Diff of /trunk/doc/diderot.tex

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	revision 42, Wed Mar 3 18:26:50 2010 UTC	revision 43, Fri Mar 12 04:28:29 2010 UTC
#	Line 40	Line 40
40	\iota & ::= & \TYbool & \text{booleans} \\	\iota & ::= & \TYbool & \text{booleans} \\
41	& \mid & \TYint & \text{integers} \\[1em]	& \mid & \TYint & \text{integers} \\[1em]
42	% \mu for memory, where images of rawtensors will live (or on disk)	% \mu for memory, where images of rawtensors will live (or on disk)
43	\mu & ::= & \TYrawten{o}{d}{\rho} & \begin{minipage}[l]{3in}\begin{flushright}	\mu & ::= & \TYrawten{\Seq{d}}{\rho} & \begin{minipage}[l]{3in}\begin{flushright}
44	tensors of order $o$ and dimension $d$,\\with coefficients of type $\rho$	tensors of order $o$ and dimension $d$,\\with coefficients of type $\rho$
45	\end{flushright}\end{minipage}\\[1em]	\end{flushright}\end{minipage}\\[1em]
46	\theta & ::= & \TYtensor{o}{d} & \begin{minipage}[l]{3in}\begin{flushright}	\theta & ::= & \TYtensor{\Seq{d}} & \begin{minipage}[l]{3in}\begin{flushright}
47	tensors of order $o$ and dimension $d$,\\with real coefficients	tensors of order $o$ and dimension $d$,\\with real coefficients
48	\end{flushright}\end{minipage}\\[1em]	\end{flushright}\end{minipage}\\[1em]
49	\tau & ::= & \iota \\	\tau & ::= & \iota \\
#	Line 61	Line 61
61	\end{figure}%	\end{figure}%
62	We use some type abbreviations for common cases:	We use some type abbreviations for common cases:
63	\begin{eqnarray*}	\begin{eqnarray*}
64	\TYreal & = & \TYtensor{0}{d} \quad\text{for any $d$} \\	\TYreal & = & \TYtensor{\epsilon} \quad\text{for any $d$} \\
65	\TYvec{d} & = & \TYtensor{1}{d}	\TYvec{d} & = & \TYtensor{d}
66	\end{eqnarray*}%	\end{eqnarray*}%
67
68	\section{Operations}	\section{Operations}
#	Line 91	Line 91
91
92	\noindent{}\point Addition:	\noindent{}\point Addition:
93	\begin{displaymath}	\begin{displaymath}
94	\BinopTy{{\odot}}{\TYtensor{o}{d}}{\TYtensor{o}{d}}{\TYtensor{o}{d}}	\BinopTy{{\odot}}{\TYtensor{\Seq{d}}}{\TYtensor{\Seq{d}}}{\TYtensor{\Seq{d}}}
95	\qquad\text{for $\odot\in\SET{{+},{-}}$}	\qquad\text{for $\odot\in\SET{{+},{-}}$}
96	\end{displaymath}%	\end{displaymath}%
97
98	\noindent{}\point Negation:	\noindent{}\point Negation:
99	\begin{displaymath}	\begin{displaymath}
100	\UnopTy{-}{\TYtensor{o}{d}}{\TYtensor{o}{d}}	\UnopTy{-}{\TYtensor{\Seq{d}}}{\TYtensor{\Seq{d}}}
101	\end{displaymath}%	\end{displaymath}%
102
103	\noindent{}\point Scalar division:	\noindent{}\point Scalar division:
104	\begin{displaymath}	\begin{displaymath}
105	\BinopTy{{/}}{\TYtensor{o}{d}}{\TYreal}{\TYtensor{o}{d}}	\BinopTy{{/}}{\TYtensor{\Seq{d}}}{\TYreal}{\TYtensor{\Seq{d}}}
106	\end{displaymath}%	\end{displaymath}%
107
108	\noindent{}\point Scalar multiplication (scalar times order-N):	\noindent{}\point Scalar multiplication (scalar times order-N):
109	\begin{displaymath}	\begin{displaymath}
110	\begin{array}{c}	\begin{array}{c}
111	\BinopTy{{?}}{\TYreal}{\TYtensor{o}{d}}{\TYtensor{o}{d}} \\	\BinopTy{{?}}{\TYreal}{\TYtensor{\Seq{d}}}{\TYtensor{\Seq{d}}} \\
112	\BinopTy{{?}}{\TYtensor{o}{d}}{\TYreal}{\TYtensor{o}{d}}	\BinopTy{{?}}{\TYtensor{\Seq{d}}}{\TYreal}{\TYtensor{\Seq{d}}}
113	\end{array}%	\end{array}%
114	\end{displaymath}%	\end{displaymath}%
115	Possible direct notation syntax (TBD): ~~ {\tt *} ~~ `` '' (space)	Possible direct notation syntax (TBD): ~~ {\tt *} ~~ `` '' (space)
#	Line 119	Line 119
119	product of 2nd-order tensors)	product of 2nd-order tensors)
120	\begin{displaymath}	\begin{displaymath}
121	\begin{array}{c}	\begin{array}{c}
122	\BinopTy{{?}}{\TYtensor{o}{d}}{\TYtensor{o}{d}}{\TYreal}	\BinopTy{{?}}{\TYtensor{\Seq{d}}}{\TYtensor{\Seq{d}}}{\TYreal}
123	\end{array}%	\end{array}%
124	\end{displaymath}%	\end{displaymath}%
125	Defined by $c = A_{\vec{i}}B_{\vec{i}}$	Defined by $c = A_{\vec{i}}B_{\vec{i}}$
#	Line 132	Line 132
132	\noindent{}\point Tensor product (aka outer product; order output is sum of orders)	\noindent{}\point Tensor product (aka outer product; order output is sum of orders)
133	\begin{displaymath}	\begin{displaymath}
134	\begin{array}{c}	\begin{array}{c}
135	\BinopTy{{?}}{\TYtensor{o}{d}}{\TYtensor{p}{d}}{\TYtensor{o+p}{d}}	\BinopTy{{?}}{\TYtensor{\Seq{d}}}{\TYtensor{\Seq{d'}}}{\TYtensor{\Seq{d}\Seq{d'}}}
136	\end{array}%	\end{array}%
137	\end{displaymath}%	\end{displaymath}%
138	Defined by $C_{\vec{i}\vec{j}} = A_{\vec{i}}B_{\vec{j}}$ \\	Defined by $C_{\vec{i}\vec{j}} = A_{\vec{i}}B_{\vec{j}}$ \\
#	Line 141	Line 141
141	\noindent{}\point Matrix Multiply	\noindent{}\point Matrix Multiply
142	\begin{displaymath}	\begin{displaymath}
143	\begin{array}{c}	\begin{array}{c}
144	\BinopTy{{?}}{\TYtensor{o}{d}}{\TYtensor{2}{d}}{\TYtensor{o}{d}} \\	\BinopTy{{?}}{\TYtensor{\Seq{d}}}{\TYtensor{d_1 d_2}}{\TYtensor{\Seq{d}}} \\
145	\BinopTy{{?}}{\TYtensor{2}{d}}{\TYtensor{o}{d}}{\TYtensor{o}{d}}	\BinopTy{{?}}{\TYtensor{d_1 d_2}}{\TYtensor{\Seq{d}}}{\TYtensor{\Seq{d}}}
146	\end{array}%	\end{array}%
147	\end{displaymath}%	\end{displaymath}%
148	Defined by $C_{i\vec{k}} = A_{ij}B_{j\vec{k}}$ and $C_{\vec{i}k} = B_{\vec{i}j}A_{jk}$	Defined by $C_{i\vec{k}} = A_{ij}B_{j\vec{k}}$ and $C_{\vec{i}k} = B_{\vec{i}j}A_{jk}$
#	Line 153	Line 153
153	with vector	with vector
154	\begin{displaymath}	\begin{displaymath}
155	\begin{array}{c}	\begin{array}{c}
156	\BinopTy{{?}}{\TYtensor{o}{d}}{\TYvec{d}}{\TYtensor{o-1}{d}} \\	\BinopTy{{?}}{\TYtensor{d'\Seq{d}}}{\TYvec{d}}{\TYtensor{\Seq{d}}} \\
157	\BinopTy{{?}}{\TYvec{d}}{\TYtensor{o}{d}}{\TYtensor{o-1}{d}}	\BinopTy{{?}}{\TYvec{d}}{\TYtensor{d'\Seq{d}}}{\TYtensor{\Seq{d}}}
158	\end{array}%	\end{array}%
159	\end{displaymath}%	\end{displaymath}%
160	Defined by $C_{\vec{i}} = A_{\vec{i}j}v_j$ and $C_{\vec{j}} = v_iA_{j\vec{j}}$ \\	Defined by $C_{\vec{i}} = A_{\vec{i}j}v_j$ and $C_{\vec{j}} = v_iA_{j\vec{j}}$ \\
#	Line 165	Line 165
165	decrease the tensor order.	decrease the tensor order.
166	\begin{displaymath}	\begin{displaymath}
167	\begin{array}{c}	\begin{array}{c}
168	\BinopTy{{?}}{\TYtensor{o}{d}}{\TYtensor{p}{d}}{\TYtensor{q}{d}}	\BinopTy{{?}}{\TYtensor{\Seq{d}}}{\TYtensor{p}{d}}{\TYtensor{q}{d}}
169	\end{array}%	\end{array}%
170	\end{displaymath}%	\end{displaymath}%
171	Defined by the conventions of Einstein summation notation. \\	Defined by the conventions of Einstein summation notation. \\
#	Line 175	Line 175
175
176	\noindent{}\point Creation from an image:	\noindent{}\point Creation from an image:
177	\begin{displaymath}	\begin{displaymath}
178	\BinopTy{\OPconvolve}{\TYkern{k}}{\TYimage{d}{\TYrawten{o}{d}{\rho}}}{\TYfield{k}{d}{\TYtensor{o}{d}}}	\BinopTy{\OPconvolve}{\TYkern{k}}{\TYimage{d'}{\TYrawten{\Seq{d}}{\rho}}}{\TYfield{k}{d'}{\TYtensor{\Seq{d}}}}
179	\end{displaymath}%	\end{displaymath}%
180
181	\noindent{}\point Scalar multiplication:	\noindent{}\point Scalar multiplication:
#	Line 208	Line 208
208
209	\noindent{}\point Differentiation:	\noindent{}\point Differentiation:
210	\begin{displaymath}	\begin{displaymath}
211	\UnopTy{\OPdiff}{\TYfield{k}{d}{\TYtensor{o}{d}}}{\TYfield{k-1}{d}{\TYtensor{o+1}{d}}}	\UnopTy{\OPdiff}{\TYfield{k}{d'}{\TYtensor{\Seq{d}}}}{\TYfield{k-1}{d'}{\TYtensor{d'\Seq{d}}}}
212	\qquad\text{for $k > 0$}	\qquad\text{for $k > 0$}
213	\end{displaymath}%	\end{displaymath}%
214

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