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Jan
22
answered Proving the reciprocal of a divergent sequence is convergent
Jan
4
comment $\left(n^n\right)_b = \left(n\right)_b\left(n\right)_b\ldots\left(n\right)_b$
May I suggest a more compact formulation of the problem? Here's my formulation: For a given integer $b > 1$, do there exist natural numbers $n$, $k$ and $\ell$ such that $n^n = n \sum_{i=0}^{k-1} b^{\ell i}$? (This seems like a very difficult problem to me, at least right now.)
Jan
4
comment $\left(n^n\right)_b = \left(n\right)_b\left(n\right)_b\ldots\left(n\right)_b$
@par What's the significance of writing the left-hand side in base $b$?
Dec
8
comment How can I visualize or understand a module in concrete terms?
It depends a lot on the ring, but the intuition of modules being a generalization of vector spaces is always there. Note that even with vector spaces, you don't get to have an arrow representation if your base field isn't $\mathbb R$. Still the intuition works in many cases. For example, the determinant, which is defined as the signed volume, becomes a lot less intuitive when the field is, say, $\mathbb Z / p\mathbb Z$, but Cramer's rule still works.
Dec
8
awarded  Caucus
Nov
10
comment How to compute the powers of $2\times2$ Markov matrices
The problem doesn't make sense as it is right now. Do you have more information?
Oct
28
answered Partial Fractions Decomposition
Oct
23
comment Relation about Disk and Sphere
I meant that $D^2 \simeq S^1$ is a comprehensible sentence. It's just a false sentence.
Oct
22
awarded  Popular Question
Oct
22
comment Relation about Disk and Sphere
$D^2 \simeq S^1$ is not true. (It's a valid expression in some sense though.) I'm not sure about what you said about interior points. Are trying to show that $S^1$ and $D^2$ are not homeomorphic?
Oct
22
comment Directs sum in exact sequences
Oh, I think I did make a serious typo in my original comment. The direct sum in this case is internal, not external.
Oct
22
comment Directs sum in exact sequences
That is the definition of the notation $\oplus$. Otherwise the statement should have been written $M = \alpha(L) + \text{ker}(r)$. Knowing that $\alpha(L) \cap \text{ker}(r) = 0$ tells you that if $m \in \alpha(L) + \text{ker}(r)$, then there exist unique $\ell \in \alpha(L)$ and $k \in \text{ker}(r)$ such that $\ell + k = m$.
Oct
22
comment Directs sum in exact sequences
There are actually two definitions of $\oplus$: the internal direct sum, and the external direct sum. In this case, it does seem like an external direct sum. The statement $M = \alpha(L) \oplus \text{ker}(r)$ simply means $\alpha(L)$ and $\text{ker}(r)$ are independent submodules of $M$ (their intersection is $0$), and they span $M$ (their sum is $M$).
Oct
22
comment Optimized way to compute L1 distance matrix
I'm not sure why computing $\| x - y \|^2$ could be slower than computing the dot products of $x$ and $y$, of $x$ and $x$, and of $y$ and $y$.
Oct
22
comment Relation about Disk and Sphere
$S^n$ is $n$-dimensional, but it is usually seen as a subset embedded in $\mathbb R^{n+1}$. Think of $D^2/S^1$ as the result of taking a 2D disc (which is $D^2$) and joining the boundary (which is $S^1$, a circle) together at a point. That should give you a sphere, which is $S^2$.
Oct
22
answered Find if this series converges and if so find its value
Oct
19
reviewed Approve Find $\lim_{x\to\infty} \frac{e^{2x}-1}{e^{2x}+1}$ and $\lim_{x\to-\infty} \frac{e^{2x}-1}{e^{2x}+1}$
Oct
19
answered Binomial Probability (Dice)
Oct
19
comment Binomial Probability (Dice)
The theoretical distribution is binomial with rate of success $1/3$ and number of trials $6$. It is usually denoted by $\text{Binomial}(1/3, 6)$. The mean is $2$ and the variance is $ 4/3$.
Oct
19
answered Which of the following is one-to-one?