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December 29

Numeric Integration

I am trying to find the numerical value of $\int \limits _{0}^{2\pi }sin(x^{x})-cos(x^{x})\,dx$

Which I know exists but it is so difficult to find.

I know this has a solution because when I ask wolfram alpha for Solve[x^x==50000 Pi,x] I get the answer x=6.42946 which is greater than 2 Pi

So the question could be partition into less than 50000 seperate integrals and the final answer could be gather by summing up all the individual integrals.

202.177.218.59 (talk) 22:25, 29 December 2013 (UTC)[reply]

Just a note: a backslash in front of trig.func. names will make them upright:

\int \limits _{0}^{2\pi }\sin(x^{x})-\cos(x^{x})\,dx

--CiaPan (talk) 06:21, 30 December 2013 (UTC)[reply]

It almost certainly does not have a closed form solution. It can be approximated in various ways. Anyway, according to Wolfram Alpha the answer is approximately 0.71554 150.203.188.53 (talk) 07:03, 30 December 2013 (UTC)[reply]

See numerical integration for instance. What methods have you tried? 150.203.188.53 (talk) 07:11, 30 December 2013 (UTC)[reply]

I am thinking about Integration by substitution. If I substitute U=x^x , maybe it can be done. 220.239.51.150 (talk) 10:37, 30 December 2013 (UTC)[reply]

Then you'll have to express x as a function of U: x = f(U), and next differentiate f to get dx = f'(U) dU. Then you replace x^x and dx in the integral to obtain expression with U and dU only and with no x in it.
However, the first step is to find f. Can you do that...? --CiaPan (talk) 10:57, 30 December 2013 (UTC)[reply]

Integration by substitution does not work. The problem is that while u=x^x is easy, find x=f(u) is next to impossible. The best I can get is

x=Log[u]/(LambertW[Log[u]])

Which makes the Integration by substitution

Sin[u]-Cos[u]/( u(1+Log[Log[u]/LambertW[Log[u]]]) )

u_lower=0^0=1

u_upper=(2 Pi)^(2 Pi)=103540.920434272

220.239.51.150 (talk) 04:00, 1 January 2014 (UTC)[reply]

Water conversion

What would 7.4kmcubed be in gallons. Could you please also show how to convert?

Thanks in advance

We are guessing water volume of Loch Ness. — Preceding unsigned comment added by 83.148.130.101 (talk) 17:29, 29 December 2013 (UTC)[reply]

Google knows Gutworth (talk) 17:38, 29 December 2013 (UTC)[reply]

Wolfram Alpga says 1 m^3 is 264.2 gallons.

(1 km)^3 is (1000 m)^3 is 1000^3 m^3

I will leave the rest to you. 202.177.218.59 (talk) 22:52, 29 December 2013 (UTC)[reply]

American, or Imperial Gallons? — Preceding unsigned comment added by 31.51.226.22 (talk) 14:37, 31 December 2013 (UTC)[reply]

When in doubt, calculate the result for both American and Imperial Gallons. 202.177.218.59 (talk) 23:30, 1 January 2014 (UTC)[reply]

For Scotland ( unless you are campaigning to become the 51st State ) use one cubic metre as equivalent to 219.969 Imperial Gallons (or 220 for a good approximation).Dbfirs 10:01, 2 January 2014 (UTC)[reply]

I believe the American gallon is known in the UK as the "wine gallon", so maybe the question is about how much wine it would take to replace Loch Ness. "I heard him then, for I had just/completed my design/to save the Menai bridge from rust/by boiling it in wine". --Trovatore (talk) 20:26, 5 January 2014 (UTC)[reply]

Until 1826, yes, it was, though it's possible that the Scots retained the old measurement for longer. The Menai Bridge is in the wrong country, by the way. I expect that the Scots would prefer to fill Loch Ness with whisky. Dbfirs 21:27, 5 January 2014 (UTC)[reply]

Christmas With Ramanujan :-)

“

I remember once going to see him for the Holidays, and remarked that the number of the upcoming year seemed to me rather a dull one, and that I hoped it was not an unfavorable omen. "No," he replied, "it is a very interesting number; it is the smallest number expressible as the product of three distinct primes, which are congruent modulo 17."

”

— 79.113.214.164 (talk) 20:49, 29 December 2013 (UTC)[reply]

That's pretty remarkable, since he would have to have said it this year for it to be correct (2014 = 2 * 19 * 53). Looie496 (talk) 02:19, 30 December 2013 (UTC)[reply]

It was actually the number of a city bus (or perhaps a taxi) Hardy was referring to. YohanN7 (talk) 11:37, 31 December 2013 (UTC)[reply]

Hardy–Ramanujan number. « Aaron Rotenberg « Talk « 07:55, 1 January 2014 (UTC)[reply]

Perhaps Ramanujan is now the OP's household god and inspires them in their dreams? ;-) Dmcq (talk) 19:27, 1 January 2014 (UTC)[reply]

Thanks for the nice riff on the Ramanujan story--2014 is already shaping up to be an interesting year. --Mark viking (talk) 19:46, 1 January 2014 (UTC)[reply]

Duo-tricenions

The Wikipedia article for quaternion says that whereas complex numbers (which we can call binions for this purpose on the model of the larger sets) have commutative multiplication, but quaternions don't.

The Wikipedia article octonion says that although quaternion multiplication is associative, octonion multiplication isn't.

Sedenion says that sedenion multiplication in turn is not alternative.

However, go to Talk:Sedenion. Someone wrote on the talk page that there are no properties of the sedenions that are not retained in the duo-tricenions.

Any proof of this statement?? (Specifically the statement that all properties of the sedenions are retained in the duo-tricenions.) Georgia guy (talk) 23:19, 29 December 2013 (UTC)[reply]

This cannot be literally true as written. If all properties of the sedonions were shared with a higher level of the Cayley–Dickson construction, then the two algebras would be isomorphic, which is clearly false. There may be some meaningful special class of properties that are identical for both, though. « Aaron Rotenberg « Talk « 07:49, 1 January 2014 (UTC)[reply]

But, are there any properties of the sedenions that are not retained in the duo-tricenions?? Georgia guy (talk) 01:52, 2 January 2014 (UTC)[reply]

Um... yes. That's just the contrapositive of what I just said. For a specific example, sixteen elements can span the sedenions but not higher-dimensional spaces. « Aaron Rotenberg « Talk « 19:00, 2 January 2014 (UTC)[reply]