Changes

{| align="center" cellspacing="6" width="90%"

{| align="center" cellspacing="6" width="90%"

| height="60" | <math>(\mathfrak{L}^\mathfrak{W})_u ~=~ \prod_{x \in X} \mathfrak{L}_{ux}^{\mathfrak{W}_x}</math>

| height="60" | <math>(\mathfrak{L}^\mathfrak{W})_x ~=~ \prod_{p \in X} \mathfrak{L}_{xp}^{\mathfrak{W}_p}</math>

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With the above picture in mind, we can visualize the computation of <math>(\mathfrak{L}^\mathfrak{W})_u = \textstyle\prod_{x \in X} \mathfrak{L}_{ux}^{\mathfrak{W}_x}</math> as follows:

With the above picture in mind, we can visualize the computation of <math>(\mathfrak{L}^\mathfrak{W})_x = \textstyle\prod_{p \in X} \mathfrak{L}_{xp}^{\mathfrak{W}_p}</math> as follows:

{| align="center" cellspacing="6" width="90%"

{| align="center" cellspacing="6" width="90%"

| valign="top" | 1.  

| valign="top" | 1.  

| Pick a specific <math>u\!</math> in the bottom row of the Figure.

| Pick a specific <math>x\!</math> in the bottom row of the Figure.

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| valign="top" | 2.

| valign="top" | 2.

| Pan across the elements <math>x\!</math> in the middle row of the Figure.

| Pan across the elements <math>p\!</math> in the middle row of the Figure.

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| valign="top" | 3.

| valign="top" | 3.

| If <math>u\!</math> links to <math>x\!</math> then <math>\mathfrak{L}_{ux} = 1,</math> otherwise <math>\mathfrak{L}_{ux} = 0.</math>

| If <math>x\!</math> links to <math>p\!</math> then <math>\mathfrak{L}_{xp} = 1,</math> otherwise <math>\mathfrak{L}_{xp} = 0.</math>

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| valign="top" | 4.

| valign="top" | 4.

| If <math>x\!</math> in the middle row links to <math>x\!</math> in the top row then <math>\mathfrak{W}_x = 1,</math> otherwise <math>\mathfrak{W}_x = 0.</math>

| If <math>p\!</math> in the middle row links to <math>p\!</math> in the top row then <math>\mathfrak{W}_p = 1,</math> otherwise <math>\mathfrak{W}_p = 0.</math>

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| valign="top" | 5.

| valign="top" | 5.

| Compute the value <math>\mathfrak{L}_{ux}^{\mathfrak{W}_x} = (\mathfrak{L}_{ux} >\!\!\!-~ \mathfrak{W}_x)</math> for each <math>x\!</math> in the middle row.

| Compute the value <math>\mathfrak{L}_{xp}^{\mathfrak{W}_p} = (\mathfrak{L}_{xp} >\!\!\!-~ \mathfrak{W}_p)</math> for each <math>p\!</math> in the middle row.

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| valign="top" | 6.

| valign="top" | 6.

| If any of the values <math>\mathfrak{L}_{ux}^{\mathfrak{W}_x}</math> is <math>0\!</math> then the product <math>\textstyle\prod_{x \in X} \mathfrak{L}_{ux}^{\mathfrak{W}_x}</math> is <math>0,\!</math> otherwise it is <math>1.\!</math>

| If any of the values <math>\mathfrak{L}_{xp}^{\mathfrak{W}_p}</math> is <math>0\!</math> then the product <math>\textstyle\prod_{p \in X} \mathfrak{L}_{xp}^{\mathfrak{W}_p}</math> is <math>0,\!</math> otherwise it is <math>1.\!</math>

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As a general observation, we know that the value of <math>(\mathfrak{L}^\mathfrak{W})_u</math> goes to <math>0\!</math> just as soon as we find an <math>x \in X</math> such that <math>\mathfrak{L}_{ux} = 0</math> and <math>\mathfrak{W}_x = 1,</math> in other words, such that <math>(u, x) \notin L</math> but <math>u \in W.</math>  If there is no such <math>x,\!</math> then <math>(\mathfrak{L}^\mathfrak{W})_u = 1.</math>

As a general observation, we know that the value of <math>(\mathfrak{L}^\mathfrak{W})_x</math> goes to <math>0\!</math> just as soon as we find an <math>p \in X</math> such that <math>\mathfrak{L}_{xp} = 0</math> and <math>\mathfrak{W}_p = 1,</math> in other words, such that <math>(x, p) \notin L</math> but <math>p \in W.</math>  If there is no such <math>p,\!</math> then <math>(\mathfrak{L}^\mathfrak{W})_x = 1.</math>

Running through the program for each <math>u \in X,</math> the only case that produces a non-zero result is <math>(\mathfrak{L}^\mathfrak{W})_e = 1.</math>  That portion of the work can be sketched as follows:

Running through the program for each <math>x \in X,</math> the only case that produces a non-zero result is <math>(\mathfrak{L}^\mathfrak{W})_e = 1.</math>  That portion of the work can be sketched as follows:

{| align="center" cellspacing="6" width="90%"

{| align="center" cellspacing="6" width="90%"

| height="60" | <math>(\mathfrak{L}^\mathfrak{W})_e ~=~ \prod_{x \in X} \mathfrak{L}_{ex}^{\mathfrak{W}_x} ~=~ 0^0 \cdot 0^0 \cdot 0^0 \cdot 1^1 \cdot 1^0 \cdot 1^1 \cdot 0^0 \cdot 0^0 \cdot 0^0 ~=~ 1</math>

| height="60" | <math>(\mathfrak{L}^\mathfrak{W})_e ~=~ \prod_{p \in X} \mathfrak{L}_{ep}^{\mathfrak{W}_p} ~=~ 0^0 \cdot 0^0 \cdot 0^0 \cdot 1^1 \cdot 1^0 \cdot 1^1 \cdot 0^0 \cdot 0^0 \cdot 0^0 ~=~ 1</math>

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