@CameronWilliams: Depends on what is meant by 'solve', I'd say. Certainly there isn't a unique answer unless other conditions are given; all the same, it is straightforward to characterize the set of solutions as $\frac{1}{2}A$ plus some anti-symmetric matrix. (i'm putting this in chat b/c i'd rather not give the answer away on the question page)
haha, so it asks to determine whether the interiors of connected sets are always connected, to which I have a counterexample in $\mathbb{R}^2$: $\overline{B}_1(0, 0) \cup B_1(2, 0)$.
The problem I'm running into is this: working in $\mathbb{R}^n$, it seems pretty difficult to rigorously prove that sets are actually connected. I don't really want to say "Obviously this is a connected set."
I mean, it seems to me that you would assume that the set were disconnected, then try to arrive at a contradiction in coming up with a separation for it. But even for simple sets like a $k$-cell in $\mathbb{R}^2$ I cannot think of how to go about this.
Working in the plane, I've defined notions of "Horizontally connected" and "vertically connected", wherein a set $S$ satisfies the former if $\{(x, y) \in \mathbb{R}^2 : y = k\} \cap S \neq \emptyset$ if and only if $k \in [a, b]$ for some interval.
Vertically connected is defined likewise, but instead $S$ is being intersected with $\{(x, y) \in \mathbb{R}^2 : x = k\}$.
I conjecture that a set $S$ is connected if and only if it is both horizontally and vertically connected.
I feel like pictures would simplify the argument, but neither am I to be honest.
This definitely works in just $\mathbb{R}$ since there is only a notion of "horizontal connectedness", and of course the only connected subsets of $\mathbb{R}$ are intervals.
So how would one show that, even simple sets like a close ball centered at the origin, are connected, especially given how few tools Rudin has afforded us at this point?
Just with Dr. Fu + Rudin knowledge, I know nothing other than a definition of connectedness and the fact that a subset $\mathbb{R}$ is connected if and only if it is an intervals.
Granted, it might be sufficient that I just give my example, but not justifying that it's actually connected bothers me. Of course, I can show the interior isn't by simply demonstrating a separation.
I think you could probably prove that the ball is connected following your lemma earlier. If you think about the interval from $(-1,0)$ to $(1,0)$ and take the union of all the connected vertical line segments along it, you can probably argue it's connected. I'm not sure what Rudin wants with this problem. You'll have all the machinery when you take 4200.
Maybe the proof would be easier using a line segment joining two distant open balls?
Then the separation would have to give you a separation of the line segment.
Ah, note #21. He has you prove every convex subset of $\Bbb R^k$ is connected.
where does the definition of product of complex numbers come from? Do we use it because it works (i.e., because it gives rise to a field) or was it chosen for some other geometric or algebraic reason?
Yes it's a course ;) Your right. No I just wanted to see if you had done :) If so, you would understand its format and maybe good tips to do well...since that's the case with standardized tests, there is a prediction know
it wasn't a grammatical mistake or something, I wanted to be nice and see if you had any questions to discuss :) and then if so first help you, and then ask my question ;)
I know how to do the (i) and I will put out the results, just in case the second part is related to the first part answers.
The C.D.F :
$$F(t) = 1 – (t – 1)^{-2}$$
$P(T>5)$ :
$$= 1 – F(5) = 1 – (1 – 4^{-2}) = 1/16$$
Now how to do the (ii) part, I don't understand. Please help. I think N ...
I'm prepping for the Math Subject Test GRE and the Putnam, and I'm taking any linear algebra questions in order to challenge myself. If you have any interesting questions, send them my way.
Hi @DanielFischer, out of interest do you know why the author would adopt such a unconventional definition of 'limit of a function', does it give some advantage in any way?
@Alex It's not so unconventional. In some parts of the world, it's the usual definition. Since the limit of a function at a point is only really interesting when the point is not in the domain, it doesn't matter much.
@EricGregor I do not understand why they bring up $f$ again, after defining $g$ (I am not asking for hints or solutions, but rather clearification about what the problem itself)
@semiclassical what I wrote is pretty much everything i know about the mathematics of the problem. the physics comes in in the following way: the dimension of the two-partite hilbertspace is n. the number m stands for the schmidt rank of some arbitrary two-dimensional entangled state in that n-dimensional hilbertspace. the p's stand for probability of different m-dimensional entangled pure states, forming together an arbitrary m-dimensional mixed state.
@semiclassical i find the problem nice, physically as well as mathematically. in the mathematical context, i find the cyclic symmetry as well as the potential connection to the shapiro inequality beautiful.
@Semiclassical correction "schmidt rank of some arbitrary two-PARTITE entangled"
@semiclassical it would be very nice :)
@semiclassical (oops, again: correction " the number m stands for the schmidt rank of some arbitrary m-dimensional two-partite entangled state in that n-dimensional hilbertspace.") now it should be correct
i have the looser criterion of down-voting answers which are not useful. something can be technically correct while still being useless/counterproductive to the reader
Right, but the ring/algebra must come equipped with the derivation, and his notation $(R,\text{blah})$ means $R$ is the ring and blah is the derivation.
@Huy: A writes down either 1 or 2 on a piece of paper, and B must guess each one. If the number A has written down is $i$ and B has guessed correctly, he gets $i$ dollars from A; if he is wrong, he pays $3/4$ dollars to A. If B randomizes his decision by guessing 1 with probability $p$ and 2 with probability $1-p$, determine his expected gain if A has written down 1, then if A has written down 2. What value of $p$ maximizes the minimum possible value of A's expected gain?
@Sawarnik: It was indeed an optional course, but I myself would never had been interested in one about things like rigorous calculus in high school, so I doubt many pupils would be.
@Sawarnik: Why not teach something more applicable to a broader spectrum of people? And not so technical.
In balance, @Huy, I've taught the Spivak course to a number of talented high school students and they have enjoyed it. But it was their entire calculus course. Your students have already taken a calculus course, so it's probably more interesting to do something different rather than embellishing with technical stuff.
@TedShifrin: Yes, but also they didn't actually finish a calculus course. I'm just teaching them differentiation right now and next semester I'll teach them integration.
@Semiclassical when you give the problem a try, even if you dont solve it, please leave some comments on your thoughts/what you found and what didnt work. would be nice, thx!
