The Proof is Trivial!

Original post by MW24595

Ach! I hadn't realized that you could apply the tools of differential geometry, particularly those properties of HyperKahler spaces, to Categories of Linear Functionals. That's good stuff, my dear friend.

I have a feeling that you can also consider this from the point of view of Fibre Bundle representational maps in Homological Algebra. Hmm, interesting.

In fact, a recent breakthrough in combinatorial geometry (Borislav, 2013) makes it possible to draw an isomorphism between theorem spaces of categories of linear functionals and Hyperkahler spaces. It's a remarkably intricate and complex argument to show it though.

Reply 2624

Sketch

Original post by Lord of the Flies

I adressed this issue in the following article, if you are interested.

Pierre Woodman :lol:

Reply 2625

Problem 427***

Evaluate

\displaystyle\int^{2\pi}_0 e^{\sin(x)}\ dx

Reply 2626

Original post by james22

Problem 427***

Evaluate

\displaystyle\int^{2\pi}_0 e^{\sin(x)}\ dx

Spoiler

(edited 10 years ago)

Reply 2627

Nebula

4

Does anyone mind if I post a question that spoils the answer to 420?

Reply 2628

Original post by matt2k8

Spoiler

I don't know, when I did this I didn't get any better than a power series (I forget exactly what my series was, but it looked similar), I was wondering if anyone could find a closed form.

Reply 2629

Original post by james22

I don't know, when I did this I didn't get any better than a power series (I forget exactly what my series was, but it looked similar), I was wondering if anyone could find a closed form.

Numerical approximations don't seem to give any hint of a candidate exact answer, but I believe that when I saw it before, that power series arose from the series for (1-x)^(-1/2) or similar.

Reply 2630

Nebula

4

Problem 428**(*):

If

S \subseteq \mathbb N

does there always exist a sequence

(x_i)

of reals such that

\displaystyle \sum_{n=1}^\infty x_n^{2s + 1}

congerges if and only if

s \in S

?

Reply 2631

Omghacklol

8

Problem 429 *
Evaluate

i^i

, given that it is real.

Reply 2632

Jooooshy

17

Problem 430**

Prove that

e

is irrational.

Problem 431*

Prove that

(Area \Delta ABC)^2 = s(s-a)(s-b)(s-c)

where

s = \frac{1}{2}(a+b+c)

(edited 10 years ago)

Reply 2633

IceKidd

Original post by Jooooshy

Problem 430**

Prove that

e

is irrational.

Problem 431*

Prove that

(Area \Delta ABC)^2 = s(s-a)(s-b)(s-c)

where

s = \frac{1}{2}(a+b+c)

i know how to do these but not bothered to type up.

Guideline 430

Assume e is rational and can be expressed as p/q where gcd(p,q)=1. Then write out the series expansion e and equate it to p/q. Then mukltiply everything by q! to get rid of the fraction on LHS.

The LHS is now an integre ((q-1)!p) and the RHS until the qth term is an integer. Now for the remained of this on the rhs which is infinite we have 1/(q+1) +1/(q+1)(q+2) and so on. This is less than the series with first term 1/(q+1) and the same common ratio as the first term. The sum of which is 1/2, so the RHS is a positive numer plus a positive number less than half.

Cleaarly the RHS is not an integer as it is the sum of two positive numbgers, one of which is an integer while the other is not. However, the LHS is an integer hence we have reached a contradiction so e must be irrational.

Guideline 431

Area of triangle is 1/2*bh. For any triangle with sides a,b,c this can be shown to be 1/2*bcsinA.

So Area=0.5bcsinA. But from the formula of cosines we know that a^2=b^2+c^2-2bcCosA, so 2bcCosA=b^2+c^2-a^2. To get sin A in a similar form we simply multiply the earlier equation by 4 to get 4Area=2bcsinA.

Aquaring and adding and using the relation sin^2A+cos^2A=1 the rest is just a case of algebraic manipulation (which can be simplified using s=(a+b+c)/2 - the semiperimeter).

Reply 2634

Jooooshy

17

Problem 432*

Prove that a real, symmetric matrix

A

has real eigenvalues

\lambda_i

Problem 433**

Prove that the only two prime Catalan numbers are

C_2

and

C_3

Also prove that

C_n = O(4^n n^{-3/2})

(edited 10 years ago)

Reply 2635

Original post by Nebula

Problem 428**(*):

If

S \subseteq \mathbb N

does there always exist a sequence

(x_i)

of reals such that

\displaystyle \sum_{n=1}^\infty x_n^{2s + 1}

congerges if and only if

s \in S

?

Idea for 428:

Spoiler

(edited 10 years ago)

Reply 2636

Original post by matt2k8

Idea for 428:

Spoiler

Not true, consider the set {3,4,5,...} and the sequence 1/n^(1/2). This converges iff s is in the set.

I think it may work for all sets with 1 in though, since increasing s should only make the series converge faster.

Reply 2637

Original post by james22

Not true, consider the set {3,4,5,...} and the sequence 1/n^(1/2). This converges iff s is in the set.

I think it may work for all sets with 1 in though, since increasing s should only make the series converge faster.

Ah yeah, my argument actually shows for any sequence, the set where it works is of the form {k, k+1, k+2 , k+3 , ...}

Reply 2638

Original post by matt2k8

Ah yeah, my argument actually shows for any sequence, the set where it works is of the form {k, k+1, k+2 , k+3 , ...}

What did you do? I've been trying to prove a more general case that if the sum of x_n converges and y_n is a sequences with each term having modulus less than the modulus of the coresponding x_n, and the sign of x_n is the sign of y_n, then the sum of y_n converges. It seems obvious but I can't get it.

Reply 2639