Mathematics - 2017-07-20 (page 4 of 4)

Akiva Weinberger

18:00

What's a thimble for even anyway

PVAL-inactive

Illustrator is $20 PER MONTH

Daminark

@PVAL proceeds to remove all pictures

Adeek

hi everyone @TedShifrin @Daminark @MikeMiller

Ted Shifrin

hi Karim

Semiclassical

Here's how I understand the term.

Akiva Weinberger

18:04

Marker on paper, scan

Semiclassical

Suppose I've got a contour integral of the form $\int_C e^{-\lambda f(z)}\,dz$

Akiva Weinberger

Profit

A great poem I just found, on the use of apostrophe's:

Semiclassical

And suppose I want that contour to come in from $\infty$ at some angle and leave at another angle. (for instance, -infty to infty on the real line)

Mike Miller

@Semiclassical Yes but I don't know what it means

Akiva Weinberger

A'postrophe's are ha'rd to pla'ce,
'But i'f you do' 'it rig'ht -
Youl'l find theyr'e fine for 'any sp'ace,
Divin'e for e'ach del'ight!'

Re'memb'er this, m'y finest' friend,
They'r'e your's to ' drop' and spill' -
'Before a word, or at the end',
Or half bet'ween at will!

'They'r'e only t'eeny tiny 'tools,
For all to u'se with 'ease -
So do you'r best' to bre'ak the rule's!
E'm'b'r'a'c'e a'p'o's't'r'o'p'h'e's!'

Semiclassical

18:06

mmkay

In order for that integral to be convergent, I need the real part of $\lambda f(z)$ to go to -infty along the two chosen directions.

Ted Shifrin

That's dedicated to @Balarka, of course, who's gone for a week.

Semiclassical

Oh. Let's say I also assume $f(z)$ to be an entire function

Vincenzo Oliva

@Semiclassical I see.

Semiclassical

(does entire say anything about the behavior at infinity? I can't remember)

Daminark

If we don't know how to do apostrophes anywhere else, we at least know how to correctly write Stoke's theorem

Akiva Weinberger

18:09

@TedShifrin A week?

Adeek

@TedShifrin I was wondering if you know any recommendation for history of geometry book ?

I would like to know little bit about how things in geometry began until now

Akiva Weinberger

S'tokes

Ted Shifrin

He went to a conference with only books ... no laptop, DogAteMy.

Semiclassical

Anyways. While that's enough to ensure that the integral is convergent, it doesn't mean that the integral necessarily is easy to compute along that path.

Akiva Weinberger

What conference? Where?

Ted Shifrin

18:10

Somewhere in India.

Diff geo, he said.

Adeek

That is pretty cool

Ted Shifrin

I don't know what you mean by geometry, Karim. And I'm no expert on history of mathematics.

Adeek

I am gonna go to complex geometry conference in Germany in October.

Semiclassical

If the imaginary part changes rapidly, for instance, it'll be hard to integrate the function numerically along that contour.

To make life easy, it's best if our integration contour has the same imaginary part throughout. in that case we can just pull that out of the integral and be left with $\int_C e^{-\text{ Re }\lambda f(z)}\,dz$

moreover, that's not impossible to do: The integrand is analytic (and therefore entire) so I can deform my contour locally without changing the integral.

So this suggests I should deform my contour to go along a line of constant imaginary part for $\lambda f(z)$.

Adeek

I just wanted to know how things advanced and how ideas came together in geometry from euclidean geometry until now @TedShifrin

Semiclassical

18:15

To make things even easier, I can choose the level set which passes through a critical point of $\lambda f(z)$ i.e. a saddle point of $\text{Im }\lambda f(z)$. In that case I'll be able to use the saddle-point approximation and get as good of an approximation as possible.

Ted Shifrin

Try Kline's Mathematical Thought, Karim — three-volume history of math.

I don't have a specific recommendation.

Semiclassical

The Lefschetz thimble is the contour which satisfies the above conditions.

Eric Silva

hi chat

Adeek

cool thanks I will try it out. It is nice reading in spare time.

Semiclassical

(I think. I'm still a bit unsure of it all.)

Ted Shifrin

18:17

hi Eric

Adeek

hi

Semiclassical

Where things become weird is when one regards that integral as a function of $\lambda$ and does analytic continuation around $\lambda=0$.

Eric Silva

@Adeek for specifically Euclidean stuff Sir Thomas Heath wrote two volumes on greek mathematics that are very good

Adeek

awesome thanks @EricSilva

Eric Silva

it's a lot of reading though and only really useful if you're really into the history

but I think they are some of the most wonderful books on any history of math, I've read them three times myself

Akiva Weinberger

18:22

OK, but what's a thimble for even anyway

Vincenzo Oliva

@Semiclassical If you had to write during the oral exams, would you have preferred to handwrite or LaTeX-type (assuming you would have had a LaTeX course in the first semester of the first undergraduate year)? Here is my original question, if you didn't see it:

Akiva Weinberger

Like, in real life

Daminark

SGA is apparently very good for history... Okay I'll stop memeing for now

Adeek

I am just interested how things began and the history of them @EricSilva Especially geometry.

Vincenzo Oliva

Can't paste it

Adeek

18:22

I love geometry

Semiclassical

I'd probably prefer just to write on the board.

Vincenzo Oliva

4

Q: Are there advantages to make students handwrite (instead of LaTeX typing) what they say in an oral exam?

In my university, in Italy, most professors want the students (at least the undergraduate ones) to handwrite what they are saying in an oral exam. If I understood correctly, this is somewhat due to needing an official document for the oral exam. Does this occur in other countries too? It's not u...

Adeek

I should learn French I would like to read EGA one day

Vincenzo Oliva

Yes, writing on the board also makes sense. You would even prefer writing on a sheet, I guess

Eric Silva

@Daminark aside from the memeing is it actually?

Daminark

18:25

Well, it's not the kind of history Karim was just talking about

Eric Silva

what kind of history

Adeek

@Daminark You can speak french right ?

SteamyRoot

@Semiclassical ooh, Lefschetz

Daminark

Like, how geometry evolved up to now from the Greeks

Eric Silva

idt a good book specifically on that honestly exists

Daminark

18:26

@Adeek a little bit, and given similarities I might be able to read some as well? If the internet is available

@Eric I mean, if there is such a book it's REALLY not SGA

Adeek

There is @Daminark

Eric Silva

sure but neither is the book i suggested

Adeek

Séminaire de Géométrie Algébrique du Bois Marie

In mathematics, the Séminaire de Géométrie Algébrique du Bois Marie (SGA) was an influential seminar run by Alexander Grothendieck. It was a unique phenomenon of research and publication outside of the main mathematical journals that ran from 1960 to 1969 at the IHÉS near Paris. (The name came from the small wood on the estate in Bures-sur-Yvette where the IHÉS was located from 1962.) The seminar notes were eventually published in twelve volumes, all except one in the Springer Lecture Notes in Mathematics series. == Style == The material has a reputation of being hard to read for a number...

Eric Silva

the best i think you could get is what Ted suggested

Adeek

It is famous in algebraic geometry

Akiva Weinberger

18:27

> A thimble is a small hard pitted cup worn for protection on the finger that pushes the needle in sewing.

Now I know

SteamyRoot

Oh

That.

Daminark

No I meant like, SGA is very much not a Euclidean geometry style historical text. I'll wait for a little bit longer before going down that rabbit hole, I think I pick up some algebra first :P

SteamyRoot

Wait, why would Lefschetz have a thimble

He had no hands

Eric Silva

It's not that hard to learn to read in french, there's a book a book called French for Reading by Sandberg written specifically for academics who want to pick up french for research purposes

Daminark

And @Eric perhaps, my point was that the recommendation I gave was more or less strictly memeing as far as I know

Eric Silva

18:29

@Daminark I am aware of what SGA is.

Adeek

@Daminark Yeah I think to do algebraic geometry properly one must have perfect commutative algebra grounding + algebra

Daminark

@Eric I was responding to Adeek

True @Adeek

Eric Silva

Also I think there are some strong sentiments from people that it isn't useful reading (I think I've heard Emerton say this for example)

Akiva Weinberger

@SteamyRoot He what

Eric Silva

@Daminark ah mmk

SteamyRoot

18:30

@AkivaWeinberger Lefschetz lost both his hands at some point.

Adeek

@EricSilva Cool I will also attend some French classes here at my university not this year though.

Alessandro Codenotti

He lost them in an accident in a factory iirc

Akiva Weinberger

@SteamyRoot Oh, wow

How did he write math

SteamyRoot

No idea...

Akiva Weinberger

I mean, I don't really know how handless people do much of anything

Daminark

18:31

@Eric Again, my intention is to wait until I pick up some algebra and then some commutative algebra and then figure out how to do algebraic geometry, so right now anything I say about it is either hearsay which I imagine is relevant in a given context, or memetics

SteamyRoot

Maybe had a secretary to write down for him or so

Akiva Weinberger

Morse code! With the stump! :D

Nah I have no idea

Alessandro Codenotti

Another "fun" fact: Morin is blind, yet he was the first guy to construct a sphere eversion

Akiva Weinberger

I wonder if there's some word processor that takes Morse code as input

Like, connected to a button or something

@AlessandroCodenotti I know, crazy

Also, apparently you can do it with a cuboctahedron as well

Eric Silva

@Daminark sure, I was just saying that there are people who are very knowledgeable about Grothendieck type stuff who think it's not the right place to go to learn Grothendieck type stuff

I know nothing obviously

Akiva Weinberger

18:33

I think he discovered that as well

Eric Silva

But I do think that's kind of hilarious

@Alessandro I love that fun fact, really makes you think about how geometric intuition works to begin with

Akiva Weinberger

I wonder what Morin's process was. Did he undo the lemmas in the proof of Smale's theorem? Or did he imagine messing with a physical ball?

SteamyRoot

@AlessandroCodenotti Well, maybe being blind helps there, though.

Adeek

That is crazy @AlessandroCodenotti I wonder how he comes up with geometrical intuition

so weird

Eric Silva

what is the dominant sense for people without sight?

SteamyRoot

18:35

Sphere eversion sounds like the kind of thing you may consider impossible because you can't "see" how it would happen.

Akiva Weinberger

He now has Morin's surface named after him

which is just the half-way point in his eversion

Alessandro Codenotti

I don't remember if it was Morin or another blind mathematician, but I remember a quote where he said he isn't at much of a disadvantage as soon as you do stuff in more than 3 dimensions

Akiva Weinberger

so a weird sphere immersion

SteamyRoot

But if you've never relied on visual representations of things, maybe you'll develop a different kind of intuition

Akiva Weinberger

@EricSilva Hearing, probably

Eric Silva

18:36

For geometry I was thinking that you could probably develop a decent intuition through touch

Like clay models might take you a long way or something

Akiva Weinberger

True for sighted people as well @EricSilva

Clay models, ball-and-stick, whatever

Eric Silva

@Akiva yeah definitely

Daminark

@AlessandroCodenotti 'tis true

Eric Silva

But sighted people also have the visual crutch so I wonder how much it would actually influence how you intuit things

if you don't have sight it might be one of your only ways of visualizing as it were

idk though, I can't even begin to imagine

Akiva Weinberger

@EricSilva Go out and get clay, or some other physical model,

Alessandro Codenotti

18:38

I always find it incredible because I need as much visual intuition as possible in geometry

Akiva Weinberger

play around with them,

Adeek

yeah that makes sense @EricSilva

Akiva Weinberger

and try to see what you learn from senses other than sight

Maybe play with your eyes closed

Adeek

yeah most of my assignments in algebraic topology last semester I relied on geometric intuition all the time

at least for homotopy stuff

even some homology stuff as well

Akiva Weinberger

The nice thing about algebraic topology is that you can think about manipulating symbols instead if the geometry gets too confusing,

and then decode the symbols at the end to get the geometric picture

(and then tweak that as well if necessary)

Adeek

18:40

yeah

Akiva Weinberger

I guess that's like visualizing something one layer up in abstraction.

SteamyRoot

You can just rely on algebraic intuition too :P

Akiva Weinberger

Smale's original proof was however-many layers up in abstraction

Adeek

@AkivaWeinberger some problems though you can't do that you need geometric intuition

Akiva Weinberger

(I haven't read it but that's my guess)

@SteamyRoot Right, that's what homology and cohomology are for :P

The "algebraic" of algebraic topology

Adeek

18:41

I don't like homology

I prefer homotopy theory

Akiva Weinberger

@Adeek Those are usually fun ones

Eric Silva

honestly the clay models might work better for algebraic topology things than visual intuition does anyway

Akiva Weinberger

I can't think of an example at the moment, though

SteamyRoot

Cohomology is more fun than homology :^)

Adeek

Cohomology is just the dual

Akiva Weinberger

18:42

The problem with clay is that it can't pass through itself :P

Adeek

haha

Eric Silva

fair point

Akiva Weinberger

Minecraft is good as well, for things that can be made of blocks

Adeek

I am gonna go back to work cya guys later

Akiva Weinberger

It's not quite physical, but it has lots of feedback that just visualization doesn't

Does anyone have a puzzle that revolves on using geometric/topological intuition?

SteamyRoot

18:44

@Adeek For some homologies

Not in general.

Eric Silva

like doing a 3D puzzle requires lots of good spatial reasoning @Akiva

like putting together 3d models of things

Akiva Weinberger

Yeah but does anyone know one that they can post in the chat, I mean

Eric Silva

hmm yeah Idk of any puzzles in that vein that aren't physical

oh man this reminds me of the wire puzzles I used to do like every day

those can get really tricky

Akiva Weinberger

Like you have two shapes that you need to separate? @EricSilva

Eric Silva

yeah

like there'd be a bunch of metal wires or pieces of complicated shapes and you'd have to untangle everything

Mike Miller

18:53

someone retired, got some free books

Eric Silva

anything good? @Mike

Akiva Weinberger

Yeah I played with some of those at a relative's house once

I guess there is a mathematical structure you could associate with such games

Mike Miller

Akiva Weinberger

If the two shapes are $A$ and $B$, then you can take the set of all pairs of rigid images of $A$ and $B$ in $\Bbb R^3$ that don't intersect each other

so it's one point per possible configuration of the puzzle, I mean

and you could put a topology on that

and the question is to analyze the connected components

Eric Silva

My mom used to work for a toy maker who used to send me 30 to do every month from when I was in second grade till about when I was 17 @Akiva, There were some that were crazy hard and had 10-20 pieces

Kirill

18:56

@TedShifrin Dear Mr. Shifrin, I cannot find the solution. If we are in $\mathbb{R}$, the function could be defined as $\frac{t^2}{4}$ for $t\ge0$ and $0$ for $t<0$. If we are not in $\mathbb{R}$, maybe there are some magical terms that involve $i$ that will do the thing.

Akiva Weinberger

Oh, wow

I need these in my life

Eric Silva

yeah there were quite a few that I could never solve, especially toward the end cause he ramped up how hard the puzzles were

I got quite decent at them after doing them every day for like ten years though

I might try to bring them all back up to Chicago with me next time I'm home

Akiva Weinberger

Clearly there should be some sort of overarching principles for making them… I don't know how I would try to analyze them mathematically

Kirill

@TedShifrin if $y'\ge0$ because of the root, then this function should be a monotonically increasing function, but I cannot find the one that will satisfy all the conditions.

Akiva Weinberger

You could try a neural-net-like thing. Give names to common patterns. Give names to patterns of those patterns.

Give names to things that are three layers up in abstraction, etc

Eric Silva

18:58

yeah idk if there's really good way to try to figure them out systematically since each puzzle is usually pretty unique, but there are some common patterns and stuff that gets ingrained in you as you solve a lot of them

Akiva Weinberger

That's what a lot of math is, right? That's why we have names for things like "cohomology"

Eric Silva

which is basically using a neural net lol

@Mike oh damn Ted's book lol

the other one looks like something I'd be into

Mike Miller

yeah i agree

Akiva Weinberger

It kinda looks like you're trying to hide a book under a comically small rug

Eric Silva

o shit i just noticed the authors

now i really wanna read it

Mike Miller

19:01

lol

Akiva Weinberger

19:59

Anyone still in here?

Paul Plummer

20:13

I am sort of here @AkivaWeinberger

Akiva Weinberger

20:40

Whoa

Solar panels and LEDs are inverses of each other

David Varela

20:50

@AkivaWeinberger This reminds me of Feynman! youtube.com/watch?v=N1pIYI5JQLE at 3:20 - 4:30 specifically

not sure if you were being more precise than this..

photosynthesis is the inverse of burning wood to create fire

Akiva Weinberger

@DavidVarela I'm looking up how they work… apparently they're built on the same principle, known as a PN-junction

which is what you get when you put what's called p-type and n-type silicon next to each other

Every silicon atom has four valence electrons, and they form nice crystals in which every silicon atom has four neighbors with which it can have covalent bonds.

If you replace some of the silicon atoms with arsenic, well, arsenic has five valence electrons

so when it's neutral you have this extra electron floating about that isn't in a bond

so that's n-type

David Varela

@AkivaWeinberger All engineering really is inspired from nature! p-type and n-type silicon sound vaguely familiar from my computer architecture class..

Akiva Weinberger

and if you replace some of the silicon atoms with boron (I think), well, boron has three valence electrons

David Varela

Which you just managed to give a concise review of, nice!

Akiva Weinberger

so you have a missing electron there, called a "hole"

(You can think of holes "moving" even though they're really the absence of stuff rather than actual stuff)

David Varela

20:58

You are making sense so far..

Akiva Weinberger

In any case, putting these next to each other ends up with a diode, and then cool stuff happens

and I probably need to watch some videos a few more times to understand them better

(The point of "diodes" is that they only let current flow through them in one direction)

David Varela

I was just about to say the one direction part lol

Akiva Weinberger

So I guess what happens is, near the boundary, some electrons and holes wander to the other side and annihilate each other

(electron + hole = electron + absence of electron = annihilation, and also some energy probably)

but you only get a small region near the boundary where this happens

David Varela

you mean energy loss correct?

Akiva Weinberger

Hm, probably

I dunno

Now, the lack of holes on the n-type side near the boundary makes it negatively charged, so more electrons don't come over

and the lack of electrons on the p-type side near the boundary makes it positively charged, so more holes don't come over

so you have this little barrier region by the boundary with no holes and no electrons, but for most of the p-type side there's lots of holes and for most of the n-type side there's lots of electrons and they can't get from one side to the other

If you apply a current from the p-type side to the n-type side, that essentially "pushes" the holes and "pulls" the electrons, making that barrier region smaller

until, with enough current, it's suddenly small enough that the electrons and holes can travel freely through the device and current flows

If you apply it in the other direction, it just makes that barrier region wider—oh, apparently it's called "depletion region"—so current doesn't flow

David Varela

21:05

ok, what happens if you apply a current in the other direction?

beat me to it again lol

Akiva Weinberger

For solar panels, each photon knocks an electron out of its place, creating an electron and a hole

David Varela

Aha! that makes a lot of sense

Akiva Weinberger

If the electron is in the depletion region and it doesn't knock into its hole again, it'll get pulled towards the n-type side

Erik M

Is there a term for a type of Markov chain where for any two states i and j, any path connecting them has equal probability?

For example, consider a three state MC with transitions from 1->2, 2->3 and 1->3. In this example, the desired property is p_{12}p_{23} = p_{13}

Akiva Weinberger

('cause there's a lack of electrons on the n-type side near the boundary, in the depletion region, so it's positively charged there and pulls electrons)

Similarly, the hole gets pulled to the p-type side, where all the other holes are.

And… uh… I guess you could join the two sides with a wire and use that to generate current. I forget.

But I think that was the basic idea?

So that's solar panels, or photo cells / photovoltaics

And I haven't watched the video on LEDs yet

but they hinted it was like the same thing but in the other way

David Varela

21:08

That's what Einstein got the Nobel prize for, no?

Akiva Weinberger

Could be, I have no idea

@ErikM No idea

David Varela

The only thing that isn't making sense is how you keep the electrons together, I feel like they would eventually repel each other..

Akiva Weinberger

Like on the p-type side?

For p-type silicon?

Remember that it's ordinarily neutral even though it has all those extra electrons

because it has arsenic atoms in it, and those have one more proton

10 mins ago, by Akiva Weinberger

Now, the lack of holes on the n-type side near the boundary makes it negatively charged, so more electrons don't come over

*p-type

10 mins ago, by Akiva Weinberger

and the lack of electrons on the p-type side near the boundary makes it positively charged, so more holes don't come over

*n-type

The n-type side has all the electrons but near the boundary they went to the other side so it has fewer electrons than usual

and the same with the p-type and the holes

David Varela

oh ok, got it

Akiva Weinberger

(N-type has electrons 'cause those are negative, p-type has holes 'cause those are positive)

David Varela

21:15

so the electrons only move in the 'depletion zone'?

where the holes and electrons are sitting next to each other

Akiva Weinberger

Well you kinda don't have any holes or electrons in the depletion zone 'cause they annihilate each other

but if you bash it with a photon, you create a hole and an electron, and if they don't re-annihilate, the electron is pulled towards the n-type side

On the n-type side all the electrons are moving around (because of heat)

They're kinda diffused

That's the reason some electrons jumped over to the other side in the first place. Diffusion

David Varela

ok, I think I got it

Akiva Weinberger

I should share with you the links to the videos I was watching

Ted Shifrin

@Kirill: It's definitely all in $\Bbb R$.

You have the idea. So now we have a solution that's 0 up to $t=0$ and then $t^2/4$ after that. Are there any other solutions with $y(0)=0$?

Akiva Weinberger

#1: How Does a Diode Work? Intro to Semiconductors (p-n Junctions in the Hood)

#2: What's a Diode? Here's why diodes conduct one way but not the other

#3: How Solar Panels Work - Convert Sunlight to Electricity in Your Own Backyard

#4: How LED's Work

(fin)

@DavidVarela

David Varela

21:23

@AkivaWeinberger thanks! I watched some of Doc's videos when I was trying not to fail physics. I bookmarked them for later, they will replace my PBS SpaceTime slot for today lol

Also, I had never made the connection between plants, diodes, and semiconductors before

Akiva Weinberger

@DavidVarela Plants?

Hey, @Typhon

> In mathematics, the resultant of two polynomials is a polynomial expression of their coefficients, which is equal to zero if and only if the polynomials have a common root (possibly in a field extension), or, equivalently, a common factor (over their field of coefficients). In some older texts, the resultant is also called eliminant.

I haven't read further in the article, but it might be an interesting puzzle to try to construct such an object without looking it up

Astyx

21:39

I've had this in one of my exams

Akiva Weinberger

There's also a related concept called a "discriminant" of a polynomial, which equals 0 iff it has a repeated root somewhere

(The discriminant of a quadratic is $b^2-4ac$, but it's not obvious what the discriminant of a cubic would be)

Semiclassical

The formal definition for higher degree is that the discriminant is the product of the differences of the roots, if memory serves

and hence vanishes if there's a multiple root

Akiva Weinberger

@Semiclassical Times some power of the leading coefficient

SteamyRoot

Also squared

Semiclassical

22:06

Yeah, I figured I was omitting something

Erik M

22:47

What are some tips when you get stuck trying to prove something?

Akiva Weinberger

yesterday, by Akiva Weinberger

"The definition of insanity is doing the same thing over and over again and expecting different results" @Mahmoud

@ErikM

In other words, if an approach isn't working, say "Maybe this is the wrong approach" or "the wrong point of view" and see if you can find a new way to start the problem

Don't stick with one approach if it doesn't seem to be going anywhere

Are you working on a specific problem? @ErikM

Erik M

@AkivaWeinberger Doesn't the persistence sometimes pay off? That is, how do tell the difference between the current approach being a dead-end, or just that you haven't spent enough time on it?

Yes, I'm working on a problem where I believe the result is true, but I don't know for sure. Every example I've written out has satisfied the claim.

Semiclassical

23:03

What's the claim?

Akiva Weinberger

@ErikM You can always return to the original approach later.

@ErikM What subject?

Erik M

It's a little complicated to write out, but I'm trying to show an adjacency matrix of a specific type of graph satisfies a property. Sorry for being vague, it would just require a lot of effort to write it out formally here.

Akiva Weinberger

Maybe think about how similar theorems are proven

(like, in graph theory)

Semiclassical

What kind of graph? (Adjacency matrices are neat)

Erik M

Good point, I have started to read other proofs. The graph I am dealing with is a hypergraph (non-uniform) and I am trying to show properties of the spectrum of its adjacency matrix. I think the problem is too complex for me, I will try to warm up by simplifying the graph and then move to the more complex case. Thank you for the tips! I'll make a post if the results turn out to be interesting

Akiva Weinberger

23:36

Random thought occurred to me in the shower yesterday

$(x-a)^2$ has its minimum at $a$, $~(x-b)^2$ has its minimum at $b$, and their sum $(x-a)^2+(x-b)^2$ has its minimum at $\frac{a+b}2$, their average.

Conjecture: that happens whenever you have a concave-up even function $f$ with a minimum at $0$: the minimum of $f(x-a)+f(x-b)$ is at $\frac{a+b}2$.

Hey, @Secret

Lucas Henrique

what is defined as $n \mathbb{Z}$?

anon

$n\Bbb Z=\{nx:x\in\Bbb Z\}$

Lucas Henrique

so what does it mean to k modulo n be in $\Bbb Z/ n\Bbb Z$?

cant understand what the quotient ring is supposed to be

Semiclassical

the equivalence relation is that a~b if n divides a-b i.e. a-b is in nZ

So 1~1+n~1+2n~1-n~1-2n...

Akiva Weinberger

Spacing is weird there

$1\sim1{+}n\sim1{+}2n\sim1{-}n\sim1{-}2n\sim\dotsb$

Semiclassical

23:50

Yeah, I'm just using the ~ character on my phone

skullpatrol

iPhone?

Semiclassical

ya

skullpatrol

cool

Akiva Weinberger

. - -* ~

Semiclassical

It's standard to denote the equivalence class mod n which k is in by $[k]_n$

But one usually omits the subscript if the modulus is fixed, and often times the brackets are also not included

The latter risks confusing elements of Z with elements of Z/n but ehhhh

skullpatrol

23:59

How did that issue you had with a timing device in your pre-med lab class turn out?

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$Mathematics$ Mathematics