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Let $R$ be a noncommutative ring with identity. The commuting graph of $R$, denoted by $\Gamma(R)$, is a graph with vertex set $R \setminus Z(R)$, and two vertices $a$, $b$ are adjacent if $a\neq b$ and $ab=ba$. Let $T=Tr(R)$ be the ring of all $2\times 2$ upper triangular matrices over $R$ and $\Gamma(T)$ be the commuting graph of $T$. In this article, we find the number of edges, cliques, clique number, and independence number of $\Gamma (T)$ when $R$ is a finite field. Moreover, we show that for the case when $R= \mathbb{Z}_{n}$ is not a field, $\Gamma (T)$ is connected with diameter 3. Some useful related results are also obtained, some examples are presented and a question is posed.

This SBIR project integrates theoretical and experimental work to enable a new generation of high-capacity, high-voltage cathode materials that will lead to high-performance, robust energy storage systems. At low operating temperatures, commercially available electrode materials for lithium-ion (Li-ion) batteries do not meet energy and power requirements for NASA's planned exploration activities. NEI Corporation, in partnership with the University of California, San Diego, has developed layered composite cathode materials that increase power and energy densities at temperatures as low as 0 degC and considerably reduce the overall volume and weight of battery packs. In Phase I of the project, through innovations in the structure and morphology of composite electrode particles, the partners successfully demonstrated an energy density exceeding 1,000 Wh/kg at 4 V at room temperature. In Phase II, the team enhanced the kinetics of Li-ion transport and electronic conductivity at 0 degC. An important feature of the composite cathode is that it has at least two components that are structurally integrated. The layered material is electrochemically inactive; however, upon structural integration with a spinel material, the layered material can be electrochemically activated and deliver a large amount of energy with stable cycling.

A method for predicting isothermal plastic fatigue life of a composite lamina is presented in which both fibers and matrix are isotropic materials. In general, the fatigue resistances of the matrix, fibers, and interfacial material must be known in order to predict composite fatigue life. Composite fatigue life is predicted using only the matrix fatigue resistance due to inelasticity micromechanisms. The effect of the fiber orientation on loading direction is accounted for while predicting composite life. The application is currently limited to isothermal cases where the internal thermal stresses that might arise from thermal strain mismatch between fibers and matrix are negligible. The theory is formulated to predict the fatigue life of a composite lamina under either load or strain control. It is applied currently to predict the life of tungsten-copper composite lamina at 260 C under tension-tension load control. The calculated life of the lamina is in good agreement with available composite low cycle fatigue data.