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Nov
10
comment Solve for positive integers: $\frac{4}{13}=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$
I know: that’s pretty much what I said in my comment.
Nov
10
revised Solve for positive integers: $\frac{4}{13}=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$
Spelling of Erdős.
Nov
10
comment Solve for positive integers: $\frac{4}{13}=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$
The greedy algorithm is guaranteed to work, but it can be very inefficient in the sense that there may be shorter expansions. But it’s surely the obvious thing to try here if one doesn’t have any cleverer ideas.
Nov
10
comment Number of spanning trees for these 2 figures
@Delvacode: You’re welcome!
Nov
10
revised How to compute the powers of $2\times2$ Markov matrices
Improved MathJax and title.
Nov
10
comment By applying term-wise differentiation and integration find the sum of the series $\sum_{k=1}^{\infty}\frac{x^k}{k}$
That’s right. You could write it up and post it as an answer to your own question.
Nov
10
comment By applying term-wise differentiation and integration find the sum of the series $\sum_{k=1}^{\infty}\frac{x^k}{k}$
You have $$x+\frac{x^2}2=\frac{x^3}3+\ldots\;;$$ what series do you get when you differentiate this term by term, just as if it were a polynomial? Once you have that, you should recognize the result as a very familiar series for which you know a closed form. Integrate that closed form to get the original function.
Nov
10
answered Number of spanning trees for these 2 figures
Nov
10
answered A Question Regarding the Origin of the Axiom of Symmetry
Nov
10
comment Limit of a converging sequence
@Scientifica: You’re welcome!
Nov
10
comment Give an example of 2 non isomorphic regular tournament of the same order
@XiaoXiaoZhen: The fact that the matrices are different isn’t enough to show that the graphs are not isomorphic, since the vertices could be permuted. I’ve not tried to verify that the graphs are non-isomorphic, so I really don’t know how hard it is to do so; it could conceivably be something that’s easiest to do on a computer, actually checking the possible maps to show that none is an isomorphism.
Nov
10
answered Limit of a converging sequence
Nov
10
comment Limit of a converging sequence
If $u_0=1$, then $u_1=\sqrt{1^2+\frac1{2^0}}=\sqrt2$, and the limit can’t be $\sqrt2$.
Nov
10
comment A question about a theorem derived from a given set of postulates
The diagonally opposite vertices of a square violate (1) unless you include the diagonals, in which case you violate (2).
Nov
10
comment Elementary topology question about bases and topologies
@FromCuba: You’re welcome!
Nov
10
comment A Question Regarding the Origin of the Axiom of Symmetry
As I understand it, the connection with Sierpiński is simply that he proved that the axiom of symmetry is equivalent to the negation of the Continuum Hypothesis. Wikipedia actually has a proof of a more general result.
Nov
10
answered Elementary topology question about bases and topologies
Nov
10
answered Density of a plate, u-sub?
Nov
10
comment Elementary topology question about bases and topologies
There are a couple of ways to define base for a topology; which one are you using?
Nov
10
comment Give an example of 2 non isomorphic regular tournament of the same order
@XiaoXiaoZhen: The top line, $111000101101101011101$, for $7$ vertices, expands into the part of an adjacency matrix above the diagonal. If the vertices are labelled $1,\ldots,7$, a $1$ in row $i$, column $j$ indicates an edge $i\to j$, and a $0$ indicates an edge $i\leftarrow j$. Here’s the matrix: $$\pmatrix{-&1&1&1&0&0&0\\ -&-&1&0&1&1&0\\ -&-&-&1&1&0&1\\ -&-&-&-&0&1&1\\ -&-&-&-&-&1&0\\ -&-&-&-&-&-&1 }$$ Each of the other two lines expands similarly, and each of the three resulting matrices gives you a tournament on $7$ vertices.