deletions | additions       diff --git a/local states.tex b/local states.tex  index dfc0642..b847e93 100644  --- a/local states.tex  +++ b/local states.tex 
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\begin{equation}  {^{1}\beta(x),^{2}\beta(x),...^{I}\beta(x)} = \sum_{H} \, m_{s}^{h} \chi_{s}(x) \chi^{h_{1}}(^{1}\beta(x)) \chi^{h_{2}}(^{2}\beta(x)) ... \chi^{h_{I}}(^{I}\beta(x))  \end{equation}  where $\chi^{h_{i}}(^{i}\beta(x))$ is that basis representation of the $i^{th}$ tuple of the local material state $^{i}\beta(x)$ . The local state index $h$ is uniquely mapped to (h_{1},h_{2},...h_{I}). It is expected that each local state space expressed by the microstructure function will require a different basis function. For example, if the local material state is expressed as a 3-tuple, EQUATION, $\{\rho, V, (\varphi_{1},\Phi,\varphi_{1})\}$,  that corresponds to the discrete phase indicator of the material (ρ), ($\rho$),  the volume fraction of an element within a phase EQUATION, phase,  and the grain orientation EQUATION, $V_{f}$,  then a different basis function will be needed for each element. Literature reports an extensive variety of basis functions.[ref] Fortunately, characteristics of the local state space inform the choice of basis functions for the ith mode of the local state, EQUATION. The following list describes all of the possible characteristics of the local state spaces:  Discrete LSS – The local state is identified by a discrete index that demarcates a particular class of material features. For example in steel, a discrete basis will uniquely index different phases of steel (e.g. martensite, austenite, pearlite). Indicator basis functions can suite this application.  Bounded, Periodic LSS – The local states on opposite boundaries exhibit similar material behavior(this sentence stinks). A local state space that identifies the angle of a material feature over [0,2π] is periodic. The cosine transform provides a basis for this type of local state space. Some new work will show an application of generalized spherical harmonics to grain orientation.