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11h
comment How many maps can exist between two sets?
In this particular problem it turns out to be easier to use induction on $m$ than on $n$. Given the functions from $m$ elements to $n$, it's easy to see how to extend each one to get a function from $m+1$ elements to $n$. It's harder, given the functions from $m$ elements to $n$, to get the functions from $m$ elements to $n+1$.
14h
comment F is a vector space and U, V, and W are subspaces of F. Prove that $U\bigcup V\bigcup W$ is a subspace of F if and only if $U,V\subset W $.
... where $F$ is a ...?
14h
comment Suppose {A} is a sequence that assumes only integer values, under what conditions does this sequence converge?
Do you mean the sequence $a_n$, or the series $\sum_{n=1}^\infty a_n$?
14h
comment Infinite closed subset of $[0, 1]$ that does not have any subset of the form $[a, b]$ for $a< b$?
Do you want examples, or do you want to know the terminology for such a set?
1d
comment What's the period of this matrix?
Your intuition doesn't match the definition. The period is not the least number of steps the process can take to return, it's the gcd of the possible number of steps the process can take to return. In this case the process can't return in 1 step, but it can return in 2, 3, .... Since $\gcd(2,3) = 1$, the period is 1.
1d
comment How to integrate $\int\limits_0^\infty e^{-a x^2}\cos(b x) dx$ where $a>0$
In the complex approach, you estimate the integral from $R$ to $R-\beta/(2a)$.
1d
comment How to integrate $\int\limits_0^\infty e^{-a x^2}\cos(b x) dx$ where $a>0$
One should be careful about the change of variables, since $\beta/(2a)$ is not real.
Feb
6
comment What is the intuitive difference between almost sure convergence and convergence in probability?
$E_1$ corresponds to $m=1$, $E_2$ and $E_3$ to $m=2$, $E_4, E_5, E_6$ to $m=3$, etc.
Feb
4
comment Solve nonlinear system of equations
@MichaelChirico Those identities are true, but I fear they are not very useful in this case.
Feb
4
comment Maximum of a sum of random variables
There are singular continuous random distributions, where the mass is concentrated on a set of Lebesgue measure $0$ (such as a Cantor set), although every single points still have measure $0$. That's not what's happening here.
Feb
4
comment Maximum of a sum of random variables
If the joint distribution of $X_1$ and $X_2$ is absolutely continuous with respect to $2$-dimensional Lebesgue measure, then it's not hard to prove that the distribution of $\max(X_1, X_1 + X_2)$ is absolutely continuous. Thus it has a density in the $L^1$ sense, which is almost everywhere the derivative of the CDF. There may be a set of measure $0$ where the CDF is not differentiable.
Feb
3
comment How to prove that unnormalized neg entropy is strongly convex with respect to 1-norm?
That inequality is clearly false. The right side grows quadratically as $x_1 \to \infty$, the left side does not.
Feb
3
comment How to check if a symmetric $4\times4$ matrix is positive semi-definite?
Rather than making substantive changes in someone else's answer, please write a comment or your own answer. Note that the matrix is supposed to be symmetric, and alick's example wasn't. I'm going to revert to the original.
Feb
3
comment Spectral radius and dense subspace
For linear operators on infinite-dimensional normed spaces, spectrum is not the same thing as the set of eigenvalues.
Feb
3
comment Spectral radius and dense subspace
Your $T$ is not a bounded operator on $L^2$. A bounded operator can't be $0$ on one dense subspace and $I$ on another dense subspace.
Feb
3
comment Spectral radius and dense subspace
In the context, subspaces are linear. What does "spectrum" even mean for operators on non-linear spaces?
Feb
3
comment Spectral radius and dense subspace
Your $E$ is not a linear subspace.
Feb
2
comment Spectral radius and dense subspace
$f = A$, I presume?
Feb
2
comment Prove that $A - B$ is closed if $A$ or $B$ is bounded (or both)
Those counterexamples are not convex. However, you can try $\{(x,y):\; x>0, y > 0, xy \ge 1\}$ and $\{(x,y): \; x < 0, y > 0, xy \le -1\}$.
Feb
2
comment How does one prove that $2\uparrow\uparrow16+1$ is composite?
Noble Mushtak : "The rest of the Fermat numbers we know are composite"?. No. We know the other Fermat numbers up to $F_{32}$, and some others, are composite. As far as we know, all sufficiently large Fermat numbers could be primes.