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Professor emeritus at Cleveland State University. I’m a set-theoretic and general topologist with an interest in combinatorics. I’m also interested in linguistics, especially historical linguistics.


7h
comment How to exhibit the set of all the limit points of this subset of $\mathbb{R}^k$?
@Saaqib: You're right; that bit was an afterthought, and I forgot about that requirement. Just let $\epsilon=\frac12\min\{\delta,\delta_0\}$, and use the point $$p_0+\langle\epsilon,\underbrace{0,0,\ldots,0}_{k-1}\rangle\;.$$ I'll fix the answer when I'm on a real computer again instead of this confounded Kindle.
7h
comment I can't understand this difference equation step
@sonicboom: My pleasure!
12h
comment How do I prove that $x^2 ≡ 3 \mod4$ has no solutions?
An integer $x$ can be congruent to $0,1,2$, or $3$ modulo $4$; those are only possibilities. What happens when you square those numbers? Do you get anything congruent to $3$ modulo $4$?
12h
revised Express σ using matrix notation
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12h
revised Express σ using matrix notation
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12h
answered Express σ using matrix notation
12h
comment Express σ using matrix notation
@Kaitlyn: No: they won’t even be compatible for multiplication. The matrix of a permutation doesn’t represent it in the same way that the matrix of a linear transformation does: applying the permutation is not multiplying by the matrix. \\ Yes, $14523$ is right.
13h
comment Alternative proof of '$I$ is maximal iff $R/I$ is a field'
@LeeKM: Yes, that’s exactly right (except that you mean that every non-zero element is a unit!).
13h
comment Express σ using matrix notation
I’m not worried about the $\LaTeX$; however, I’m not sure what you mean by the first and second parts. Are we still talking about the matrix form of $\sigma$? Because its top line is just $12345$, and its bottom line starts $145$, not $143$.
13h
comment Express σ using matrix notation
@Kaitlyn: Neither of the $3$-cycles affects $1$, so it goes to itself. The first $3$-cycle leaves $2$ alone — i.e., sends it to itself — and the second then sends it to $4$. The first sends $3$ to $4$, and the second then sends $4$ to $5$, so the composition sends $3$ to $5$. Can you finish it from there to complete the matrix form, which so far has $\binom{1\;2\;3}{1\;4\;5}$?
13h
comment Express σ using matrix notation
Before you can do anything else, you need to know in which order you apply the cycles: do you apply first $(3,4,5)$ and then $(2,4,5)$, or do you apply first $(2,4,5)$ and then $(3,4,5)$. Both orders are fairly common, so you’ll have to tell us what your convention is. \\ I can tell you right off, though, that your matrix notation is not correct: it doesn’t specify what happens to $1$, and it sends $2$ and $3$ to the same place, $4$, meaning that it doesn’t represent a permutation.
13h
answered Alternative proof of '$I$ is maximal iff $R/I$ is a field'
13h
revised Find the domain, co-domain and range of a function
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13h
comment Find the domain, co-domain and range of a function
@user3554599: That does turn out to be the case, though you do have to verify it by showing that if $a\ne 1$, then the equation $g(x)=a$ has a solution.
13h
revised Find the domain, co-domain and range of a function
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13h
answered Find the domain, co-domain and range of a function
13h
revised Find the domain, co-domain and range of a function
MathJax
14h
comment Example of a P.I.D. that is not a Euclidean domain?
You’ll find a rather well-known example here.
15h
comment Hasse diagram question about relations
@SayfT: You’re welcome. (I have to admit that I can’t shed any better light on $I$: this is not something that I’ve seen in this form before.)
15h
answered Hasse diagram question about relations