User talk:LBehounek

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[edit] Welcome

Hey Libor, Looks like you know what you are doing, but usually new users are given "welcome" messages and the like, and I didn't want you to be an exception. Anyway, thanks a lot for all of your help to wikipedia so far. I'm excited seeing all the work you are doing, and to see your to do list on your user page. There are a few mathematicians around, some of whom understand fuzzy logic a little, so don't hesitate to ask if you have any questions. The talk pages at Wikipedia:WikiProject Mathematics are a good place to look for help specific to how wikipedia approaches math. But again, you seem to know whats up. Let me know if you have any questions. And here is a formulaic welcome, just in case you want some links to the basics.

Welcome!

Hello, LBehounek, and welcome to Wikipedia! Thank you for your contributions. I hope you like the place and decide to stay. Here are some pages that you might find helpful:

I hope you enjoy editing here and being a Wikipedian! Please sign your name on talk pages using four tildes (~~~~); this will automatically produce your name and the date. If you need help, check out Wikipedia:Questions, ask me on my talk page, or place {{helpme}} on your talk page and ask your question there. Again, welcome!  Smmurphy(Talk) 03:34, 19 February 2007 (UTC)

Thank you for welcoming me to Wikipedia, and thanks for your kind words. I've been lurking about for some time (made some 20 smaller edits to English and Czech Wikipedia during the year before my registration), but since Wikipedia has become so immensely useful (esp. after it reached several hundred thousand articles), I feel obliged to contribute more significantly — and I do enjoy it a lot. LBehounek 10:54, 19 February 2007 (UTC)

[edit] capacity =? fuzzy membership function

I'm wondering if you had a definition of a fuzzy membership function handy and could verify that these two things are the same. A capacity in decision theory is simply a function from the set of subsets of some set into [0,1] which is monotone and is normalized (ie \empty \rightarrow 0, \Omega \rightarrow 1). Clearly this is a generalization of a probability measure, where the probability axiom of countability is weakened. Would you say that this is the same as a membership function? I put this into the membership function, so that I could write an article on the Choquet integral. Perhaps there is a fuzzy integral parallel of this. If so, don't hesitate to pitch in or give me a hint where to look. Thanks, Smmurphy(Talk) 03:49, 25 March 2007 (UTC)

A membership function on a set X is just any mapping X → [0, 1]. (The definition in the article on membership functions a bit unclear and I'm going to make a small edit right now to make it clearer; unfortunately I am currently too busy for larger edits which should be made there.) So capacities are a special kind of membership functions (capacities are defined on powersets and must be normalized and monotone; membership functions need not satisfy any of these conditions; often they are not even required to take values in [0, 1], but just in any fixed lattice or poset). In any case, your text on capacities as an important example of membership functions is definitely correct.
Since capacities are special membership functions (on the powerset of Ω), a capacity can be viewed as a fuzzy set of subsets of Ω. Because of this connection to fuzzy sets, capacities are also called fuzzy measures. Wikipedia contains a stub on fuzzy measure theory, which may be a suitable place for some info on capacities (you may consider moving a part of your text there, leaving just one or two sentences with a link in the article on membership functions). -- LBehounek 20:00, 27 March 2007 (UTC)
OK thanks, that makes sense. The leads in both articles are a bit off, then. I guess I'm not going to work on it for a couple days at least, but I'm taking a course in decision theory, and I'll probably try to work on that stuff soon. Do you think that there should be seperate articles for fuzzy measure, fuzzy measure theory, and capacity? Best, Smmurphy(Talk) 04:18, 28 March 2007 (UTC)
Yes, I think it will be good to have three separate articles, after enough contents in which they should differ is written (until then, I'd prefer redirects). But surely the three topics are different, since the agenda of fuzzy measure theory is not just the notion of fuzzy measure itself; and even though capacities coincide with fuzzy measures as mathematical structures, they surely have different motivation, applications, history, references, etc. Looking forward to your edits of capacity-related articles, LBehounek 17:00, 28 March 2007 (UTC)
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