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3h
answered Why the Ramsey number $R(2,4)$ is not equal to $2$?
14h
comment $ N_{1} $ , $ N_{2} $ are minimal normal subgroups of $ G $, $ G/N_{1}\cap N_{2} \cong G $?
This is the standard way of showing that a subgroup is normal: take an $n\in N_1\cap N_2$ and a $g\in G$. If we can show that $gng^{-1}\in N_1\cap N_2$, no matter which $n$ or $g$ we choose, then we have shown that $N_1\cap N_2$ is normal. Once that is done, minimality and non-equality of $N_1,N_2$ will therefore imply that $N_1\cap N_2=\{e\}$, the trivial group, which is what we want.
15h
comment $ N_{1} $ , $ N_{2} $ are minimal normal subgroups of $ G $, $ G/N_{1}\cap N_{2} \cong G $?
Hint: show that $N_1\cap N_2$ is normal.
1d
comment How can I parametrize $|x|+|y|=1$
@Alfrefft I knew that for the $x(t)$ part, I needed a function that went from $1$, down to $-1$ and back to $1$ again, and it needed to be straight line segments. The $y(t)$ part needed to go from $0$ to $1$, down to $-1$ and back up to $0$. Therefore, I knew that the parametrization had to be primarily linear, with some absolute values in there for the turning points Thus I played around with $x(t) = |t-2|$ and $y(t) = |t-3|-|t-1|$ until I got what I wanted. Using WA to help in the plotting didn't hurt either.
1d
revised How can I parametrize $|x|+|y|=1$
deleted 14 characters in body
1d
answered How can I parametrize $|x|+|y|=1$
1d
comment Give an example of a function $f :X \to Y$ which is sequential continuous but not continuous where $X$ and $Y$ are some topological spaces.
The example I've seen is taking $X = \omega_1 + 1$ where $\omega_1$ is the first uncountable ordinal and $Y = \Bbb R$. You then set $$f(\alpha) = \cases{0 & if $\alpha < \omega_1$\\ 1 & if $\alpha = \omega_1$}$$This is a sequentially continuous function, but not a continuous one, if $X$ is endowed with the order topology and $Y$ with the standard one.
1d
asked Global section on product of sheafifications
2d
answered how to get XAU in usd formula if i knew the USD value
2d
comment Commutative rings of order $p^3$.
It is not difficult to find all abelian groups of a given finite order. If the order you want is $p^3$, then there are $3$ distinct abelian groups (up to isomorphism). In each case, there are at least $2$ associative, commutative and distributive operations you can consider for multiplication. One is element-wise multiplication, and one is the trivial multiplication (all products are $0$). I don't know whether there are more.
2d
comment Can the determinant of an integer matrix with a given row be any multiple of the gcd of that row?
Might this not follow by induction by expanding the determinant along the second row?
Aug
26
comment Can someone check my answers on group permutation and answer part (g)
@kane Yes. In my opinion you could even leave it out of the disjoint cycle notation, but that might be contested.
Aug
26
comment A discontinuous function $f: X \rightarrow Y$ satisfying: for each closed ball $B$ of $Y, f^{-1}(B)$ is closed in $X$
If $f^{-1}(B) \subseteq X$ were closed for any closed set $B \subseteq Y$, then the function is continuous. So you need to make sure that there are many closed sets that aren't balls. As Daniel Fischer points out in the above comment, putting the discrete metric (all distances that aren't $0$ are $1$) on $Y$ works nicely.
Aug
26
comment Distinct elements of relations
Which four elements is that?
Aug
26
comment Distinct elements of relations
Can you come up with examples of elements of $R$? Non-examples? Is $1R5$? How about $3R9$?
Aug
26
comment How to solve equation to the third power
1) A solution to the equation that is a real, positive number. 2) It's complicated. The true answer is $$\frac13\left(1 + \sqrt[3]{\frac{29-3\sqrt{93}}{2}} + \sqrt[3]{\frac{29+3\sqrt{93}}{2}}\right)$$ 3) It's complicated.
Aug
26
answered Can someone check my answers on group permutation and answer part (g)
Aug
26
comment Can someone check my answers on group permutation and answer part (g)
I've looked through a-f and they seem correct.
Aug
26
comment Is this “truncating” matrix function well known?
@picasso that would not necessarily be enough, since it's not obvious just from that which columns are deleted. If you say "define the map $\pi_r:M_{n\times n}\to M_{n\times r}$ for $r\leq n$ by projecting onto the first $r$ columns", or something along those lines, then you should be good to go.
Aug
26
comment When SVD has repeated singular values?
The symmetric matrix $\left(\begin{smallmatrix}1&-1\\-1&1\end{smallmatrix}\right)$ doesn't have repeated eigenvalues.