mathwiz

About the calc question: sure. y=1. All the points are local maxima.

Ben Golub

strict local maxima! now it's harder, isn't it

sr6622

Ben, why don't you give an interview and see what happens...

Adcom: "So, what other schools are you applying too?
Ben: "Umm...I'm applying to that other, better tech school out in Cali. I think you might have heard of it."
Adcom: "I believe you are referring to Caltech..."
Ben: "Damn straight...."

[takes out Caltech flag and runs around room...]

Ben Golub

Hahaha. I love MIT, I would do no such thing. And if we have a flag, I haven't seen it ;-)

RobbCarr

Hehe, I thought of f(x)=1 as well however I made the assumption that you were referring to strict local maxima...eh in this case I would have to say yes...but I dont really have anything rigorous...but...of course the best place to start would be to consider the devils staircase...define...the middle third of the interval a,b not as being equal to ((f(b)-f(a))/2... but create a function such that...l=(2a+b)/3 m = (2b+a)/3 and so that f(l)=f(m)=(f(b)-f(a))/2>x ergh I hope I explained this well...I dont think I really did, basically instead of drawing a flat line for each ititeration of the devils staircase...you draw a parabola so that the end points are local maximas, so basically you have a function so that its set of local maxima are the cantor set. And it therefore has Cardinality c...because the cantor function has cardinality c...I suppose you could generalize this result a bit to say that any the set of local maxima of any function over the interval 0,1 has at most cardinality c.

RobbCarr

Actually in hindsight thats fairly rigorous...the end needs a little more explination...but eh

Ben Golub

Is it clear that this function is continuous?

RobbCarr

...Mmmm...making a small change...where I have f(b)-f(a)/2>x
make that >x for all x in l,m...and eh...I am fairly sure its continuous...but im not exactly sure, I guess I was kind of taking that for granted...and wow fast! Mm...ill write up something...better...in LaTeX tommorow and make a pretty post script file, however right now it is 2:44....and I slept for three hours last night (Number theory problem set...which I thought was due next week was actually due today[EPGY])...soooooo sleep time.

Ben Golub

: ) good luck.

RobbCarr

Well...www.wired-designs.net/localMaxima.pdf is a little better...I suppose its still not rigorously continuous...mmm...once I have some caffeine in my system I will do a little more...its become a bit of a personal vendetta

Ben Golub

It is not clear that the set of local maxima is indeed the Cantor set, nor do I understand the sentence "create a continuous function m = stuff, n = stuff and such that f(m) = f(n) = (f(b)−f(a))/2 > x for all x in (m, n)" (i.e. the sentence is nonsense as written).

Hint: it is hard to prove something false

RobbCarr

Are you sure it isnt true?...Eh I will rewrite it in a bit...I am not wonderful with mathematical writing...well writing in general, I have difficulties putting down what I am thinking of...mm...again basically the idea of that sentence is instead of an ititeration of the devils staircase being flat (constant) we create a function such that each step is a parabola with the endpoints being local maxima...eh I can work on it a bit after I finish school (online school). However I think it is very clear that it is the cantor set, as the cantor set operates on removing the middle thirds of the interval [0,1] and continue ad infinitum, the cantor set consists of all points not removed. In this case we have turned the middle third of an interval (a,b) into a parabola such that the endpoints are local maxima so the endpoints would be the points which would be left after a complete runthrough of the cantor set. http://en.wikipedia.org/wiki/Cantor_set also, it is possible we have a confusion on the definition of the devils staircase...Wikipedia and google both tell me different things for that, so I am guessing that there are multiple definitions.

RobbCarr

Maybe a better summary would almost be that...you are constructing the cantor set from the devils staircase by turning the middle third into a parabola such that the end points are strict local maxima and turning the middle third of the two parts left into a parabola again such that the endpoints are local maxima and continuing ad infinitum.

Ben Golub

Is every point in the Cantor set the endpoint of some interval removed during the construction? It seems you would have to show at least that, and I don't think it's true.

See if you can give the other way a whack -- try to prove no such function exists. Recall that a strict local maximum is a point x such there is some interval (a,b) containing x with f(x) > f(y) for all y in (a,b).

RobbCarr

Agh! what a horrible oversight on my part, consider x=1/4, part of the cantor set but not of the construction because as you remove thirds...eh oh well...and I definetly wasnt considering that a proof as much as...a heuristic argument I am fairily sure for it to be rigorous I would have to show that but eh...I am not certain it is true, clearly because the cantor set removes open sets, (1/3,2/3) from [0,1] etc etc...so the endpoints are obviously there but again something like 1/4 is part of the cantor set yet not covered in the construction...Oh well it was an interesting attempt. My original thought was actually no there was not...as far as proving that it does not exist....

Statement: Given a finite function of a Real Variable F, the set of points at which it assumes a strict local maxima are at most enumerable.

Proof: Consider the set A for the points at which the the function F a ssumes a strict local maximum, Let a sub n denote for all positive integers n the set of points such that, F(j)<F(x) holds for each points J not equal to X on the interval, ((x-1)/2,(x+1)/2). Clearly the set A sub n is isolated and therefore at most enumerable. However we know by definition that A = Sigma_n A sub N, it follows that the set A is at most enumerable.