The Proof is Trivial!

Problem 252*

Can anyone find a nice answer to,

\displaystyle\int\sqrt{\frac{ \sin(x+\alpha) }{\sin(x-\alpha)}}\,dx

?

Reply 1761

Mladenov

1

Original post by jack.hadamard

Thanks!

I should have thought of that. What happens if we drop the condition on the sets being finite? I tried to adapt the argument and construct such sets, whose union is not a line, but didn't succeed. In a more systematic approach, I tried to use subsets of rational, irrational and mixed points (various combinations) in the plane, but again it didn't work. Can you construct such sets?

There is a theorem due to Borwein, if I am not mistaken, which states: Let

A

and

B

be disjoint, countable, compact subsets of

\mathbb{C}

. Then, either

A \cup B

lies on a line, or there exists a line through two points of one of the sets which does not intersect the other.

The proof is based on the following lemma, and Cantor-Bendixon theorem.

Let

a

be an element of

A^{(\alpha)}

, where

\alpha

is a given countable ordinal. Suppose also that

a_{1},a_{2} \in A

and

a,a_{1},a_{2}

are not collinear. Then, either the line through

a

and

a_{1}

, or the line through

a

and

a_{2}

, intersects

B^{(\alpha)}

.

Reply 1762

Jkn

11

Original post by bogstandardname

Problem 252*

Can anyone find a nice answer to,

\displaystyle\int\sqrt{\frac{ \sin(x+\alpha) }{\sin(x-\alpha)}}\,dx

?

Does one exist? :lol:

I've reduced it to an equation that gives an extremely messy solution which I have absolutely no intention of writing down!

Let

u^2=\frac{\sin(x+\alpha)}{\sin(x-\alpha)}

,

\displaystyle \Rightarrow \int \sqrt{\frac{\sin(x+\alpha)}{\sin(x-\alpha)}} \ dx = -2 \sin2\alpha \int \frac{u^2}{u^4-4(\cos2 \alpha )u^2+1} \ du

Whilst I would be interested to see if the solution can be miraculously simplified from the horrible mess that integral gives you (after using a variety of tan substitution), I am reminded of the following quote:

"The mathematician’s patterns, like the painter’s or the poet’s must be beautiful; the ideas like the colours or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics." - G. H. Hardy, A Mathematician's Apology, 1941.

(edited 10 years ago)

Reply 1763

jack.hadamard

4

Original post by Mladenov

There is a theorem due to Borwein...

Very good! Here is the paper (number 23). I think I made my way through those being closed sets (with the given condition), so if we add to the assumptions the sets being countable and bounded, the result follows from Heine-Borel. :smile:

Reply 1764

MW24595

Original post by bogstandardname

Problem 252*
Can anyone find a nice answer to,

\displaystyle\int\sqrt{\frac{ \sin(x+\alpha) }{\sin(x-\alpha)}}\,dx

?

Original post by Jkn

x

Haha, nice.
But then again, sometimes even some of the craziest algebraic slogs yield some quaint conclusions, eh?

I managed to simplify the integral to:

\sqrt{\cos{2\alpha}} \displaystyle \int \sqrt{1+\tan{2\alpha} \cot{x}} \ dx

Let

\beta = \tan {2\alpha}

,

\Rightarrow = \sqrt{\cos{2\alpha}} \displaystyle \int \sqrt{1+\beta \cot{x}} \ dx

Although, I think the integral now looks a fair bit neater, I must say it remains just as recalcitrant.
Let's see how it goes.

Reply 1765

Jkn

11

Original post by MW24595

Haha, nice.
But then again, sometimes even some of the craziest algebraic slogs yield some quaint conclusions, eh?

I managed to simplify the integral to:

\sqrt{\cos{2\alpha}} \displaystyle \int \sqrt{1+\tan{2\alpha} \cot{x}} \ dx

Let

\beta = \tan {2\alpha}

,

\Rightarrow = \sqrt{\cos{2\alpha}} \displaystyle \int \sqrt{1+\beta \cot{x}} \ dx

Although, I think the integral now looks a fair bit neater, I must say it remains just as recalcitrant.
Let's see how it goes.

That is a far more beautiful form though I believe my form is closer to the solution. Basically, I have it, but it's an excrescence :lol:

Reply 1766

Felix Felicis

13

Original post by TheMagicMan

Time for the most inelegant solution on this thread

Original post by Jkn

*notation barrier* :lol:

Where can topology be applied btw? Does it link well with number theory, calculus and/or theoretical physics? (would be apprehensive about taking the module if it did not link well with these specific areas)

Number theory isn't my thing, really (hence why I rarely post solutions! :wink:

). Although topological methods will probably be of great use; it is a very unifying field of mathematics. Calculus is just baby analysis; where topology is very much important (any book on complex analysis worth any attention shall discuss the topology of the complex plane). In particular, some of the important results of complex integration rely on the Heine-Borel theorem.

Theoretical physics? theoretical anything.. Topology led to knot theory, which led to understanding the mechanisms by which DNA clones itself! Immediate applications of point-set topology include the Hairy-Ball Theorem (any tangential vector field on a sphere must have a vanishing point), which gets its name as it proves you cant comb a ball of hair so that there is no cowlick. Who cares? It also implies there is always a storm on the Earth.. and for purely topological reasons! Very nice way to establish what I would imagine is an important meteorological result. Or the fixed-point theorems; if you stir a cup of fluid, no matter how weird you wish to make the motion, there will always be a particle on the surface which is stationary at any given time.

Such theorems have their novelty, whilst not being so serious, but topology is a very useful tool in theoretical physics; You can find entire texts on the applications of topology to physics. Particularly in understanding magnetic fields, relativity and quantum theory, since they are all modelled by strange spaces and continuous functions on them - the very point of topology.

Reply 1768

Dirac Spinor

10

Original post by Jkn

*notation barrier* :lol:

Where can topology be applied btw? Does it link well with number theory, calculus and/or theoretical physics? (would be apprehensive about taking the module if it did not link well with these specific areas)

3 Fields medals (I think) have been awarded for topological quantum field theory. Ed Witten is the master of this area.

Reply 1769

MW24595

Original post by FireGarden

Topology led to knot theory, which led to understanding the mechanisms by which DNA clones itself! Immediate applications of point-set topology include the Hairy-Ball Theorem (any tangential vector field on a sphere must have a vanishing point), which gets its name as it proves you cant comb a ball of hair so that there is no cowlick. Who cares? It also implies there is always a storm on the Earth.. and for purely topological reasons! Very nice way to establish what I would imagine is an important meteorological result. Or the fixed-point theorems; if you stir a cup of fluid, no matter how weird you wish to make the motion, there will always be a particle on the surface which is stationary at any given time.

I've heard of the Hairy-Ball Theorem before. I read a tiny, lay-man's exposition of it in "The Universe in a Nutshell."
When, you actually just read it, it doesn't seem that impressive, but well, I really liked how you put across the storm-analogy. That seems gorgeous.
Similarly with the fixed-point theorems.

Where could I go to learn a tad about this stuff? :colone:

Reply 1770

FireGarden

14

Original post by MW24595

I've heard of the Hairy-Ball Theorem before. I read a tiny, lay-man's exposition of it in "The Universe in a Nutshell."
When, you actually just read it, it doesn't seem that impressive, but well, I really liked how you put across the storm-analogy. That seems gorgeous.
Similarly with the fixed-point theorems.

Where could I go to learn a tad about this stuff? :colone:

I have already suggested what is my favourite book for getting into (and getting quite deep) into topology: http://www.amazon.co.uk/Essential-Topology-Springer-Undergraduate-Mathematics/dp/1852337826/ref=sr_1_1?ie=UTF8&qid=1372090595&sr=8-1&keywords=essential+topology

The theorems I mention above are stated and proved in/by chapter 6. I suggest this book for its discursive nature, lesser prerequisites, whilst sacrificing no rigour.

Reply 1771

jack.hadamard

4

Original post by Mladenov

...

I have run into some problems trying to play with this in other spaces. I noticed something worrying. Clearly, if

A

and

B

are both compact, then they must both be closed by Heine-Borel. Now, notice that

A

also has at least one limit point, call it

a \in A

, by the analog of Bolzano-Weierstrass. Taking this point, every open set (think of an open disk, since we are in Euclidean plane) that contains it also contains another, distinct point of

A

. Hence, this open set also contains a point of

B

, by hypothesis, which implies that

a

is in the closure of

B

. But,

A

and

B

are disjoint and, therefore,

B

is not closed. So, if two sets satisfy the condition that between any two points of the same set there is a point of the other set (being disjoint and countable), then they cannot both be compact?

Reply 1772

ukdragon37

14

Original post by ukdragon37

I can't believe it.... ten thousand words of category theory, done and submitted. I never thought I'd be able to :cry:

Now to get some sleep. :sleep:

Turned out the examiners actually quite liked my dissertation. They are raving, I tell you :bird:

Reply 1773

shamika

16

Original post by ukdragon37

Turned out the examiners actually quite liked my dissertation. They are raving, I tell you :bird:

Many congratulations, I assume that means you got a distinction? Watch out Cornell :smile:

Reply 1774

ukdragon37

14

Original post by shamika

Many congratulations, I assume that means you got a distinction? Watch out Cornell :smile:

Yes, fortunately. Thanks!

Reply 1775

Mladenov

1

Original post by jack.hadamard

Very good! Here is the paper (number 23).

Exactly, I was unable to find it.

Original post by jack.hadamard

...

Well, if you are still interested, here is what I think.

There is actually nothing worrying. In the infinite case,

A

and

B

cannot satisfy these conditions simultaneously (closure, boundedness, countability, and that between any two points of the same set there is a point of the other set) as Borwein supposes and derives a contradiction.
By the way, in the infinite case, we do not suppose that the segment joining two points of one of the sets contains a point of the other set (which is trivial, as you have noticed already), but rather - we suppose that the line joining two points of one of the sets contains a point of the other.

As you mentioned other spaces, there is much more general result it this direction.
In any real locally convex Hausdorff linear topological space, we pick an arbitrary collection (finite) of pairwise disjoint, and compact sets. Then, if this union is not a subset of a single straight line, there exists a hyperplane which cuts exactly two of the sets in this collection, and does not intersect the convex hull (convex envelope) of the complement of the union of the sets which it cuts with respect to the set of the accumulation points of the union of all sets.

Let us back in the Euclidean plane. We pick an arbitrary finite collection of pairwise disjoin sets. Suppose that at least one of them is infinite.
Claim:
If any line intersecting two of the sets intersects at least one more set, then all sets are subsets of a single line.
This claim is true iff the sets are countable and compact.

Here is some reading on this problem: 1,2,3.

I was wondering, what can we say in the case when we deal with a finite collection of finite sets, say, in the Euclidean plane?

(edited 10 years ago)

Reply 1776

jack.hadamard

4

Original post by Mladenov

In the infinite case,

A

and

B

cannot satisfy these conditions simultaneously (closure, boundedness, countability, and that between any two points of the same set there is a point of the other set) as Borwein supposes and derives a contradiction.

Precisely. What I meant is that Borwein's theorem does not quite apply to this specific problem, but I realised it works when we substitute the (weaker) condition the point to be on the line and not simply the segment. I tried to mimic the real line, in

\mathbb{R}^2

, consisting of

\mathbb{Q}

and

\mathbb{R}\setminus\mathbb{Q}

(neither of which is closed) having the property that between any two points of the same set there is a point of the other set. Other examples can be constructed by considering intervals of the real line. I have to study this problem more.

As you mentioned other spaces, there is much more general result it this direction.

Very interesting. I haven't found the time to read what is in Borwein's paper yet, but I will do. After that, I will be interesting in reading more about this. Thanks!

If any line intersecting two of the sets intersects at least one more set, then all sets are subsets of a single line. This claim is true iff the sets are countable and compact.

That seems rather nice, too.