Given a closed surface Sg of genus g, a mapping class f in \MCG(Sg) is said to be pseudo-Anosov if it preserves a pair of transverse measured foliations such that one is expanding and the other one is contracting by a number \lambda(f). The number \lambda(f) is called a stretch factor (or dilatation) of f. Thurston showed that a stretch factor is an algebraic integer with degree bounded above by 6g-6. However, little is known about which degrees occur. Using train tracks on surfaces, we explicitly construct pseudo-Anosov maps on Sg with orientable foliations whose stretch factor \lambda has algebraic degree 2g. Moreover, the stretch factor \lambda is a special algebraic number, called Salem number. Using this result, we show that there is a pseudo-Anosov map whose stretch factor has algebraic degree d, for each positive even integer d such that d≤g. Our examples also give a new approach to a conjecture of Penner.

Recently, many natural disasters have occurred (e.g., the 2011 tsunami in Japan). In response to those disasters, Wireless Sensor Networks have been proposed to improve their detection level. This new technology has two main challenges which are routing and topology control where their multi-dimensional dilations need to be improved/balanced. Related to those metrics, the packet delivery factor also needs to be improved/guaranteed. This thesis presents the design of new routing protocols, referred to as: 1) the 3-D Sensing Sphere close to the Line:Smallest Angle to the Line (SSL:SAL) protocol, 2) the 3-D Randomized Sensing Spheres (RSS) protocol, and 3) the SSL:SAL version 1 and version 2 (i.e., SSL:SALv1 and SSL:SALv2, respectively). Through simulations, these protocols are shown to balance/improve the multi-dimensional dilations metrics which also include new bandwidth metrics. The balance/improvement is achieved over some existing position-based protocols. In addition, packet delivery is guaranteed mathematically for new and existing protocols. Furthermore, some experimental evidences are gathered regarding the delivery rate impact on the multi-dimensional metrics. The thesis also proposes a new set of 2-D and 3-D graphs, so called: 1) the Derived Circle version 1 (DCv1) graphs and 2) the Derived Sphere (DSv1) graphs. The new approaches improve the multi-dimensional dilations over some existing graphs. In addition, connectivity, rotability, fault tolerance properties are achieved. Lastly, the thesis develops a framework that combines routing protocols and graphs in fully covered regions. Some experimental evidences demonstrate the improvement of the multi-dimensional metrics and the packet delivery rate for the routing protocols based on the DSv1. This is compared to the routing protocols based on an existing graph. Furthermore, based on either the proposed or existing graphs, some important findings are demonstrated for routing in terms of multi-dimensional metrics and packet delivery rate. Among those findings, the proposed protocol and an exiting protocol have higher delivery rates compared to another existing protocol. Furthermore, the proposed graph improves the multi-dimensional metrics for the proposed and existing protocols over another existing protocol for low communication ranges.