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Comments:
<0> moin
<1> I just have to pick one
<2> hellop: I'd say that inf *can* be a limit, but it *cannot* be a "point" ;)
<1> it doesn't really matter which.. imho
<3> if we're talking about real numbers here (which I believe we are)
<4> hellop: you should probably read an intro to topology, it will possibly clarify this stuff for you
<1> int-e yes R
<4> or make them even more confusing :)
<5> heh
<2> Should one say "limit *at* infinity"?
<3> same for lim {x -> oo} ... that's also a special notation with its own definition.
<4> if you're talking about R, then infinity isnt in R, so the question "is it a limit point" isnt well-posed for your previous definition
<1> _llll_ great sounds good to me.
<4> so you can either change the definition, or move to a bigger set which does include infinity
<0> just a quick question reguarding the poincare conjuncture, isnt is possible to place a loop on a torus which retracts to a point? i mean as long as its not winding around the torus directly. or is the conjucture stating that any possible loop has to be retractable to a point?
<4> bourbaki: you're tlaking about the fundamental group of the tourus, which is Z(+)Z, that isnt the same thing as the poincare conjecture at all...
<1> So for this one: If A is infinite, then A has a limit point. I can say no, e.g. positive integers.
<0> i thought that the conjutrue says that a l***o around every sphere homoemorphic object is when pulled retracting to a point
<1> because an open interval cannot contain infinity
<1> I prefer to think that inf can be a "limit point" or a limit like in Calc 1 the teacher says, Whats the limit? You say pos infinity. and it's correct.
<6> Aldaron, "goes to" sounds better to my ears
<7> bourbaki: No, it says that the *only* 3-manifold that has trivial fundamental group is the 3-sphere. But note that a 3-sphere is not an ordinary sphere; an ordinary sphere is a 2-sphere.
<0> hellop well im just referring to what the wikipedia page says maybe im not correct on the stuff here though :)
<2> Volatile: yeah.. Y goes to inf when x..
<4> but a torus isnt homoemorphic to a 3-sphere
<0> sure
<0> but what i mean is i can place such a string on a torus as stated on the wikipedia page such as its retracting to a point
<6> homoemorphic souds like something naughty...
<4> yes but that isnt really relevent
<0> http://en.wikipedia.org/wiki/Poincar_conjecture
<4> on any space you can find a loop that is contractible to a point, just take a constant loop
<0> _llll_ ok so i guess the "intuitive" description is a bit fragged
<0> right
<1> Volatile how about homomorphic bijection which is onto U
<4> i think their description is fine, but you're reading it backwards
<6> hellop, hehe
<0> ah i see it says each loop
<8> How does on take an nth order cotangent of something?
<4> they say "if *all loops can be contracted*, then we have a 3-sphere"
<0> all possible loops on the manifold have to be able to bend to a point
<0> yes i just saw it :) sry
<6> hellop, I see a whole new dimension of math now...
<6> s/of/to/
<0> i bet that this is also interesting for m theorists isnt it?
<4> possibly, although they tend to want manifolds of higher dimensions than 3
<0> ah right but it might be interesting to apply some of these ideas to branes and such
<4> apparnetly the central question in M-theory is "what does M stand for?"
<0> the way the "rubberbands"/ strings would behave on the brane would tell you something about the brane itself
<0> hehe :) true
<5> It's interesting to note this case was the only one remaining.
<0> if you have a plane
<0> and you have a simple mapping into R^2 for this plane
<0> which also enables you to have a simple distance function on this mapping
<5> what does mean "simple"?
<0> is it then possible if you want to have a hole in the plane
<0> ie the plane is just like this
<0> plane(x,y) = (x,y,0)
<8> Is cot^2(a) really equal to (cot(a))^2?
<0> and if i want to have a hole in the plane
<0> then i could puncture a hole in there and widen the hole
<3> heath: sadly, that's a common notation.
<8> int-e: Which one?
<3> heath: using trig^n(x) for trig(x)^n where trig is a trigonometric function
<0> as long as this hole is starlike i could just use the bending of the hole to bend the path i had (the shortest direct path) around the hole to get the shortes path around the hole right?
<8> int-e: But are they equal or not?
<6> hellop, it's simply a shorthand notation
<6> eh
<6> heath
<8> int-e: Or maybe I should say: Is sine squalred theta equal to the sine of theta squared?
<6> trig^n(x) is another way of writing (trig(x))^n
<8> Ok.
<3> isn't that what I wrote?
<6> and is not neccesari equal to trig(x^n)
<8> So my next question is, how does one calculate something with a fractional exponent?
<6> int-e, yes, but not in the same exact words
<3> the bad thing about it is that it clashes with the f^n(x) = f(f...f(x)...) notation, which is different.
<6> ah
<8> int-e: That was my confusion.
<6> well, that's life, I guess
<9> and even worse with f^(-1)
<6> like orthogonal matrices, yes?
<3> heath: sin(sin(x)) does not equal sin(x)^2, but sin^2(x) usually means sin(x)^2
<3> Catfive: oh yes.
<8> int-e: Got it.
<10> Is it easy to find the extrema of f(x,y) = sin(x/y)?
<3> yes
<10> A bit too short for me :)
<3> you asked a yes-or-no question
<3> what did you expect?
<10> Ok. How?
<3> what are the maximum and minimum values of sin? are they realized? where?
<10> The usual terminology of extremum denotes the point(s) where the function is locally minimum or maximum.
<10> And of course, the value(s).
<3> I know.
<10> Sorry, I was unsure.
<3> I think the problem is easy as long as you avoid using calculus. if you use calculus it becomes somewhat awkward but still quite doable.
<3> (where using calculus means taking the two partial derivatives and setting both to 0)
<4> % D[Sin[x/y],x]
<11> _llll_: Cos[x/y]/y
<3> % D[Sin[x/y],y]
<11> int-e: -((x*Cos[x/y])/y^2)
<3> could be worse
<3> but just solving f(x,y)=1 and f(x,y)=-1 is easier.
<10> int-e, I'm not clear why you can avoid calculus obviously.
<3> delta: because you know that -1 <= sin(x) <= 1.
<10> int-e, are you looking for absolute extremum or every extremum?
<3> I was looking for global extrema.
<10> Which is an easuer question.
<10> Not that difficult though, thx for your help :)
<3> ok, I would use a little bit of calculus then, and look at df / dx to show that the other points aren't local extrema either
<10> heh
<3> I can still avoid looking at df / dy, which is nice :)
<12> delta : ***embling coirse problems ? or exams ?
<3> delta: some problem like this might be awarding people who haven't forgotten the basic definition of maximum and minumum though :)
<10> kmh, who knows? :)
<10> int-e, that would be unfontunate :)
<12> delta : isn't that obvious ?
<10> No idea.
<12> lol
<10> ;)
<12> well unless you are not in state of mental confusion, i'd ***ume you know (i.e. who = you)
<3> delta: I've seen a few people who, when trying to find extrema, would always start with taking derivatives ...
<12> int-e : automatic reflexes
<12> int-e : haha they (in)properly conditioned by their teacher
<12> and it's is a side effect of problem sets, that only deal with the material just covered
<10> int-e, I've seen a few ppl who, when trying to find extrema, look for global ones only.
<3> delta: If I attended your course I knew that you wanted local extrema as well. But I'm used to that qualifier being stated explicitely.
<10> Ok, ok :)
<10> maple is a surprising software.
<10> % help
<11> delta: "See http://documents.wolfram.com/v5/ for detailed Mathematica help."
<3> in a good way or in a bad way? *chuckles*
<12> int-e : did you see cale mathematica plot a few days back ?
<3> no
<12> the sin(x+Pi/2)-cos(x) one ?
<3> no, I didn't but it probably looked spiky due to rounding errors?
<12> Plot[Sin[x+Pi/4]-Cos[x],{x,540,560}]
<12> try that one loos amazing :)
<12> loos = looks
<3> haha
<12> nice isn't it ?:)
<10> int-e, very bad but maybe there is a magic to apply.
<3> very interesting.
<3> (I'm seing a sin-like wave with amplitude about 3/4)
<12> if you enforce an Eval of the complete function first you get the expected plot
<12> int-e : yes it is quite a deceiving plot (in particular if one ignores/overlooks the scale)
<3> did you mean pi/2?
<3> evil kmh
<12> oh yes
<12> sorry
<12> i constantly mistype it
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