational uncertainties. In this paper, a new algorithm called DNSS is developed to constrain both translation and rotation at the same time by introducing a dual-normal space. With a new definition of the normal space, the algorithm complexity of DNSS is the same as that of NSS, and it can be readily implemented in all types of application that are currently using ICP. The experimental results show that DNSS has better efficiency, quality, and reliability than both normal-based and feature-based methods have