The Proof is Trivial!

Reply 1800

jack.hadamard

4

Original post by FireGarden

...

Which university do you go to?

Reply 1801

FireGarden

14

Original post by jack.hadamard

Which university do you go to?

Exeter. I just finished my second year.

Reply 1802

jack.hadamard

4

Original post by FireGarden

Exeter. I just finished my second year.

I have a friend who does CS with Maths there. :smile:

The seaside is nice, isn't it?

Reply 1803

Jkn

11

Original post by FireGarden

Exeter. I just finished my second year.

Ah, just a few dozen miles south of me!

As it happens, I think there were 3 or 4 people from my old school who went on to do maths at Exeter and they would have have just finished their second year! Small world! :eek:

Reply 1804

FireGarden

14

Original post by jack.hadamard

I have a friend who does CS with Maths there. :smile:

The seaside is nice, isn't it?

Oh cool

. I wonder if they're in my year; there's only, like, 4 of them! The beach is really nice! I live about a 10 minute drive away, and makes a nice place for a break :smile:

Original post by Jkn

Ah, just a few dozen miles south of me!

As it happens, I think there were 3 or 4 people from my old school who went on to do maths at Exeter and they would have have just finished their second year! Small world! :eek:

Again, I wonder if I know them! I'm at least familiar with somewhere about a third to a half of my year in total, so there's a good chance :smile:

.

Reply 1805

jack.hadamard

4

What book(s) do you use for examples in Topology?

Reply 1806

Felix Felicis

13

Problem from ages ago :L

Solution 168 (2)

This was a slog but the Beta and Gamma functions are sexy :sexface:

\displaystyle \int_{0}^{ \frac{\pi}{2}} \ln \sin x \ln \cos x dx

Let

t = \sin^{2} x :

\displaystyle \int_{0}^{ \frac{\pi}{2}} \ln \sin x \ln \cos x dx = \frac{1}{8} \int_{0}^{1} \frac{ \ln t \ln ( 1 - t)}{\sqrt{t} \sqrt{1 - t}} dt

Now, consider

I = \displaystyle\int_{0}^{1} t^{ a - 1/2} (1-t)^{b - 1/2} dt = \text{B} (a + 1/2, b + 1/2) = \dfrac{ \Gamma(a + 1/2) \Gamma ( b + 1/2)}{ \Gamma ( a + b + 1) }

Differentiating under the integral sign, we get:

\displaystyle \frac{ \partial^{2} I}{ \partial a \partial b} = \int_{0}^{1} \ln t \ln (1-t) t^{a - 1/2} (1- t)^{b - 1/2} dt

which is x8 of the desired integral for a, b = 0

Furthermore, we have:

\displaystyle\begin{aligned} \frac{ \partial^{2} I}{ \partial a \partial b} &= \frac{ \partial^{2}}{ \partial a \partial b} \text{B} ( a + 1/2, b + 1/2) \\ \\ &= \frac{ \partial^{2}}{ \partial a \partial b} \frac{ \Gamma (a + 1/2) \Gamma (b + 1/2) }{ \Gamma ( a + b + 1)} \bigg|_{a = 0, b = 0} & (*) \\ \\ & = \text{B} \left( 1/2 , 1/2 \right ) \cdot \left( \left[ \psi (1/2) - \psi (1) \right]^{2} - \psi ' (1) \right) \\ & = 4 \pi \ln^{2} 2 - \frac{ \pi^{3}}{6} \\ & \Rightarrow \int_{0}^{ \frac{ \pi}{2}} \ln \sin x \ln \cos x dx = \frac{ \pi}{2} \ln^{2} 2 - \frac{ \pi^{3}}{48} \end{aligned}

stuff

(edited 10 years ago)

Reply 1807

FireGarden

14

I use Essential Topology, published by Springer. It's a very friendly book for an otherwise tersely written subject, though it lacks no rigour. The first 6 chapters are on general point set topology, and the remainder on algebraic topology.

Topology of surfaces, knots and manifolds is also recommended by my university, though I haven't read it properly I've had a glance in the library, and it seemed more basic, and focused (as the title implies anyway) much more on geometry. For me, Topology is more the study of continuous functions than geometry (especially since my first exposure was within an Analysis course!). The power of topology in geometry is entirely due to its powerful language to deal with continuity, which allows one methods to 'play with' shapes and surfaces, with topological invariants and the like to be analogous to geometric properties. Bear this in mind! for if, like me, you are intrigued and drawn in by strange geometry, then (if you haven't already seen it) you will undoubtedly be let down by the abstract definition of a topology which will seem to lead nowhere geometrical.

Reply 1808

jack.hadamard

4

Original post by FireGarden

...

I'm not interested in Algebraic Topology at present. I meant more like a book that provides examples in General Topology. For example, if I was to attempt showing that a closed set must contain at least one limit point and didn't succeed in a while, then to go and look up examples of topologies in search of a counterexample. I can sometimes make up counterexamples myself, but it gets progressively harder as you go along and, effectively, a waste of time.

Reply 1809

Hasufel

16

Problem 255

what are the largest basins of attraction for the roots of

y=x^{3}-2x

?

(this might be a simple one for most, though)

(edited 10 years ago)

Reply 1810

Mladenov

1

Problem 256***

Let

E

be a

\sigma

-locally compact space,

R

- equivalence relation on

E

such that the set

C

determined by

R

in

E \times E

is closed. Show that

E/R

is separable.
Does the claim hold true if

E

is not

\sigma

-compact?

Reply 1811

Jkn

11

Original post by Felix Felicis

Spoiler

Absolutely brilliant! :biggrin:

(PRSOM!) So impressive to spot the possibility of differentiating under the integral sign twice, truly beautiful! Had you seen this technique used in such a way before? (if you have I would appreciate a link! :smile:

)

Reply 1812

Felix Felicis

13

Original post by Jkn

Absolutely brilliant! :biggrin:

(PRSOM!) So impressive to spot the possibility of differentiating under the integral sign twice, truly beautiful! Had you seen this technique used in such a way before? (if you have I would appreciate a link! :smile:

)

I made a thread inquiring about differentiating under the integral sign twice wrt 2 parameters a while ago because there was a question about it on this video which I watched when I first learned about the technique :biggrin:

I remember when I first saw the integral that LotF set ages ago and I tried the substitution I did and got to the stage of lnt ln(1-t)/root(t) root(1-t) but couldn't progress any further so I just gave up - I spent all of yesterday learning about the Beta and Gamma functions then it occurred to me I could differentiate the gamma function to evaluate this :biggrin:

Reply 1813

und

OP

16

All credit for the long-awaited update to the OP goes to Lord of the Flies. :smile:

Now that STEP is over I might give some of the questions a go. Just hoping that they're reasonable!

Reply 1814

MathsNerd1

17

Would anyone be able to supply me with a few questions that require DUTIS to solve them as its a new technique I've recently learnt and would like to try and apply it to a few questions to see how I handle them, thanks in advance for any questions anyone might suggest to me. :smile:

Reply 1815

Lord of the Flies

19

Well, thanks to my computer literacy, it turns out the updated list I made for the OP actually sends all the question numbers to a single question (140) and all the solutions to a single solution (Felix's solution to that problem) - so the OP is actually not updated yet.

:sigh:

Reply 1816

Hasufel

16

Original post by MathsNerd1

Would anyone be able to supply me with a few questions that require DUTIS to solve them as its a new technique I've recently learnt and would like to try and apply it to a few questions to see how I handle them, thanks in advance for any questions anyone might suggest to me. :smile:

DUTIS?

Reply 1817

MathsNerd1

17

Original post by Hasufel

DUTIS?

Differentiation under the integral sign.