Cabling sequences of tunnels of torus knots

This is the second of three papers that refine and extend portions of our earlier preprint, "The depth of a knot tunnel." Together, they rework the entire preprint. The theory of tunnel number 1 knots that we introduced in "The tree of knot tunnels" yields a parameterization in which each tunnel is described uniquely by a finite sequence of rational parameters and a finite sequence of 0's and 1's, that together encode a procedure for constructing the knot and tunnel. In this paper we calculate these invariants for all tunnels of torus knots.Comment: 17 pages, 12 figures, to appear in Algebraic and Geometric Topolog

Charge transfer electrostatic model of compositional order in perovskite alloys

We introduce an electrostatic model including charge transfer, which is shown to account for the observed B-site ordering in Pb-based perovskite alloys. The model allows charge transfer between A-sites and is a generalization of Bellaiche and Vanderbilt's purely electrostatic model. The large covalency of Pb^{2+} compared to Ba^{2+} is modeled by an environment dependent effective A-site charge. Monte Carlo simulations of this model successfully reproduce the long range compositional order of both Pb-based and Ba-based complex A(BB^{'}B^{''})O_3 perovskite alloys. The models are also extended to study systems with A-site and B-site doping, such as (Na_{1/2}La_{1/2})(Mg_{1/3}Nb_{2/3})O_3, (Ba_{1-x}La_{x})(Mg_{(1+x)/3}Nb_{(2-x)/3})O_3 and (Pb_{1-x}La_{x})(Mg_{(1+x)/3}Ta_{(2-x)/3})O_3. General trends are reproduced by purely electrostatic interactions, and charge transfer effects indicate that local structural relaxations can tip the balance between different B-site orderings in Pb based materials.Comment: 15 pages, 6 figure

Cosine Similarity Measure According to a Convex Cost Function

In this paper, we describe a new vector similarity measure associated with a convex cost function. Given two vectors, we determine the surface normals of the convex function at the vectors. The angle between the two surface normals is the similarity measure. Convex cost function can be the negative entropy function, total variation (TV) function and filtered variation function. The convex cost function need not be differentiable everywhere. In general, we need to compute the gradient of the cost function to compute the surface normals. If the gradient does not exist at a given vector, it is possible to use the subgradients and the normal producing the smallest angle between the two vectors is used to compute the similarity measure

Routing, Localization And Positioning Protocols For Wireless Sensor And Actor Networks

Wireless sensor and actor networks (WSANs) are distributed systems of sensor nodes and actors that are interconnected over the wireless medium. Sensor nodes collect information about the physical world and transmit the data to actors by using one-hop or multi-hop communications. Actors collect information from the sensor nodes, process the information, take decisions and react to the events. This dissertation presents contributions to the methods of routing, localization and positioning in WSANs for practical applications. We first propose a routing protocol with service differentiation for WSANs with stationary nodes. In this setting, we also adapt a sports ranking algorithm to dynamically prioritize the events in the environment depending on the collected data. We extend this routing protocol for an application, in which sensor nodes float in a river to gather observations and actors are deployed at accessible points on the coastline. We develop a method with locally acting adaptive overlay network formation to organize the network with actor areas and to collect data by using locality-preserving communication. We also present a multi-hop localization approach for enriching the information collected from the river with the estimated locations of mobile sensor nodes without using positioning adapters. As an extension to this application, we model the movements of sensor nodes by a subsurface meandering current mobility model with random surface motion. Then we adapt the introduced routing and network organization methods to model a complete primate monitoring system. A novel spatial cut-off preferential attachment model and iii center of mass concept are developed according to the characteristics of the primate groups. We also present a role determination algorithm for primates, which uses the collection of spatial-temporal relationships. We apply a similar approach to human social networks to tackle the problem of automatic generation and organization of social networks by analyzing and assessing interaction data. The introduced routing and localization protocols in this dissertation are also extended with a novel three dimensional actor positioning strategy inspired by the molecular geometry. Extensive simulations are conducted in OPNET simulation tool for the performance evaluation of the proposed protocol