am I understanding the question correctly if I read my task is to write an expression for a particle which travels around the circle starting infinitly close to (-1,0) and ends it's journy infinitly close to point (-1,0)?
How many homomorphism are there from $ Z $ to $Z_{n}$? I know that the kernel would be $nZ$, and then by first theorem on isomorphism, $ Z/nZ$ will be isomorphic to $Z_{n}$ and the number of automorphisms on $Z_{n}$ is $/phi (n) }, but the answer given is $n$. Where am i wrong?
There's a theorem in my lecture notes that says, "All piecewise continuous curves [in a closed and bounded interval] are rectifiable". Isn't that wrong? Counterexample: $f(x) = x \sin(1/x)$ for $x \ne 0$ and $0$ for $x = 0$
I was able to prove this for piecewise continuously-differentiable but I don't see in what sense of the term piecewise continuous does this theorem hold
I did talk to the professor about this, and he is quite adamant. He asked me to think about the infinite broom and topologist's sine curve (which quite literally is a counterexample) and come up with an answer
Sorry, I didn't mean to say that it's a counterexample
Can anyone tell me if my working is ok ? I want to show $ \int ( V * f (x) - V * g(x) ) f(x) dx \leq C W_2^2(f,g) $ for distributions f and g. and $W_2$ the wasserstein metric
@AndrewMicallef fields medalist and one-time employer of me as a grader for his abstract algebra class Richard Borcherds does some of the details of this calculation around the 6 minute mark of youtube.com/watch?v=JZKDmTIFR7A
@AndrewMicallef it doesn't matter. he wants you to parameterise the circle less $(-1,0)$ by a scalar parameter $t$. however, the picture gives a very strong hint of one straightforward way of doing so.
it does say 'length' and not signed length. maybe people are uncomfortable with t being larger than 1. all of this reminds me of a funny comment on the dot product
i did yesterday. that first vaccine shot really did me in for about 24 hours. now, i'm OK.
i've just set up a fairly unpleasant phone call for tomorrow. in my day job i play-act as somebody who is constantly raising objections, waving red flags in front of bulls, and looking for trouble. this is not who i am.
@leslietownes Don't feel too bad. My son passed out right after and they hadn't made sure his car was in park. He drifted into the car ahead of him and tapped bumpers. They really should have made sure he was ready, especially since he had sounded uneasy.
i did see someone pass out at the vaccination site. needles are a weird thing. my wife is not that good with them.
they recommended that we sit for 15 minutes after the vaccine. i high-tailed it out of there like the jerk that i sometimes pretend to be. but then my arm began to hurt and the brain fog descended. it is better now.
@leslietownes I didn't even know that the first injection had been given. The person who gave it to me said "all done" and I was amazed. The second shot was given by a guy who was less gentle and I felt a bit of pressure and heard the squoosh of the hypodermic, but didn't feel any pain. My wife felt a bit on her first injection, but she had Moderna and got "Covid-arm".
@copper.hat At CSUN, I had to wait 15 minutes and they checked on you several times. My wife went to Hansen Dam and they told us to wait in a side parking lot. No one checked on us and we left with no one checking. I guess if you have a seizure, you have to go get someone to help.
at albany i stood around so they could see if something was going on, at the coliseum they lined all the cars up for each 15 mins group, so i guess if you didn't move they would check. hard problem with low frequency potentially disasterous events.
I am not consistent. It depends very much on my take of how hard they are trying.
In giving, I give to those who cannot help and to those who are willing to try. Perhaps selfish, but there you are. I will not help those who can help themselves and are not willing to try.
It is a tough problem to deal with the range of possibilities out there. At some point one needs to make choices purely from an energy perspective and those choices inevitable impact some.
And energy includes mental setup and thought processes which may be limited by many factors.
I am going to stop pontificating before I make Ted shout...
If you have 6 projections $(x,y,0),(x,y,1),(x,1,z),(x,0,z),(1,y,z),(0,y,z)$ for $x,y,z \in (0,1)$ and you are given 6 congruent vector fields on the projection planes, how do you construct a vector field in $(0,1)^3?$ Would taking an average of the 6 projections to construct a vector for $(x,y,z)$ work?
@TedShifrin Ah ha, there was an ambiguity... i think? But now I think I have seen this one before. Though I wasnt paying close enough attention to how the answer was derived (dont tell me, this is for lunch time math)
To be clear, $t$ could be any scalar according to @copper.hat, but I am going to think of it as the height of the intersection of the bolded lineseg and the yaxis
Let $X_1,...,X_n$ be the incomes of $n$ person chosen at random from a certain population. Each $X_i$ has the Pareto density $$f(x,\theta)=c^{\theta}\theta x^{-(1+\theta)}, x>c$$ where $\theta>1, c>0$.
I have obtained th mean income $\mu$ in terms of $\theta$ as:
$$E(X)=\int_1^\infty xf(x)dx=\int...
@AkivaWeinberger The distribution function of $Z = \min\{X^2, Y^2\}$ is given by $$F_{Z}(z) = \mathbb{P}(\min(\{X^2, Y^2\} \leq z) = 1 - \mathbb{P}(\min(\{X^2, Y^2\} > z) = 1 - \mathbb{P}(X^2 > z \wedge Y^2 > z)$$
No, it would be and. If you know that the minimum of two numbers is greater than a value, then it follows that both numbers must be greater than that same value
Dammit, ahwell...also what are you doing using inkscape for your graphics @TedShifrin. I would gladly offer my services as an amatuer artist to render bueatifull geometry drawings in tikz/latex
It makes sense from an intuitive sense as well that you'd want to max it rather than min it, because squaring (and cubing) makes things from 0 to 1 go lower, so you need something to make it go back higher
@Andrew: I did the graphics for all four of my books using (Mathematica and) Adobe Illustrator. However, Illustrator no longer runs on my computer and I'm not willing to pay $30 a month or whatever outrageous amount Adobe demands. So if I need to change my graphics, I need to do so in Inkscape. In this case, Leslie and copper convinced me that I should have a one-sided arrow rather than two-sided in that figure, so I changed that.
It took me 10 years to get good with Illustrator, and this is the first thing I actually had to use Inkscape to do. First have to save the .eps file as .pdf and then import it to Inkscape, blah, blah, blah.
So what do we have here... the density is $$f_{Z}(z) = 2(1-\sqrt{z}) \cdot \dfrac{1}{2\sqrt{z}} = \dfrac{1-\sqrt{z}}{\sqrt{z}} = z^{-1/2} - 1$$ for $z \in (0, 1)$.
My guess is that this is some sort of generalized beta distribution.
The divergence of the harmonic series is familiar: the partial sums of positive integer reciprocals grow without bound. A less familiar but still well-known result is Kempner's series: If we only use integers whose base-10 expansion contains a 9, then the sum of reciprocals converges. The intuti...
I miss my teaching (as you can usually tell in here), but I'm relieved that I retired almost 6 years ago before COVID and before bureaucracy in GA got even worse.
The tricky thing is that I had to go deep into the college website where these pages are stored to remove my old version so that the new version would have the same name (rather than .0 or .1), because I have links to that page on MSE, for example. Technology can be a pain.
I changed the date. What else changed?
Something had to change in the TeXLive implementation of stuff, as I ran into running head issues with my index.
And the typesetting of the answer section added some space. I didn't change the file at all. But I ended up with a longer section there.
