« first day (3150 days earlier)   
00:00 - 17:0017:00 - 22:00

5:13 PM
where is ted
i have a BONE to pick with him
Okay I have to go.
BYE <3
5:31 PM
You left us boneless
Q: Upper Derivatives and Chebyshev's Inequality

user193319Recall Chebyshev's Inequality: Let $f$ be a nonnegative measurable function on $E$. Then for any $\lambda > 0$, $$m \{x \in E \mid f(x) \ge \lambda \} \le \frac{1}{\lambda } \int_{E} f$$ Here is a theorem in my book: Let $f$ be an increasing function on the closed, bounded interval $[...

1 hour later…
6:43 PM
Question with a bounty:
Q: Upper Derivative and Increasing Function on a Compact Interval

user193319 Definition. For a real valued function $f$ and an interior point $x$ of its domain, the uppper derivative of $f$ at $x$ denoted by $\overline{D}f(x)$ is defined as follows: $$\overline{D}f(x)=\lim_{h\rightarrow0}\left[ \sup \left \{\frac{f(x+t)-f(x)}{t}: 0<|t|\leq h \right \} \right]$$ I am ...

7:18 PM
Hi @Balarka
Hi @Ted, @Alessandro
I've started appreciating the theory of harmonic functions quite a bit
They're the best functions after polynomials
Hi again, a @Balarka, and hi, demonic @Alessandro
There are deep connections between harmonic functions and Markov processes
7:25 PM
I only know discrete Markov processes ...
I didn't know about this story at all until very recently
@TedShifrin There's an analogue in the discrete setup as well, you just replace the Laplacian with the discrete Laplacian.
Urgh when the professor started talking about Markov processes I stopped going to the probability II classes
They're beautiful!
That's the whole point of probability
Eh dunno, it was mostly because the first part was measure theoretic probability and then it became very computational and applied with Markov processes
So basically the story goes as follows. If you have an open subset $U \subset \Bbb R^n$ with Lipschitz boundary (basically having a cone with sufficient solid angle at any point on the boundary which does not intersect $U$ anywhere except at the vertex point suffices), say, then the Dirichlet problem $\Delta F = 0$ with $F|\partial U = f$ has the solution $F(x) := \Bbb E[f(B_\tau)|B_0 = x]$
where $\{B_t\}$ is a Brownian motion on $\Bbb R^n$ with initial distribution $B_0$ and hitting time at $\partial U$ being $\tau$. The expectation is taken with respect to the measure $\nu_x$ called the "hitting distribution at $\partial U$" or the "harmonic measure" where for any Borel subset $A \subset \partial U$, $\nu_x(A) = \Bbb P[B_\tau \in A]$, i.e., probability that the Brownian motion escapes $U$ through $A$.
For any $y \in \partial U$ you can try to solve the Dirichlet problem distributionally as well; take $f = \delta_y$ to be the "delta function at $y$". Then $F(x) = f(B_\tau = y|B_0 = x)$ is the probability density that the Brownian motion starting at $x$ escapes $\partial U$ from $y$.
This is the "Green's function" that keeps appearing everywhere in physics
Should have used a different notation for pdf since I was already using $f$ as boundary condition for the Dirichlet problem
7:39 PM
still pretty clear
If I look up Green's function, will I find this, or do you have a reference?
Let f(x)=x[x]. If x is not an integer then what is f'(x)?
I got [x].
I think people use Green's function in a way general context, where if they have some sufficiently nice linear differential operator $\mathcal{L}$, to solve the equation $\mathcal{L} f = g$ they solve for $\mathcal{L} f = \delta_y$ first, the solution to which they call the Green's function and denote by $f(x) = G(x, y)$, and then the general solution to the original thing is obtained by convolving $g$ with this $G$.
I always wanted to learn some probability theory
So, uh, as reference, I'd recommend this book by Gregory F. Lawler called "Random Walks and the Heat Equation" that I'm reading
It's an undergrad level book though
It still gets there pretty quickly
7:45 PM
yeah its very nice
Thanks :)
Someone please help me with the question I asked.
You're right @MrAP
@KarlKronenfeld Haven't seen you in a long time. How's it going?
@BalarkaSen Been good. Staying sufficiently busy with math. :) How about you?
7:48 PM
Cool! Well, I'm in my second half of first year in undergrad. The first half was somehow more exciting :S
Still learning a thing or two. I attended some workshop on probabilistic methods in negative curvature a week ago
I did it using the chain rule. $f'(x)=x\frac{d[x]}{dx}+[x]\frac{dx}{dx}=0+[x]=[x]$. Was I right to use the chain rule?
That's where I learnt this harmonicity vs processes business from. Now trying to pick up the classical story
Nice @BalarkaSen
@MrAP Product rule, you mean? I suppose. You could also derive it directly from the limit formulation of derivatives.
The latter is somehow more straightforward in this case.
Oops. Product rule.
Hi chat
7:59 PM
I think that is more complicated @KarlKronenfeld. I did it by that method just now and found that using product rule is much simpler provided you know the differentiation of $[x]$ is 0 at non-integer points and undefined at integer points.
8:27 PM
How do you know that @MrAP? The limit method is 3 steps by my count.
How did you find it more straighforward than the method using the product rule?
The method using the product rule is actually just 2 steps.
Provided you know the differentiation of [x] for the two possible cases.
8:45 PM
How do you know the derivative of [x]? (That 's what I was asking)
You have to remember that as a result as I said earlier.
For integer values of x, differentiation of [x] is undefined while for non-integer values of x, differentiation of [x] is 0.
BTW, can you help me with this:
For four proper fractions $a, b, c, d$ X writes $a+ b + c >3(abc)^{1/3}$. Y also added that $a + b + c> 3(abcd)^{1/3}$. Z says that the above inequalities hold only if a, b,c are positive.
(a) Both X and Y are right but not Z.
(b) Only Z is right
(c) Only X is right
(d) Neither of them is absolutely right.
Hey everyone!
9:02 PM
00:00 - 17:0017:00 - 22:00

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