Mixing
Mapping and conservation law
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What is a general map from a point sitting in one dimensional space, to a set of points sitting in two dimensional space, to a set of points sitting in three dimensional space, to a set of points embedded in a torus sitting in four dimensional space? This is what I have but I'm not sure if it's correct notation. The parentheses are to show that those mappings inside the parentheses occur before the mapping to the torus.
The motivation for this comes from physics and viewing these points as fixed points in space throughout all transformations, which lends itself to a conservation of momentum law for these points, which can be thought of as particles. The last embedding onto the torus is to provide a ring structure to the solution space. Physicists are not necessarily known as tidy mathematicians so I thought I'd seek help from the pros.
$ F : ((Bbb R to Bbb R^2) to Bbb R^3) hookrightarrow Bbb T^4 $.
Thanks.
euclidean-geometry transformation
asked Aug 6 at 19:58
George Thomas
51416
 |Â
show 3 more comments
up vote
1
down vote
favorite
What is a general map from a point sitting in one dimensional space, to a set of points sitting in two dimensional space, to a set of points sitting in three dimensional space, to a set of points embedded in a torus sitting in four dimensional space? This is what I have but I'm not sure if it's correct notation. The parentheses are to show that those mappings inside the parentheses occur before the mapping to the torus.
The motivation for this comes from physics and viewing these points as fixed points in space throughout all transformations, which lends itself to a conservation of momentum law for these points, which can be thought of as particles. The last embedding onto the torus is to provide a ring structure to the solution space. Physicists are not necessarily known as tidy mathematicians so I thought I'd seek help from the pros.
$ F : ((Bbb R to Bbb R^2) to Bbb R^3) hookrightarrow Bbb T^4 $.
Thanks.
euclidean-geometry transformation
asked Aug 6 at 19:58
George Thomas
51416
Is it an embedding? If so, how can "one point sitting in one dimensional space" map to "a set of points sitting in two dimensional space"?
– md2perpe
Aug 6 at 20:31
The last transformation is an embedding, the first three are supposed to be structure preserving maps, so maybe I could just get rid of the $ Bbb R $ and start from $Bbb R^2$
– George Thomas
Aug 6 at 20:35
So, $mathbb R^3 to mathbb T^4$ is en embedding. Such are often written usinghookrightarrow, a combination of an arrow and the subset symbol: $mathbb R^3 hookrightarrow mathbb T^4.$
– md2perpe
Aug 6 at 20:39
ah thank you @md2perpe I will edit the question to reflect that
– George Thomas
Aug 6 at 20:40
Do you have names/symbols (like $f$) for the first two mappings?
– md2perpe
Aug 6 at 20:43
 |Â
show 3 more comments
up vote
1
down vote
favorite
up vote
1
down vote
favorite
What is a general map from a point sitting in one dimensional space, to a set of points sitting in two dimensional space, to a set of points sitting in three dimensional space, to a set of points embedded in a torus sitting in four dimensional space? This is what I have but I'm not sure if it's correct notation. The parentheses are to show that those mappings inside the parentheses occur before the mapping to the torus.
The motivation for this comes from physics and viewing these points as fixed points in space throughout all transformations, which lends itself to a conservation of momentum law for these points, which can be thought of as particles. The last embedding onto the torus is to provide a ring structure to the solution space. Physicists are not necessarily known as tidy mathematicians so I thought I'd seek help from the pros.
$ F : ((Bbb R to Bbb R^2) to Bbb R^3) hookrightarrow Bbb T^4 $.
Thanks.
euclidean-geometry transformation
asked Aug 6 at 19:58
George Thomas
51416
What is a general map from a point sitting in one dimensional space, to a set of points sitting in two dimensional space, to a set of points sitting in three dimensional space, to a set of points embedded in a torus sitting in four dimensional space? This is what I have but I'm not sure if it's correct notation. The parentheses are to show that those mappings inside the parentheses occur before the mapping to the torus.
The motivation for this comes from physics and viewing these points as fixed points in space throughout all transformations, which lends itself to a conservation of momentum law for these points, which can be thought of as particles. The last embedding onto the torus is to provide a ring structure to the solution space. Physicists are not necessarily known as tidy mathematicians so I thought I'd seek help from the pros.
$ F : ((Bbb R to Bbb R^2) to Bbb R^3) hookrightarrow Bbb T^4 $.
Thanks.
euclidean-geometry transformation
asked Aug 6 at 19:58
George Thomas
51416
edited Aug 6 at 20:41
asked Aug 6 at 19:58
George Thomas
51416
asked Aug 6 at 19:58
George Thomas
51416
asked Aug 6 at 19:58
George Thomas
51416
51416
Is it an embedding? If so, how can "one point sitting in one dimensional space" map to "a set of points sitting in two dimensional space"?
– md2perpe
Aug 6 at 20:31
The last transformation is an embedding, the first three are supposed to be structure preserving maps, so maybe I could just get rid of the $ Bbb R $ and start from $Bbb R^2$
– George Thomas
Aug 6 at 20:35
So, $mathbb R^3 to mathbb T^4$ is en embedding. Such are often written usinghookrightarrow, a combination of an arrow and the subset symbol: $mathbb R^3 hookrightarrow mathbb T^4.$
– md2perpe
Aug 6 at 20:39
ah thank you @md2perpe I will edit the question to reflect that
– George Thomas
Aug 6 at 20:40
Do you have names/symbols (like $f$) for the first two mappings?
– md2perpe
Aug 6 at 20:43
 |Â
show 3 more comments
Is it an embedding? If so, how can "one point sitting in one dimensional space" map to "a set of points sitting in two dimensional space"?
– md2perpe
Aug 6 at 20:31
The last transformation is an embedding, the first three are supposed to be structure preserving maps, so maybe I could just get rid of the $ Bbb R $ and start from $Bbb R^2$
– George Thomas
Aug 6 at 20:35
So, $mathbb R^3 to mathbb T^4$ is en embedding. Such are often written usinghookrightarrow, a combination of an arrow and the subset symbol: $mathbb R^3 hookrightarrow mathbb T^4.$
– md2perpe
Aug 6 at 20:39
ah thank you @md2perpe I will edit the question to reflect that
– George Thomas
Aug 6 at 20:40
Do you have names/symbols (like $f$) for the first two mappings?
– md2perpe
Aug 6 at 20:43
Is it an embedding? If so, how can "one point sitting in one dimensional space" map to "a set of points sitting in two dimensional space"?
– md2perpe
Aug 6 at 20:31
Is it an embedding? If so, how can "one point sitting in one dimensional space" map to "a set of points sitting in two dimensional space"?
– md2perpe
Aug 6 at 20:31
The last transformation is an embedding, the first three are supposed to be structure preserving maps, so maybe I could just get rid of the $ Bbb R $ and start from $Bbb R^2$
– George Thomas
Aug 6 at 20:35
The last transformation is an embedding, the first three are supposed to be structure preserving maps, so maybe I could just get rid of the $ Bbb R $ and start from $Bbb R^2$
– George Thomas
Aug 6 at 20:35
So, $mathbb R^3 to mathbb T^4$ is en embedding. Such are often written using
hookrightarrow, a combination of an arrow and the subset symbol: $mathbb R^3 hookrightarrow mathbb T^4.$– md2perpe
Aug 6 at 20:39
So, $mathbb R^3 to mathbb T^4$ is en embedding. Such are often written using
hookrightarrow, a combination of an arrow and the subset symbol: $mathbb R^3 hookrightarrow mathbb T^4.$– md2perpe
Aug 6 at 20:39
ah thank you @md2perpe I will edit the question to reflect that
– George Thomas
Aug 6 at 20:40
ah thank you @md2perpe I will edit the question to reflect that
– George Thomas
Aug 6 at 20:40
Do you have names/symbols (like $f$) for the first two mappings?
– md2perpe
Aug 6 at 20:43
Do you have names/symbols (like $f$) for the first two mappings?
– md2perpe
Aug 6 at 20:43
 |Â
show 3 more comments
2 Answers
2
active
oldest
votes
up vote
1
down vote
accepted
You need more brackets to make the notation unambiguous. Suppose we write something like
$F: big ( (mathbb R to mathbb R^2) to mathbb R^3big) to mathbb T^4 $.
That could be reasonably interpreted to mean $F$ takes as input a function $f:(mathbb R to mathbb R^2) to mathbb R^3 $ and each $F(f)$ is an element of $mathbb T^4$. But what does $f:(mathbb R to mathbb R^2) to mathbb R^3 $ mean? It means $f$ takes as input a function $alpha: mathbb R to mathbb R^2$ and each $f(alpha)$ is an element of $mathbb R^3$.
Exercise: What does $G: big ( mathbb R to (mathbb R^2 to mathbb R^3) big) to mathbb T^4 $ mean?
A slightly more common notation would be to write something like $hom(X,Y)$, or $textMor (X,Y)$ or $textfun(X,Y)$ or $F(X,Y)$ for the set of all maps from $X$ to $Y$, or $C(X,Y)$ for the set of all continuous maps. In that case the above example is written:
$F in C , (C ,(C,(mathbb R, mathbb R^2),, mathbb R^3),, mathbb T^4)$.
Personally I'd stick to the earlier notation. It is easier on the eyes and also resembles function definition syntax from some programming languages.
answered Aug 6 at 20:23
Daron
4,3581923
ah i see, thank you
– George Thomas
Aug 6 at 20:25
Is this really what you want, @GeorgeThomas? Do you want a function taking a function? I interpret your question as asking for a chain of mappings, one from 1D to 2D, then one from 2D to 3D, and then finally one from 3D to the 4D torus: $mathbb R to mathbb R^2 to mathbb R^3 to mathbb T^4.$
– md2perpe
Aug 6 at 20:35
Hmm, yes it is a chain of mappings... I guess I need the actual functions to act as transformations to go from one mapping to the next, right?
– George Thomas
Aug 6 at 20:39
It's hard deriving those
– George Thomas
Aug 6 at 20:39
add a comment |Â
up vote
1
down vote
If I understand correctly you have first a mapping $mathbb R stackrelflongrightarrow mathscr P(mathbb R^2),$ where $mathscr P$ denotes power set, and then a chain of mappings
$mathbb R^2 stackrelglongrightarrow mathbb R^3 stackreliota hookrightarrow mathbb T^4$ acting on each element in $f(t)$ for $t in mathbb R.$
answered Aug 6 at 21:13
md2perpe
6,02511022
add a comment |Â
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
You need more brackets to make the notation unambiguous. Suppose we write something like
$F: big ( (mathbb R to mathbb R^2) to mathbb R^3big) to mathbb T^4 $.
That could be reasonably interpreted to mean $F$ takes as input a function $f:(mathbb R to mathbb R^2) to mathbb R^3 $ and each $F(f)$ is an element of $mathbb T^4$. But what does $f:(mathbb R to mathbb R^2) to mathbb R^3 $ mean? It means $f$ takes as input a function $alpha: mathbb R to mathbb R^2$ and each $f(alpha)$ is an element of $mathbb R^3$.
Exercise: What does $G: big ( mathbb R to (mathbb R^2 to mathbb R^3) big) to mathbb T^4 $ mean?
A slightly more common notation would be to write something like $hom(X,Y)$, or $textMor (X,Y)$ or $textfun(X,Y)$ or $F(X,Y)$ for the set of all maps from $X$ to $Y$, or $C(X,Y)$ for the set of all continuous maps. In that case the above example is written:
$F in C , (C ,(C,(mathbb R, mathbb R^2),, mathbb R^3),, mathbb T^4)$.
Personally I'd stick to the earlier notation. It is easier on the eyes and also resembles function definition syntax from some programming languages.
answered Aug 6 at 20:23
Daron
4,3581923
ah i see, thank you
– George Thomas
Aug 6 at 20:25
Is this really what you want, @GeorgeThomas? Do you want a function taking a function? I interpret your question as asking for a chain of mappings, one from 1D to 2D, then one from 2D to 3D, and then finally one from 3D to the 4D torus: $mathbb R to mathbb R^2 to mathbb R^3 to mathbb T^4.$
– md2perpe
Aug 6 at 20:35
Hmm, yes it is a chain of mappings... I guess I need the actual functions to act as transformations to go from one mapping to the next, right?
– George Thomas
Aug 6 at 20:39
It's hard deriving those
– George Thomas
Aug 6 at 20:39
add a comment |Â
up vote
1
down vote
accepted
You need more brackets to make the notation unambiguous. Suppose we write something like
$F: big ( (mathbb R to mathbb R^2) to mathbb R^3big) to mathbb T^4 $.
That could be reasonably interpreted to mean $F$ takes as input a function $f:(mathbb R to mathbb R^2) to mathbb R^3 $ and each $F(f)$ is an element of $mathbb T^4$. But what does $f:(mathbb R to mathbb R^2) to mathbb R^3 $ mean? It means $f$ takes as input a function $alpha: mathbb R to mathbb R^2$ and each $f(alpha)$ is an element of $mathbb R^3$.
Exercise: What does $G: big ( mathbb R to (mathbb R^2 to mathbb R^3) big) to mathbb T^4 $ mean?
A slightly more common notation would be to write something like $hom(X,Y)$, or $textMor (X,Y)$ or $textfun(X,Y)$ or $F(X,Y)$ for the set of all maps from $X$ to $Y$, or $C(X,Y)$ for the set of all continuous maps. In that case the above example is written:
$F in C , (C ,(C,(mathbb R, mathbb R^2),, mathbb R^3),, mathbb T^4)$.
Personally I'd stick to the earlier notation. It is easier on the eyes and also resembles function definition syntax from some programming languages.
answered Aug 6 at 20:23
Daron
4,3581923
ah i see, thank you
– George Thomas
Aug 6 at 20:25
Is this really what you want, @GeorgeThomas? Do you want a function taking a function? I interpret your question as asking for a chain of mappings, one from 1D to 2D, then one from 2D to 3D, and then finally one from 3D to the 4D torus: $mathbb R to mathbb R^2 to mathbb R^3 to mathbb T^4.$
– md2perpe
Aug 6 at 20:35
Hmm, yes it is a chain of mappings... I guess I need the actual functions to act as transformations to go from one mapping to the next, right?
– George Thomas
Aug 6 at 20:39
It's hard deriving those
– George Thomas
Aug 6 at 20:39
add a comment |Â
up vote
1
down vote
accepted
up vote
1
down vote
accepted
You need more brackets to make the notation unambiguous. Suppose we write something like
$F: big ( (mathbb R to mathbb R^2) to mathbb R^3big) to mathbb T^4 $.
That could be reasonably interpreted to mean $F$ takes as input a function $f:(mathbb R to mathbb R^2) to mathbb R^3 $ and each $F(f)$ is an element of $mathbb T^4$. But what does $f:(mathbb R to mathbb R^2) to mathbb R^3 $ mean? It means $f$ takes as input a function $alpha: mathbb R to mathbb R^2$ and each $f(alpha)$ is an element of $mathbb R^3$.
Exercise: What does $G: big ( mathbb R to (mathbb R^2 to mathbb R^3) big) to mathbb T^4 $ mean?
A slightly more common notation would be to write something like $hom(X,Y)$, or $textMor (X,Y)$ or $textfun(X,Y)$ or $F(X,Y)$ for the set of all maps from $X$ to $Y$, or $C(X,Y)$ for the set of all continuous maps. In that case the above example is written:
$F in C , (C ,(C,(mathbb R, mathbb R^2),, mathbb R^3),, mathbb T^4)$.
Personally I'd stick to the earlier notation. It is easier on the eyes and also resembles function definition syntax from some programming languages.
answered Aug 6 at 20:23
Daron
4,3581923
You need more brackets to make the notation unambiguous. Suppose we write something like
$F: big ( (mathbb R to mathbb R^2) to mathbb R^3big) to mathbb T^4 $.
That could be reasonably interpreted to mean $F$ takes as input a function $f:(mathbb R to mathbb R^2) to mathbb R^3 $ and each $F(f)$ is an element of $mathbb T^4$. But what does $f:(mathbb R to mathbb R^2) to mathbb R^3 $ mean? It means $f$ takes as input a function $alpha: mathbb R to mathbb R^2$ and each $f(alpha)$ is an element of $mathbb R^3$.
Exercise: What does $G: big ( mathbb R to (mathbb R^2 to mathbb R^3) big) to mathbb T^4 $ mean?
A slightly more common notation would be to write something like $hom(X,Y)$, or $textMor (X,Y)$ or $textfun(X,Y)$ or $F(X,Y)$ for the set of all maps from $X$ to $Y$, or $C(X,Y)$ for the set of all continuous maps. In that case the above example is written:
$F in C , (C ,(C,(mathbb R, mathbb R^2),, mathbb R^3),, mathbb T^4)$.
Personally I'd stick to the earlier notation. It is easier on the eyes and also resembles function definition syntax from some programming languages.
answered Aug 6 at 20:23
Daron
4,3581923
edited Aug 6 at 20:30
answered Aug 6 at 20:23
Daron
4,3581923
answered Aug 6 at 20:23
Daron
4,3581923
answered Aug 6 at 20:23
Daron
4,3581923
4,3581923
ah i see, thank you
– George Thomas
Aug 6 at 20:25
Is this really what you want, @GeorgeThomas? Do you want a function taking a function? I interpret your question as asking for a chain of mappings, one from 1D to 2D, then one from 2D to 3D, and then finally one from 3D to the 4D torus: $mathbb R to mathbb R^2 to mathbb R^3 to mathbb T^4.$
– md2perpe
Aug 6 at 20:35
Hmm, yes it is a chain of mappings... I guess I need the actual functions to act as transformations to go from one mapping to the next, right?
– George Thomas
Aug 6 at 20:39
It's hard deriving those
– George Thomas
Aug 6 at 20:39
add a comment |Â
ah i see, thank you
– George Thomas
Aug 6 at 20:25
Is this really what you want, @GeorgeThomas? Do you want a function taking a function? I interpret your question as asking for a chain of mappings, one from 1D to 2D, then one from 2D to 3D, and then finally one from 3D to the 4D torus: $mathbb R to mathbb R^2 to mathbb R^3 to mathbb T^4.$
– md2perpe
Aug 6 at 20:35
Hmm, yes it is a chain of mappings... I guess I need the actual functions to act as transformations to go from one mapping to the next, right?
– George Thomas
Aug 6 at 20:39
It's hard deriving those
– George Thomas
Aug 6 at 20:39
ah i see, thank you
– George Thomas
Aug 6 at 20:25
ah i see, thank you
– George Thomas
Aug 6 at 20:25
Is this really what you want, @GeorgeThomas? Do you want a function taking a function? I interpret your question as asking for a chain of mappings, one from 1D to 2D, then one from 2D to 3D, and then finally one from 3D to the 4D torus: $mathbb R to mathbb R^2 to mathbb R^3 to mathbb T^4.$
– md2perpe
Aug 6 at 20:35
Is this really what you want, @GeorgeThomas? Do you want a function taking a function? I interpret your question as asking for a chain of mappings, one from 1D to 2D, then one from 2D to 3D, and then finally one from 3D to the 4D torus: $mathbb R to mathbb R^2 to mathbb R^3 to mathbb T^4.$
– md2perpe
Aug 6 at 20:35
Hmm, yes it is a chain of mappings... I guess I need the actual functions to act as transformations to go from one mapping to the next, right?
– George Thomas
Aug 6 at 20:39
Hmm, yes it is a chain of mappings... I guess I need the actual functions to act as transformations to go from one mapping to the next, right?
– George Thomas
Aug 6 at 20:39
It's hard deriving those
– George Thomas
Aug 6 at 20:39
It's hard deriving those
– George Thomas
Aug 6 at 20:39
add a comment |Â
up vote
1
down vote
If I understand correctly you have first a mapping $mathbb R stackrelflongrightarrow mathscr P(mathbb R^2),$ where $mathscr P$ denotes power set, and then a chain of mappings
$mathbb R^2 stackrelglongrightarrow mathbb R^3 stackreliota hookrightarrow mathbb T^4$ acting on each element in $f(t)$ for $t in mathbb R.$
answered Aug 6 at 21:13
md2perpe
6,02511022
add a comment |Â
up vote
1
down vote
If I understand correctly you have first a mapping $mathbb R stackrelflongrightarrow mathscr P(mathbb R^2),$ where $mathscr P$ denotes power set, and then a chain of mappings
$mathbb R^2 stackrelglongrightarrow mathbb R^3 stackreliota hookrightarrow mathbb T^4$ acting on each element in $f(t)$ for $t in mathbb R.$
answered Aug 6 at 21:13
md2perpe
6,02511022
add a comment |Â
up vote
1
down vote
up vote
1
down vote
If I understand correctly you have first a mapping $mathbb R stackrelflongrightarrow mathscr P(mathbb R^2),$ where $mathscr P$ denotes power set, and then a chain of mappings
$mathbb R^2 stackrelglongrightarrow mathbb R^3 stackreliota hookrightarrow mathbb T^4$ acting on each element in $f(t)$ for $t in mathbb R.$
answered Aug 6 at 21:13
md2perpe
6,02511022
If I understand correctly you have first a mapping $mathbb R stackrelflongrightarrow mathscr P(mathbb R^2),$ where $mathscr P$ denotes power set, and then a chain of mappings
$mathbb R^2 stackrelglongrightarrow mathbb R^3 stackreliota hookrightarrow mathbb T^4$ acting on each element in $f(t)$ for $t in mathbb R.$
answered Aug 6 at 21:13
md2perpe
6,02511022
answered Aug 6 at 21:13
md2perpe
6,02511022
answered Aug 6 at 21:13
md2perpe
6,02511022
answered Aug 6 at 21:13
md2perpe
6,02511022
6,02511022
add a comment |Â
add a comment |Â
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Is it an embedding? If so, how can "one point sitting in one dimensional space" map to "a set of points sitting in two dimensional space"?
– md2perpe
Aug 6 at 20:31
The last transformation is an embedding, the first three are supposed to be structure preserving maps, so maybe I could just get rid of the $ Bbb R $ and start from $Bbb R^2$
– George Thomas
Aug 6 at 20:35
So, $mathbb R^3 to mathbb T^4$ is en embedding. Such are often written using
hookrightarrow, a combination of an arrow and the subset symbol: $mathbb R^3 hookrightarrow mathbb T^4.$– md2perpe
Aug 6 at 20:39
ah thank you @md2perpe I will edit the question to reflect that
– George Thomas
Aug 6 at 20:40
Do you have names/symbols (like $f$) for the first two mappings?
– md2perpe
Aug 6 at 20:43