Mathematics - 2018-02-10 (page 1 of 3)

Much of the money made by aggressive investors is mainly from exploiting other speculators. Some of these people (e.g. Simons) have very serious math/models on their side. Others cheat. Be careful.

MatheinBoulomenos

12:48 AM

@Daminark you like finite group theory, right? Take a look at this: math.stackexchange.com/a/2643818/348926

this took my quite some time

Ted Shifrin

WAYYYYYY too long.

MatheinBoulomenos

I wouldn't say it's elegant

Ted Shifrin

I hope someone reads it carefully, but it shan't be I.

MatheinBoulomenos

It looks longer than it is, conceptually, because I use the same tricks a lot

Ted Shifrin

Mark Bennet's question is a good one. Why is this an interesting question?

Mike Miller

12:53 AM

@PVAL a homeomorphism can't give a counterexample to my officemates question - take $\phi_Y$ to be the inverse of the homeomorphism. Aka you can undo the nastiness you started with.

I told him that's too hard for me now

PVAL-inactive

I don't understand the question then.

oh i see

Faust

morning @KasmirKhaan

Mike Miller

I find it plausible but too complicated to want to think about it

Ted Shifrin

<---- never wants to think again

Faust

@TedShifrin you should retire then

Ted Shifrin

12:58 AM

From the world? Good idea.

Maneesh Narayanan

-1

Q: Problem 2.1 Griffith-Introduction to Electrodynamics

Reference: Introduction to Electrodynamics-Griffith (a) It is easy to see that net force is zero. (b) Only one force will be experienced to $Q$ due to the charge just diametrically opposed to the removed charge. (c) The answer is given that it is zero. How is it coming? No chance for diametri...

PVAL-inactive

Yeah I don't think left composes to a simplicial map is anywhere near as natural as conjugate to a simplicial map.

Faust

you could move into a nursing home

Maneesh Narayanan

Please help me the mathematical part

PVAL-inactive

So I'm less interested after reading the question correctly.

Ted Shifrin

12:59 AM

Gee, thanks, Faust. I spent enough time in one visiting my mom. I'll probably end up in one eventually.

Faust

your the one who doesnt want to think!

Maneesh Narayanan

Zarathustra solved by using nth root of unity idea in complex plane.

Kasmir Khaan

Hi yall

MatheinBoulomenos

Hi @Kasmir

Mike Miller

@PVAL I think it's for the purpose of analysis (he does a lot of quasiconformal stuff) and not really topology

Maneesh Narayanan

1:00 AM

But problem is in $R^2$

Ted Shifrin

That's the same thing, @Maneesh.

We've discussed things like this in here recently.

So which part are you asking about?

Hi Kasmir

Kasmir Khaan

@MatheinBoulomenos haha nice solution, i put that on my fave and upvoted it :D i have to read it later tho =p

@TedShifrin Ted :D

It looks like a full lecture mathein well done

Ted Shifrin

@Maneesh: The sum of the vectors to the vertices of any regular $n$-gon is always zero. What kind of symmetry does the regular $n$-gon have?

@Kasmir: It'll probably take you a week of complete study for that.

Kasmir Khaan

So mean ?

Ted Shifrin

No, I took one glance at the ten-page answer and closed it.

MatheinBoulomenos

1:03 AM

@Kasmir long and laborious arguments are not necessarily a good thing. But I have to admit, it was fun

Maneesh Narayanan

n rotation and n flipping about axis@TedShifrin

Kasmir Khaan

Anyway does manifolds and stokes has something in commen?

MatheinBoulomenos

Stokes is a theorem about manifolds

Ted Shifrin

Of course, @Kasmir. You didn't watch my lectures.

Kasmir Khaan

I watched those lecture about closeness Ted

Ted Shifrin

1:03 AM

@Maneesh: OK, think about rotational symmetry. Whatever the sum of the vectors is, what happens when you rotate by $2\pi/n$?

Kasmir Khaan

and i watched when I took analysis

Ted Shifrin

There are dozens of lectures on differential forms and Stokes's Theorem, and a few on topology applications, @Kasmir.

Kasmir Khaan

I am not taking it yet, just wanted to ask, saw that Word many times =p

"manifold"

Maneesh Narayanan

orientation doesnot change @TedShifrin, same regular n-gon.

Kasmir Khaan

@MatheinBoulomenos are you good at analysis as much as algebra? or you like algebra more? =p

Maneesh Narayanan

1:05 AM

sir, I have small doubt. Let me type @TedShifrin

Kasmir Khaan

well maneesh shows respect that is good :D

PVAL-inactive

manifold - in mechanical sense, first as "pipe or chamber with several outlets," 1884, see manifold (adj.); originally as manifold pipe (1857), with reference to a type of musical instrument mentioned in the Old Testament.

That isn't so far away from what I think about as manifolds, though I guess everythings 2-d.

Maneesh Narayanan

@KasmirKhaan what do you mean?

Kasmir Khaan

Hmm ._. just today i realised that F(b) - F(a) formula to get an area

is just spceial case of stokes , even tho i knew about stokes for a year now almost :D

@ManeeshNarayanan You address Ted as sir , that is good

Ted Shifrin

Why is it good?

Maneesh Narayanan

1:08 AM

I have seen his lecture videos in youtube.

Kasmir Khaan

because in old Culture Ted

the teacher has high status

I like to keep that

Ted Shifrin

I don't think names and titles confer respect. They're just words. I've been disrespected by people that called me Sir.

Kasmir Khaan

It is first the most important job, second the most hard one, also alot of work to make someoen understand

Hmm well that is a good Point ._.

I take back what I said sorry -.-

Lets talk math :D

PVAL-inactive

Actually 1884 is probably after Riemann's paper...

Ted Shifrin

ROFL

Kasmir Khaan

1:10 AM

Ted do you have a geometric meaning of modules?

I know its a generalization of vector spaces

but I dont quite have a "feel " for it yet ._.

Maneesh Narayanan

@TedShifrin sorry, I didn't mean to disrespect you.

Ted Shifrin

There are geometric interpretations of certain types of modules (that would show up in algebraic geometry or manifold theory or ....) ... but for a random module, I probably don't.

PVAL-inactive

There's the whole quasicoherent sheaves are vector bundles thing.

Ted Shifrin

@Maneesh: You didn't do any such thing. I'm just saying I don't necessarily like being called "Sir."

PVAL-inactive

That seems like a geometric interpretation of modules.

Kasmir Khaan

1:12 AM

@TedShifrin okay thanks ! :)

Ted Shifrin

Sure, @PVAL. They are included in my vague comment. :)

Kasmir Khaan

@PVAL-inactive well thanks to you as well =p

Maneesh Narayanan

@TedShifrin ok

Ted Shifrin

@Maneesh: At any rate, so you rotate the polygon and it returns to its original position. What about the sum of the vectors?

Kasmir Khaan

Ted why dont you do seminars ? :D

Ted Shifrin

1:13 AM

Huh? @Kasmir

Kasmir Khaan

I mean visit colleges and do some talks

one could learn alot from that

PVAL-inactive

@Ted Isn't the thing I said essentially true though for like all modules over a fixed ring?

Ted Shifrin

I'm a retired bum, Kasmir. I did that a little bit during my employed years.

But I don't know how to interpret an arbitrary ring geometrically, PVAL.

PVAL-inactive

Over an affine scheme, modules should correspond to quasicoherent sheaves which "should" correspond to line bundles.

Maneesh Narayanan

Zarastustra solved the problem solved in complex plane,How can we convert the problem in to $\mathbbR^2$?By isomorphism(Here, Identity). right?@TedShifrin

Kasmir Khaan

1:15 AM

Retired maybe ! bum NO ! :D

had to google that word

Maneesh Narayanan

He deleted his answer.

Ted Shifrin

Don't worry about Zaratustra's solution, @Maneesh.

He's using complex multiplication. And I'm talking rotation. It's the same thing in this case.

Maneesh Narayanan

@TedShifrin sum of vectors means, vectors pointing towards origin.right?

Ted Shifrin

@Maneesh: That's fine. I was going from the origin to the points, but it makes no difference (other than a total negative sign).

Semiclassical

hi chat

skullpatrol

1:20 AM

\o

Ted Shifrin

hi Semiclassic

Maneesh Narayanan

By triangle law of vector addition, it is equal to the sum of vectors of sides. Right?

@TedShifrin

Ted Shifrin

Not quite, Maneesh.

But go back to my question about rotating $2\pi/n$. What happens to the sum?

Semiclassical

not gonna lie, i'm glad it's the weekend. i think i've got a cold so I'm appreciating the chance to sleep in

Ted Shifrin

Just make sure you don't have the 'flu, Semiclassic. It's still raging.

Semiclassical

1:23 AM

yeah, fingers crossed on that

Kasmir Khaan

flu came extremly strong this year

those bastards who use lots of antiobiotic ruined it for -.-

Maneesh Narayanan

@TedShifrin How rotation and sum is related. I don't understand. Can you give hints?

Ted Shifrin

What happens to the vectors to the vertices when you rotate?

MatheinBoulomenos

@KasmirKhaan I like algebra more

Semiclassical

I don't actually know how the symptoms for flu vs. common cold compare, come to think of it

google time i guess

Ted Shifrin

1:25 AM

No, Kasmir, I don't think this is a matter of antibiotics at all. The vaccine just was not including the right strains.

Kasmir Khaan

@MatheinBoulomenos Nice , I too like algebra more so ill spend more time Learning it properly :D

Maneesh Narayanan

vector 1 goes to vector 2 , vect 2 to 3, so on. I am not able to think further. Please help me@TedShifrin

PVAL-inactive

flu is generally fever + muscle weakness I think.

Kasmir Khaan

hmm it could be that too Ted =p

Semiclassical

@TedShifrin yeah, it's always a bit of a guess on their part as far as what to put in the vaccine

Maneesh Narayanan

1:26 AM

vector one goes to vector 13

Ted Shifrin

no, 13 goes to 1.

Oh, well, it depends how you're numbering.

I number counterclockwise.

PVAL-inactive

@KasmirKhaan influenza is a virus so I doubt that.

Kasmir Khaan

@PVAL-inactive i thought by the Word flu he meant when someone gets sick ,like hmm i dont know the Word in english

Ted Shifrin

@Maneesh: So I have $v_1+v_2+\dots+v_{13}$ and I rotate and I have $v_2+v_3+\dots+v_1$. What does that tell you?

@Kasmir: influenza

Kasmir Khaan

what am talking about is caused by bacteria

so aint that ><

Ted Shifrin

1:28 AM

Right.

Kasmir Khaan

My sister is a doc so i know some stuff =p

Ted Shifrin

But I totally agree that antibiotics have been overprescribed and overused and we're gonna pay for that.

Kasmir Khaan

yepp , bacteria never really dies

then it get immunity from that antibiotic so year generation grows stronger

Ted Shifrin

immunity

Kasmir Khaan

japan is worst Place so far in the world

yes ><

Ted Shifrin

1:30 AM

well, Trompolini and N. Korea will blow us all up soon enough.

Kasmir Khaan

lets hope that those "presidents" find a peaceful solutions

PVAL-inactive

I'm not sure I believe there is a peaceful solution to NK's problems.

Antonios-Alexandros Robotis

oh great, everyone is still here :)

Semiclassical

one isn't hoping for good solutions as much as "least bad" ones

Maneesh Narayanan

@TedShifrin By commutativity both are same. How to manipulate this?

Ted Shifrin

1:32 AM

Nope, I've evaporated.

Antonios-Alexandros Robotis

It'd be great if heads of state had to fight to the death to decide wars.

Or at least some sort of MMA/boxing match.

Kasmir Khaan

@Antonios-AlexandrosRobotis :D

I knew you were cool

Antonios-Alexandros Robotis

You'd see a lot more peace.

Ted Shifrin

@Maneesh: It's commutativity plus associativity, but no one would question that it's obviously the same. So what vector(s) stay(s) the same when you rotate through an angle $2\pi/n$?

Antonios-Alexandros Robotis

:P

Ted Shifrin

1:33 AM

Or pieces, @Antonios.

Antonios-Alexandros Robotis

Come on, Trump vs. Kim Jong Un in a cage match? LOL

PVAL-inactive

@Semiclassical If there is no peaceful one, I think that one in which NK was as little militarily developed as possible might be the "least bad" one.

Kasmir Khaan

we need some handicap on kim because of age difference

Ted Shifrin

I root for both to die.

PVAL-inactive

I'd rather rest the fate of the free world on the strongest armed forces in the world than on Trump's fitness.

Ted Shifrin

1:35 AM

Certainly not on his intellect or sanity.

Antonios-Alexandros Robotis

"Free world." :P

let's not make any unreasonable assumptions here

Ted Shifrin

Our Trompolini fan is blissfully absent.

PVAL-inactive

Certainly free in comparison to NK.

Ted Shifrin

Anyhow, back to math(s).

Semiclassical

I'd rather the fate of so many people not rest on people making smart decisions about use of nuclear weapons

Maneesh Narayanan

1:36 AM

@TedShifrin no vectors stay the same. how associativity help me?

Ted Shifrin

All of Trompolini's advisers are resigning because their wives allege physical abuse.

Antonios-Alexandros Robotis

I imagine that's the case. It is mildly disturbing to note that you can never be sure that the info. you've heard re. NK isn't wrong. (or at least I can't be sure). Granted, I know which reality I'd place my money on.

Ted Shifrin

@Maneesh: That's not true. Associativity allows you to talk about the sum without parentheses.

Maneesh Narayanan

mmm

PVAL-inactive

@TedShifrin It's probably difficult coming back to the real world after advising Trump all day.

Ted Shifrin

1:37 AM

Indeed, but presumably these problems precede their recent post. I believe that's so.

@Maneesh: In case my answer was ambiguous, I said you need to be careful about "no vectors stay the same."

Antonios-Alexandros Robotis

A friend of mine got in to arguably the top 4 PhD's in the US

pretty crazy.

Ted Shifrin

That happens quite often (I mean over the years). Good for her/him.

Antonios-Alexandros Robotis

Yeah she's quite bright. Doing PDE stuff.

Ted Shifrin

Great :)

Antonios-Alexandros Robotis

alright, time to bang out this rep theory pset. Ping me if needed.

PVAL-inactive

1:49 AM

Does she have a real result as an undergrad>

?

Ted Shifrin

In my day, no one had anything like that.

PVAL-inactive

I'd think those kind of people usually have their choice.

Antonios-Alexandros Robotis

@PVAL-inactive she's a master's student and apparently her thesis in PDE is of interest to the profs in the schools she's applied to

that probably helped.

Ted Shifrin

High GREs, strong recs, and — to be honest — targeted minority ... is going to make a huge difference.

Antonios-Alexandros Robotis

yeah. Targeted minority definitely helps. On the flip side, she is international asian, which certainly does not.

Ted Shifrin

1:50 AM

Top 4 = Chicago, Berkeley, Harvard, MIT, Princeton [some subset of 4]

Maneesh Narayanan

@TedShifrin I am not getting.

Ted Shifrin

I didn't mean to sound disparaging. I'm sure she's great.

Antonios-Alexandros Robotis

Anyone who knows anything about admissions in general knows that's how it goes.

Ted Shifrin

@Maneesh: There is a vector that stays the same when you rotate ...

Antonios-Alexandros Robotis

Right or wrong, it's how it is.

I do feel for some friends of mine in undergrad who had serious imposter syndrome issues because they thought they were quota acceptances.

Certainly, they were not.

anyway, reps time

Ted Shifrin

1:52 AM

It's a really tough situation. Bubye.

PVAL-inactive

No idea in general but it least in my area Stanford is miles ahead of the places you listed.

Ted Shifrin

Yes, Stanford is in the top 10, universally. But there's this long-standing list ranking departments. Perhaps it's changed since I last looked.

In geometry, none of those places is at the top any more.

Maneesh Narayanan

By Rotation $2\pi/13 v_1 \to v_2 \to v_3 \to v_4 \to v_5 \to v_6 \to... v_1 $ .

similarly other rotation.

Ted Shifrin

Sure, @Maneesh. So the sum doesn't change.

So what must the sum be?

Maneesh Narayanan

@TedShifrin yes , from commutativity. but How it is zero. I am not getting. I tried. I failed.

Ted Shifrin

2:03 AM

You haven't correctly answered my question yet: What vector stays the same when you rotate (by any $\theta\ne 0, 2\pi$, etc.) around the origin?

Maneesh Narayanan

Let me write the case of all rotation@TedShifrin and find the stabilizer of each point.

Maneesh Narayanan

2:21 AM

at $4\pi/13$ rotation, $v_{12}$ is not changing. at $6\pi/13$ rotation, $v_6$ is not changing. at $8\pi/13$ rotation, $v_4$ is not changing. How it is going to help me? @TedShifrin

Maneesh Narayanan

2:34 AM

at $10\pi/13$ rotation, $v_3$ is not changing. at $12\pi/13$ rotation, $v_5$ is not changing. at $14\pi/13$ rotation, $v_2$ is not changing. at $16\pi/13$ rotation, $v_11$ is not changing. @TedShifrin

still I am not getting. Let me try today. If I have doubt, May I ask you tomorrow?

Daminark

2:55 AM

Hot take: we should actually start using $x(f)$ instead of $f(x)$

Because function composition is gonna be way less off-putting

Semiclassical

3:15 AM

so you prefer reverse polish notation

actually, reverse polish is worse than that

0celo7

"If E is a regular set in R^d"

excellent precision

Daminark

Lol I'm not gonna ask for that for stuff like operations (based on wikipedia's shtick about polish business), I'm happy with infix

It's more just like, I'd rather not have to start reading right to left for a while to think about function composition, breaks the flow and also doesn't map how you'd draw it in a diagram

0celo7

@XanderHenderson yo

@XanderHenderson do you know a condition for when a Radon measure is really $f\, d\mathcal L^n$?

for $f$ some function?

Xander Henderson

4:00 AM

isn't everything Radon? Like, for all practical purposes?

I mean, you can construct things that are not, but you have work for it, no?

(and disregard counting measure on $\mathbb{R}$; that's just cheating...)

0celo7

@XanderHenderson ok, when is a generic measure in the sense of Henderson $f\, d\mathcal L^n$?

Xander Henderson

heh

off the top of my head, I don't know

presumable $\mathcal{L}$ is Lebesgue measure?

0celo7

yes

Xander Henderson

Again, I don't know off the top of my head. I might work with pathological things, but they aren't so pathological that they aren't Radon

in a metric space, balls should have finite measure, I suppose

0celo7

@XanderHenderson not really

something like $\frac{1}{x}\, dx$ doesn't do that

idk, the paper is really vague...they say the measure "only depend on $r$" (the radius), which really makes no sense

Xander Henderson

4:11 AM

remind me what the definition of a Radon measure is?

shouldn't it be like, locally finite or somethign?

0celo7

oh yeah...

Xander Henderson

$\frac{\mathrm{d}x}{x}$ isn't really locally finite, is it? Isn't zero a problem there?

0celo7

well, maybe this isn't a radon measure

idk, it's late

Xander Henderson

again, everything nice is Radon.

Everything else is crap.

but it is late, and I don't math good when late

me no brain

0celo7

idk, it's either dealing with this crap or with this other paper that uses a maximum principle when there's no sign to justify using it

0celo7

4:29 AM

Ahhhhh

This paper must have been written by a grad student who can't into writing

Secret

5:32 AM

O, that's new: The operation is not closed

Let $x_1,x_2$ be the tumor cells, and let $v$ be the vaccine. We then compute $vx_1$ only

Since $v$ is a vaccine, it will map anything from the set of tumor cells $x_i$ to the set of dead tumor cells $d_i$

So obviously, $vx_1 = d_1$

But what happens here is unusual:

We instead have $vx_1+x_2 = d_1 + d_2$

and "$vx_2$" never existed

Xander Henderson

uh...

16 hours ago, by 0celo7

I think Secret has finally snapped

Secret

This is one reason why immunology and sometimes the mechanism of drugs are interesting, because it gives me ideas on thinking about unusual algebraic structures

@XanderHenderson not this time: yesterday, Dattlier made me snap because of some weird persistent duplicated post of his problems

But me expressing stuff as if they are algebra, they are part of my normal routine in the chat

Xander Henderson

Then, perhaps, the "finally" is unwarranted. You have always been, and always shall be, in a perpetual state of "having been snapped."

Secret

I have a lot of weird ideas, I just don't have time to organise, research and convert them all into mainstream language that is easy to comprehend

most of the time, as I read more, some ideas disappear because the mainstream literature already covered them

and then the focus will be on said mainstream literature since it provide the established framework for me to test my ideas

In the past, we (the chat and I) have discussed about a host of weird ideas such as tensor-like objects with more than 2 types of indices, notation for brackets in nonassociative algebras, division by zero algebras, properties of some nonlinear maps and so on. The chat usualyl give me useful guides and advice on how to research further and expanding upon

so I am not really that lunatic as some would think

Secret

5:51 AM

If $d_1,d_2 = 0$, then the above equation reduces to $vx_1+x_2 = 0$ meaning that these two elements are additive inverses to each other (which is a rather ordinary result)

Semiclassical

@Secret question leeches are exhausting

Secret

I don't know there's a term for that

Secret

Anyway, here's the above idea in english:

The way these researchers approach to cancer in making that vaccine, can well be a problem solving technique that can be applied to some stubborn proofs and theorems in maths:

For example, suppose you have a result you want to prove but the scope is huge, counterexamples are highly nontrivial or computationally expensive, with a lot of dead ends, thus it seems as if the proof is actively trying to prevent itself being proved (or the other possibility is that the proof is simply not true and a counterexample exists). Then instead of trying to find a fresh path on trying to lead to a proof or its counterexample, perhaps one thing that can be tried is to understand is

why the dead ends are dead ends, and then nudge from there, thus overcoming the blockade that leads to the proof or counterexample

Btw, the video is here: youtube.com/watch?v=PLcm4U2E4yM from a medicinal perspective, it sounds promising

In general, whn I solve problems, I am kinda like an experimentalist, by trying to parametrise the problem. So if a certain approach does not work, I can simply switch to another parameter and try again. Like many programming languages, I also like to vectorise my problem solving. Therefore, if I can express some parameters as a list, then I can prove multiple pathways at the same time in one equation. This is one reason I like to solve systems of equations together in vector form, instead of

approaching them one at a time

Ted Shifrin

6:12 AM

@Maneesh: You're making this way too complicated. Just rotate $2\pi/13$. The sum of the vectors does not change. Therefore ... ?

Rick

7:18 AM

How do I find the sum of all 4 digit formed by using the digits 1, 0, 2 and 3?

-1

Q: How do I find the sum of all 4 digit formed by using the digits 1, 0, 2 and 3?

I don't understand how to proceed here, I thought adding the greatest and smallest would give some sort of pattern, but that doesn't work here. Can someone please help

combinatorics

I asked it a long time back, but I didn't understand it

Maneesh Narayanan

7:35 AM

I got one idea, I don't know whether it is true or false. $v_1+v_2+...v_{13}= \left[ {\begin{array}{cc}
\cos 2\pi/13 & \sin2\pi/13\\
-\ sin2\pi/13& \cos2\pi/13 \\
\end{array} } \right](v_1+v_2+...+v_{13}$)

where $v_i=\left[ {\begin{array}{cc}
x_i \\
y_i \\
\end{array} } \right]$ on the i-th corner of 13-gon

$(I+P)(v_1+v_2+...v_{13})=O$

Secret

You got a rotation matrix, and all your vectors that point to the vertices of the 13-gon are starting from the origin. What happens when you rotate all 13 vectors about the origin with the angle of 2pi/13 ?

Maneesh Narayanan

determinant of $P$ non zero

so, sum of vectors equall to zero.

Am I correct?

we know $I+P \ne O$

@TedShifrin

@Secret landed upon the same n-gon. right?index will be changed.

Secret

yes, you end up with the same set of vectors, just with their indices permuted, thus you don't really change the sum of all of them

Maneesh Narayanan

@Secret why it is then zero? Are my arguments correct?

Abcd

@Rick What is the answer?

Maneesh Narayanan

7:54 AM

@Secret I think some theory is missing from my mind. Can you help me to fill that?. I have been struggling for 2 days.

Secret

6 hours ago, by Ted Shifrin

You haven't correctly answered my question yet: What vector stays the same when you rotate (by any $\theta\ne 0, 2\pi$, etc.) around the origin?

That vector is not one of the 13 vectors given

Put it in another way, if you have the whole $\Bbb{R}^2$ plane and you rotate it about the origin, which vector will remain the same (indices and all)?

Maneesh Narayanan

ok. Now it is clear. Thank you @Secret

Secret

@TedShifrin I think your method suggested to Maneesh only works for rotations in 2D vector spaces (which is the case we have here). Had the question asked for a distribution of charge in a 13-gon prism, then since you have an axis of rotation being invariant, there is no way to conclude that the sum of all vectors is zero since the zero vector in such case is not the only invariant vector under rotation

I am not sure what else is needed to guarenteee the sum of vectors is zero conclusion for higher dimensional systems like these, though. Probably that is a reason why we have stokes theorem

Having said that, I found this problem solving method interesting (for cases which it applies). I never thought about reasoning an expression by relating its invariance under a group action to the unique element that is invariant under the same group action

Tim The Enchanter

Hey chat, quick question. I'm trying to prove the following statement.

Suppose (For discrete random variables $X$ and $Y$ such that their expectations are finite) $P(X \ge Y) = 1$. Then $EX \ge EY$; moreover, $EX = EY$ if and only if $P(X = Y) = 1$.

Now the text I'm using starts with:
Let $Z = X - Y$. Since $P(Z \ge 0) = P(X \ge Y) = 1$, the values $z$, that $Z = X - Y$ assumes must all be nonnegative.

I don't see why this implication follows, as it seems to me that the range of $Z$ could contain $-1$ and countably many other nonnegative values such that $P(Z \ge 0) = 1$ ( Such as perhap…

Also, the variables are all discrete.

Abcd

8:13 AM

Hi @Rick? I have answered your question. Does my answer match the one given in your answer sheet?

Rick

@Abcd, Hi, the answer given in 38,664

YOUSEFY

8:37 AM

Hi, look at this inequality: (it is from Math Olymbiad) :ibb.co/dG4L87, I want to check with you. I applied induction proof. P(1) is ok. Now, for induction step I consider a=b=c, then show that by induction that P(k+1) is also true (using infinity rules). I just have one problem which is when a, b, and c are different. Now, my question is that: Is it Ok to use induction proof to show this inequality true when a, b and c are different?

Abcd

@Rick Done! Now you may see my answer.

Kaustabha Ray

For the Euclidean distance metric if we do not take the square root, and compare two such distances only on the basis of squared distances would it result in the same had we compared the Euclidean distances with the square root?

Tuki

Hello what latex notation is used when you have computed indefinite integral and you want to insert bounds ?

Maneesh Narayanan

@TedShifrin Thank you very much.

user243301

Many thnaks @KirylPesotski , then I preferred to delete the question since I've the information of previous comment and yours, then I try to search or calculate it. My apologizes. Isn't required a response of this comment and good week.

user2219896

10:49 AM

@YOUSEFY Induction seems to be hard in this case. as the inequality must hold for all reals. Sometimes you can prove something holds for all rational numbers if you do induction for example first by nominator and then by denominator but you need some theorem which proves that if theorem holds for all rationals, it holds for all reals.

I have difficulties on probability problem. Suppose we have one virus. On every turn, virus either dies with probability 1/3, does nothing with probability 1/3 and splits up to two viruses with probability 1/3. What is the probability that the virus population become extinct at some point? Can this be solved by Markov chains although the number of viruses can be arbitrary?

Secret

11:33 AM

$x_0 = 1$

$x_1 = x_0 (\frac{1}{3} * 0 + \frac{1}{3} * 1 + \frac{1}{3} * 2)$

$x_2 = \frac{x_1}{3} (0+1+2) $

uh, that cannot be right, the virus population never rise...

Leaky Nun

11:50 AM

Cauchy's integral formula is so beautiful

Secret

that one from complex analysis?

Leaky Nun

$$f(a) = \frac 1 {2i\pi} \oint_\gamma \frac {f(z)} {z-a} \ \mathrm dz$$

for any holomorphic function $f:\Bbb C \to \Bbb C$

@Secret yes

Secret

well, that basically illustrates nicely on how complex differentiation is integration in disguise

flatronka

12:08 PM

hi guys, I was wondering why there is no such thing of multiplying a scalar with a point in euclidean space? There are tons of definition of multiplying a scalar with a vector but not with a point. hmmm

Secret

a vector and a point are interchangeable in the context of vector spaces, since a point is basically a vector (0,0,0) -> (x,y,z)

flatronka

uhhh, right, thanks for the quick clear answer @Secret

User432477438

Sir, may I ask help about Legendre symbol?

Secret

the above also explains why you cannot freely translate a point, unlike how a vector in flat space you can keep it unchanged after translation

since the tail for a position vector always rest on the origin, while all other vectors are compute by the difference of its two coordinates

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