Wikipedia:Reference desk/Archives/Mathematics/2017 June 11

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June 11

Showing partial products converge uniformly to 0

Let $B_{n}(z)={\frac {{\bar {a}}_{n}}{|a_{n}|}}{\frac {a_{n}-z}{1-{\bar {a}}_{n}z}}$ where $|a_{n}|<1,\sum (1-|a_{n}|)=\infty$. I want to show that $B_{1}(z)\cdots B_{n}(z)$ converges to zero uniformly on the closed disk $\{|z|\leq r\}$, for r<1. I know $|B_{n}(z)-1|\geq {\frac {1-r}{2}}(1-|a_{n}|)$ so the product cannot converge pointwise to a nonzero value, and it does converge to zero since $|B_{n}(z)|<1$ always so $\prod ^{n}|B_{i}(z)|$ converges. How do I know that the convergence is uniform?--46.117.104.173 (talk) 02:20, 11 June 2017 (UTC)

It's fairly easy to show that $|B_{n}(z)|=1$ when $z=1$, and $|B_{n}(0)|=|a_{n}|<1$, so $|B_{n}|$ maps the unit disk to itself. I'm thinking you can put a bound on $|B_{n}(z)|$ in terms of $|z|$ and $|a_{n}|$. I'll try to fill in more details as I work them out, but I wanted to put some sort of response here before too much time goes by. The terminology here is a bit confusing since normally if $B_{1}(z)\cdots B_{n}(z)=0$ you say the product diverges; see infinite product. --RDBury (talk) 10:13, 12 June 2017 (UTC)

Because the sequence of partial products is uniformly bounded and holomorphic, it forms a normal family in the disc, which converges pointwise to zero. Hence the convergence is also uniform on compact subsets, by normality. This uses the following general fact, which is easy to prove by contradiction: if $f_{n}$ is a normal family of continuous complex-valued functions that converges pointwise on a compact metric space, then $f_{n}$ converges uniformly. Sławomir Biały (talk) 13:26, 12 June 2017 (UTC)

What I had in mind was a bit more elementary; it wasn't that much harder to prove uniform convergence than pointwise, even without using compactness. First, for $|a|<1$ and $|b|<1$,

${\frac {|a+b|}{|1+{\bar {a}}b|}}\leq {\frac {|a|+|b|}{1+|a||b|}}$

I'm not sure if this is a well-known inequality but it's not too hard to proof. (I'll fill in details upon request.) This implies

$|B_{n}(z)|\leq {\frac {|a_{n}|+|z|}{1+|a_{n}||z|}}$

Also, ${\frac {|a_{n}|+x}{1+|a_{n}|x}}$ is an increasing function by calculus, so if $|z|<r$ then

$|B_{n}(z)|\leq {\frac {|a_{n}|+r}{1+|a_{n}|r}}$.

This is independent of $z$, so if you can show

$\prod _{i=1}^{\infty }{\frac {|a_{n}|+r}{1+|a_{n}|r}}=0$

then the original product converges to 0 uniformly. But

$\left|1-{\frac {|a_{n}|+r}{1+|a_{n}|r}}\right|=|1-r|{\frac {1-|a_{n}|}{1+r|a_{n}|}}\geq {\frac {|1-r|}{2}}|1-|a_{n}||$

which is a divergent series. This seems to be more or less the same argument used to proof that the product converges to 0 pointwise, as seen in the original post. --RDBury (talk) 15:40, 12 June 2017 (UTC)

Adding apples and oranges

I know you can't add two different variables - apples and oranges - together. But in real life, you can consider them both fruits and add them. In computer science, I think you can declare two variables, add them, and assign to third variable. Though, I think the arithmetic analogue is that the unit is the same? Anyway, is it possible to assign two different things, apples and oranges, to the same variable in mathematics? Or do you have to assign a constant to a variable (apples=8, oranges=3) to add them and get 11 (which represents fruits)? Or does that involve transforming the units "apples" and "oranges" to "fruits", which is non-mathematical? 50.4.236.254 (talk) 11:02, 11 June 2017 (UTC)

Let A be the set of apples, and let O be the set of oranges. By Zorn's Lemon, there exists a frutomorphism $f:A\to O$. So if we have x apples and y oranges, we can simply take their sum to be $f(x)+y$ oranges. --Deacon Vorbis (talk) 13:43, 11 June 2017 (UTC)

Alternatively, you could use the Banach-Tarski pearadox to slice the apples up into pieces, and reassemble the pieces into oranges, thus eliminating the problem of adding apples and oranges. Sławomir Biały (talk) 14:05, 11 June 2017 (UTC)

You could also look at this as a direct sum, which allows one to formally add two quantities that don't necessarily admit a natural addition operation between each other.--Jasper Deng (talk) 18:18, 12 June 2017 (UTC)

You convert number of apples to number of fruit by multiplying by Costermonger's constant, which is a universal constant of nature with the value C_c = 1 fruit/apple. The number of oranges is similarly converted and then added to obtain a total with units of fruit. This is analogous to calculating the total weight of your groceries, except that in that case the numerical values of the conversion constants (4.5 oz/apple and 6 oz/orange), are not unity, and so their presence is less easily overlooked. --catslash (talk) 00:42, 13 June 2017 (UTC)