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Jul
20
comment Prove: $D_{8n} \not\cong D_{4n} \times Z_2$.
Why do you believe there is no $D_4$?
Jul
20
comment $x,y \in G$, $o(x) = 2, o(y) = 2, o(xy) = k, k\geq 3$. Show that $D_k \cong \langle x,y\rangle$
James: do you understand what a homomorphism of groups is? This isn't the place for a basic course on group theory, I'm afraid. Perhaps you should try the math.SE chat? You could likely find good interactive help there.
Jul
20
comment $x,y \in G$, $o(x) = 2, o(y) = 2, o(xy) = k, k\geq 3$. Show that $D_k \cong \langle x,y\rangle$
Note that you only need to show that the homomorphism(s) hold on the relations, because it will follow automatically for all the group elements once you've shown that - why?
Jul
20
comment $x,y \in G$, $o(x) = 2, o(y) = 2, o(xy) = k, k\geq 3$. Show that $D_k \cong \langle x,y\rangle$
$D_k=\{1, r, \ldots, r^{k-1},s,sr,\ldots,sr^{k-1}$ isn't a definition of $D_k$ because it doesn't explain what the group operation is; how do $r^3$ and $sr^5$ multiply, for instance? Once you've fully specified the presentation of $D_k$ in terms of your $r$ and $s$, you either need to find an isomorphism in both directions, or show a homomorphism in one direction and show that they're of the same size (since this will automatically imply that it's 1-1 and onto).
Jul
20
revised $x,y \in G$, $o(x) = 2, o(y) = 2, o(xy) = k, k\geq 3$. Show that $D_k \cong \langle x,y\rangle$
edited title
Jul
19
comment Simple algebra, but causing arguments in a home school co-op discussion. $6^2/2(3)+4$
Don't forget $\dfrac{6^2}{2(3)+4}=3.6$. The short version is that the written form isn't enoguh to specify a unique value; no answer is unambiguously correct.
Jul
19
answered Convergent conjecture: What is the proof?
Jul
19
comment Convergent conjecture: What is the proof?
If you mean 'exponential' in the sense that $K(n)$ (note that $K$ doesn't have to be defined anywhere but the integers) must grow at least exponentially - i.e., $K(n)\geq e^{Cn}$ for some constant $C$ and all sufficiently large $n$ - then not only this is true but in fact you can show that $K(n)$ is super exponential - the previous statement is true for all constants $C$.
Jul
18
comment Branching Paths Problem
For all rational angles (that is, rational multiples of $\pi$), always turning in one direction should sketch out a (possibly star) polygon that will always take you back to the origin. This is easy to see for angles of the form $\frac\pi n$ but I believe it's relatively straightforward for all angles of the form $\frac pq\pi$.
Jul
18
comment Stanford math qual: Abelian groups $G$ satisfying $0\to \Bbb{Z} \oplus \Bbb{Z}/3\Bbb{Z} \to G \to \Bbb{Z} \oplus \Bbb{Z}/3\Bbb{Z} \to 0$
How are you performing your row reduction? That transformation (from the first matrix to the second) seems off to me; in particular, I don't see how any operations can change the 3 in the top left corner.
Jul
18
comment Prove that $\displaystyle \binom{n}{r} \leq \displaystyle \binom{n}{\lfloor{\frac{n}{2}}\rfloor} $ is true
People pointing out other questions: if you know the question is a duplicate, flag it as a duplicate!
Jul
17
comment Generalizing the Fibonacci sum $\sum_{n=0}^{\infty}\frac{F_n}{10^n} = \frac{10}{89}$
Your assertion on the generating function is very easy to prove, not worth hunting a reference for; just compute $S(z)-(zS(z)+z^2S(z)+\cdots+z^kS(z))$ term-by-term, using the generating relation for $n\gt k$ and the explicit definition for $n\leq k$.
Jul
17
comment How do I find the minimum size of a generating set of a group?
To supplement this, note that the question 'given a presentation of a group, what is the size of its smallest generating set?' is recursively unsolvable in general, since the question of whether the group is trivial is reducible to this question.
Jul
16
comment Theory of real numbers and using functions
Why can't you define a function from $\mathbb{R}\mapsto\mathbb{R}$ using the same logical formalism? A function is a set of ordered pairs $\{\langle a, b\rangle:a,b\in\mathbb{R}\}$ such that any two distinct pairs have a different $a$; all of these statements are formalizable in the usual set theory.
Jul
16
comment Prove that $\displaystyle\sum_{n=1}^{\infty}a_n< \infty $ imply $\lim_{x \rightarrow \infty} \frac{1}{x} \sum_{n \leq x} n a_n=0$
How do you dervive $\lim_{N\to\infty}(\sum_{n=1}^{N-1}A_n)/N=a$? (Not saying that it's not correct, but that that is a non-trivial result in its own right.)
Jul
15
answered Upper bound of digit sum of powers
Jul
15
comment Upper bound of digit sum of powers
(Also, I've tweaked your notation a bit, adding a parameter where one was needed and changing your exponent variable from $l$ to $m$ to avoid confusion with $b_l$; please feel free to refine as you see fit)
Jul
15
revised Upper bound of digit sum of powers
Tweaked notation a bit
Jul
15
comment Upper bound of digit sum of powers
You can show that $b_l\gt 1$ for similarly obvious reasons (what are the numbers $n$ with $d(n)=1$?) and using similar techniques it should be pretty straightforward to push past this to $b_l\gt 2$ and even to $b_l\gt 3$. Beyond that, though, things get substantially hairy.
Jul
15
awarded  Nice Answer