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tutorial/chap5.html

@@ -324,7 +324,7 @@ <h5>5.5-1 <span class="Heading">Non-trivial cup product</span></h5>
 
 <h5>5.5-2 <span class="Heading">Explicit homology generators</span></h5>
 
-<p>It could be desirable to obtain explicit representatives of the persistent homology generators that "persist" through a significant sequence of filtration terms. There are two such generators in degree <span class="SimpleMath">1</span> and one such generator in degree <span class="SimpleMath">2</span>. The explicit representatives in degree <span class="SimpleMath">n</span> could consist of an inclusion of pure cubical complexes <span class="SimpleMath">Y_n ⊂ X_10</span> for which the incuced homology homomorphism <span class="SimpleMath">H_n(Y_n, Z) → H_n(X_10, Z)</span> is an isomorphism, and for which <span class="SimpleMath">Y_n</span> is minimal in the sense that its homotopy type changes if any one or more of its top dimensional cells are removed. Ideally the space <span class="SimpleMath">Y_n</span> should be as "close to the original dataset" <span class="SimpleMath">X_0</span>. The following commands first construct an explicit degree <span class="SimpleMath">2</span> homology generator representative <span class="SimpleMath">Y_2⊂ X_10</span> where <span class="SimpleMath">Y_2</span> is homotopy equivalent to <span class="SimpleMath">X_10</span>. They then construct an explicit degree <span class="SimpleMath">1</span> homology generators representative <span class="SimpleMath">Y_1⊂ X_10</span> where <span class="SimpleMath">Y_1</span> is homotopy equivalent to a wedge of two circles. The final command displays the homology generators representative <span class="SimpleMath">Y_1</span>.</p>
+<p>It could be desirable to obtain explicit representatives of the persistent homology generators that "persist" through a significant sequence of filtration terms. There are two such generators in degree <span class="SimpleMath">1</span> and one such generator in degree <span class="SimpleMath">2</span>. The explicit representatives in degree <span class="SimpleMath">n</span> could consist of an inclusion of pure cubical complexes <span class="SimpleMath">Y_n ⊂ X_10</span> for which the incuced homology homomorphism <span class="SimpleMath">H_n(Y_n, Z) → H_n(X_10, Z)</span> is an isomorphism, and for which <span class="SimpleMath">Y_n</span> is minimal in the sense that its homotopy type changes if any one or more of its top dimensional cells are removed. Ideally the space <span class="SimpleMath">Y_n</span> should be "close to the original dataset" <span class="SimpleMath">X_0</span>. The following commands first construct an explicit degree <span class="SimpleMath">2</span> homology generator representative <span class="SimpleMath">Y_2⊂ X_10</span> where <span class="SimpleMath">Y_2</span> is homotopy equivalent to <span class="SimpleMath">X_10</span>. They then construct an explicit degree <span class="SimpleMath">1</span> homology generators representative <span class="SimpleMath">Y_1⊂ X_10</span> where <span class="SimpleMath">Y_1</span> is homotopy equivalent to a wedge of two circles. The final command displays the homology generators representative <span class="SimpleMath">Y_1</span>.</p>
 
 
 <div class="example"><pre>

tutorial/chap5.txt

@@ -315,7 +315,7 @@
   incuced  homology homomorphism H_n(Y_n, Z) → H_n(X_10, Z) is an isomorphism,
   and  for which Y_n is minimal in the sense that its homotopy type changes if
   any  one or more of its top dimensional cells are removed. Ideally the space
-  Y_n should be as "close to the original dataset" X_0. The following commands
+  Y_n  should  be  "close to the original dataset" X_0. The following commands
   first  construct an explicit degree 2 homology generator representative Y_2⊂
   X_10  where  Y_2  is  homotopy  equivalent  to  X_10. They then construct an
   explicit  degree 1 homology generators representative Y_1⊂ X_10 where Y_1 is

tutorial/chap5_mj.html

@@ -327,7 +327,7 @@ <h5>5.5-1 <span class="Heading">Non-trivial cup product</span></h5>
 
 <h5>5.5-2 <span class="Heading">Explicit homology generators</span></h5>
 
-<p>It could be desirable to obtain explicit representatives of the persistent homology generators that "persist" through a significant sequence of filtration terms. There are two such generators in degree <span class="SimpleMath">\(1\)</span> and one such generator in degree <span class="SimpleMath">\(2\)</span>. The explicit representatives in degree <span class="SimpleMath">\(n\)</span> could consist of an inclusion of pure cubical complexes <span class="SimpleMath">\(Y_n \subset X_{10}\)</span> for which the incuced homology homomorphism <span class="SimpleMath">\(H_n(Y_n,\mathbb Z) \rightarrow H_n(X_{10},\mathbb Z)\)</span> is an isomorphism, and for which <span class="SimpleMath">\(Y_n\)</span> is minimal in the sense that its homotopy type changes if any one or more of its top dimensional cells are removed. Ideally the space <span class="SimpleMath">\(Y_n\)</span> should be as "close to the original dataset" <span class="SimpleMath">\(X_0\)</span>. The following commands first construct an explicit degree <span class="SimpleMath">\(2\)</span> homology generator representative <span class="SimpleMath">\(Y_2\subset X_{10}\)</span> where <span class="SimpleMath">\(Y_2\)</span> is homotopy equivalent to <span class="SimpleMath">\(X_{10}\)</span>. They then construct an explicit degree <span class="SimpleMath">\(1\)</span> homology generators representative <span class="SimpleMath">\(Y_1\subset X_{10}\)</span> where <span class="SimpleMath">\(Y_1\)</span> is homotopy equivalent to a wedge of two circles. The final command displays the homology generators representative <span class="SimpleMath">\(Y_1\)</span>.</p>
+<p>It could be desirable to obtain explicit representatives of the persistent homology generators that "persist" through a significant sequence of filtration terms. There are two such generators in degree <span class="SimpleMath">\(1\)</span> and one such generator in degree <span class="SimpleMath">\(2\)</span>. The explicit representatives in degree <span class="SimpleMath">\(n\)</span> could consist of an inclusion of pure cubical complexes <span class="SimpleMath">\(Y_n \subset X_{10}\)</span> for which the incuced homology homomorphism <span class="SimpleMath">\(H_n(Y_n,\mathbb Z) \rightarrow H_n(X_{10},\mathbb Z)\)</span> is an isomorphism, and for which <span class="SimpleMath">\(Y_n\)</span> is minimal in the sense that its homotopy type changes if any one or more of its top dimensional cells are removed. Ideally the space <span class="SimpleMath">\(Y_n\)</span> should be "close to the original dataset" <span class="SimpleMath">\(X_0\)</span>. The following commands first construct an explicit degree <span class="SimpleMath">\(2\)</span> homology generator representative <span class="SimpleMath">\(Y_2\subset X_{10}\)</span> where <span class="SimpleMath">\(Y_2\)</span> is homotopy equivalent to <span class="SimpleMath">\(X_{10}\)</span>. They then construct an explicit degree <span class="SimpleMath">\(1\)</span> homology generators representative <span class="SimpleMath">\(Y_1\subset X_{10}\)</span> where <span class="SimpleMath">\(Y_1\)</span> is homotopy equivalent to a wedge of two circles. The final command displays the homology generators representative <span class="SimpleMath">\(Y_1\)</span>.</p>
 
 
 <div class="example"><pre>