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4h
comment Solving a $2$ variable recurrence
@ak0817: You’re welcome!
4h
answered What are the two disjoint closed sets that cannot be separated by two disjoint open neighborhoods in the Ellentuck topology?
6h
comment Is there no difference in symbols between the floor and the ceiling of x?
Floor: $\Bigg\lfloor\text{thing}\Bigg\rfloor$. Ceiling: $\Bigg\lceil\text{thing}\Bigg\rceil$. Square brackets: $\Bigg[\text{thing}\Bigg]$. Note the placement of the horizontal hooks, which may be more obvious in these expanded versions.
6h
comment Property of Nowhere Dense Sets
@LordVader007: It’s defined in the statement of the lemma: it is some discrete subset of $U$ with the property that every point of $U$ is contained in the $\epsilon$-ball centred at some point of $D_\epsilon(U)$. The lemma is simply the assertion that for each $\epsilon$ and $U$ such a set exists.
11h
comment The role of visualization and intuition in graduate and postgraduate math and developing it
Topology is not necessarily visual; it depends on the topologist. Some people think primarily in visual terms, some primarily in what I might call more algebraic terms, and some occupy various points between the extremes. I’m a topologist who does not use a lot of visualization.
11h
answered Proving following regular expressions equal to one another?
1d
answered If two nested open sets have the same nonempty boundary, are they the same set?
1d
comment Choosing an abstract algebra text
@astudent: The first-year sequence was MATH $741$ and $742$ as listed here; I already knew quite a lot of the material on the basis of Herstein. I think that I took one other algebra course, probably $745$. I’m a set-theoretic topologist, so most of my courses were in topology and logic.
1d
comment Choosing an abstract algebra text
@astudent: Herstein covers everything that I’d expect an undergraduate math major to know; D&F covers some topics that I consider graduate level. It’s possible that my view of the matter is dated: my undergraduate degree is from $1969$. On the other hand, it’s from a very good school, and I had no trouble stepping into the standard first-year graduate algebra course at Univ. of Wisc.-Madison that year.
1d
answered Property of Nowhere Dense Sets
1d
comment Property of Nowhere Dense Sets
@Hagen: Let $$D=\bigcup_{n\in\Bbb N}\left\{(k+i)2^{-n}:k\in\Bbb Z\right\}\;;$$ $D$ is discrete, and its set of accumulation points is the real axis.
1d
answered Finding common ranking of contestants in dance competition
1d
comment Question regarding prefix codes
@Evinda: You’re welcome! Sorry I couldn’t be more helpful.
1d
comment Showing the group in $\Bbb R$
@IdiotfromPrinceton: You’re very welcome.
1d
answered Combinatorial proof of summation of $\sum_{k = 1}^{n-1} {n \choose k}= 2^1 + 2^2 + 2^3 +\ldots+ 2^{n-1}$
1d
comment Combinatorial proof of summation of $\sum_{k = 1}^{n-1} {n \choose k}= 2^1 + 2^2 + 2^3 +\ldots+ 2^{n-1}$
Exactly as I made it the first time I corrected it.
1d
revised Combinatorial proof of summation of $\sum_{k = 1}^{n-1} {n \choose k}= 2^1 + 2^2 + 2^3 +\ldots+ 2^{n-1}$
edited title
1d
revised Combinatorial proof of summation of $\sum_{k = 1}^{n-1} {n \choose k}= 2^1 + 2^2 + 2^3 +\ldots+ 2^{n-1}$
Fixed MathJax.
1d
comment Showing the group in $\Bbb R$
@IdiotfromPrinceton: The identity can’t be $-1$: that’s not even in the set. When you solve $xe+x+e=x$ for $e$, you first get $xe+e=0$, then $(x+1)e=0$. Since $x$ can’t be $-1$ and can be any other real number, this implies that $e$ must be $0$. Check: $x*0=x\cdot 0+x+0=x$, and $0*x=0\cdot x+0+x=x$. Now $x^{-1}$ must satisfy $x*x^{-1}=0$, or $x\cdot x^{-1}+x+x^{-1}=0$. Solve this for $x^{-1}$ in terms of $x$.
1d
comment Question regarding prefix codes
@Evinda: I know just a little about it, I'm afraid: it’s not at all my field. I can’t really tell you any more than you could get from the Wikipedia article on it. You would definitely want a discrete mathematics course some combinatorics, and you’ll also need some abstract algebra; beyond that I couldn’t say.