Mixing
Trying to arrange sums in neat way
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I am trying to write the following summations as compactly as possible using summation and enumeration:
$$fracy_11+y_2+y_3+y_4+y_5+fracy_21+y_3+y_4+y_5+fracy_31+y_4+y_5+fracy_41+y_5 leq 1$$
$$fracy_1+y_21+y_3+y_4+y_5+fracy_2+y_31+y_4+y_5+fracy_3+y_41+y_5 leq 1$$
$$fracy_1+y_2+y_31+y_4+y_5+fracy_2+y_3+y_41+y_5leq 1$$
$$frac y_1+y_2+y_3+y_41+y_5leq 1$$
I wrote it as following:
$$sum_k=1^4 frac y_k1+y_k+1+...+y_5 leq 1$$
$$sum_k=1^3 frac y_k+y_k+11+y_k+2+...+y_5 leq 1$$
$$ sum_k=1^2 frac y_k+y_k+1+y_k+21+y_k+3+...+y_5 leq 1$$
$$ frac sum_k=1^4 y_k1+y_5 leq 1$$
Is there any enumeration to combine the $4$ summations?
calculus summation
asked 1 hour ago
Dontknowanything
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up vote
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I am trying to write the following summations as compactly as possible using summation and enumeration:
$$fracy_11+y_2+y_3+y_4+y_5+fracy_21+y_3+y_4+y_5+fracy_31+y_4+y_5+fracy_41+y_5 leq 1$$
$$fracy_1+y_21+y_3+y_4+y_5+fracy_2+y_31+y_4+y_5+fracy_3+y_41+y_5 leq 1$$
$$fracy_1+y_2+y_31+y_4+y_5+fracy_2+y_3+y_41+y_5leq 1$$
$$frac y_1+y_2+y_3+y_41+y_5leq 1$$
I wrote it as following:
$$sum_k=1^4 frac y_k1+y_k+1+...+y_5 leq 1$$
$$sum_k=1^3 frac y_k+y_k+11+y_k+2+...+y_5 leq 1$$
$$ sum_k=1^2 frac y_k+y_k+1+y_k+21+y_k+3+...+y_5 leq 1$$
$$ frac sum_k=1^4 y_k1+y_5 leq 1$$
Is there any enumeration to combine the $4$ summations?
calculus summation
asked 1 hour ago
Dontknowanything
1,334518
add a comment |Â
up vote
5
down vote
favorite
up vote
5
down vote
favorite
I am trying to write the following summations as compactly as possible using summation and enumeration:
$$fracy_11+y_2+y_3+y_4+y_5+fracy_21+y_3+y_4+y_5+fracy_31+y_4+y_5+fracy_41+y_5 leq 1$$
$$fracy_1+y_21+y_3+y_4+y_5+fracy_2+y_31+y_4+y_5+fracy_3+y_41+y_5 leq 1$$
$$fracy_1+y_2+y_31+y_4+y_5+fracy_2+y_3+y_41+y_5leq 1$$
$$frac y_1+y_2+y_3+y_41+y_5leq 1$$
I wrote it as following:
$$sum_k=1^4 frac y_k1+y_k+1+...+y_5 leq 1$$
$$sum_k=1^3 frac y_k+y_k+11+y_k+2+...+y_5 leq 1$$
$$ sum_k=1^2 frac y_k+y_k+1+y_k+21+y_k+3+...+y_5 leq 1$$
$$ frac sum_k=1^4 y_k1+y_5 leq 1$$
Is there any enumeration to combine the $4$ summations?
calculus summation
asked 1 hour ago
Dontknowanything
1,334518
I am trying to write the following summations as compactly as possible using summation and enumeration:
$$fracy_11+y_2+y_3+y_4+y_5+fracy_21+y_3+y_4+y_5+fracy_31+y_4+y_5+fracy_41+y_5 leq 1$$
$$fracy_1+y_21+y_3+y_4+y_5+fracy_2+y_31+y_4+y_5+fracy_3+y_41+y_5 leq 1$$
$$fracy_1+y_2+y_31+y_4+y_5+fracy_2+y_3+y_41+y_5leq 1$$
$$frac y_1+y_2+y_3+y_41+y_5leq 1$$
I wrote it as following:
$$sum_k=1^4 frac y_k1+y_k+1+...+y_5 leq 1$$
$$sum_k=1^3 frac y_k+y_k+11+y_k+2+...+y_5 leq 1$$
$$ sum_k=1^2 frac y_k+y_k+1+y_k+21+y_k+3+...+y_5 leq 1$$
$$ frac sum_k=1^4 y_k1+y_5 leq 1$$
Is there any enumeration to combine the $4$ summations?
calculus summation
calculus summation
asked 1 hour ago
Dontknowanything
1,334518
asked 1 hour ago
Dontknowanything
1,334518
edited 52 mins ago
asked 1 hour ago
Dontknowanything
1,334518
asked 1 hour ago
Dontknowanything
1,334518
asked 1 hour ago
Dontknowanything
1,334518
1,334518
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2 Answers
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If I'm not mistaken, the following should be an equivalent expression:
$$forall i in 1,2,3,4: sum_k=1^5-i fracsum_j=k^k+i-1 y_j1+sum_j=k+i^5 y_j leq 1.$$
It is obviously way less clear and intuitive than the original, though.
answered 34 mins ago
Bastián Núñez
314
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+1. Please, do the same at the denominator. :)
– farruhota
17 mins ago
Missed it. Thanks!
– Bastián Núñez
7 mins ago
add a comment |Â
up vote
0
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What you gain it compactness you lose (massively) in comprehsion but:
$frac y_11 + y_2 + ... + y_5 + .... = sum_k=1^4 frac y_k1+sum_j=k+1^5 y_jle 1$
And $fracy_1+y_21+y_3+y_4+y_5+fracy_2+y_31+y_4+y_5+fracy_3+y_41+y_5 = sum_k=1^5-1frac y_k + y_k+11+sum_j=k+2^5 y_jle$
And then you get
$[sum_k=1^5-mfrac sum_i=k^k+m-1 y_i1+sum_j=k+m^5 y_j le 1]|_m=1^4$
But really, if a were a reader and I came across that... I'd hit you.
answered 28 mins ago
fleablood
61.7k22678
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
3
down vote
If I'm not mistaken, the following should be an equivalent expression:
$$forall i in 1,2,3,4: sum_k=1^5-i fracsum_j=k^k+i-1 y_j1+sum_j=k+i^5 y_j leq 1.$$
It is obviously way less clear and intuitive than the original, though.
answered 34 mins ago
Bastián Núñez
314
New contributor
Bastián Núñez is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
+1. Please, do the same at the denominator. :)
– farruhota
17 mins ago
Missed it. Thanks!
– Bastián Núñez
7 mins ago
add a comment |Â
up vote
3
down vote
If I'm not mistaken, the following should be an equivalent expression:
$$forall i in 1,2,3,4: sum_k=1^5-i fracsum_j=k^k+i-1 y_j1+sum_j=k+i^5 y_j leq 1.$$
It is obviously way less clear and intuitive than the original, though.
answered 34 mins ago
Bastián Núñez
314
New contributor
Bastián Núñez is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
+1. Please, do the same at the denominator. :)
– farruhota
17 mins ago
Missed it. Thanks!
– Bastián Núñez
7 mins ago
add a comment |Â
up vote
3
down vote
up vote
3
down vote
If I'm not mistaken, the following should be an equivalent expression:
$$forall i in 1,2,3,4: sum_k=1^5-i fracsum_j=k^k+i-1 y_j1+sum_j=k+i^5 y_j leq 1.$$
It is obviously way less clear and intuitive than the original, though.
answered 34 mins ago
Bastián Núñez
314
New contributor
Bastián Núñez is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
If I'm not mistaken, the following should be an equivalent expression:
$$forall i in 1,2,3,4: sum_k=1^5-i fracsum_j=k^k+i-1 y_j1+sum_j=k+i^5 y_j leq 1.$$
It is obviously way less clear and intuitive than the original, though.
answered 34 mins ago
Bastián Núñez
314
New contributor
Bastián Núñez is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited 8 mins ago
answered 34 mins ago
Bastián Núñez
314
New contributor
Bastián Núñez is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
answered 34 mins ago
Bastián Núñez
314
answered 34 mins ago
Bastián Núñez
314
314
New contributor
Bastián Núñez is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
Bastián Núñez is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Bastián Núñez is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
+1. Please, do the same at the denominator. :)
– farruhota
17 mins ago
Missed it. Thanks!
– Bastián Núñez
7 mins ago
add a comment |Â
+1. Please, do the same at the denominator. :)
– farruhota
17 mins ago
Missed it. Thanks!
– Bastián Núñez
7 mins ago
+1. Please, do the same at the denominator. :)
– farruhota
17 mins ago
+1. Please, do the same at the denominator. :)
– farruhota
17 mins ago
Missed it. Thanks!
– Bastián Núñez
7 mins ago
Missed it. Thanks!
– Bastián Núñez
7 mins ago
add a comment |Â
up vote
0
down vote
What you gain it compactness you lose (massively) in comprehsion but:
$frac y_11 + y_2 + ... + y_5 + .... = sum_k=1^4 frac y_k1+sum_j=k+1^5 y_jle 1$
And $fracy_1+y_21+y_3+y_4+y_5+fracy_2+y_31+y_4+y_5+fracy_3+y_41+y_5 = sum_k=1^5-1frac y_k + y_k+11+sum_j=k+2^5 y_jle$
And then you get
$[sum_k=1^5-mfrac sum_i=k^k+m-1 y_i1+sum_j=k+m^5 y_j le 1]|_m=1^4$
But really, if a were a reader and I came across that... I'd hit you.
answered 28 mins ago
fleablood
61.7k22678
add a comment |Â
up vote
0
down vote
What you gain it compactness you lose (massively) in comprehsion but:
$frac y_11 + y_2 + ... + y_5 + .... = sum_k=1^4 frac y_k1+sum_j=k+1^5 y_jle 1$
And $fracy_1+y_21+y_3+y_4+y_5+fracy_2+y_31+y_4+y_5+fracy_3+y_41+y_5 = sum_k=1^5-1frac y_k + y_k+11+sum_j=k+2^5 y_jle$
And then you get
$[sum_k=1^5-mfrac sum_i=k^k+m-1 y_i1+sum_j=k+m^5 y_j le 1]|_m=1^4$
But really, if a were a reader and I came across that... I'd hit you.
answered 28 mins ago
fleablood
61.7k22678
add a comment |Â
up vote
0
down vote
up vote
0
down vote
What you gain it compactness you lose (massively) in comprehsion but:
$frac y_11 + y_2 + ... + y_5 + .... = sum_k=1^4 frac y_k1+sum_j=k+1^5 y_jle 1$
And $fracy_1+y_21+y_3+y_4+y_5+fracy_2+y_31+y_4+y_5+fracy_3+y_41+y_5 = sum_k=1^5-1frac y_k + y_k+11+sum_j=k+2^5 y_jle$
And then you get
$[sum_k=1^5-mfrac sum_i=k^k+m-1 y_i1+sum_j=k+m^5 y_j le 1]|_m=1^4$
But really, if a were a reader and I came across that... I'd hit you.
answered 28 mins ago
fleablood
61.7k22678
What you gain it compactness you lose (massively) in comprehsion but:
$frac y_11 + y_2 + ... + y_5 + .... = sum_k=1^4 frac y_k1+sum_j=k+1^5 y_jle 1$
And $fracy_1+y_21+y_3+y_4+y_5+fracy_2+y_31+y_4+y_5+fracy_3+y_41+y_5 = sum_k=1^5-1frac y_k + y_k+11+sum_j=k+2^5 y_jle$
And then you get
$[sum_k=1^5-mfrac sum_i=k^k+m-1 y_i1+sum_j=k+m^5 y_j le 1]|_m=1^4$
But really, if a were a reader and I came across that... I'd hit you.
answered 28 mins ago
fleablood
61.7k22678
answered 28 mins ago
fleablood
61.7k22678
answered 28 mins ago
fleablood
61.7k22678
answered 28 mins ago
fleablood
61.7k22678
61.7k22678
add a comment |Â
add a comment |Â
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