Find the expectation of the random variable $X^2$ when $X$ is a uniform random variable in $(0,1).$
I tried solving the problem as follows:
$X$ is a uniform random variable (URV) implies that, if $f_X$ is the probability density function of $X$ then, $f_X:\Bbb R\to \Bbb R$ such that $f_X(x)=1,x\i...
I am reading a proof of the fact that if $E\subset\mathbb R^n$ is bounded, then it is totally bounded. It suffices here to prove that $T=[-b,b]^n\supset E$ is totally bounded, as any subspace of a totally bounded space is totally bounded. For this, let $\epsilon>0$. Then the balls $B(\epsilon j,\epsilon)$ cover $T$, where $j=(j_1,\ldots,j_n)$ ranges over all integral lattice points of $\mathbb R^n$ which satisfy $\epsilon |j_i|\leq 2b$, $1\leq i\leq n$.
(Recall, an integral lattice point is a point whose coordinates are integers.)
Since there are only finitely many such lattice points, $T$ is totally bounded, which completes the proof.
Question: How can I verify that the balls $B(\epsilon j,\epsilon)$ cover $T$? In particular, why the condition $\epsilon |j_i|\leq 2b$, $1\leq i\leq n$?
@psie Here's my reasoning so far: if $x\in T$, then $x_i\in [-b,b]$ and $\epsilon j_i\in [-2b,2b]$, $1\leq i\leq n$. Now I want to show $d(x,\epsilon j)<\epsilon$, but I am not sure how to do that. I suspect one would like to consider a third point and use the triangle inequality somehow. Not sure.
@psie It'd be easier if you draw a picture. For example, consider [-5,5] in R. Say you want to cover it by balls of radius 1/3. Now if you just put such a ball at each of the points -5,-4,...,4,5 then it won't cover the whole interval. So you need more points where you can place a ball.
And about the 1st question, suppose you have a point in R^2, and you place a ball of radius 1 at that point (the point is the center of the ball ). Can you claim that the ball contains an integral point?
Yes, though you have to argue that the nearest lattice point from any point in the plane is at a distance of at most $\sqrt{2}\over 2$
Anyway they are doing the same thing here, they are reducing the balls from radius 1 to $\epsilon$ but they are also adding more points
Another approach would be to consider the biggest cube that lies inside the ball of radius $\epsilon$ and then cover the space with such cubes and then replace each cube by a ball
ok, hmm, I need to think about this some more probably. I still have trouble seeing how some $x\in T$ is contained in $B(\epsilon j,\epsilon)$ for some $j\in\mathbb R^2$ satisfying $\epsilon |j_i|\leq 2b$.
@SoumikMukherjee ok, for example, if $b=1$, $n=2$, $\epsilon=1.5$ and $x=(0.5,0.5)$, then it should be contained in the ball $B(\epsilon j,\epsilon)$ where $j=(0,1)$. Using the triangle inequality and $j$ as the third point (and the max metric) $$d(x,\epsilon j)\leq d(x,j)+d(\epsilon j,j)=\max\{|0.5-0|,|0.5-1|\}+\max\{|0-0|,|1.5-1|\}=1<\epsilon,$$but does this procedure generalize?
I don't see yet where I specifically used $\epsilon |j_i|\leq 2b$.
@SoumikMukherjee probably, but to show that $T$ is included in $\bigcup_{j\in I} B(\epsilon j,\epsilon)$ where $I=\{j\in\mathbb R^n:\epsilon |j_i|\leq 2b\}$ we simply need to show that $x\in T$ is included in some ball
There are a number of theorems or lemmas or mathematical ideas that come to be known as eponymous
tricks, a term which in this context is in no sense derogatory.
Here is a list of 11 such tricks (the last of which I learned at MO):
the Whitney trick
the DeTurck trick
the Cayley trick
the Rabinow...
I have this integral: $\int_D \frac{y \sin\sqrt{x^2+y^2}}{x^2+y^2} \text{dx dy}$ , on the domain D of the first quadrant delimited by the x axis and by the circle of radius 1 and centers O (0,0) and C (1,0)
Given $\varepsilon > 0$, the vector $\frac{\varepsilon}{2}(1, 1, ..., 1)$ will be distance $\geq \sqrt{n}\frac{\varepsilon}{2}$ away from each vector of the form $\varepsilon j$ where $j$ is an point on the integer lattice.
so for $n\geq 4$ it won't belong to any ball of the form $B(\varepsilon j, \varepsilon)$
I'll be systematic here, I think it can help.
D Let $S$ be any subset of $\Bbb R^n$. Given $\epsilon >0$, we say that $N$ is an $\epsilon$-net for $S$ if the set of open balls
$$B_\epsilon(N)=\{B(x,\epsilon):x\in N\}$$
covers $S$. That is, the set of open balls of radius $\epsilon$ centered at...
I'm pretty sure one just has to go for $B(\frac{\varepsilon j}{\sqrt{n}}, \varepsilon)$ instead, the inequality $\varepsilon |j_i|\leq 2b$ may be changed too
Using polar coordinates, you should have $\theta \in [0,\pi/2]$ and $0\le r \le \min\{2\cos \theta,1\}$
you can split $\int_0^{\pi/2}=\int_0^{\pi/3}+\int_{\pi/3}^{\pi/2}$
to find the bounds, express $D$ in polar coordinates. $y\ge 0 \implies r\sin \theta \ge0$, i.e. $\theta \in [0,\pi]$. $x^2+y^2\le 1\implies 0\le r \le 1$
I find rudin a bit "uninspiring" in that, I tend to look into books like, eg. Shirali & Vasudeva- Metric spaces, or Searcoid Metric spaces for intuition, which Rudin is devoid of. But that said, Rudin definitely does teach the techniques quite well (albeit in a few pages). I kind of like that about it
@nickbros123 I dislike baby Rudin because I think that it is overly terse and fails to be a pedagogical book---it is a reference, not something you are meant to learn out of. There is no intuition, and some of the stuff he does is, in my opinion, just backward. There are lots of better books for the material. Spivak and Apostal each have books, for example. I kind of like Tao's approach. Even Folland's undergrad analysis text is better (in my opinion---and I am not a huge fan of Folland).
@SineoftheTime That question is nonsense, I think. University professors don't read books to be knowledgeable about what they teach (or, at least, that is not all that they do).
I would expect someone teaching calculus to at least be knowledgeable in real analysis. I would also expect them to know some diffy q and linear algebra (rigorously; beyond just being able to do computations). They should know where the ideas taught in calculus come from, how to prove them, and how to prepare students for future classes.
Moreover, I would want a professor to have read several books on single variable calculus before teaching the class. I want them to have considered the approaches of different books, and made an active choice to choose a book which best reflects how they want to teach the class.
@SineoftheTime Already, that makes no sense to me. In the US, there are 2 year degrees (associates and masters), 4 year degrees (bachelors), and phds (which typically take at least 4 years, depending on field).
So I don't know of a program of study for which 3 years is normative.
As such, I doubt the advice I could give you would apply to your situation, as you clearly don't come from the US, and so things are going to be very, very different.
Ah. Sure. But it is also worth noting that European schools tend to start tracking students into college prep classes much earlier than in the US. So you miss out on some of the liberal arts in favor of specializing much earlier (high school students in a college preparatory track miss out on some gen ed, but get more specialized training younger).
there is no real "college preparation" but for example the university I used to go did organize some classes for high school students to prepare them for university
but I didn't even participate in those classes since I didn't know they exist
so yeah, I don't know how it's in other countries but at least in here it's not like we have some kind of preparation class mentality
(If you really want to know the answer to the question, you can look through the list of people who have earned a badge for voting in the last week, and determine whether or not my name is among them.)
@XanderHenderson Just curious, as you asked most of the questions that were asked to the candidates in the comment section, so whether any candidate seemed eligible to you( ofc I won't ask who you have voted for, if any)
@Shaun no need to take a look. We found basic errors... -__- anyhow back on my ArrowGlue code for now, and I'll put twin primes on pause. It takes a lot of mental energy to work on it and doing it all day just isn't productive ^_^
I do however like the approach of prime-power equations, because there's a lot of group theory, just have to find the right framing of it, because there are actually many ways to frame it.
For example the minimal element > 1 in the group of units modulo $p_n\#$ is most definitely the residue $p_{n+1}$. But take some certain subgroups of the group of units modulo primorial, and how do you compute the "smallest non-trivial residue" in the group? That is essentially what I'm trying to answer. Or at least show that it can't be > $p_{n+1}^2 - 2$ and then we're done; twin primes would be proven.
@Jakobian I think indeed you have found an error in the book. Thanks. Folland has another proof which is correct I think, but I'd really like to amend this proof; you suggest the balls $B(\frac{\varepsilon j}{\sqrt{n}}, \varepsilon)$. Under which condition on the components $j_i$ would they work? Grateful for any explanation.
@jasmine because if you define $(a+\mathfrak{m})\cdot (b+\mathfrak{m}^3) := ab+\mathfrak{m}^3$ then this might not be well-defined since if $a'\in\mathfrak{m}, b'\in \mathfrak{m}^3$ then by the same definition $(a+a'+\mathfrak{m})(b+b'+\mathfrak{m}^3) = ab+a'b+\mathfrak{m}^3$ where the term $a'b$ isn't necessarily in $\mathfrak{m}^3$
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@psie I think $\frac{\varepsilon}{2\sqrt{n}}|j_k|\leq b$ should work, but you can always go through the argument and check
@JakobianIf $a' \in m$ then $a'\in m^3$ because ideal $m$ contain $m^3$ .This implies $a'b \in m^3$. I don't understand why the term $a'b$ isn't necessarily in $m^3.$
I'll be systematic here, I think it can help.
D Let $S$ be any subset of $\Bbb R^n$. Given $\epsilon >0$, we say that $N$ is an $\epsilon$-net for $S$ if the set of open balls
$$B_\epsilon(N)=\{B(x,\epsilon):x\in N\}$$
covers $S$. That is, the set of open balls of radius $\epsilon$ centered at...
