logo
IRC Archive / Freenode / #math / 2010 / February / 05 / 6
oxeimon
I've spent over 3 hours on this
and that's today
I feel like I just don't have good intuition for functional analysis so far
vinay
oxeimon: beyond me, sorry. perhaps someone else can help you.
oxeimon: or perhaps you can help me. i don't know.
oxeimon: do you have a moment to take a look at my problem?
kingfishr
Sorry to spam, but I'm going to bump...Can someone give me slight prodding ? I'd like to show that if G is a finite group such that for n \in (naturals) there are at most n elements x such that x^n=e then G is cyclic.
Kasadkad
kingfishr, do you mean for all n?
kingfishr
yep sorry left that word out
Kasadkad
:\
oh
vinay, it's sort of a confusing question
vinay
Kasadkad: yeah.. i don't really understand it
Kasadkad: do you have any ideas?
BrainDeadGebril
kingfishr: where does this question come from?
Kasadkad
vinay, well i guess they're saying to show that if you have a rational function or an infinite series in z, z*, then differentiating formally with respect to z or z* is the same as using those partials they defined
so check dz/dz = 1, dz*/dz* = 1, and that the product rule and quotient rule and such still work
kingfishr
BrainDeadGebril, friend's hw...I guess he figured it out, but I can't stop thinking about it :\
vinay
Kasadkad: but how do "a" and "b" come into play?
Kasadkad
eh i think they're just saying what i said
OxE6
I like "c" and "d" better
Kasadkad
"differentiate formally with respect to z or z*"
vinay
i think that the point is to prove that z and z* behave as independent real variables
Kasadkad
i don't know what that means
BrainDeadGebril
hmm
kingfishr: ok, let me try to come up with something: if |G| is prime we are done, if not, then say |G|=pq
consider H={g^p | g belongs to G} as a subset of G
then there are at most p identity elements
sparr
Is there any database of polyhedra more suited to searching/indexing than the polyhedra family (archimedean, johnson, catalan, etc) pages on wikipedia and mathworld?
BrainDeadGebril
then consider remaining at least G-p elements, put them to the power q, there can be at most q additional identities, so that we have |G|-p-q non-identity elements though all elements are to the power |G| which forces them to be identities.
so we must have |G|=p+q as well as p,q | |G|
oh right, I see, sorry missunderstood statment in the problem arrived at contradiction that it is a group:)
I think I saw that in Herstein though
kingfishr
BrainDeadGebril, that might very well be the text they're using...they changed in the two years since I took it
tmorton
Anyone around familiar with basis polynomials for lagrange interpolation?
i'm trying to figure out why if you add all the basis polynomials, you always get 1
Kasadkad
kingfishr: take g in G with maximal order n
hm
ah
then <g> is the only cyclic subgroup of G with order n
so it's a normal subgroup
maybe it doesn't work
i wanted to say G/<g> will still have your property
so it would be cyclic by induction
BrainDeadGebril
proof is simplier
as i remember how I done it, but I still can be wrong:)
Kasadkad
but i don't like that idea anymore, i didn't even use that g had maximal order
BrainDeadGebril
aha, I found it
but it assumes it's abelian
kingfishr
screw it...I'll ask my friend later
I'm so tired of thinking about it
i hate not knowing though
BrainDeadGebril
I am awake for 24 hours so I am not of very help either
very much of*
asn
I have trouble understanding basic sequence convergence (|a_n -a| < x , etc.). Anyone knows some good reading on it (probably with graphic representation)?
freeboot
by x, you mean some epsilon?
oxeimon
sorry to spam, but I'm still stuck on this
if X is a normed vector space, and X* is separable, then X is separable...why?
http://mathbin.net/41759
asn
freeboot: (exactly, I just didn't have a greek epsilon here :P)
(and I thought that the absolute value would give what I meant away)
b4ry0n
asn: try this: http://en.wikibooks.org/wiki/Real_Analysis/Sequences it's some important stuff put in a nutshell...
asn
b4ry0n: okie, will read it! thanks
b4ry0n
but a real analysis book is of course more helpful
CESSMASTER
asn: Real Mathematical Analysis, by Charles Pugh, has a lot of illustrations
asn: also Calculus, by Michael Spivak
freeboot
just imagine a box that starts at some N and has a side from a+e to a-e and then stretches off to infinity. Convergence just says for any e you want, you can find an N so that all the points after N are in that box
asn
b4ry0n: I'm reading from a 'real analysis book' (my university one) it just that the formal definition of sequence convergence confuses me (shallow as it may sound, it probably is because it introduces many variables in inequalities)
Robba
How do I show that 1-x^2 <= e^(-x^2) <= 1/(1+x^2) ?
asn
freeboot: yes, that's what http://explainingmaths.wordpress.com/2008/12/12/quantifier-packaging-when-teaching-convergence-of-sequences/ explains, by introducing "absorption".
freeboot
Robba: did you try calculus?
__penguin__
what's calculus
Robba
I dont think I'm allowed to use the series expansion.
freeboot
no. first and second derivative
asn
but still when I see convergence in 'real' examples, I really don't understand where the n_o and epsilon's come from.
CESSMASTER: will try to find a copy of them, they seem interesting
freeboot
e^(-x^2) +x^2 -1 ... find the minimum of that. if it's > 0 then the equality holds
b4ry0n
asn: actually there is an awesome algebra and calculus book by R. Wuest. with lots of proofs and examples Unfortunately it is in german, and i'm not sure, if there is a translation...
CESSMASTER
asn: spivak is widely used, a copy should be easy to find
asn
"Calculus On Manifolds: A Modern Approach To Classical Theorems Of Advanced Calculus " ?
Robba
Spivak has another book called "Calculus"
freeboot
1+x <= e^x is true so you could just try that and save yourself some extra algebra
asn
yep found it. thanks.
CESSMASTER
asn: no, just Calculus
Robba
thanks, freeboot
__penguin__
what's calcĂșlus
asn
found it! I'll read from there then
CESSMASTER: Robba: does it also have convergence examples?
Robba
Yes.
asn
like, find if a_n = (3*n + 5)/2n converges?
nice, thanks.
Robba
There's a whole chapter on infinite sequences and series
oxeimon
is it true that any continuous functional on a closed subspace of a normed vector space achieves its minimum and maximum on that space?
tmorton
Anyone around familiar with basis polynomials for lagrange interpolation?
asn
thanks Robba, CESSMASTER and freeboot :)
tmorton
i'm trying to figure out why if you add all the basis polynomials together, you always get 1
CESSMASTER
oxeimon: it need not achieve a maximum or minimum at all
oxeimon
well if the functional is continuous, then it's bounded
right?
actually this seems ot suggest it does
well hold on hmm
okay, nevermind, I generalized too much
but it holds if the function is a norm
:-D
BrainDeadGebril
kingfishr: but doesn't that condition give just that number of solutions to x^n=I for n<|G| is |G|-Euler Totient(|G|) by just simply counting them? That provides us with Euler Totient(|G|) of generating elements, which is kinda enough!
(by counting over all divisors of |G|)
oxeimon
CESSMASTER: any chance you could take a look at my problem? o.o
CESSMASTER
i will probably look at it and immediately fall asleep
oxeimon
lol
CESSMASTER
but link it
oxeimon
well, in case you don't....http://mathbin.net/41759
freeboot
I completely forget how any of that goes.
CESSMASTER
surprise it is too late to be doing functional analysis
good night
oxeimon
:-(
night
freeboot
night
oxeimon
thanks anyway
kingfishr
BrainDeadGebril, I don't follow the beginning...how does the condition lead to the number of solutions being |G| - phi(|G|)?
BrainDeadGebril
inclusion-exclusion over divisors
vinay
would this be the place for formal language theory questions? or is there a better channel for that?
BrainDeadGebril
I am sorry, I gtg to a lecture, coincidentally on groups rings and modules
kingfishr
BrainDeadGebril, thanks, I'll think about it
enjoy
vinay, ask...I'm so much better at that than algebra :)
vinay
hehe, ok
here's the problem:
Teknomancer
morning
vinay
Say that string x is a prefix of string y if a string z exists where xz = y and that x
is a proper prefix of y if in addition x != y.
n each of the following parts we define
an operation on a language A. Show that the class of regular languages is closed
under that operation.
NOPREFIX(A) = { w in A|no proper prefix of w is a member of A}
it seems that it would be wise to start with a DFA that recognizes A for this problem rather than tatking the regex route, right?
kingfishr
vinay, pretty sure, yeah
vinay
alright... so we start with the DFA. now we want to construct a DFA such that the goal states represent all the ways to generate A without generating a prefix of A first, right?
oh, this is really easy with an NFA isn't it?
we start with an NFA with one start state and one goal state. we then remove any outward arrows from the goal state
then we're done, right?
kingfishr
vinay, yep :)
vinay
awesome :)
kingfishr
vinay, it's no fun if you solve it by yourself
vinay
that was far easier than i was making it
hehe, well i've got 2 more if you wanna have some fun :)
don't feel obligated to work on this one - i haven't thought about it much myself yet. but if you're interested, the next one is NOEXTEND(A) = { w in A|w is not the porper prefix of any string in A}
kingfishr
vinay, I see the solution...you shouldn't have any problem with that one.
vinay
the gears are turning...
ah.. start w/ NFA w/ 1 goal state, remove any arrows that start in the goal state and point back into the goal state
that does it :)
kingfishr: here's the last one, which is one i've actually thought about a bit and haven't yet come up with a solution. Let A be any language. Define DROP-OUT(A) to be the language containing all strings that can be obtained by removing one symbol from a string in A.
asn
wee, finally I understood convergence! Having a nice book makes a total difference.
vinay
Thus, DROP-OUT(A) = {xz | xyz in A where x, z in Epsilon*, y in Epsilon}. Show that the class of regular languages is closed under the DROP-OUT operation
kingfishr
vinay, sigma* right...
vinay
whoops, i'm sorry
tired :). yes. x, z in Sigma*, y in Sigma
iSchool
hey
vinay
in epsilon* would be quite different huh :P
iSchool
Fiend two numbers who's sum is 7 and who's product is a maximum
kingfishr
vinay, you would have defined the language with just the empty string, so it would make the problem pretty easy :)
vinay
kingfishr: hehe yes
kingfishr
vinay, again easy
vinay, wait...no my solution doesn't work. Thinking...
vinay, ok got it. Let me know if you have trouble.
vinay
kingfishr: could you give me a hint, perhaps? i've thought about this one and nothing is coming yet
kingfishr
vinay, regular languages are closed under union
vinay
so... do we union together all the possible ways to create A, dropping out one character?
meaning we drop out one character each time we have a concatenation, keep things the same every time we have a star
kingfishr
vinay, I think that works. Remember the first and last ones are special cases...I was thinking about it in terms of NFAs
vinay
yeah.. this was kinda the line of thought that i went down, but it gets messy
i.e. if we have (a U b)*
hm
maybe i need to think about this in NFA land
take our NFA, remove an arrow. take the NFA, remove a different arrow. union all these together
but i don't think that works...
insert "do this for all arrows in the NFA" before "union all these together"
kingfishr
vinay, you have to do more than remove arrows
vinay
oh.. insert epsilon move
hm.. i'm not sure if that works, either...
kingfishr
vinay, oh hmm you're right...loops complicate things
vinay
yeah...
when we have a loop, we want a no-op, right?
DROP-OUT(a*) = DROP-OUT(a*)
er
DROP-OUT(a*) = a*, i think
kingfishr
vinay, yes
vinay
DROP-OUT(a) = DROP-OUT(epsilon)
DROP-OUT(ab) = DROP-OUT( (a U b ))
kingfishr
vinay, DROPOUT(a) = epsilon
vinay
err.. i did it again
both times, remove the drop-out on the right hand side
kingfishr
vinay, yep yep
vinay
DROP-OUT(a U b) = epsilon
all of these things only hold true for a, b primitives, though. not a, b regex
i wonder if we can generalize these to regex?
DROP-OUT(R*) = R*, but we can't do the same thing for concatenation
kingfishr
vinay, that's not actually true
vinay
kingfishr: which part?
kingfishr
vinay, DROPOUT(w*) = w*DROPOUT(w)w*...i think
Nece228
hi, how to how much is -2sin135 ?