serious-discussion

[deep mango]

Hmmm, this is cool

Oh yeah

I remember why but only by contour integration techniques

This is how you compute the fourier transform of a gaussian in fact

Complete square and use that the imaginary shifts dont matter

[hollow sundial]

you're both right. i found it cool because the idea of "shifting normal distribution by a number" extends to complex values since we don't typically use complex numbers in probability theory

and of course the value is the $\sqrt{\pi}$

[fathom swallow]

riemann

[crystal stream]

i'm guessing you integrate around a rectangle with vertices {-L, -L+ia, L, L + ia}

and show the integral over the smaller sides vanishes as L goes to infinity

[oblique mirage]

isn't this a recursive definition?

since the dot product is |a||b|cos(theta)?

[leaden torrent]

not if you define the dot product another way

say through the sum of elementwise products

or what have you

[uncut socket]

I'm pretty sure that may mean a times b, not a dot b

[oblique mirage]

can it be defined just as |a||b|?

[leaden torrent]

"it"?

[oblique mirage]

that's derived from the dot product definition

[oblique mirage]

the dot product

[leaden torrent]

thats not an inner product

so no

[oblique mirage]

but it appears that the book thinks it's fine?

[leaden torrent]

that is not what its doing

thats the sum of elementwise products i was mentioning

not |a||b|

[oblique mirage]

ohh

so it has 2 definitions

[leaden torrent]

if $\mathbf{a} = (a_1, a_2, \dots, a_n)$ and $\mathbf{b} = (b_1, b_2, \dots, b_n)$, then $\mathbf{a} \cdot \mathbf{b} = \sum_{i=1}^{\text{n}}a_ib_i$

[fathom swallow]

Namington

[oblique mirage]

right

[leaden torrent]

i would consider this a more fundamental definition than the cosine one

personally

since sums and products are a lot simpler than... however you define cosine

[oblique mirage]

but if I need to find angle between 2 vectors it seems that I need to use both

[leaden torrent]

no?

or like

you just use the formula given in the image

[oblique mirage]

having 2 definitions just makes it more confusing

[leaden torrent]

$\theta = \cos^{-1}\left(\frac{\mathbf{a} \cdot \mathbf{b}}{\abs{\mathbf{a}}\abs{\mathbf{b}}}\right)$

[fathom swallow]

Namington

[leaden torrent]

i'd consider it 1 definition and 1 fact personally

but whatever

a lot of things in mathematics have multiple ways you could write them

you could write $\abs{x}$ as $\sqrt{x^2}$, for example

[fathom swallow]

Namington

[leaden torrent]

you could write $\cos(\theta)$ as $\sin(\frac{\pi}{2} - \theta)$

[oblique mirage]

but that doesn't change the value of the definition

[fathom swallow]

Namington

[oblique mirage]

adding cosine does

[leaden torrent]

im not sure what "value of the definition" means

[oblique mirage]

so say I have 2 vectors a and b

I will get a different value of their dot product depending on which definition of dot product I use

[leaden torrent]

no you wont

assuming you did it correctly

[oblique mirage]

oh?

[leaden torrent]

$\sum_{i=1}^{n}a_ib_i = \abs{\mathbf{a}}\abs{\mathbf{b}}\cos\theta$ is a fact

[fathom swallow]

Namington

[leaden torrent]

you can try to prove it, its not hard if youre familiar with basic proof techniques

[oblique mirage]

okay then that makes a lot more sense

thanks 🙂

[leaden torrent]

https://proofwiki.org/wiki/Cosine_Formula_for_Dot_Product

??

i mean okay, you could phrase it as the dot product having 2 separate definitions and the fact is that those definitions are equal

whatever

my point is that one can prove the equivalence

youll never get 2 different values

so what are you adding?

since i said exactly this above

[neat lintel]

What does it mean by "turning point of a function"? Can we say that for this graph, turning point is x = -1, x = 0 and x = 3?

Function is f(x) = x/(x+1)(x-3)

[leaden torrent]

see #❓how-to-get-help for more detailed help.

but afaict that graph has no turning points

[neat lintel]

Okay

[neat lintel]

i have a question to all the mathematicians out here, how do you understand a course script with bunch of formulas that are totally new to oneself?

and i am not talking about the simple stuff e.g: (signum function)

the way I understand things is I visualize them in my head but once it gets complicated it gets very confusing

[leaden torrent]

do problems until the definition sets in

[neat lintel]

well I got a lot of content to practice for, im pretty sure 24 hours not going to suffice lol

[ancient flame]

lmao

[deep mango]

[cinder roost]

Do you think it's possible to submit my 3 hour stats exam in LaTeX? Never used it before, have a few months, and have to submit my other (24 hour) using LaTeX.

[deep mango]

Oof

Uhh

My advice would be to start doing your homeworks in it

(and my other advice is to come to my latex workshop tomorrow or watch the recording when I make it :p see #events )

[cinder roost]

and see how fast I get? my typing speed isn't too shabby, around 100 wpm and I don't practice, but working with a markdown is very different to raw sentence typing speed

[deep mango]

Yeah

Like if you find that you pick it up quickly then you don't need to do hw in it the whole sem

But the best way to practice it is by forcing yourself to use it for real stuff

[cinder roost]

yup, I think so too, thanks for the advice

[deep mango]

Cause then you know your use cases for these particular classes

[cinder roost]

I'll probably use it right now, see how long it takes me to do past paper examinations closer to the exam and if I can complete the exams within time then should be good

[deep mango]

Yeah

[cinder roost]

Also worth noting we get like 20 minutes extra to upload answers, LaTeX would reduce "upload time"

[deep mango]

I recommend just using an account on overleaf.com, it's so much easier than trying to set up latex locally (at least for now)

And overleaf also has lots of great tutorials

[cinder roost]

I've set it up locally, side by side refresh on save with vscode

[deep mango]

Oh ok

Thats good

The overleaf tutorials are still very helpful

[cinder roost]

yup, I just don't know how to practice it

[deep mango]

Yeah thats what I have too

[cinder roost]

do I do my homework sheets in it or do I copy a document?

[deep mango]

Doing homework is more productive I think

Like

[cinder roost]

2 birds with one stone, yea

I hear you

[deep mango]

Copying something someone else has written is going to get real boring real quick

And you'll be spending a lot of time typing at first, which is great when doing homework because it means you have some time to process the problems better in your head. At least it helped me to be forced to slow down a little

[cinder roost]

Yup

Thanks man

[neat lintel]

anyone here good with exponential and logarithmic functions?

[rotund steppe]

Nope

[bright hill]

Nah

[neat lintel]

No

[jovial ember]

Nein

[neat frost]

Unyes

[ancient flame]

no

[kindred ore]

im pretty bad with them

[mint patio]

I can solve exponential and logarithmic functions yeah

[fair mural]

[ancient flame]

👀

[deep mango]

[robust osprey]

Wise man

[odd yew]

2 pigeons one hole!

[gentle bay]

Yes!

Pigeonhole.

[frail sail]

That does not sound right

Now I won’t be able to use php ☹️

[quaint loom]

Are there any kind of resources I could look into that would improve my general mathematical thinking in a way that common highschool math doesn't? Like, something related to one's thinking / approach / process if that makes sense?

[surreal sapphire]

maybe an intro proofs book

or just an introductory book that a topic that is not covered in highschool

[scenic inlet]

Introduction to Mathematical Thinking by Devlin. I think he has an accompanying course on Coursera as well.

[hollow sundial]

#help-1 message

https://giphy.com/gifs/girl-life-car-3ov9jWu7BuHufyLs7m

[sick burrow]

starting to wonder if I don't actually hate geometry and was just traumatized by my class in highschool

or maybe traumatized is too strong a word but that class was fucking awful

[thorn brook]

what type of geometry?

[sick burrow]

just normal highschool geometry

[surreal sapphire]

geometry is peak math

[sick burrow]

congruence rules with no justification given, two column "proofs", the usual

[thorn brook]

what kinds of "geometries" are there even?

[fair mural]

here we go

[thorn brook]

is algebraic geometry considered geometry?

[surreal sapphire]

yes

thats why its called geometry

[thorn brook]

what more stuff is there in geometry?

[sick burrow]

idk but I have a god tier quote from my professor for the good geometry class I'm in now

"If you asked me what is abstract algebra I could answer in a second! It's uhh... the study of structured sets. <continues about groups and rings and fields and other algebra stuff for significantly more than a second>... and I would be right:"

[surreal sapphire]

if you want a broader perspective on things that more closely resemble geometry, you can look into the erlangen program

there was this historical issue that we suddenly had non-euclidean geometry

and felix klein tried to reconcile what "a geometry" is

[thorn brook]

oh daim that's pretty cool

[devout nacelle]

What is geometry

[thorn brook]

triangle.

[fair mural]

square

[sick burrow]

when donut does not equal ONE COFFIS CUP

[neat lintel]

the book im reading has some theorem without a proof. Under the statement of the theorem it says "Proof: Look at exercise 2". I look at the exercise 2: "Prove theorem 8"

[fair mural]

impossible

[surreal sapphire]

its a homogenous space with a transitive action by a lie group acting as the symmetry group of the geometry

[fair mural]

[crystal stone]

HS geometry doesn't count

[fair mural]

are you aware you just

used geometry

[crystal stone]

Get a copy of Euclid

[sick burrow]

I mean I know it shouldn't

[fair mural]

in the definition of geometry

[sick burrow]

and I have a copy of Euclid

that I never read

[crystal stone]

lol

I got the green lion press edition

and I love it

I'm teaching geometry at an after school program

[fair mural]

vectors are elements of a vector space moment

[devout nacelle]

That I can understand some aspects of this sentence comforts me

[sick burrow]

highschool geometry shouldn't count but it still might have turned me off of the field

which is kinda sad when you think about it

[crystal stone]

Differential Geometry is pretty cool

[thorn brook]

is topology considered geometry btww?

[crystal stone]

"rubber sheet geometry"

[thorn brook]

it do be stretchy

[crystal stone]

There's a field of topology called geometric topology

[sick burrow]

I'm trying to see how much of the sentence I understand

[surreal sapphire]

the wikipedia page has examples

[thorn brook]

wtf is that all about?

[crystal stone]

Geometric Topology is a hard subject with hard problems, but not too many people work in them

[surreal sapphire]

that are easier to understand

[sick burrow]

a lie group is a group that's a topological space such that multiplication is continuous?

[crystal stone]

Knots, 3 manifolds, 4 manifolds, surfaces in them, etc.

Projective geometry is fun to talk to kids about

Because the parralel postulate fails

[fair mural]

lines (great circles)

[surreal sapphire]

a bit more, you want the group to also be a real smooth manifold and multiplication (and inversion) smooth

[devout nacelle]

[fair mural]

[devout nacelle]

Hmm, would computational geometry be considered a subset of Euclidean geometry?

[surreal sapphire]

probably?

i mean you can write algorithms to compute stuff in other geometries

[thorn brook]

I didn't even know computational geomtry was a thing lmao

cat theory is geometry because you draw arrows

[devout nacelle]

[fair mural]

melting brain

[surreal sapphire]

in the examples above, all spaces are just subspaces of some R^n and the lie groups are some matrix subgroups it seems

[devout nacelle]

Isn't there some vague notion of algebra and geometry being "dual" in a certain abstract sense

[surreal sapphire]

this is also the only case i am somewhat familiar with

[devout nacelle]

[surreal sapphire]

in the sense of algebraic geometry, yes

in general no idea

[devout nacelle]

What does the duality mean in algebraic geometry?

[surreal sapphire]

you often get some functor between "geometric" and "algebraic" categories

[crystal stone]

It's not really that abstract, circles are given by the polynomial x^2 + y^2 - r^2 = 0

[surreal sapphire]

in some cases an equivalence of categories

like

[crystal stone]

So you can play dumb algebra games on those types of things

[surreal sapphire]

affine algebraic varieties (zero sets of polynomials) are isomorphic to finitely generated reduced k-algebras

[devout nacelle]

I see

[surreal sapphire]

finitely generated reduced k-algebras is k[x_1, ..., x_n] modulo some ideal

so essentially

[crystal stone]

Algebraic Varities are cool to learn and think about if you haven't heard of them Manan

They can be motivated at a low level

[surreal sapphire]

the functions on the variety determine the variety completely

[crystal stone]

http://www.math.lsa.umich.edu/~wfulton/CurveBook.pdf

[surreal sapphire]

in other cases that arent as simple as affine varieties you get similar results

[devout nacelle]

Fulton

[crystal stone]

This is like a standard intro to algebraic curves

That can get you up to speed

Pretty quickly

On what the basics are. It's not exceedingly difficult, but there are a lot of details and subtleties

[surreal sapphire]

gathmann also has notes on algebraic curves: https://www.mathematik.uni-kl.de/~gathmann/class/curves-2018/curves-2018.pdf

i will remain gathmann shill

[crystal stone]

That's fine. I took a 16 week grad course on Fulton's algebraic curves

Prof had a proclivity for long computational problems

So we'd be factoring degree 6 or 8 polynomials over different fields

[devout nacelle]

[crystal stone]

Every class started out with a quiz

Good times

Even the first day, it was

"State the first isomorphism theorem and draw the diagram"

[ancient flame]

bruh

that sounds terrible

[surreal sapphire]

ye some prof here does this all day

and builds his research career on this

he says he spent the last 20 years understanding chapter 1 of hartsthorne and motivate schemes

[static loom]

oh god

[devout nacelle]

[thorn brook]

[static loom]

laughing out loud

[ancient flame]

laughing loudly

[thorn brook]

lol

[fair mural]

lol

sounds about right

not books but khan academy is good for high school math

so it’s good to use it

[steep mountain]

if thats confusing you need to restudy what limits really mean

[frozen merlin]

if you approach from the right, it's +infinity

but if you approach from the left, it's -infinity

[steep mountain]

you wont reach 0 you're just coming very close

[frozen merlin]

no, 1/0 is illegal

you could be put away for a very long time

[fair mural]

straight to prison

no sympathy

i mean

forgive me

but that’s kinda right

but it’s wrong

because it can’t have two values

[frozen merlin]

if it makes you feel better, you can extend the real line/complex plane by adding a point which you call ∞, and defining 1/0 = ∞, 1/∞ = 0

[fair mural]

,w abs(1/0)

[fathom swallow]

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[crystal stone]

yeah, algebraic curves is a lot more interesting than it first sounds to be

There's a lot of detail you can just go into

[hollow sundial]

In the complex plane $i \infty$ is a useful concept

[fathom swallow]

riemann

[hollow sundial]

you learn something new everyday

https://math.stackexchange.com/questions/2280052/what-is-imaginary-infinity-i-lim-limits-x-to-infty-x-i-infty

[frozen merlin]

me neither, but it makes sense

[vast surge]

Sorry if this is a dumb question, but as I'm looking at my options for the summer, I've become terribly afraid that I won't get into an REU, for the third year in a row. Does having no REUs hurt grad school applications at all?

I thought it might hurt a little bit but I'd be fine, but then I saw someone on r/math say that they didn't want to do a third REU and that they were afraid that would just tank their chances and now I'm terrified I have basically no chance.

[static loom]

I wasn't under the impression that they really mattered tbh

[hollow sundial]

no singular thing matters all that much. it's the "totality" of your application

I didn't have a great GPA, above average math GRE, but i had two REUs, and great letters of recommendations, so i got into an applied math phd program

graduate programs take a diversity of students as well. they don't just select the top GPA students.

[vast surge]

That's a massive relief.

[hollow sundial]

also there are over 200 universities around the world that offer phds in math. apply big and far

[bronze pelican]

I also didn't get into any traditional REUs despite applying for them freshman, sophomore and junior year. I did get into polymath REU but thats was an online opportunity created in the pandemic, open to basically anyone who applied and doesnt pay student any stipend

So like, getting into REUs is not the important part. What's important is you letters of recommendation and the mathematical work you can show off in your application

[vast surge]

Okay. So, basically, I should focus on getting my profs to like me and write something interesting in my thesis?

[bronze pelican]

If you don't get into an REU, ask a prof at your school for a research opportunity or independent study to do over the summer and write something to show for your work

[bronze pelican]

Yeah

[vast surge]

I'm doing two independent studies this semester, but yeah, if I don't get into an REU I'll start begging my profs to give me a summer job. My school's REU wasn't renewed for this year, so last year that wasn't an option because my professors were all working on their REUs.

[long matrix]

@vast cipher i probably overemphasizing ppl needing to draw pic but when did u find them useless lol 😂

Like I don't feel there are many hard problems you don't need a pic (i guess there's sometimes plain algebra/differentiation)

[vast cipher]

well it made sense there

honestly cool problem

mb im dumb but

[quaint loom]

does anyone else sometime feel they understand a subject well, and subsequently do well on "difficult" questions but when it comes to an exam you just perform less consistently and well, worse?

[long matrix]

I never can do rlly well on exams. Time managing in exams is a pain, and I generally prefer more time to think

You're meant to skip parts you're stuck on... doesn't sit well with me

[quaint loom]

Well the issue is that you're graded off of those exams, not off of some general ability and knowledge that's there in one's "default state"

[frozen venture]

If you present the natural numbers ( in terms of symbols 1, 2, 3 etc) and their addition and multiplication to very smart aliens, would they be able to reverse engineer what they represent?

[quaint loom]

Haven't we done this ourselves with archaeology?

[leaden torrent]

schoolchildren do it, so as long as their biology is sufficiently similar to ours, sure

[frozen venture]

Well, I mean you literally write out all the symbols, all the expression a + b = c, and all the expressions ab=c and nothing else

No illustrations involving groups of concrete objects allowed

[quaint loom]

a + b = c

can mean many things

to humans

lol

[frozen venture]

I mean all expressions "like" 1 + 2 =3

In an infinite list

[quaint loom]

what do you mean by all expressions

[frozen venture]

and you gave no more info or context\

like literally a giant list of all additions written out

imagine a literal infinite piece of paper

[quaint loom]

so you just mean a lot of addition?

[leaden torrent]

are you being intentionally obtuse jonatan

[quaint loom]

No

I don't get what he's referring to

[leaden torrent]

the set of valid strings ${a + b = c | a, b, c \in \mathbb{N}}$

[fathom swallow]

Namington

[frozen venture]

Yeah ^

[leaden torrent]

"valid" meaning correct

(so no 4 + 4 = 6 or whatever)

[quaint loom]

well if they're ordered then I'd presume an intelligent species would be able to decode the value of the characters with a list that wouldn't have to be very long at all.
Well, it doesn't have to be ordered nvm that

as long as there's a pattern ( pattern being they're valid ) and the species is "intelligent"

[long matrix]

1+1+1=3
1+1=2
1+2=3

maybe theyll notice something

[fair mural]

implying they even notice what the heck the addition symbol does

[quaint loom]

Presumably they would share our operations and have the same base arithmetic? highly likely they wouldn't use base 10 though

[long matrix]

2 fingers

base 2

wait that would be bad

[quaint loom]

lol

[long matrix]

might not be able to understand base 10 if thats the case

[quaint loom]

we understand base 2 though

If they're more knowledgeable or around the same mark as us then base 12 would be neat and maybe more widely used throughout the universe?

Idk if species would go from say base 8 to base 12, base 10 to base 12 or just settle for what came naturally to them, there not being much of a point in making a change to that

[fast ivy]

Maybe they never learned how to count and just do math over weird rings or something

that would be cool

[quaint loom]

rings?

oh i'm out of my depth and there's a language barrier i see

heh

This? https://en.wikipedia.org/wiki/Ring_(mathematics)

[fast ivy]

Yeah, that's what I was talking about

[leaden torrent]

huh thats a weird way for the wiki article to introduce them

the way i think about rings is not as "a generalization of fields"

like yes, thats true, in the same way that field extensions are actually "backwards" subfield relations

but i dont think thats a good way to think about them

pedagogically or mathematically

[quaint loom]

I've not studied any abstract algebra so this is all very abstract to me

[leaden torrent]

the tl;dr is that mathematicians use the term "ring" to refer to things that behave "like the set of integers"

[tender tulip]

Z

[leaden torrent]

with integers, we can add them, subtract them, and multiply them

and these operations are fairly well-behaved

a + (b + c) = (a + b) + c and similar for multiplication, a + b = b + a, you get the idea

[hollow sundial]

Are most rings isomorphic to some direct sum or product of $\mathbb{Z}$ ?

[leaden torrent]

so its useful to try and identify the "basic rules" that give us a structure that "acts like the integers"

[fathom swallow]

riemann

[leaden torrent]

and we call these "rings"

well, specifically we call them "commutative rings"

a general ring does not have ab = ba

[tender tulip]

SpanDeX

[leaden torrent]

but thats the gist

[quaint loom]

oh

[hollow sundial]

ah ok. what's an example of a non-commutative ring

[leaden torrent]

square matrices of a fixed size

[quaint loom]

that makes it abstract?

[leaden torrent]

you can add them, subtract them, and multiply them, but AB is not always equal to BA

[hollow sundial]

bitch ass matrices

[leaden torrent]

the quaternions are another example you mightve heard of

[leaden torrent]

the "abstract" just means that we study structures by looking at their "properties" rather than specific examples

we call these "properties" axioms

[hollow sundial]

i'm definitely more familiar with matrices. don't know why it didn't occur to me since I spent so much time with the GOE/GUE/GSE class of matrices

[leaden torrent]

so if i want to prove something about a ring, im not allowed to introduce natural numbers like 15 and 27 because most rings dont "look like" the integers

[quaint loom]

are these axioms then?

[leaden torrent]

but i am allowed to use the "rules" of rings

ie the ring axioms

[leaden torrent]

yes, here's a full list

basically:

• addition is really nice and "reversible"
• multiplication is nice, though not necessarily as nice as addition
• addition and multiplication are "compatible", specifically being related by the distributive property

without distributivity we just have 2 random unrelated operations on the same set

if your multiplication is as nice as addition, we call that a "field"

the integers are not a field since we cant "reverse" multiplication of integers

we can in some specific cases, like 4 / 2 makes sense, but 3 / 2 does not

[deep mango]

is almost as nice *

[leaden torrent]

okay yeah, 0 exists

but whatever

[deep mango]

Still cant invert 0

[hollow sundial]

bitch ass 0

[leaden torrent]

the rationals are considered a field

since multiplication commutes (ab = ba) and we can divide by any element (except 0) no matter what

so in a sense, rings are a "generalization" of fields

they're fields but "less restrictive"

[quaint loom]

what are you reversing by doing 3 / 2 tho

[leaden torrent]

but i dont like this framing

[leaden torrent]

thats the point: nothing, as far as integers go

but for rational numbers, this would be a "reverse" of the process of multiplying 3/2 (aka 1.5) by 2

[quaint loom]

ah I see

[leaden torrent]

the more formal way of phrasing this is:

for any nonzero element a, there exists an element a⁻¹ such that a * a⁻¹ = 1

so for example, in the rational numbers, for any x, we can simply multiply by 1/x

and get 1

since x * 1/x = x/x = 1 (as long as x isn't 0)

but in the integers we cant do this

2 * ???? = 1

you cant multiply 2 by an integer to get 1

you can multiply it by 1/2, but that aint an integer

aside: this process of taking an existing mathematical structure, "forcing" it to follow another "rule", and adding elements until it does is a very common way of "creating" mathematical objects

the complex numbers, for example, are gained by "taking" ℝ and "forcing" any polynomial in ℝ to have a root

the fastest way to do this is to add an element i such that i² = -1

(this is the zero of the polynomial x² + 1)

it turns out that this is "enough", in that all you need to do is add this element i and make i compatible with the field operations

and you get the complex numbers

and every polynomial has a root

(this is the fundamental theorem of algebra)

[long matrix]

witchcraft

[leaden torrent]

theres a funny ring-theoretic way of phrasing this fact

$\mathbb{C} \cong \mathbb{R}[x]/(x^2 + 1)$

[fathom swallow]

Namington

[quaint loom]

Does R stand for ring

[fast ivy]

R stands for the real numbers

[long matrix]

What's the bijection? 🤔

[quaint loom]

Oh

[leaden torrent]

R is common notation for a ring, but ℝ (as in my image) means the real numbers

the "double stroke R"

[fast ivy]

R[x] is the ring of polynomials over R

[leaden torrent]

$a + bi \mapsto a + bx$

[fathom swallow]

Namington

[leaden torrent]

idk what youre expecting lmao

[long matrix]

I learnt all this before but forgot it

makes sense

aha thats so good. i^2 -> x^2 === -1

[quaint loom]

Are there rings you can’t do + - * / operations on?

[fast ivy]

Well, +, - and * always make sense in a ring, although division is a problem because there might non invertible elements in a ring or even zero divisors.

Which is stuff we don't want in order to properly define division.

[tiny marten]

Yeah you define the ring as having a group structure plus another usually less nice operation

And we think of them as an + and a ×

But lots of rings have weird versions of operations that dont seem like either addition or multiplication of integers or reals or whatever

You can do the same with vector spaces, the probability simplex can be made into a vector space

A kinda weird one

Where componentwise multiplication followed by normalization is vector addition

They can be absurdly weird tho, and stretch the analogy to +,× to a degree

[frozen venture]

What does /(x^2 +1) mean here?

Like I see youre letting X^2 + 1 =-

=0 i mean

[leaden torrent]

quotient by x² + 1, i.e. partition ℝ[x] into equivalence classes by the equivalence relation a ~ b iff a - b = k(x² + 1)

[frozen venture]

but what does the notation mean formally

[leaden torrent]

and consider the structure of those equivalence classes

[frozen venture]

gotcha

is k an integer?

[leaden torrent]

no, im abusing notation slightly

x² + 1 generates an ideal

so really we're saying a ~ b iff a-b in the ideal generated by x² + 1

[long matrix]

its like modular arithmetic

this is 'mod x^2 + 1'

[mystic mountain]

how did you type the squared thing without latex? x.x

[leaden torrent]

i use software called wincompose

in theory you could also just memorize a numpad code but

wincompose lets you set up more intuitive shortcuts

² is rightalt → ^ → 2 for me

→, incidentally, is rightalt → - → >

[mystic mountain]

oh I see…

[stray kite]

wincompose is good

[neat lintel]

does anyone know which subjects a compsci degree entails?

or even a math degree

it may vary

[neat lintel]

but for me it had digital logic, databases, operating systems, networking, computer architecture, compiler design, algorithms, theory of computation, cryptography, ML, graphics, a few software engineering ones and some more..

is that really a compsci degree?

yeah

from my understanding math is what most of the degree is in here

that must be a more theory focused one

what you're describing would be software engineering in here

yep

does it have any https://www.youtube.com/watch?v=BvWefB4NGGI

[neat lintel]

there's also a bunch of math, but it leans more towards the 'engineering' side

i see

[dawn bridge]

you should look up the website of whatever university u are applying to

different uni will often have different courses

[frank verge]

Are these two same or is there any difference?

[dawn bridge]

log(x^2) is the top one

its not the same

[frank verge]

Bottom is wrong and top is the right one?

[deep mango]

The bottom one is incorrect.

[dawn bridge]

bottom one is an equation in x

top one is an identity

[deep mango]

The top one is also an equation in x, it's just true for all x

[dawn bridge]

yes

[static loom]

help is this one correct $\sqrt[\sqrt{2}]{x} = \sqrt{x}^{\sqrt{2}}$

[fathom swallow]

Merosity

[burnt canyon]

1+1 = 2

[neat lintel]

I mean

[fair mural]

help is this correct $\sqrt[1]{x} = \sqrt{x}^1$

[dawn bridge]

i thought left hand side would be the same as $x^{\frac{1}{\sqrt{2}}}$

[fathom swallow]

bm

quantum

[fair mural]

joke is ruined

[neat lintel]

Lol wut

[fair mural]

had to edit

[static loom]

yeah it is

[dawn bridge]

yeh so its correct

[deep mango]

For which $t$'s are the solution to $\sqrt[t]{x} = \sqrt{x}^t$ extremal?

[fathom swallow]

ryc

[deep mango]

oh it's always just 1.

[dawn bridge]

define extremal for me

[deep mango]

lmao

[static loom]

heh fun idea though

[deep mango]

max or min bm

how about this

[fair mural]

huh

[deep mango]

For which $t$'s are the solution to $2\sqrt[t]{x} = \sqrt[2]{x}^t$ extremal?

[fathom swallow]

ryc

[fair mural]

bruh

what does that mean

[dawn bridge]

i hate these square root symbols

[deep mango]

or maybe i should have put a 3

since no one puts 2 in the root

[static loom]

maybe put some fixed arbitrary s

[deep mango]

anyway 2^(1/(t/2 - 1/t)) = x, so idk how to maximize / minimize this thing

seems nasty

i guess since 2^x is monotone

yeah if you let the exponent vary it's just s^(1/(t/s - 1/t))

t/(t^2/s - 1)

i don't want to differentiate this

it has a nice max and min though which is pleasant

[static loom]

,calc derivative("s^(1/(t/s - 1/t))", "t")

[fathom swallow]

Result:

-(s ^ (1 / (t / s - 1 / t)) / (t / s - 1 / t) ^ 2 * log(s) * (1 / s + t ^ (-2)))

[deep mango]

[vestal tapir]

s and t are both parameters

[deep mango]

oh, also im wrong

there's no nice max and min

it's only nice if you replace the - with a +

what a silly conversation

[static loom]

ahaha

[fair mural]

it’s all silly

[static loom]

how dare you, this is my preferred form of conversation

here's something useless:

[fair mural]

silly is fun

[static loom]

$$(\ln x)^n = \ln(x^{(\ln x)^{n-1}})$$

so you can keep repeating this inside to get an ugly thing lol

oh I put one too many )

[fathom swallow]

Merosity

[fair mural]

haha

what

funny because it’s true

yet it looks like nonsense

[static loom]

$(\ln x)^3 = \ln (x^{\ln(x^{\ln x})})$

[fathom swallow]

Merosity

[static loom]

idk does anything come out of taking the limit n-> infinity or something

$$y=\ln (x^{\ln(x^{\ln x^{\dots}})})$$ $$y=\ln(x^y)$$ and so uhh... $e^y=x^y$ so $x=e$ and $y=?$

[fathom swallow]

Merosity

[static loom]

that's awkward

[toxic schooner]

$\text{applying properties of logs} \ y=\ln(x^y)=y\cdot \ln(x)$

[fathom swallow]

Shyshu of the Golden Flower✓

[toxic schooner]

so ig it works for any value of y?

[sleek wing]

Yeah this looks right

Sub x=e and then try and take a limit of a sequence of nestled ln memes I suppose is the way to contain

*continue

That sounds like a pain though

[static loom]

I guess if you're taking it as a limit $\lim_{n \to \infty} (\ln e)^n = 1$

[fathom swallow]

Merosity

[toxic schooner]

[neat lintel]

i have finished my mustard pisces

3x+5y+6x+3y=9x+8y

[vagrant sundial]

Hi, does anybody have resources for algebra word problem exercises? (Except khan academy)... I wanted to start solving 3-4 such problems a day as a way to not forget what i learn from math, but khan academy ones don't really have much range in terms of word problems i think

[gentle bay]

AoPs.

Mathisfun.com.

Paul's Notes.

These are all great.

[vagrant sundial]

that was fast, thank you

[gentle bay]

Lol.

Yeah.

[vagrant sundial]

i was actually already checking out aops since i found some reddit posts about there being some exercises in the forums but i can't understand where they are

like, precisely

[gentle bay]

Community.

And books.

https://artofproblemsolving.com/community/c89

Here.

[vagrant sundial]

Oh, thank you x2

[cinder flume]

guys

can I get away with

not studying inequalities

ever

[rotund steppe]

no

I think half of my analysis class was spam triangle inequality

[cinder flume]

with just the stuff in analysis?

[rotund steppe]

don't you use triangle inequality a bunch?

for metric space stuff

[cinder flume]

everyone does that

[rotund steppe]

yes so there you go lol

[cinder flume]

yes of course

but

[rotund steppe]

you used inequality

[cinder flume]

I didnt study

inequalities for that

like

in a book

[rotund steppe]

no one studies inequalities specifically

[cinder flume]

I just read in analysis and

[cinder flume]

oh

[rotund steppe]

yes

[cinder flume]

I saw books on

inequalities

like cauchy schwartz

masterclsas

[rotund steppe]

probably for olympiad math or something

[cinder flume]

okay

I dont know where they use advanced inequalities

but today I read in analysis book that "inequalities are used very much in fields other than analysis"

[neat lintel]

ngmi

[sleek wing]

mf can't add fractions and is now baby raging

[neat lintel]

crypto ngmi

ngmi crypto nft ngmi no money ngmi

[sleek wing]

look bro... I'm gonna be a DOCTOR yeah?!??!

[neat lintel]

crypto and math is together

[sleek wing]

yeah?!?!?!?!?!? so dont fuck with me or I'll have the state of rwanada on yo ass

the entire nation's army is at my disposal

[neat lintel]

nft ngmi youre gonna be poor ngmi

[sleek wing]

it's a doctorate in maths

[neat lintel]

🤦

[sleek wing]

not a stupid medical doctor

aka crankery

[neat lintel]

crypto dumb ngmi

[sleek wing]

sounds like someones JEALOUSSSS

[neat lintel]

sure?

[sleek wing]

👂 can you hear the sounds of jealousy kids?

[neat lintel]

no monet ngmi

[sleek wing]

see stopped responding cause he knows I called em out

epic winnn

[neat lintel]

wow based!??

[sleek wing]

oh yeah I don't deny that

ty boss

ok stay safe don't talk to strangers do drugs

[mint patio]

Wtf is happening here

Can someone explain

[sleek wing]

nope

[mint patio]

I have this implicit equation (x is the independent variable)

So I used Wolfram to solve for y

(y = v and t = x xd)

So into Desmos it goes right

Purple is the implicit equation and green is the explicit

What's causing me to lose this middle section?

[sleek wing]

looks like a missing minus sign in one of the exponents

[mint patio]

wha

Nope that's not it

It's not in Wolfram's equation and even still when I tried putting in a minus the graphs still don't line up

Strange

What's even weirder is that

The next part of the problem needs me to take lim t->infty

And when I do that I get the right answer, 100, whether I use the implicit or explicit equation

So wtf is happening here LOL

[sleek wing]

purple line looks like a tanh type beat

hmm

[mint patio]

never in my LIFE have I heard something more zoomer than this

😭

[sleek wing]

I'm barely even a zoomer

[brave hollow]

but you talk like one

[sleek wing]

ok boomer

[frozen venture]

Why does Russell's paradox lead us to declare that there can't be a set of all sets?

[fresh nova]

quick simple question for somebody. Linear equations aren't functions by default, only becoming functions if specifically stated?

[deep mango]

yeah

[fresh nova]

thank u

[honest veldt]

gentlemen

[fair mural]

gentleman

[odd narwhal]

Gentle

[honest veldt]

i now know that a 2-sphere is a homeomorphic to R^2 local to a point, but what is actually called the 2-manifold here? is it the entire 2-sphere or is it that neighborhood around the point?

[fair mural]

man i gotta pick up the pace

[frozen merlin]

the sphere is the manifold

[odd narwhal]

A manifold is a thing that is locally homeomorphic to R^n

So the sphere is the manifold

[honest veldt]

Goootcha gotcha gotcha, alright thank you

also I read an example about a 3-sphere, I'm not really supposed to be trying to visualize this in any way, right?

[odd narwhal]

You can't

It's only embeddable in R^4

[honest veldt]

Ah, I see, no shot then

I tried looking it up and all I found were a bunch of colored squiggles

that's when I called it a day

[odd narwhal]

If you want, you can think of it like this: if you were on the surface of a 3-sphere, if you moved long enough in any direction, you would end up where you started. Also, if you took a 3d cross section of a 3-sphere, you'd get a 2-sphere

[honest veldt]

suffices to be convinced that it's locally homeomorphic to R^3 then

3d cross section

[odd narwhal]

You could be living on a 3-sphere and you'd never know

[honest veldt]

Alright I guess I can kinda imagine what it could look like if I could imagine it now

That's enough for me

[frozen merlin]

in the same way that passing a 2-sphere through a plane would give you a series of circles (1-spheres), passing a 3-sphere through space would give you a series of 2-spheres

[honest veldt]

Ooooooh, now that I can imagine

Yeah I get it now, many thanks andman

[frozen merlin]

😎

[uncut socket]

@long matrix Do you know/have you done cover up method for solving partial fractions?

[long matrix]

yeah

its just subbing in x

so factors are 0

iirc

[uncut socket]

Sorta, like $\frac{x^2 +4x-2}{ x(x^2-4)} = \frac{A}{x} + \frac{Bx + c}{x^2 -4}$, you multiply $\frac{x^2 +4x-2}{ x(x^2+4)}$ by x and plug in x = 0, to get A. Same thing for Bx + C, you plug in x = 2, -2

[fathom swallow]

dldh06

[long matrix]

yh well i never used it

works for factors that have no real roots i assume?

You plug in say x^2 + 1 = 0? or am i misunderstanding

[uncut socket]

It works for complex roots too

[long matrix]

witchcraft to most

haha

complex stuff in integral

[uncut socket]

Repeated roots is tough though, because there's a derivative method and it takes so long to do

[frozen venture]

How is the contradiction derived?

[deep mango]

using another axiom, you can then define the subset from russell's paradox.

[leaden torrent]

$X \in X$ and $X \not \in X$

[fathom swallow]

Namington

[leaden torrent]

this is inherently contradictory

since these conditions are negations of each other

("not in" isnt a proper relation in and of itself, it just means, well, "not in")

[frozen venture]

Without separation, you can't define the set that leads to the paradox?

[leaden torrent]

wdym

separation is in ZF which doesnt have russells paradox

[frozen venture]

Without axiom of separation

[leaden torrent]

if you use a stronger form of separation - namely unrestricted comprehension - then you can define the set in russell's paradox

which is, well, where the contradiction comes from

thats (part of the reason) why we need to restrict our comprehension

which is what the axiom schema of separation does

[frozen venture]

Gotcha

[leaden torrent]

in particular, the axiom schema of separation forces you to quantify "with respect to another set" when defining a set

so {x | x not in x} isnt valid since our set definitions need to be of the form {x in y | P(x)}

and the set of all sets does not exist

[frozen venture]

By the way, what's the difference between a paradox and a contradiction?

[leaden torrent]

"paradox" is an informal term that usually means "something contradictory" but can also just mean "something counterintuitive"

(the "banach-tarski paradox" is an example of the latter)

[frozen venture]

Is Russell's paradox a contradiction? Or something more subtle?

[leaden torrent]

in the context of set theory though, it almost always just means "something contradictory"

yeah it just gives a contradiction

if you have a set S = {x | x not in x}, then is S in S?

• if S is in S, then S is not in S
• if S is not in S, then S satisfies "S not in S" and so S is in S

in either case, we have a contradiction since set membership is a binary

you're either in a set or you're not

there's no "both" or "kinda sorta"

but S is both in S and not in S

[frozen venture]

We have the negation of S is in S OR S is not in S

We've contradicted the law of excluded middle

Is this correct?

[leaden torrent]

no, the contradiction arises even without LEM

its more fundamental than that

let P(x) be the statement "x is in S"

[frozen venture]

I'm afraid I don't understand the underlying logical system well enough

[leaden torrent]

then "x is not in S" is the negation of P(x)

so we have P(S) AND NOT P(S)

this just cant happen

if you lack LEM, its possible to have neither P(S) nor NOT P(S)

but its always impossible to have both

[frozen venture]

What bothers me is that it feels like there is a chain of implications that doesn't stabilize

This doesn't feel the same as other contradictions I've seen

[leaden torrent]

i don't really see what you mean.

[frozen venture]

Is there something more subtle/nuanced about this contradiction when compared to common ones?

[leaden torrent]

no

[frozen venture]

P implies not P implies P implies not P....

Forever

[leaden torrent]

i dont see why thats a problem

??

like okay first off

your bracketing is ambiguous

[frozen venture]

It's not of the form P and not P

[leaden torrent]

P implies (Q implies R) is a very different statement from (P implies Q) implies R

so im not really sure what youre saying

[frozen venture]

I think I understand now

I had to get rid of all extraneous details in my mind of what's a valid argument and what isn't

[long matrix]

P => Q and Q => R
(!P or Q) and (!Q or R)
(!P and !Q) or (!P and R) or (Q and R)

P => (Q => R)
!P or !Q or R

(P => Q) => R
!(!P or Q) or R
(P and !Q) or R

=====

I was just randomly curious

[frozen venture]

Suppose P. This implies P and not P. Suppose not P. This implies P and not P. In every case, we are led to a contradiction.

Right?

[west sundial]

P and not P is always false :o

[long matrix]

how does P => (P and !P)

[frozen venture]

Then how do you ever do a proof by contradiction if you're not allowed to derive P and not ap

And not P

Isn't that the definition of a proof by contradiction

[long matrix]

oh ur saying IF P => (P and !P) we have a contradiction.

I mean... sure

[frozen venture]

Yeah

This convo is making me think I should take a course in logic lol

[leaden torrent]

this is a sketch of the formal argument:

∀x (¬P(x, x) ↔ P(x, S))        [definition of the set S, here P is membership]
∀x ¬(P(x, x) ∧ ¬P(x, x))       [this is the law of non-contradiction, which does not rely on LEM]
---
¬P(S, S) ↔ P(S, S)              [from the assumption, with x = S]
P(S, S) → ¬P(S, S)              [previous line]
P(S, S) → P(S, S)               [obvious]
P(S, S) → (P(S, S) ∧ ¬P(S, S))  [previous 2 lines]
¬(P(S, S) ∧ ¬P(S, S))           [law of noncontradiction]
¬P(S, S)                        [modus tollens on the previous 2 lines]
P(S, S)                         [from the very first line of proof and the previous line]
¬P(S, S) ∧ P(S, S)              [previous 2 lines]
⊥                               [law of noncontradiction]

no LEM necessary

just modus tollens and the law of noncontradiction

both of which are constructively valid (ie dont require LEM)

[long matrix]

Hmmm I am in the habit of writing => between every line in a proof (when I really mean the 1st). Seems like I should be using something like therefore instead unless they are <=> 🤔

[leaden torrent]

if you dont quite follow my syntax, P(x, y) means x ∈ y

so P(S, S) is S ∈ S

we can "trim the fat" a bit if we just represent this fact with the symbol P instead of P(S, S):

¬P ↔ P             [from the assumption, with x = S]
P → ¬P             [previous line]
P → P              [obvious]
P → (P ∧ ¬P)       [previous 2 lines]
¬(P ∧ ¬P)          [law of noncontradiction]
¬P                 [modus tollens on the previous 2 lines]
P                  [from the very first line of proof and the previous line]
¬P ∧ P             [previous 2 lines]
⊥                  [law of noncontradiction]

does that argument make sense to you?

[frozen venture]

That was dope

Thank you so much

I still want to understand logic more deeply but I feel closer now

So, our original assumption that there was a set of all sets led to a contradiction

Correct?

[leaden torrent]

our original assumption was that there was a set {x | x not in x}

"set of all sets" doesnt immediately lead to a contradiction, but within ZF it does

because of how ZF's axiom schema of separation is phrased

ZF allows you to construct sets of the form {x in y | P(x)}

where P{x} is some statement about x

if there is a set of all sets (lets call it U for universal set), we then can construct {x in U | x not in x}

and get the exact same problem

there ARE actually consistent ways to construct a theory with a "set of all sets" that dont lead to a contradiction, but you have to make compromises

(for example, you have to make sure your theory doesn't prove Cantor's theorem)

(this is surprisingly hard)

its generally way easier to just say there's no set of all sets and be done with it

we can still talk about a "collection" of all sets, we just cant call it a "set"

[the notion of "class" was invented partially for this reason]

[as well as "universe", see e.g. grothendieck universes]

[frozen venture]

Why did you say that was a proof sketch? Is it possible to make it more rigorous than that?

[leaden torrent]

i just didnt list the formal rules i used

like i said "obvious" instead of a proper derivation for P → P

a proper derivation would look something like (depending on proof system):

blahblahblah
| P    [begin a subproof by assuming P]
| -----
| P    [restatement of assumption on line 2]
P → P  [implication introduction on lines 2-4]

of course, in practice no one works in this level of formality outside of very specific contexts

[mortal igloo]

You need that to prove that P implies P? interesting

[leaden skiff]

what's cantor theorem?

[leaden torrent]

|powerset(S)| is strictly greater than |S|

[leaden skiff]

wait what really? you can have that be false?

[toxic schooner]

[fathom swallow]

TheBedrockBird#1739's Avatar

Click here to view the image.

[midnight iron]

why elementary logic assertions look the same

Q is a necessary condition for P, then why isn't it a sufficient one?

[steep mountain]

i assume you know that but lets define both 1st
P-->Q or Q is necessary for P means P cant occur without Q being true (Q is necessary)
P is sufficient if P occuring means Q occurs (its sufficient) but Q can occur without P
but one can exist without the other
a common example
x is rational is sufficient to x being real but not necessary
on the other hand x is real is necessary to x being rational but not sufficient (x real does not mean its rational)
do correct me on anything

[mild nest]

helping out on here is interesting

because you're trying to explain things that are so well internalised

that it becomes hard to explain

does anyone have tips for this?

i've been struggling to do this in person as well: i tried to teach my sister in 7th grade, what a quadratic equation is

it didn't go well

[unborn trellis]

if she knows what a linear equation is, it's the same thing but with an additional term

the conceptual gap comes in how you solve them

[mild nest]

that's not how i did it

she does know what a linear equation is but not what a linear function is

so i decided to teach her about functions

but then i also had to teach her what a set was

and then i started saying stuff about linear functions and plotting them and how they make a line and how the slope comes into play

and then i said here's another random type of function

it's called a quadratic, and the thing you get with the graph is not a line it's called a parabola

looking back

that was bad because there's no motivation to introduce this "random type of function"

i'm just shoving information in her head

[unborn trellis]

to me the logical progression is that now you're solving equations where the slope is no longer constant

instead of mx, it's (mx)*x = mx^2, where slope is now proportional to x

so naturally, instead of a straight line, it will curve

[mild nest]

hmmm

[fringe needle]

Lol @mild nest this is me when I try to explain anything to my mom

I’ll start going off on tangents and so she quickly gets lost

It’s important to stay on track: you don’t have to worry too much about skipping over some of the details as long as they understand the general idea

The details can come after

[neat lintel]

i feel like if i put more effort in math earlier my chess game would be much better

or maybe even vice versa

[deep mango]

[fair mural]

good at math means good at chess i see

[sleek wing]

god I love chess

[neat lintel]

both are mainly about pattern recognition and test your memory and visualization

not that hard to believe both have transferrable skills

[deep mango]

there might be something to this, but you'd get much better at math by spending whatever time you spent studying chess on studying math

i'm terrible at chess and i'm a better mathematician for it

[analog salmon]

What's that one rule called, where if you have sets $A$ and $B$, $|A\cup B|=|A|+|B|-|A\cap B|$?

[fathom swallow]

cgodfrey

[analog salmon]

I thought it was law of excluded middle but turns out it's not that lmao

[surreal sapphire]

inclusion-exclusion

[analog salmon]

🙏 god bless

[bright hill]

Is the proof of the chain rule hard?

[fair mural]

no

[bright hill]

It's rigorously proven in real analysis, am I correct?

[rose dock]

yes

[bright hill]

Ok, was just checking

[acoustic linden]

Is 1 a Complex number ? As 1 + 0i

[sly vale]

yes

[acoustic linden]

thanks

[sly vale]

R is a subset of C

[iron osprey]

Has anyone here in grad school for pure maths done undergraduate research in an applied subject which led to a journal submission?

I'm wondering if this is the norm since i've only seen one or two people produce publications for pure maths in undergrad during my studies

[tiny marten]

What are you wondering about, if people go this route anecdotally?

I am at least applying for pure math grad school but all my work is in applied, including on some stuff I'm trying to submit to a conference and another prolly published

Tho I will likely go with a CS department in the end

I dont think it's a big issue to publish in applied, publishing at all is very good

And like, my "applied work" rn is a geometric theory of an algorithm using metric spaces and ergodic theory it's not too shabby

I dont think anyone will be worried that its applied

[iron osprey]

Yes

[tiny marten]

You can do some really cool stuff with applied it doesnt just gotta be like

Something that wont seem abstract

[iron osprey]

The thing I am doing is not abstract, though it won't be my only researct

Research*

[tiny marten]

Oh well it's mostly about quality of the work anyways

[iron osprey]

I see

[tiny marten]

But like, it's probably not your only evidence for you being interested in X field

[iron osprey]

The situation is a bit odd

As it's like

[tiny marten]

And is probably just a merit in this situation

[iron osprey]

Well, it's quantum chemistry so it's totally out of left field

[tiny marten]

Oh, huh

[iron osprey]

It was just something I wanted to help out with and was useful for

Surely it won't do harm though

[tiny marten]

I think that having the experience would reflect well

You get to explain your interest in your desired field too, so it's basically just research experience

The more the better but having some closer to your field is optimal

I think that a lot of people just wanna make sure you have been busy

Some departments or individuals are more particular

[deep mango]

good ping

[deep mango]

yeah we did this but it only led to a paper submission after 1.5 years of work

(the research program we were in only lasted a summer, we kept working for an extra year very consistently and just submitted in november)

also like

both of us are in grad school this year

[iron osprey]

Ah, in a research program. That's interesting that you'd have set out to do that from the start

[deep mango]

we only submitted after graduating

[iron osprey]

This is not at another institution, it's with a professor i met in sophomore year

[deep mango]

we never intended to write a paper, we just had cool results and wanted to improve them

[iron osprey]

So this is interesting actually

[deep mango]

and we kept improving them over and over

until they were publishable

[iron osprey]

Very cool. Did you publish with the faculty at the institution you did the program at?

[deep mango]

yeah our program was at our uni so it was with profs we knew from beforehand. it was a fellowship we applied to

[iron osprey]

Assuming it was elsewhere

Ah ok

[deep mango]

and we published with 2 profs and a grad student

we did the theoretical work and simulations, the grad student ran benchmarking to test our proposed modifications to the relevant algorithm

anyway

you should really rarely expect to actually publish in undergrad

it's much more reasonable to just expect to have a paper which is maybe submitted and is posted on arxiv

publishing something takes a long time

[tiny marten]

yeah

getting a paper arxiv-able is probably fine in math, pure or applied (and note that this is actually a lot harder than just writing some research paper you wanna show an admission committee/particular prof)

apparently people do it all the time in the stuff i do, they just get it on arxiv and try to get it into a conference

but that's like the CS influence i guess

like i dont think i'll probably publish in undergrad actually, the work i have in mind will take a while just to get it "presentable"

and it's certainly not pure math

it's a very good idea tho i do believe it's publishable

it sorta unites some stuff from hilbert convex geometry and machine learning

[neat lintel]

like perelman

[iron osprey]

I would like to see it

[iron osprey]

This

[tiny marten]

it's some cool stuff

i was sorta doing a bunch of things near it then found another thing and realized it's an ideal geometry for my research

[iron osprey]

I'm mostly interested in information geometry in neuroscience & statistical learning theory

[tiny marten]

yeah information geometry is interesting

i like it a lot at first it's not clear what is even going on

but you get these cool statistical properties

it's possible that the principle i am using extends to other ML algorithms but that's speculation, though it may for any that are specifically defined using a probability simplex

what's really cool is when you can use the probability simplex as a vector space, in my case it keeps coming back to that

it's not too crazy, tho i'm still not really sure what a line is in this space

[iron osprey]

What is a probability simplex?

Just a simplex homeomorphic to a probability manifold?

Googled, saw, neat

[tiny marten]

yeah just a kinda weird simplex bounded away from the origin on R^n

[iron osprey]

I guess the points on the faces are the vectors, with components being the event weightings?

[tiny marten]

you can make it into a vector space with the uniform distribution as the 0 vector

it's actually in the interior! although you can make it compact by including the face boundaries

[iron osprey]

Well, i mean the n-faces

But ye

[tiny marten]

oh i see

[iron osprey]

Oh strictly interior

[tiny marten]

something weird happens if you add the boundary tho

i'm not sure if it's still a vector space i gotta play with it more

but it is still a metric space, which is what i needed

[iron osprey]

Ye

Last i was reading on manifold theory was partitions of unity

Have had to focus on topology lately

[tiny marten]

yeah i have found in this stuff that the devil is in the details

you gotta keep track of some pretty subtle topological stuff

[iron osprey]

It's kind of weird how literally everything i'm reading in every subject rn (algebra, geometry, topology) is talking about extending objects with an extra dimension

Not weird that they all have this topic

Just weird that i ended up reading that at the same time in each subject

[tiny marten]

yeah i had that happen with complex projective geometry

i was suddenly reading three books that all covered it

[quaint loom]

How should I go about memorising how to derive/prove a fair bit of the trigonometric formulas ie Cos(a+b) = cosACosB + SinASinB and so on forth

[leaden skiff]

eulers formula

that's like, really the only way

for context eulers formula is this

$e^{ix} = \cos x + i\sin x$

[fathom swallow]

Ninja II

[warped ore]

somebody pls solve it ( 47a^2 + 81b^3 )^2

[warped ore]

"^" is the power

[rotund steppe]

ok we have the server's greatest minds on your problem now Devil. We will get back to you in a week with the solution.

[toxic schooner]

Lmfao

[tender tulip]

Some have geometric interpretations, others use Euler’s formula

But many abide by binomial coefficient rules just with complex numbers

If you’re familiar with them at least to some extent then you should be able to remember the basic sine or cosine ones

essentially cos(narccos(x)) is literally just the alternating even powers and binomial coefficients

[barren smelt]

math is scary

[robust osprey]

No

Jk

It’s dead a*s scary

[sleek wing]

no it's cool

[cinder flume]

solve the equation

"solve"1

[quaint loom]

Is that more uh, optional than sketching out a bunch of unit circles or something of that nature?

[leaden skiff]

there's other ways to derive it but trust me, they wont make sense

the other ones are incredibly ugly and don't explain anything

i was in your boat once lol

[quaint loom]

Yeah, I'm pretty sure our teacher wants us to like, plot out various angles on a unit circle and then do all that stuff like that, as we haven't covered complex numbers yet. Is that way of going about it what you're referring to with ugly and not explaining anything?

[leaden skiff]

that's just verification no?

like one way i saw it proved without eulers identity was to consider some arbitrary angle theta and 2theta and do a lot of geometry

and it made no sense

like maybe you'll believe it's true

but it wont make more sense, or help you remember

like at this stage, all you can do is accept it as true

and then after you learn about complex numbers, understand why it's true

[quaint loom]

The direct translation of our term for the process is "derivation"

[leaden skiff]

derive it yourself?

[quaint loom]

no