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A Summer of Maths

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I haven't done any maths for like 2 weeks and I already feel rusty :/
Reply 41
What is the derivative of y=x^x?

What is the derivatie and inverse of y=x^x^x^x^... (an infinite string of x's)

Here x^x^x=x^(x^x) not (x^x)^x

Also for what values of x does y exist?
(edited 11 years ago)
Original post by Brit_Miller
I have a question but not an answer as I don't know how to do it. Hopefully someone can show how.

Let f:f: R\mathbb{R} x\mathrm{x} R\mathbb{R} β†’\rightarrow R\mathbb{R} x\mathrm{x} R\mathbb{R} beΒ givenΒ by\mathrm{be\ given\ by} f(x,y)=(x+y3,x5)f(x,y) = (x + y^3, x^5)

ProveΒ that\mathrm{Prove \ that} f f isΒ bijectiveΒ andΒ findΒ itsΒ inverse.\mathrm{ is \ bijective \ and \ find \ its \ inverse.}


I have never done anything like this so the following is possibly wrong. Even if it is correct I'm sure there's a much prettier way of doing it:

Spoiler

(edited 11 years ago)
Original post by Brit_Miller
I have a question but not an answer as I don't know how to do it. Hopefully someone can show how.

Let f:f: R\mathbb{R} x\mathrm{x} R\mathbb{R} β†’\rightarrow R\mathbb{R} x\mathrm{x} R\mathbb{R} beΒ givenΒ by\mathrm{be\ given\ by} f(x,y)=(x+y3,x5)f(x,y) = (x + y^3, x^5)

ProveΒ that\mathrm{Prove \ that} f f isΒ bijectiveΒ andΒ findΒ itsΒ inverse.\mathrm{ is \ bijective \ and \ find \ its \ inverse.}


{*} Additional:

Spoiler



{**} A solution:

Spoiler

(edited 11 years ago)
Original post by jack.hadamard
{*} Additional:

Spoiler



{**} A solution:

Spoiler



Nice - and informative too! I guess I was completely off the map on this one. :rolleyes:
Original post by Lord of the Flies
Nice - and informative too! I guess I was completely off the map on this one. :rolleyes:


You can add some bits to it, and fix some, but in general I don't think you were off the map at all. (not that I am much on the map, but anyway)

For instance, RΓ—R≑R2\mathbb{R} \times \mathbb{R} \equiv \mathbb{R}^2, so that (x,y)∈R2(x,y) \in \mathbb{R}^2 is a point in the plane.
The function is a mapping from points on the plane to points on the real plane; i.e. f:R2β†’R2f: \mathbb{R}^2 \to \mathbb{R}^2.
Original post by jack.hadamard
{*} Additional:

Spoiler



{**} A solution:

Spoiler



Nicely done!
Is anyone interested in summing up

βˆ‘n=0∞n!(n+x)!\displaystyle \sum_{n=0}^{\infty} \frac{n!}{(n + x)!}

where xx is a positive integer? Ideas how we can do this?
Reply 48
Original post by jack.hadamard
Is anyone interested in summing up

βˆ‘n=0∞n!(n+x)!\displaystyle \sum_{n=0}^{\infty} \frac{n!}{(n + x)!}

where xx is a positive integer? Ideas how we can do this?


Without doing any mathematics, I think that should diverge. For large n the value of x can be ignored so n!/(n+x)! is approximately 1 so it must diverge.

EDIT: After doing some maths, it seems that the above reasoning is flawed, the terms do indeed tend to 0 so it may converge.
(edited 11 years ago)
Reply 49
Original post by james22
Without doing any mathematics, I think that should diverge. For large n the value of x can be ignored so n!/(n+x)! is approximately 1 so it must diverge.


Take x = 2, then n!/(n+2)! = 1/(n+1)(n+2) which converges.
Original post by jack.hadamard
Is anyone interested in summing up

βˆ‘n=0∞n!(n+x)!\displaystyle \sum_{n=0}^{\infty} \frac{n!}{(n + x)!}

where xx is a positive integer? Ideas how we can do this?

Spoiler

Reply 51
subscribing.....around 2 in the night I tend to get bored :tongue:
Reply 52
Original post by Lord of the Flies

Spoiler



Sorry I wasn't clear, there are meant to be infinite x's.

As a side question to it, for what values of x is f(x) defined?
Original post by james22
Sorry I wasn't clear, there are meant to be infinite x's.

As a side question to it, for what values of x is f(x) defined?


Ah. In any case my working is wrong, I stupidly misread the question!...

... which makes the question more difficult, but more interesting! Hm...
(edited 11 years ago)
Reply 54
Original post by jack.hadamard
{*} Additional:

Spoiler



{**} A solution:

Spoiler


Nice

The following result could of been quoted to make your answer alot shorter (though it's always good practice to do things straight from the definitions)
Related exercise:
Let f:X→Y,g:Y→Xf: X \rightarrow Y, g:Y \rightarrow X
g is said to be a left [or right] inverse of f if (g∘f)(x)=x,[(f∘g)(x)](g \circ f)(x)=x, [(f \circ g)(x)] respectively


Show that f is surjective iff it has a right inverse
Show that f is injective iff it has a left inverse

Hence a map is bijective iff it has an (left and right) inverse. (an g is said to be an inverse iff g is both a left and right inverse.)

Using this, you only need to verify that your inverse is an inverse :tongue:.
(edited 11 years ago)
{*} Question:

The polynomial x2+1x^2 + 1 is irreducible over R\mathbb{R}.

i) By completing the square, show that x4+1x^4 + 1 is not irreducible over the set of real numbers.

Hence, derive the Sophie Germain algebraic identity

x4+4y4 ≑ (x2+2xy+2y2)(x2βˆ’2xy+2y2)x^4 + 4y^4\ \equiv\ (x^2 + 2xy + 2y^2)(x^2 - 2xy + 2y^2)

by starting from the left-hand side.


ii) Evaluate βˆ‘k=1n4k4k4+1\displaystyle \sum_{k=1}^{n} \frac{4k}{4k^4 + 1}


{**} Required:

Spoiler

Okay, I have another question which I'm sure is relatively simple and someone will know (sorry for using the thread without answers, but it's a nice place to ask questions!)

Consider this 2nd order differential equation:

yβ€²β€²+2tyβ€²+t2y=2,y'' + 2ty' + t^2y = 2, y(0)=1,y(0) = 1, yβ€²(0)=βˆ’1y'(0) = -1

Write this as a system of 1st order equations with appropriate initial conditions.
Original post by Brit_Miller
Okay, I have another question which I'm sure is relatively simple and someone will know (sorry for using the thread without answers, but it's a nice place to ask questions!)

Consider this 2nd order differential equation:

yβ€²β€²+2tyβ€²+t2y=2,y'' + 2ty' + t^2y = 2, y(0)=1,y(0) = 1, yβ€²(0)=βˆ’1y'(0) = -1

Write this as a system of 1st order equations with appropriate initial conditions.


Let z=y'.

Also quite easy: How many (real) solutions does (x+101)16+x2=0(x + 101)^{16} + x^2 = 0 have? Knowledge required: GCSE & below.
(edited 11 years ago)
Reply 58
In an office, at various times during the day, the boss gives the secretary a letter to type, each time putting the letter on top of the pile and types it. There are nine letters to be typed during the day, and the boss delivers them in the order 1,2,3,4,5,6,7,8,9. While leaving for lunch, the secretary tells a colleague that letter 8 has already been typed, but says nothing about the morning's typing. The colleague wonders which of the nine letters remain to be typed after lunch and in what order they will be typed. Based on the above information, how many such after-lunch typing orders are possible? (That there are no letters to be typed is on of the possibilities).

No ugrad knowledge required beyond combinations/ permutations.
(edited 11 years ago)
Original post by electriic_ink
Let z=y'.

Also quite easy: How many (real) solutions does (x+101)16+x2=0(x + 101)^{16} + x^2 = 0 have? Knowledge required: GCSE & below.


Thanks

(and none surely?)

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