deletions | additions       diff --git a/local states.tex b/local states.tex  index b847e93..1de067e 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, $\{\rho, V, (\varphi_{1},\Phi,\varphi_{1})\}$, that corresponds to the discrete phase indicator of the material ($\rho$), the volume fraction of an element within a phase, phase $\varphi$,  and the grain orientation $V_{f}$, 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 $i^{th}$  mode of the local state, EQUATION. $\chi^{h}(^{i}\beta(x))$.  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.  Bounded, Non-periodic LSS – I need help with this description. Legendre polynomials are an example of a basis function for this application. Volume fraction is the example here.