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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, 5 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.


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 $p^{-1}+q^{-1}=u^{-1}+1$?
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.