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12:22 AM
@TedShifrin are you hinting at the Euler theorem saying that you can write down rotations in 3 dimensions as rotations in 2 dimensions along a new axis?

@TedShifrin is the following complete nonsense or does it make sense? Define $R_x = x \times \nabla_x$. Define $R_x^2 = R_x \cdot R_x$. Then, is $R_x^2 = -x -(I - x x^T)^2$?

i think to get all the coefficients you need 4x4.
if by conic you mean quadric surface or whatever and not conic section.

12:40 AM
I assumed conic referred to a curve. To expect an online calculator for high dimensions seems preposterous.
Define notation, JoeShmo.

1:29 AM
@TedShifrin ugh.

what do you mean?

Do you mean that $\left\langle u,R_x^2u\right\rangle=\|R_xu\|^2$ This is my question. Ted's may be different.

I'm actually not entirely sure what the author means.. half of why I'm asking :\ I thought there might be a standard definition.. he's using physics notation here that I'm not familiar with

that's the mistake

yeah well..

1:38 AM
Also, I'm not sure what $\nabla_x$ means; how does it differ from $\nabla$?

me neither. presumably the gradient w.r.t to $x$

I usually think of that as $x\cdot\nabla$
unless there is another variable that we're not seeing.

well there's a time variable, but time is a real number, not a vector..
he writes $x \otimes x$ in place of $x x^T$ (why is this necessary?!)

Then I don't see a reason to separate the $t$ from the $x$ since we don't often take the gradient including time
@JoeShmo it's a different way of writing it, I think

yeah... but why
the outer product..

1:42 AM
Don't ask me to explain the way physicists do math.

to me that notation indicates a bilinear form
but I'm no physicist

@Thorgott $xx^T$ would give a rank $1$ matrix, so a degenerate bilinear form.
I'm guessing here

yeah.. I think it's beside the point here
SDEs, Fokker-Planck is the context here, FWIW
the identity above concerns the brownian motion on the 2-sphere in 3-space

2:02 AM
@robjohn I actually like Common Core and have (unofficially) taught it to little kids.
My point (which of course you realized) is that you can do such additions without ever carrying a $1$ ... so you are free to walk more haphazardly.

@JoeShmo I have some background in Brownian Motion. What's the text?

@robjohn Where did the differentiation disappear to? Product rule and all. I have no clue what this means.

common core looks good on it's face. people aren't understanding the point behind it because they were taught -math- algorithms in school

I wonder if you're using Ito differentiation.

am I using ito differentiation? there is an ito sde

2:06 AM
I decided trying to help with K-12 math education was pointless when I saw a group of math education students sobbing to a Abstract Algebra professor about how they thought LaTeX was too hard to learn
@JoeShmo What page?

not in the preview :-)

I might have access to this through my uni; I can look up the page that way

page 172 bottom of the page, page 173 top of the page
brownian motion on the sphere. take $n=3$
question $(8.2)$ on page 193 is the one I'm looking at right now, but it doesn't really matter.. I don't understand his notation

Ask me in a year, lol. I guess this will be one text I'll have to read, since I just finished measure-theoretic probability

ohh.. hah
thanks anyway :-)

2:16 AM
I've used Ito stuff in the actuarial world, which is completely devoid of formalism

don't read that. find a different text. this one isn't all that great. but it's got excellent applied math recipes

Thanks for the tip
I've skimmed it briefly, but it looks decent

what probability book did you read

I just finished taking the class; it was a year long. If I'm being honest, they're all terrible.

in the USA?

2:23 AM
I finished my master's degree last Spring, thought about maybe doing a PhD, took one year of measure theory
Yes, I'm in the US
There are some that aren't as bad as others - e.g., Resnick's A Probability Path, Klenke's Probability Theory, and Athreya and Lahiri's Measure Theory and Probability Theory
But they are all bad in their own ways

interesting
try Durrett

Yeah, I'm considering re-learning this material with Durrett as my primary text
I did get an A both semesters. But it was a struggle, to say the least.

what doesn't kill you..

Was not a fan of the fact that my professor would casually drop functional analysis at us (not a prereq for the course).
Right, I completely get it

and was the course on limit theorems?

2:26 AM
My BS is in mathematics, and my MS was in statistics
We covered the CLT toward the end, but nothing more than that. I am aware of the existence of the Lindeberg and Lyapunov CLTs.

interesting. then what did you do for a year?
what did you go over?

Lol, that's a good question
Well

measure theory?
fatou, dominated, bounded convergence theorems? that sorta thing?

We spent the first semester going through $\sigma$-algebras, rings, algebras, and the like - then the Kolmogorov extension theorem, some stuff about $L^p$ spaces, inequalities, measurability, etc
Yeah
I was rather disappointed that we didn't see the probability analogues of those concepts until toward the end of the first semester

there's no probability analogues

2:29 AM
Lots of time on atomic and nonatomic measures

probability starts with limit theorems

Well, here's what I mean
We didn't even define what a random variable was until toward the end of the first semester

funny, but meh.. not the end of the world
sounds like it was a course in measure theory
pretty thorough course, at that

I had a colleague who took the same course with the same professor, except they used Durrett

2 semesters and all

2:30 AM
My colleague had the same opinion as what you expressed
We ended the second semester at ergodicity
and stationary processes

cool, well you might like Durrett

The text we were told to use was Dudley's text. Terrible textbook IMO for learning for the first time.

It's an excellent book, and comes recommended by everybody

Yeah, from the times I skimmed Durrett over the last year, it looked like that text seems to move straight to the point

also doesn't hurt to keep Stein's book on measure theory around, of course
he's an excellent writer, and he's got excellent proofs IIRC
yeah don't dwell too much over the philosophy of measure theory

2:34 AM
Ah, you're talking about Stein and Shakarchi. I haven't read the analysis text yet. I think if I start on that series, I might actually try to follow it linearly and actually learn Fourier stuff

Durrett is a pretty dense text. I would not use it as primary first time learning material.

Durrett I'm using for re-learning this material
I'm honestly debating about whether I should try to learn complex analysis and functional analysis before trying to learn probability again

nyet
not particularly relevant

Functional analysis is unnecessary. You don't need much complex analysis except for a few things.

This professor gave us homework problems that used the complex Gamma function, and in lecture, when he was talking about things like MGF convergence implying convergence of distribution functions, or the Krein-Milman Theorem, or the Riesz-Markov Theorem... gosh, I was lost

2:42 AM
complex is good to know, but its not exciting (although beauty is in the eye of the beholder..)

Thank you both for convincing me off that path for now

2:55 AM
krein milman in a probability course???

@copper.hat Yeah, I was not pleased
I couldn't follow that lecture at all or why it was even brought up

seems unnecessary.
unless there was some advanced topic

It was brought up during that one lecture, and I don't know if it ever came up again
I looked back at my notes from that lecture
Apparently this was the end result:
In mathematics, Choquet theory, named after Gustave Choquet, is an area of functional analysis and convex analysis concerned with measures which have support on the extreme points of a convex set C. Roughly speaking, every vector of C should appear as a weighted average of extreme points, a concept made more precise by generalizing the notion of weighted average from a convex combination to an integral taken over the set E of extreme points. Here C is a subset of a real vector space V, and the main thrust of the theory is to treat the cases where V is an infinite-dimensional (locally convex Hausdorff...
i.e., Choquet's Theorem
Hate to say I got an A in both semesters of this class, and I don't really understand that theorem

loosely it is saying that for any $c \in C$ a compact convex set that there is a probability measure $p$ supported on the extreme points of $C$ such that $c = \int \omega dp$ .
krein milman follows from this.

But yeah, that was the first lecture where I honestly thought I'd fail the class

3:06 AM
not sure how that would come up unless you were dealing with some particular situations.
i'm a fan of minimal to do the trick.

I think it came up on one homework assignment question, but if I recall correctly, I ended up solving that problem a different way
Can't remember what problem that was off the top of my head

Choking theorem

i mean, you will never know everything you need.

@Bhavay when you equate coefficient of xy =0 you get the value of angle thetha.

3:33 AM
i don't trust any theory named after a guy whose first name is Gustave.
you have to draw the line somewhere
there, i've said my smart thing for the day

that's an extreme point

how was the 7am bike ride Copper?

it was delightful, it was yesterday morning, we cycled from a place called Ross to (more or less) the top of Mt Tamalpais in Marin County north of SF. Incredible views at times.
and back down, I should add :-). Albeit, downhill on a mountain bike is not relaxing for me.

I would've been relishing the downhill portion of the ride. Guess you're not one for the adrenaline rush part of it...

It must've been after I finished my undergrad - I feel so out of touch with what is going on with the world, lol. I've been watching videos on "gen-Z slang:" it is mostly nonsense to me.
I don't think I was into slang much at all as a teenager

3:46 AM
@dc3rd not any more, i've come off the bike too many times and am no longer adept at rolling out of a fall :-)

what gen-z slang you learn Clarinetest? We're probably in the same age range, I'm not a gen-z'er either

I'm 29. Something about "lit" and "sus" lol

math is lit

lol....."lit" is gen-z, "sus" that is from the millenial era

I still remember the days of AIM and MSN Messenger
and phpBB boards

3:48 AM
you're beyond needing to show "war" scars from your trips @copper.hat
AIM and MSN Messenger were the ish.....thrilling days.

plus i've popped off a normal bike on icy hills in ireland too many times :-)

well I could see how that would have an effect on you not wanting to ride down hills....

any numerical analysts in the hizouse?

my cycling friend is patient.
plus my back tyre was skidding a large percentage of the downhill!

gotta keep that back hand brake ready

3:57 AM
modulation is key
and it's important to use the front brake
a lot of novice riders rely too much on the back brake

that would be me. I don't think I've ever used the front brake in my life.
I'm also working towards getting a mountain bike for the summer.

you see the front brake can give you 3,4x the braking power
and the rear brake is really just there to keep you stable when using the front brake
now if you are exerting braking forces so high on one wheel that your wheel is locking
that's not somehow "safer" because it's the back brake

interesting....so what do you do in a high speed scenario? put your weight to the back of the bike and use the back brake and then add in the front brake to balance things out?

your weight distribution should not change
a bicycle is optimized to perform with a certain weight distribution
that is a big fctor to bicycle fit and bicycle frame geometry
cyclists lean in to corners (left and right) but do not really manipulate front to back loading
(I'm not counting bmx tricks, artistic cycling etc. here)

Let's say I have 5 flavours of ice cream of which chocolate is the most selling and the remaining 4 have uniformly varying demand. What's the probability of a customer choosing chocolate then?
Can I say that P(Choc) + 4*P(others) = 1?

4:09 AM
the reason the front brake works better is that the same or more weight is on the front wheel as the front brake is applied
in a high speed scenario, you apply as much front brake as you can and the rear to balance with the caveat of not locking the wheels ever
locking the wheels (skidding) = no bueno
braking power is actually lowered once your wheels are locked

locking the wheels on the front is where you end up flying over the bar....
whole new approach to bike riding presented to me today.....interesting.

not necessarily you can skid sideways a little in a hard turn for example

Weirdly my back brakes locking makes me fall off the seat and onto the rod xD

but let's say we are just talking about the rear wheel for the sake of argument, locking it reduces braking force
if you are fliritng with locking it learn to modulate the force you apply so you don't
in essence, do not slam the brakes

If you turn corners with locked back brakes it's like drifting right?

4:13 AM
it is counterproductive
yes cyclists call that skidding

In cars do brakes lock while drifting?
Or do they still rotate but 90 degrees to the motion?

i should add that i use a \$350 specialised that had a slight buckled front rim, so the front braking has an odd modulation that tends to skid easily if you are pushing things. my friend is always nagging me to spend $$on kit, but i am good with my cheap bike :-) I was looking into getting a Trek 850. Had one when I was a kid. Think it has a good enough balance to do mountain biking and having to use in the city. well having wheels that are true, rims in order, good tires etc. is just optimal safety imho not fancy kit i had a nice bridgestone mb5 but it got stolen, that was a good compromise 4:16 AM @AbhigyanChattopadhyay This is probably digging too far into the weeds, but there is a difference between estimating probabilities using data, versus making assumptions about the probabilities in general. You could assume such a model, yes. But whether such a model is compatible with the demand you've seen empirically is an entirely different question. well, i just go down more slowly :-) I like how the name of the company is Specialized, but you refuse to use the "z" in this scenario Copper old habits :-) proud Irishmen. i communicate with a lot of folks in ireland/uk, so my spelling is all over the place :-) india too 4:18 AM I have the same happen with me between friends in europe, america, and the caribbean. o's and u's get tossed in whatever the flavour of the day is sometimes i throw them in just as a potential convo starter :-) i bike mainly for exercise, so having top notch gear is not a biggie. i do maybe 2-5 km of really steep downhill per month, so just going slow is an acceptable tradeoff. i usually spent more that the cost of the bike yearly replacing chainset & wheels @Clarinetist Let's just assume this model is true Then what would be the probability of any new customer choosing Chocolate @AbhigyanChattopadhyay If you assume the model is true and it extends to any customer choosing a particular flavor, nothing else to it. It's true, it stands at is. I just want to assume this model and do a Monte Carlo simulation So I need to generate random values for the other flavours And I need to know what factor I should multiply them by Then it's just an algebra problem from there. You have constants p and q between 0 and 1 for which p + 4q = 1. What are you aiming to simulate? 4:23 AM monte carlo for this sounds like a bulldozer for a mole hill @AlexandruIonut Fair enough Given how the simple the problem sounds like it should be, I would be inclined to agree @Clarinetist See I think I'm making a mistake in that equation The probabilities of the remaining flavours aren't equal They're uniformly distributed Okay, let me get this straight So you have this "chocolate" flavor 4:25 AM and these other four flavors, which are apparently uniformly distributed in some interval *whose probabilities are uniformly distributed in some interval, that is That's right, and chocolate always has the highest demand. So what are you aiming to do, simulate the chocolate probability? Not exactly, it's part of a bigger question I'll send it here So I need to essentially find various probabilities using monte carlo, such as probability of getting each flavour, even getting a single chocolate at all, etc I realise that the P+4Q equation is completely wrong Wow, this is complicated @Rover Thank you. 4:29 AM @Clarinetist Ikr... So I was breaking it down xD the problem is time varying which does not suit a simple mc approach Yeah, I had a feeling it's a Markov-Chain-ish approach, but yeah, the time varying business throws me off @copper.hat That's true, I intend to simulate it over a few iterations and then take that as a day, then move to the next day, etc. it is not clear what the point is? i'll pay full price to skip this problem and just get 7 mini-scoops of kesar pista. 4:32 AM @leslietownes xD me too I have a feeling this is some advanced probability material I haven't learned yet. I wonder if it's a queueing theory problem (note: I know nothing about this topic). @copper.hat We want to find the probability of having all 5 flavours in the cone. That's one target Besides that, a number of others too x^2+3\sqrt(3)x-84=0 Well, good luck. This is a tough one. Any faster technique to middle term factorize this? 4:34 AM I really need to take a more extensive probabilistic modeling class some day. Statistical modeling is all very similar at the end of the day. @Clarinetist Me too, by the looks of it... I've taken courses in basic Statistics and Probability and this stuff is flying over my head does it have a stationary distribution? @copper.hat Yes I would think it does Otherwise it gets overcomplicated But the daily values change As in the distribution is uniform but it takes on different values from it daily i don't have time to look at it now but that is not clear to me Do you think if I post this on Main I may get some answers? Maybe without using Monte Carlo 4:37 AM I would give it a shot if I were you. Cool, on it BTW I just finished probability at the PhD level and I can't solve this. What a shame. Lol. I need to dive in to probabilistic modeling. Never took a class which required me to know about Markov Chains, though I know how they work through actuarial experience Woahhhh it's that hard??? I have to talk to my prof then xD I'm just a Bachelors student in Engineering, not even Math If they have you learning this in a probabilistic modeling class, it makes COMPLETE sense to bring it up I haven't taken such a class yet I've gone through bachelor's, master's, and PhD-level probability, primarily focusing on the theory, but none of my classes covered Markov chains (which is stupid) @Clarinetist Ahh ok. So I take it that some Markov Chain approach will work? 4:41 AM in a passed life, I did a master's in applied math for stochastic simulation I can't say for sure this is something that can be attacked with programming + monte carlo simulation @copper.hat would know better than I do essentially simulating the ice cream shop and after many iterations of the simulated ice cream shop, you can have some data it is not something they want you to attack analytically this is empirical stuff @AlexandruIonut That's exactly what I'm attempting to do 4:43 AM well you need a pseudorandom number generator and some basic probability to come up with your samples and then do the programming work to simulate the icecream shop @AlexandruIonut Yeah, my first problem arises on what value to take for Chocolate probability and the rest Like the rest will be scaled down by a factor you can have that as a variable too this is typical in such academic exercises it feels underspecified to me Oh. But then my answer will not be a deterministic probability xD but i should not get involved, i can't commit right now 4:45 AM Like it'll depend on that variable distribution of customers for example? Ah I just realised, the probability of getting all 5 flavours is 0 isn't it? Because no matter what you have 2x 3 unpopular flavours And 1 more Daiyum Or am I missing something? problem looks sketch, as my daughter would say sus, even @copper.hat it is underspecified by design to get kids to dick around 4:49 AM good use of the paralance of the day you two.....y'all would fit in well with the clique Haha this website is all formal and all right Imma try to fit in @AlexandruIonut i missed that part of the discussion :-) what formula would you use to factor x^2 +7x + 10 @RajorshiKoyal? that's what i get for procrastinating... I break 7 into 5 and 2. 4:52 AM E look at the bold lettering I wrote. @RajorshiKoyal i said formula @AbhigyanChattopadhyay tinyurl.com/cfqcvpc bookmark the "start chatjax" link Sorry I didn't read description properly and click on it to render mathjax (-b +- sqrt(b^2-4ac))/2a 4:55 AM yes. @RajorshiKoyal you're not going to get a "nice" answer though, but it is the answer @RajorshiKoyal write the expression inside two$$ ok Thanks no, just one$

yes
the plus minus symbol can be written as $\pm$

$\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

4:57 AM
nice

what the frac?
frac is gen-z latex

what the sqrt

Lol why gen-z

what the int
what the sum

I mean it's been around right?

4:59 AM
just rambling as olf folks are wont to do
yes, i'm kidding :-) there as a mention of gen-z lingo a short while ago
and some folks make fun of my archaic latex

:pat-pat:

if it ain't broke...

hand typesetting :D

No wonder books were so expensive. And yet these days books are still expensive
They use that stupid glossy paper that won't let you write on it

5:27 AM
i liked the paper used in older books. modern paper is thicker and books are huge

yes lol

1 hour later…
6:28 AM
One question... Are the arrival times of a Bernoulli process (i.e. number of trials till success) independent Pascal variables?

1 hour later…
7:40 AM
Suppose that it’s not known that order of a subgroup of a group divides order of the group. How can we find all subgroups of symmetric group of order 3 $S_3$?
$S_3$= set of bijections from $\{1,2,3\}$ onto itself.
So $S_3=\{(1),(123),(132),(12),(23),(31)\}$, where $(1)$ represents identity map. I observe that $(123)^2=(132)$ and $(12)^2=(23)^2=(31)^2=(1)$ and that $(132)^2=(1)$
$(1)$ and $S_3$ are subgroups of $S_3$. $\{(1),(12)\}$ and similarly two more. $\{(1),(123),(132)\}$ and $\{(1),(132)^2\}$
How do I show that these are all the subgroups? @leslietownes

5 hours later…
1:06 PM
If $X$ is a path connected space and $A\subset X$ is a path connected subspace, then $H_0(A)\to H_0(X)$ induced by inclusion $A\to X$ is always injective?

1:32 PM
@barista they're both Z and generator is sent to generator
so it's just an isomorphism

@LeakyNun are you in grad school yet?

@LeakyNun How do you know it sends generator to generator?

oh, planning on it?

@barista what's the generator?

1:34 PM
@LeakyNun in Z or H_0(A)?
1 you mean
I'm worrying the zero map case

in H0 of any path connected space

A point

Hello everyone. Sorry if the question is too basic.

I have a finite field $\mathbb{F}_q$ and with $k,n\in\mathbb{N}$ we define $C\subseteq \mathbb{F}_q^n$ as a $k$-dimensional linear subspace, calling $C$ a linear code.

Now we can also say that the basis for $C$ provides an isomorphism $\mathbb{F}_q^k \to C$. Does this mean that $\text{Span}(\text{Basis}(C))=\mathbb{F}_q^k$?

Can we say that for $C\subseteq\mathbb{Z}_2^8$ with $\{(1,0,0,0,0,0,0,0),(0,1,0,0,0,0,0,0),(0,0,1,0,0,0,0,0),(0,0,0,1,0,0,0,0)\}$ providing an isomorphism to $\mathbb{Z}_2^4$? That is, we are saying in this case $k=4 Or should I say a coset? it's the coset of a point (where we interpret the point as a 0-simplex), yes now where does the inclusion map that to 1:44 PM It maps to the same point and the coset of that point also generate H_0(X) so maps generator to generator you mean indeed Thank you all 2:12 PM A point O is a centre of circle circumscribed about a triangle ABC.Then, OAvector(sin2A)+OBvecto(sin2B)+OCvector(sin2C) is equal to ..? time to learn mathjax @copper.hat For representing vectors I am not getting mathjax What's it ? you could try$\vec{OA}$if that is what you are asking. i have never used this symbol and just did a web search for it. you can do same. 2:34 PM Ok So, A point O is a centre of circle circumscribed about a triangle ABC.Then,$\vec{OA}sin2A+\vec{OB}sin2B+\vec{OB}sin2C$is equal to..? Hello 👋 I have a non empty group G which has no proper subgroups (that is its only subgroups are identity group and G), how to prove that order of G is prime without using concept of order of element. Lagrange theorem and Cauchy Theorem are not allowed. I have already proven that such a group will always be cyclic. Order will be prime is what I am stuck at Any hints please? 2:50 PM @Koro well you know how cyclic groups look like I thought along these lines: Let order of G be m and for contradiction suppose that m is composite. And therefore there exist some primes$p_1,p_2, \cdots, p_k$such that$m=p_1p_2\cdots p_k$. Following which I claimed that$\{e,a,…, a^{p_1-1}\}$is a subgroup but then got stuck 🥲 If I can prove$ \{e,a,…, a^{p_1-1}\}\$ is a subgroup then I am done. But I’m having difficulty here ☹️

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