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The Proof is Trivial!

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Original post by Smaug123
I am sad that there doesn't seem to be a nice proof of this :frown:


Oh there definitely is. In fact it is one of the cleverest proofs I know, and it is about 3 lines long.
Original post by Lord of the Flies
Oh there definitely is. In fact it is one of the cleverest proofs I know, and it is about 3 lines long.


There's an infinite-descent proof of the form "suppose this fails; then we prove that there is an infinite number of points", but I don't think that proof is very nice :frown: do you have a different one in mind?
Problem 562*

Factorise x2n+1−1x^{2n+1}-1 as a product of real linear and quadratic polynomials.
Write x2n+x2n−1+...+x+1x^{2n}+x^{2n-1}+...+x+1 as a product of real quadratic polynomials.
Solution 562

Spoiler

(edited 8 years ago)
Original post by Smaug123
There's an infinite-descent proof of the form "suppose this fails; then we prove that there is an infinite number of points", but I don't think that proof is very nice :frown: do you have a different one in mind?


I think we are referring to the same idea then, though in your version contradiction/descent seems like a superfluous addition. It is more "do X, if A: end, if B: do Y and restart with X. There are finitely many points, this must halt, done".

I personally think the idea is extremely clever. Granted drawing lines isn't too pretty, but the idea is very hard to come by from the statement - not to mention that by nature of the problem, it feels overoptimistic to expect a proof that does not draw lines and look at properties of said lines. Different tastes I guess! :biggrin:
Original post by Lord of the Flies
I think we are referring to the same idea then, though in your version contradiction/descent seems like a superfluous addition. It is more "do X, if A: end, if B: do Y and restart with X. There are finitely many points, this must halt, done".

I personally think the idea is extremely clever. Granted drawing lines isn't too pretty, but the idea is very hard to come by from the statement - not to mention that by nature of the problem, it feels overoptimistic to expect a proof that does not draw lines and look at properties of said lines. Different tastes I guess! :biggrin:


whats the proof then
Original post by newblood
...


Spoiler



It is surprising, at least to me, that this statement was only first proved in 1943/44.
This **** crayyyyyyyyy
Problem 563*

Define a family of polynomials by:

T0(x)=1 T_0 (x) = 1, T1(x)=x T_1 (x) = x

Tn+1(x)=2xTn(x)−Tn−1(x) for n≥1T_{n+1} (x) = 2xT_n(x) - T_{n-1}(x) \text{ for } n \geq 1


Prove that deg⁡(Tn(x))=n \deg (T_n (x)) = n and find the coefficient of xnx^n in Tn(x)T_n (x)

Form and prove a conjecture about the significance of Tn(cos⁡θ)T_n (\cos \theta)



Problem 564*

Suppose g(x) g(x) and f(x)f(x) are two nth order polynomials such that g(x)=f(x)g(x) = f(x) at n+1 distinct real values of x. Prove that f(x)=g(x)f(x) = g(x).
Original post by 16Characters....
Problem 563*

Define a family of polynomials by:

T0(x)=1 T_0 (x) = 1, T1(x)=x T_1 (x) = x

Tn+1(x)=2xTn(x)−Tn−1(x) for n≥1T_{n+1} (x) = 2xT_n(x) - T_{n-1}(x) \text{ for } n \geq 1


Prove that deg⁡(Tn(x))=n \deg (T_n (x)) = n and find the coefficient of xnx^n in Tn(x)T_n (x)

Form and prove a conjecture about the significance of Tn(cos⁡θ)T_n (\cos \theta)



Problem 564*

Suppose g(x) g(x) and f(x)f(x) are two nth order polynomials such that g(x)=f(x)g(x) = f(x) at n+1 distinct real values of x. Prove that f(x)=g(x)f(x) = g(x).


I always thought the Chebysheff polynomials woukld be a good theme for a STEP question; they certainly have a lot of odd properties.

564 looks like another good 'not obvious unless you can prove it' type of question. I'll try both of these today.

Posted from TSR Mobile
Original post by Krollo
I always thought the Chebysheff polynomials woukld be a good theme for a STEP question; they certainly have a lot of odd properties.

564 looks like another good 'not obvious unless you can prove it' type of question. I'll try both of these today.

Posted from TSR Mobile


Agreed. This was sort of the reason I was researching them actually. I had just done STEP II 1999 Q3 and was looking for a bit of practice with other "weirdly defined" polynomials.
Original post by 16Characters....
Agreed. This was sort of the reason I was researching them actually. I had just done STEP II 1999 Q3 and was looking for a bit of practice with other "weirdly defined" polynomials.


Have you done the question on bernoulli polynomials (87/3/10)?

Posted from TSR Mobile
Original post by 16Characters....
Agreed. This was sort of the reason I was researching them actually. I had just done STEP II 1999 Q3 and was looking for a bit of practice with other "weirdly defined" polynomials.


There's a small section on orthogonal polynomials in some tripos courses, such as methods and numerical analysis.
The questions are a little limited though, but if you're interested it may be worth a look.
Original post by Krollo
Have you done the question on bernoulli polynomials (87/3/10)?

Posted from TSR Mobile


Nope, barely touched any of the pre-1993 papers yet but I'll have a look later in the week thanks.

Original post by joostan
There's a small section on orthogonal polynomials in some tripos courses, such as methods and numerical analysis.
The questions are a little limited though, but if you're interested it may be worth a look.


Thanks for the suggestion.
Solution 563*
(Please let me know of mistakes/improvements, Im only a 'learner'!). I also apologise for the lack of LaTeX as I wasnt planning on sharing it.
FullSizeRender.jpg
Maybe it was a bit of an overkill for the first part, but Im never too sure when it says to prove something which seems obvious.

Problem 565*
By considering the results from Problem 563, show that
Unparseable latex formula:

\displaystyle\int^1_{-1} \frac{T_{n}(x)T_{m}(x)}{\sqrt{1-x^{2}}}\ dx =[br] 0 for\ n\not=m[br]\pi for\ n=m=0, [br]\frac{\pi}{2}\[br] for n=m\not=0

Solution 565
Try Q7 here for a similar question.

Spoiler

(edited 8 years ago)
Original post by 16Characters....
Solution 565
Try Q7 here for a similar question.

Spoiler



Lol yep i know what you mean, it it took a long while to type up the question to start with :tongue: So are you a current student or graduate?
Original post by EnglishMuon
Lol yep i know what you mean, it it took a long while to type up the question to start with :tongue: So are you a current student or graduate?


Year 13 lol, Joostan suggested I look at a Numerical Analysis worksheet because me and Krollo were discussing the Chebyshev polynomials. That question is the only one I am brave enough to even attempt!
(edited 8 years ago)
Original post by 16Characters....
Year 13 lol, Joostan suggested I look at a Numerical Analysis worksheet because me and Krollo were discussing the Chebyshev polynomials. That question is the only one I am brave enough to even attempt!


Ah nice! Do you have any links to these worksheets? I've been having great fun on these polynomial questions. I first thought they were Bernoulli polynomials but it just turns out they have many cool properties in common. But yeah I should've know u were year 13, seen u around in the camb. Offers holders thread me thinks:wink:
Original post by EnglishMuon
Ah nice! Do you have any links to these worksheets? I've been having great fun on these polynomial questions. I first thought they were Bernoulli polynomials but it just turns out they have many cool properties in common. But yeah I should've know u were year 13, seen u around in the camb. Offers holders thread me thinks:wink:


Here are the Camb example sheets.

Also as Krollo recommended on the previous page, STEP III 1987 Q10 is about the Bernoulli poylnomials.
(edited 8 years ago)

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