Mathematics - 2021-04-11 (page 1 of 2)

now I can't unsee the image of Father Ted smacking a blackboard with a cigarrete hanging out and a bottle of whisky "Don't unfactor a fecking perfect factorisation

Ted Shifrin

12:09 AM

Well, this Ted has never smoked anything and finds it repellent.

Andrew Micallef

hah hah

StudySmarterNotHarder

Don't smoke :)

Andrew Micallef

I'm only on your second lecture Ted, but I feel I learned something all the same, I am using a lot more english in my notes to my self

Ted Shifrin

Oh, cool. :) Well, it does get quite hard later on, but as much as it helps you, I'm very pleased.

Andrew Micallef

also I feel I missed an injoke somewhere, "and then I say to myself:" ausidence responds "self"

Ted Shifrin

12:12 AM

That's one of my patented schticks ... but I had to encourage the class to do that at first. Then they did it without prompting.

I ask myself ...

Andrew Micallef

This is where I was before Father Ted chimed in anyhow :
$$A^2 \left(
\frac{b^2}{b^2 -2ab + a^2}
-\frac{2bx}{b^2 -2ab + a^2}
+\frac{x^2}{b^2 -2ab + a^2}
\right)
$$

Ted Shifrin

I'm trying to teach people learning to ask themselves questions :)

Andrew Micallef

ahh it is a good one

HereToRelax

why did she leave

Ted Shifrin

Always, always, always pull out constants from integrals and derivatives.

Andrew Micallef

12:13 AM

I feel I will adopt it

do I need to pay royalties if I do?

Ted Shifrin

I'll give you a discount 'cuz you're a newbie.

HereToRelax

No use scihub

Andrew Micallef

neat,

anyway, I really should get back to it, I've been on this book for nearly a month and am still on page 16

(mostly because I keep lurking here in my alotted study time)

HereToRelax

@AndrewMicallef average if u ask me

Andrew Micallef

I'm average in most respects (had an interview recently where that was my response to the question "what is your superpower?")

@TedShifrin gonna cite that message and go back a few steps.

HereToRelax

12:18 AM

I interviewed for a facebook software dev position once

dc3rd

I've finally managed to drag myself to my computer after getting sick this morning (I made an egg white omlette and everything went south after I ate it)............I thought about the question from last night this morning before my calamitous meal. I got a solution and to me it feels like the right reasoning....@TedShifrin

HereToRelax

and everyone that interviewed me had beginner english

dc3rd

True/ False: all interior points of $\bar{S}$ are points of $S$?

Claim: False.

Consider the set $[a,b]$. The closure of this set can be defined as $\overline{[a,b]} = (a,b) \cup \{a,b\}$ as in the frontier points are explicitly $a$ and $b$ individually. So if $x \in \overline{[a,b]}$ then it is in either $(a,b)$ or $\{a,b\}$. If $x \in (a,b)$ it is trivially an interior point. So now examining the case for the set of frontier points $\{a,b\}$, going back to the definition of an interior point of a set, it says that a point $x$ is an interior point if there exists an $\epsilon > 0$ such …

HereToRelax

and I messed up the parameters in a recursion

those two things were completely unrelated

Andrew Micallef

HereToRelax

12:23 AM

I don't know when things are beautifully factored

I feel like it's just practice

your brain rewires to know when a certain form is more useful

although I may be saying nonsense

Andrew Micallef

I'm sure it comes with practice

I feel like that is a big part of it, you do this stuff enough and you just start to see when things are similar to an easier problem you have already solved in the past

HereToRelax

yeah for sure

I think uberchad Von Neumann had a phrase

Andrew Micallef

uberchad

lol

what?

HereToRelax

"Young man, in mathematics you don't understand things. You just get used to them.”

should I not use that word?

Andrew Micallef

I just think Albert Camus, not John Von Neuman

HereToRelax

12:29 AM

I read that guys famous book in high school

it's pretty dope

the foreigner or something

Andrew Micallef

I'm sure there were uberchad mathemeticians, I just don't know of any

HereToRelax

idk I read it in spanish

Well Von Neumann definitely classifies as one

Andrew Micallef

I'm not sure if it counts, but the only badass mathamaticians story I can think of is that annectode of pythagoras telling the centurion to get out of his sun.

I need not to go down a rabbit hole of digging up badass von neuman moments ant solve some integrals

HereToRelax

Von Neumann allegedly hosted a ton of parties in his house when he worked at the IAS

and he had high level clearence

Andrew Micallef

IAS?

HereToRelax

12:31 AM

Institute for Advanced Study

The Institute for Advanced Study (IAS), located in Princeton, New Jersey, in the United States, is an independent postdoctoral research center for theoretical research and intellectual inquiry. It has served as the academic home of internationally preeminent scholars, including Albert Einstein, J. Robert Oppenheimer, Hermann Weyl, John von Neumann, and Kurt Gödel, after they immigrated to the United States. It was founded in 1930 by American educator Abraham Flexner, together with philanthropists Louis Bamberger and Caroline Bamberger Fuld. Although it is close to and collaborates with Princeton...

Andrew Micallef

I get a department of the australian taxation office when i searched

HereToRelax

that's probably cuz your a money launderer

Andrew Micallef

allegedly

HereToRelax

I promise your honor, the rolex is fake

Andrew Micallef

Hey man, don't dis the imitations. Seriously brand naming is basically theft anyhow

HereToRelax

12:34 AM

i bought a rolex

now my yatch goes faster

robjohn

12:45 AM

@HereToRelax don’t

HereToRelax

but wouldn't that require a breakthrough?

I wasn't considering asking anything

The last combinatorics question was almost a psq

but the dude added that he wanted a solution with probabilistic method and posted the official solution

It appears that was enough to save it from psq status

Also, no one added a probabilistic method solution

Andrew Micallef

1:00 AM

so I have this:
$$
\frac{ -A^2(a-b)^3}{3(b-a)^2}
$$
can I do this?
$$
\frac{ -A^2(a-b)^3}{3(b-a)^2} = \frac{ A^2(b-a)^3}{3(b-a)^2} = \frac{ A^2(b-a)}{3}
$$

HereToRelax

The first step is completely legit, and the second step is also completely legit (but if $(b-a)= 0$ then I'm not sure how legit it is).

Well I don't know if that helps

but it looks good to me

the expression on the right makes sense for more cases so I guess you may have to take that into account

Andrew Micallef

oh, okay, $a$ and $b$ are both given as positive constants, but I take your point

that helped

HereToRelax

you're a motor mechanic?

Andrew Micallef

uh huh

filter spinner

HereToRelax

Will I be able to put my car on neutral gear

when all cars are electric

I like when the engine is not connected to the wheels

it feels dope

Andrew Micallef

1:07 AM

if it still has a mechanical transmission, sure

HereToRelax

well yeah

but will that happen?

Andrew Micallef

what a mechanical trans + electric motor?

HereToRelax

well it doesn't have to be mechanical

Andrew Micallef

yeah, all toyota hybrids do that

HereToRelax

Oh that's dope

I like how the car feels when the motor isn't locked to the wheels

Andrew Micallef

1:09 AM

(granted not full electric, but I can imagine they must have in Tesla,s given they have some models with a single motor)

that is a weird thing to enjoy

but good on ya :P

HereToRelax

really?

I feel the car is super predictable

and it just goes with the flow

like it feels free

Andrew Micallef

I dunno it is pretty predicable aother wise as well imho

HereToRelax

Maybe if you're a genius

Andrew Micallef

they aren't cosmic mysteries

HereToRelax

well that's true

but I'm not experienced enough to now exactly how much the engine is going to slow the car when I downshift

Andrew Micallef

1:11 AM

that is why you use your clutch

also that is a bad way of driving unless you are in a big rig

brakes are cheap to replace, engine components not so much

use your brakes not the engine to slow yourself

I used to drive like you do, then I became a mechanic :P

HereToRelax

well my car is a 1998 civic

but I don't brake with the engine

Andrew Micallef

use your brake

it's not so much that it 'breaks' just that you apply unnecessry levels of force decreasing the lifespan and wearing down components that are timeconsuming and expensive to replace

Also, wasting fuel

HereToRelax

what wastes fuel?

Andrew Micallef

engine braking

HereToRelax

I thought the fuel injector was mostly controled by the fuel pedal

gas pedal

Andrew Micallef

1:14 AM

the engine speeds up when you engine brake

so more cycles

so more fuel

HereToRelax

it doesn't speed up does it?

Andrew Micallef

you are using the compression in the engine to slow yourself

HereToRelax

it would be going faster if it wasnt locked to the wheels

Andrew Micallef

it does if youu drop a gear to slow down

dont take my word for it, do the experiment

watch the tacho when you drop a gear

HereToRelax

When you put the clutch in it revs freely doesnt it?

and then when it gets put into gear it slows down

Andrew Micallef

1:16 AM

sure, over time

HereToRelax

well I always double clutch like I should I don't like to granny shift

Andrew Micallef

initially when you disengage the clutch if the transmission is going faster than the engine the tansmission drives the engine

HereToRelax

oooh

that makes sense

Andrew Micallef

that is what slows you down

HereToRelax

ooh

Andrew Micallef

1:18 AM

the engine trying to get back to the lower speed

HereToRelax

I guess I'm dumb

yeah ur right my bad

Andrew Micallef

np

I used to think like you do

Now I realise that automatics are actually a really great idea (no fun, but way better in every other regard)

HereToRelax

but automatics feel so wrong

but I think that's because of the throttle curve of new cars

or something

my grandpa has an automatic car and the throttle is super unprecise

Andrew Micallef

Really really off topic, but that stuff that ires rev heads is actually a fantastic idea from the engineering POV

basically the engine is full of components that wear out on a per cycle basis, so if you can keep revs low you maximise economy everwhere, but that hurts pple who like to hear the roar of the machine :P

HereToRelax

So cars that feel terrible are actually really dope?

Andrew Micallef

1:22 AM

uh huh

HereToRelax

how tragic

Andrew Micallef

I'm an old man, so that excites me more (the economics)

HereToRelax

I don't like that the motors that use magnetic wierd fields need a computer

DC motors seem easier

Andrew Micallef

uh, They all use computers

just in different ways

HereToRelax

but some of the computers are dumb

Andrew Micallef

1:24 AM

hahah

HereToRelax

I prefer dumb computers

Andrew Micallef

I like my computers dumb too

hidden states are a nightmare to diagnose

but economics is more powerfull than fun, and smart computers are generally more economical

HereToRelax

I guess that makes sense but I don't like it

leslie townes

2:40 AM

When you give instructions to your evil robot, don't forget to add, "but not me."

Andrew Micallef

All my evil robots obey asimovs laws

user863565

dude what's this field called?

$\Bbb{F}_2:={0,1}$ it got addition operator which gives 0+0=0, 0+1=1, 1+0=0 but 1+1=0?

1=-1 ?

multiplication operator table is as follows: 0*0=0,0*1=0,1*0=0 and 1*1=1

is it okay to say 1=-1?

it's bit abstract and odd for me to digest

leslie townes

2:57 AM

yeah, 1=-1 in that field

-1 is just a thing that you can add to 1 and get zero.

if 1+z = 0 then you can call z by -1. even if z happens to also be 1.

sometimes in this context people use other integer notations in relation to fields. for example by "5" they mean not the integer 5, but whatever 1+1+1+1+1 happens to be in the field. which in your field is also 1.

there's a map from the integers into any field given by sending the integer n (when positive) to the n-fold addition of 1 with itself, and similarly with additive inverses. and sometimes people just write "n" in the field when they mean the image of the integer n under the map.

so you get equations like 1=-1 and 3=5=1 and stuff. it's not integer arithmetic, it's arithmetic in the context of that addition operator.

barista

3:14 AM

I already know the ideal correspondence between $R$ and $R/I$ but why this implies there is an ideal correspondence between f(A) and ideal of $A$ which contains $ker(f)$? where $f:A\to B$ is a ring homomorphism

Thorgott

isomorphism theorem

barista

Oh, right

user863565

3:28 AM

oops others are not element of the field sorry

it makes sense now

Ted Shifrin

@dc3rd I hope you're feeling better. Food poisoning from an egg white omelet seems unlikely, but ... I don't know if other people commented, but for the example of $S=(a,b)$, the statement that an interior point of $\bar S$ is in $S$ is correct. I don't know why you keep trying to prove that it's wrong there. I told you that you needed to find more inventive examples.

Howdy, @Thor.

@Andrew Define "old man."

Andrew Micallef

4:27 AM

@TedShifrin hmm, I'll pass. But in the context of that conversation, old enough to not get excited by loud revving cars

love_sodam

4:43 AM

0

Q: Integral ring homomorphism induces a closed map

I tried to understand this answer. As far as I understood, as $f(A)\subset B$ is an integral extension, $f(A)/(f(A)\cap I)\subset B/I$ is also integral extension. Then he use lying over theorem. In what way? What prime ideal in $f(A)/(f(A)\cap I)$ should I use?

commutative-algebra

can somebody answer this?

leslie townes

think of how troubling it would be if the answer were no.

love_sodam

What do you mean

leslie townes

i'm joking (i don't know the answer) but the premise of the joke is, what if you somehow stumbled on a gap in all of mathematics and nobody could answer it.

that would be a real "uh oh" moment, i think.

you literally meant, can somebody present in the chat right now answer this, which for the moment, the answer appears to be no. that is less troubling than what i was contemplating.

somebody should hop on it. it seems like the kind of thing that a person somewhat more experienced than myself in algebra would be able to have insight into.

the lure of imaginary internet points will draw someone into posting an answer. that usually happens when i'm writing an answer to a linear algebra question.

maybe 1/4 of the time i think 'no, that's not too good of an answer, i'll still post mine.' 3/4 of the time i upvote and hit cancel or escape or whatever it is.

copper.hat

dude, what are you smoking?

leslie townes

currently i am under the influence of ginger turmeric tea.

copper.hat

4:57 AM

omg, 911 now

leslie townes

which isn't tea at all, but i can't bring myself to say 'tisane.' i didn't have that kind of upbringing.

copper.hat

new vocab word for me

leslie townes

if it says tea on the box it's tea, i don't care about biological definitions. it can legally be called tea.

copper.hat

went to an r&b session at the riggers loft in richmond this afternoon

went w some friends which was nice.

insane prices for tiny plates of over prepared snacks.

leslie townes

i have yet to go out with friends. many of my closest friends are high risk and we're not all vaccinated. i'm still leery of public places but probably that is just a psychological complex of mine.

copper.hat

5:01 AM

im in desperate need of social activity

we cycled back which was nice

about 10mi

leslie townes

that sounds like a nice day out. and a bit of uphill on the back end of that i imagine.

copper.hat

tiny bit. i went for a proper ride earlier this morning to get my jollies out

exercise is my drug

and at the moment that means cycling

leslie townes

i use herbal tea, and a tiny bit of birding. there's a duck pond near by, i take my daughter there, she bothers the geese while i just sit there. it's therapeutic.

copper.hat

that is nice. does she feed them bread thereby sentencing them to a lifetime of dependency?

so i used to get lectured

leslie townes

she does not. we sneer at the people who do.

there's a tree with a hollowed out hole in it and often there is an egyptian goose inside. it's usually the highlight of the trip.

copper.hat

5:04 AM

we would have been the subject of your derision...

very nice. the suez goose.

leslie townes

it's funny, wikipedia says they like to rest in hollows of trees. it's cool when the birds behave according to the manual.

copper.hat

i used to like to sketch them.

very therapeutic, plus you notice all sorts of detail that would pass by otherwise

leslie townes

copper.hat

very nice!

leslie townes

he's there every weekend.

most people don't know about him because he's somewhat high off the ground and just minding his own business. sometimes we tell people about him. if they don't seem like the type of degenerates who will attempt to feed him and create a lifetime of dependency.

i joke. i do not have strong feelings unless someone leaves a feeder out when there is a feeder-spread disease going around. an interest and appreciation of birds will do more good than harm, even if you toss them the odd bit of bread every now and again.

i sometimes, rarely, give our crows peanut butter.

Andrew Micallef

5:18 AM

Yo @leslietownes I want to share more nightmare formatting with you

leslie townes

i will steel myself.

copper.hat

i used to feed robins & wrens growing up. sometimes the robins would land on my hand and eat.

Andrew Micallef

gofile.io/d/zsBu8t

its a pdf with no paginations

so one long scroll

leslie townes

i love watching robins fight in the early spring. there's nothing pleasant about it. things really kick off. from such innocuous looking birds.

andrew my initial thought on formatting is this looks great. i wonder about the use of color, if it's supposed to convey information or if it's just a sytlistic choice. some people are very deliberate users of color.

Andrew Micallef

I'm just a sucker for colours I guess

leslie townes

5:22 AM

in the definition i think you mean A((b-x)/(b-a)). with parentheses around the inside bit so you get A(1) when x = a.

Andrew Micallef

(I haven proofed below the part where I cheated to sketch the function also)

leslie townes

that's mostly clear from the computation but not clear in the initial definition. you have to scratch your head for a minute.

Andrew Micallef

yeah I do mean that (or at least I use that definition to find A)

leslie townes

or at least i did.

maybe it's just i have a hair scratching disease.

Andrew Micallef

I should revise that

I don't want to scratch my hairs out in the futuer

I think colouring stuff is fun, in this case I used it because I couldn't figure out how to get eqrefs working in jupyter, so I wanted it to be clear which part of the equation I was dealing with in each pass, without being able to cross reff the normal way.

also I cheated and used wolfram alpha for the algebra at the end, not sure how bad I should feel about that

leslie townes

5:26 AM

there are a few people who use it to indicate things that are changing, so you can see pieces of expressions being substituted into other expressions somewhat more clearly. i find it effective but i understand it, if i didn't understand it maybe it would be distracting.

i'd worry about teaching out of it and then having a student ask me if something should be red or blue.

Andrew Micallef

yeah, that I think can be dangerous, I guess the trick is not to overdo it

I think sparing use of colour is the way to go

leslie townes

i did have a convention in one course where i'd write things that were useful slogans and thoughts to have but not the basis of formal reasoning or computation in light blue. sometimes you want to speak to students in a different register. it sort of worked.

Andrew Micallef

what was the course? and why only sort of worked?

leslie townes

it was linear algebra. it only sort of worked because some people still took those remarks as some kind of coded messages about what the real stuff was supposed to be. i think they forgot about the color system.

in the end i don't think it helped the stronger students and it may have annoyed some of the weaker ones. i only did it once.

Andrew Micallef

fair enough, I find semantic highlighting a nice touch, but I can see how it can be distracting / can be read too much into by someone who isn't fully following

leslie townes

5:31 AM

i think quite a lot of expository choices are made for the benefit of the top 20% of the class because the 80% may be unreachable under time constraints.

i'm pontificating again.

i'll take off my pope hat.

Andrew Micallef

That is a shame for the bottom 80%, (I think I was in that cohort, though it didn't help that I took linear algebra in the same year as the first iphone came out, and sat with a group of people who wanted to show off all the cool stuff you could do on your phone...)

leslie townes

i was bottom 80% in my first linear algebra class. i was mostly tired. i had an unworkable living situation with people who were up all night.

i wouldn't have noticed colors on the board. i remember literally falling asleep in class, it was the only place i could get some quiet. i used the instructor as a form of white noise.

i sat in the back, the lecture hall was large enough that i expect he didn't notice.

if he did, i apologize. i was annoyed when students would do that when i taught. i'd just tell them to go home.

have a friend take notes, catch up on sleep, come back on wednesday.

copper.hat

did you know that the european 911 is 112?

HereToRelax

yeah

066 in mexico

leslie townes

isn't it sometimes 999?

copper.hat

5:40 AM

in ireland 999 works too

leslie townes

i should work this out before i go to europe. seems like it is good to know.

HereToRelax

in mexico none work

one time a dude pulled out a taser on me and I called them to tell them and they hung up on me

leslie townes

haha

HereToRelax

It was pretty funny tbh

copper.hat

i never used it in ireland. the police were 30+mins away at night so we took care of stuff ourselves.

leslie townes

5:42 AM

we've had stuff like that. someone broke into our house while we were home, the cops couldn't be bothered. one time my wife was assaulted while walking our baby, again, nothing. but if some teen is looking in susipcious in an alleyway they show up.

in my hometown my dad's coworker had a police scanner on most of the time, you'd get the weirdest reports. people do bother the police about the strangest things.

HereToRelax

For some reason I thought it was more of a third world problem

copper.hat

technically ireland is part of the 3rd world

HereToRelax

oh

leslie townes

it is highly variable in the USA. to the point that it might depend on the side of the street you live on.

copper.hat

@HereToRelax the use of the term has changed over the last 80 yrs

HereToRelax

5:45 AM

yeah I guess it might depend

copper.hat

the first world referred to uk, usa, etc, the second world were the axis powers and the third world the non aligned nations

HereToRelax

sometimes I feel like mexico isn't third world

but it's only for very short periods of time

leslie townes

i haven't consulted the more recent definitions. with rising stratification of wealth the concept seems fairly useless now.

HereToRelax

very true

copper.hat

the convo was getting too mathematical, i had to lob a bomb in

leslie townes

5:47 AM

in los angeles you can go from some of the nicest housing and office buildings to an enormous tent city in about five minutes.

thanks for that.

copper.hat

where were we

leslie townes

Psi is real so is equal to its conjugate. also, it's sometimes in blue, For Some Reason.

copper.hat

coloUrs should be used sparingly

sortof watching galaxy quest right now

its brilliant

leslie townes

i remember liking it very much.

copper.hat

sad thing is my wife & i watched it years ago and i am enjoying it again as if i had never seen it :-)

leslie townes

6:03 AM

if you luck out eventually you get to the point where you get to meet new people every day.

copper.hat

the delights of aging...

yearning4pi

Don't mind my clones.

love_sodam

7:22 AM

If $R$ is a local ring then $R^\times \to (R/I)^\times$ induced by $R\to R/I$ is surjective?

I is an ideal of R

Sha Vuklia

10:12 AM

Is it true if you have a Dynkin system $D$, and a subset $A\subset D$ that is closed under finite intersections, then $\sigma(A)\subset D$? (here $\sigma(A)$ is the sigma-algebra of $A$)

This set $A$ has more properties, but I don't know if I need these

user863565

10:32 AM

Bit china coin

love_sodam

11:07 AM

0

Q: Some basic fact of homogeneous polynomial

I was actually reading some proof of Noether's normalization lemma but in the proof, it uses some basic property (at least the proof stated like that) of homogeneous polynomial. Suppose $k$ is an infinite field and $A$ be a finitely generated $k$-algebra. Let $x_1,...,x_n$ generate $A$ as a $k$-...

abstract-algebra commutative-algebra solution-verification

'the homogeneous part of highest degree' can have several terms right? like in $f(x,y) = x^3y^3+x+x^6$ then it would be $x^3y^3+x^6$

Let $f(x_1,...,x_n)=0$ where $f$ is a polynomial of $n$ variable over a field $k$. Let $F$ be the homogeneous part of highest degree in $f$. Then there exists $\lambda_1,...,\lambda_{n-1}\in k$ such that $F(\lambda_1,...,\lambda_{n-1},1) \neq 0$

The proof of this is

because if $\lambda_i$ does not exist, then we have $F(x_1,...,x_{n-1},x_n) = x_n^{\text{deg}\ F}F(x_1,...,x_{n-1},1) = 0$ by homogeneity.

I don't understand what is this. I mean how x_n^{\text{deg}\ F} can come out?

Srijan M.T

11:29 AM

when we talk about the graph of af(x+b/2a) = a(x+b/2a)^2(Dont need value of y axis right now so not written )
This is Q.E graph using completing the square method
So , first I draw graph of x^2
Then , af(x)
Then shift
If you notice , the value of turning point I,e -b/2a changes.
It is Sth else now.
How is that possible
Turning point changes to -b/2a^2

Sha Vuklia

(What I asked is true. I had forgotten about a result which gives this)

Srijan M.T

user863565

12:21 PM

Dumbbell Nan Kilo Moteru Opening Karaoke - IRL

►

(removed)

loch

2:04 PM

the answer wrote deg_nF there and I suspect the intended meaning of that is 'lowest power of x_n which divides all terms in the expression'

or in other words it's just factoring the largest power of x_n you possibly can

love_sodam

@loch But the resulting should be x_n^{deg F}F(x_1,..,x_{n-1},1) so every $x_n$ should come out

leslie townes

4:23 PM

my daughter's singing about her imaginary friend again.

interesting mental math exercise in how this could be possible. many commenters are convinced that it isn't. twitter.com/TheoShantonas/status/1380536139954655237