What is the size of these discrete function sets?












2












$begingroup$


I'm interested in the set $F_n$ of functions $f:{-1,1}^nto{-1,1}$ which can be represented as $xmapstotext{sign}(acdot x)$ for some $ainmathbf{R}^n$. What is the size of $F_n$? Is there some discrete encoding which is 'better' then to write down the complete lookup table?



My idea is to look at the number of connected components in $mathbf{R}^nsetminus(cup_x{x}^bot)$, since if $a,a'$ represent different functions $f,f'$, say $f(y)neq f'(y)$, then I can't go from $a$ to $a'$ without crossing the border $acdot y=0$. But I still don't know how many such components there are.



[EDIT] I just tried to plot the planes ${x}^bot$ in $mathbf{R}^3$ (w.l.o.g. $x_1=1$ since ${x}^bot={-x}^bot$), there seem to be $14$ partitions and they seem to correspond to the faces and edges of the $[-1,1]^3$-cube. But in even dimensions the picture needs to be different since the edges of the $[-1,1]^{2n}$-cube lie directly on the planes






linear-algebra discrete-mathematics







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$endgroup$

















    2












    $begingroup$


    I'm interested in the set $F_n$ of functions $f:{-1,1}^nto{-1,1}$ which can be represented as $xmapstotext{sign}(acdot x)$ for some $ainmathbf{R}^n$. What is the size of $F_n$? Is there some discrete encoding which is 'better' then to write down the complete lookup table?



    My idea is to look at the number of connected components in $mathbf{R}^nsetminus(cup_x{x}^bot)$, since if $a,a'$ represent different functions $f,f'$, say $f(y)neq f'(y)$, then I can't go from $a$ to $a'$ without crossing the border $acdot y=0$. But I still don't know how many such components there are.



    [EDIT] I just tried to plot the planes ${x}^bot$ in $mathbf{R}^3$ (w.l.o.g. $x_1=1$ since ${x}^bot={-x}^bot$), there seem to be $14$ partitions and they seem to correspond to the faces and edges of the $[-1,1]^3$-cube. But in even dimensions the picture needs to be different since the edges of the $[-1,1]^{2n}$-cube lie directly on the planes






    linear-algebra discrete-mathematics







    share|cite|improve this question











    $endgroup$















      2












      2








      2





      $begingroup$


      I'm interested in the set $F_n$ of functions $f:{-1,1}^nto{-1,1}$ which can be represented as $xmapstotext{sign}(acdot x)$ for some $ainmathbf{R}^n$. What is the size of $F_n$? Is there some discrete encoding which is 'better' then to write down the complete lookup table?



      My idea is to look at the number of connected components in $mathbf{R}^nsetminus(cup_x{x}^bot)$, since if $a,a'$ represent different functions $f,f'$, say $f(y)neq f'(y)$, then I can't go from $a$ to $a'$ without crossing the border $acdot y=0$. But I still don't know how many such components there are.



      [EDIT] I just tried to plot the planes ${x}^bot$ in $mathbf{R}^3$ (w.l.o.g. $x_1=1$ since ${x}^bot={-x}^bot$), there seem to be $14$ partitions and they seem to correspond to the faces and edges of the $[-1,1]^3$-cube. But in even dimensions the picture needs to be different since the edges of the $[-1,1]^{2n}$-cube lie directly on the planes






      linear-algebra discrete-mathematics







      share|cite|improve this question











      $endgroup$




      I'm interested in the set $F_n$ of functions $f:{-1,1}^nto{-1,1}$ which can be represented as $xmapstotext{sign}(acdot x)$ for some $ainmathbf{R}^n$. What is the size of $F_n$? Is there some discrete encoding which is 'better' then to write down the complete lookup table?



      My idea is to look at the number of connected components in $mathbf{R}^nsetminus(cup_x{x}^bot)$, since if $a,a'$ represent different functions $f,f'$, say $f(y)neq f'(y)$, then I can't go from $a$ to $a'$ without crossing the border $acdot y=0$. But I still don't know how many such components there are.



      [EDIT] I just tried to plot the planes ${x}^bot$ in $mathbf{R}^3$ (w.l.o.g. $x_1=1$ since ${x}^bot={-x}^bot$), there seem to be $14$ partitions and they seem to correspond to the faces and edges of the $[-1,1]^3$-cube. But in even dimensions the picture needs to be different since the edges of the $[-1,1]^{2n}$-cube lie directly on the planes






      linear-algebra discrete-mathematics




      linear-algebra discrete-mathematics






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      share|cite|improve this question













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      share|cite|improve this question








      edited Dec 3 '18 at 16:06







      fweth

















      asked Dec 2 '18 at 1:00









      fwethfweth

      1,148711




      1,148711






















          1 Answer
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          2












          $begingroup$

          EDIT: My previous rough idea about orthants was completely off of the mark. I need to go rethink everything.



          I have provided a link to a set of slides in the comments below, but I think I'm starting to get out of my depth. Someone who knows better than I should swoop in and save the day.






          share|cite|improve this answer











          $endgroup$









          • 1




            $begingroup$
            Hey, thanks a lot for the input! I just tried to plot the planes ${x}^bot$ in $mathbf{R}^3$ (w.l.o.g. $x_1=1$ since ${x}^bot={-x}^bot$), but the resulting components don't look like orthants to me, some of them are enclosed by four planes but some only by three. Or am I confused about this?
            $endgroup$
            – fweth
            Dec 2 '18 at 3:25






          • 1




            $begingroup$
            @fweth Oh man, what I wrote was truly a half baked idea if there ever was one. My very first line is obviously false; these are not spokes of a coordinate system as there are WAY too many of them, even after accounting for negatives. Sorry, back to the drawing board for me...
            $endgroup$
            – FranklinBash
            Dec 2 '18 at 3:36






          • 1




            $begingroup$
            Don't worry, I'm glad you shared your intuition :)
            $endgroup$
            – fweth
            Dec 2 '18 at 4:17






          • 2




            $begingroup$
            @fweth I found an interesting set of slides about counting regions of $mathbb{R}^n$ that have been separated by hyperplanes. They even discuss what to do with hyperplanes that are not in general position, which is what we have here. It also looks a bit complicated. physics.csusb.edu/%7Eprenteln/papers/hyperintro.pdf
            $endgroup$
            – FranklinBash
            Dec 2 '18 at 4:21











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          1 Answer
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          active

          oldest

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          1 Answer
          1






          active

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          active

          oldest

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          active

          oldest

          votes









          2












          $begingroup$

          EDIT: My previous rough idea about orthants was completely off of the mark. I need to go rethink everything.



          I have provided a link to a set of slides in the comments below, but I think I'm starting to get out of my depth. Someone who knows better than I should swoop in and save the day.






          share|cite|improve this answer











          $endgroup$









          • 1




            $begingroup$
            Hey, thanks a lot for the input! I just tried to plot the planes ${x}^bot$ in $mathbf{R}^3$ (w.l.o.g. $x_1=1$ since ${x}^bot={-x}^bot$), but the resulting components don't look like orthants to me, some of them are enclosed by four planes but some only by three. Or am I confused about this?
            $endgroup$
            – fweth
            Dec 2 '18 at 3:25






          • 1




            $begingroup$
            @fweth Oh man, what I wrote was truly a half baked idea if there ever was one. My very first line is obviously false; these are not spokes of a coordinate system as there are WAY too many of them, even after accounting for negatives. Sorry, back to the drawing board for me...
            $endgroup$
            – FranklinBash
            Dec 2 '18 at 3:36






          • 1




            $begingroup$
            Don't worry, I'm glad you shared your intuition :)
            $endgroup$
            – fweth
            Dec 2 '18 at 4:17






          • 2




            $begingroup$
            @fweth I found an interesting set of slides about counting regions of $mathbb{R}^n$ that have been separated by hyperplanes. They even discuss what to do with hyperplanes that are not in general position, which is what we have here. It also looks a bit complicated. physics.csusb.edu/%7Eprenteln/papers/hyperintro.pdf
            $endgroup$
            – FranklinBash
            Dec 2 '18 at 4:21
















          2












          $begingroup$

          EDIT: My previous rough idea about orthants was completely off of the mark. I need to go rethink everything.



          I have provided a link to a set of slides in the comments below, but I think I'm starting to get out of my depth. Someone who knows better than I should swoop in and save the day.






          share|cite|improve this answer











          $endgroup$









          • 1




            $begingroup$
            Hey, thanks a lot for the input! I just tried to plot the planes ${x}^bot$ in $mathbf{R}^3$ (w.l.o.g. $x_1=1$ since ${x}^bot={-x}^bot$), but the resulting components don't look like orthants to me, some of them are enclosed by four planes but some only by three. Or am I confused about this?
            $endgroup$
            – fweth
            Dec 2 '18 at 3:25






          • 1




            $begingroup$
            @fweth Oh man, what I wrote was truly a half baked idea if there ever was one. My very first line is obviously false; these are not spokes of a coordinate system as there are WAY too many of them, even after accounting for negatives. Sorry, back to the drawing board for me...
            $endgroup$
            – FranklinBash
            Dec 2 '18 at 3:36






          • 1




            $begingroup$
            Don't worry, I'm glad you shared your intuition :)
            $endgroup$
            – fweth
            Dec 2 '18 at 4:17






          • 2




            $begingroup$
            @fweth I found an interesting set of slides about counting regions of $mathbb{R}^n$ that have been separated by hyperplanes. They even discuss what to do with hyperplanes that are not in general position, which is what we have here. It also looks a bit complicated. physics.csusb.edu/%7Eprenteln/papers/hyperintro.pdf
            $endgroup$
            – FranklinBash
            Dec 2 '18 at 4:21














          2












          2








          2





          $begingroup$

          EDIT: My previous rough idea about orthants was completely off of the mark. I need to go rethink everything.



          I have provided a link to a set of slides in the comments below, but I think I'm starting to get out of my depth. Someone who knows better than I should swoop in and save the day.






          share|cite|improve this answer











          $endgroup$



          EDIT: My previous rough idea about orthants was completely off of the mark. I need to go rethink everything.



          I have provided a link to a set of slides in the comments below, but I think I'm starting to get out of my depth. Someone who knows better than I should swoop in and save the day.







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited Dec 2 '18 at 4:35

























          answered Dec 2 '18 at 2:58









          FranklinBashFranklinBash

          1012




          1012








          • 1




            $begingroup$
            Hey, thanks a lot for the input! I just tried to plot the planes ${x}^bot$ in $mathbf{R}^3$ (w.l.o.g. $x_1=1$ since ${x}^bot={-x}^bot$), but the resulting components don't look like orthants to me, some of them are enclosed by four planes but some only by three. Or am I confused about this?
            $endgroup$
            – fweth
            Dec 2 '18 at 3:25






          • 1




            $begingroup$
            @fweth Oh man, what I wrote was truly a half baked idea if there ever was one. My very first line is obviously false; these are not spokes of a coordinate system as there are WAY too many of them, even after accounting for negatives. Sorry, back to the drawing board for me...
            $endgroup$
            – FranklinBash
            Dec 2 '18 at 3:36






          • 1




            $begingroup$
            Don't worry, I'm glad you shared your intuition :)
            $endgroup$
            – fweth
            Dec 2 '18 at 4:17






          • 2




            $begingroup$
            @fweth I found an interesting set of slides about counting regions of $mathbb{R}^n$ that have been separated by hyperplanes. They even discuss what to do with hyperplanes that are not in general position, which is what we have here. It also looks a bit complicated. physics.csusb.edu/%7Eprenteln/papers/hyperintro.pdf
            $endgroup$
            – FranklinBash
            Dec 2 '18 at 4:21














          • 1




            $begingroup$
            Hey, thanks a lot for the input! I just tried to plot the planes ${x}^bot$ in $mathbf{R}^3$ (w.l.o.g. $x_1=1$ since ${x}^bot={-x}^bot$), but the resulting components don't look like orthants to me, some of them are enclosed by four planes but some only by three. Or am I confused about this?
            $endgroup$
            – fweth
            Dec 2 '18 at 3:25






          • 1




            $begingroup$
            @fweth Oh man, what I wrote was truly a half baked idea if there ever was one. My very first line is obviously false; these are not spokes of a coordinate system as there are WAY too many of them, even after accounting for negatives. Sorry, back to the drawing board for me...
            $endgroup$
            – FranklinBash
            Dec 2 '18 at 3:36






          • 1




            $begingroup$
            Don't worry, I'm glad you shared your intuition :)
            $endgroup$
            – fweth
            Dec 2 '18 at 4:17






          • 2




            $begingroup$
            @fweth I found an interesting set of slides about counting regions of $mathbb{R}^n$ that have been separated by hyperplanes. They even discuss what to do with hyperplanes that are not in general position, which is what we have here. It also looks a bit complicated. physics.csusb.edu/%7Eprenteln/papers/hyperintro.pdf
            $endgroup$
            – FranklinBash
            Dec 2 '18 at 4:21








          1




          1




          $begingroup$
          Hey, thanks a lot for the input! I just tried to plot the planes ${x}^bot$ in $mathbf{R}^3$ (w.l.o.g. $x_1=1$ since ${x}^bot={-x}^bot$), but the resulting components don't look like orthants to me, some of them are enclosed by four planes but some only by three. Or am I confused about this?
          $endgroup$
          – fweth
          Dec 2 '18 at 3:25




          $begingroup$
          Hey, thanks a lot for the input! I just tried to plot the planes ${x}^bot$ in $mathbf{R}^3$ (w.l.o.g. $x_1=1$ since ${x}^bot={-x}^bot$), but the resulting components don't look like orthants to me, some of them are enclosed by four planes but some only by three. Or am I confused about this?
          $endgroup$
          – fweth
          Dec 2 '18 at 3:25




          1




          1




          $begingroup$
          @fweth Oh man, what I wrote was truly a half baked idea if there ever was one. My very first line is obviously false; these are not spokes of a coordinate system as there are WAY too many of them, even after accounting for negatives. Sorry, back to the drawing board for me...
          $endgroup$
          – FranklinBash
          Dec 2 '18 at 3:36




          $begingroup$
          @fweth Oh man, what I wrote was truly a half baked idea if there ever was one. My very first line is obviously false; these are not spokes of a coordinate system as there are WAY too many of them, even after accounting for negatives. Sorry, back to the drawing board for me...
          $endgroup$
          – FranklinBash
          Dec 2 '18 at 3:36




          1




          1




          $begingroup$
          Don't worry, I'm glad you shared your intuition :)
          $endgroup$
          – fweth
          Dec 2 '18 at 4:17




          $begingroup$
          Don't worry, I'm glad you shared your intuition :)
          $endgroup$
          – fweth
          Dec 2 '18 at 4:17




          2




          2




          $begingroup$
          @fweth I found an interesting set of slides about counting regions of $mathbb{R}^n$ that have been separated by hyperplanes. They even discuss what to do with hyperplanes that are not in general position, which is what we have here. It also looks a bit complicated. physics.csusb.edu/%7Eprenteln/papers/hyperintro.pdf
          $endgroup$
          – FranklinBash
          Dec 2 '18 at 4:21




          $begingroup$
          @fweth I found an interesting set of slides about counting regions of $mathbb{R}^n$ that have been separated by hyperplanes. They even discuss what to do with hyperplanes that are not in general position, which is what we have here. It also looks a bit complicated. physics.csusb.edu/%7Eprenteln/papers/hyperintro.pdf
          $endgroup$
          – FranklinBash
          Dec 2 '18 at 4:21


















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