User:Jon Awbrey/WORDPRESS - Revision history

Jon Awbrey: /* Fourier Transforms of Boolean Functions */

2013-06-05T03:28:19Z

Fourier Transforms of Boolean Functions

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Jon Awbrey: /* Table A6. Df Expanded Over Ordinary Features {x, y} */

2013-06-05T03:02:20Z

Table A6. Df Expanded Over Ordinary Features {x, y}

Jon Awbrey: /* Table A5. Ef Expanded Over Ordinary Features {x, y} */

2013-06-05T03:00:21Z

Table A5. Ef Expanded Over Ordinary Features {x, y}

Jon Awbrey: /* Table A5. Ef Expanded Over Ordinary Features {x, y} */

2013-06-05T02:56:23Z

Table A5. Ef Expanded Over Ordinary Features {x, y}

Jon Awbrey: /* Table A5. Ef Expanded Over Ordinary Features {x, y} */

2013-06-05T02:54:24Z

Table A5. Ef Expanded Over Ordinary Features {x, y}

Jon Awbrey: /* Table A4. Df Expanded Over Differential Features {dx, dy} */

2013-06-05T02:53:28Z

Table A4. Df Expanded Over Differential Features {dx, dy}

Jon Awbrey: /* Table A3. Ef Expanded Over Differential Features {dx, dy} */

2013-06-05T02:52:34Z

Table A3. Ef Expanded Over Differential Features {dx, dy}

Jon Awbrey: /* Table A2. Propositional Forms on Two Variables */

2013-06-05T02:51:19Z

Table A2. Propositional Forms on Two Variables

Jon Awbrey: /* Table A1. Propositional Forms on Two Variables */

2013-06-05T02:50:06Z

Table A1. Propositional Forms on Two Variables

Jon Awbrey: /* Fourier Transforms of Boolean Functions */

2013-06-04T03:54:11Z

Fourier Transforms of Boolean Functions

@@ Line 835: / Line 835: @@
 \\
 \hline
-\end{tabular}&amp;fg=000000$
+\end{tabular}
 </pre>

@@ Line 568: / Line 568: @@
 $latex
 \begin{tabular}{|c|c||c|c|c|c|}
-\multicolumn{6}{c}{Table A6. \(\mathrm{D}f\) Expanded Over Ordinary Features \(\{x, y\}\)} \\
+\multicolumn{6}{c}{Table A5. \(\mathrm{E}f\) Expanded Over Ordinary Features \(\{x, y\}\)} \\
 \hline
 &amp;
 \(~~~~~~~~ f ~~~~~~~~\)&amp;
-\(~~\mathrm{D}f|_{ x\;y }~~~\)&amp;
+\(~~\mathrm{E}f|_{ x\;y }~~~\)&amp;
-\(~~\mathrm{D}f|_{ x~(y)}\,~~\)&amp;
+\(~~\mathrm{E}f|_{ x~(y)}\,~~\)&amp;
-\(~~\mathrm{D}f|_{(x)~y }\,~~\)&amp;
+\(~~\mathrm{E}f|_{(x)~y }\,~~\)&amp;
-\(~~\mathrm{D}f|_{(x)(y)}\,~\)
+\(~~\mathrm{E}f|_{(x)(y)}\;~\)
 \\
 \hline\hline
@@ Line 588: / Line 588: @@
 \(f_{1}\)&amp;
 \((x)(y)\)&amp;
-~~d\(x\)~~d\(y~~\)&amp;
+~d\(x\)~~d\(y~\)&amp;
-\;d\(x\)~(d\(y\))~&amp;
+~d\(x\)~(d\(y\))&amp;
-~(d\(x\))~d\(y~~\)&amp;
+(d\(x\))~d\(y~\)&amp;
-((d\(x\))(d\(y\)))
+(d\(x\))(d\(y\))
 \\
 \(f_{2}\)&amp;
 \((x)~y~\)&amp;
-\;d\(x\)~(d\(y\))~&amp;
+~d\(x\)~(d\(y\))&amp;
-~~d\(x\)~~d\(y~~\)&amp;
+~d\(x\)~~d\(y~\)&amp;
-((d\(x\))(d\(y\)))&amp;
+(d\(x\))(d\(y\))&amp;
-~(d\(x\))~d\(y~~\)
+(d\(x\))~d\(y~\)
 \\
 \(f_{4}\)&amp;
 \(~x~(y)\)&amp;
-~(d\(x\))~d\(y~~\)&amp;
+(d\(x\))~d\(y~\)&amp;
-((d\(x\))(d\(y\)))&amp;
+(d\(x\))(d\(y\))&amp;
-~~d\(x\)~~d\(y~~\)&amp;
+~d\(x\)~~d\(y~\)&amp;
-~~d\(x\)~(d\(y\))~
+~d\(x\)~(d\(y\))
 \\
 \(f_{8}\)&amp;
 \(~x~~y~\)&amp;
-((d\(x\))(d\(y\)))&amp;
+(d\(x\))(d\(y\))&amp;
-~(d\(x\))~d\(y~~\)&amp;
+(d\(x\))~d\(y~\)&amp;
-\;d\(x\)~(d\(y\))~&amp;
+~d\(x\)~(d\(y\))&amp;
-~~d\(x\)~~d\(y~~\)
+~d\(x\)~~d\(y~\)
 \\
 \hline
 \(f_{3}\)&amp;
 \((x)\)&amp;
-d\(x\)&amp;
+ d\(x\) &amp;
-d\(x\)&amp;
+ d\(x\) &amp;
-d\(x\)&amp;
+(d\(x\))&amp;
-d\(x\)
+(d\(x\))
 \\
 \(f_{12}\)&amp;
 \( x \)&amp;
-d\(x\)&amp;
+(d\(x\))&amp;
-d\(x\)&amp;
+(d\(x\))&amp;
-d\(x\)&amp;
+ d\(x\) &amp;
-d\(x\)
+ d\(x\)
 \\
 \hline
 \(f_{6}\)&amp;
 \( (x,y) \)&amp;
-(d\(x\), d\(y\))&amp;
+ (d\(x\), d\(y\)) &amp;
-(d\(x\), d\(y\))&amp;
+((d\(x\), d\(y\)))&amp;
-(d\(x\), d\(y\))&amp;
+((d\(x\), d\(y\)))&amp;
-(d\(x\), d\(y\))
+ (d\(x\), d\(y\))
 \\
 \(f_{9}\)&amp;
 \(((x,y))\)&amp;
-(d\(x\), d\(y\))&amp;
+((d\(x\), d\(y\)))&amp;
-(d\(x\), d\(y\))&amp;
+ (d\(x\), d\(y\)) &amp;
-(d\(x\), d\(y\))&amp;
+ (d\(x\), d\(y\)) &amp;
-(d\(x\), d\(y\))
+((d\(x\), d\(y\)))
 \\
 \hline
 \(f_{5}\)&amp;
 \((y)\)&amp;
-d\(y\)&amp;
+ d\(y\) &amp;
-d\(y\)&amp;
+(d\(y\))&amp;
-d\(y\)&amp;
+ d\(y\) &amp;
-d\(y\)
+(d\(y\))
 \\
 \(f_{10}\)&amp;
 \( y \)&amp;
-d\(y\)&amp;
+(d\(y\))&amp;
-d\(y\)&amp;
+ d\(y\) &amp;
-d\(y\)&amp;
+(d\(y\))&amp;
-d\(y\)
+ d\(y\)
 \\
 \hline
@@ Line 663: / Line 663: @@
 \((~x~~y~)\)&amp;
 ((d\(x\))(d\(y\)))&amp;
-~(d\(x\))~d\(y~~\)&amp;
+((d\(x\))~d\(y\)~)&amp;
-\;d\(x\)~(d\(y\))~&amp;
+(~d\(x\)~(d\(y\)))&amp;
-~~d\(x\)~~d\(y~~\)
+(~d\(x\)~~d\(y\)~)
 \\
 \(f_{11}\)&amp;
 \((~x~(y))\)&amp;
-~(d\(x\))~d\(y~~\)&amp;
+((d\(x\))~d\(y\)~)&amp;
 ((d\(x\))(d\(y\)))&amp;
-~~d\(x\)~~d\(y~~\)&amp;
+(~d\(x\)~~d\(y\)~)&amp;
-~~d\(x\)~(d\(y\))~
+(~d\(x\)~(d\(y\)))
 \\
 \(f_{13}\)&amp;
 \(((x)~y~)\)&amp;
-\;d\(x\)~(d\(y\))~&amp;
+(~d\(x\)~(d\(y\)))&amp;
-~~d\(x\)~~d\(y~~\)&amp;
+(~d\(x\)~~d\(y\)~)&amp;
 ((d\(x\))(d\(y\)))&amp;
-~(d\(x\))~d\(y~~\)
+((d\(x\))~d\(y\)~)
 \\
 \(f_{14}\)&amp;
 \(((x)(y))\)&amp;
-~~d\(x\)~~d\(y~~\)&amp;
+(~d\(x\)~~d\(y\)~)&amp;
-\;d\(x\)~(d\(y\))~&amp;
+(~d\(x\)~(d\(y\)))&amp;
-~(d\(x\))~d\(y~~\)&amp;
+((d\(x\))~d\(y\)~)&amp;
 ((d\(x\))(d\(y\)))
 \\
@@ Line 691: / Line 691: @@
 \(f_{15}\)&amp;
 &amp;
 &amp;
 &amp;
 &amp;
-−
 \\
 \hline

@@ Line 568: / Line 568: @@
 $latex
 \begin{tabular}{|c|c||c|c|c|c|}
-\multicolumn{6}{c}{Table A5. \(\mathrm{E}f\) Expanded Over Ordinary Features \(\{x, y\}\)} \\
+\multicolumn{6}{c}{Table A6. \(\mathrm{D}f\) Expanded Over Ordinary Features \(\{x, y\}\)} \\
 \hline
 &amp;
 \(~~~~~~~~ f ~~~~~~~~\)&amp;
-\(~~\mathrm{E}f|_{ x\;y }~~~\)&amp;
+\(~~\mathrm{D}f|_{ x\;y }~~~\)&amp;
-\(~~\mathrm{E}f|_{ x~(y)}\,~~\)&amp;
+\(~~\mathrm{D}f|_{ x~(y)}\,~~\)&amp;
-\(~~\mathrm{E}f|_{(x)~y }\,~~\)&amp;
+\(~~\mathrm{D}f|_{(x)~y }\,~~\)&amp;
-\(~~\mathrm{E}f|_{(x)(y)}\;~\)
+\(~~\mathrm{D}f|_{(x)(y)}\,~\)
 \\
 \hline\hline
@@ Line 588: / Line 588: @@
 \(f_{1}\)&amp;
 \((x)(y)\)&amp;
-~d\(x\)~~d\(y~\)&amp;
+~~d\(x\)~~d\(y~~\)&amp;
-~d\(x\)~(d\(y\))&amp;
+\;d\(x\)~(d\(y\))~&amp;
-(d\(x\))~d\(y~\)&amp;
+~(d\(x\))~d\(y~~\)&amp;
-(d\(x\))(d\(y\))
+((d\(x\))(d\(y\)))
 \\
 \(f_{2}\)&amp;
 \((x)~y~\)&amp;
-~d\(x\)~(d\(y\))&amp;
+\;d\(x\)~(d\(y\))~&amp;
-~d\(x\)~~d\(y~\)&amp;
+~~d\(x\)~~d\(y~~\)&amp;
-(d\(x\))(d\(y\))&amp;
+((d\(x\))(d\(y\)))&amp;
-(d\(x\))~d\(y~\)
+~(d\(x\))~d\(y~~\)
 \\
 \(f_{4}\)&amp;
 \(~x~(y)\)&amp;
-(d\(x\))~d\(y~\)&amp;
+~(d\(x\))~d\(y~~\)&amp;
-(d\(x\))(d\(y\))&amp;
+((d\(x\))(d\(y\)))&amp;
-~d\(x\)~~d\(y~\)&amp;
+~~d\(x\)~~d\(y~~\)&amp;
-~d\(x\)~(d\(y\))
+~~d\(x\)~(d\(y\))~
 \\
 \(f_{8}\)&amp;
 \(~x~~y~\)&amp;
-(d\(x\))(d\(y\))&amp;
+((d\(x\))(d\(y\)))&amp;
-(d\(x\))~d\(y~\)&amp;
+~(d\(x\))~d\(y~~\)&amp;
-~d\(x\)~(d\(y\))&amp;
+\;d\(x\)~(d\(y\))~&amp;
-~d\(x\)~~d\(y~\)
+~~d\(x\)~~d\(y~~\)
 \\
 \hline
 \(f_{3}\)&amp;
 \((x)\)&amp;
- d\(x\) &amp;
+d\(x\)&amp;
- d\(x\) &amp;
+d\(x\)&amp;
-(d\(x\))&amp;
+d\(x\)&amp;
-(d\(x\))
+d\(x\)
 \\
 \(f_{12}\)&amp;
 \( x \)&amp;
-(d\(x\))&amp;
+d\(x\)&amp;
-(d\(x\))&amp;
+d\(x\)&amp;
- d\(x\) &amp;
+d\(x\)&amp;
- d\(x\)
+d\(x\)
 \\
 \hline
 \(f_{6}\)&amp;
 \( (x,y) \)&amp;
- (d\(x\), d\(y\)) &amp;
+(d\(x\), d\(y\))&amp;
-((d\(x\), d\(y\)))&amp;
+(d\(x\), d\(y\))&amp;
-((d\(x\), d\(y\)))&amp;
+(d\(x\), d\(y\))&amp;
- (d\(x\), d\(y\))
+(d\(x\), d\(y\))
 \\
 \(f_{9}\)&amp;
 \(((x,y))\)&amp;
-((d\(x\), d\(y\)))&amp;
+(d\(x\), d\(y\))&amp;
- (d\(x\), d\(y\)) &amp;
+(d\(x\), d\(y\))&amp;
- (d\(x\), d\(y\)) &amp;
+(d\(x\), d\(y\))&amp;
-((d\(x\), d\(y\)))
+(d\(x\), d\(y\))
 \\
 \hline
 \(f_{5}\)&amp;
 \((y)\)&amp;
- d\(y\) &amp;
+d\(y\)&amp;
-(d\(y\))&amp;
+d\(y\)&amp;
- d\(y\) &amp;
+d\(y\)&amp;
-(d\(y\))
+d\(y\)
 \\
 \(f_{10}\)&amp;
 \( y \)&amp;
-(d\(y\))&amp;
+d\(y\)&amp;
- d\(y\) &amp;
+d\(y\)&amp;
-(d\(y\))&amp;
+d\(y\)&amp;
- d\(y\)
+d\(y\)
 \\
 \hline
@@ Line 663: / Line 663: @@
 \((~x~~y~)\)&amp;
 ((d\(x\))(d\(y\)))&amp;
-((d\(x\))~d\(y\)~)&amp;
+~(d\(x\))~d\(y~~\)&amp;
-(~d\(x\)~(d\(y\)))&amp;
+\;d\(x\)~(d\(y\))~&amp;
-(~d\(x\)~~d\(y\)~)
+~~d\(x\)~~d\(y~~\)
 \\
 \(f_{11}\)&amp;
 \((~x~(y))\)&amp;
-((d\(x\))~d\(y\)~)&amp;
+~(d\(x\))~d\(y~~\)&amp;
 ((d\(x\))(d\(y\)))&amp;
-(~d\(x\)~~d\(y\)~)&amp;
+~~d\(x\)~~d\(y~~\)&amp;
-(~d\(x\)~(d\(y\)))
+~~d\(x\)~(d\(y\))~
 \\
 \(f_{13}\)&amp;
 \(((x)~y~)\)&amp;
-(~d\(x\)~(d\(y\)))&amp;
+\;d\(x\)~(d\(y\))~&amp;
-(~d\(x\)~~d\(y\)~)&amp;
+~~d\(x\)~~d\(y~~\)&amp;
 ((d\(x\))(d\(y\)))&amp;
-((d\(x\))~d\(y\)~)
+~(d\(x\))~d\(y~~\)
 \\
 \(f_{14}\)&amp;
 \(((x)(y))\)&amp;
-(~d\(x\)~~d\(y\)~)&amp;
+~~d\(x\)~~d\(y~~\)&amp;
-(~d\(x\)~(d\(y\)))&amp;
+\;d\(x\)~(d\(y\))~&amp;
-((d\(x\))~d\(y\)~)&amp;
+~(d\(x\))~d\(y~~\)&amp;
 ((d\(x\))(d\(y\)))
 \\
@@ Line 691: / Line 691: @@
 \(f_{15}\)&amp;
 &amp;
 &amp;
 &amp;
 &amp;
-−
 \\
 \hline

@@ Line 697: / Line 697: @@
 \\
 \hline
-\end{tabular}&amp;fg=000000$
+\end{tabular}
 </pre>

@@ Line 559: / Line 559: @@
 \\
 \hline
-\end{tabular}&amp;fg=000000$
+\end{tabular}
 </pre>

@@ Line 838: / Line 838: @@
 <blockquote>
-<p>The problem is concretely about Boolean functions $latex {f}$ of $latex {k}$ variables, and seems not to involve prime numbers at all.  For any subset $latex {S}$ of the coordinates, the corresponding Fourier coefficient is given by:</p>
+<p>The problem is concretely about Boolean functions <math>f\!</math> of <math>k\!</math> variables, and seems not to involve prime numbers at all.  For any subset <math>S\!</math> of the coordinates, the corresponding Fourier coefficient is given by:</p>
-<p align="center">
+<p align="center"><math>\hat{f}(S) = \frac{1}{2^k} \sum_{x \in \mathbb{Z}_2^k} f(x)\chi_S(x)\!</math></p>
-<p>where <math>\chi_S(x)\!</math> is <math>-1\!</math> if <math>\sum_{i \in S} x_i\!</math> is odd, and <math>+1\!</math> otherwise.</p>
+<p>where <math>\chi_S(x)\!</math> is <math>-1\!</math> if <math>\textstyle \sum_{i \in S} x_i\!</math> is odd, and <math>+1\!</math> otherwise.</p>
 </blockquote>