Mixing
A (simple?) matter of notation
$begingroup$
I am currently working on some power series problems, so I deal with the sequence of their (generally complex) coefficients $a_i$, $iinmathbb{N}$, denoted in the sequel as $langle a_nrangle$ following reference [1]. A sequence is simply a function
$$
mathbb{N}ni nmapsto a_ninmathbb{C}tag{1}label{1}
$$
Given a finite/infinite subset $S$ of the range of $langle a_nrangle$, I need to analyze the indexes $n$ of each member $a_nin S$. And now my question arises: does there exists a standard notation for the inverse image of $S$ under the function expressed by eqref{1}?
Having read many textbooks on real and complex analysis, power series and so on, I did not find anything on such matter: however, perhaps specialists in combinatorics or other similar field are more customary with such concepts so I decided to ask this question in order to not introduce in my work useless notations.
[1] Emanuel Fisher (1983), "Intermediate Real Analysis", Springer Verlag.
sequences-and-series index-notation
asked Sep 19 '18 at 5:59
Daniele TampieriDaniele Tampieri
2,2922922
$endgroup$
add a comment |
$begingroup$
I am currently working on some power series problems, so I deal with the sequence of their (generally complex) coefficients $a_i$, $iinmathbb{N}$, denoted in the sequel as $langle a_nrangle$ following reference [1]. A sequence is simply a function
$$
mathbb{N}ni nmapsto a_ninmathbb{C}tag{1}label{1}
$$
Given a finite/infinite subset $S$ of the range of $langle a_nrangle$, I need to analyze the indexes $n$ of each member $a_nin S$. And now my question arises: does there exists a standard notation for the inverse image of $S$ under the function expressed by eqref{1}?
Having read many textbooks on real and complex analysis, power series and so on, I did not find anything on such matter: however, perhaps specialists in combinatorics or other similar field are more customary with such concepts so I decided to ask this question in order to not introduce in my work useless notations.
[1] Emanuel Fisher (1983), "Intermediate Real Analysis", Springer Verlag.
sequences-and-series index-notation
asked Sep 19 '18 at 5:59
Daniele TampieriDaniele Tampieri
2,2922922
$endgroup$
1
$begingroup$
$a^{-1}_S$ or $a^{-1}(S)$ usually
$endgroup$
– Holo
Sep 19 '18 at 6:09
1
$begingroup$
How about $langle a_n rangle^{-1}(S)$? $langle a_n rangle$ denotes a function, so the pre-image notation seems suitable here.
$endgroup$
– E-mu
Jan 16 at 21:13
$begingroup$
@E-mu: nice comment. Why don't you post it as an answer?
$endgroup$
– Daniele Tampieri
Jan 16 at 21:15
$begingroup$
This post is slightly dated and also has an accepted answer, so I am a bit tentative about posting an answer :-). I am also not sure that this notation is standard.
$endgroup$
– E-mu
Jan 16 at 21:27
1
$begingroup$
@E-mu: well, even if I accepted Fred's answer, I can upvote yours too. However, if you fear to be downvoted by other members, I'll respect your choice.
$endgroup$
– Daniele Tampieri
Jan 16 at 21:31
add a comment |
$begingroup$
I am currently working on some power series problems, so I deal with the sequence of their (generally complex) coefficients $a_i$, $iinmathbb{N}$, denoted in the sequel as $langle a_nrangle$ following reference [1]. A sequence is simply a function
$$
mathbb{N}ni nmapsto a_ninmathbb{C}tag{1}label{1}
$$
Given a finite/infinite subset $S$ of the range of $langle a_nrangle$, I need to analyze the indexes $n$ of each member $a_nin S$. And now my question arises: does there exists a standard notation for the inverse image of $S$ under the function expressed by eqref{1}?
Having read many textbooks on real and complex analysis, power series and so on, I did not find anything on such matter: however, perhaps specialists in combinatorics or other similar field are more customary with such concepts so I decided to ask this question in order to not introduce in my work useless notations.
[1] Emanuel Fisher (1983), "Intermediate Real Analysis", Springer Verlag.
sequences-and-series index-notation
asked Sep 19 '18 at 5:59
Daniele TampieriDaniele Tampieri
2,2922922
$endgroup$
I am currently working on some power series problems, so I deal with the sequence of their (generally complex) coefficients $a_i$, $iinmathbb{N}$, denoted in the sequel as $langle a_nrangle$ following reference [1]. A sequence is simply a function
$$
mathbb{N}ni nmapsto a_ninmathbb{C}tag{1}label{1}
$$
Given a finite/infinite subset $S$ of the range of $langle a_nrangle$, I need to analyze the indexes $n$ of each member $a_nin S$. And now my question arises: does there exists a standard notation for the inverse image of $S$ under the function expressed by eqref{1}?
Having read many textbooks on real and complex analysis, power series and so on, I did not find anything on such matter: however, perhaps specialists in combinatorics or other similar field are more customary with such concepts so I decided to ask this question in order to not introduce in my work useless notations.
[1] Emanuel Fisher (1983), "Intermediate Real Analysis", Springer Verlag.
sequences-and-series index-notation
sequences-and-series index-notation
asked Sep 19 '18 at 5:59
Daniele TampieriDaniele Tampieri
2,2922922
asked Sep 19 '18 at 5:59
Daniele TampieriDaniele Tampieri
2,2922922
asked Sep 19 '18 at 5:59
Daniele TampieriDaniele Tampieri
2,2922922
asked Sep 19 '18 at 5:59
Daniele TampieriDaniele Tampieri
2,2922922
asked Sep 19 '18 at 5:59
Daniele TampieriDaniele Tampieri
2,2922922
2,2922922
1
$begingroup$
$a^{-1}_S$ or $a^{-1}(S)$ usually
$endgroup$
– Holo
Sep 19 '18 at 6:09
1
$begingroup$
How about $langle a_n rangle^{-1}(S)$? $langle a_n rangle$ denotes a function, so the pre-image notation seems suitable here.
$endgroup$
– E-mu
Jan 16 at 21:13
$begingroup$
@E-mu: nice comment. Why don't you post it as an answer?
$endgroup$
– Daniele Tampieri
Jan 16 at 21:15
$begingroup$
This post is slightly dated and also has an accepted answer, so I am a bit tentative about posting an answer :-). I am also not sure that this notation is standard.
$endgroup$
– E-mu
Jan 16 at 21:27
1
$begingroup$
@E-mu: well, even if I accepted Fred's answer, I can upvote yours too. However, if you fear to be downvoted by other members, I'll respect your choice.
$endgroup$
– Daniele Tampieri
Jan 16 at 21:31
add a comment |
1
$begingroup$
$a^{-1}_S$ or $a^{-1}(S)$ usually
$endgroup$
– Holo
Sep 19 '18 at 6:09
1
$begingroup$
How about $langle a_n rangle^{-1}(S)$? $langle a_n rangle$ denotes a function, so the pre-image notation seems suitable here.
$endgroup$
– E-mu
Jan 16 at 21:13
$begingroup$
@E-mu: nice comment. Why don't you post it as an answer?
$endgroup$
– Daniele Tampieri
Jan 16 at 21:15
$begingroup$
This post is slightly dated and also has an accepted answer, so I am a bit tentative about posting an answer :-). I am also not sure that this notation is standard.
$endgroup$
– E-mu
Jan 16 at 21:27
1
$begingroup$
@E-mu: well, even if I accepted Fred's answer, I can upvote yours too. However, if you fear to be downvoted by other members, I'll respect your choice.
$endgroup$
– Daniele Tampieri
Jan 16 at 21:31
1
1
$begingroup$
$a^{-1}_S$ or $a^{-1}(S)$ usually
$endgroup$
– Holo
Sep 19 '18 at 6:09
$begingroup$
$a^{-1}_S$ or $a^{-1}(S)$ usually
$endgroup$
– Holo
Sep 19 '18 at 6:09
1
1
$begingroup$
How about $langle a_n rangle^{-1}(S)$? $langle a_n rangle$ denotes a function, so the pre-image notation seems suitable here.
$endgroup$
– E-mu
Jan 16 at 21:13
$begingroup$
How about $langle a_n rangle^{-1}(S)$? $langle a_n rangle$ denotes a function, so the pre-image notation seems suitable here.
$endgroup$
– E-mu
Jan 16 at 21:13
$begingroup$
@E-mu: nice comment. Why don't you post it as an answer?
$endgroup$
– Daniele Tampieri
Jan 16 at 21:15
$begingroup$
@E-mu: nice comment. Why don't you post it as an answer?
$endgroup$
– Daniele Tampieri
Jan 16 at 21:15
$begingroup$
This post is slightly dated and also has an accepted answer, so I am a bit tentative about posting an answer :-). I am also not sure that this notation is standard.
$endgroup$
– E-mu
Jan 16 at 21:27
$begingroup$
This post is slightly dated and also has an accepted answer, so I am a bit tentative about posting an answer :-). I am also not sure that this notation is standard.
$endgroup$
– E-mu
Jan 16 at 21:27
1
1
$begingroup$
@E-mu: well, even if I accepted Fred's answer, I can upvote yours too. However, if you fear to be downvoted by other members, I'll respect your choice.
$endgroup$
– Daniele Tampieri
Jan 16 at 21:31
$begingroup$
@E-mu: well, even if I accepted Fred's answer, I can upvote yours too. However, if you fear to be downvoted by other members, I'll respect your choice.
$endgroup$
– Daniele Tampieri
Jan 16 at 21:31
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
Denote the function in (1) by $a$, hence $a: mathbb N to mathbb C$ and $a(n)=a_n$.
Then ${n in mathbb N: a_n in S}=a^{-1}(S)$.
answered Sep 19 '18 at 6:05
FredFred
46.9k1848
$endgroup$
$begingroup$
so the advice is to use the standard notation for generic functions. This is somewhat unsatisfactory as the fact that we are dealing with a sequence is "momentarily" put in the shade: however, from the point of view of economy of thought and simplicity, this is probably the best solution, so +1.
$endgroup$
– Daniele Tampieri
Sep 19 '18 at 6:59
add a comment |
$begingroup$
If we use the notation $langle a_n rangle$ to denote a sequence $a$, where for each $n in mathbb{N}$, $a_n := a(n)$, it seems fitting to write the pre-image of $S subseteq mathbb{C}$, $ a^{-1}(S),$ as $langle a_n rangle ^{-1}(S).$
Going from what you mentioned under Fred's post, one advantage of this notation is that we are not hiding the fact that $a$ is a sequence. I am not sure whether writing the pre-image this way is standard, however.
answered Jan 16 at 21:49
E-muE-mu
787417
$endgroup$
add a comment |
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
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votes
$begingroup$
Denote the function in (1) by $a$, hence $a: mathbb N to mathbb C$ and $a(n)=a_n$.
Then ${n in mathbb N: a_n in S}=a^{-1}(S)$.
answered Sep 19 '18 at 6:05
FredFred
46.9k1848
$endgroup$
$begingroup$
so the advice is to use the standard notation for generic functions. This is somewhat unsatisfactory as the fact that we are dealing with a sequence is "momentarily" put in the shade: however, from the point of view of economy of thought and simplicity, this is probably the best solution, so +1.
$endgroup$
– Daniele Tampieri
Sep 19 '18 at 6:59
add a comment |
$begingroup$
Denote the function in (1) by $a$, hence $a: mathbb N to mathbb C$ and $a(n)=a_n$.
Then ${n in mathbb N: a_n in S}=a^{-1}(S)$.
answered Sep 19 '18 at 6:05
FredFred
46.9k1848
$endgroup$
$begingroup$
so the advice is to use the standard notation for generic functions. This is somewhat unsatisfactory as the fact that we are dealing with a sequence is "momentarily" put in the shade: however, from the point of view of economy of thought and simplicity, this is probably the best solution, so +1.
$endgroup$
– Daniele Tampieri
Sep 19 '18 at 6:59
add a comment |
$begingroup$
Denote the function in (1) by $a$, hence $a: mathbb N to mathbb C$ and $a(n)=a_n$.
Then ${n in mathbb N: a_n in S}=a^{-1}(S)$.
answered Sep 19 '18 at 6:05
FredFred
46.9k1848
$endgroup$
Denote the function in (1) by $a$, hence $a: mathbb N to mathbb C$ and $a(n)=a_n$.
Then ${n in mathbb N: a_n in S}=a^{-1}(S)$.
answered Sep 19 '18 at 6:05
FredFred
46.9k1848
edited Sep 19 '18 at 7:54
answered Sep 19 '18 at 6:05
FredFred
46.9k1848
answered Sep 19 '18 at 6:05
FredFred
46.9k1848
answered Sep 19 '18 at 6:05
FredFred
46.9k1848
46.9k1848
$begingroup$
so the advice is to use the standard notation for generic functions. This is somewhat unsatisfactory as the fact that we are dealing with a sequence is "momentarily" put in the shade: however, from the point of view of economy of thought and simplicity, this is probably the best solution, so +1.
$endgroup$
– Daniele Tampieri
Sep 19 '18 at 6:59
add a comment |
$begingroup$
so the advice is to use the standard notation for generic functions. This is somewhat unsatisfactory as the fact that we are dealing with a sequence is "momentarily" put in the shade: however, from the point of view of economy of thought and simplicity, this is probably the best solution, so +1.
$endgroup$
– Daniele Tampieri
Sep 19 '18 at 6:59
$begingroup$
so the advice is to use the standard notation for generic functions. This is somewhat unsatisfactory as the fact that we are dealing with a sequence is "momentarily" put in the shade: however, from the point of view of economy of thought and simplicity, this is probably the best solution, so +1.
$endgroup$
– Daniele Tampieri
Sep 19 '18 at 6:59
$begingroup$
so the advice is to use the standard notation for generic functions. This is somewhat unsatisfactory as the fact that we are dealing with a sequence is "momentarily" put in the shade: however, from the point of view of economy of thought and simplicity, this is probably the best solution, so +1.
$endgroup$
– Daniele Tampieri
Sep 19 '18 at 6:59
add a comment |
$begingroup$
If we use the notation $langle a_n rangle$ to denote a sequence $a$, where for each $n in mathbb{N}$, $a_n := a(n)$, it seems fitting to write the pre-image of $S subseteq mathbb{C}$, $ a^{-1}(S),$ as $langle a_n rangle ^{-1}(S).$
Going from what you mentioned under Fred's post, one advantage of this notation is that we are not hiding the fact that $a$ is a sequence. I am not sure whether writing the pre-image this way is standard, however.
answered Jan 16 at 21:49
E-muE-mu
787417
$endgroup$
add a comment |
$begingroup$
If we use the notation $langle a_n rangle$ to denote a sequence $a$, where for each $n in mathbb{N}$, $a_n := a(n)$, it seems fitting to write the pre-image of $S subseteq mathbb{C}$, $ a^{-1}(S),$ as $langle a_n rangle ^{-1}(S).$
Going from what you mentioned under Fred's post, one advantage of this notation is that we are not hiding the fact that $a$ is a sequence. I am not sure whether writing the pre-image this way is standard, however.
answered Jan 16 at 21:49
E-muE-mu
787417
$endgroup$
add a comment |
$begingroup$
If we use the notation $langle a_n rangle$ to denote a sequence $a$, where for each $n in mathbb{N}$, $a_n := a(n)$, it seems fitting to write the pre-image of $S subseteq mathbb{C}$, $ a^{-1}(S),$ as $langle a_n rangle ^{-1}(S).$
Going from what you mentioned under Fred's post, one advantage of this notation is that we are not hiding the fact that $a$ is a sequence. I am not sure whether writing the pre-image this way is standard, however.
answered Jan 16 at 21:49
E-muE-mu
787417
$endgroup$
If we use the notation $langle a_n rangle$ to denote a sequence $a$, where for each $n in mathbb{N}$, $a_n := a(n)$, it seems fitting to write the pre-image of $S subseteq mathbb{C}$, $ a^{-1}(S),$ as $langle a_n rangle ^{-1}(S).$
Going from what you mentioned under Fred's post, one advantage of this notation is that we are not hiding the fact that $a$ is a sequence. I am not sure whether writing the pre-image this way is standard, however.
answered Jan 16 at 21:49
E-muE-mu
787417
answered Jan 16 at 21:49
E-muE-mu
787417
answered Jan 16 at 21:49
E-muE-mu
787417
answered Jan 16 at 21:49
E-muE-mu
787417
787417
add a comment |
add a comment |
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1
$begingroup$
$a^{-1}_S$ or $a^{-1}(S)$ usually
$endgroup$
– Holo
Sep 19 '18 at 6:09
1
$begingroup$
How about $langle a_n rangle^{-1}(S)$? $langle a_n rangle$ denotes a function, so the pre-image notation seems suitable here.
$endgroup$
– E-mu
Jan 16 at 21:13
$begingroup$
@E-mu: nice comment. Why don't you post it as an answer?
$endgroup$
– Daniele Tampieri
Jan 16 at 21:15
$begingroup$
This post is slightly dated and also has an accepted answer, so I am a bit tentative about posting an answer :-). I am also not sure that this notation is standard.
$endgroup$
– E-mu
Jan 16 at 21:27
1
$begingroup$
@E-mu: well, even if I accepted Fred's answer, I can upvote yours too. However, if you fear to be downvoted by other members, I'll respect your choice.
$endgroup$
– Daniele Tampieri
Jan 16 at 21:31