Mathematics - 2017-06-05 (page 3 of 5)

@Araske did I not tell you? He was with Calegari and was like "Oh yeah the guy is a good lecturer, you should take a class of his, even though he doesn't know much math"

Araske

lol yeah

i've never met th guy but he sounds interesting

SBM

Hello

Ted Shifrin

Is Araske someone known?

Araske

that's pretty savage as professors go

yeah we both go to UC

Daminark

They're a friend of mine as well at UC

SBM

6:42 AM

I'm looking to improve my linear algebra.

Ted Shifrin

Damn, 3 now?

Araske

we're multiplying

Daminark

Yup, overrunning the chat

Ted Shifrin

Good goal, SBM.

Daminark

@SBM good life choice is good

Araske

6:43 AM

linear algebra is great

Abhishekstudent

Why should one of the dets vanish @TedShifrin

SBM

@Daminark Any resource recommendations to learn about and practice linear algebra?

Ted Shifrin

Because one column is a linear combo of the other two.

Daminark

What are you aiming for with it?

Balarka Sen

I should know more matrix analysis

Abhishekstudent

6:44 AM

linear combo?

SBM

@Abhishekstudent linear combination

Ted Shifrin

Linear combination

Daminark

@Balarka analysis in the generic sense of analyzing? Or like, real analysis but with matrices?

Balarka Sen

calculus with matrices. the computational stuff

Daminark

Ah

Eh

Ted Shifrin

6:45 AM

you skipped it all in my book, a Balarka!

Balarka Sen

the things i have to do for a living, yes

@Ted Your multivariable text has stuff like that?

Daminark

Laci has a sort of book draft online based on how he taught it in the REU, and he definitely had a more matrix focus

Abhishekstudent

Say two planes are parallel and one of the dets vanish @TedShifrin

Balarka Sen

@Daminark link me!

Abhishekstudent

What will it signify geometrically?@TedShifrin

Daminark

6:46 AM

people.cs.uchicago.edu/~laci/linalgbook.dir/book.pdf

Basically, he gave an integrated treatment of more vector space/linear transformation-y stuff and matrix stuff, focused on making sure we understood how to bounce between one and the other

Ted Shifrin

If two planes are parallel, there are no solutions. This means the vector is NOT a linear comb of the columns of A.

Balarka Sen

thanks man

Daminark

As opposed to just having matrices as a side note

No problem!

Ted Shifrin

so one of the dets is for sure nonzero.

Ok, I have to board my flight. See you all another day.

Balarka Sen

Safe flight.

Fargle

6:49 AM

Have fun!

Araske

Have a nice flight!

Daminark

See you @Ted, have a good flight!

And for the rest of the room: is there a way to sort of... I dunno, "get an intuition for measure theory"?

SBM

@TedShifrin Have a happy journey

measure theory?

Daminark

What I mean is something to the effect of, some manipulations I've been seeing in certain proofs feel like black magic

Easy to justify in hindsight but why would anyone think to do it, you know?

I've started to get better at recognizing when Vitali or Besicovitch is needed, and also at choosing density points

But that it's getting more doable feels like it's due to pattern matching as opposed to, I dunno, "having insight"

Fargle

@Daminark "Young man, in mathematics you don't understand things. You just get used to them." -von Neumann

That strikes me as comforting coming from one of the most prodigious minds of the modern era.

Daminark

6:55 AM

That is true

Araske

i mean sure

Daminark

(Though for Marianna all of measure theory is probably trivial)

Araske

but to some extent you gotta 'see' it to make it really work for you

Avery

@Shog9 thanks for resolving this issue.

While you haven't issued any bans, your message seems to have caused the offending users to accept & apologise. So yeah, thanks.

Secret

7:10 AM

> "Young man, in mathematics you don't understand things. You just get used to them." -von Neumann

I argue that there exists maths where understanding is possible

e.g. linear algebra can (mostly) be visualised with geometric intuition

(though if one is talking about the theorems, it takes more creativity)

Araske

Linear Algebra over $\Bbb C$ is more difficult

arctic tern

?

Fargle

@Araske At least you always have eigenvalues, though.

Araske

to visualise anyway

true

also I find kernel/image/basis arguments rather visual as well

...actually, I haven't yet found a theorem or property I couldn't visualise in some way

Daminark

For what it's worth I wasn't quite going for seeing that theorems were true in hindsight, almost smelling their presence before you know for sure that you should use them

So, define $Mf(x) = \frac{1}{\mu(B(x,r))}\int_{B(x,r)} |f|d\mu$. Then you want to show that $f\mapsto Mf$ is a bounded operator on $Lp$ for $p > 1$

One of the lemmas in this proof is that $\mu(\{x: Mf(x) > t\}) \le \frac{c}{t}\int_{\{x:|f(x)| > \frac{t}{2}\}} |f| d\mu$

Proving it is fine, you just define $f_1$ to be $f$ where it's less than $\frac{t}{2}$ and $f_2$ elsewhere, so that $f = f_1 + f_2$

Then $t < Mf(x) \le Mf_1(x) + Mf_2(x)$, but since $\|f_1\|_{\infty} \le \frac{t}{2}$, we have that $Mf_2(x) > \frac{t}{2}$ (both statements are almost everywhere), you get that except on for a null set, $\{x:Mf(x) > t\} \subset \{x:Mf_2(x) > \frac{t}{2}\}$

Then you use the weak 1-1 estimate and get $\mu(\{x:Mf(x) > t\}) \le \mu(\{x:Mf_2(x) > \frac{t}{2}\}) \le \frac{2c}{t}\int_{x:|f(x)| > \frac{t}{2}} |f|d\mu$

Like sure, this checks out and all, but what?

xoxox

7:59 AM

hello

SBM

@xoxox hey

xoxox

I'm trying to approximate E[X] where X ~ Geo(1/2) with two methods: once I just "throw the coin", yields ~2, good. But the second method using CDF gives me 1.44

using -Math.log2(1-Math.random())

I don't see what I'm doing wrong

CDF is: 1-(1/2)^k

for Geo(1/2)

the value of k should be >= 1, but the code does not guarantee that

xoxox

8:20 AM

the solution apparently is to ceil the value

going

9:35 AM

I'm studying to be a school teacher and trying to up my maths game by improving my math skills by tackling some basic problems every day

Today I'm trying to work out radius of a circle based upon chord length and height of circle

sorry, that is height of the segment

my assumption is this height sits at a right angle directly in the middle of the chord

so, i've pulled out my compass etc. etc. and drawn up a circle of radius 6cm, a chord length of 8 which is giving me a segment height of 1.42cm

Using either of the formulas in this question math.stackexchange.com/questions/564058/…

I can't get a radius of 6cm

There are 2 different formulas, in the one with 7 votes and the one with 3 votes

SBM

oh

going

For the one voted 7 votes, it provideds CQ=(4*h squared)+(ℓ squared) over 8*h

if h is 1.42 and ℓ is 8

I get (4 x 1.42 x 1.42) + 64 divided by (8 x 1.42)

approx 72 divided by 11.36

About 6.34

SBM

um

going

Apologies, I don't know how to correctly post formulas in here :(

SBM

no worries

going

9:46 AM

The other formula is ((1/2 the chord length squared) + segment height squared) over 2 * segment height

SBM

Why not try deriving something and then use it

going

Using the other formula I get 18.0164/2.84 which is also 6.34 (approx)

@SBM - Sorry not sure what you mean by deriving something

SBM

Why not draw a diagram and figure out the formula for yourself first?

That way you'd get exactly what you need

going

I've drawn the diagram. Unfortunately my trigonometry is nowhere near strong enough to derive the formula myself.

Astyx

10:06 AM

@going Do you know mathjax ?

SBM

@going oh I guess it might not even require trigonometry a lot

Abcd

10:25 AM

I have tan alpha= 4/3 . How do I find alpha?

Daminark

Hey @Alessandro and @Astyx!

Alessandro Codenotti

Hi @Dami

going

@sb

Daminark

I can't tell if you're coming or @going! :P

But yeah how's everything doing @Alessandro?

Alessandro Codenotti

@Daminark I'm definitely leaving after this one

going

10:27 AM

@SBM - I'm only going to be teaching 5 to 10 year olds. Not much of an excuse. But im trying to improve

Daminark

Noo :(

Alessandro Codenotti

I'm studying algebraic topology and complex analysis for an exam

Daminark

Good luck

SBM

@AlessandroCodenotti All the best

Daminark

I've got to study for real analysis

SBM

10:30 AM

@going Oh, I see. But this doesn't appear to be complicated either.

Daminark

Though I think I've more or less got it down. It's only on the stuff from the last few weeks and I think I've materialized most of the main proofs except for the dual of $L_p$

Abcd

tan alpha= 4/3 . How do I find alpha?

Alessandro Codenotti

I'll have real analysis in July

SBM

@Abcd Inverse trig.

plus the period

Daminark

Ah

Abcd

10:31 AM

@SBM Can you please explain?

Daminark

It's today for me :P

It'll be nice to be done sooner though

Like, this Saturday everything will be complete

Abcd

@SBM ?

SBM

@Abcd $$\tan \alpha = \frac{4}{3} \implies \alpha = n\pi + \arctan \left( \frac{4}{3} \right)$$

Can you figure out why?

@Abcd The tangent function is periodic

being periodic you can't expect to make an inverse function unless you restrict the domain

Abcd

How do I enable mathjax on mobile??

Daminark

Oh by the way @SBM I don't think I ever ended up answering you about linear algebra.

SBM

10:36 AM

tinyurl.com/cfqcvpc

Abcd

@SBM I am unable to read that

SBM

@Abcd check the link

Daminark

But yeah so, what are you hoping to do with the subject? Is there something specific you wish to do with it, gain a more theoretical perspective, etc

SBM

@Daminark Right now; I do not have much idea about linear algebra, that is why I wish to learn more about it.

Daminark

That's fair

So, this book followed the curriculum I learned it with the very first time: people.cs.uchicago.edu/~laci/linalgbook.dir/book.pdf

Though the book itself doesn't really prove much, just kinda leaves things to exercises

(As the professor did sometimes :P)

For reference this was written almost as lecture notes without many details

Manolis Lyviakis

10:40 AM

Good Day to you guyz

Daminark

The first book I worked out of was Hoffman and Kunze's Linear Algebra

It's a bit old school, harsh, and does things with more of an eye toward algebra than anything else, but it's good if you've got the time

SBM

@ManolisLyviakis you too have a good day yourself

@Daminark I see.

Daminark

Anyway, I'm now gonna pull a Balarka-lite and have a screwed sleep schedule

SBM

@Daminark is it too late there?

Daminark

It's almost 6AM

My original plan was to just stay up all night studying for analysis, then go to sleep after add-dropping into bio, logic, and complex analysis until it gets closer to time for math

That's cutting it a bit close, though, so instead I'll just sleep the next 2-3 hours, wake up in time for add-drop, then go back to sleep, so this gives me panic review time before the exam

Balarka Sen

10:52 AM

lol

Daminark

Your influence is spreading!

Balarka Sen

you really shouldn't have done this. i try to fix my sleep schedule at stress hours

go get sleep

Daminark

Part of it is that I'm fasting, no food or water all day. Taking a 6:30PM exam like that is merp. At least shifting sleep like this will help lessen that blow somewhat

But yeah anyway, good morning!

Astyx

Hi @Dami

Daminark

Lolol, hibye :P

Astyx

10:54 AM

hibye I guess

Terrible timing as usual

Balarka Sen

Ah, I see. Well, I am not sure if that's good; I personally would not recommend fasting and sleep-wrecking simultaneously.

But good luck!

Astyx

Why the fasting ?

SBM

@BalarkaSen Neither would I. Though all the best @Daminark

Balarka Sen

I am guessing a religious/cultural custom, @Astyx.

SteamyRoot

Ramadan ?

Astyx

10:58 AM

Makes sense

Sha Vuklia

hi guys

$$\begin{aligned}
\left.{E\sim\rho\sim Q\\
V\sim E}\right\}
\end{aligned}$$

anyone any idea how this extra enter got in?

SBM

@ShaVuklia hello

Sha Vuklia

I'm trying to make these curly brackets

SBM

easy

Sha Vuklia

and I'm kind of doing it "by hand", using \left and \right

alright tell me @SBM

Balarka Sen

11:01 AM

irrelevant but wow, good song

SBM

\[f(x) = \begin{cases} a + b^2 \\ a - k^2 \end{cases}\]

Sha Vuklia

no

SBM

like that?

Sha Vuklia

that's not what i'm asking

i need brackets at the right side

SBM

Oh equation system

Sha Vuklia

11:01 AM

yea

Balarka Sen

(I guess it would have made better sense if I read 1984)

Astyx

@BalarkaSen Maybe, yeah :p

SBM

George Orwell writes well.

Sha Vuklia

oh

i've figured it out

i need to add "aligned" to it

SBM

ok

Sha Vuklia

11:02 AM

and then all works out

SBM

yes

$$\left. \begin{align} a = b + c \\ c = d + e \end{align} \right\}$$

That

Balarka Sen

I at least know who Big Brother is, so :P

SBM

@BalarkaSen Interesting.

Balarka Sen

I know enough of a thing or two about some major 20th century literature that I can pretend I have read them.

and get away

SBM

@BalarkaSen Maybe.

Display name

11:31 AM

-1

Q: Logical Question based on digits.

Being given a 9 digit number $N$ in which digits MAY be repeated. Consider another single digit no. $X$. Now if we remove any digit from $N$ and multiply it by $X$, the sum of digits of the product is the same in all possible cases. Also whatsoever digit we remove, on the same position in the pro...

recreational-mathematics

the question interests me, i dont know how to approach it.how will it be solved

the question is closed so just help me out here.

Abcd

11:50 AM

@SBM I dont know whats arctan

SBM

@Abcd It's basically a function which returns the angle for the tangent function in the range $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right] $

Abcd

@SBM :( Sorry. I am in 11th class. Havent studeid that stuff yet

studied*

SBM

So suppose $\tan x = \frac{1}{\sqrt{3}}$

let that be y

that means $\arctan y = \frac{\pi}{6}$

In general for $x \in \left[-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right]$ and $\tan x = k$
$\arctan k = x$

@Abcd

Abdullah UYU

Can anyone help me to find $\vec{AO}\cdot\vec{OC}$? Noting that $\measuredangle{BAC}=90^{\circ}$

SBM

use trigonometry

Abdullah UYU

12:03 PM

to find vector's length?

SBM

12:15 PM

have you found the length?

Abdullah UYU

yeah and i proceed.

did you use tirgonometry to find lengths?

SBM

12:30 PM

I haven't even touched your question.

mathvc_

Hey guys. I have a question

hatcher used this fact without proof

consider a map from S^n to S^n of degree m

and its Mapping cylinder

M_f

why H_n(M_f,S^n) is Z/mZ?

I guess its easy but i dont get it

Abcd

@SBM Sorry. Is there any other way? i cant understand this

SBM

@Abcd 'What did you not understand?

@Abcd You were like asking given $\tan x$ find $x$, right?

Abcd

@SBM This step

yes

tan x = 4/3 . What is the value of x in degrees

?

SBM

@Abcd You know that $\tan$ is a function that can give the same value for different parameters, right?

Abcd

12:43 PM

@SBM No. How? What do you mean?

SBM

@Abcd $$ \tan 180^{\circ} = \tan 0^{\circ} = 0$$

Display name

Plz any1 see my question

SBM

right?

Abcd

@SBM I didnt know this

SBM

You should learn that first before jumping into inverse trigonometry.

Semiclassical

12:46 PM

geometrically, the point is that a line through the origin has two intersections with the origin

It's not so strange when you think of how tangent is defined: as the ratio of sine to cosine.

That'll be positive if both cos and sin are positive, but also if they're both negative

mathvc_

Any idea?

Semiclassical

Speaking without any significant knowledge, I presume it comes down to what it means to have a degree-m mapping from the n-sphere to itself

Also something-something relative homology

mathvc_

I know its meaning

n-th homology group of n-sphere is Z. the degree of f is the multiplication of the induced homomorphism of f from Z to Z

Abcd

1:02 PM

@SBM from where should I learn? any links?

SBM

@Abcd It should be there in your 11th grade syllabus

BAYMAX

1:23 PM

Is there any symbolic equation checker

hey @Semiclassical

You might be using like this in your work too

Akiva Weinberger

Think of a number from 1 to 100

Is it 23?

BAYMAX

I thought of it , now?

Ah,Why so serious! it is not 23!

N. Younger

1:49 PM

Hello everybody! I am new here! Can I ask about a mathematical problem in this chat room?

Astyx

Yes, as long as you don't ask to ask

Sha Vuklia

haha

N. Younger

:-) I have a understanding problem about generating functions

My questions is related to the following problem: math.stackexchange.com/questions/2306756/…

I am trying to understand the given answer

Why is a replacement of zeros the same as the equation (3)?

Transcript for

$Mathematics$ Mathematics