Math Is Fun Forum

Yes, but would you agree that the clip itself is hilarious?

In any case

“Have I Got News For You” is meant to be a funny comedy show. Lighten up, dude!

Thanks.

This is one of those arithmetic–geometric hybrid series.

Planck’s radiation law

The intensity per unit wavelength

of the radiation of a black body of absolute temperature

is given by

[align=center]

[/align]
where

is Planck’s constant,

is the speed of light in a vacuum, and

is Boltzmann’s constant.

From this, both Stefan’s law (which states that the intensity of the radiation of a black body is proportional to the 4[sup]th[/sup] power of its absolute temperature) and Wien’s displacement law (which states that the maximum wavelength of the radiation of a black body is inversely proportional to its absolute temperature) can be derived.

To derive Stefan’s law, hold

constant and evaluate

. For Wien’s displacement law, set

.

http://z8.invisionfree.com/DYK/index.php?showtopic=867

It’s not difficult. You just need to apply the formula

three times.

Subgroups of

have orders 1, 2, 3, 4, 6, 8, 12, or 24. Obviously there is only one subgroup (up to isomorphism) of each of the orders 1, 2 and 3. There is only one subgroup of

of order 24, namely

, and one subgroup of

of order 12, namely

.

Which leaves 4, 6 and 8. As

does not have any cyclic subgroup of order 6 (if it did it would have to have either a 6-cycle or a disjoint product of a 2-cycle and a 3-cycle, neither of which is possible) all subgroups of order 6 are thus isomorphic (to

). As for 8, notice that

. Hence every subgroup of order 8 is a Sylow 2-subgroup; therefore all subgroups of order 8 are conjugate and therefore isomorphic (to the dihedral group

).

has two nonisomorphic subgroups of order 4, namely

(cyclic) and

(the Klein Viergruppe).

I think what you meant was that a constant function is neither strictly increasing nor strictly decreasing.

Well, aren’t constant functions just fascinating? They are both increasing and decreasing, yet neither strictly increasing nor strictly decreasing. Amazing how we take so many things for granted, only to find, on closer inspection, that we have been mistaken all along.

Wow, they sure look nasty! (At first sight, at any rate.)

About constant functions …

You realize that this is wrong? In fact, a Constant Function is both increasing and decreasing (but not strictly) along its whole domain.

Think about it.

The inequality holds for

as well, in which case the method will not be trivial for

.

And note that there is an apostrophe in That’s.