@BalarkaSen Umm, is this also for hyperbolic 3-manifolds with conformal boundaries? I mean manifolds with nonempty boundaries which is not necessarily toroidal.
Hello there. In some lecture notes I'm reading on probability, the author keeps talking about an additive functional. I wonder, what is an additive functional? For example, the author writes the notion of volume is an additive functional and that of a probability is also an additive functional. I'm familiar with linear functional, but this requires a normed linear space, and is certainly a different setting. Any ideas what they could mean by this?
@Anacardium Take $z\in U$ with $\rho(z) = 0$. Since $z$ is not a local minimum of $\rho$, we need to be able to find a sequence $z_n$ such that $z_n\to z$ and $\rho(z_n)<0$
@psie It would be helpful to know the source of that image, and (in particular) the text directly preceding it.
In general, a functional is just a function from some space into $\mathbb{R}$ (or $\mathbb{C}$). As you note, you see linear functionals a lot in linear algebra and functional analysis.
My guess is that the authors here are thinking of probability measures as functionals (which is a not-uncommon approach), and are asking for additivity in the usual sense for measures (e.g. $\mu(A\cup B) = \mu(A) + \mu(B)$ for disjoint $A$ and $B$).
@Jakobian: Oh! Now I am getting the point you are using the second condition $(d \rho)_z \neq 0$ for all $z \in U$ to ascertain that $z$ cannot be a local minima. That's what I was missing.
This shows that such a $z$ should be in $\overline {G}.$
@psie as xander suggests, they mean simply a function whose codomain is the reals (perhaps also whose domain is an algebra of sets, from looking at other uses of the term). it would have been better practice for the author to have defined this usage (looking more generally at grigoryan's notes his mathematical english is somewhat idiosyncratic; for example, he refers to a "coin tossing" for what more native speakers would call a "coin toss")
as another example, the page before the examples of 'functionals' the author states "but now we can recall the familiar from the elementary mathematics notions, which are all particular cases of measure:" and this is not at all how you should phrase that in english, unless you are trying to sound like a german spy
i once went to a talk by a prof who had lived at least 40 of his 60+ years in russia before moving to a country in south america, and he combined "stage russian" grammar with substituting words that he didn't know in english with their spanish equivalents, and the whole thing felt like a vaudeville routine that would get anybody canceled if it wasn't real
In my first year of college, I tried to take German. But the German prof was married to one of the Russian profs, and he spoke perfect Russian. For whatever reason, whenever I couldn't come up with a word in German, I would switch to Russian, and then he would respond in Russian, and we would have whole conversations in Russian.
While the rest of the class looked confused. He would then usually end the conversation by saying something to the effect of "Okay, it's cool that you can Russian, but this is a German class. Can you try that, please?"
After learning Python this month (and utilising it for some numerical algorithms), I've realised how grossly I underestimated the convenience of a computer for computations, especially in linear algebra. I used to just skip computations if they got too cumbersome, but I see more opportunities now. Am really glad I finally got over my programming phobia.
@ShaVuklia During my masters program, I took a year of numerical analysis, during which we did a lot of work in C (not C++, plain ol' ANSI C) and FORTRAN. I can't say that I have ever used those skills, but the classes were a lot of fun, and gave some nice insights into how a computer does what it does.
@XanderHenderson Glad it was fun. I tried taking a CS course (Datastructures) some years ago where we had to program in C, but it was just a disaster for me, as I was trying to rush it. After that experience I didn't dabble in programming anymore (even though previously I'd enjoyed an introductory course in Java). Am glad to be back, though I'll stay in my lane with just simple Python scripts :'D
Its not about the words themselves, or the grammar, though grammar is totally different too. Which is also something that confused me a lot in high school.
The fact that the teacher treated us like we are supposed to know the language didn't help and only discouraged me from learning it
maybe that can be put under bad education
I never actually sit in my room and studied something... other than math. That was never part of my routine
Maybe thats why I feel like I'm being too soft for myself
To study for $k\in\Bbb R^+$ the punctual, absolute and uniform convergence in $\Bbb R$ of the following
series of functions:
$$\sum_{n=1}^\infty\frac{\cos n^k x}{1+n^k}$$
One has obviously, for each $x\in\Bbb R$:
$$\left|\frac{\cos n^k x}{1+n^k}\right|\leq\dfrac1{1+n^k}\qquad\forall x\in\Bbb R,$$...
@leslietownes cool that you found the author :) indeed it is from Grigoryan's notes (he has some good notes on ODEs, and therefor I looked up his probability notes). But additive functional is certainly an unusual name for, what I believe he means, an additive set function. Whatever. Oddities like this used to get me down a lot, but not anymore, since I make'em myself...
