If I am not mistake, $C(\{1,...,n\})$ denotes all of the continuous functions from $\{1,...,n\}$ to $\Bbb{C}$. I read somewhere that every automorphism on $C(\{1,...,n\})$ is a composition with a homeomorphism on $\{1,...,n\}$. My question is, a composition with a homeomorphism and what else? Als...
Nice ... I always did that stuff at the end of the first semester at UGA, but very few instructors ever had time to cover it.
(You may not recall that, thanks to me and much to the ire of most of your algebra colleagues, my book did rings first and group theory second semester.)
The book covers the proof that those primes that can be written as a sum of two squares are precisely those congruent to $1$ mod $4$, and from there, it is a very small step to counting solutions in general
I had sound rationale for doing rings first. (The main motivation was the math education majors taking only the first semester ... and what they needed to understand to be better high school teachers. ... But then I realized $R/I$ is far easier for commutative rings than wrestling with normal subgroups, so why not do the stuff with more structure first. It's against Bourbaki, but who cares?)
@TedShifrin Also, students have been working with multiplication and addition their whole lives so it seems intuitive to start with all the usual "stuff" and then strip away structure to get groups
actually @ManeeshNarayanan asked this first part of question and I was trying to help,we did for odd order matrices but what about even order we are thinking
for #9 i have that $\delta = \min(1, 6\epsilon)$. we have that $|\frac{2x+1}{x+3} - 1| = |\frac{2x+1}{x+3} - \frac{x+3}{x+3}| = |\frac{x-2}{x+3}| < |\frac{6\epsilon}{6}| = \epsilon$
apperntly my word comprehension and sentence comprehension fall in the 8th and 9th percentile for the population but my ability to understand essays is in the 92 percentile which i find utterly confusing :S
At UGA students would get extra time and other accommodations for that if they university disability folks verified the problems, @Faust. That is tough going for math.
Morning everyone! https://imgur.com/a/lCKSI My professor's answer (in the picture) is wrong, I believe. The Standard Deviation is supposed to be $\sqrt{10}$. But as you can see, he just used the number $10$ as the SD. Is he correct or is it a mistake?
@TedShifrin apparently its really common for people with Autism that they can't understand short sets of instructions cause we see to many possible contexts for the sentence or word. but given enough sentences together we are able to narrow down what the intended instructions actually are
@TedShifrin the rest of my tutoring for calc I went really well! i was able t explain derivatives and limits and continuity and differntiability and lots of things related to to tangent lines really well +)
@Faust: I've been repeatedly reminding my precalculus students to draw the $30^\circ$-$60^\circ$-$90^\circ$ triangle and double it to get an equilateral triangle. Perhaps that'll help your student. Yeah, everyone cops out and allows calculators too much.
i just told him when your evaluating it a good idea when your done is to check the derivative and sub in the values of x to see if its right and he was mind blown they were the same thing =)
What is the difference in usage between maximum and maximal? When would you use one or the other?
Maximum can be a noun or an adjective:
This is the maximum it can be set to.
This is the maximum value.
whereas "maximal" is always an adjective:
This is the maximal value.
Is this ...
@TedShifrin do you happen to have any homomorphism between groups questions or categorizing abelian groups in terms of cyclic ones lieing around in your pockets i can have?
@Faust: Consider $G = \left\{\begin{bmatrix}a&b\\0&c\end{bmatrix} : ac\ne 0, b\in\Bbb R\right\}\subset GL(2,\Bbb R)$. Let $H$ be the subgroup of such matrices with $a=c=1$ and let $K$ the the subgroup of such matrices with $b=0$. Prove that $H$ is a normal subgroup of $G$ and that $G/H \cong K$.
This doesn't quite fit what you said, @Faust, but it's important. Suppose $G$ is a group, $a\in G$, and the order of $a$ is $n$. Prove that $\{k\in\Bbb Z: a^k = e\} = \langle n\rangle\subset\Bbb Z$. Let $d=\gcd(\ell,n)$. Prove that the order of $a^\ell$ is $n/d$.
@Meow: The point of mathematics is to write clear, correct sentences, not flurp around. That's one of the reasons I wrote that handout — to give my students a paradigm to follow, not to ignore.
@Baymax: Right. As I said, that's the definition I give and most books give. Nonsingular means row rank n (for square matrix, of course, in both cases).
@Faust In that case, to start: Let $G$ be a cyclic group and consider the set $H$ of all homomorphisms from $G$ to $\mathbb{C}^*$ as a group with pointwise multiplication. Show that $G\cong H$.
What is the difference in usage between maximum and maximal? When would you use one or the other?
Maximum can be a noun or an adjective:
This is the maximum it can be set to.
This is the maximum value.
whereas "maximal" is always an adjective:
This is the maximal value.
Is this ...
