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oxeimonI've spent over 3 hours on this
oxeimonand that's today
oxeimonI feel like I just don't have good intuition for functional analysis so far
vinayoxeimon: beyond me, sorry. perhaps someone else can help you.
vinayoxeimon: or perhaps you can help me. i don't know.
vinayoxeimon: do you have a moment to take a look at my problem?
kingfishrSorry 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.
Kasadkadkingfishr, do you mean for all n?
kingfishryep sorry left that word out
Kasadkadvinay, it's sort of a confusing question
vinayKasadkad: yeah.. i don't really understand it
vinayKasadkad: do you have any ideas?
BrainDeadGebrilkingfishr: where does this question come from?
Kasadkadvinay, 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
Kasadkadso check dz/dz = 1, dz*/dz* = 1, and that the product rule and quotient rule and such still work
kingfishrBrainDeadGebril, friend's hw...I guess he figured it out, but I can't stop thinking about it :\
vinayKasadkad: but how do "a" and "b" come into play?
Kasadkadeh i think they're just saying what i said
OxE6I like "c" and "d" better
Kasadkad"differentiate formally with respect to z or z*"
vinayi think that the point is to prove that z and z* behave as independent real variables
Kasadkadi don't know what that means
BrainDeadGebrilkingfishr: ok, let me try to come up with something: if |G| is prime we are done, if not, then say |G|=pq
BrainDeadGebrilconsider H={g^p | g belongs to G} as a subset of G
BrainDeadGebrilthen there are at most p identity elements
sparrIs there any database of polyhedra more suited to searching/indexing than the polyhedra family (archimedean, johnson, catalan, etc) pages on wikipedia and mathworld?
BrainDeadGebrilthen 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.
BrainDeadGebrilso we must have |G|=p+q as well as p,q | |G|
BrainDeadGebriloh right, I see, sorry missunderstood statment in the problem arrived at contradiction that it is a group:)
BrainDeadGebrilI think I saw that in Herstein though
kingfishrBrainDeadGebril, that might very well be the text they're using...they changed in the two years since I took it
tmortonAnyone around familiar with basis polynomials for lagrange interpolation?
tmortoni'm trying to figure out why if you add all the basis polynomials, you always get 1
Kasadkadkingfishr: take g in G with maximal order n
Kasadkadthen <g> is the only cyclic subgroup of G with order n
Kasadkadso it's a normal subgroup
Kasadkadmaybe it doesn't work
Kasadkadi wanted to say G/<g> will still have your property
Kasadkadso it would be cyclic by induction
BrainDeadGebrilproof is simplier
BrainDeadGebrilas i remember how I done it, but I still can be wrong:)
Kasadkadbut i don't like that idea anymore, i didn't even use that g had maximal order
BrainDeadGebrilaha, I found it
BrainDeadGebrilbut it assumes it's abelian
kingfishrscrew it...I'll ask my friend later
kingfishrI'm so tired of thinking about it
kingfishri hate not knowing though
BrainDeadGebrilI am awake for 24 hours so I am not of very help either
BrainDeadGebrilvery much of*
asnI have trouble understanding basic sequence convergence (|a_n -a| < x , etc.). Anyone knows some good reading on it (probably with graphic representation)?
freebootby x, you mean some epsilon?
oxeimonsorry to spam, but I'm still stuck on this
oxeimonif X is a normed vector space, and X* is separable, then X is separable...why?
asnfreeboot: (exactly, I just didn't have a greek epsilon here :P)
asn(and I thought that the absolute value would give what I meant away)
b4ry0nasn: try this: it's some important stuff put in a nutshell...
asnb4ry0n: okie, will read it! thanks
b4ry0nbut a real analysis book is of course more helpful
CESSMASTERasn: Real Mathematical Analysis, by Charles Pugh, has a lot of illustrations
CESSMASTERasn: also Calculus, by Michael Spivak
freebootjust 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
asnb4ry0n: 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)
RobbaHow do I show that 1-x^2 <= e^(-x^2) <= 1/(1+x^2) ?
asnfreeboot: yes, that's what explains, by introducing "absorption".
freebootRobba: did you try calculus?
__penguin__what's calculus
RobbaI dont think I'm allowed to use the series expansion.
freebootno. first and second derivative
asnbut still when I see convergence in 'real' examples, I really don't understand where the n_o and epsilon's come from.
asnCESSMASTER: will try to find a copy of them, they seem interesting
freeboote^(-x^2) +x^2 -1 ... find the minimum of that. if it's > 0 then the equality holds
b4ry0nasn: 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...
CESSMASTERasn: spivak is widely used, a copy should be easy to find
asn"Calculus On Manifolds: A Modern Approach To Classical Theorems Of Advanced Calculus " ?
RobbaSpivak has another book called "Calculus"
freeboot1+x <= e^x is true so you could just try that and save yourself some extra algebra
asnyep found it. thanks.
CESSMASTERasn: no, just Calculus
Robbathanks, freeboot
__penguin__what's calcúlus
asnfound it! I'll read from there then
asnCESSMASTER: Robba: does it also have convergence examples?
asnlike, find if a_n = (3*n + 5)/2n converges?
asnnice, thanks.
RobbaThere's a whole chapter on infinite sequences and series
oxeimonis it true that any continuous functional on a closed subspace of a normed vector space achieves its minimum and maximum on that space?
tmortonAnyone around familiar with basis polynomials for lagrange interpolation?
asnthanks Robba, CESSMASTER and freeboot :)
tmortoni'm trying to figure out why if you add all the basis polynomials together, you always get 1
CESSMASTERoxeimon: it need not achieve a maximum or minimum at all
oxeimonwell if the functional is continuous, then it's bounded
oxeimonactually this seems ot suggest it does
oxeimonwell hold on hmm
oxeimonokay, nevermind, I generalized too much
oxeimonbut it holds if the function is a norm
BrainDeadGebrilkingfishr: 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!
BrainDeadGebril(by counting over all divisors of |G|)
oxeimonCESSMASTER: any chance you could take a look at my problem? o.o
CESSMASTERi will probably look at it and immediately fall asleep
CESSMASTERbut link it
oxeimonwell, in case you don't....
freebootI completely forget how any of that goes.
CESSMASTERsurprise it is too late to be doing functional analysis
CESSMASTERgood night
oxeimonthanks anyway
kingfishrBrainDeadGebril, I don't follow the does the condition lead to the number of solutions being |G| - phi(|G|)?
BrainDeadGebrilinclusion-exclusion over divisors
vinaywould this be the place for formal language theory questions? or is there a better channel for that?
BrainDeadGebrilI am sorry, I gtg to a lecture, coincidentally on groups rings and modules
kingfishrBrainDeadGebril, thanks, I'll think about it
kingfishrvinay, ask...I'm so much better at that than algebra :)
vinayhehe, ok
vinayhere's the problem:
vinaySay that string x is a prefix of string y if a string z exists where xz = y and that x
vinayis a proper prefix of y if in addition x != y.
vinayn each of the following parts we define
vinayan operation on a language A. Show that the class of regular languages is closed
vinayunder that operation.
vinayNOPREFIX(A) = { w in A|no proper prefix of w is a member of A}
vinayit seems that it would be wise to start with a DFA that recognizes A for this problem rather than tatking the regex route, right?
kingfishrvinay, pretty sure, yeah
vinayalright... 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?
vinayoh, this is really easy with an NFA isn't it?
vinaywe start with an NFA with one start state and one goal state. we then remove any outward arrows from the goal state
vinaythen we're done, right?
kingfishrvinay, yep :)
vinayawesome :)
kingfishrvinay, it's no fun if you solve it by yourself
vinaythat was far easier than i was making it
vinayhehe, well i've got 2 more if you wanna have some fun :)
vinaydon'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}
kingfishrvinay, I see the shouldn't have any problem with that one.
vinaythe gears are turning...
vinayah.. start w/ NFA w/ 1 goal state, remove any arrows that start in the goal state and point back into the goal state
vinaythat does it :)
vinaykingfishr: 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.
asnwee, finally I understood convergence! Having a nice book makes a total difference.
vinayThus, 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
kingfishrvinay, sigma* right...
vinaywhoops, i'm sorry
vinaytired :). yes. x, z in Sigma*, y in Sigma
vinayin epsilon* would be quite different huh :P
iSchoolFiend two numbers who's sum is 7 and who's product is a maximum
kingfishrvinay, you would have defined the language with just the empty string, so it would make the problem pretty easy :)
vinaykingfishr: hehe yes
kingfishrvinay, again easy
kingfishrvinay, my solution doesn't work. Thinking...
kingfishrvinay, ok got it. Let me know if you have trouble.
vinaykingfishr: could you give me a hint, perhaps? i've thought about this one and nothing is coming yet
kingfishrvinay, regular languages are closed under union
vinayso... do we union together all the possible ways to create A, dropping out one character?
vinaymeaning we drop out one character each time we have a concatenation, keep things the same every time we have a star
kingfishrvinay, I think that works. Remember the first and last ones are special cases...I was thinking about it in terms of NFAs
vinayyeah.. this was kinda the line of thought that i went down, but it gets messy
vinayi.e. if we have (a U b)*
vinaymaybe i need to think about this in NFA land
vinaytake our NFA, remove an arrow. take the NFA, remove a different arrow. union all these together
vinaybut i don't think that works...
vinayinsert "do this for all arrows in the NFA" before "union all these together"
kingfishrvinay, you have to do more than remove arrows
vinayoh.. insert epsilon move
vinayhm.. i'm not sure if that works, either...
kingfishrvinay, oh hmm you're right...loops complicate things
vinaywhen we have a loop, we want a no-op, right?
vinayDROP-OUT(a*) = DROP-OUT(a*)
vinayDROP-OUT(a*) = a*, i think
kingfishrvinay, yes
vinayDROP-OUT(a) = DROP-OUT(epsilon)
vinayDROP-OUT(ab) = DROP-OUT( (a U b ))
kingfishrvinay, DROPOUT(a) = epsilon
vinayerr.. i did it again
vinayboth times, remove the drop-out on the right hand side
kingfishrvinay, yep yep
vinayDROP-OUT(a U b) = epsilon
vinayall of these things only hold true for a, b primitives, though. not a, b regex
vinayi wonder if we can generalize these to regex?
vinayDROP-OUT(R*) = R*, but we can't do the same thing for concatenation
kingfishrvinay, that's not actually true
vinaykingfishr: which part?
kingfishrvinay, DROPOUT(w*) = w*DROPOUT(w)w*...i think
Nece228hi, how to how much is -2sin135 ?
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