Wikipedia:Reference desk/Archives/Mathematics/2014 April 8

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April 8

One Last Coincidence

${\begin{cases}\displaystyle ~\int _{0}^{1}{\Big (}1-{\sqrt[{^{\frac {1}{2}}}]{x}}{\Big )}^{^{1-{\frac {1}{2}}}}dx~=~{\frac {\pi }{4}}\\\\\displaystyle ~\int _{0}^{1}{\Big (}1-{\sqrt[{x}]{x}}{\Big )}^{1-x}dx~\approx ~{\sqrt {\frac {\pi }{4}}}~\approx ~\int _{0}^{1}{\Big (}1-{\sqrt[{^{\frac {1}{3}}}]{x}}{\Big )}^{^{\frac {1}{3}}}dx\end{cases}}$

As usual, my question is whether there might not be something deeper to these `coincidences`. — 79.113.255.148 (talk) 00:29, 8 April 2014 (UTC)

Lorentz group: Exponential mapping onto?

Hi!

Is the map exp:so(3;1) → SO⁺(3;1) onto?

I know it is for the rotation subgroup. I also know that every LT Λ can be written as Λ = BR where B is a pure boost and R is a pure rotation. Then Λ = e^Ke^J, where K and J are suitable "generators" of boosts and rotations respectively.

Is it generally true then that Λ = e^M for some M ∈ so(3;1)?

The answer is definitely affirmative near the identity by the Baker-Campbell-Hausdorff formula. M is then a bracket series in K and J (not K + J).

How about far from the identity? YohanN7 (talk) 11:29, 8 April 2014 (UTC)

Yes, the exponential map is surjective in this case. Whenever the Lie group is compact, the exponential map is a surjection onto the identity component. I don't know of an easy proof of this, but you can use the Hopf–Rinow theorem to see it. In the case of the Lorentz group $\mathrm {O} (3,1)$, the connected component containing the identity is $\mathrm {SO} ^{+}(3,1)$. --Sam^Talk 12:51, 8 April 2014 (UTC)

I believe your conclusion may be correct, but the reason you give isn't, because the Lorentz group is not compact. YohanN7 (talk) 13:22, 8 April 2014 (UTC)

The onto character of exp in the case of SO(3) and other compact classical groups is easiest seen utilizing the fact that the group elements are all conjugate to matrices of a special form. The representatives of these conjugacy classes are then explicitly seen to be one-parameter subgroups. YohanN7 (talk) 13:35, 8 April 2014 (UTC)

The same reasoning will apply for the Lorentz group. It's a quotient of the complex semi simple group SL(2,C), and in that group you can use the Jordan decomposition. (It's probably true for all complex semisimple connected groups for basically the same reason, but I haven't really thought about the details). Sławomir Biały (talk) 13:52, 8 April 2014 (UTC)

The exponential map for $\mathrm {SL} _{2}(\mathbb {C} )$ is not surjective, for instance the matrix $q={\begin{pmatrix}-1&1\\0&-1\end{pmatrix}}$ is in Jordan normal form and is not in the image of the exponential map. --Sam^Talk 14:12, 8 April 2014 (UTC)

Oh, of course you're right. My argument only works if you have nonzero trace. Sławomir Biały (talk) 15:19, 8 April 2014 (UTC)

Oops, sorry about that YohanN7! I was too hasty. Yes, another proof for compact groups is to use that every element is contained in a Cartan subgroup, and that these are all conjugate. This is still not helpful in this instance though.

Here's an argument. The exponential map $\exp \colon {\mathfrak {sl}}_{2}(\mathbb {C} )\to \mathrm {SL} _{2}(\mathbb {C} )$ is nearly surjective: $\exp \colon {\mathfrak {sl}}_{2}(\mathbb {C} )\to \mathrm {PGL} _{2}(\mathbb {C} )=\mathrm {SL} _{2}(\mathbb {C} )/\{\pm \mathrm {I} \}$ is surjective. From this the result follows for $\mathrm {SO} ^{+}(3,1)\cong \mathrm {PGL} _{2}(\mathbb {C} )$ because of the exceptional isomorphism $\mathrm {SL} _{2}(\mathbb {C} )\cong \mathrm {Spin} (3,1)$. I don't know how you would see it in more generality. --Sam^Talk 14:10, 8 April 2014 (UTC)

My purpose is to come with a reliable statement regarding this in Representation theory of the Lorentz group. Either of a "reliable source" or a simple proof would do. I haven't seen the statement (or its negation) being made anywhere in the literature.

Would you be able to come up with something less general without explicit or implicit isomorphisms? The exponential mapping is (in this context), after all, a map from a Lie algebra to the connected group generated by it, not to another isomorphic group or some quotient. Consider, for reference, the proof below for compact classical groups:

Consider a compact classical group K. Let g ∈ K be arbitrary. Then g = hch^-1 where h ∈ K and c = e^X with X ∈ k, the Lie algebra of K (conjugation properties of compact classical groups). Now, by following the definition of exp, he^Xh^-1 = e^hXh-1 and hXh^-1 ∈ k by using properties of the adjoint representation of K.

A proof like this, I could shorten/translate to words and give ample references to the literature/internal links for each step. (Unfortunately, it doesn't apply.) Proofs using hidden isomorphisms aren't convincing. YohanN7 (talk) 14:58, 8 April 2014 (UTC)

Let π: SL(2, C) → PGL(2, C) be the quotient map. Could I then easily prove that for each p ∈ PGL(2, C), there is at least one (exactly one would be nice) element of the fiber π^-1(p) that is in the image of exp: sl(2,C) → SL(2, C)? Your answers above suggests that this is the case, except perhaps for "ease" of proof. That would solve the problem (using some isomorphisms). YohanN7 (talk) 16:47, 8 April 2014 (UTC)

I'm not sure what you're asking. Once you know that the exponential map $\exp \colon {\mathfrak {sl}}_{2}(\mathbb {C} )\to \mathrm {PGL} _{2}(\mathbb {C} )$ is surjective, the rest follows. That it is surjective for $\mathrm {PGL} _{n}(\mathbb {C} )$ follows from the fact that it is so for $\mathrm {GL} _{n}(\mathbb {C} )$, by the Jordan normal form considerations mentioned above.

There can be either 1 or 2 elements in the image of $\exp \colon {\mathfrak {sl}}_{2}(\mathbb {C} )\to \mathrm {SL} _{2}(\mathbb {C} )$ in a fiber, e.g. $\{\pm \mathrm {I} \}$. --Sam^Talk 17:46, 8 April 2014 (UTC)

I don't know how to prove the more general result for $\mathrm {SO} (n,1)$, so I don't know how to avoid using accidental isomorphisms. --Sam^Talk 17:48, 8 April 2014 (UTC)

Thanks, with the GL(n, C)-argument, there are no loose ends, but that was new. At any rate, I have found an argument using SL(2, C) and its conjugacy classes only (that I can source, Rossman). If I can prove that

$p={\begin{pmatrix}1&-1\\0&1\end{pmatrix}}$

is in the image of exp, then I'm done because qp^-1 = -I so p ~ q under π. (Note that I've taken the liberty of naming your matrix of above.) The rest is exactly as in the compact case. YohanN7 (talk) 18:47, 8 April 2014 (UTC)

And it looks like

$p=exp{\begin{pmatrix}0&-1\\0&0\end{pmatrix}}.$ YohanN7 (talk) 18:57, 8 April 2014 (UTC)

Just to explain my "problem" with (hidden) isomormhisms. With these, for a full proof, one needs to display (at least imagine) a commutative diagram, at the worst one has to prove that it commutes. In the present case, with notation as above, there is a theorem. If π: G → H is a homomorphism, then its differential φ: g → h is a homomorphism and π(exp(X) = exp(φ(X)). The lhs as a whole is surjective. Therefore the exp on the rhs (not the same exp as on the lhs) is surjective. Since the conjugacy class represented by q above was the only problematic class, I'm now completely satisfied. YohanN7 (talk) 20:25, 8 April 2014 (UTC)

Expressing 2.5cm in mm?

How will you do that? Please explain me, step by step how to do it (everywhere I ask, I get too much of a short answer which doesn't help me understand the way needed for this simple calculation). thank you. 79.177.27.113 (talk) 18:38, 8 April 2014 (UTC)

Converting between metric units is designed to be "easy," but it can be tedious to explain every little step. Metric conversion doesn't exist as an article, and conversion of units thinks that converting between metric prefixes is too east to mention! So, here's an explanation, hope it helps.

First, you should be familiar with the meter, whose symbol is 'm', and the metric prefix system, which explains how powers of ten are communicated with words like "centi-" or "deci-" and so on. So, 2.5 cm is 2.5 centi meters and centi- means "one one-hundredth", which is (2.5 meters)/100. You could also think of it as (2.5 meters) * 0.01, which is the same thing written a little differently. Now, if you want it it milli meters, you look and see that "milli-" means a factor of 1/1000, or 0.001. Now, we can convert directly to millimeters by just "moving the decimal point" - but that can be tricky for unfamiliar prefixes. So the safest way is to convert cm to m, then m to mm. In this case, $2.5cm=2.5m/100=0.025m$. Now 1 m = 1000 mm, so we can multiply a unit in meters by 1000 to get a unit in mm. So, 0.025 m = 0.025 X 1000 mm = 25 mm.

Finally, we have $2.5cm=2.5m/100=0.025m=25mm$ If that explanation doesn't make sense, here's a few other tutorials on the topic [1] [2]. SemanticMantis (talk) 19:07, 8 April 2014 (UTC)

If you know there are 10 mm in a cm, you can use the unit multiplication method:

2.5 cm x 10 mm = ?
         cm

The cm's cancel out:

2.5 cm x 10 mm = 25 mm
         cm

StuRat (talk) 23:20, 9 April 2014 (UTC)

What's happened to my earlier reply on this topic? And is other stuff missing too? HiLo48 (talk) 23:48, 9 April 2014 (UTC)

I don't see it listed in your contribution history. So, unless there's a bug or an Admin redacted it, that means you weren't able to save it. StuRat (talk) 23:58, 9 April 2014 (UTC)

Here it is:

"Year 7 students (12 to 13 years old) in Australia are simply told "To convert centimetres to millimetres, just multiply by 10." HiLo48 (talk) 01:29, 9 April 2014 (UTC)"

HiLo48 (talk) 01:17, 10 April 2014 (UTC)