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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, 7 months
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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.


Dec
3
comment Proving uniform convergence of an integral-defined function on compact sets
Ah, then you will have to adjust the limits in the integral accordingly to take into account the domain. But note for instance that if $f$ is supported on $K\subset\mathbb{R}$, then $f_{\epsilon}$ is supported on a subset $K_{\epsilon}\subset K$. The specific range of $f$ is irrelevant, only that it allows for $\int_{\mathbb{R}} f=1$ and that $f\geq0$.
Dec
2
comment Proving uniform convergence of an integral-defined function on compact sets
Actually, since uniform convergence is usually harder to demonstrate than pointwise convergence, a good starting point would be to prove pointwise convergence first, even if restricted to compact sets. I elaborated on my answer a bit; hopefully it will get you started.
Dec
1
comment Proving uniform convergence of an integral-defined function on compact sets
In the statement of your problem, does $f$ have compact support?
Dec
1
comment Orders of growth of typical sequences
Are you using the $\ll$ symbol to mean $P(n)\ll Q(n)$ if there is an absolute constant $C$ such that $P(n)\leq CQ(n)$ or that $P(n)/Q(n)\leq C$ for $n\geq N$ for some $N$ sufficiently large; or is it notation for one quantity being "much smaller" than the other? Also, are the numbers $p_{j},q_{j}$ and $a_{j},b_{j}$ arbitrary pairs with no relation between the indices?
Nov
20
comment Sum to infinity of the sum 1/n^2
Look to the right as you type this question and you'll see a ton of similar questions w/ answers. There's even one with literally dozens of solutions.
Nov
4
comment Resource on Pathwise Computations Involving Brownian Motion
Thanks - I edited my question to take some of this into account.
Oct
26
comment Proving that if $f$ is Riemann integrable and $1/f$ is bounded then $1/f$ is Riemann integrable
If you wanted to make it all workout to $\epsilon$ at the end, then just refine $P$ (if necessary) so that $|U(P,f)-L(P,f)|<(m_{f})^{2}\epsilon.$ But understand that this is not necessaty, since in general a quantity bounded by $C\epsilon$ for a constant that does not depend on $\epsilon$ is as good as being bounded by $\epsilon$ since both upper bounds can be made arbitrary small by sending $\epsilon\to0$.
Oct
26
comment Proving that if $f$ is Riemann integrable and $1/f$ is bounded then $1/f$ is Riemann integrable
No - consider the case where $(m_{f})^{-2}<1$.
Oct
26
comment Proving that if $f$ is Riemann integrable and $1/f$ is bounded then $1/f$ is Riemann integrable
Not sure what you mean.
Oct
14
comment 2. Differential equation with initial condition
$\int(y-1)^{-1}\;dy\neq\ln(y-1)$ without further specifying $y$. Note that with $y=-3$ we get $\ln(-4)$ which is undefined.
Oct
10
comment When computing the CDF from a PDF, why is the integral bound a different variable? $F(x) =\int_{-\infty}^x f(t)\,dt$
The properties of a function remain the same whether you use $t,x,y,\alpha,u,etc.$ as the name for the dependent variable. It is strictly a matter of choice/convenience and to keep things clear for the exposition. $t$ in this case has nothing to do with time.
Oct
10
comment Evaluating $ \sum\frac{1}{1+n^2+n^4} $
What do you mean evaluate? This sum has no easily attainable closed form, but it evidently converges by comparing to $a_{n}=n^{-2}.$ To get a sense of the challenges you face in evaluating this sum analytically, see math.stackexchange.com/questions/8337/… for the simpler case $\sum n^{-2}.$
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 $p^{-1}+q^{-1}=u^{-1}+1$?