deletions | additions       diff --git a/A Generic Framework to Quantify Materials Science Information.tex b/A Generic Framework to Quantify Materials Science Information.tex  index de6f18a..7309e4d 100644  --- a/A Generic Framework to Quantify Materials Science Information.tex  +++ b/A Generic Framework to Quantify Materials Science Information.tex 
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1.3 Local States and Local States Spaces of Material Dependent Information  Independent of the spatial domain is the n-tuple of material features at each position x; a tuple is an ordered sequence of values. Each tuple in the sequence describes a local material state with values constrained are by a bounded space called the local state space; the local state space is the complete set of possible local states for a material feature. It is possible for one or many local material states to be extracted from the same model; several models can be combined to form the ordered set of local material states for the same sample volume. The microstructure function intrinsically accommodates multiple local state spaces. This can be demonstrated by expanding the Eq. xx as follows  {(_ ^1)β (x),..,(_ ^I)β (x)} (_ ^1)β (x),..,(_ ^I)β (x)=∑_H▒〖m_s^h χ_s (x) χ^(h_1 ) ((_ ^1)β (x))⋯χ^(h_I ) ((_ ^I)β(x) ) 〗 EQUATION  where χ^(h_i ) ((_ ^i)β ) EQUATION  is that basis representation of the ith tuple of the local material state (_ ^i)β(x) . The local state index h is uniquely mapped to (h_1,…,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, {ρ,V_f,(φ_1,Φ,φ_1 )}, that corresponds to the discrete phase indicator of the material (ρ), the volume fraction of an element within a phase (V_f ), and the grain orientation (φ_1,Φ,φ_1 ), 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, χ^h ((_ ^i)β ). 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. 
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In many experiments and some simulations, the material information extracted by a model may be uncertain or partial. A partial dataset will contain empty data points. An uncertain dataset will contain confidence indices statistical information corresponding to the uncertainty of each data point. A dataset can be both partial and uncertain. These types of datasets may arise from poor boundary conditions or numerical instabilities in the simulation. In experiments, the resolution of the detector may impose epistemic uncertainty in the local material information. For example, in Electron Back Scattered Detection (EBSD) the confidence index at each position that sampled is provided as an output from the model. The grain boundaadries in the material are often recorded as low confidence parameters. Figure yy(a) shows an EBSD scan of 7xxx series Aluminum and Figure yy(b) is a map of the confidence indexes associated with each position. It can be seen in this figure that there are regions of both high and low certainty.   A model may provide the confidence of each data point extracted within the volume. Using the same basis function that partitions the spatial domain, the digitized weight of uncertainty for each partition is derived from  w_ ^i (x)=∑_H▒〖 w_s^i χ_s (x) χ^i (w_ ^i (x)) 〗 EQUATION  where w_ ^i (x) is a weight associated with the confidence of the ith mode of the local material state sampled described by the weight basis χ^i (w_ ^i (x)) at x provided by the model. w_s^i is the spatially resolved weighting signal. The confidence of the sampled information is incorporated into the microstructure function as   β^i n(x,β^i )=w_s^i ∑_H▒〖m_s^h χ_s (x) χ^h (β^i (x) β^i ) 〗 EQUATION  The weights are bounded between zero and one. Partial data points will assume a weight of zero and will be completely ignored from the analysis of the sample volume. Meanwhile, a weight of one indicates complete information that and assumes total confidence in the local state information that is sampled. A dataset is partial if ∃s,i such that w_s^i=0. Any weights between zero and one indicate uncertain data points.(Figure yy(b))