Dustan Levenstein
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 1d comment About the $1$ of ring My guess is the answer is yes, with a proof similar to that given in this problem - adjoin a unit, and use Artin-Wedderburn to obtain a system of idempotents modulo nilpotents. My guess is you'll be able to extract an idempotent for $A$ somehow. 1d comment About the $1$ of ring not sure why someone has downvoted this question. It doesn't look trivial to me. 2d comment Minimum number of marked squares on $n × n$ board Glad to have helped! :) 2d comment If a function maps an input to its inverse, is it bijective? Presuming you're talking about $X = \mathbb R$, this function isn't defined at $x=0$. If you take that point out, by letting $X = \mathbb R-\{0\}$, $f(x) = x^{-1}$, then you do get a bijection, and $f$ is its own inverse. However, there are many other kinds of bijections very different from this one. 2d comment Minimum number of marked squares on $n × n$ board Fair enough. It seems I'm not allowed to remove my downvote unless you edit this post, for some reason. If I had the option, I would remove this downvote, but not the one on your other answer, because there you do claim this value of $N$ is optimal. Apr 30 revised Minimum number of marked squares on $n × n$ board added 2 characters in body Apr 30 comment Minimum number of marked squares on $n × n$ board I've addressed both questions in my own answer. Apr 30 revised Minimum number of marked squares on $n × n$ board avoiding usage of "marked" as in the problem statement. Apr 30 comment Minimum number of marked squares on $n × n$ board @JuanSebastianLozano, @ Jens, I've added an answer for your questions. Apr 30 answered Minimum number of marked squares on $n × n$ board Apr 29 comment Proving a subgroup is a basis for a space Your proof of linear independence and spanning is correct. The dimension follows, because there are 3 elements in your basis. Also, fyi, subset is preferable to subgroup - the latter means something completely different in mathematics. Apr 28 comment Minimum number of marked squares on $n × n$ board @Jens Note that jwsiegel has updated their answer to include a reference to the closed form solution. You might want to move your accepted answer over to the correct answer... Apr 28 comment Minimum number of marked squares on $n × n$ board Well, Juan's answer isn't optimal. This answer is only asymptotically optimal, but for large $n$ it does improve on Juan's. Apr 27 comment Minimum number of marked squares on $n × n$ board @Jens The problem portion of the board with this layout is the border, which consists of $4(n-1)$ squares. When $n$ is large, this number of squares is tiny relative to the total of $n^2$ squares. That's why, in the limit, you obtain the optimal result that one can cover the board by marking at most "just over" a quarter of the squares. Apr 26 comment Question regarding countable ordinals Yeah, my thinking was backwards. Apr 26 comment Question regarding countable ordinals Do you know whether or not $\phi(\beta)<\beta$ for every $\beta \in \omega_1$? Apr 26 reviewed Approve Proof the existence of a specific linear transformation. Apr 18 awarded Nice Answer Apr 17 comment $RS$ is a subgroup if $RS=SR$ That's certainly understandable; just something to keep in mind. Have a good one. :) Apr 17 comment How to prove that a set exists in ZFC? Somebody may have already said this: technically in ZFC the only axiom that states, without qualifiers, that a set exists is the axiom of infinity. This is enough, because you have the axiom of subsets which guarantees the empty set exists, and then all the axioms for combining sets you have already proven exist to produce new sets.