Yes, isomorphic vector spaces have isomorphic complexifications.
If you're asking how to "implement $\mathbf{R}^{+} \otimes \mathbf{C}$" without unwrapping the isomorphism with the usual vector space structure on $\mathbf{R}$, I don't see a way to do this offhand. At first glance it's tempting to think of the complexification as $\mathbf{C}^{\times}$ with complex multiplication as "addition", and exponentiation as "scalar multiplication". The fatal problem is, the complex exponential mapping is not injective (unlike the real exponential mapping), so to make sense of the operations you end …
Also, I think i have pretty solid evidence that the AI overlords are well in control of my locality
Went to the supermarket to stock up on surplus Chocolate Holiday Bunnies, to find the supermarket had exactly stocked the right quantity and had no significant excess to be marked down
oh, yeah, no, I do have a partner who gets easter chocs from her PhD supervisor, but otherwise all chocolate has had to be self purchased in this household ;P
I think It's all the inane questions I am bothering strangers with
Also it sorta serve me right too. She suggested we stock up on chocolate before the weekend and my response was : Nah, they are bound to be overstocked and will have to mark all these things down in a couple of days
little did I know there was a highly calibrated statistical engine in control of stocktake
Like, I have always liked the idea of coffee. The smell and the sight of one, but I drink coffee diluted down to the 1:100 with milk and generous helping of sugar
and while I like the idea of maths, for the longest time, it was really only in the form of pop-maths books and youtube videos.
Ok I think I get it now. I was trying to go from a Dedekind cut to a sequence of rationals that converges to that cut. I guess if you don't start with something like a cut, you have to use a notion of a sequence that doesn't talk about what it converges to
hey, so I've noticed that $\mathbf{a}\times \mathbf{b} = \gamma(\mathbf{a}\wedge \mathbf{b})$, where $\gamma\colon \bigwedge^2(\mathbb{R}^3) \to \mathbb{R}^3$ gets cyclic basis elements to the next one (i.e. $\mathbb{e}_i\wedge \mathbb{e}_{i+1} \mapsto \mathbb{e}_{i+2}$, indices mod $3$)
is there a easy way to prove that $\mathbf{a}\cdot \left(\mathbf{a}\times \mathbf{b} \right) = 0$ using exterior algebra?
obviously you want to get something like $\mathbf{a}\wedge \mathbf{a} \wedge \mathbf{b} = \mathbf{0}$
@robjohn I want to buy your stackexchange game account
user435118
10:14
The volume of a container can be found using the formula v = 16x + 4/x where x> 0. Use calculus to figure out the minimum value of V. Can I have some help with that please?
@AndrewMicallef I've been using Python for a few years. Eg, stackoverflow.com/a/33715116/4014959 It's fine for numerical work, but it runs slower than languages like C++ & Fortran which get compiled to machine code. Depending on what you're doing there are libraries like Numpy that can speed up calculations on large arrays of numbers.
If you need more precision than standard 64 bit floats (aka IEEE-754 binary64) then there's the standard decimal module, and if you want more functions than decimal provides, there's the 3rd party lib mpmath.
Catastrophic cancellation may be an issue with that function you (Andrew) posted earlier that has a subtraction in the denominator. Sometimes you can fix that sort of thing by rearranging the calculation, and checking that a & b aren't too close if you have (a-b) in the denominator. And sometimes you can fix it by simply using numbers with higher precision.
@LucasHenrique Yesterday, I was messing around with calculating cosines of ridiculously large integers that are close to odd multiples of pi/2 . They were behaving as expected until the numbers got around 10^15 or so, then I started getting stupid answers. Fortunately, I was able to fix it by going to higher precision. See my comment here for details: math.stackexchange.com/q/4086857/207316
I guess I should also mention SageMath, which is a gigantic framework of mathematical stuff built on top of Python. And SageMathCell, which lets you run Sage anywhere.
It's very handy to be able to write a Sage / Python script on my phone, and then compress it into a URL that fits into a comment or chat message.
@AndrewMicallef Ah, I see you've posted in the SO Python room. Andras is a quantum mechanics professor, and he knows a fair bit about using Python for numeric work. He's also worked a fair bit with Matlab and Fortran.
there's a good movie about banks accepting money for the purpose of lending. "it's a wonderful life."
some people see sentimental aspects in its plot but it is mostly about the need for deposit insurance, meaningful capital requirements, and banking regulation generally.
the problem with 'it's a wonderful life' is in the alternate timeline, pottersville is supposed to look like this hellhole, but it looks like a whole lot more fun than bedford falls.
it's one of my favorite movies. my wife has never seen it. she resists seeing stuff that is too out there. she thinks that nothing that airs every year on TV can be any good.
I haven't owned a TV for years. But I have binges of watching music videos on YouTube once a week or so. I have pretty broad musical tastes, which I guess confuses the YouTube algorithm a bit. ;)
oh, you should see a serious man. it is a treat. it's very funny, but it's also something more than that. i don't know how to describe it. the main character is just doomed. he is helpless in an indifferent universe. how can i sell a movie better than that
oh brother where art thou is the first movie i enjoyed george clooney in. he is also great in 'burn after reading.'
brad pitt is also amazing in that. he was somehow a great actor the whole time. i missed it.
@leslietownes Fictional mathematicians are often fairly broken characters, unless the author happens to be a mathematician or scientist, as discussed here: kasmana.people.cofc.edu/MATHFICT/links.php
@leslietownes I'll bear that in mind. I'd like to see more Coen brothers stuff. They seem to know what they're doing.
yeah, that's the problem. there is a lot of romanticization of what is basically mental illness. also romanticization of the 'genius' stereotype. i knew a few people who worked with john nash, during his illness. there was nothing great about any of it. prevented me from enjoying 'a beautiful mind' although it was a well made movie.
sometimes romantic stereotypes prevent people from getting help, or acting in obvious ways that they would act if 'genius' wasn't in the mix.
Yeah. It can be pretty disheartening & depressing when somone goes through a psychotic breakdown, especially when they're brilliant. And especially when you know them well.
We occasionally get some pretty way-out theories posted on the Physics & Astronomy stacks. Sometimes, they're probably just the result of an over-active imagination combined with misinformation and lack of actual scientific knowledge. But sometimes I get a strong feeling that the OP is delusional, and should seek professional help. But what can you do? They're hardly likely to see a psychiatrist or even just a regular doctor on the advice of some random internet person.
a lot of 'cranks' (loaded term but i'll go with it) are harmless, unless they are focusing on their hobby horse to the exclusion of earning a living, or supporting any people that they are responsible for. which tends not to be the case. they tend to be OK. but sometimes i do wonder.
i did see a few people break down during grad school. the 20s are a time when stuff tends to come up. none of it was pleasant and it probably would have been caught sooner if they were working at home depot or something and not in a math department.
Hello, I have a question about using Ito's lemma to solve SDEs: solve $dX_t=X_tW_tdt+dW_t$, $X(0)=X_0$, where $W_t$ is the standard brownian motion. I try to use $exp(\int_0^t W_sds$ as the integrating factor, but I cannot get the $dW_t$ term
if $X$ is compact, path connected and locally path connected, and $Y$ is a path connected, locally path connected ,hausdorff space, is every map $f : (X,x_0) \rightarrow (Y,y_0)$ such that $f_{\ast} \pi_1(X,x_0) = \mathbb{Z} \subset \pi_1(Y,y_0)$ necessarily homotopic to a map that factors through $S^1$?
where by factor through $S^1$ I mean of the form $X \rightarrow S^1 \rightarrow Y$
i've mentioned this before. some of the TAs in my graduate program would go through this whole ritual of "oh, i'm sorry, did you think log meant base 10 log? it's base e log. everybody knows that." they tended to have been educated at private schools where maybe sensitivity to context and perspective was less important than acquiring badges of rites of passage.
@leslietownes certainly not universally... but the odds on this person having been raised on "log without a base is implicitly log_100" rather than it just being a typo... I'm just looking for a sanity-check. (I only got 4 hours' sleep last night, so maybe I'm being extra-cautious!)
you can make 10*9*8 ordered selections of three people from a group of ten people. but disregarding order, there will be 3! copies of every selection that you make.
Google Calculator is pretty awesome. It understands a huge number of different units. But sometimes it doesn't get that you're trying to do a calculation, so it just attempts to do a normal search.
A few months ago, I noticed that Google Calculator uses the wrong value for a light year. I tried to complain about it, but the value's still wrong. :( A light year should use the Julian year of 365.25 days, but instead they use the mean tropical year of ~365.242199 days. Try (1 light year)/(1 light day)
Definitions — As defined by the International Astronomical Union (IAU), a light-year is the distance that light travels in vacuum in one Julian year (365.25 days) ...
one time i emailed tumblr about how their API seemed to allow submitting posts from different dates, so you could do like the diary of samuel pepys or something on tumblr, dating everything with the real dates in the 1600s. but in practice they rejected any dates before unix time zero and it really messed a blog up.
someone wrote back and was like "uh, yeah, we'll totally get to this." they never got to this.
no, they didn't. i understand they were owned by yahoo, which ran out of money. i get why they didn't fix it. but it would have been a simple fix.
their documentation suggested that something was possible that wasn't possible. i knew when i was writing the email that whoever had been responsible for that implementation had probably been fired years ago.
Date handling in programs can be painful, depending on what your libraries provide. Sometimes, you just have to ignore the standard library, and write your own date handling stuff from scratch, if someone else hasn't already done that.
Lots of systems can't cope with leap seconds correctly, even though the standard C library has provision for them. Google copes with leap seconds by "smearing" them out. developers.google.com/time/smear
I guess the most famous example of date handling ineptitude is the Excel leap year bug: it considers the year 1900 to be a leap year because Lotus 1,2,3 had that bug, and when Excel was originally developed they decided to preserve the bug for compatibility reasons.
For more info about timekeeping than any sane person should ever need or want, see this excellent page by Lick Observatory astronomer, Dave Allen: ucolick.org/~sla/leapsecs in particular, A brief history of time scales.
this is exactly the kind of stuff i like to see on the internet. it reminds me of when everything was simpler. the 90s. it was just pages like that and not corporate crap and troll farms.
i'm loving the HTML 1.0. or whatever. feels like home.
@leslietownes I used to frequent a (now defunct) science forum, where each day one of the top contributors would post a Samuel Pepys diary entry for that day. I don't think he managed to post the whole thing, though.
@leslietownes :) On his home page, he warns that if you send him email that isn't plain text it will be ignored.
it's actually good to edit it out. there's quite a lot in his unedited diary about him masturbating. i don't think that needs to be reproduced. except for scholars, of course.
Hey all, I am using Math SE Chat for the first time, this is such a cool place! I plan on self-studying undergrad probability theory from the books (1) Feller (2) Ross (3) Grimmet Stirzaker over the next 7-8 months. The idea is to set weekly goals work on a good mix of problems.
I would like to ask, if anybody out here is interested to co-study, we can divvy up the problems and discuss our solutions/challenges.
i had some other good probability books. bruno de fanetti. somewhere above our garage i have all of this finitely additive, not countably additive probability theory. some of the books are autographed, although not to me.
one time in a final exam in measure theory in graduate school i proved a finitely additive version of the theorem and then took a limit. my professor told me it was an insane way to solve the problem. there was a simpler way to do it. i spent way too much exam time on it. he thought it was funny, though.
it has a section called "how to gamble if you must." everything is connected. there is some analysis of horse racing and card games. there is less roulette than you might expect.
you may be right about that. there is some surprisingly good stuff on youtube, but it tends to depend on the people who are making it. and a whole lot of trash.
probability theory is also very easy to present "well" without giving any indication of what the subject is about. my wife has an MS in statistics and half of her professors were reading from some script and running code written by their advisors. it was not flattering to them but they made the experience seem easy.
there's a constant struggle in instruction. you always want to make things simple. but sometimes the simplest explanation requires a lot on the part of the reader to read the actual words of the explanation, and maybe a more complicated explanation requires less. i'm cynical about all of this.
i only taught for about 8 years, but i felt like i was constantly fighting against, the impulse to just sell a product to students, as opposed to actually having them engage with it. i don't think professional incentives in the US incentivize the latter over the former, which is maybe a problem.
but i don't know, i skipped out before it got too real
almost all postdocs in my college education were absolute trash at instruction, and it was clear that this had absolutely never been considered as a criterion of job-worthiness. and as someone potentially interested in academia, i may have internalized that message.
i think it helps to find people who can explain things simply and in multiple ways. they are the best resources.
when i was in graduate school i tried to focus 1200 to 1700 on my research and reserve the rest for stuff that was not maybe necessarily connected to a research goal.
this was before i had a kid. now i wake up at 0600 and i just limp through the first half of the day.
@AndrewMicallef Filing taxes is a moderately complex process in the USA. I am self employed so it adds some more complexity, and my children often need returns too.
i was next to a guy on a plane about a year ago, worked in the entertainment industry. every one of his projects was a separate form. it was like 20 forms!
@Quasar, Want an exciting story about Probability theory and betting, read the article done by Bloomberg about Bill Benter: bloomberg.com/news/features/2018-05-03/…
@Quasar you can always ask me if you have a problem, I may or may not be able to answer something useful. It's good practice for me
@Mike there's a formula on wiki that solves linear equations. If you want a hint: you need to take the stochastic integral of that exponential, and choose the right boundaries of integration in the exponent
