Add the beginning of the final paper.

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+#LyX 1.6.8 created this file. For more info see http://www.lyx.org/
+\lyxformat 345
+\begin_document
+\begin_header
+\textclass amsart
+\use_default_options false
+\begin_modules
+theorems-ams
+eqs-within-sections
+figs-within-sections
+\end_modules
+\language english
+\inputencoding auto
+\font_roman ae
+\font_sans default
+\font_typewriter default
+\font_default_family default
+\font_sc false
+\font_osf false
+\font_sf_scale 100
+\font_tt_scale 100
+
+\graphics default
+\paperfontsize default
+\spacing single
+\use_hyperref false
+\papersize letterpaper
+\use_geometry false
+\use_amsmath 1
+\use_esint 1
+\cite_engine basic
+\use_bibtopic false
+\paperorientation portrait
+\secnumdepth 3
+\tocdepth 3
+\paragraph_separation skip
+\defskip bigskip
+\quotes_language english
+\papercolumns 1
+\papersides 1
+\paperpagestyle default
+\tracking_changes false
+\output_changes false
+\author "" 
+\author "" 
+\end_header
+
+\begin_body
+
+\begin_layout Title
+Spline3
+\end_layout
+
+\begin_layout Author
+Michael Orlitzky
+\end_layout
+
+\begin_layout Example*
+First, will will define our grid 
+\begin_inset Formula $G$
+\end_inset
+
+ to be over a 
+\begin_inset Formula $3\times3\times3$
+\end_inset
+
+ cube having grid size 
+\begin_inset Formula $h=1$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Example*
+To test the reproduction of trilinears, we will take the function (per Sorokina
+ and Zeilfelder, p.
+ 88),
+\end_layout
+
+\begin_layout Example*
+\begin_inset Formula \[
+f\left(x,y,z\right)=1+x+xy+xyz\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Example*
+and compute its value at the 27 points, 
+\begin_inset Formula $\left(0,0,0\right),\left(0,0,1\right),\dots\left(2,2,2\right)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Example*
+\begin_inset listings
+inline false
+status open
+
+\begin_layout Plain Layout
+
+sage: x,y,z = var('x,y,z')
+\end_layout
+
+\begin_layout Plain Layout
+
+sage: f = 1 + x + x*y + x*y*z
+\end_layout
+
+\begin_layout Plain Layout
+
+sage: f(x=0, y=0, z=0)
+\end_layout
+
+\begin_layout Plain Layout
+
+1
+\end_layout
+
+\begin_layout Plain Layout
+
+sage: f(x=1, y=0, z=0)
+\end_layout
+
+\begin_layout Plain Layout
+
+2
+\end_layout
+
+\begin_layout Plain Layout
+
+sage: f(x=2, y=0, z=0)
+\end_layout
+
+\begin_layout Plain Layout
+
+3
+\end_layout
+
+\begin_layout Plain Layout
+
+sage: f(x=0, y=1, z=0)
+\end_layout
+
+\begin_layout Plain Layout
+
+1
+\end_layout
+
+\begin_layout Plain Layout
+
+sage: f(x=1, y=1, z=0)
+\end_layout
+
+\begin_layout Plain Layout
+
+3
+\end_layout
+
+\begin_layout Plain Layout
+
+sage: f(x=2, y=1, z=0)
+\end_layout
+
+\begin_layout Plain Layout
+
+5
+\end_layout
+
+\begin_layout Plain Layout
+
+sage: f(x=0, y=2, z=0)
+\end_layout
+
+\begin_layout Plain Layout
+
+1
+\end_layout
+
+\begin_layout Plain Layout
+
+sage: f(x=1, y=2, z=0)
+\end_layout
+
+\begin_layout Plain Layout
+
+4
+\end_layout
+
+\begin_layout Plain Layout
+
+sage: f(x=2, y=2, z=0)
+\end_layout
+
+\begin_layout Plain Layout
+
+7
+\end_layout
+
+\begin_layout Plain Layout
+
+sage: f(x=0, y=0, z=1)
+\end_layout
+
+\begin_layout Plain Layout
+
+1
+\end_layout
+
+\begin_layout Plain Layout
+
+sage: f(x=1, y=0, z=1)
+\end_layout
+
+\begin_layout Plain Layout
+
+2
+\end_layout
+
+\begin_layout Plain Layout
+
+sage: f(x=2, y=0, z=1)
+\end_layout
+
+\begin_layout Plain Layout
+
+3
+\end_layout
+
+\begin_layout Plain Layout
+
+sage: f(x=0, y=1, z=1)
+\end_layout
+
+\begin_layout Plain Layout
+
+1
+\end_layout
+
+\begin_layout Plain Layout
+
+sage: f(x=1, y=1, z=1)
+\end_layout
+
+\begin_layout Plain Layout
+
+4
+\end_layout
+
+\begin_layout Plain Layout
+
+sage: f(x=2, y=1, z=1)
+\end_layout
+
+\begin_layout Plain Layout
+
+7
+\end_layout
+
+\begin_layout Plain Layout
+
+sage: f(x=0, y=2, z=1)
+\end_layout
+
+\begin_layout Plain Layout
+
+1
+\end_layout
+
+\begin_layout Plain Layout
+
+sage: f(x=1, y=2, z=1)
+\end_layout
+
+\begin_layout Plain Layout
+
+6
+\end_layout
+
+\begin_layout Plain Layout
+
+sage: f(x=2, y=2, z=1)
+\end_layout
+
+\begin_layout Plain Layout
+
+11
+\end_layout
+
+\begin_layout Plain Layout
+
+sage: f(x=0, y=0, z=2)
+\end_layout
+
+\begin_layout Plain Layout
+
+1
+\end_layout
+
+\begin_layout Plain Layout
+
+sage: f(x=1, y=0, z=2)
+\end_layout
+
+\begin_layout Plain Layout
+
+2
+\end_layout
+
+\begin_layout Plain Layout
+
+sage: f(x=2, y=0, z=2)
+\end_layout
+
+\begin_layout Plain Layout
+
+3
+\end_layout
+
+\begin_layout Plain Layout
+
+sage: f(x=0, y=1, z=2)
+\end_layout
+
+\begin_layout Plain Layout
+
+1
+\end_layout
+
+\begin_layout Plain Layout
+
+sage: f(x=1, y=1, z=2)
+\end_layout
+
+\begin_layout Plain Layout
+
+5
+\end_layout
+
+\begin_layout Plain Layout
+
+sage: f(x=2, y=1, z=2)
+\end_layout
+
+\begin_layout Plain Layout
+
+9 
+\end_layout
+
+\begin_layout Plain Layout
+
+sage: f(x=0, y=2, z=2) 
+\end_layout
+
+\begin_layout Plain Layout
+
+1
+\end_layout
+
+\begin_layout Plain Layout
+
+sage: f(x=1, y=2, z=2)
+\end_layout
+
+\begin_layout Plain Layout
+
+8 
+\end_layout
+
+\begin_layout Plain Layout
+
+sage: f(x=2, y=2, z=2)
+\end_layout
+
+\begin_layout Plain Layout
+
+15
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Example*
+We are most interested in the 
+\begin_inset Quotes eld
+\end_inset
+
+interior
+\begin_inset Quotes erd
+\end_inset
+
+ cube centered at 
+\begin_inset Formula $\left(1,1,1\right)$
+\end_inset
+
+ with datum 
+\begin_inset Formula $4$
+\end_inset
+
+.
+ We can enter the data above into a list,
+\end_layout
+
+\begin_layout Example*
+\begin_inset listings
+inline false
+status open
+
+\begin_layout Plain Layout
+
+sage: g = [[[ f(x=a, y=b, z=c) for a in range(0,3) ] for b in range(0,3)
+ ] for c in range(0,3) ]
+\end_layout
+
+\begin_layout Plain Layout
+
+sage: g
+\end_layout
+
+\begin_layout Plain Layout
+
+[[[1, 2, 3], [1, 3, 5], [1, 4, 7]],
+\end_layout
+
+\begin_layout Plain Layout
+
+ [[1, 2, 3], [1, 4, 7], [1, 6, 11]],
+\end_layout
+
+\begin_layout Plain Layout
+
+ [[1, 2, 3], [1, 5, 9], [1, 8, 15]]]
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Example*
+although the list will be indexed by 
+\begin_inset Formula $\left(z,y,x\right)$
+\end_inset
+
+ so we define a function to access it by 
+\begin_inset Formula $\left(x,y,z\right)$
+\end_inset
+
+,
+\end_layout
+
+\begin_layout Example*
+\begin_inset listings
+inline false
+status open
+
+\begin_layout Plain Layout
+
+sage: def grid(x,y,z):
+\end_layout
+
+\begin_layout Plain Layout
+
+....:     return g[z][y][x]
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Example*
+and define directional functions according to Sorokina and Zeilfelder, p.
+ 81.
+\end_layout
+
+\begin_layout Example*
+\begin_inset listings
+inline false
+status open
+
+\begin_layout Plain Layout
+
+sage: def I(x,y,z):
+\end_layout
+
+\begin_layout Plain Layout
+
+....:     return grid(x,y,z)
+\end_layout
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\begin_layout Plain Layout
+
+sage: def F(x,y,z):
+\end_layout
+
+\begin_layout Plain Layout
+
+....:     return grid(x-1, y, z)
+\end_layout
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\begin_layout Plain Layout
+
+sage: def B(x,y,z):
+\end_layout
+
+\begin_layout Plain Layout
+
+....:     return grid(x+1, y, z)
+\end_layout
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\begin_layout Plain Layout
+
+sage: def L(x,y,z):
+\end_layout
+
+\begin_layout Plain Layout
+
+....:     return grid(x, y-1, z)
+\end_layout
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\begin_layout Plain Layout
+
+sage: def R(x,y,z):
+\end_layout
+
+\begin_layout Plain Layout
+
+....:     return grid(x, y+1, z)
+\end_layout
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\begin_layout Plain Layout
+
+sage: def T(x,y,z): 
+\end_layout
+
+\begin_layout Plain Layout
+
+....:     return grid(x, y, z+1)
+\end_layout
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\begin_layout Plain Layout
+
+sage: def D(x,y,z): 
+\end_layout
+
+\begin_layout Plain Layout
+
+....:     return grid(x, y, z-1)
+\end_layout
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\begin_layout Plain Layout
+
+sage: def FL(x,y,z):
+\end_layout
+
+\begin_layout Plain Layout
+
+....:     return grid(x-1, y-1, z)
+\end_layout
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\begin_layout Plain Layout
+
+sage: def FR(x,y,z):
+\end_layout
+
+\begin_layout Plain Layout
+
+....:     return grid(x-1, y+1, z)
+\end_layout
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\begin_layout Plain Layout
+
+sage: def FD(x,y,z):
+\end_layout
+
+\begin_layout Plain Layout
+
+....:     return grid(x-1, y, z-1)
+\end_layout
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\begin_layout Plain Layout
+
+sage: def FT(x,y,z):
+\end_layout
+
+\begin_layout Plain Layout
+
+....:     return grid(x-1, y, z+1)
+\end_layout
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\begin_layout Plain Layout
+
+sage: def BL(x,y,z):
+\end_layout
+
+\begin_layout Plain Layout
+
+....:     return grid(x+1, y-1, z)
+\end_layout
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\begin_layout Plain Layout
+
+sage: def BR(x,y,z):
+\end_layout
+
+\begin_layout Plain Layout
+
+....:     return grid(x+1, y+1, z)
+\end_layout
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\begin_layout Plain Layout
+
+sage: def BD(x,y,z):
+\end_layout
+
+\begin_layout Plain Layout
+
+....:     return grid(x+1, y, z-1)
+\end_layout
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\begin_layout Plain Layout
+
+sage: def BT(x,y,z):
+\end_layout
+
+\begin_layout Plain Layout
+
+....:     return grid(x+1, y, z+1)
+\end_layout
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\begin_layout Plain Layout
+
+sage: def LD(x,y,z):
+\end_layout
+
+\begin_layout Plain Layout
+
+....:     return grid(x, y-1, z-1)
+\end_layout
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\begin_layout Plain Layout
+
+sage: def LT(x,y,z): 
+\end_layout
+
+\begin_layout Plain Layout
+
+....:     return grid(x, y-1, z+1)
+\end_layout
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\begin_layout Plain Layout
+
+sage: def RD(x,y,z): 
+\end_layout
+
+\begin_layout Plain Layout
+
+....:     return grid(x, y+1, z-1)
+\end_layout
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\begin_layout Plain Layout
+
+sage: def RT(x,y,z): 
+\end_layout
+
+\begin_layout Plain Layout
+
+....:     return grid(x, y+1, z+1)
+\end_layout
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\begin_layout Plain Layout
+
+sage: def FLD(x,y,z):
+\end_layout
+
+\begin_layout Plain Layout
+
+....:     return grid(x-1, y-1, z-1)
+\end_layout
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\begin_layout Plain Layout
+
+sage: def FLT(x,y,z):
+\end_layout
+
+\begin_layout Plain Layout
+
+....:     return grid(x-1, y-1, z+1)
+\end_layout
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\begin_layout Plain Layout
+
+sage: def FRD(x,y,z):
+\end_layout
+
+\begin_layout Plain Layout
+
+....:     return grid(x-1, y+1, z-1)
+\end_layout
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\begin_layout Plain Layout
+
+sage: def FRT(x,y,z):
+\end_layout
+
+\begin_layout Plain Layout
+
+....:     return grid(x-1, y+1, z+1)
+\end_layout
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\begin_layout Plain Layout
+
+sage: def BLD(x,y,z): 
+\end_layout
+
+\begin_layout Plain Layout
+
+....:     return grid(x+1, y-1, z-1)
+\end_layout
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\begin_layout Plain Layout
+
+sage: def BLT(x,y,z):
+\end_layout
+
+\begin_layout Plain Layout
+
+....:     return grid(x+1, y-1, z+1)
+\end_layout
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\begin_layout Plain Layout
+
+sage: def BRD(x,y,z):
+\end_layout
+
+\begin_layout Plain Layout
+
+....:     return grid(x+1, y+1, z-1)
+\end_layout
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\begin_layout Plain Layout
+
+sage: def BRT(x,y,z):
+\end_layout
+
+\begin_layout Plain Layout
+
+....:     return grid(x+1, y+1, z+1)
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Example*
+Next, we define the coefficients for the cube centered on 
+\begin_inset Formula $\left(1,1,1\right)$
+\end_inset
+
+ based on these directional functions.
+\end_layout
+
+\begin_layout Example*
+\begin_inset listings
+inline false
+status open
+
+\begin_layout Plain Layout
+
+sage: c0030 = (1/8)*( I(1,1,1) + F(1,1,1) + L(1,1,1) + T(1,1,1) +
+\end_layout
+
+\begin_layout Plain Layout
+
+                      LT(1,1,1) + FL(1,1,1) + FT(1,1,1) + FLT(1,1,1) )
+\end_layout
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\begin_layout Plain Layout
+
+sage: c0003 = (1/8)*( I(1,1,1) + F(1,1,1) + R(1,1,1) + T(1,1,1) +
+\end_layout
+
+\begin_layout Plain Layout
+
+                RT(1,1,1) + FR(1,1,1) + FT(1,1,1) + FRT(1,1,1) )
+\end_layout
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\begin_layout Plain Layout
+
+sage: c0021 = (5/24)*(I(1,1,1) + F(1,1,1) + T(1,1,1) + FT(1,1,1)) +
+\end_layout
+
+\begin_layout Plain Layout
+
+              (1/24)*(L(1,1,1) + FL(1,1,1) + LT(1,1,1) + FLT(1,1,1))
+\end_layout
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\begin_layout Plain Layout
+
+sage: c0012 = (5/24)*(I(1,1,1) + F(1,1,1) + T(1,1,1) + FT(1,1,1)) +
+\end_layout
+
+\begin_layout Plain Layout
+
+              (1/24)*(R(1,1,1) + FR(1,1,1) + RT(1,1,1) + FRT(1,1,1))
+\end_layout
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Example*
+We can see what the constant values are now:
+\end_layout
+
+\begin_layout Example*
+\begin_inset listings
+inline false
+status open
+
+\begin_layout Plain Layout
+
+sage: c0030
+\end_layout
+
+\begin_layout Plain Layout
+
+17/8
+\end_layout
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\begin_layout Plain Layout
+
+sage: c0003
+\end_layout
+
+\begin_layout Plain Layout
+
+27/8
+\end_layout
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\begin_layout Plain Layout
+
+sage: c0021
+\end_layout
+
+\begin_layout Plain Layout
+
+61/24
+\end_layout
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\begin_layout Plain Layout
+
+sage: c0012
+\end_layout
+
+\begin_layout Plain Layout
+
+71/24
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_body
+\end_document