Any I deas on how solve this? It's the first question of an ENS of Paris oral examination...

42 man!

A few years ago, sci would have solved this in 10 minutes

But now this board is so full of biologists and psychologists that it really lost all meaning

>implying an ecologist didn't btfo lifetimes worth of mathematicians.

link?

I still believe it is better to be a Kayne West than a STEM scientist but anyway...

It literally does not pay to solve those riddles

so why don't you do it?

which would be a correct deduction, yes

I didn't say something (you know it anyway)

You believe wrong. Life is about truth and beauty, not about sex and money you ingorant fuck
Also, it literally does. ENS students are paid about 1100 euros a month during their time at the school, and when they decide to go in the private sector are usually extremely succesful.

tell me more about white priviledge

uh?

no but really i'm disappointed sci

the stipend of 1000 european shekels is given to students from the international selection

actually only French student get it, I reckon

Did you mean School of Exercise & Nutritional Sciences?

But jokes aside, on how many major scientific events do you see non-whites, non-Japanese?

ens.fr/admission/selection-internationale/

yea but they have to reimburse afterwards while frenchmen can work 6 years for the gvt (including getting a phd) and then its ok

very few asians at the very top in France, there are many in classes prépa (so sort of prestigious), but very few in the big ones. Very top (ens paris, polytechnique, mines, centrale) is basically white. the only noticeable minority i'd say is jews (a lot in my class and bretty good), maybe morrocans who come to france because their education system is shit but that's only because they already take the top of morrocans to start with

I have a theory that every non-country (so most of them) should completely disband their higher educational system and spend the money on debt, whores and beer.

It is self-deception, more or less.

youtube.com/watch?v=QauoO0j9Y9Y

Try to ask here OP : forum.prepas.org/viewforum.php?f=3&sid=548aa9e0abec8c9ff970827fa9273b46

Or you can ask on AoPS.

Geometrically, we can think of this as saying that for any sequence of moves [math]\varepsilon_1, \cdots, \varepsilon_k\in \{ \mathrm{forward},\mathrm{backward} \}[/math], there exists an angle [math]\theta[/math] such that starting from the origin and making the corresponding moves in the complex plane will let you escape from the circle of radius [math]\sqrt{n}[/math].

Perhaps try proving by induction?

Op is obviously French and that forum is filled with people who are into the schools that have the OP question as part of their entrance exam

solve what?
that is not a question, that is two matheatical statements in the form of an equation and an inequality.

You have to ask something in order for it to be a question

It's pretty obvious he's asking for a proof of the inequality.

my math teacher roams these places he's gonna be disapointed in me if he sees i cant solve something :(

But i'm probably going to anyway. you guys are disapointting with your shitty propositions
sci is shit desu

why do you expect somebody help you for free?
just because you asked?

welcome to the real world

people here won't help me because they aren't able to, not because they don't want to. just look at the shitty proposition like 'try induction'. quit being stupid please

they could solve this but they have no incentives

prove it. or maybe is 'prove inf f(x) >= a by proving that for all x f(x) >= a' an interesting idea to you?

Si t'as déja passé les concours il s'en fout
Si tu entres en MP y'a pas de honte a bloquer la-dessus (même si tu as fini ta MP d'ailleurs)

Je rentre en mp* ouais.
j'ai trop de mal pour ces exos 'sans thèmes' de ce genre, ou des trucs en arithmétique elementaire et ca m'angoisse pour ulm
t'en penses quoi ?

all righty, I know how to solve but I am too lazy to actually type it, where is my 1000 eur stipend?

then you get to try to solve the second question:
get an upper bound on alpha_n strictly lower than n

Perso pour l'arithmétique j'avais des problèmes en sup (pas fait spé math au lycée) mais en spé tu mélanges ça avec de l'algébre (des groupes) et ça passe bien mieux, en tout cas pour moi.

Et les exos "sans thème" de ce genre j'ai toujours trouvé ça difficile aussi, le truc en général c'est de trouver l'astuce/par ou passer, c'est en particulier galère pour les inégalités ou égalités à prouver comme ça sans rien parce que tu peux aborder ça de 25 façons (Topologiquement, espaces euclidiens, analyse et inégalité de convexité/concavité) et c'est assez emmerdant en oral quand tu es en temps limité. Après si tu cherches de manière pertinente sans forcément tomber sur le bon truc l'examinateur te réoriente et tu n'es pas pénalisé.

Je sais même pas si tu peux résoudre l'exo en OP avec juste le programme de sup pêh.

ouais mais j'ai fait l'algèbre de spé donc je pense que ca passe qd même
mais ouais je pense comme toi, j'aime pas les astuces de derrière les fagots... mais tu as sûrement raison pour le coup de l'examinateur

t'as eu quoi toi ?

en fait j'ai rien branlé en sup parce que ma classe etait pas super donc je ressentais pas trop de difficulté et maintenant je culpabilise/stress pour ulm a cause d'exos de ce genre

J'ai passé CCP et Centrale que j'ai eu (sauf centrale Paris). Je suis parti en magistère après donc j'ai pas passé les oraux par contre.
T'as fait quelle partie de l'algèbre de spé ?
Si c'est pas ton prof qui t'as donné cet exo je ne m'embêterai même pas avec. Il te manque sûrement un bout du programme. Pour tes vacances essaie d'être au taquet sur le programme de sup, lis les oeuvres de philo (oui je sais c'est con mais si tu ne les as pas lues tu prends au moins -10 à ta note de philo qui a un coeff pas dégueu...), et surtout ne t'inquiète pas trop, ton niveau augmentera énormément entre fin de sup et fin de spé.

ah ok je vois
algèbre générale et reduction j'ai fait parce que c'est bien cool je trouve
merci pour les conseils :)

Il te manque toute la topologie et la partie sur les espaces préhilbertiens/ espaces normés faite en spé, j'ai pas trop réfléchi à l'exo en OP mais je pense que c'est pas possible sans

You still want a solution?

yup

ok je vois

[eqn]| \sum_k \epsilon_k e^{i k \theta} | ^2 = \sum_{k,l} \epsilon_k \epsilon_l e^{i \theta (k-l)} =
\sum_{k,l} \epsilon_k \epsilon_l e^{i \theta (k-l)}+ \sum_{k} \epsilon_k^2 = \sum_{k,l} \epsilon_k \epsilon_l e^{i \theta (k-l)} + n [/eqn] now take sup then inf. Should work i think.

ah last sum is k not equal to l

What are we taking the infimum over? I don't understand the set you've written. Is e_n just oscillating between -1 and 1 as n increases?

thx i'll try to do the rest
i really suck at calculating things though fml

yes

So is e_1=1 or is e_1=-1? It's confusing, you should have written e_n=(-1)^n or e_n=(-1)^(n+1), depending on which it is.

>tmw you finally see a cool problem on Veeky Forums and it was solved 15 minutes ago
>fml

Anyways, OP, for future reference, the intuition of this solution comes from the sqrt(n). And when you square a sum, it is a common analytical technique to split the sums into same indices and different indices. The most important thing about solving problems isn't getting the solution. It's understanding how you get to the solution.

[math] \displaystyle
| \sum_k \epsilon_k e^{i k \theta} | ^2 = \sum_{k,l} \epsilon_k \epsilon_l e^{i \theta (k-l)} = \sum_{k,l} \epsilon_k \epsilon_l e^{i \theta (k-l)}+ \sum_{k} \epsilon_k^2 = \sum_{k \neq l} \epsilon_k \epsilon_l e^{i \theta (k-l)} + n
[/math]

The infimum is taken over all possible combinations of -1s and 1s.

yes thanks for that
but try the second question then (the actual difficulty in the problem): get an upper bound on alpha_n finer than n

[eqn](\varepsilon_i,\dots,\varepsilon_n) \in (\pm 1)^n [/eqn]
Thats some shitty notation there senpai.

>Why e^iθ(k−l) and not e^iθ(k+l) ?

absolute value of a number squared is number times complex conjugate which doesnt affect the -1 or +1 but give a - in the exponent

not really no. should have been curcly brackets but I don't have them on my keyboard. but maybe it's french-only notation, bu then again bourbaki notation rules

Right, thanks !

Do you know the answer by any chance? Its a kind of cool problem. I think i can get n^(3/4) as upper bound, but unsure.

I don't know the answer but I know the guy who got this at his oral examination and he told me he was told to find the finest possible, with an indication. I can give the indication, it will just have simple french but should be understandable just a sec

I speak some french. Need to read some papers from time to time. Those frogs refuse to publish in english...

Nice well there it is

top line shouldn't be there

merci je ne parle pas francais at all

for intuition: It's a lot like parking

Ok. I think I can prove 3/4 as upper bound exponent. But I use some more advanced stuff from number theory called large sieve. Look into that. Original paper is in French if I recall. Bombieri le grand cribe. Asterique 18. Or Iwaniec Kowalski Analytic number theory. Off to the gym - greetings from Veeky Forums

looked it up, can't say i got all of it (fourier transforms are 2nd or 3rd year for us, for now i've only done handwavey-ly it in physics) but seems nice

thanks

Why's that?
What would you use instead?

In the civilized world we use [math]\left( \varepsilon_k \right)_{k \,\in\, [\![1,\, n]\!]} \,\in\, \left\{-1,\, 1\right\}^n[/math]

Why not just [math]\epsilon_k \in {-1,1} [/math]?

because that doesn't fit into the usual infimum notation, where you minimize over one element

I'm having some trouble finishing from here. How do we show that [math] \displaystyle \sum_{k \neq l} \epsilon_k \epsilon_l e^{i \theta (k - l)} \geq 0 [/math]

sorry but as frenchmen we have authority on notational matters

...

more like because it makes absolutely no sense

trivial result for basic analysis

i'm sure if you think about it a bit you'll be able to deduce the solution to this question.

you have my blessings

Pair them up?

I had tried pairing them up but got stuck at [math] \displaystyle \sum_{k \neq l} \epsilon_k \epsilon_l \cos \theta (k - l) \geq 0 [/math]. Sorry for these dumb questions, I haven't studied analysis before.

met le sujet complet au lieu de baigner dans l'autisme.

You don't need to separate out the terms where k=l.
Here's how to solve.

First square the absolute value term as the other user suggested. The square of the aboslute value is monotonically increasing preserving order, so [math]\alpha_n[/math] is also just squared.
[math]\displaystyle \alpha_n^2 = \underset{(\epsilon_1,\dots,\epsilon_n) \in \{-1,1\}^n }{\inf} \underset{ \theta \in [0,2 \pi] }{\sup} \left| \sum_{k=1}^n \epsilon_k e^{i k \theta} \right|^2 [/math]

Then some algebra. Don't know why I bothered to write all this out.
[math]\displaystyle \left| \sum_{k=1}^n \epsilon_k e^{i k \theta} \right|^2[/math]
[math]\displaystyle \left( \sum_{k=1}^n \epsilon_k e^{i k \theta} \right) \left( \sum_{k=1}^n \epsilon_k e^{-i k \theta} \right)[/math]
[math]\displaystyle \left( \sum_{k=1}^n \epsilon_k \cos(k \theta) + i \sum_{k=1}^n \epsilon_k \sin(k \theta) \right) \left( \sum_{k=1}^n \epsilon_k \cos(k \theta) - i \sum_{k=1}^n \epsilon_k \sin(k \theta) \right) [/math]
[math]\displaystyle \left( \sum_{k=1}^n \epsilon_k \cos(k \theta) \right)^2 + \left( \sum_{k=1}^n \epsilon_k \sin(k \theta) \right)^2 [/math]
[math]\displaystyle \sum_{k=1}^n \sum_{l=1}^n \epsilon_k \epsilon_l \cos(k \theta) \cos(l \theta) + \sum_{k=1}^n \sum_{l=1}^n \epsilon_k \epsilon_l \sin(k \theta) \sin(l \theta) [/math]
[math]\displaystyle \sum_{k=1}^n \sum_{l=1}^n \epsilon_k \epsilon_l \cos((k-l) \theta) [/math]

Now we can establish an inequality by noting the supremum is greater than the average.
The average of [math]\cos(n \theta)[/math] over [math]\theta \in [0,2 \pi][/math] is [math]\mathrm{sinc}(2 \pi n)[/math] so all but the [math]l=k[/math] term of inner sum vanish. All the epsilons are just -1 or 1 so
[math]\displaystyle \underset{ \theta \in [0,2 \pi] }{\sup} \left| \sum_{k=1}^n \epsilon_k e^{i k \theta} \right|^2 >= \sum_{k=1}^n \epsilon_k^2 = n[/math]
The infimum of a finite number of a constant is obviously just the constant so [math]\alpha_n >= \sqrt{n}[/math]

I see now. The averaging technique at the end was what I was missing. Thanks for the explanation user.

C'est le sujet complet, pour le bien de la baise
C'est un exo d'oral pas math D pêh

ching chang chong
take your pedophile language back to

Lol Americans....xD

I like this new meme.

I'm the one with the seperation solution.
Indeed you can do it that way. Seperating gieves the n and one can easily argue that the sup over the remeining 'offdiagonal' terms is grater or equal to one since it takes values bigger or equal than 0. Its a pretty cool exercise. I think ill put it on a number theory exercise sheet.

tu qoi mec ?

OP here
I don't think the exercise was meant to be solved with your averaging method since no student here knows the expression for the mean of cos(nx), nor even the existence thereof.

Also if you're actually writing exercise sheet I can give you very high quality problems from the books which we use to prepare for the ENS

How do you solve it without averaging? And isn't finding the mean of cos(nx) a basic technique in integral calculus?

If its not too much to ask. I would love to take a look at some problems. Actually ill be assistent for algebraic topology but i have major influence on the analytic number theory sheet :)