Mixing
Generalise to any dimension some notation
$begingroup$
I would like your help to generalise to any dimension and in the most simple way the following piece of notation (written for dimension $3$).
Step 1: Consider the 3 dimensional random vector $epsilonequiv (epsilon_0, epsilon_1, epsilon_2)$ with support the 3d Euclidean space $mathbb{R}^3$.
Step 2: Consider the set $mathcal{A}$ of all possible unordered pairs of elements from the set ${epsilon_0,epsilon_1, epsilon_2}$, i.e.,
$$
mathcal{A}equiv Big({epsilon_1,epsilon_0}, {epsilon_2, epsilon_0}, {epsilon_1, epsilon_2} Big)
$$
Take the difference between the two components of each element in $mathcal{A}$ and store them in a vector $Delta epsilon$, i.e.,
$$
Delta epsilon equiv (epsilon_1-epsilon_0, epsilon_2-epsilon_0, epsilon_1-epsilon_2)
$$
Step 3: Write down the support of $Delta epsilon$, i.e.,
$$
mathcal{S}equiv {(a,b,c)in mathbb{R}^3 text{ s.t. } cequiv (a-b)}
$$
The notation that I'm struggling to generalise to any dimension is the one in Step 2, that, in turn, is crucial for Step 3. Indeed, there are many ways to represent $mathcal{A}$: we could set
$$
mathcal{A}equiv Big({epsilon_1,epsilon_0}, {epsilon_2, epsilon_0}, {epsilon_1, epsilon_2} Big)
$$
as above, but also
$$
mathcal{A}equiv Big({epsilon_0,epsilon_1}, {epsilon_2, epsilon_0}, {epsilon_2, epsilon_1} Big)
$$
or
$$
mathcal{A}equiv Big({epsilon_2, epsilon_0},{epsilon_2, epsilon_1}, {epsilon_1,epsilon_0} Big)
$$
and many more. Different representation of $mathcal{A}$ leads to different definitions of $Delta epsilon$ and in turn to different definitions of $mathcal{S}$. Any representation of $mathcal{A}$ is fine with me, but I want to notationally transmit the idea that when once the reader has fixed a certain representation of $mathcal{A}$, then the definitions of $Delta epsilon$ and $mathcal{S}$ unambiguously follow.
combinatorics permutations notation combinations
asked Jan 10 at 12:18
STFSTF
831420
$endgroup$
add a comment |
$begingroup$
I would like your help to generalise to any dimension and in the most simple way the following piece of notation (written for dimension $3$).
Step 1: Consider the 3 dimensional random vector $epsilonequiv (epsilon_0, epsilon_1, epsilon_2)$ with support the 3d Euclidean space $mathbb{R}^3$.
Step 2: Consider the set $mathcal{A}$ of all possible unordered pairs of elements from the set ${epsilon_0,epsilon_1, epsilon_2}$, i.e.,
$$
mathcal{A}equiv Big({epsilon_1,epsilon_0}, {epsilon_2, epsilon_0}, {epsilon_1, epsilon_2} Big)
$$
Take the difference between the two components of each element in $mathcal{A}$ and store them in a vector $Delta epsilon$, i.e.,
$$
Delta epsilon equiv (epsilon_1-epsilon_0, epsilon_2-epsilon_0, epsilon_1-epsilon_2)
$$
Step 3: Write down the support of $Delta epsilon$, i.e.,
$$
mathcal{S}equiv {(a,b,c)in mathbb{R}^3 text{ s.t. } cequiv (a-b)}
$$
The notation that I'm struggling to generalise to any dimension is the one in Step 2, that, in turn, is crucial for Step 3. Indeed, there are many ways to represent $mathcal{A}$: we could set
$$
mathcal{A}equiv Big({epsilon_1,epsilon_0}, {epsilon_2, epsilon_0}, {epsilon_1, epsilon_2} Big)
$$
as above, but also
$$
mathcal{A}equiv Big({epsilon_0,epsilon_1}, {epsilon_2, epsilon_0}, {epsilon_2, epsilon_1} Big)
$$
or
$$
mathcal{A}equiv Big({epsilon_2, epsilon_0},{epsilon_2, epsilon_1}, {epsilon_1,epsilon_0} Big)
$$
and many more. Different representation of $mathcal{A}$ leads to different definitions of $Delta epsilon$ and in turn to different definitions of $mathcal{S}$. Any representation of $mathcal{A}$ is fine with me, but I want to notationally transmit the idea that when once the reader has fixed a certain representation of $mathcal{A}$, then the definitions of $Delta epsilon$ and $mathcal{S}$ unambiguously follow.
combinatorics permutations notation combinations
asked Jan 10 at 12:18
STFSTF
831420
$endgroup$
$begingroup$
The problem is not the definition of $mathcal A$, but the definition of $Deltaepsilon$. The sets ${x_0,x_1}$ and ${x_1,x_0}$ are the same set, so $Deltaepsilon$ is ill-defined. (Of course, you can fix this by just taking the absolute value of the difference, and mentioning in a footnote that the sign doesn't matter here.)
$endgroup$
– Deusovi
Jan 10 at 17:55
$begingroup$
Thanks. I see the point. Still, any definition of $Delta epsilon$ is OK with me. But then how do I write $mathcal{S}$ in general?
$endgroup$
– STF
Jan 10 at 19:05
add a comment |
$begingroup$
I would like your help to generalise to any dimension and in the most simple way the following piece of notation (written for dimension $3$).
Step 1: Consider the 3 dimensional random vector $epsilonequiv (epsilon_0, epsilon_1, epsilon_2)$ with support the 3d Euclidean space $mathbb{R}^3$.
Step 2: Consider the set $mathcal{A}$ of all possible unordered pairs of elements from the set ${epsilon_0,epsilon_1, epsilon_2}$, i.e.,
$$
mathcal{A}equiv Big({epsilon_1,epsilon_0}, {epsilon_2, epsilon_0}, {epsilon_1, epsilon_2} Big)
$$
Take the difference between the two components of each element in $mathcal{A}$ and store them in a vector $Delta epsilon$, i.e.,
$$
Delta epsilon equiv (epsilon_1-epsilon_0, epsilon_2-epsilon_0, epsilon_1-epsilon_2)
$$
Step 3: Write down the support of $Delta epsilon$, i.e.,
$$
mathcal{S}equiv {(a,b,c)in mathbb{R}^3 text{ s.t. } cequiv (a-b)}
$$
The notation that I'm struggling to generalise to any dimension is the one in Step 2, that, in turn, is crucial for Step 3. Indeed, there are many ways to represent $mathcal{A}$: we could set
$$
mathcal{A}equiv Big({epsilon_1,epsilon_0}, {epsilon_2, epsilon_0}, {epsilon_1, epsilon_2} Big)
$$
as above, but also
$$
mathcal{A}equiv Big({epsilon_0,epsilon_1}, {epsilon_2, epsilon_0}, {epsilon_2, epsilon_1} Big)
$$
or
$$
mathcal{A}equiv Big({epsilon_2, epsilon_0},{epsilon_2, epsilon_1}, {epsilon_1,epsilon_0} Big)
$$
and many more. Different representation of $mathcal{A}$ leads to different definitions of $Delta epsilon$ and in turn to different definitions of $mathcal{S}$. Any representation of $mathcal{A}$ is fine with me, but I want to notationally transmit the idea that when once the reader has fixed a certain representation of $mathcal{A}$, then the definitions of $Delta epsilon$ and $mathcal{S}$ unambiguously follow.
combinatorics permutations notation combinations
asked Jan 10 at 12:18
STFSTF
831420
$endgroup$
I would like your help to generalise to any dimension and in the most simple way the following piece of notation (written for dimension $3$).
Step 1: Consider the 3 dimensional random vector $epsilonequiv (epsilon_0, epsilon_1, epsilon_2)$ with support the 3d Euclidean space $mathbb{R}^3$.
Step 2: Consider the set $mathcal{A}$ of all possible unordered pairs of elements from the set ${epsilon_0,epsilon_1, epsilon_2}$, i.e.,
$$
mathcal{A}equiv Big({epsilon_1,epsilon_0}, {epsilon_2, epsilon_0}, {epsilon_1, epsilon_2} Big)
$$
Take the difference between the two components of each element in $mathcal{A}$ and store them in a vector $Delta epsilon$, i.e.,
$$
Delta epsilon equiv (epsilon_1-epsilon_0, epsilon_2-epsilon_0, epsilon_1-epsilon_2)
$$
Step 3: Write down the support of $Delta epsilon$, i.e.,
$$
mathcal{S}equiv {(a,b,c)in mathbb{R}^3 text{ s.t. } cequiv (a-b)}
$$
The notation that I'm struggling to generalise to any dimension is the one in Step 2, that, in turn, is crucial for Step 3. Indeed, there are many ways to represent $mathcal{A}$: we could set
$$
mathcal{A}equiv Big({epsilon_1,epsilon_0}, {epsilon_2, epsilon_0}, {epsilon_1, epsilon_2} Big)
$$
as above, but also
$$
mathcal{A}equiv Big({epsilon_0,epsilon_1}, {epsilon_2, epsilon_0}, {epsilon_2, epsilon_1} Big)
$$
or
$$
mathcal{A}equiv Big({epsilon_2, epsilon_0},{epsilon_2, epsilon_1}, {epsilon_1,epsilon_0} Big)
$$
and many more. Different representation of $mathcal{A}$ leads to different definitions of $Delta epsilon$ and in turn to different definitions of $mathcal{S}$. Any representation of $mathcal{A}$ is fine with me, but I want to notationally transmit the idea that when once the reader has fixed a certain representation of $mathcal{A}$, then the definitions of $Delta epsilon$ and $mathcal{S}$ unambiguously follow.
combinatorics permutations notation combinations
combinatorics permutations notation combinations
asked Jan 10 at 12:18
STFSTF
831420
asked Jan 10 at 12:18
STFSTF
831420
asked Jan 10 at 12:18
STFSTF
831420
asked Jan 10 at 12:18
STFSTF
831420
asked Jan 10 at 12:18
STFSTF
831420
831420
$begingroup$
The problem is not the definition of $mathcal A$, but the definition of $Deltaepsilon$. The sets ${x_0,x_1}$ and ${x_1,x_0}$ are the same set, so $Deltaepsilon$ is ill-defined. (Of course, you can fix this by just taking the absolute value of the difference, and mentioning in a footnote that the sign doesn't matter here.)
$endgroup$
– Deusovi
Jan 10 at 17:55
$begingroup$
Thanks. I see the point. Still, any definition of $Delta epsilon$ is OK with me. But then how do I write $mathcal{S}$ in general?
$endgroup$
– STF
Jan 10 at 19:05
add a comment |
$begingroup$
The problem is not the definition of $mathcal A$, but the definition of $Deltaepsilon$. The sets ${x_0,x_1}$ and ${x_1,x_0}$ are the same set, so $Deltaepsilon$ is ill-defined. (Of course, you can fix this by just taking the absolute value of the difference, and mentioning in a footnote that the sign doesn't matter here.)
$endgroup$
– Deusovi
Jan 10 at 17:55
$begingroup$
Thanks. I see the point. Still, any definition of $Delta epsilon$ is OK with me. But then how do I write $mathcal{S}$ in general?
$endgroup$
– STF
Jan 10 at 19:05
$begingroup$
The problem is not the definition of $mathcal A$, but the definition of $Deltaepsilon$. The sets ${x_0,x_1}$ and ${x_1,x_0}$ are the same set, so $Deltaepsilon$ is ill-defined. (Of course, you can fix this by just taking the absolute value of the difference, and mentioning in a footnote that the sign doesn't matter here.)
$endgroup$
– Deusovi
Jan 10 at 17:55
$begingroup$
The problem is not the definition of $mathcal A$, but the definition of $Deltaepsilon$. The sets ${x_0,x_1}$ and ${x_1,x_0}$ are the same set, so $Deltaepsilon$ is ill-defined. (Of course, you can fix this by just taking the absolute value of the difference, and mentioning in a footnote that the sign doesn't matter here.)
$endgroup$
– Deusovi
Jan 10 at 17:55
$begingroup$
Thanks. I see the point. Still, any definition of $Delta epsilon$ is OK with me. But then how do I write $mathcal{S}$ in general?
$endgroup$
– STF
Jan 10 at 19:05
$begingroup$
Thanks. I see the point. Still, any definition of $Delta epsilon$ is OK with me. But then how do I write $mathcal{S}$ in general?
$endgroup$
– STF
Jan 10 at 19:05
add a comment |
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$begingroup$
The problem is not the definition of $mathcal A$, but the definition of $Deltaepsilon$. The sets ${x_0,x_1}$ and ${x_1,x_0}$ are the same set, so $Deltaepsilon$ is ill-defined. (Of course, you can fix this by just taking the absolute value of the difference, and mentioning in a footnote that the sign doesn't matter here.)
$endgroup$
– Deusovi
Jan 10 at 17:55
$begingroup$
Thanks. I see the point. Still, any definition of $Delta epsilon$ is OK with me. But then how do I write $mathcal{S}$ in general?
$endgroup$
– STF
Jan 10 at 19:05