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bio website mathtm.blogspot.com
location University of California Los Angeles, CA
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visits member for 2 years, 2 months
seen Jul 16 at 15:11

I am a recent graduate in applied mathematics at UCLA and currently trying to break into the quantitative investment/trading industry while also continuing to pursue advanced graduate-level mathematics as both a hobby and career necessity.


Jun
22
comment Computing the norm of operator when space is equipped with sup norm and $L^1$ norm
For the example sequence of continuous functions of $L1$ norm $1$ we have $|\phi(f_{n})|\to\infty$ as $n\to\infty$ by construction of the Dirac functions outlined above. Do you see this at least? Therefore, since $||\phi||\geq|\phi(f_{n})|$ for all $n$, the claim follows.
Jun
22
comment Computing the norm of operator when space is equipped with sup norm and $L^1$ norm
I'm not sure why you're being asked such a question if you don't follow (b). Dirac functions to which I am referring are nonnegative bump functions that spike near the origin and approach 0 elsewhere (rapidly), but such that their total integral is always $1$. In particular, they belong to the class of functions you must consider in evaluating the $\sup$ appearing in the definiton of $||\phi||$.
Jun
22
comment Computing the norm of operator when space is equipped with sup norm and $L^1$ norm
If you followed hint you would quickly see $||\phi||=\infty$ since the example sequence of sample norms in the hint is unbounded.
Jun
22
comment Computing the norm of operator when space is equipped with sup norm and $L^1$ norm
See my edit for (a). I made a slight oversight and you are correct about your comment.
Jun
6
comment (Ito lemma proof): convergence of $\sum_{i=0}^{n-1}f(W(t_{i}))(W(t_{i+1})-W(t_{i}))^{2}.$
Thanks Saz! I'll have to check this textbook out. You make a lot of references to it in your answers.
May
29
comment If $f$ and $g$ are integrable functions on $I=[a,b]$ and if $h(x):=\inf(f(x), g(x))$ for all $x \in I$, prove that $h$ is integrable at $I$.
$P=\{x_{0},x_{1},\ldots,x_{n}\}$. It could be that, say, $P_{1}=\{x_{0},x_{3},x_{5},x_{11},\ldots,x_{n-1}\}$ and $P_{2}=\{x_{1},x_{2},x_{4},\ldots,x_{n}\}$. Then $P_{1}$ and $P_{2}$ are partitions of $I$ and they are sub-partitions of $P$ in the sense that $P=P_{1}\cup P_{2}$. The index sets in this example are $I_{1}=\{0,3,5,11,\ldots,n-1\}$ and $I_{2}=\{1,2,4,\ldots,n\}.$ The manner in which the sub-partitions are obtained depend on $f$ and $g$ in accordance to the construction in the solution above. Perhaps the notation of $I_{j}$ is confusing-it has nothing to do with the interval $I$
May
14
comment Bound on uniform norm of convolution of $L^p$ functions
Oh. I didn't even know that notation was still used!
May
13
comment Bound on uniform norm of convolution of $L^p$ functions
Don't you need $p^{-1}+q^{-1}=u^{-1}+1$?
May
12
comment How do I convert the limit definition of differentiability to different variables?
Because $h$ is just a displacement vector (from the point in question, $(x_{0},y_{0})$), and you send it to $0$. Being a vector, it has coordinates. Call them $(x,y)$, or anything you wish.
May
9
comment Variation of parameters: $y''-y'-2y=4e^{-t}$
See extended answer above.
May
9
comment Variation of parameters: $y''-y'-2y=4e^{-t}$
Well isn't that the question of the day; have you even attempted to compute it on your own?!
May
9
comment Variation of parameters: $y''-y'-2y=4e^{-t}$
For example, $v_{1}$ (with my convention) is $-\int\frac{e^{2t}4e^{-t}}{3e^{t}}\;dt=-\frac{4}{3}\int\;dt=\frac{4}{3}t$.
May
9
comment Variation of parameters: $y''-y'-2y=4e^{-t}$
Yes. And just proclaiming you can't do it doesn't help us to help you. I don't understand what's so difficult here; you're just computing integrals at this point, something you should have had lots of practice in before taking your current ODE course.
May
9
comment Questions about surface integrals and an example problem
What do you mean? It's just notation for integrals; you're not literally multiplying anything by $dS$. $\int_{S}(x+y)\;dS$ just means integrate $(x+y)$ over $S$ with respect to the surface measure $dS$. To reduce matters to an ordinary iterated Riemann integral, you parameterize the surface and pull back $S$ to the parameter domain $D$, and along the way $dS$ becomes $|\phi_{u}\times\phi_{v}|\;du\;dv$. This can all be made rigorous, and indeed, if you looked in your calculus text book you would find a sketch of the proof.
May
9
comment Variation of parameters: $y''-y'-2y=4e^{-t}$
Your answer is fine. It depends on what you take to be $y_{1}$ and $y_{2}$; permuting them in the definition of $W$ will only change the sign, since $W$ is a determinant. But once you decide on your labeling, you must be consistant with the definitions for $v_{1}$ and $v_{2}$ (see the derivation of $v_{1}$ and $v_{2}$ in your text book).
May
9
comment Variation of parameters: $y''-y'-2y=4e^{-t}$
$W[e^{-t},e^{2t}](t)=2e^{-t}e^{2t}-(-e^{-t}e^{2t})=2e^{t}+e^{t}=3e^{t}$.
May
8
comment Let $f$ be a real-valued function on $[a,b]$ such that $f(x) = 0$ for all $x \neq c_1,…,c_n$. Prove that $f \in R[a,b]$ with $\int_a^b f = 0$
Once $\delta$ is small enough, the $c_{n}$ do lie in separate sub-intervals.
Apr
29
comment Left sided limit of an Riemann-Stieltjes integral
Yes, that's fine, since $\alpha$ is left continuous at $b$ and $|f|$ is bounded by the assumption $f\in\mathscr{R}(\alpha)$.
Apr
28
comment What is $\int e^{-\frac{(y-\mathbf{x}^T\mathbf{\theta})^2}{2}}d\mathbf{\theta}$?
It's the same thing. You're just taking one more step.
Apr
28
comment Need to find infimum S!!
^ essentially, yes. And using your calculator to plug in sample values is not an acceptable argument. It is sometimes helpful for giving you intuition and an idea of what the answer should be, but in this situation you need to apply some additional reasoning.