Mathematics - 2021-05-10 (page 1 of 2)

BigSocks

00:22

@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?

Joe Shmo

@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 $R_x^2 = -x -(I - x x^T)^2$?

leslie townes

i think to get all the coefficients you need 4x4.

if by conic you mean quadric surface or whatever and not conic section.

Ted Shifrin

00:40

I assumed conic referred to a curve. To expect an online calculator for high dimensions seems preposterous.

Define notation, JoeShmo.

robjohn

01:29

@TedShifrin ugh.

Joe Shmo

what do you mean?

robjohn

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

Joe Shmo

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

Thorgott

that's the mistake

Joe Shmo

yeah well..

robjohn

01:38

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

Joe Shmo

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

robjohn

I usually think of that as $x\cdot\nabla$

unless there is another variable that we're not seeing.

Joe Shmo

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?!)

robjohn

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

Joe Shmo

yeah... but why

the outer product..

robjohn

01:42

Don't ask me to explain the way physicists do math.

Thorgott

to me that notation indicates a bilinear form

but I'm no physicist

robjohn

@Thorgott $xx^T$ would give a rank $1$ matrix, so a degenerate bilinear form.

I'm guessing here

Joe Shmo

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

Ted Shifrin

02:02

@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.

Clarinetist

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

Ted Shifrin

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

Joe Shmo

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

Clarinetist

I wonder if you're using Ito differentiation.

Joe Shmo

am I using ito differentiation? there is an ito sde

this is the text -- google.com/books/edition/Applied_Stochastic_Analysis/…

Clarinetist

02:06

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?

Joe Shmo

not in the preview :-)

Clarinetist

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

Joe Shmo

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

Clarinetist

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

Joe Shmo

ohh.. hah

thanks anyway :-)

Clarinetist

02:16

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

Joe Shmo

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

Clarinetist

Thanks for the tip

The one I have sitting in my shelf is amazon.com/Brownian-Martingales-Stochastic-Calculus-Mathematics/…

I've skimmed it briefly, but it looks decent

Joe Shmo

haven't read this one

what probability book did you read

Clarinetist

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

Joe Shmo

oh are you an undergrad?

in the USA?

Clarinetist

02:23

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

Joe Shmo

interesting

try Durrett

Clarinetist

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.

Joe Shmo

what doesn't kill you..

Clarinetist

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

Joe Shmo

is your background in mathematics?

and was the course on limit theorems?

Clarinetist

02:26

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.

Joe Shmo

interesting. then what did you do for a year?

what did you go over?

Clarinetist

Lol, that's a good question

Well

Joe Shmo

measure theory?

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

Clarinetist

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

That's about right

I was rather disappointed that we didn't see the probability analogues of those concepts until toward the end of the first semester

Joe Shmo

there's no probability analogues

Clarinetist

02:29

Lots of time on atomic and nonatomic measures

Joe Shmo

probability starts with limit theorems

Clarinetist

Well, here's what I mean

We didn't even define what a random variable was until toward the end of the first semester

Joe Shmo

funny, but meh.. not the end of the world

sounds like it was a course in measure theory

pretty thorough course, at that

Clarinetist

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

Joe Shmo

2 semesters and all

Clarinetist

02:30

My colleague had the same opinion as what you expressed

We ended the second semester at ergodicity

and stationary processes

Joe Shmo

cool, well you might like Durrett

Clarinetist

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

Joe Shmo

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

Clarinetist

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

Joe Shmo

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

get straight down to business - analysis is about inequalities

Clarinetist

02:34

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

copper.hat

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

Clarinetist

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

Joe Shmo

nyet

not particularly relevant

copper.hat

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

Measure theory is helpful.

Clarinetist

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

Joe Shmo

02:42

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

Clarinetist

Thank you both for convincing me off that path for now

copper.hat

02:55

krein milman in a probability course???

Clarinetist

@copper.hat Yeah, I was not pleased

I couldn't follow that lecture at all or why it was even brought up

copper.hat

seems unnecessary.

unless there was some advanced topic

Clarinetist

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:

Choquet theory

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

copper.hat

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.

Clarinetist

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

copper.hat

03:06

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.

Clarinetist

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

Euler 2

Choking theorem

copper.hat

i mean, you will never know everything you need.

Rover

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

leslie townes

03:33

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

copper.hat

that's an extreme point

dc3rd

how was the 7am bike ride Copper?

copper.hat

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.

dc3rd

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

Clarinetist

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

copper.hat

03:46

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

dc3rd

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

Clarinetist

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

Alexandru Ionut

math is lit

dc3rd

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

Clarinetist

I still remember the days of AIM and MSN Messenger

and phpBB boards

dc3rd

03:48

you're beyond needing to show "war" scars from your trips @copper.hat

AIM and MSN Messenger were the ish.....thrilling days.

copper.hat

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

dc3rd

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

Alexandru Ionut

any numerical analysts in the hizouse?

copper.hat

my cycling friend is patient.

plus my back tyre was skidding a large percentage of the downhill!

dc3rd

gotta keep that back hand brake ready

Alexandru Ionut

03:57

modulation is key

and it's important to use the front brake

a lot of novice riders rely too much on the back brake

dc3rd

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.

Alexandru Ionut

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

dc3rd

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?

Alexandru Ionut

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)

Abhigyan Chattopadhyay

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?

Alexandru Ionut

04:09

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

dc3rd

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.

Alexandru Ionut

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

Abhigyan Chattopadhyay

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

Alexandru Ionut

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

Abhigyan Chattopadhyay

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

Alexandru Ionut

04:13

it is counterproductive

yes cyclists call that skidding

Abhigyan Chattopadhyay

In cars do brakes lock while drifting?

Or do they still rotate but 90 degrees to the motion?

copper.hat

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 :-)

dc3rd

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.

Alexandru Ionut

well having wheels that are true, rims in order, good tires etc. is just optimal safety imho

not fancy kit

copper.hat

i had a nice bridgestone mb5 but it got stolen, that was a good compromise

Clarinetist

04:16

@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.

copper.hat

well, i just go down more slowly :-)

dc3rd

I like how the name of the company is Specialized, but you refuse to use the "z" in this scenario Copper

copper.hat

old habits :-)

dc3rd

proud Irishmen.

copper.hat

i communicate with a lot of folks in ireland/uk, so my spelling is all over the place :-)

india too

dc3rd

04:18

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

copper.hat

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

Abhigyan Chattopadhyay

@Clarinetist Let's just assume this model is true

Then what would be the probability of any new customer choosing Chocolate

Clarinetist

@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.

Abhigyan Chattopadhyay

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

Clarinetist

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?

Alexandru Ionut

04:23

monte carlo for this sounds like a bulldozer for a mole hill

Abhigyan Chattopadhyay

@AlexandruIonut Fair enough

Clarinetist

Given how the simple the problem sounds like it should be, I would be inclined to agree

Abhigyan Chattopadhyay

@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

Clarinetist

Okay, let me get this straight

So you have this "chocolate" flavor

Abhigyan Chattopadhyay

Yes

Clarinetist

04:25

and these other four flavors, which are apparently uniformly distributed in some interval

*whose probabilities are uniformly distributed in some interval, that is

Abhigyan Chattopadhyay

That's right, and chocolate always has the highest demand.

Clarinetist

So what are you aiming to do, simulate the chocolate probability?

Abhigyan Chattopadhyay

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

Clarinetist

Wow, this is complicated

Bhavay

@Rover Thank you.

Abhigyan Chattopadhyay

04:29

@Clarinetist Ikr...

So I was breaking it down xD

copper.hat

the problem is time varying which does not suit a simple mc approach

Clarinetist

Yeah, I had a feeling it's a Markov-Chain-ish approach, but yeah, the time varying business throws me off

Abhigyan Chattopadhyay

@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.

copper.hat

it is not clear what the point is?

leslie townes

i'll pay full price to skip this problem and just get 7 mini-scoops of kesar pista.

Abhigyan Chattopadhyay

04:32

@leslietownes xD me too

Clarinetist

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).

Abhigyan Chattopadhyay

@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

Rajorshi Koyal

$x^2+3\sqrt(3)x-84=0$

Clarinetist

Well, good luck. This is a tough one.

Rajorshi Koyal

Any faster technique to middle term factorize this?

Clarinetist

04:34

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.

Abhigyan Chattopadhyay

@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

copper.hat

does it have a stationary distribution?

Abhigyan Chattopadhyay

@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

copper.hat

i don't have time to look at it now but that is not clear to me

Abhigyan Chattopadhyay

Do you think if I post this on Main I may get some answers?

Maybe without using Monte Carlo

Clarinetist

04:37

I would give it a shot if I were you.

Abhigyan Chattopadhyay

Cool, on it

Clarinetist

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

Abhigyan Chattopadhyay

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

Clarinetist

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)

Abhigyan Chattopadhyay

@Clarinetist Ahh ok.

So I take it that some Markov Chain approach will work?

Alexandru Ionut

04:41

in a passed life, I did a master's in applied math for stochastic simulation

Clarinetist

I can't say for sure

Alexandru Ionut

this is something that can be attacked with programming + monte carlo simulation

Clarinetist

@copper.hat would know better than I do

Alexandru Ionut

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

Abhigyan Chattopadhyay

@AlexandruIonut That's exactly what I'm attempting to do

Alexandru Ionut

04:43

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

Abhigyan Chattopadhyay

@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

Alexandru Ionut

you can have that as a variable too

this is typical in such academic exercises

copper.hat

it feels underspecified to me

Abhigyan Chattopadhyay

Oh. But then my answer will not be a deterministic probability xD

copper.hat

but i should not get involved, i can't commit right now

Abhigyan Chattopadhyay

04:45

Like it'll depend on that variable

copper.hat

distribution of customers for example?

Abhigyan Chattopadhyay

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?

copper.hat

problem looks sketch, as my daughter would say

leslie townes

sus, even

Alexandru Ionut

@copper.hat it is underspecified by design to get kids to dick around

dc3rd

04:49

good use of the paralance of the day you two.....y'all would fit in well with the clique

Abhigyan Chattopadhyay

Haha this website is all formal and all right

Imma try to fit in

copper.hat

@AlexandruIonut i missed that part of the discussion :-)

dc3rd

what formula would you use to factor $x^2 +7x + 10$ @RajorshiKoyal?

copper.hat

that's what i get for procrastinating...

Rajorshi Koyal

I break 7 into 5 and 2.

Euler 2

04:52

E

dc3rd

look at the bold lettering I wrote. @RajorshiKoyal i said formula

Euler 2

@AbhigyanChattopadhyay tinyurl.com/cfqcvpc

bookmark the "start chatjax" link

Abhigyan Chattopadhyay

Sorry I didn't read description properly

Euler 2

and click on it to render mathjax

Rajorshi Koyal

(-b +- sqrt(b^2-4ac))/2a

dc3rd

04:55

yes. @RajorshiKoyal you're not going to get a "nice" answer though, but it is the answer

Euler 2

@RajorshiKoyal write the expression inside two $$

Rajorshi Koyal

ok

Thanks

dc3rd

no, just one $

Euler 2

yes

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

Abhigyan Chattopadhyay

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

Euler 2

04:57

nice

copper.hat

what the frac?

frac is gen-z latex

Euler 2

what the sqrt

Abhigyan Chattopadhyay

Lol why gen-z

Euler 2

what the int

what the sum

Abhigyan Chattopadhyay

I mean it's been around right?

copper.hat

04:59

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

Abhigyan Chattopadhyay

:pat-pat:

copper.hat

if it ain't broke...

Euler 2

before latex was made

hand typesetting :D

Abhigyan Chattopadhyay

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

Euler 2

books.google.co.in/…

copper.hat

05:27

i liked the paper used in older books. modern paper is thicker and books are huge

Euler 2

yes lol

Abhigyan Chattopadhyay

06:28

One question... Are the arrival times of a Bernoulli process (i.e. number of trials till success) independent Pascal variables?

Koro

07:40

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

barista

13:06

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?

Leaky Nun

13:32

@barista they're both Z and generator is sent to generator

so it's just an isomorphism

Joe Shmo

@LeakyNun are you in grad school yet?

Leaky Nun

no

barista

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

Joe Shmo

oh, planning on it?

Leaky Nun

@barista what's the generator?

barista

13:34

@LeakyNun in Z or H_0(A)?

1 you mean

I'm worrying the zero map case

Leaky Nun

in H0 of any path connected space

barista

A point

Tangoed

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…

barista

Or should I say a coset?

Thorgott

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

barista

13:44

It maps to the same point and the coset of that point also generate H_0(X) so maps generator to generator you mean

Thorgott

indeed

barista

Thank you all

Rover

14:12

A point O is a centre of circle circumscribed about a triangle ABC.Then, OAvector(sin2A)+OBvecto(sin2B)+OCvector(sin2C) is equal to ..?

copper.hat

time to learn mathjax

Rover

@copper.hat For representing vectors I am not getting mathjax

What's it ?

copper.hat

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.

Rover

14:34

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..?

Koro

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?

Leaky Nun

14:50

@Koro well you know how cyclic groups look like

Koro

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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