Mathematics

2:09 AM

Hi! I have a question in Elementary abstract algebra, I have a question that says: Let a and b be elements of group G, and let ord(a)=m and ord(b)=n, then: If a and b commute, then ord(ab) is divisor of lcm(m,n).
Now to prove it, let q be any nonnegative integer number, then (ab)^q = e, then since they commute, then a^q.b^q=e (the proof continues to show that a is not inverse of b, therefore a^q=e and b^q=e and it implies that they must have a common divisor which is lcm(n,m) ). Now, my question: it seems for me that if ab commute or not, then it is always true. I don't understand why commu…

it is in the first question here: math.wisc.edu/~mstemper2/Math/Pinter/Chapter10E

Leaky Nun

@user777 "then since they commute, then a^q.b^q=e"

user777

2:24 AM

what if I say: if they don't commute, then a^q.b^q=e

Note that the order of ab is the same as order of ba.

Ted Shifrin

If they don’t commute, is $(ab)^2=a^2b^2$?

CaptainAmerica16

3:47 AM

@TedShifrin I know it's super simple, but I feel like I've had an epiphany. I did this in like five minutes and it matched the answer given by the author. When I started something like this would take me a really long time to just wrap my head around. It's starting to all make sense ;-;

Leaky Nun

4:15 AM

@CaptainAmerica16 congratulations

CaptainAmerica16

4:25 AM

lol, I came to reply to skull but he deleted the comment.

@LeakyNun Tanks

skullpatrol

I thought you left, pal

nvm

Knight

8:12 AM

@LeakyNun You there?

If I got a limit something like this $$ \lim_{y\to 0} \frac{y}{\sqrt{\sin^2 y}}$$ can I remove the square root, and write $$ \lim_{y\to 0} \frac{y}{\sin y}$$ ?

MathStudent

10:48 AM

@Knight I think so. I wouldn't see a reason why you couldn't

MathStudent

11:00 AM

A theorem starts with Assume $X_1,...,X_n$ are i.i.d. according to q.m.d. family $\{P_{\theta}, \theta \in \Omega\}$}. Now if I want to check if this theorem is applicable to the logistic regression model, do I need to check whether the family of likelihoods that you usually use in a logistic regression model is q.m.d. or do I need to check it for the joint density of $(X,Y)$?

feynhat

@Knight No

MathStudent

11:27 AM

@feynhat Looks like he's right, @Knight. Here's the solution to the problem with steps: symbolab.com/solver/limit-calculator/…

feynhat

$\sqrt{x^2} = |x|$ not $x$.

MathStudent

@feynhat Yeah you're right, thanks. By definition. I didn't think about the possibility that $sin y$ could be negative.

MathStudent

11:57 AM

I think it's quite strange though because I would rewrite $\sqrt{x^2}={x^2}^{\frac{1}{2}} = x^{2*\frac{1}{2}} = x^1 = x$. So that means that rewriting it like that is wrong even though usually you could always rewrite $\sqrt{x} = x^{\frac{1}{2}}$, right? Not doubting that $\sqrt{x^2} = |x|$, just wondering what it means for rewriting it like that in general

Thorgott

12:14 PM

it means that $(x^y)^z$ and $x^{yz}$ need not be the same in general

MathStudent

True. I wonder if this is only the case for $y=2$ and $z=\frac{1}{2}$ or if there are other exceptions and/or if there is a general rule for when it is the same and when it isn't necessarily.

Thorgott

the identity holds for real $y,z$ and real, positive $x$

it also always holds whenever $y,z$ are integers, of course

MathStudent

12:38 PM

Sounds like that could be right. Thanks.

Regarding my own question above about applying the theorem: I think that I probably need to check it for the family that "you usually use" because the statement of the theorem is about the likelihood ratio statistic which depends on the family and I want to check it for the likelihood ratio statistic with that "usual family"

Knight

12:57 PM

@Thorgott Did you mean something like this $$ \sqrt {(-x)^2} =( \sqrt{-x} )^2 $$ the RHS is not possible if $x$ is positive therefore, we should always write $\sqrt{ |x|^2} $ ?

@MathStudent Thanks for taking interest in my question :-)

Knight

1:18 PM

You there?

Thorgott

It appears as if you answered your own question

VJ123

1:36 PM

@Thorgott Hey thanks so much for your reply on my post. I've looked into the Henstock-Kurzweil integral, but it seems slightly beyond me. So far, from your earlier help and some further research, I've gathered the following. (1) For Fubini's theorem to apply, the function must be Lebesgue integrable - that is, the integral of the absolute value of the function over the rectangle must be finite, and the function must be measurable.

@Thorgott (2) All Riemann integrable functions are Lebesgue integrable except for conditionally convergent improper Riemann integrals.

@Thorgott So if you had one of these improper integrals in your double integrals, how would you know? And when you did, would you be able to switch the order of integration or not?

@Thorgott And how does Fubini's theorem for improper integrals tie in with all this?

@Thorgott Sorry for the barrage of questions. I'm just trying to piece all the information I have together, and there's a lot!

Archer

2:37 PM

Anyone into inferential statistics here?

MathStudent

@Archer I am somewhat

Hi @BalarkaSen ! I am user403640 (changed my username), the one that had asked the question about the limsup that involved an MLE on Monday and you suggesting using the proof of asymptotic normality of the MLE, remember?

I am wondering if you had tried it out and/or seen my messages regarding it?

*suggested

Archer

0

Q: How are the random variables $Y_i$s identically distributed in this example?

I am reading Probability and Statistics (8th ed.) by Devore and cannot understand some aspects of example 6.5 from page 246: For point estimation we require that the random variables have identical distribution. I don't see how the random variables $Y_i$s are identically distributed here....

Please give this a look.

MathStudent

3:05 PM

@Archer I am not an expert but I think you don't need to require for the random variables to be identically distributed. Also, there is not just one 'point estimation method'.

E.g. maximum likelihood estimation is a method of point estimation and Wikipedia (en.wikipedia.org/wiki/Maximum_likelihood_estimation) says that you look at

the joint probability distribution of the random variables { y 1 , y 2 , … } {\displaystyle \left\{y_{1},y_{2},\ldots \right\}} {\displaystyle \left\{y_{1},y_{2},\ldots \right\}}, not necessarily independent and identically distributed

.

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$Mathematics$ Mathematics