What is the size of these discrete function sets?
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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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add a comment |
$begingroup$
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
$endgroup$
add a comment |
$begingroup$
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
$endgroup$
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
edited Dec 3 '18 at 16:06
fweth
asked Dec 2 '18 at 1:00
fwethfweth
1,148711
1,148711
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1 Answer
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$begingroup$
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.
$endgroup$
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...
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– FranklinBash
Dec 2 '18 at 3:36
1
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Don't worry, I'm glad you shared your intuition :)
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– fweth
Dec 2 '18 at 4:17
2
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@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
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– FranklinBash
Dec 2 '18 at 4:21
add a comment |
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1 Answer
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$begingroup$
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.
$endgroup$
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
add a comment |
$begingroup$
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.
$endgroup$
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
add a comment |
$begingroup$
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.
$endgroup$
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.
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
add a comment |
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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