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I am interested in knowing if there is a name for this process:

Suppose I have positive reals $a_1,a_2,\ldots, a_k, b_1,b_2,\ldots, b_m$ such that $\sum_{i=1}^k a_i = \sum_{j=1}^m b_j.$

Then, I can come up with a sequence $c_1,c_2,\ldots, c_l$ where $k,m\leq l\leq k+m$ recursively as follows starting with $c_0 = 0.$

$$c_r = \min\left\{\min_{t: \sum_{i=1}^t a_i > c_{r-1}} \sum_{i=1}^t a_i, \min_{s: \sum_{j=1}^s b_j > c_{r-1}} \sum_{j=1}^s b_j \right\}$$

The $c_r$'s are an ordered list of partial sums.

Let us also define $d_r = c_r - c_{r-1},$ the difference sequence.

Is there a name for this construction of $c_r$'s or $d_r$'s?

I am working on a paper on network flows where I have to split flows in this particular way. It is easy to see what I mean from a picture but cumbersome to understand from the equation. If there is a well-known name for this, I could use it. Thanks.

Hedonist
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    Hey. I edited the question to make it clear. Is it clear now? – Hedonist Jan 09 '14 at 00:44
  • Thanks, the edit is helpful. Perhaps you might add some motivation and background as well. Why are you interested in this sequence? – j.c. Jan 09 '14 at 01:04
  • Hi j.c. I am working on a paper on network flows where I have to split flows in this particular way. It is easy to see what I mean from a picture but cumbersome to understand from the equation. If there is a well-known name for this, I could use it. Thanks. – Hedonist Jan 09 '14 at 01:09
  • That would be useful information to include when asking the question! See e.g. http://meta.mathoverflow.net/a/883/353 – j.c. Jan 09 '14 at 01:14
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    Are you just taking the union of the sets of partial sums? If not, please give an example. – Douglas Zare Jan 09 '14 at 02:09
  • Douglas: Yes, it is a list of set of partial sums, except that it is ordered. – Hedonist Jan 09 '14 at 06:07

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