The Proof is Trivial!

Scroll to see replies

Ach, this thread is overloaded with calculus!

Here is an interesting problem I encountered not too long ago: it is tough in my opinion, but the solution is very short.

Problem 454**/***

Consider

\vartheta:\;\mathbb{R}\to \wp(\mathbb{R})

where

\vartheta (x)

finite and

x\not\in \vartheta(x)

for all

x

. The question is whether there exists

S\subset\mathbb{R}

s.t.:

S\cap \vartheta(S)= \varnothing

and

|S| =\mathfrak{c}

Reply 2761

henpen

Original post by Lord of the Flies

Ach, this thread is overloaded with calculus!

Here is an interesting problem I encountered not too long ago: it is tough in my opinion, but the solution is very short.

Problem 454**/***

Consider

\vartheta:\;\mathbb{R}\to \wp(\mathbb{R})

where

\vartheta (x)

finite and

x\not\in \vartheta(x)

for all

x

. The question is whether there exists

S\subset\mathbb{R}

s.t.:

S\cap \vartheta(S)= \varnothing

and

|S| =\mathfrak{c}

Well, if you don't need

\vartheta

to be injective,

\displaystyle \vartheta(x)=\begin{cases} \{0\} & \text{ if } x \in \mathbb{R} \setminus \{0\} \\ [br]\{1 \} & \text{ if } x= 0[br]\end{cases}

and

S= \mathbb{R} \setminus \{0\}\subset \mathbb{R}

. Obviously,

\vartheta(S)= \{0\}

, so

S \cap \vartheta(S)= ( \mathbb{R} \setminus \{0\} ) \cap \{0\}= \emptyset

, and

| \mathbb{R} \setminus \{0\} | = \aleph_1

. Thus there is some

\vartheta

for which this property holds.

I guess I've made a mistake or misinterpreted the question.

(edited 10 years ago)

Reply 2762

Lord of the Flies

Original post by henpen

...

Indeed you have misunderstood the question: it asks whether for any

\vartheta

there exists such an

S

, not whether you can find both

S

and

\vartheta

satisfying the conditions (as an aside:

\aleph_1

is the successor cardinal of

\aleph_0

and nothing more. It seems you believe

\aleph_1=\mathfrak{c}

; this is an undecidable assertion).

Reply 2763

Noble.

Since this thread loves integration:

Problem 455 **/***

Let

J_0(x) = \dfrac{2}{\pi} \displaystyle\int_0^{\frac{\pi}{2}} \cos(x\cos\theta) \ d\theta

. Show that

\displaystyle\int_0^{\infty} J_0(x)e^{-ax} \ dx = \dfrac{1}{\sqrt{1+a^2}}

, for

a > 0

Problem 456 **/***

For

b > a > 1

find the value of

\displaystyle\int_1^{\infty} \dfrac{x^{-a}-x^{-b}}{\log(x)} \ dx

To make it easier, you may assume both the Tonelli and Fubini theorem hold for both.

Reply 2764

james22

Original post by Lord of the Flies

Ach, this thread is overloaded with calculus!

Here is an interesting problem I encountered not too long ago: it is tough in my opinion, but the solution is very short.

Problem 454**/***

Consider

\vartheta:\;\mathbb{R}\to \wp(\mathbb{R})

where

\vartheta (x)

finite and

x\not\in \vartheta(x)

for all

x

. The question is whether there exists

S\subset\mathbb{R}

s.t.:

S\cap \vartheta(S)= \varnothing

and

|S| =\mathfrak{c}

Original post by henpen

Well, if you don't need

\vartheta

to be injective,

\displaystyle \vartheta(x)=\begin{cases} \{0\} & \text{ if } x \in \mathbb{R} \setminus \{0\} \\ [br]\{1 \} & \text{ if } x= 0[br]\end{cases}

and

S= \mathbb{R} \setminus \{0\}\subset \mathbb{R}

. Obviously,

\vartheta(S)= \{0\}

, so

S \cap \vartheta(S)= ( \mathbb{R} \setminus \{0\} ) \cap \{0\}= \emptyset

, and

| \mathbb{R} \setminus \{0\} | = \aleph_1

. Thus there is some

\vartheta

for which this property holds.

I guess I've made a mistake or misinterpreted the question.

Solution 454

As stated in the original problem, the solution is very short.

A very short solution does not allow time to consider different cases.

Threfore the answer is either yes or no for all such functions.

Since we have found a function for which the answer is yes, the answer must be yes for all such functions.

Reply 2765

james22

Problem 457**

Find all automorphisms of the real numbers

i.e. find all functions

f:\mathbb{R} \rightarrow \mathbb{R}

such that

f(x+y)=f(x)+f(y)

and

f(xy)=f(x)f(y)

Now do the same of the complex numbers***

Reply 2766

Smaug123

Original post by james22

Problem 457**

Find all automorphisms of the real numbers

i.e. find all functions

f:\mathbb{R} \rightarrow \mathbb{R}

such that

f(x+y)=f(x)+f(y)

and

f(xy)=f(x)f(y)

Now do the same of the complex numbers***

Done now - thanks, LotF!

Spoiler

(edited 10 years ago)

Reply 2767

Lord of the Flies

Original post by Smaug123

...

Hint:

f(x)\geq 0

for

x\geq 0

. Two possible routes now: prove that a solution to Cauchy's eq. satisfying either 1.

f

increasing or 2. there exists a real interval on which

f

is bounded below [the former case is easier to deal with in my opinion], must be

f=ax

Reply 2768

user2020user

Spoiler

Original post by Flauta

Problem 439**

Find

\displaystyle\int^{\pi}_0 \frac{d \theta}{\sin \theta + \csc \theta}

Nothing particularly ground breaking, just thought the graph looked pretty cool

Solution 439**

I haven't seen the graph yet, but Wolfram agrees with my final result :smile:

Spoiler

Original post by Lord of the Flies

...

Loving your new avatar :wink:

(edited 10 years ago)

Reply 2769

james22

Original post by Smaug123

Done now - thanks, LotF!

Spoiler

Nice solution. Although you are wrong about the automorphisms of the complex numbers. It was a bit of a trick question because you have found all the contructablle automorphisms, but the set of automorphisms of

\mathbb{C}

is the same size as the power set of the reals.

EDIT: If you assume that the automorphisms must be continuous, you have them all.

EDIT 2: Your mistake with the complex numbers bit is you assume that the restriction of an automorphism of

\mathbb{C}

\mathbb{R}

is an automorphism of

\mathbb{R}

, this is not true. In your proof for

\mathbb{R}

you assumed that y^2>0 for all y, but this may not be true in

\mathbb{C}

(edited 10 years ago)

Reply 2770

Smaug123

Original post by james22

\mathbb{C}

\mathbb{C}

\mathbb{R}

is an automorphism of

\mathbb{R}

, this is not true. In your proof for

\mathbb{R}

you assumed that y^2>0 for all y, but this may not be true in

\mathbb{C}

Ah, that did twinge in my mind at the time, but I dismissed it because I was thinking about increasing-ness :frown:

I loved the problem, though!

Reply 2771

james22

Original post by Smaug123

Ah, that did twinge in my mind at the time, but I dismissed it because I was thinking about increasing-ness :frown:

I loved the problem, though!

Try and find one that is not continuous (hint: you need the axiom of hoice, in particular zorns lemma).

Reply 2772

Indeterminate

Problem 458***

For

u, v >0

show that

\displaystyle \int^{\infty}_{-\infty} \dfrac{u\cos vx - x\sin vx}{(u^2+x^2)^2} dx = \dfrac{\pi}{2u^2e^{uv}}.

(edited 10 years ago)

Reply 2773

newblood

Original post by Lord of the Flies

Well done

I'll type it out for you (credit to Izzy in the OP):

\displaystyle\begin{aligned}\int_1^n \frac{n}{1+x^n}\,dx &=\int_{1}^n \frac{n}{x^n}\left(\frac{1 }{1+(1/x)^n}\right)dx\\&=\int_1^n\frac{n}{x^n}\left(1-\frac{1}{x^n}+\frac{1}{x^{2n}}-\cdots\right)dx\\&=n\int_1^n \frac{1}{x^n}-\frac{1}{x^{2n}}+\frac{1}{x^{3n}}-\cdots\,dx\\&=n\left[\frac{1}{n-1}-\frac{1}{2n-1}+\cdots\right]-\underbrace{n\left[\frac{n^{1-n}}{n-1}-\frac{n^{1-2n}}{2n-1}+\cdots\right]}_{\to\,0}\end{aligned}

Hence the limit is

\ln 2

Replacing

n

with

\pi

obviously makes no difference.

how does one evaluate the last line to get ln2?

Reply 2774

newblood

Original post by Khallil

Spoiler

Solution 439**

I haven't seen the graph yet, but Wolfram agrees with my final result :smile:

Spoiler

Loving your new avatar :wink:

nice solution though i think a more basic one would be to change the sin^2 in the denominator into 1-cos^2 and let u=cosu, then a little partial fractions and were done.

Reply 2775

user2020user

Original post by newblood

nice solution though i think a more basic one would be to change the sin^2 in the denominator into 1-cos^2 and let u=cosu, then a little partial fractions and were done.

I noticed that way but I really wanted to bash some of the algebra out in the integrand with the Weierstrass sub :tongue:

Reply 2776

newblood

Anyone like stats? Here are a couple of questions i particularly liked that id done recently [not very hard, especially for someone whos done any uni stats or S3/4 even]:

Problem 459**
A fair coin is tossed n times. Let un be the probability that the sequence of tosses never has two consecutive heads. Show that u_n =0.5u_(n-1)+0.25u_(n-2) .Find also u_n

Problem 460*
The radius of a circle has the exponential distribution with parameter lambda. What is the probability density function of the area of the circle.