What is this mathematical/set notation?
May 11, 2005 2:16 AM   Subscribe

Two mathematical syntax questions--one easy, one astoundingly maybe not so complex

Ok, here are the guts of the easy one:
What does this mean?

l?{l:l>BID}
(or if that doesn't show up)
l (epsilon) {l:l >BID}

the epsilon is the typical "element L is a member of the following set"...BID is the bypass ID (related to the problem)--I presume it's an integer. L is also an integer...but what is the colon, exactly? I'm used to seeing a " | " in that place for a "such that", but this is throwing me for a loop. Would the significance change if it were a semicolon? The paper does have it that way once, and with a colon all other times (I'm used to random typos in academic papers these days)...


Ok, second question.
If one were to come across the following in an equation, how would you interpret it?

constant*( Max ( 0, ( variable/constant ) - constant ))
This is all within a quadruple summation, so there are actually a lot of them. The only reasonable suggestion I've heard yet is that it has something to do with setting the derivative equal to zero--finding the max, etc.

It's from this paper, if it helps:
Efficient Multilevel MINLP Strategies for Solving Large Combinatorial Problems in Engineering, by A. Sorsak, S. Kravanja, and Z. Kravanja, Computers and Engineering.
posted by hototogisu to Science & Nature (8 answers total)
 
The first question: yes, it's the same thing.

I don't understand your second question. Do you not understand the max operator? This merely specifies that the multiplier is always less than or equal to 0. If (x/k)-k > 0, then 0 is replaced.
posted by grouse at 3:11 AM on May 11, 2005


That makes sense, grouse--I'd just never seen it written like that (or at all, really).
I'm a 3rd year in a graduate course in system optimization, and I don't quite have some of the more math-stuff down yet.

Thanks a lot!
posted by hototogisu at 3:33 AM on May 11, 2005


It looks like functional notation. There's a branch of programming languages where algorithms can be 'proved'. This is the notation that they are commonly represented in.

For example, "x [epsilon] {x:x>y}" means that, within the context of the algorithm, there exists at least one x, where x is greater than y.

This can be used in conjunction with "x [foxhead] {x:x>y}", which means that, within the context of the algorithm, every x is greater than y.

Don't know the proper name for 'foxhead'. It looks like a capital V with a horizontal bar through it. Or an inverted capital A, where the crossbar extends past the downstrokes.

Within the clause (the bit after the colon), any combination of conditionals can be used. This allows very complex systems to be described.
posted by veedubya at 4:13 AM on May 11, 2005


grouse, I'd express Max as something like-

x(y) [epsilon] {x,y:x>y}

Which (hopefully - it's been a long time) would mean, give x if x is greater than y, or else give y.
posted by veedubya at 4:18 AM on May 11, 2005


That "foxhead" is an inverted capital A. It means For All.
posted by stopgap at 7:57 AM on May 11, 2005


veedubya: Isn't ∀ called "for all", which is its meaning?

My understanding was that mathematicians felt like saving some blackboard time, and thus invented a bunch of symbols as abbreviations for common snippets of text.
posted by breath at 3:06 PM on May 11, 2005


Re the first question, I often see the set notation written with a colon if a vertical bar isn't handy. I don't think it means anything different, but I've always been under the impression that the vertical bar is the real notation and the colon is just a typographical approximation.

backwards E and upside-down A are qualifiers: the existential qualifier ("there exists an X such that ...") and the universal qualifier ("for all X, it is true that ...").
posted by hattifattener at 8:56 PM on May 11, 2005


stopgap and breath, you're right, and I'm embarassed. I took that stuff for two semesters, and it never occurred to me that it was actually an upside-down A. I thought it just looked like an upside-down A.

I must have been asleep during that lecture.
posted by veedubya at 1:22 AM on May 12, 2005


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