Reputation
1,266
Top tag
Next privilege 2,000 Rep.
Edit questions and answers
Badges
6 15
Impact
~14k people reached

  • 0 posts edited
  • 0 helpful flags
  • 184 votes cast
Jul
18
comment Describe the Riemann surface for $w=z^2-1$.
@Potato As requested, I have unaccepted your original and accepted this one instead! Thanks, Potato.
Jul
10
comment Assume a die is rolled repeatedly. Find the markov matrix $P$ for the random variable of the time until the next $6$.
This is what I suspected. Thanks!
Jun
25
comment Evaluate $\lim_{\alpha \to \infty} e^{-t\sqrt{\alpha}}(1-\frac{t}{\sqrt{\alpha}})^{-\alpha}$
Slick! I wish I came to this myself but thank you.
May
6
comment Evaluate $\int_0^1 \int_\sqrt{y}^1 \int_0^{x^2+y^2} dz dx dy$.
Oh I see, so there is in fact no factor of $3$ that I am missing? The answer is indeed $26/105$?
Apr
25
comment Equality in Minkowski's theorem
What is the bar notation above $g(x)$? I'm a little confused by your exposition on the first paragraph about at least one of the two functions attaining the value 0 or "pointing in the same direction". Are you assuming they are complex-valued?
Apr
23
comment Why does Fubini's theorem not hold/apply to this function?
I have that book and will take a look. Thank you.
Apr
22
comment Why does Fubini's theorem not hold/apply to this function?
Is $c \times c$ in your last sentence the counting measure? Do you have any good literature on that?
Apr
18
comment Let $1 \leq p <\infty$ and $f \in L^p(\mathbb{R})$. Prove $\lim_{x \to \infty} \int_x^{x+1} f(t) dt = 0$.
Is there an error in this solution? Not sure why it has a downvote as it looks correct to me.
Apr
12
comment Show $(\partial^2 z / \partial x \partial y)^2 = \frac{\partial^2z}{\partial x^2} \cdot \frac{\partial^2z}{\partial y^2}$ for $z=ax + yf(a)+\phi(a)$.
Thanks for clearing up my understanding.
Apr
12
comment Show $(\partial^2 z / \partial x \partial y)^2 = \frac{\partial^2z}{\partial x^2} \cdot \frac{\partial^2z}{\partial y^2}$ for $z=ax + yf(a)+\phi(a)$.
Don't we also have that $y$ is a function of $x$ and vice versa when implicitly differentiating? (i.e. $y(x)$ so requires the product rule when differentiating $yf'(a)$?)
Apr
12
comment Demand $z=x+y$ and $x^2/4 + y^2/5 + z^2/25 = 1$. What is the maximum value of $f(x,y,z) = x^2+y^2+z^2$?
Thanks for this. What is the best way to solve the system of equations by hand? At first glance, it seems quite messy.
Apr
11
comment What is the shortest/longest distance from $9x^2 + 4y^2 = 36$ to $(5,5)$?
This is a good answer also. Although, I see we want to set $d/dx (d^2) =0$ and solve for $t$. What is the best way to do this by hand? It again seems cumbersome.
Apr
8
comment Let $g$ be a non-negative measurable function. Show $\int g(x)^p d\mu = \int_0^\infty p t^{p-1} m_g(t) dt$.
It cannot be done with this assumption that $g(0)=0$?
Apr
5
comment Prove $X$ is a complete vector space $\iff$ $\Big[\quad\sum_{n=1}^\infty \| x_n \| \implies \sum_{n=1}^\infty x_n$ converges$\quad\Big]$
This is indeed the proof I've seen. Thanks for the reference.
Apr
5
comment Show that a system of equations can be solved in terms of $x,y,z$ (Rudin, ex 9.19)
Thanks. I understand this part of the question. What I'm not sure about is why x=(-9-z)/z if u=3. Is that the answer you get as well?
Apr
5
comment What are the inverse function and inverse Jacobian of $f(x,y,z) = \left(\frac{ix^2 + hy^2}{2}, \frac{jy^2+kz^2}{2}, \frac{mx^2 + nz^2}{2} \right)$?
Question about your solution: Say I want to make restrictions on (x,y,z) such that $f(x,y,z)$ is injective and surjective throughout its entire domain (not just locally). Is it enough to restrict each of $x,y,z$ to either it's positive or negative part such that there are 8 possible domains where $f$ is injective and surjective throughout?
Apr
4
comment What are the inverse function and inverse Jacobian of $f(x,y,z) = \left(\frac{ix^2 + hy^2}{2}, \frac{jy^2+kz^2}{2}, \frac{mx^2 + nz^2}{2} \right)$?
sure thats fine
Apr
3
comment What are the inverse function and inverse Jacobian of $f(x,y,z) = \left(\frac{ix^2 + hy^2}{2}, \frac{jy^2+kz^2}{2}, \frac{mx^2 + nz^2}{2} \right)$?
Thanks. I will digest this but from what I can see, your approach is to find the inverse Jacobian first because it seems easier. So, there are six different values for $g$ -- I assume this is because of the squares of $x,y,z$ in $f$. What method did you use to find $g$? Substitution?
Apr
3
comment What are the inverse function and inverse Jacobian of $f(x,y,z) = \left(\frac{ix^2 + hy^2}{2}, \frac{jy^2+kz^2}{2}, \frac{mx^2 + nz^2}{2} \right)$?
For an invertible matrix? Yes. Or are you saying there's a closed for for the inverse of the function? I think it's finding an explicit form for the inverse of the function that is my real problem.
Apr
3
comment What is needed to apply the inverse function theorem to $f(x,y,z) = \left(\frac{ax^2 + by^2}{2}, \frac{cy^2+dz^2}{2}, \frac{ex^2 + fz^2}{2} \right)$?
Right. I meant if we restrict the domain to $D$ in my previous comment AND we have that $\alpha\gamma\varphi + \beta\delta\epsilon \neq 0$, then $f$ is injective and surjective on $D$.