serious-discussion

[cyan goblet]

heh

ok yeah anyways

noted i'll take a look into it thanks

[velvet dagger]

This is the main book I know in this sorta category

Basically it's designed as an intro to proofs/integrated treatment of linear algebra and multivariable calculus

There's also Hubbard and Hubbard but... don't

If you find others they may be better but yeah that's imo what you'd do if you wanna do Spivak CoM type stuff along with linear algebra. Otherwise you just do a linear algebra book outright first, in which case I like Hoffman-Kunze, nowadays probably use Treil or Friedberg-Insel-Spence

[cyan goblet]

✍🏼 no hubbard

yeah im doing H+K rn

[velvet dagger]

I'm not sure how much material is contained in Spivak vs Shifrin tbh, nor do I know exactly how much linear algebra is assumed in Spivak but not implicitlyish done in chapter 1

[cyan goblet]

btw wdym "nowadays"

is H+K outdated or something?

[velvet dagger]

It's, let's say somewhat "old school" but not out of date. Mostly that it has that weird frontloaded chapter on linear equations, and the typesetting is an oof

[cyan goblet]

agreed on typesetting

feels like one big paragraph visually

[velvet dagger]

I kinda learned out of it myself. Kinda because, I had a 5 week class of sorts on linear algebra and graph theory summer after my first year

Which was a very certain angle on the material

Then in my analysis class we did a ton of problems from Hoffman-Kunze

[cyan goblet]

oh interesting

i feel like HK does some handwaving at parts and is super thorough in other parts

[velvet dagger]

Does it ever handwave?

[cyan goblet]

i think it may just be my ignorance

sometimes it'll be like "x, so clearly yz" and im kinda scratching my head

[velvet dagger]

I don't remember much of handwaving occurring there, maybe the first chapter tbh because that shit's a mess

[cyan goblet]

i do think it's just my ignorance

[velvet dagger]

Ah, yeah I guess that kinda thing happens tru

I guess mentally I make a distinction between "clearly" in the sense of "it is now obvious what to do from here", which is more the author making a call that it goes without saying

And "clearly" in the handwavy sense 😛

[cyan goblet]

yeah i think hk uses more of the former defn of clearly

kinda like

"this is true, and it's simple to prove why" sorta "clearly"

[neat lintel]

I'm stuck on trying to prove four lemma

:(

[brittle socket]

Lee's books

[ripe wasp]

i switched from treil to hk and the main difference i noticed

is that

it focuses on row vectors

idk if thats the norm or if columns vectors are more popular

but

it just uses rows for everything

[neat lintel]

@ripe wasp We like to think of actions as acting on the left

And if we were to put a matrix on the left of a vector, matrix multiplication forces it to be a column vector

I think that's the only reason

When you write a function you write f(x)

When you write a linear transformation using a matrix you do Ax

So it's about the conventions I think

[neat lintel]

I've proved one part of

https://tenor.com/view/its-something-gif-7339093

[devout nacelle]

@supple flame

[toxic schooner]

@bright hill

[deep mango]

[bright hill]

dami just checked the table of content once and based his entire opinion off of that lmfao

I have my own criticism of hubberd and hubberd ofc (like using abstract vector spaces without properly introducing them and introducing the characteristic polynomial 2 whole chapters after eigenstuff are introduced) but it's honestly pretty nice

I also appreciate chapter 0 which really helped as a total beginner (not too terse not too boring)

it also has some pretty fun exercises so that's nice

granted I haven't reached manifold stuff yet but it served me well so far

(also did I mention it's highly recommended by 3b1b?)

[long epoch]

Is hubbard that bad?

I was planning on using it for vector calc

If so, any better recs for vector calc?

[bright hill]

it's not bad

[neat lintel]

well it's good to do that when recommending someone a book just to know what they'll learn, but saying it's bad off of it is kind of yeah

questionable

[charred mortar]

Ngl I kinda do the same when selecting books for myself

But that’s because I’m lzy

[harsh fiber]

Guys what jobs involve really high level of mathematics, like Real Analysis, Abstract Algebra, Differential Geometry/topology, Functional Analysis, etc

[fast ivy]

Being a mathematician

[harsh fiber]

right, i also meant outside of that

sorry i wasnt specific

[fair mural]

physics but that’s kinda obvious

[fair mural]

no they were just giving a bad answer lol

[harsh fiber]

i also mean outside of that as well 😂

[fair mural]

oof

[harsh fiber]

ah

[fast ivy]

Ok

To give a more serious answer

Some STEM fields use some of these areas of mathematics in specific ways.

For instance

[harsh fiber]

I heard machine learning engineers use alot of math, but it isnt higher than Calc III and Linear Algebra, though idk myself

[fast ivy]

Differential Geometry has some uses in statistics via information geometry.

[harsh fiber]

oh wow

interesting

in Stats?

that i didnt know

[fast ivy]

Yeah

In fact

Information geometry is quite a big field rn

[harsh fiber]

I heard studying protein synthesis and genetics in biology, requires alot of math

[fast ivy]

Statistics also uses some functional analysis and measure theory.

[harsh fiber]

ill be sure to search this up, anything else?

[harsh fiber]

is functional analysis a graduate level course or a 4th year course in university?

is it also possible to take a graduate level math course during undergrad?

[fast ivy]

It is possible, in fact, I am doing this rn

But I guess it depends on your department.

[harsh fiber]

do you recommend it? or is it too difficult for the average math major?

[surreal sapphire]

i think a few grad classes are expected of those undergrads who want to go to grad school?

[fast ivy]

I don't really know, it depends a lot on your background and the courses that are being offered.

For instance

[harsh fiber]

Oh that is interesting, I'll be sure to ask my guidance counsellors. Though none of them really know much about math, or even physics. Most did bio and chem

[fast ivy]

I am mostly taking grad courses on topics I am somewhat already a bit familiar with.

[surreal sapphire]

also i dont think that a functoinal analysis class is (necessarily) a grad class

it used to be a standard 2nd year undergrad class at least in europe

[harsh fiber]

Ah, tyty, i appreciate the info

[surreal sapphire]

(and still is in many places)

[harsh fiber]

the thing is, that functional analysis at my university is in the 6000 level courses, so that means it graduate level. But I've never really heard of people doing functional analysis in 2nd year

that's crazy

[surreal sapphire]

americans smh

[harsh fiber]

canadians* but yea

It's really slow here

[fast ivy]

Functional Analysis is usually a PhD course in Brazil too

[surreal sapphire]

well

[fast ivy]

~brazilians smh~

[surreal sapphire]

if you want measure theory as a prereq you can do a better class

[harsh fiber]

I heard people in other countries doing real analysis in high school

[surreal sapphire]

and i guess thats quite prereq heavy then

with at least a year of real analysis and then a semester of measure theory or something

[fast ivy]

Yeah, at IMPA and the uni I am in measure theory is a prerequisite.

Measure theory is an undergrad class tho

[fast ivy]

Germans

[harsh fiber]

then I got a question about Physics, what jobs outside of being physicist, and outside of academia, is available for them?

[surreal sapphire]

i never heard of actual real analysis in highschool

except in specific schools

[harsh fiber]

should someone pursue, engineering physics, applied physics, or medical physics instead of regular physics?

I've heard that being said before

[surreal sapphire]

IT and banks/insurances hire physics grads

[harsh fiber]

oh yea i heard this before. I'm assuming its because the critical thinking skills and math skills developed by physics majors

[bright hill]

aleph is not a greek letter

[surreal sapphire]

yeah

[cyan goblet]

yeah ik aleph isnt greek

[surreal sapphire]

good math background and ability to learn new stuff fast

[cyan goblet]

i asked why we use aleph, a non-greek letter, as opposed to just another greek letter

[harsh fiber]

ah tyty, ill be sure to research the market related to them

[bright hill]

ah

[harsh fiber]

do you think engineering R&D hire physicsts?

or do they just hire Eng PHDs?

[surreal sapphire]

i know some people working for automobile industry in R&D

there are engineers of all fields but also physicists

[harsh fiber]

oh wow, interesting

do you think R&D in those industries pay well?

[surreal sapphire]

ye, better than academia

[harsh fiber]

I dont know anyone personally in r&d since im kinda new

[fair mural]

just wondering, why exactly would a physics graduate be of interest to a bank?

[harsh fiber]

I heard wages are like 17 dollars an hour for post doc

thats really low

[surreal sapphire]

they know math and how to learn things fast

[harsh fiber]

in USD ofc

[fast ivy]

They usually learn a ton of math that is useful for doing simulations and such.

[surreal sapphire]

i mean the specific skills arent that important

[harsh fiber]

makes sense

[surreal sapphire]

lots of banks and insurance companies really need people now

they are willing to train on the job, but more math background is better

and doing some kind of "hard" major that teaches you how to learn stuff fast

[harsh fiber]

ah makes sense

[limber thunder]

physicists are like, freakishly good at working with data and computer tools

as well as math modelling and numerics

at least that's the impression I get, many of my friends are physicists

so they have a skill set that's particularly well suited for these companies

[stark charm]

Happens in USA also

Rare albeit

[bright hill]

bruh

I thought the first chapters were decent

does it get worse?

[stark charm]

Have you heard of eigenchris on youtube.com?

[bright hill]

that's understandable coz it's not really a linear algebra text

[stark charm]

They give good video series for introduction to vector calculus if I remember. Or its differential geometry

[bright hill]

it's a calculus text that introduces what it needs of linear algebra

hubberd says it's calc until the appendix iirc

[stark charm]

What is tensor calculus?

Oh it just looks like operations on tensors

[harsh fiber]

I dont doubt it. Im in Canada, and it’s unheard of from my perspective

[stark charm]

including transport and contractions

[harsh fiber]

The highest I’ve heard of is those “IB programs”

[stark charm]

Oh yeah those also

[harsh fiber]

And idk what they do there

Cuz I’ve never done it

I heard the physics is hard in that program

[bright hill]

despite that, do you think it's a no no?

coz I wanna get back to it after rudin

[queen crown]

Hey

I'm feeling so demotivated rn I feel like I'm forgetting all the maths I know

[bright hill]

that's a relief

yeah, hubberd is obviously not perfect

but I don't think any text is

[neat lintel]

spivak's calculus on manifolds is perfect

[bright hill]

really?

I remember hearing some criticism concerning it

[clear parcel]

Exercise increases scores on everything

[neat lintel]

t

[harsh fiber]

I do have a non HW question, was substitution and elimination methods for math, originally from Linear Algebra?

[leaden torrent]

that kind of depends on what you mean by "linear algebra"

[harsh fiber]

Ummmmm

The introductory uni course

[leaden torrent]

we knew how to solve systems of linear equations before we had terms like "vector" and "matrix" and stuff

if that's what you mean

but historically "linear algebra" didn't resemble what we think of today

it wasnt until Grassmann really that it took its modern form

[teal charm]

Hello.

[harsh fiber]

Oh wow, interesting. That is fascinating. I’ll be sure to search up the name.

[leaden torrent]

in any case, techniques similar to substitution can be found in really old texts

[harsh fiber]

That makes sense, math has evolved quite a lot these past centuries

[leaden torrent]

in fact, something like substitution predates variables, not just linear algebra

diophantus outlines how to use what we'd call "substitution" to solve a certain diophantine equation

[harsh fiber]

Variables? Like the branch in mathematics (ex. Complex variables)?

[leaden torrent]

no, like the idea of representing unknowns with symbols.

[harsh fiber]

Ah

[harsh fiber]

That’s a new name, I’ll be sure to check it out as well

[leaden torrent]

arabic mathematicians (as in, from the Arabic golden age) as well had a very well-developed theory of solving linear equations

[harsh fiber]

Wow. That is a mouth full, impressive!

[leaden torrent]

but certainly nothing quite like "linear algebra" as we know it

[harsh fiber]

Mhm

My class starts in 20 mins

[leaden torrent]

all that said, it is certainly possible to think of "substitution" and "elimination" in terms of linear algebra

[harsh fiber]

Ill be on do not disturb in 10-15 mins

[leaden torrent]

gaussian elimination is basically just elimination, but written with matrices.

and it's how computers typically solve linear systems (with some optimizations)

[harsh fiber]

Oh yea I saw this on yt hoping to get into that soon.

Preferably this class or the next

Are rules in calculus, ex. The product rule considered a “mathematical operation” like Addition is?

[hushed turret]

differentiation is an operation

the product rule is an equation this operation satisfies

much like commutativity
a + b = b + a
is an equation addition satisfies

[brittle socket]

[harsh fiber]

Would integration be another operation?

[charred mortar]

Yeah

[harsh fiber]

Wait so we have Addition, Subtraction, Multiplication, Division, Exponents, Brackets, Differentiation And Integration, then what’s after Integration?

Does Guassian Elimination count?

And substitution?

Are there any more operations after this?

[next schooner]

Infinitely many

[harsh fiber]

Could you list a couple more? Maybe some in 3rd or 4th year university level?

[next schooner]

Sure

Operations on what, though?

On functions?

[harsh fiber]

Yes, that works

[next schooner]

So the most common other binary operations on functions are probably composition and convolution

I'm not personally aware of any 3-nary operations that see common use on "functions", per se

There's many defined in linear algebra which are applied to vectors, but their definitions extend nicely to functions also

[mild nebula]

This is a very broad topic. Basically everything in math is realized as an operator somewhere

[next schooner]

Inner/outer products

What they said.

I mean really an operator is just a function viewed in a particular way

So asking what operators there are is like asking what functions there are

[harsh fiber]

Im gonna search that up rn

[harsh fiber]

Oh wow, interesting

Is there any new operation that happens in Real Analysis?

[next schooner]

I mean

Your question doesn't really make sense

Thinking of things in this way is not necessarily going to be useful

Like, anything you "do" to any mathematical object can be viewed as an "operation"

[harsh fiber]

Actually that makes sense

I just checked Wikipedia on what an operator is

Now I see why my question doesn’t make sense

[next schooner]

That's not to say that operators aren't interesting. Indeed, operator theory is a whole thing. Perhaps functional analysis would be your jam!

[harsh fiber]

Operators are used to change “elements” in a given “space”

[next schooner]

Exactly!

So differentiation, for example, turns functions (elements in the space of functions) into other functions

[harsh fiber]

I really want to take it in 4th year. But it’s a 5000 or 6000 level course, which means it graduate not undergraduate. However some guys told me here it’s possible to do grad courses during undergrad depending on the circumstances.

[next schooner]

But there's uncountably many ways to turn functions into other functions

It's just that differentiation is quite a useful one with nice properties 🙂

[harsh fiber]

Hm, makes sense to me.

[next schooner]

Haha yeah you'd dig functional analysis it seems

Unfortunately I cannot advise on university-related stuff (assuming you're not in the UK, anyway)

Things in the states work very differently from over here

[harsh fiber]

It’s also pretty cool tbh, I liked the differentiation optimization problems during high school calculus, like the ones relating to area or finance. Like max volume or max profit. That was really cool, especially on a graph.

[next schooner]

Yup, calculus is magic

[harsh fiber]

Works differently in the states, UK and here in Canada haha.

[next schooner]

My research is applied mathematics, which is basically a huge train crash between functional, numerical, and stochastic analysis

[harsh fiber]

Ill havé to ask my advisors in grad courses

[next schooner]

So calculus is my bread and butter

I'd be surprised if you flat-out weren't allowed to take graduate courses at undergrad

[harsh fiber]

Oh yea, I’m doing a major in applied mathematics, today was my 2nd day.

[next schooner]

Certainly it was permitted when I was an undergrad

[next schooner]

Awesome!

[harsh fiber]

Yea I heard guys here saying it would be strange to be denied it.

It was nice meeting you but I have to sleep now

Ill keep functional analysis in my mind tho

Tyty for thé advice

[next schooner]

No problemo, any time

[harsh fiber]

yo guys, if the order is Pre-Calculus, Calculus I, Calculus II, Multivariate Calculus/Calculus III, then what is after Calc III/Multivariate Calc?

[leaden torrent]

"along the calculus track" the typical next courses are differential equations

ODEs then PDEs

but in practice theres a lot of branching that happens at that point

a student of mathematics would start analysis then (if they havent already)

a CS student would be more likely to take stats

etc

[olive kayak]

for me it was underaduate analysis/linear algebra

[sleek wing]

Lin alg after multi var

[deep mango]

Linear algebra should DEF come before ODEs here even though it's not calculussy

[neat lintel]

but all math is calculus

all math is same, calculus is math

[deep mango]

When I listed all of my math classes I determined that I had taken 32 calculus classes

or something like that

and like 10 non-calculus math classes

[hushed turret]

that sounds like a lot

[deep mango]

it was more than i was expecting

[torn willow]

I think I will be prepared for calc when I study it

[bright hill]

[fair mural]

me, having learned both mvc and differential equations before LA

[next schooner]

yeah it's not particularly uncommon to learn it that way around

[sleek wing]

[next schooner]

also the way round I learned things haha

[bright hill]

but

like

[bright hill]

why?

linear algebra isn't even that bad

[next schooner]

sure, but it's not strictly required for learning the techniques of mvc

it's how curricula are generally laid out in the UK, in fact

not in an undergraduate math course

but in engineering etc

indeed, I have engineering friends who never did LA at all

[bright hill]

[torn willow]

What do those people do for a living

[bright hill]

how tf do you survive in STEM majors without LA?

[torn willow]

The only engineering that I can imagine doesn't involve la is computer engineering lmao

[torn willow]

Just do cs

[bright hill]

not all of cs tho

I imagine

[torn willow]

I mean there are cs programs which don't even touch la

I think

[next schooner]

depends what you consider LA I suppose

like, you don't need to know about rank-nullity to plug PDEs into ansys

[torn willow]

True

[sleek wing]

Once more the algebra gang are oppressed by the analysis crew

[toxic schooner]

[deep mango]

[harsh fiber]

Ok, I’ve never heard this before. Thanks for telling me. I’m assuming courses like real analysis, complex analysis, etc.

[harsh fiber]

WOAH! I thought it was mandatory for ALL accredited engineering programs

[next schooner]

Again, depends what you consider to be like, pure LA material

[fair estuary]

Is there a formula to turn an infinite sum into an infinite product?

I want this formula as an infinite product

[bright hill]

I very much doubt there is a single formula to turn every sum to a product

but I think some sums can be turned to products

[fair estuary]

Is there one for the formula I posted?

[bright hill]

i don't know

that's a hard question

[brave hollow]

exponentiate

[bright hill]

lmao

but then it's not really a sum-turned-product

it's just a product-that's-connected-to-a-sum

[fair estuary]

@deep mango bro, do you have any ideas?

[grizzled grove]

Use exponential function

[neat lintel]

W server

[fair estuary]

Can you give me an example?

[tender tulip]

Dirty

What was your last math class

[fair estuary]

I think I took a Calc 2 at ucsd as my last one

I don't remember it though

[neat lintel]

btw are you a CS undergrad, Dirty ?

[fair estuary]

I graduated a long time ago. Health science.

[neat lintel]

ahh

[pine finch]

what would be an appropriate channel to post optimization questions using matlab

[earnest pike]

i'm trying to think of how to ask a question

but can't describe what i want

[pine finch]

like im looking at the early university and advanced and dont see anything for optimization

[earnest pike]

matlab should be as optimized as you can go already

[pine finch]

umm

its not so simple

[earnest pike]

i'm a coder my dude

if you did your matlab correctly there shouldn't be a whole lot of logic code meaning you're just running calcs

change the formula you're using

[pine finch]

optimization isnt so simple

there's no just 1 simple formula for these things

[earnest pike]

you mean increasing processing speed right?

or do you mean something else

[pine finch]

nooooo

[earnest pike]

oh.

[pine finch]

mathematical optimization

[earnest pike]

in the programming world when you say optimize that is usually what that means

that's not specific enough for anybody to be able to help you

[pine finch]

so stuff like this

but this is like the basics

[earnest pike]

like you want to simpligy an algo?

for display?

ooor?

[pine finch]

my group and I are working on problem 1 rn

[earnest pike]

and?

[pine finch]

well we have 2 different codes rn and both give separate answers

[earnest pike]

for the same inputs?

[pine finch]

yes

and neither look remotely correct

[earnest pike]

then either one is wrong or both are

[pine finch]

we are under the impression both are

[earnest pike]

then you can assume both are

wait you want me to teach you how to program?

oh no that's a rabbit hole i'm not going down rn

[pine finch]

no we get the programming we are just mostly debugging and want to make sure we have the theory bit correct

[earnest pike]

and what's your theory?

[pine finch]

sorry let me rephrase

not theory but method

[earnest pike]

method == function?

or series of events leading up to an answer

[pine finch]

this is what I came up with. basically you take the maximum value of bi - b*ai (so a scalar - a vector) then set the inequality for t and minimize t. minimizing t here would basically mean it's = to f1

[pine finch]

this

my code doesn't even give u any x* values

(optimal x)

[earnest pike]

and it gives you the wrong answer

[pine finch]

yeah

[earnest pike]

honestly if i had a formula that complex i wouldn't bother writing it myself

i'd use a library

[pine finch]

yeah we have a library for that

[earnest pike]

then use it

[pine finch]

I am

that's part of the cvx library

[earnest pike]

does that function give you a wrong answer?

[pine finch]

well it might not even be the right function to use which is what we are starting to think

that's where we need the help with

[earnest pike]

then do the trial and error method

just use as many cvx functions as you can until it gives you an appropriate result

[pine finch]

yeah but I care less about the answer more about the why

cuz if I don't know the why ill fail my exam in 6 weeks

[earnest pike]

sorry dude this is really nowhere near my forteih

i'm not a methematician i'm a programmer

[green scroll]

yeah no one asked you

[earnest pike]

i mean if you use the debugger tool and already know what each val should be at each step you can just attribute it to that specific line number where it breaks apart

[ancient flame]

I've come across this situation occasionally (but I cant really remember any specific scenarios atm): basically if you're trying to determine whether something is equal to, less than, or greater than another thing, how do you represent it notationally?

like ig let's say the two 'things' are polynomials P(x) and Q(x) and we need to do some work to determine that

I cant put an equal sign since idk if they're equal nor can I put a > or < so what would I do

[next blade]

why are you solving for t when it's a given value? why are you summing over 1:12 and not using matrices like you're told? where are your subject to constraints to CVX?

[pine finch]

its not given

the question is find the optimal value for t and optimal x

[next blade]

oh i see for the second part this is indeed the case - so i take it you already wrote something for the bit before "extend your script"?

[pine finch]

yeah the stuff in problem 0

but that's given

its just copy and paste to get your matrices

and the 1:12 is a range from 1 to 12

since the f1 is i from range 1 to m (m is 12 and is the number of rows for the matrices)

but yeah I think the issue might be that it's taking the sum of all those values for the different f1

[next blade]

i don't see the script in problem 0 deciding if a solution to the inequality for a given t exists? or are we talking about different things?

[pine finch]

oooh shoot ur right

also we got a message from our professor to not use the minimize function and code it ourselves so I'm gonna go to sleep and attempt this in the morning

[pine finch]

but if we are able to find an approximate lowest value for t that means it exists

and that's less coding xd

but also probably less marks

[next blade]

well really you need two things: a check to see if a feasible solution exists (and i've forgotten all of my linear programming, but that should be relatively simple)

[pine finch]

but thank you for helping me realize this I need to fix that

[next blade]

and then you'll just e.g. bisect t down to the smallest t where the problem is feasible

and that's that

[pine finch]

yeah

that would basically be where t is = to f1 no?

[pine finch]

that's just farkas lemma.

I think

yeah im pretty sure it is

[harsh fiber]

guys, I just want to say

I really appreciate everyones help

I just started uni, and I'm understanding the pre calculus work that i forgot

especially unit circle stuff

I think I'm off to a good start

[mint patio]

Glad to hear that

[harsh fiber]

tyty

[mint patio]

Keep it up friend

[harsh fiber]

i definitely will

tyty

[covert nest]

can a mod give me helper role if i am asking in the wrong place dm me where to ask

[mint patio]

Yeah they can just give em a second 🙂 if you don't have it in a bit you can ping an active mod

[deep mango]

I'll give you the role if you're okay with receiving helper pings.

[sick burrow]

you get like an obscene number of them

I remember I was given the help role against my will a while back for saying that dogs are better than cats and I got like 40 pings a day

[cyan goblet]

that's a genius punishment actually

[sacred arch]

The server icon is very good because torus

[fair estuary]

Am I allowed to ping helpers in room 21 if my helpers don't know how to do it?

It's been 4.5 hours so far :(

[surreal kindle]

#rules ^^

Also, what timezone are you in?

In Europe its morning on a work day

so there is a high % you wont get help atm

[bright hill]

only 40?

lmao

[toxic schooner]

i remember i had days with well over 100 pings

[neat lintel]

Hi can anyone help me on help 6 pls, its been almost an hour and i still cant get help

[mint patio]

wtf happened to probability

class was so easy and now it makes no sense

[deep mango]

screams

[olive kayak]

for introductory course

hard part is counting

[deep mango]

probability is hard but so fun

[olive kayak]

for analytic course

well...good luck

[deep mango]

i had to answer so many questions about martingales

[bright hill]

I should do probability

[deep mango]

they're so good

i love martingales

[olive kayak]

btw what even is martingale

isn't it that gambling strategy

[deep mango]

gambling strategy is kind of a way to phrase it

[mint patio]

It is an introductory course, idk why I can't understand it

[olive kayak]

my friend explained in the simple way that it is a fair game

super/sub martingales are unfair ones

[mint patio]

Just basic cringe engineering probability and statistics

[bright hill]

the only probability we learned this year in HS was like, "how many numbers can you make with 7 different digits"

which isn't even probability

[olive kayak]

btw do you study stochastic differential equations?

[sleek wing]

SDEs my beloved

[deep mango]

a martingale is a random process where knowing information from the past doesn't help you to make decisions for the future. this is kind of a loose definition, but it can be written formally using conditional expectations. the amount you gain at any step, given full knowledge of what happened before, should be on average 0.

[olive kayak]

wait

i've learnt similar concept in game theory

maybe I'm just confusing those two

[deep mango]

so that's kind of why it describes a fair betting strategy: you are very very rarely going to lose all your money because if the expected outcome is 0, then eventually you'll regain everything you have spent.

(and hopefully you'll randomly rise above 0 at some point - then you can cash out)

[deep mango]

i am studying them right now in a class

but i'm hoping to learn a lot more!

[olive kayak]

oh

i've heard those are nightmarishy hard

I wish you good luck

[deep mango]

They're very fun!

I really like PDEs and the interactions with PDEs are super cool

[deep mango]

a simple example of a martingale is a random walk: keep a tally X(n), with X(0) = 0, and at every step decide randomly whether to add 1 to X(n) or to subtract 1 from X(n) with probability 1/2. This new number is X(n+1).

The point is that at every step, the average change is 0: you either add or subtract 1 with equal chance.

But martingale theory lets you take this simple walk and answer complicated questions about it: for example, if a, b > 0, what's the probability that X(n) hits a before it hits -b?

[cyan goblet]

this sounds like some rigorous definition of mutually exclusive

[deep mango]

This is hard to calculate without proving facts about martingales. But once you've proven some facts, it becomes easy.

[olive kayak]

hmm sounds interesting

[deep mango]

In this case, there's a theorem called the "optional stopping theorem" that says that the expectation of a martingale stopped at a random time (under certain conditions) is the same as E X(0). (the rules of martingales tell us that E X(n) = E X(0) for all n, this is like a version of that where n is allowed to depend on a random event)

For example, one such random time is "the least n so that either X(n) = a or X(n) = -b"

Well, what's E X(this time)? either X(n) is a, and that happens with probability P(X(n) hits a first). Or X(n) is -b, and that happens with probability P(X(n) hits -b first) = 1 - P(X(n) hits a first).

Then 0 = E X(0) = E X(this time) = a P(X(n) hits a first) + b(1 - P(X(n) hits a first)). And we can solve for the probability.

We find that P(X(n) hits a first) = b/(a + b). If b is bigger, it's way more likely for X(n) to hit a first. if b is small and a is big, it's more likely for X(n) to hit b first. If they're equal, it's 50/50.

A theorem about martingales helped us to get a really clean answer for a problem that would have a very very messy solution had we tried to use combinatorics to count the number of paths hitting a before -b.

And that theorem isn't super hard to prove, either

[olive kayak]

don't they need graduate level of real analysis>

?

[sleek wing]

I saw a very similar result in my finanical maths class and I don't think you need real analysis to prove it

[deep mango]

knowing measure theory is helpful for doing a lot of this but you definitely don't need measure theory to prove the optional stopping theorem!

you can honestly just kinda... write it down and it works

the thing you do need to do is check that the probability that X(n) never hits a OR -b is 0

that takes a tiny bit of work

[sick burrow]

Can someone explain how simply-connected closed 3-manifold equal ONE THRIS SPHERE

[fast ivy]

[fair mural]

one what sphere?

[split glade]

This sounds a lot like stuff from my Markov Chains class last year, are these a special case of markov chains or vice versa?

[deep mango]

They're closely related to markov chains, but not exactly the same thing.

Or like

There's no inclusion either way

[errant ridge]

lol that's the poincaré conjecture

[sick burrow]

yes...

that was the joke

[errant ridge]

oh nvm am stoopid

[sick burrow]

it's a reference to this post

[bright hill]

so this random just asked a question and became a meme?

[velvet dagger]

DarQ tryna find out how rando question equal ONE MEM

But yeah it's just the phrasing was so funny

[sleek wing]

LOLLL

[empty stratus]

I can now say that I have officially improved the American Mathematical Society through my membership in it.

[velvet dagger]

Dope

[empty stratus]

Apparently, none of the AMS' discord servers had TeXit.

Thanks to me letting them know about it, now they do.

[bright hill]

based

[empty stratus]

😄

[harsh fiber]

man i'm doing linear algebra, stuff is really enjoyable

[clear parcel]

yeah it’s pretty interesting, rn though my teacher is throwing a bunch of different topics at us, hard to see how it’s all connected

[fast ivy]

[tender tulip]

AP Physics 1 algebra based exam being bell

[mellow estuary]

FROM?

[vivid halo]

did you give your talk?

[empty stratus]

Saturday.

[vivid halo]

ah

[empty stratus]

I let them know about this via e-mail.

And they have been profuse in their thanks. 😎

[vivid halo]

yes they tend to do that

good luck!

[mellow estuary]

yoooo how?

[empty stratus]

Apparently, none of the AMS' discord servers had TeXit.
Thanks to me letting them know about it, now they do.

[vivid halo]

oh nice!

I didn't know they had discords

[mortal igloo]

Me neither

[mellow estuary]

you great at math?

[mortal igloo]

Should I be in the AMS discord?

[empty stratus]

I'm better than some people, but not better than everyone.

The discord is for the particular conference I'm attending.

[mortal igloo]

Oh

[empty stratus]

Which is a virtual conference.

[mellow estuary]

accurate answer digestable

[gilded hare]

@knotty willow @long parcel sorry for the ping, but I think I figured out what was going wrong (with the sound example). I forgot to adjust the speed of sound relative to the person moving. Once I did that, everything works out.

[ancient flame]

took the ap physics exam today

it was

interesting

I think I got a 3

[gilded hare]

im too scared to take the ap physics rn

i took AP calc BC and stats, might do physics in grade 11 or 12

[bright hill]

rip

[ancient flame]

YOU TAKE AP CALC BC IN 10TH????

[bright hill]

is that gud?

[gilded hare]

yea

[ancient flame]

that's like

insane

im a senior taking ap calc ab

a lot of people don't even get to take calc in high school

[gilded hare]

im from canada so it was exam only, thats why im scared for physics

[ancient flame]

oh

no course??

yeah don't take ap physics

[gilded hare]

yea i self taught

[ancient flame]

even with the course it is hard

[bright hill]

I know a 10th grader that's doing rudin rn

[ancient flame]

43% pass rate!

[ancient flame]

WTF

[bright hill]

lmao

[ancient flame]

these kids

[gilded hare]

bro wtf

[bright hill]

lmao

[gilded hare]

rudins actually pain

[bright hill]

actually

I disagree

the only pain I've seen is from chapter 2

chapter 2 is literally

[gilded hare]

like actual rudin

or is he doing baby rudin?

[bright hill]

baby rudin is actual rudin

[gilded hare]

it is

bruh

oh yea i have a question for a book recommendation

[bright hill]

[gilded hare]

should i do shilov lin algebra

cause my teacher said i should, but my other teacher said no

[bright hill]

there are a lot of good books on lin alg

if I were you I'd do artin (but that's because I also want to do abstract algebra soon)

gimme a sec

[gilded hare]

oooo ive done some group theory which i liked

[bright hill]

you should read this #book-recommendations message

but if you want to do multivariable calc, ignore the above and do hubberd and hubberd

[gilded hare]

well, for multivariable i did Munkres

[bright hill]

does that mean...

you did calc 3 before lin alg?

[deep mango]

Plenty of people do that

Anyway

[bright hill]

ye

[gilded hare]

yea

[bright hill]

but that's shtupid

[deep mango]

Linear Algebra Done Wrong is the best book

[bright hill]

you shouldn't do that

[gilded hare]

i didnt have access to anything that wasnt given to me from my teacher

cause books are expensive

[deep mango]

And the PDF is free online

[hollow sundial]

the price of textbooks are too damn high

https://giphy.com/gifs/rent-sdlih3BPUik1y

[deep mango]

https://www.math.brown.edu/streil/papers/LADW/LADW.html

[bright hill]

the secret ingredient is ||crime||

[deep mango]

(the name is a joke about the book Linear Algebra Done Right, which is popular and a bit weird)

[gilded hare]

lmao legend

[deep mango]

You could also use MIT OpenCourseWare. They have a series of lecture videos on linear algebra by Gilbert Strang. These come with problems and exams.

They're excellent but they aren't quite as proofy as the linear algebra books we're recommending.

[gilded hare]

yea i watched some of those, i just wanted something more concrete

[bright hill]

ryc, what did you use for intro anal?

[deep mango]

My class used Rudin and Pugh simultaneously

But mostly rudin

I like Pugh better for reading purposes

[hollow sundial]

Analysis really deserves two books. Rudin for main and a simpler one with examples for supplementary

maybe this one https://link.springer.com/book/10.1007/978-1-4614-6271-2

[bright hill]

neat

also, are the harder questions left last?

[deep mango]

In what

[bright hill]

rudin

[deep mango]

Yeah I think so

[bright hill]

hmmm

I should review chapter 2's exercises then lel

[sick burrow]

I wonder what percentage of Americans could do 2/3 + 1/4

[deep mango]

I don't even know if I can do that

[sick burrow]

same tbh

what even is a number

[bronze pelican]

I ate a chocolate croissant when I didnt need to

[deep mango]

[sick burrow]

skill issue

[deep mango]

I want a chocolate croissant!

Jealous

[sick burrow]

ryc I feel like we have a similar appreciation for food

[deep mango]

I don't know what to have for dinner

[sick burrow]

food is honestly so good

[deep mango]

I have lots of opinions on food

Ready to get mad Gamma

Last night I had cacio e pepe for the first time

From a fancy restaurant

[long epoch]

Just add the numbers in the numerator and add the numbers in the denominators ez

[deep mango]

Someone else was paying so I decided to take the risk and go for the fancy cheese pasta

[bronze pelican]

Ryc u should take me to a fancy restaurant

[sick burrow]

like it's crazy to run into people who don't seem to understand that eating good food is one of the best things about being alive

[deep mango]

What I got was very very very good fancy cheese pasta. But I was underwhelmed.

[bronze pelican]

◉◡◉

[sick burrow]

don't even know what that is

[deep mango]

[deep mango]

The implication being that I will pay for our meal?

[bronze pelican]

No

[deep mango]

Oh ok

Then sure we could go to a fancy restaurant

[sick burrow]

oh it's just fancy cheese pasta

[deep mango]

But I don't know what exactly

[deep mango]

Yeah!

[sick burrow]

see the about pasta is

[bronze pelican]

I just never go to fancy restaurants

[deep mango]

I don't really go to fancy restaurants much either

[long epoch]

Should have said maybe so could have possibly got a free meal too 😭

[deep mango]

This was just an event

[sick burrow]

good spaghetti and meatballs is so hard to beat

[bronze pelican]

Ah ok

[deep mango]

Ikr like

Why didn't I just get the lasagna

[bronze pelican]

What event

[sick burrow]

fancy restaurants are overrated

[deep mango]

Or some really good pizza

[ancient flame]

I love lasaña

[bronze pelican]

I gotta try once

[sick burrow]

like there are so many fantastic restaurants near where I live

it's honestly absurd

[deep mango]

It was an event with my apartment building, the grad students have an RA who holds events sometimes

[bronze pelican]

Ah cool

[sick burrow]

and you can get a meal at any one of them for 10-20 dollars

[deep mango]

Fancy restaurants are sort of overrated but

[sick burrow]

a really fucking good meal

[deep mango]

There are also some really good ones

[sick burrow]

yeah

[deep mango]

That have stuff that you can't really get elsewhere

[sick burrow]

there's a couple great Indian places near where I live

[cyan goblet]

just because they have a star from a tire company isnt gonna get me to waste my tires going there

[ancient flame]

2 exams down ✅ one left to go

[sick burrow]

and those are on the pricy side

but they're really good

[cyan goblet]

i have uhhh

two final exams

left

[deep mango]

I was very mad the other day because my favorite thai place raised the price of their lunch special by 2 dollars

[ancient flame]

but idk how I feel about the ap physics exam

[deep mango]

I also have 2 final exams left

Well

They're both not exams

One is a writeup on hodge theory

[modest rune]

I'm going to start adding new challenge problems to the server that are just lemmas I can't figure out

[cyan goblet]

multi calc and physics e&m

[ancient flame]

I don't have any finals bc im a senior!

[deep mango]

Hello Max

[modest rune]

can we make the bot give back certain roles that it doesnt

[sick burrow]

what's funny though is they're not any better than the 10-20 dollar meal restaurants, just different

[deep mango]

the other is a talk on a nonlinear functional analysis inequality

I don't know

Maybe

[modest rune]

being able to leave and rejoin the server at will would be nice without bothering people

[deep mango]

I thought that was like a

[modest rune]

it preserves honorable for example

[deep mango]

if people get muted then they have to ask back for roles thing

[modest rune]

but not advanced

[deep mango]

or kicked

[sick burrow]

linearly independent but same magnitude one could say

[deep mango]

i agree with that in most cases

[sick burrow]

I went to a fancier restaurant a while back

[modest rune]

ive had some dope expensive indian food

[sick burrow]

got an insanely good fried chicken and gravy dish

but it was only like 18 dollars

[deep mango]

Nice

[bright hill]

[sick burrow]

just give me mod

[bright hill]

I would love to code smth for the server

but alas, finals are creeping up

[sick burrow]

and have Max ping me whenever he rejoins the server

this is a flawless plan with absolutely no downsides

actually being a mod on this server seems big annoying

[bright hill]

lel

[sick burrow]

like reserve moderator seems great since you get all the power of mod but without the responsibility, and honorable seems great since you get all the clout of reserve moderator but with none of the power, but actual mod seems afwul

I'd rather not get pinged 50 times a day thank you very much

[bright hill]

if no one volunteers by june 20th I promise I'll do it

(or at least try )

[sick burrow]

anyways time to finish this essay

[deep mango]

the infrastructure for this is already implemented

the roles just need to be added

[fervent pebble]

so we're actually doing proofs in math lmfao

which is pretty cool

also bruhhh white name

im so sad

:((

this is a tragedy

i didnt even get emeritus

[deep mango]

Deserved big time

[fervent pebble]

facts i defo deserve emeritus

that's y u should give it to me

ryc what if i told you you are my fav hawk

[bright hill]

ryc be like:

[deep mango]

I'm not a hawk. I'm a pigeon.

[fervent pebble]

wait

hol up

when did this happen

#chill message

liar

[deep mango]

Well done

You called my bluff

[cyan goblet]

i thought you were a lobster

[deep mango]

But if you really cared about me then you would have said "no ryc, you're not a pigeon. dont be silly."

And that's all I needed to know.

[deep mango]

I'm not a lobster!!

Lobsters dont even like

[fervent pebble]

oiiii i didn't want to be disrespectful just in case you had suddenly decided to become a pigeon

[deep mango]

Think

I bet

[fervent pebble]

the fact that i thought abt ur feelings shows i really cared about you

[deep mango]

Lmao

Yes I suppose it did

Anyway I dont have control over emeritus

[fervent pebble]

BRUH

scam

fucking scammed

[cyan goblet]

well ricey

[fervent pebble]

by a pigeon

[sleek wing]

Ryc u r a pigeon but ur my favourite pigeon

[fervent pebble]

a mfin pigeon

[cyan goblet]

aka mr jordan peterson fan

[vivid halo]

every time I see ryc's lobster emoji I almost mistake him for a jbp stan

[deep mango]

[cyan goblet]

https://theconversation.com/psychologist-jordan-peterson-says-lobsters-help-to-explain-why-human-hierarchies-exist-do-they-90489

[deep mango]

Ok ok

[cyan goblet]

boom

[deep mango]

Ill change it

Fuck

[cyan goblet]

your idol, your god, your icon has proven you wrong

[bronze pelican]

Ryc, you are a pigeon. You are pidgeonYC

[vivid halo]

being associated with jbp is the final straw and what gets him to change it

[deep mango]

@supple flame fuck you for making me out to be a lobster

[fervent pebble]

crabs are better than lobsters anyways tbh

[sleek wing]

Lobster boyyyyyy lobster booyyyyyy

[vivid halo]

convergent evolution towards crab

[cyan goblet]

we share common ancestor with lobsters right?

[fervent pebble]

yes

[vivid halo]

not really

[fervent pebble]

but also technically we share common ancestors w/ everything

so

[vivid halo]

yeah

the only point that jbp is making in the lobster thing is like

lobsters have the same kind of serotonin signaling that we have

[cyan goblet]

oh i didnt even read it lol i just saw that it was by jbp and about lobsters

[fervent pebble]

lmfao

[sleek wing]

Lobster.... Lobby... lobbying the government... think about it

[fervent pebble]

wew supporting jbp

canceled??

[sleek wing]

Mess with the lobster get the claws

[cyan goblet]

oh shit

that's a good line

[sleek wing]

https://tenor.com/view/memes-meme-random-captions-caption-gif-23557955

These things are probably lobsters

[fervent pebble]

pursuit of happiness by kid cudi is so good

[bronze pelican]

Whata the fu k is this

[cyan goblet]

lobsters

probably

[deep mango]

I don't like them

[stark charm]

We all hopefully share common ancestors

It would be weird to think life didnt start from single group of organisms

[stark charm]

Crawfish

Crayfish*

I meant crawdaddies

[bronze pelican]

Wew lobsters tbh

[stark charm]

You know what I find interesting

People thinking issac newton was a virgin?

Its one of the most useless facts. I think he just had a lot of haters

[near fox]

Newton was a very private and introverted individual and was not that comfortable in a lot of social settings.

[stark charm]

Im calling cap

[leaden torrent]

he was also almost definitely a virgin

[stark charm]

Its possible that these are his personality traits

But really

[leaden torrent]

some historians suspect he was asexual but in practice it's easier to just say "he's an asshole" and move on

because he was definitely an asshole

dude abused the shit out of his position in the academy

[stark charm]

Imagine 200+ years from now people are writing history about you. You most definitely will have haters

[leaden torrent]

stomped on some of his personal enemies' careers

for really petty reasons

like "once wrote a letter to leibniz"

[stark charm]

Maybe this is where virgin meme started?

idk how common kissing and telling was back then

[leaden torrent]

premarital sex wasn't illegal but was considered very scandalous back then

an upper class man like newton would certainly not publicly indulge

[stark charm]

Im skeptical

[leaden torrent]

privately, perhaps but we have no reason to think he'd be interested in that

[stark charm]

?

Unless he stated explicitly his sexuality?

I think it would be safe to assume he is heterosexual

[leaden torrent]

"sexuality" as we understand it was like... not a thing back then

also i dont see why anything i said assumes he's heterosexual?

or not?

i dont see why that came up

[stark charm]

Its a counterpoint to having interest in sex

[leaden torrent]

??

[light needle]

Gays don’t have sex

[leaden torrent]

i think he's implying straights don't have sex

actually

im not sure

[stark charm]

So gay sex was more acceptable back then?

I assumed it wasn’t

[leaden torrent]

no?

i never said that

but being gay was like

burned-at-the-stake tier crime

why did this even come up

i never mentioned sexuality besides mentioning off-hand that he could've been asexual

[stark charm]

Thats what I thought

[stark charm]

The reason I bring it up is because you said

So privately he would like you said

[leaden torrent]

i was referring to premarital sex with that message

[stark charm]

Oh

[leaden torrent]

my point was that, if he had sex, he certainly would not admit it

[stark charm]

That makes sense

[leaden torrent]

it would be very scandalous

since he was unmarried

[stark charm]

Oh okay. That was my misunderstanding thats why I brought up sexuality

[leaden torrent]

that said, we have no salacious love letters or anything in the historical archive, and given his behaviour, it's pretty unlikely he visited the local brothel or what-have-you

so it's safest to assume he just genuinely never had sex

whether it be because he wasn't interested in it, or no one was interested in him

[stark charm]

I just find it suspicious

There are lots of people in history

Why is the fact that Issac Newton is a virgin so publicized. Im assuming its just haters

[leaden torrent]

i think because it's funny.

[stark charm]

Also he lived pretty long too

So thats why I am doubtful. Asexual is a possible answer and it would be interesting if that was the case

But also think about it from a personal level

If you became famous what proof do others have of you having sex in the future?

[leaden torrent]

fwiw this fact is relevant historically

[stark charm]

Me personally have zero written proof. And I know if I had haters they would write about that

[leaden torrent]

mathematical dynasties were very common back then

[stark charm]

Dynasties?

[leaden torrent]

given newton's corruption, he couldve easily given his kids a leg up

yet he had no children

so such a thing never emerged

contrast with e.g. the gauss family, which had dozens of mathematicians across the generations

[but one very famous one]

[stark charm]

I didnt know Gauss fucked

Oh bruh there are a lot of Gausses

[stark charm]

I see this as a really good point

But he was also an asshole so maybe this extends to progeny

Or did women have a good reason for accusing/claiming sex out of marriage?

[primal seal]

This all goes without mentioning: a relationship with a woman would undoubtedly take away time he almost certainly would’ve preferred spending doing research

Darwin is noted to have contemplated this trade-off as well

[charred mortar]

From what I’ve read of Newton’s life it seems like that may sort of extend to friendships in general

Did he have anyone in his life that he didn’t treat poorly?

[torn willow]

Well we will never know

https://www.newtonproject.ox.ac.uk/his-personal-life

His childhood was tragic. His dad died before his birth and his mother left him when he was 3

It's literally Doofenshmirtz's backstory

[stark charm]

Anyone want to watch WNBA with me?

[near fox]

Which teams do you follow

[stark charm]

Seattle Storm

[long parcel]

ohhhh it makes sense now

[frigid lark]

Johan Sebastian Bach had 20 children and was still a very productive composer 😂... personally for me the best composer.

[errant ridge]

there was a mathematician who was a murder

he did really good research in complex analysis

after he killed his family and while he was in a mental asylum

not kidding

https://en.wikipedia.org/wiki/André_Bloch_(mathematician)

andre bloch 💀

[bright hill]

I mean

it is

but jeez yohan

lmfao

[arctic grove]

[ivory mason]

Hello

I want to learn more about probability theory and stochastic process, but I'm a little bit lost, like where should I start after finishing the basic of statistics, linear regression

Do you have some book to recommend ?

[deep mango]

You cant expect me to remember things...

In a month or so I will forget again

I barely even know who Jordan Peterson is

Lol

Yes

Anyway this is all the stuff I knew

[arctic grove]

[ivory mason]

[olive kayak]

god self studying textbooks with no solutions are annoying

like it seems to have good review but nobody seems to post their solutions or use it

[brittle socket]

Who's that

Oh btw hi Chalk Bird

[vast surge]

I'd contend that Bernie Sanders might be centre-left, if we want to be particularly generous.

[odd narwhal]

You should start with a measure theoretic probability book if you wanna get to stochastics. Klenke is quite fine imo

Iirc it teaches you the measure theory you need in the framework of probability theory

It might be a bit of a jump from statistics in terms of rigour

[torn willow]

Well I have done like 4 entire books without solutions now smh

Make that 5

[olive kayak]

I mean how'd you pull that off?

you need some ways to check your progress/skills at least

[torn willow]

I just assumed I was good enough

I guess olympiad experience helped

[charred mortar]

At some point you get good at knowing whether your answers are right or not

Before that though you’d ideally be able to check with a professor/teacher

[bright hill]

[olive kayak]

the problem is I kinda don't have access to them currently

maybe I should use a book with solutions alongside

[bright hill]

this server can help

[torn willow]

Well What's the problem you face exactly

Like for me I just checked if my solutions made sense

[bright hill]

having someone tell you if your solution is right or wrong and pointing out what mistakes you did is invariably more important then having the correct solution at hand

[torn willow]

Fair point

[olive kayak]

oh okay

maybe I should learn latex

[bright hill]

you should definitely learn latex

it'll just make your life SO much better as a mathematician

(or anyone that uses math notation for that matter)

[charred mortar]

You can always ask in this discord or on stackexchange

Alternatively get good at googling

[bright hill]

this server is the best option imo