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?

edited 52 mins ago

asked 1 hour ago

Dontknowanything

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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?

edited 52 mins ago

asked 1 hour ago

Dontknowanything

1,334518

add a commentÂ |Â

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?

edited 52 mins ago

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

edited 52 mins ago

asked 1 hour ago

Dontknowanything

1,334518

edited 52 mins ago

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

add a commentÂ |Â

2 Answers
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up vote
3
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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.

edited 8 mins ago

answered 34 mins ago

BastiÃ¡n NÃºÃ±ez

314

New contributor

+1. Please, do the same at the denominator. :)
â€“Â farruhota
17 mins ago

Missed it. Thanks!
â€“Â BastiÃ¡n NÃºÃ±ez
7 mins ago

add a commentÂ |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

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2 Answers
2

active

oldest

votes

2 Answers
2

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.

edited 8 mins ago

answered 34 mins ago

BastiÃ¡n NÃºÃ±ez

314

New contributor

+1. Please, do the same at the denominator. :)
â€“Â farruhota
17 mins ago

Missed it. Thanks!
â€“Â BastiÃ¡n NÃºÃ±ez
7 mins ago

add a commentÂ |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.

edited 8 mins ago

answered 34 mins ago

BastiÃ¡n NÃºÃ±ez

314

New contributor

+1. Please, do the same at the denominator. :)
â€“Â farruhota
17 mins ago

Missed it. Thanks!
â€“Â BastiÃ¡n NÃºÃ±ez
7 mins ago

add a commentÂ |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.

edited 8 mins ago

answered 34 mins ago

BastiÃ¡n NÃºÃ±ez

314

New contributor

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.

edited 8 mins ago

answered 34 mins ago

BastiÃ¡n NÃºÃ±ez

314

New contributor

edited 8 mins ago

answered 34 mins ago

BastiÃ¡n NÃºÃ±ez

314

New contributor

answered 34 mins ago

BastiÃ¡n NÃºÃ±ez

314

answered 34 mins ago

BastiÃ¡n NÃºÃ±ez

314

New contributor

BastiÃ¡n NÃºÃ±ez 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. :)
â€“Â farruhota
17 mins ago

Missed it. Thanks!
â€“Â BastiÃ¡n NÃºÃ±ez
7 mins ago

add a commentÂ |Â

+1. Please, do the same at the denominator. :)
â€“Â farruhota
17 mins ago

Missed it. Thanks!
â€“Â BastiÃ¡n NÃºÃ±ez
7 mins ago

+1. Please, do the same at the denominator. :)
â€“Â farruhota
17 mins ago

Missed it. Thanks!
â€“Â BastiÃ¡n NÃºÃ±ez
7 mins ago

add a commentÂ |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 commentÂ |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 commentÂ |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

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

add a commentÂ |Â

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