book-recommendations

[limber tiger]

These are the texts assigned to us. Just looking if there is more you guys recommend

[devout sphinx]

I like Dummit and Foote but my personal favorites are Jacobson Basic Algebra and Algebra by Mclane

Dummit and Foote is amazing as a reference though

It can be adapted to all kinds of curriculum

[limber tiger]

Ah okay. Nicee

[vital bane]

wait wdym?

how

[median creek]

you may want to read this post @limber tiger

[limber tiger]

Ah okay. Didn't checked pinned messages. Apologies

[gray gazelle]

I would definitely recommend matej bresar: undergraduate algebra, its a beautiful book that tries to develop the group , ring etc. theory simultaneously, as far as possible. This way the key concepts are really made clear. This was the authors philosophy when writing it, he says the students understand material substantially better since he started teaching it this way.

[limber tiger]

Ahh okay. I'll take a look too

[thin hollow]

Putnam prep books

[timber mesa]

courses waste too much time on point-set tbh

and on top of that give a really unmotivated exposition of it

[gray gazelle]

Proofs/Real Analysis
Putnam & Beyond
IMO Compendium (Some crossover in these competitions, but the problem solving ability is what you need the most.)
Abstract Algebra (atleast up to group theory)
Eucilidean and Non-Euclidean Geomerty (I recommend Hartshorne.)

Edit:

1. Expanding on this I think AoPS’s free forums is a good idea to see some insight in problem-solving abilities and strategies being used to tackle Putnam-leve problems.
2. I do believe you will also need some discrete mathematics as some number theory and advanced topics as such can be found in the Putnam.
3. Overall, these are prep books as you’ve asked for, but the primary thing you should be doing after you’re somewhat confident is Doing Problems

Serge Lang’s “Algebra” or Dummit & Foote

[limber tiger]

Ahh okay. I think my friends recommended the grad text on lang s algebra

[blazing canopy]

Serge Lang's book is not suitable for introduction

[gray gazelle]

I would agree with this, but I ended up learning much more in 10 pages of Lang’s Algebra than I did in 50 pages of DF.

[blazing canopy]

Well Dummit and Foote spends a lot of time on lengthy examples and exercises, I don't think it's too bad. But even D&F I wouldn't say is a good intro book

[cursive orbit]

it also takes more time and effort to read through 10 pages of Lang than 50 of DF

[gray gazelle]

I do think this makes it more managable: https://math.berkeley.edu/~gbergman/.C.to.L/.

After sufficiently checking and attempting a difficult problem, you can also refer to this for the first six chapters, the solutions don’t have too many errata: https://sites.google.com/view/kellervandebogert/notesresources/lang-algebra

[solemn rover]

to be clear are you discussing "Geometry: Euclid and Beyond" by Hartshorne?

[gray gazelle]

Yes. Apologies.

[solemn rover]

Ok, understood. That was not the first book that came to mind when I saw "Hartshorne" and it took me a minute to put it together.

[lapis sundial]

apply schemes to plane geometry

[hearty steppe]

Physical phenomena are all modeled as dynamical systems

Learn dynamical systems then maybe you have another way of mastering physics at the best of your capacity

And it’s an honest angle to approach physics because everything in classical mechanics and quantum mechanics can be broken down into state spaces

[gray gazelle]

It would be quite dismaying to touch on one of the hardest mathematical textbooks for one of the hardest subjects when you’re simply trying to prepare for a friendly math competition 🤣.

[hearty steppe]

Also everything can be analyzed pretty much* by using a lebesgue measure. Now let’s do some fancy stuff with strange attractors and Ergodic behavior, I think this is where the light is aiming in the bottomless tunnel

If a whole system is an amalgamation of persisting state spaces, we know it must be possible that if we can take a lebesgue measure between all possible paths of state space points, then all possible paths of persisting behavior will be visited based on time averaging converging behavior. This behavior is what we call Ergodic.

It breaks my brain everytime I try to think about it

Ergodic is not random but I fancy the idea of random behavior evolving into Ergodic behavior if the state space paths allow convergence to occur at some instance

The trippy part is this does not necessarily mean that all possible outcomes are going to eventually happen that belong to some sort of state space associated with a distributed weight of probabilities (or literally just our sample space be it continuous or discrete)

At least I have not seen where this is proven, given many people may misinterpret what Ergodic behavior means.

Sorry I’m not a mathematician, I just like math. Unless you guys deem me an honorary mathematician, then I shall call myself a mathematician 😂

Now we can take everything I just mentioned and make free body diagrams in quantum and classical world. Alright now we can have fun

Oh btw this was my way of fitting dynamical systems into the picture of physics using my angle

[grave thorn]

serge lang as introduction lmao

[grave thorn]

wdym everything reduces to a lebesgue measure?

i can think of a lot of things that dont come down to a lebesgue measure

[grave thorn]

10 pages of lang is just arrows

arrows everywhere

[gray gazelle]

holds the same energy

[hearty steppe]

I’m thinking in terms of physical phenomenal and statistical representations. Maybe there are some very specific things I’m not considering

Otherwise non-measurable sets seem extremely specific and abstracted away from reality into math land

https://math.stackexchange.com/questions/2419384/is-there-any-example-of-a-non-measurable-set-whose-proof-of-existence-doesnt-ap

Yea seems like I think I’m on the ball here. This stuff gets pretty abstract but I mean ok fine not everything is lebesgue measurable

[grave thorn]

for instance lots of QFT reduces to studying Gaussian measure on infinite dimensional hilbert spaces (on which no lebesgue measure exists)

or other examples

like dirac measures

[hearty steppe]

I actually like the example in this Wikipedia article, now this makes sense https://en.m.wikipedia.org/wiki/Non-measurable_set

[grave thorn]

im not referring to measurability

im referring to the statement "everythinng reduces to a lebesgue measure"

which may not always exist

and in fact ergodic theory is abstract

and has nothing to do w/ lebesgue measures

[hearty steppe]

Oh I haven’t really got to QFT yet I missed this remark. Interesting

Correct Ergodic theory does not rely on lebesgue measures but a lot of Ergodic behavior is lebesgue measurable. Well you know what you got me thinking perhaps this is irrelevant

[grave thorn]

in fact the ergodic theorem is really a generalization of the law of large numbers, which is itself based on a space which you want to be abstract

[hearty steppe]

Cuz that whole remark on QFT is not something I considered, mainly cuz I’m not at QFT yet and i might need another 6 months or more to get there

I haven’t had time to progress in my physics texts recently

[grave thorn]

I dont do QFT/have ever taken a course but its pretty well known that gaussian measure theory is how you formulate a lot of the problems there

[hearty steppe]

Ahh ok

[grave thorn]

(ofc if you use a physics text they will abuse the gaussian measure)

[hearty steppe]

Physics texts always abuse math

😂

[grave thorn]

for some reason a lot of the texts start by saying "lebesgue measure on C0([0,1])" lmao

[hearty steppe]

Yea and it makes you take measures too literally as being pretty much lebesgue based but I kinda had a funny feeling that QFT land is not going to play nicely with this.

Also negative probability distributions are not always that simple to deal with are they? Even in QFT land? There goes our baby lebesgue out with the bath water

[grave thorn]

(a) lebesgue measure is not a probability measure on R, (b) negative has no "classical" interpretation (iirc it arises as a purely mathematical thing as a consequence of a very abstract approach to probability via algebras of random variables (a consequence of monotone class theorem))

[hearty steppe]

Oh wait right it’s a borel measure

I am not quite sure what this means

[grave thorn]

what does "borel on (0,1) mean?" (0,1) is by def a borel set, idk what you mean

[hearty steppe]

Not quite sure what that means* now I guess that’s where I got it from

[grave thorn]

Borel spaces arent complete - but the completion of the Borel algebra on R is the lebesgue measurable sets

[hearty steppe]

Ahh ok yea that is a little abstract I guess

Books toss jargon around very loosely

Why would a physicist want to care about a non-measurable set though

[grave thorn]

Mathematicians don't care about null sets

Unless you work in logic

[forest sleet]

but the cantor set

physicists actually care about that too

though idk if there's much for non-measurable sets

[hearty steppe]

I could definitely look more into what is going on with non-measurable cantor sets but at the same time I don’t really feel like I am motivated to really understand why

[storm moth]

guys sorry to interrupt with an off topic message, but I cant find the general channel

or an off topic channel, it's 6 am please note that 😄

[heady ember]

Do you realise that this channel is book recs right

[storm moth]

bruh chill man dont get mad, read the message I just sent 🤦

[topaz rune]

Yeah I mean correctly formalising everything to the gills is just way too much work for applications

Distracts from the main problem too

[heady ember]

i wasn't mad

And wdym by you can't find an offtopic channel? Did you hide it or smt lol

Spivak perhaps?

Since you seem to want something 'more for mathematicians'

If you want something more rigorous than that, it's probably in the realm of analysis

[lime sapphire]

There's no harm in using stewarts

Do you need to learn the content for a course you're taking at university? For fun? High school?

Either way feel free to use stewarts for now if all you need is to learn about maxima and minima and turning points

A whole book like spivak is quite rigorous and assumes a level of mathematical maturity so save it for when you're ready to take an intro to analysis

Well make sure you know what you're getting yourself into

As in

Do you know how to write proofs?

Cool

Should be fine with spivak then

Maybe

Idk

I think its a hard book for beginners usually

[pliant stream]

Try marsden?

[lime sapphire]

Its fine as long as you're going to learn some form of rigorous calculus later on

[grave thorn]

[gray gazelle]

Rigorous calculus is a waste of time in my opinion if you have a good grasp on school Calculus, learn proofs well (some linear algebra would be helpful) and go straight to real analysis. No point going in a medium where it is not quite easy, but not quite hard.

[hollow shore]

what really is the distinction between rigorous calculus and real analysis?

[lime sapphire]

You do realise real analysis is a form of rigorous calculus

[hollow shore]

I think they are more or less the same?

[pliant stream]

As in there are texts that are "rigorous calculus"

The most famous ones being spivak and apostol

Depends what you think calculus is

If you think of calculus as using derivatives and integrals, and assuming everything is sufficiently nice, then yeah

[grand thistle]

pretty sure rigorous calculus hides a lot of the topology

[pliant stream]

i mean calculus is entirely based on the real line

So not really sure how that's useless to calculus

[grand thistle]

never seen spivak or apostol mention "compactness" or something

[forest sleet]

Idk nonmeasurable sets but for cantor sets they show up in hofstaedter butterfly in condensed matter

[pliant stream]

I mean you don't need to know the construction of the real line to learn calculus

It's enough to assume the existence of least upper bound property/ completeness of R

i mean most people don't care about the construction its not so interesting

Analysis or calculus

nobody is like wow completion of a metric space via Cauchy sequences is so cool and interesting!

I really want to prove R has the lub property!

So even for the purposes of doing analysis, it's not exactly useful

Even rudin has dedekind cuts in the appendix and cauchy sequences completion in an exercise

[gray gazelle]

Rigorous calculus normally refers to mid-ground book such as Spivaks or Apostol. Most books would say you require “Real Analysis,” not “Rigorous Calculus”

Analysis is a superset of Calculus. There is no reason to address to it using “Rigorous Calculus”

[lime sapphire]

I still don't see the reason for such pedantry

[pliant stream]

I mean it is rigorous calculus lol

[gray gazelle]

No, it is not. Topology is not covered in any calculus book.

[pliant stream]

Okay?

[lime sapphire]

^

[pliant stream]

Topology also is not usually covered in real analysis courses

and who cares?

[lime sapphire]

It is actually but usually not in intro courses

And i agree

Who cares

[pliant stream]

What if introducing topological notions of the real line makes it easier to talk about calculus?

and it does

so why exactly does the existence of topological notions exclude the notion of rigorous calculus? It doesn't, it's completely arbitrary

and when people ask what's analysis, what do you say? You just tell thwm it's proving calculus

[gray gazelle]

It makes the discussion completely different.

[pliant stream]

Realistically speaking, what topic is covered in a first semester analysis class that is not calculus? The only one really is uniform convergence

[lime sapphire]

Unnecessary debate tbh

[pliant stream]

Realistically speaking, how many people even take more than one semester of analysis?

[lime sapphire]

Just allow it to be calculus and move on

[gray gazelle]

Not to mention, Calculus is mostly an American thing. Analysis is the only form of “calculus” in quite a bit of other countries.

[sage python]

Yeah Wintlumi isn't correct here

[hearty steppe]

Huh wdym?

Are we talking about Douglas Hofstadter here or you are talking about some other guy with the name Hofstaedter 😂

[sage python]

Specifically regarding calculus being an American term

Honestly nobody has a clear idea of what the distinction is. Some people say oh if you're doing things rigorously it's analysis

Others, and I'm kinda in this category but not enough to really care, say that calculus is the subset of topics in analysis concerned with computations of derivatives and integrals and whatnot

[pliant stream]

It's ubiquitous to discuss convergence of integrals and series in calculus

[sage python]

So you can do things with or without rigor

[pliant stream]

I mean it's not like people just handwave stuff here. They still say by comparison with, or by root test, or so and so. It's still entirely justified

[fickle whale]

imo the difference is the level of rigor, which will naturally affect what topics you bring up. But I don't care enough to have a discussion on that

[pliant stream]

And if you think about it, in an analysis class, when you talk about series, what is the homework? It's identical to the hw you'd get on series in calculus

The only difference in analysis is your lecturer will probably prove the tests

[sage python]

To be fair Andrew things aren't always uniform

[forest sleet]

https://en.wikipedia.org/wiki/Hofstadter's_butterfly yeah sorry was on mobile earlier

they don't mention cantor set on the wiki page but for non-research articles mentioning it there's https://phweb.technion.ac.il/~odim/hofstadter.html

[hearty steppe]

Ok it is Douglas Hofstadter. He’s an interesting character and a lot of cognitive scientists provide some interesting insights into mathematics and other areas

Don Hoffman is very interesting

You should read his book, I didn’t regret it still to this day

Also I am a fan of Godel Escher Bach still but I’m not referring to that book, I’m referring to this one https://www.worldcat.org/title/1054001128

Trust me the content of the book is as good as the ballsy wording and cover art

[tawny crater]

is godel escher bach good?

i always dismissed it as pop-math

hm good to know

it's quite a large book so i didn't want to even pick it up

[sharp latch]

It’s not a math book it’s more cognitive science/philosophy and how math relates to them

Pretty good so far I’m only a bit in though

[royal perch]

Hi all,
I'm looking for a rigorous treatment of the basic operators used in computer graphics (so for example, scalar projection, cross product, orthonormal bases),
does anybody know where I can find them? Especially for those first two...

Actually I think i'll try to go with exercises and problems in linear algebra for the short term 🙂

[timber mesa]

unis over here do the same as euro unis but call it calculus anyway

[timber mesa]

yeah it seems like you'd want a linear algebra book

[royal perch]

🙂
I just need axioms otherwise my brain gets so confused

I don't have geometric intuition whatsoever

past two dimensions at least

and not for a lack of trying 😆

[pliant stream]

Apostol calculus vol 1 has a nice section on what you’re looking for

It’s continued in volume 2, but I think most of what you’re looking for can be found towards the end of volume 1

[hearty steppe]

It’s good and it’s poppy sure but it’s not easy to read and it’s very easy for math and physics newbs to misinterpret the content

I think it is a nice trippy exploration into recursion theory and how deep down the rabbit hole we can go with it (but not everything is recursive)

The problem is it’s not a math or physics book. It’s a conjecture based book like Wolfram’s A New Kind of Science which, I also suggest reading but only when you spent some time developing foundations in math and physics

[brittle breach]

the first chapter of linear algebra done wrong.

[storm sage]

do you guys recommend Schaum's Outline of Advanced Calculus, Third Edition?

and Schaum's Outline of Linear Algebra, Sixth Edition

[still kayak]

Hi. Im reading Knuth Art of programming. At the beginning of the book he expresses the algorithm as a tuple. Why are there two letters of 'p' in the last picture

[gray gazelle]

Wrong place to ask #❓how-to-get-help

[quaint pine]

I highly recommend convex analysis by Tyrell Rockafellar.

[quaint pine]

Both are highly recommended.

[sand frost]

I've seen the openstax books recommended a lot of times here
Their pdfs are (legally) free
https://openstax.org/subjects/math

[dense wren]

Anyone familiar with John lee’s manifold trilogy? Is it a good choice for self study?

[heady ember]

A h y e s, Lee manifolds

[drowsy thicket]

Has anyone read the OpenStax Calculus books?

If so, were they good?

Apparently the senior author was Gilbert Strang

[sudden kindle]

I have used the books for calc courses

They r fine

[drowsy thicket]

thx

[grand thistle]

i've asked something similar to this before, but im asking again because im curious what other people have to say. im about to finish baby rudin and my group/field theory class in the next two months, and i'm wondering where i should go next. i have a few options that i'm considering and honestly i'd be pretty happy learning any of them, but i want to do the one that would be most applicable and relevant in other fields of math instead of just a very niche knowledge of one thing. rn my list of books i think are pretty cool are

• Stein and Shakarchi, Complex Analysis
• Grimmett Stirzaker, Probability Theory and Random Processes/Jacod and Protter, Probability Essentials
• Munkres, Analysis on Manifolds (just analysis on R^n im pretty sure)
• Ireland and Rosen, Introduction to Modern Number Theory
  which one do you guys think i should do? My background when i start it would probably be analysis of one real variable, linear algebra at the level of halmos, group theory up to first iso, very basic field theory (like extensions and stuff i think). someone on another server also recommended me Riehl's category theory in context, which i think might be cool to learn, but not sure if it's a good idea since it might be too abstract for me

[gray jungle]

i think you should study what you find the most interesting , but you can consider measure theory.

[gray gazelle]

Stein and shakarchi: complex analysis is currently my favorite mathematical book, so definitely take a look

[analog lava]

i would say read more group theory and learn algebra uptill galois theory and then learn about complex analysis

plus i think topology is something fundamental you have to learn

once ur done do whatever u like

stein is good

i read it

dont think baby rudin is enough for point-set topology tbh

[pliant stream]

Can somebody recommend a source for improper integrals, and uniform convergence of integrals? In particular, Dirichlet’s test and Leibniz rule should be discussed. Thank you!

[brittle breach]

abbot understanding analysis

[pliant stream]

Abbott covers this?

Abbott certainly does not

[timber mesa]

Abbott has a very short section about improper integrals and another about differentiation under the integral (assuming that's what you mean by Leibniz rule?)

generally the exposition about integrals is a bit lacking and the book focuses on other things

[timber mesa]

have you checked Spivak and Rudin?

[pliant stream]

Well I’m really hoping for differentiation under the integral where the integral is improper

And apparently this is where uniform convergence of improper integrals comes in

I’ll check spivak, but Rudin doesn’t

I mean if it’s an tucked away as an exercise and not part of the main text I guess that’s okay too

[grand thistle]

does complex analysis need galois theory or is it just like general knowledge i should have?

[analog lava]

No stein doesnt need galois theory

Baby rudin should be enough

Galois theory is p like cool and hot

like there is no reason not to learn it

And algebraic number theory needs galois theory

[grand thistle]

ah that’s fair

also, what if i know like zero multi variable calculus?

do u think i can learn along the way?

[pliant stream]

That’s a pretty good reason not to learn it

i don’t think it’s realistic to say just to learn stuff because it exists and why not

[fickle whale]

I do

[analog lava]

yes ofc

idk why everyone thinks u need multivariable calc tbh

just know what a partial derivative is lmao

complex analysis at steins leevel is basically corollaries of cauchys integral formula and residues

so yeah

[gray jungle]

i do

[pliant stream]

lol okay

[stark wing]

Does anyone know any math books that start with basic algebra and goes up to Calculus or higher levels of math?

[gray gazelle]

Doubt it

[pliant stream]

The biggest reason not to learn it is it’s not the most employable skill

[fickle whale]

if someone markets a skill as employable it's because that's the only redeeming aspect

[gray gazelle]

Any good book on metric space topology?
I need one that covers some stuff like open/closed sets, limits, sequences and continuity, compactness and connectedness

[grand thistle]

i mean i eventually have to learn it anyway

[pliant stream]

do you?

i don't 🤷‍♀️

[topaz zinc]

Can someone recommend me a good plane geometry and solid geometry book? Thanks!!!

[karmic crater]

Chapter 2 in baby Rudin

[hearty steppe]

Then chapter 2 of Brin and stuck, topological dynamics!

Go from basic topology to topological dynamics!

No but that’s a chapter I am probably starting next week. Feel free to join me learning about topological dynamics

[rain stream]

Let's suppose I have no trouble understanding basic calculus and I can understand everything in Stewart's calculus. Should I try Apostol? Is there an 'advanced'calc book you'd suggest?

[main walrus]

I would like to familiarise myself with the basic properties of holomorphic functions and measure theory in order to be able to study more topics in functional analysis. I would prefer to have some options that briefly but rigorously cover these topics and cover many essential theorems that will be required to study topics such as spectral theory, function spaces of holomorphic functions, L^p spaces and be able to use the Bochner integral in a comfortable manner.

I would prefer to follow the Weierstrass approach if possible for Complex Analysis resources and be able to integrate functions taking values in a Banach space if possible. I am willing to look at other options for these though if the topics are covered very quickly while still in detail.

I am comfortable with vector-valued differential calculus, know how to work with power series, have a good enough knowledge of topology and some very basic knowledge of functional analysis covering Hahn-Banach theorems, some theorems following from the Baire Category theorem and have an idea of what weak topologies and topological vector spaces are.

At the moment I am considering the following resources:

Measure Theory - Chapters VI and VII of Lang’s Real and Functional Analysis
Complex Analysis - Chapters 10-17 of Rudin’s Real and Complex Analysis

I would appreciate any suggestions or advice regarding this.

I will also mention that I am familiar with many basic results about Banach spaces and am able to integrate continuous functions from a compact interval of real numbers into a Banach space using that they can be uniformly approximated by step maps and would prefer proofs that allow the statements to hold in functions from an open subset of the complex numbers into a complex Banach space. But this is not essential.

[blazing canopy]

"Briefly but rigorously" is not something I've really seen out of a good analysis book but if you want to be comfortable with Bochner integration then Lang's analysis book covers this just fine; Rudin's complex analysis book is also fairly brief compared to some other standard books on complex functions. I should also mention that in the context of functions taking values in separable Banach spaces, Bochner integration and strong measurability are all pretty much just as nice as regular integrability and measurability on R.

[dense wren]

Help

[trim solstice]

Can anybody suggest a good introductory text on algebraic geometry.

[sage python]

@dense wren Lee Topological is imo better than Munkres as an intro to topology tbh. Smooth manifolds does everything and thoroughly, but as a result is disgustingly long. Riemannian idk

@trim solstice well this is the classic question isn't it. Feels like it depends on what you're doing

The sorta starting point in AG is the idea of an affine algebraic set, just a subset of k^n (k is a field) which is equal to the vanishing locus of some ideal of n-variable polynomials

You play around and realize the geometry of these guys is tied to the algebraic properties of the ideal, and really of the k-algebra of polynomial functions on that set.

[distant bobcat]

best textbook for learning linear algebra?

[heady ember]

Look in pinned

[trim solstice]

I just know basic algebra like group ring field and point set topology.

[sage python]

Then various algebraic and geometric considerations lead you to want to replace these notions with more general ones, this is sheaf/scheme theory

So to that end there are a few things you can do. You can focus on varieties, and within that you can ask if you want sheaf theory or just projective varieties. Or you can go to the more general schemes

So good you answered your background, now here are some relevant questions

What about algebraic geometry sounds appealing? Also, do you have a particular interest in complex analysis and/or manifold topology? Separately, do you have a particular interest in number theory?

[trim solstice]

I kind of liked algebra more than topology. I don't know anything about algebraic geometry so it would be nice if there is some text which tells me what this subject is all about and why do we care and stuff.

[sage python]

Okay this is good to know

So to give my thought process out loud. There's a book by Neeman which is kinda in the vein of complex AG (builds up to GAGA)

But that might not quite be your favorite

Since you want to see why we care I think it's good to have something that emphasizes cool theorems

How do you like number theory btw?

This isn't gonna weigh in big time at this point for me but it's prob relevant in some capacity

[trim solstice]

Haven't had a major encounter with the subject yet.

Only number theory maybe I know of is due to discrete math or algebra

[sage python]

Gotcha

Basically what I'm thinking is like

The early cool theorems in the subject to me are like

Bezout, 27 lines, Riemann-Roch

Also small stuff like "5 points uniquely determine a conic" (does that sound cool to you at all?)

[trim solstice]

👍

[sage python]

Try Gathmann's notes

https://www.mathematik.uni-kl.de/~gathmann/de/alggeom.php

He has a few versions which is kinda unfortunate

The old one has the most but the fact that he wrote new ones to replace makes me wonder if like, that one had too many errors or otherwise leaves something to be wanted

For you I think the 2014 version might actually be best

Doesn't quite get to Riemann-Roch but to be fair that's a bit harder

nGroupoid and Chmonkey probably have good commentary as well but that's my take as someone who's currently a relative outsider

[trim solstice]

Thanks will take a look into it. Also at the lee's book which you recommended to someone earlier.

[sage python]

Lee is a different topic but yeah

[main walrus]

Thank you for this information!

[versed magnet]

Do you know any easy to read book, to linear algebra? My lecturer told me to use transcripts of Aleksiej Kostrikin books, but i'm out after 5 pages. I'd be nice, if it taught me about basics od algebra, and cover bilinear forms

[unique ice]

@versed magnet read Carl D. Meyer’s Matrix analysis and applied linear algebra. Do all the problems they have solutions anyway, study them. It doesn’t cover bilinear forms.

I wonder which book on linear algebra covers bilinear forms

[foggy relic]

most standard linear algebra boks

[unique ice]

Example?

[foggy relic]

Treil

[vital bane]

not necessarily a book, but has anyone come across a "analysis in general metric spaces" lecture series on youtube? if you do please do let me know (by that I mean, a lecture series on that topic, not with that exact name)

[hybrid chasm]

Does anyone know the prereq material for arithmetic moduli of elliptic curves? (I’m looking into it and also cornell/silvermans diophantine geometry book [which seems more accessible])

[heady ember]

Look in pinned for Dami's opinions on an assortment of LA books

In general though, Friedberg, Insel, Spence has always been a well-regarded choice by many around here

[grand thistle]

i like halmos

[remote ginkgo]

both shilov and kostrikin/manin

but mostly shilov

[clear mist]

Hey guys, have any recommendations for trigonometry with proof, problems and solution?

[heady ember]

uh

Khan Academy?

[clear mist]

Thx, gotta try it then

How about coordinate geometry guys, just saw khan acamedy and others mostly no proof, especially the conic section?

[fierce hedge]

Are you talking about Lee's Introduction to Topological Manifolds?

[sage python]

Yeah

[sand frost]

👋 Any recommendations (videos are good too) about the formalisation of the rules used when solving inequalities (especially with various functions)?

[orchid mortar]

Do you mean algebraic inequalities?

If you mean very exciting general inequalities then you are talking about general optimisation problems which can be very difficult, and even if not difficult, admit no simpler closed form

[sand frost]

Yes I'm talking about very exciting inequalities, do you have anything to recommend?

[orchid mortar]

Not really, unless you have some structure to the inequalities involved

For example $f(x) \leq a, \forall x\in\mathcal{X}$ imposes the condition $\max_{x\in\mathcal{X}}f(x) \leq a$, but this is just repetition rather than progress

[hasty eagle]

草w

[sand frost]

I mean, I don't really have a general structure of inequalities I want to learn how to solve, to make up an example I'd just throw together some pesky functions

Well, thanks anyways

[orchid mortar]

Unfortunately I don't think civilisation has developed far enough to handle pesky functions, but we are doing well for linear inequalities

For a start in that you can look at https://transp-or.epfl.ch/books/optimization/html/index.html

[sand frost]

I'll check it out, thank you

[hearty steppe]

This is going to be a silly request but how about a book on the mathematics of fidgets, puzzles, dexterity toys?

And also can you guys help me design some

I have a couple really cool ideas

I call them tactile reinforcement aides

https://www.aroundsquare.com/products/deco-untitled-1

Here’s example

It is mathematically designed

Maybe just learn more dynamic systems theory?

Combinatorics toys for everyone who helps me

Or like I send you an STL to print for yourself. The condition would be prototype

Solve your hard math problems today

https://tenor.com/view/the-math-seems-to-check-gif-20186998

[hearty steppe]

I am incorporating survey data from use of these prototypes as well for my research on compulsive behavior dynamics

[topaz plaza]

if I wanted to learn calculus from calculus books I want to know how would a calculus book help me?

[pliant stream]

What are you trying to ask?

[clear mist]

It gives you proof, problems to solve, overall increases your skill

[gray gazelle]

Where can I find a geometry text book online?

[dense hamlet]

https://www.google.com/search?client=firefox-b-1-d&q=geometry+text+book+online

one of the first few results will work

You don't need a textbook though, just use khan academy

[fierce hedge]

Has anyone read, "Topology and groupoids" by Ronnie Brown? The table of contents seems pretty good

[dapper root]

@golden locust did

[golden locust]

Shamrock.

I liked it

Would recommend

Very idiosyncratic though. Would it be your first topology book?

[fierce hedge]

Nah, my first topology book was Munkres. I am revising topology for some competitive exams

[golden locust]

Okay cool, I think it'd be good then!

I hesitate to recommend it as a first reference but if you've seen some stuff already it should be fine

[fierce hedge]

any particular reason for that?

[golden locust]

it's just weird

It defines a topology in terms of a neighborhood basis and not open sets

It defines paths to have arbitrary lengths and not just domain [0, 1]

It uses additive notation for composition of paths in a groupoid

[fierce hedge]

arbitrary length paths ?!?? wot! additive notation instead of the usual composition? Is there a rationale behind this

[golden locust]

oh it uses ]a, b[ notation for open intervals

I told you it's weird lmao

[fierce hedge]

I thought that's just a Fr*nch thing

[golden locust]

haha

So yeah weird book

But good

imo

[fierce hedge]

Plus, saying You seem to ignore my book first published in 1968 and of which the 3rd edition is entitled "Topology and groupoids" see http://pages.bangor.ac.uk/~mas010/topgpds.html in comment section definitely doesn't help

[golden locust]

lmfaoooooo

Yeah he's uh

very active at plugging his work on mse/mo

[fierce hedge]

This was not even mse/me. This was a blog post summarising topology book threads

Oh btw, how much metric spaces does his book do?

[golden locust]

none

basically none

I'm not sure it even defines a cauchy sequence

It talks a tiny bit about them at the very start

I don't believe so

[fierce hedge]

Who's even the intended audience for this book !? Person willing to learn groupoids !?

ugct wanting to learn group theory?

[dapper root]

Homotopy theorists

Or something

[fierce hedge]

I presume that it's just the old notation or something

[fierce hedge]

Alright then

Yeah, the book is very old like 1968 old so maybe this notation was more common then

[golden locust]

My impression is just that ronald brown is a weird dude

[fierce hedge]

I just bought an algebra book published by Dover and the page quality is weird, like the pages aren't that thin but you can kinda see the black ink from other side making it a bit harder to read. Anyone else experienced this? Is this true for all Dover books?

[tepid prairie]

Now you mention it I can see it but I definitely have never had a issue reading them.

[fierce hedge]

Like the pages aren't that thin but I can clearly see what's there in the next page. It makes it more than just a bit annoying

I might return this since it wasn't that cheap also

[tepid prairie]

I love all my Dover books except one. I could never return them.

[fierce hedge]

Which one would that be?

[tepid prairie]

Advanced Calculus of Several Variables by C. H. Edwards, Jr, it's not really the books fault I just don't understand all the linear mapping language it uses and don't want to invest time into that stuff yet.

[clear mist]

Hey guys, any good coordinate geometry book, with proof especially conic section?

[timber mesa]

there's one by Lehmann but it's a bit dated

I thought it was good but it's the only one I know or ever read about plane geometry and such

[sudden kindle]

First chapter of Geometry by Brannan Esplen and Gray

[gray gazelle]

Thank you

[clear mist]

Ok thx, gotta try it

[viscid citrus]

Has anyone here used the Sheldon Ross Probability book and can recommend it?

[hazy elk]

I've used it, its pretty easy to read. (maybe slightly less fun than Blitzstein tho). Covers intro probability reasonably well, exercises are pretty basic tho.

Can't really complain, does what it's meant to do.

[viscid citrus]

Ok thanks!

[hearty steppe]

Blitzstein… wow I can play around with that name and come up with so many funny phrases

Pop culture made me do it! Ahhh!!!!

[royal perch]

Anyone have a good book rec for types of the CS sort viewed mathematically and especially algebraically?

In Patrick Cousot's Principles of Abstract interpretation, he builds types through a series of Galois Connections between dcpos (starting with ``jaques herbrand domain" which abstracts properties of expression trees to expression trees containing variables, choosing the least under a subsumption order subject to the constraint that all common terms are represented by the same variable, then moving on to abstractions of environments, where environments in his book give values to syntactic variables; he embeds in one of the adjoints here the algorithm for inferring hindley milner types) but I kind of wonder if the essence can be distilled even further

[inland dawn]

Did you try Knuth’s The Art Of Computer Programming?

[tranquil vine]

Can anybody recommend an ODE book to me?

[royal perch]

TaoCP has types?

[inland dawn]

Oh my bad didn’t read it properly, so you wanted type theory

Yeah I haven’t read any books on it so can’t suggest sorry

[tawny copper]

Nice texts for a first course in probability?

Thanks

[pliant stream]

Grimmett and Stirzaker

[waxen elbow]

hey ! any good reference for graph theory ? a pdf or a book, at intro level, which had a nice depth

[inland dawn]

https://link.springer.com/book/10.1007/978-0-387-79711-3

[waxen elbow]

thank you !

[gray gazelle]

My favourite book https://59clc.files.wordpress.com/2011/01/real-and-complex-analysis.pdf

[bright crow]

johnny the walrus by matt walsh

that book so fine

[gray gazelle]

Yo I have a year to prepare for iB math HL AA

what should I use

[tight basin]

any book rec for Probability Theory and Mathematical Statistics

[grave thorn]

At what level

[tight basin]

currently reading Wackerly's Mathematical Statistics and its pretty wordy

like uni

undergrad

[grave thorn]

Ross, Haigh for probability

[tight basin]

appreicated 🙏

[pliant stream]

Grimmett and stirzaker

Casella and Berger for math stats

[tight basin]

isnt Casella and Berger notoriously harder to get through?

[inner token]

Personally I found degroot approachable and had no complaints, I know there's a PDF and solutions online

[pliant stream]

Idk if people think it’s notoriously hard

Pretty approachable to me

[grave thorn]

The person was asking for an undergraduate book

[pliant stream]

Casella and Berger is undergraduate for sure

[grave thorn]

Cassela and Berger is (a) not too bad but (b) (in most places!) Grad level

[pliant stream]

In what places? In biostatistics programs?

I mean a text commonly used for first semester graduate courses can definitely be used for undergrads

[grave thorn]

That's my point - it is a first year grad book

Probably best to go w/ another option

[pliant stream]

I mean biostats first year courses use it

And those people have taken what, like linear algebra and calculus?

it’s probably really hard for them, that’s true, but it should be okay for undergrad math majors!

[grave thorn]

Stop using "biostatistics" to mean "easy"

Still, for a first course and not that much exposure to stats, best to stick w/ other books

[pliant stream]

I didn’t say it’s east lol

you misinterpreted what I said

[grave thorn]

This is for the same reason if someone asks for a first course on probability, you don't give a text that uses measure theory

Its not that it's very hard or smth - it's just not suited for a first course

(for someone who has only taken calculus and the like)

[pliant stream]

Okay I mean my school uses Wasserman’s book for a first course in math stats at the undergrad level

So if you want a true “undergrad” math stats book then there’s that

Except a “graduate level” course at my school also has Wasserman in the syllabus so

anyways, it’s a pointless distinction in my opinion

[pliant stream]

Actually we use both c&b and Wasserman for undergrad math stats

[tranquil vine]

Hi, any reference for Ordinary Differential Equations?

[zealous light]

At what level?

[tranquil vine]

University

[zealous light]

I tend to like Teschl or Perko for ODEs

[tranquil vine]

Can you send the link

[pliant stream]

I think by what level they meant

What’s your background?

have you taken a first course in analysis? Only calculus? etc.

And what are you looking for in a book?

[zealous light]

Yeah

[pliant stream]

Otherwise recommendations are likely to not be helpful at all

[zealous light]

https://www.mat.univie.ac.at/~gerald/ftp/book-ode/ode.pdf
Yeah I can send this, but this is Teschl's book. As Andrew mentioned, without knowing your background, it's hard, but take a look if this is what you're looking for

[tranquil vine]

I am studying BSc Mathematics in college. I have taken Single and Multiple Variable Calculus. I need a book which concentrate on ODEs and PDEs

[zealous light]

Oh

If you're comfortable with analysis, the book above should still be good for ODEs

I can't really think of a good ODEs book at a lower level though

[tranquil vine]

oh

okay

[zealous light]

I'm sure someone else has a better suggestion

[pliant stream]

Well mit has ocw for ode

Why not just follow with that?

(They also list the book they use and should have assigned readings/problems)

[tranquil vine]

helpful

[faint oak]

Do you recommend Strang's Calculus for learning Calc?
https://ocw.mit.edu/ans7870/resources/Strang/Edited/Calculus/Calculus.pdf
It teaches from an engineer's perspective with plenty of applications and I like the index of contents a lot but want to get your opinions before starting to learn from it
I seem to have problems with learning math purely theoretically (because I'm not able to visualize things) so I'm thinking of using this book instead of Spivak calculus

[orchid mortar]

Not a good yardstick. Terrible books for undergraduates.

[orchid mortar]

It's actually not, unless you can prove this statement in an actual pedagogical setting

[wet holly]

Does anyone have any good book recommendation for symetrical polinomions with formula demonstration?

[vital bane]

simmons diff eq book is really good

it's not necessarily entirely rigorous, it's a nice balance of rigor and applications

[coral narwhal]

I liked coddington intro to ode

[gray gazelle]

Anybody have some recommendations on books that focus more on developing mathematical thinking skills and intuition? So like, not really a work textbook but I guess a more pleasure-reading book that gives you some insight into developing thinking for math? I'll be taking college calculus soon and I wanted to just read something passively in the meantime.

[warm glen]

you could try "how to study as a math major" by alcock but that isn't specifically for calculus

[inner token]

Strogatz has a book like that I think. More a math appreciation book than anything

"infinite powers" it's called

[timber mesa]

Strogatz is a good writer for that kinda stuff

[brittle breach]

has anyone read Rudin functional analysis

[grave thorn]

imo good book

Though brezis is better if you want PDE applications

[timber mesa]

I've read some of it (the first chapter about TVS and some of the latter stuff about Banach algebras) and I think it's pretty good, certainly leans more to the theoretical side of things

my friend says it's good to read both Rudin and Brezis in parallel

Rudin FA is very well written imo

[forest sleet]

Teschl has an ode book

I haven't looked at it though

I think it's graduate studies in math series

It's free on his website though

[inner token]

Give Teschl a try

[zealous light]

I can personally recommend teschl

[sage python]

I've heard of Teschl, Perko, and Arnold

[violet lagoon]

Anyone please tell me book for imo

[sage kelp]

What’s a good book for introductory geometry? Maybe something like high school geometry

[gray gazelle]

I would reccomend

https://www.amazon.com/Everything-Need-Geometry-Notebook-Notebooks/dp/1523504374/ref=asc_df_1523504374/?tag=hyprod-20&linkCode=df0&hvadid=459709609434&hvpos=&hvnetw=g&hvrand=1870324246383217227&hvpone=&hvptwo=&hvqmt=&hvdev=c&hvdvcmdl=&hvlocint=&hvlocphy=9005349&hvtargid=pla-919464574960&psc=1

its pretty helpful for review and a foundational understanding

[granite viper]

are you familiar with middle school geo?

[sage kelp]

Not sure what that entails, but so any suggestion is appreciated

Like I’m looking to practice some geometry exercises involving congruence, similarity in the style of GRE questions. They say it’s high school material, so that’s why I thought of that. But I’m dissatisfied with the official material, so I’m looking for something else

[granite viper]

i wouldn't get an entire book just for that. khan academy should do the trick.

in what way are you dissatisfied?

[sage kelp]

I would like to practice more problems like the ones that are in the official prep tests. Sometimes it takes me a while to realize I need to use similarity or congruence, but the number of exercises that they provide is limited

[granite viper]

if it's only those two specifically. consider using https://www.khanacademy.org/math/geometry

see: congruence and similarity

if you want a complete and proper understanding of highs school geometry, consider
Geometry: A High School Course by Serge Lang

[sage kelp]

Great, thanks @granite viper. I'll check Khan Academy and maybe Lang's book later

[sage kelp]

Thank you! I will check it out as well

[manic cairn]

I recommend Euclidean geometry for mathematical Olympiads (EGMO) by Even Chen, if you’re just doing Euclidean geometry in high school

[granite viper]

bro i recommended serge lang and ur username is serge lang

cmon...

[manic cairn]

after Lang, obviously

You can’t start egmo without any Euclidean geometry in practice

[granite viper]

idk i'd move past euclidean asap

just remember the basics like what a circle means, and that the angles in a triangle are 180 and you good

[manic cairn]

I would too, but if you want to be really good at it, egmo is the book

[granite viper]

respect

[granite viper]

what's egmo?

[manic cairn]

Euclidean geometry in mathematical olympiads

The book I recommended

[sage kelp]

lol

[violet lagoon]

Euclidean geometry from evan chan

[normal reef]

Hi, I'm looking for a good exercise book on Calculus I, with hard problems and solutions.

Any suggestions?

[heady ember]

"Hard"
Try Spivak then

[normal reef]

ok @heady ember maybe I can do with something simpler...just spam titles to excersise books

[heady ember]

What do you not like about Spivak? Just curious btw

[lime sapphire]

Man said just curious

Yk its because spivak is rlly not for someone without mathematical maturity

[heady ember]

It depends. If you're willing to try questions for hours at a time with no progress in sight, then you trying Spivak out for a bit should be fine.

With that said, i was just genuinely curious.
(Not to mention that understanding that can help people to give better book recs)

Also, mathematical maturity is gained at some point. So it could be a good thing in helping one learn in a bit more detail about calculus, as well as have as relatively challenging questions to hone one's problem solving skills.

Of course, i absolutely don't think it is for everyone. Especially if one is new to calculus. And if one is already familiar with calculus, they might otherwise wish to try a real analysis text like Schroder.

[warm mantle]

What's mathematical maturity ? How does done become mathematically mature ?

[faint nebula]

After studying math for 18 years then you are of age and mathematically mature

[warm mantle]

Damn

It takes that long ?

[hearty steppe]

Mathematical maturity is just the time it takes you to develop enough refinement to work through a textbook of a certain level of understanding in terms of how deep in the woods you want to go with understanding math

I will say this varies depending on what you are trying to do.

I don’t really go through a math book like someone who is a PhD would. I have problems interacting with the way that most higher level math book exercises even try to challenge my understanding of concepts.

I think if you want to go the pure math route where your just focusing on the abstractions of a subject area your into, it will be important to be able to know how to tackle those exercise problems formally but the real question is how much free time do you have?

I spent about over 2 years developing rigor before I got to the big boy books. Then again I know that’s not enough if I want to really just do math, cause I struggle at times with formal presentation and I am terrible at interpreting and approaching the exercise problems

It’s not that I can’t do the exercise problems it’s just not the way I can engage with the material properly that suits my needs

So yes I am limited in my ability to engage with the material in that respect but everyone is human

Therefore I might misinterpret information sometimes and that’s why I talk in the channels here

[wooden anchor]

Can I asked that there any kind of subject in pure math that more focus on theory ( abstraction) thinking ( like philosophy) than just doing some messy computational and messy plug and chuck number into equation ? ( Sorry i kinda new to pure math and not so good with English)

[stray veldt]

modern mathematics is never about plug and chug

there are messy computations, but probably not in the way you think

[grave thorn]

every branch of pure math

and every branch of applied math

@wooden anchor logic is pretty close to philosophy

[hearty steppe]

I think it’s important to just appreciate mathematics and explore it for its beauty if you really want to spend time learning math.

Then you might be able to become a clever person one day and help the world in some way.

If I were to believe in god, I would call it math.

Just like Euler did

Math is just that captivating

Newton probably felt the same way

[bitter raptor]

Anyone know of a book that approaches quantum mechanics from a mathematical standpoint while introducing and motivating the relevant concepts along the way (assuming prereqs of one year of undergrad)

Seems like much to ask for cause I know rep theory and PDEs are important in qm

[hearty steppe]

I like Griffiths

But it’s not an easy book

[bitter raptor]

The one I’m finding is on electrodynamics?

[hearty steppe]

Physics books kinda half ass their math

[bitter raptor]

Oh nvm

[bitter raptor]

Yeah I’m looking for a book that treats it rigorously

I’ll give Griffiths a look though thanks

[hearty steppe]

My goal is to work through the first four chapters of Griffiths for now in terms of QM

And the last chapter might be QFT stuff so maybe take a look at that

[bitter raptor]

Thanks 👌

[hearty steppe]

Yea I’m on chapter 2. The exercises seemed kinda poorly motivated tbh

I can recommend Wald’s relativity book if you want some pretty cool physics exercises tho. That Lorentz contraction problem is fun to think about in chapter 1

[next storm]

Good books on set theory logic and model theory?

[faint nebula]

Everybody read Berserk

[tawny copper]

Nice video lectures on logic?

[tawny copper]

see the pinned message of Clerk

[still panther]

I think mathematical maturity is when you first and foremost see ideas and motivations behind proofs and definitions, not literal words on the page

[uncut zealot]

Does anyone know like a good undergrad diffeqs book with a chapter on bessel functions? When I took diffeqs we didn't cover them, and a problem I'm working on now requires me to know a bit about them.

[timber mesa]

I really like Kreider's Introduction to Linear Analysis, seems to have a chapter on boundary value problems involving Bessel functions

the rest of it is your typical ODE book but aimed at math undergrads, with a focus on the theory and proofs

[digital prism]

Does anyone have experience with aops books? I want to get started with one but I don't know which one to get. I know algebra 1

[remote sparrow]

https://www.amazon.com/What-Quantum-Field-Theory-Mathematicians/dp/1316510271/ref=sr_1_1?keywords=talagrand+quantum+field+theory&qid=1662929555&s=books&sprefix=talagrand%2Cstripbooks%2C142&sr=1-1

https://www.amazon.com/Quantum-Theory-Mathematicians-Graduate-Mathematics/dp/146147115X

talagrand's book (blue) was inspired partly by the springer book

[remote sparrow]

https://www.youtube.com/playlist?list=PLjJhPCaCziSRSUtQiTA_yx5TJ76G_EqUJ

[gray gazelle]

Good calc I textbooks?

[timber mesa]

<@&268886789983436800>

[gray gazelle]

you are welcome

good luck

[sick wasp]

I was looking for books about combination, geometry, algebra and number theorey

I want to participate in our national olympiad

[dense wren]

Does anyone do all the exercises in a book when they are self studying, im talking late undergrad/beginning graduate level

[lapis sundial]

Doing all exercises sounds potentially very unreasonable (depending on the book) 👀

[remote sparrow]

generally stewart is recommended. old editions can be found very cheaply and they're pretty much all identical.

there are tons of alternatives though

[dapper root]

I did all the exercises in Matsumura

Because I am insane

Most ppl never do all exercises in a book

[earnest fable]

does matsumura not have many exercises

or are you just off the deep end

[dapper root]

There’s like 200-something

But it’s like, a 300 page graduate book on commutative algebra, so most people don’t even read the whole thing lol

[gray gazelle]

Oh wait he didn't leave

My discord is just bugging out

[sick wasp]

Any recommendations?

[foggy relic]

wtf

which one?

[dapper root]

The new one (Commutative Ring Theory)

Old one barely has any lol

[foggy relic]

how long did that take you lmfao

[dapper root]

1.5 months

I was really really depressed and did it like 16 hours a day for like 6 weeks straight to distract myself lol

That was last summer

[sage python]

New Matsumura any% speedrun

[dapper root]

I might be WR holder

[foggy relic]

mental health go brrr

[grave thorn]

[dapper root]

It probably got me into Columbia so ¯_(ツ)_/¯

[foggy relic]

oh worth it then

[grave thorn]

their grad analysis class one year was taught by a guy whose tests were literally impossible

thats all i know about columbia

[dapper root]

Lmao

Analysis here is historically piss easy

[grave thorn]

lol

[dapper root]

Like that’s a class u take if u want a free class haha

[tough zealot]

because lets be honest
commalg without alggeo be boring

[dapper root]

I like it :)

But for most ppl yeah lmfao

[tough zealot]

I dunno
it seems like a pretty dry subject
kinda like measure theory (probability excluded) and pointset topology
I can at least recognize Im biased to like those two a bit more

[grave thorn]

measure theory is fun 😡

pointset topology is fun😡

[tough zealot]

its ok

its fun when you integrate fairly exotic things like Lie groups imo

[grave thorn]

measure theory and pointset topology arent really tools you use much in lie group theory

haars theorem maybe

[tough zealot]

pointset I do fin fun, but I can recognize there is a good analogy between its dryness and say commalg

[tough zealot]

I gave it more as an example on what you can do with it

when talking about Sobolev spaces and their applications to PDEs or even memes like the Henstock-Kurzweil integral it seems way more interesting than like proving Radon-Nykodym, Hahns Decomposition, Fubini-Tonelli or even Riesz-Kakutani Representation

[grave thorn]

why are you personally attacking me 😭

[tough zealot]

that said, thats a basic course in measure theory but I cant say I know more

[forest sleet]

wow measure theory is a cool topic

even without probability

and radon-nikodym had a neat proof iirc

[grave thorn]

measure theory is the cornerstone of "modern" analysis

its so much more than just probability

[forest sleet]

maybe fubini-tonelli proof was kinda dry

lebesgue differentiation and maximal functions is definitely not dry

[tough zealot]

yeah, the monotone class lemma is definitely yikers

[timber mesa]

just what I was gonna say lmao

[grave thorn]

as in boring?

[timber mesa]

[tough zealot]

yeah

it has its purpose

but boring

[grave thorn]

well it can be made interesting by looking at its corollaries

[timber mesa]

its just that the constructions in measure theory generally have a lot of steps (that are routine once you know them, but still can get messy and it's just tedious)

[grave thorn]

for instance one leads to an algebraic approach to probability

[tough zealot]

just liie pointset and commalg

I think the analogy is decent

[timber mesa]

yeah

[forest sleet]

sure some beginning measure theory things can be tedious but there's a lot of cool results that aren't super tedious

[tough zealot]

like the proofs in commalg are soo boring

krulls principal ideal theorem and noether normalization are good examples

[timber mesa]

same goes for point-set topo and comm alg so I guess Fractal's analogy also applies

[forest sleet]

you can skip the tedious parts

[timber mesa]

like one of the cool theorems in point-set is Tietze's

[forest sleet]

I actually have no recollection of anything like monotone class thm or proof of fubini-tonelli thm from my measure theory class

and i'm pretty sure i attended that class

[timber mesa]

well that's the standard technique to prove it as far as I'm aware

[tough zealot]

from what I remember one of my favorites is urysohn metrization

[timber mesa]

that one too

[forest sleet]

i guess I just mean you don't need to go through all the technical stuff to get to fun things like Radon-Nikdoym or lebesguie differentiation

oh probably fubini-tonelli was not in my first class with measure theory

I can't remember

[tough zealot]

caratheodory is also fairly boring
the idea is cool and all tho

[grave thorn]

Haar>>>>>caratheodory

too bad Haar makes incomplete spaces

[tough zealot]

one of the funniest memes to me was how the borel SA in R^n is strictly smaller than the n-product of the Lebesgue SA which is strictly smaller than the Lebesgue SA in R^n

[faint nebula]

Yeah that was pretty funny

[tough zealot]

yeah, good one buddy

[gray gazelle]

Are there notable intro texts for getting into semigroup/monoid theory?

[grand thistle]

is measure theory for probability analogous to point set topology for intro real analysis?

like kinda the thing everyone has to do because it makes ur life a hell of a lot easier, but really dry

if so, what’s a text that incorporates it well and is still suitable for a first look at both measure theory and probability?

i don’t want to repeat my experience with point set for measure theory, since it seems really cool

[forest sleet]

i'm of the opinion that measure theory is interesting by itself

though defining lebesgue measure on R can be kinda tedious

for just measure theory, there is https://www.mat.univie.ac.at/~gerald/ftp/book-ra/index.html

which is free legally

if your goal is probability, then a standard grad text will usually have the first chapter be the measure theory you need for probability

so it will get you to probability a lot faster than going through a real analysis measure theory text

[grand thistle]

sum me up, henri

[grand thistle]

i’ve got three texts i’m looking at, which are grimmett stirzaker, williams probability with martingales and jacod protter probability esstenials

are those any good?

i think williams may be a bit too difficult

so i’ll use that as reference

[forest sleet]

lemme ping @grave thorn

[grand thistle]

my background should be baby rudin by the time i start probability

[forest sleet]

since I didn't get into probability until many years after taking measure theory so I never read the first chapters on measure theory

[grand thistle]

ah yeah, he recommended me jacod/protter and it looked pretty good

[placid pollen]

Cool you guys are talking about my interests. I have the Williams book on Probability and Martingales on my Amazon cart but idk I also have a spectral theory book there. I just know I like where modern analysis seems to be going I'm in undergrad

[placid pollen]

Literally everything there I want to know and see where to branch to

[forest sleet]

he also has a functional analysis book for free https://www.mat.univie.ac.at/~gerald/ftp/book-fa/fa.html

[grave thorn]

In some way yes, but not really

As washingbear said measure theory is more than just probability

[forest sleet]

and a QM book that does spectral theory

[gray gazelle]

wheres that huge book that a group of people literally tried to fit all math inside of

the napkin project or something?

[placid pollen]

Thursday I had the girlfriend of a friend turn my Indian Rudin's RCA to Hardcover. She works restoring books. Feel like doing the same with more analysis texts. Maybe Rudin FA, or does anyone here know about another good cheap paperback Modern Analysis book

[forest sleet]

if you have a printer they're all cheap 😅

springer was having a 50% off sale and they have a lot of paperback mycopy versions that are cheap

if your uni has springer access

[grand thistle]

how’s RCA?

do you think it’s a good book

[remote ginkgo]

it's the best intro to measure theory yeah

fastest intro to integrating

[placid pollen]

It's hard to me to describe because I haven't read many books. I find it easier to read given that I have other books around, Rudin becomes the main thing I want to read because it's minimalistic and tries to push the general without losing being slightly talkative. Feels to me like Kolmogorov Theory of Functions or Paul Halmos style

[grand thistle]

well that sounds great because i really like halmos’ finite dim vector spaces

and also rudin pma is going pretty great so far

so i think i might consider RCA after i do PMA?

paired with a probability text for measure theory motivation

[remote ginkgo]

so like

the only motivation you really need

is the lebesgue-stieltjes integral

you know how you can write down the CDF for a random variable in a way that generalizes to both discrete and continuous vars?

like

say

[hasty eagle]

Internal Revenue Service

[remote ginkgo]

measure theory lets you do this in a rigorous strict way

and chapters 1-2 of RCA tell you what all you need to understand the above in the general setting of lebesgue theory

i will say you may find it beneficial to read a little munkres topology

[placid pollen]

I recommend it but it's not going to be like reading PMA where all concepts are within reach of the book itself. When you feel there's a gap between PMA and RCA it's because there kind of is; do go back to skim through a general topology book with an objective from RCA. And the same do for a naive set theory book. For example, to understand why borel sets exist (or something like that which has to do with the power set). Why do some sets exist? Check Paul Halmos book for set theory.

But I say, go back from a more advanced book. Don't lose yourself in the fundaments book because the motivation is lacking. Analysis is dense with motivation

So in that sense I like RCA more even though I understand less of it as of now, I can't stand books that are understandable but uninsteresting. I'd rather be in a harder book that makes me dig for other books. Makes me feel like I'm hunting for understanding I need for the other book

PMA makes me feel like I'm just stupid instead, you don't see much behind that would be an excuse for your current lack of understanding

[humble mirage]

hi, i’m a freshman in hs taking geometry & algebra 2, any book recommendations to self-study/review concepts i learn in class?

[placid pollen]

This might actually be a larger property of book styling. When the author makes it look self-contained... There I will be more afraid. Nowhere to run

[grand thistle]

hmm i see thank you for your opinion that was very insightful

looks like RCA seems like a good choice

ofc after going through some set theory and topology

[placid pollen]

Yes but you need to know what you will be looking for in those. Try my plan of scouting a hard book for objectives and then going back for what you need

[remote ginkgo]

this is the correct approach

[grand thistle]

alright sounds great i’ll be doing that once i complete PMA first 8 chapters

[sage python]

@forest sleet wait those books look kinda goated ngl

[forest sleet]

yeah I really like Teschl's books plus he makes them free online which is nice

I use his QM book for spectral theory every time I forget it

and his measure theory + integration coverage is very clear and covers everything you need

the proofs are nice

like always clear and well-written

[sage python]

Nice

I know he has an ODE book actually

Which along with Perko and Arnold

[forest sleet]

less terse than rudin

[hearty sluice]

we're using it in my ode class actually

teschls ode book

[sage python]

Is in that category of actually serious treatment of ODEs

[hearty sluice]

as a complement to the HORRIBLE textbook we're using

[zealous light]

I used Teschl for my ODEs course last year alongside Perko, and I enjoyed it

3rd time I've mentioned Teschl ODEs in a week lmao

[remote sparrow]

what's a book that treats multivariable calculus similarly to velleman's Calculus: A Rigorous First Course?

yes, proofs, but not necessarily super into the nitty-gritty of analysis

i've heard hubbard and hubbard is a good choice

p.s. does hubbard and hubbard also make a good choice for a multivariable analysis book? its appendix contains all of the more complicated proofs with some problems.

[novel solar]

Is linear algebra done right good for first introduction to linear algebra

[pseudo zodiac]

no. the author even says in the introduction that it's intended as the text for a second course in linear algebra; it will probably feel unmotivated since it starts with linear transformations and barely touches matrices. otoh it is great if you've already seen some linear algebra

[gray gazelle]

I think it is great, if you are doing a math degree. In our uni, the linear algebra class (we.called it algebra 1) started with linear transformations just as well, and I now think it's the best possible approach for someone interested in the theory. So I would recommend it, though my favorite book on the topic is lax: linear algebra and its applications

And nobody here had any linear algebra experience before taking the class

[heady ember]

From what people here say, its handling of determinants and characteristic polynomials is absolute trash. It is regularly shit on here because of that. Other than that, the rest of the contents are fine.

Lol you can look at Dami's remarks here

[pseudo zodiac]

neat!

[karmic thorn]

I would say that some mathematical maturity (that comes from a semester long proof-based course) should be enough to dig into Axler. That said, the problems are relatively more challenging (I see this as a big plus point but you should be prepared to spend some time and effort on them) and there is some nonstandard presentation of the material (with respect to determinants and related stuff, as has been pointed out; the merit of this is contestable).

[sudden kindle]

Axler is the best la book period. Just read first 6 chapters

[karmic thorn]

Yeah, the exposition and presentation are probably one of the best for any math textbook I've worked through so far (this is a small set but still something).

[sudden kindle]

Learn exterior algebra and determinants elsewhere, like analysis on manifolds by munkres and abstract algebra by dummit and foote

Ideally u will cone into contact with linear algebra multiple times, u don't have to learn everything in your first pass

[remote ginkgo]

i really hated it

i also hated analysis on manifolds by munkres and abstract algebra by dummit and foote

[karmic thorn]

Anything in particular that you disliked?

[remote ginkgo]

sec lemme finish getting ready for commute

the only book by axler i have liked is his measure theory book

and the only one by munkres i've liked was his topology

i just do not like how they write purely in terms of how they organize info and write prose

it's antagonistic to my brain

adversarial writing

like they're intentionally trying to make the information less straightforwardly presented than it is elsewhere

i hypothesize they do this on purpose somehow, but i dont know why

much bigger fan of shilov or strang or something

[karmic thorn]

Weird, I always felt Axler's book is far more straightforward and clean in terms of how he organises the book, things seem to follow a sound order

[remote ginkgo]

most people ive asked havent really had that opinion

[karmic thorn]

I haven't seen much of Strang and probably none of Shilov so I can't compare

[remote ginkgo]

some do ofc but

the material is generally considered better presented elsewhere

[karmic thorn]

To each one their own Yamin is right to say that this is a topic one has to keep seeing over and over in different places anyway

[remote ginkgo]

ive seen it dozens of times but those treatments are just really bad

really alien to me that people rec these books

everything else is fine, these are just seemingly unreadable

it's like i live in topsy turvy land

[sage python]

His organization isn't particularly screwy, again the main considerations with Axler are

The thing that's objectively bad is how he handles determinants/characteristic polynomials.

A human being who writes this down and doesn't immediately projectile vomit either does not understand characteristic polynomials or eats cereal with toothpaste instead of milk and enjoys it

[heady ember]

[sage python]

Other than that, the main consideration is that he only works over R and C. If you're a CS/combo/algebra type this is suboptimal. It's probably good to learn how to do things over e.g. finite fields. Also relevant point is he de-emphasizes the matrix and linear equations pov.

Basically this has the angle that "linear algebra is finite dimensional functional analysis"

[pearl token]

Which book might be an appropriate choice when I finish reading this book (done right), if I'm interested in vector spaces over a finite field?

[placid pollen]

So like Paul Halmos but worse

[sudden kindle]

Whacko

[tawny crater]

Wait why? Whats the differences for finite fields?

[gray gazelle]

Any modern/contemporary, rigorous treatments of trigonometry, trigonmetric identities, etc?

Interested in various series, identities, and algebraic structures based on trigonometric functions. Recognizing how that taps into complex numbers and maybe even broader structures like quarternions, etc, looking for a later undergrad / early grad treatment.

[stray veldt]

this is a big question, but there are a bunch of things different
you will often use the fact that R is char 0, ordered, or move to C (which is algebraically closed) and use the fact C = R[i]

for example the theory of inner product spaces breaks completely without order

you have to use a more general theory

then char != 0 doesnt allow you to divide whenever you want and especially char 2 often needs special treatment

[gray gazelle]

Hey people, is Basic Mathematics by Lang a good book? - for an 11th grader that doesn’t take physics class.

[tawny crater]

True, it does make it far more complicated

[tame tree]

Which LA book would you recommend?

For first pass through

Finishing book of proof then looking for a good la book

[sage python]

Probably either Linear Algebra Done Wrong

Or Friedberg Insel Spence

[gray jungle]

go the full masochist route , roman. just in case, this is sarcasm

[tame tree]

Difference between LADW and LADR?

The latter gets flames for determinants I know that

Friedberg insel spence sounds good too

[placid pollen]

Just get the PDF of those books and compare the key theorems and the different proofs, the different definitions, try some exercises. I can't find linear algebra interesting by itself though, I sleeped through the course and then actually cared when I needed it and did just fine going back and reading the theorems and proofs I needed

LADW is more like Friedberg. LADR is like Paul Halmos Finite Dimensional Vector Spaces

[remote sparrow]

it's sort of an intro to proof book. a good portion is devoted to "precalculus" topics, but there is some basic number theory, geometry, and basic ideas about vectors (don't think he goes whole hog on linear algebra though).

you can take a look at the foreword and table of contents and see if it's right for you

[remote sparrow]

john baez recommended meckes and meckes' linear algebra book. free alternatives would be linear algebra by jim hefferon or linear algebra by robert beezer.

both hefferon and beezer's books have print copies that are very cheap too

a book that i haven't read but looks interesting would be linear algebra: theory, intuition, code

print copy is pretty cheap

[fierce hedge]

I was looking at some LA reference books and someone on mse had this comment on LA Done Right

Honestly, Axler's book, in my opinion, lacked any sense of "common sense" when presented to students with no formal analytical algebraic background. Even now, when I read back through that book, I see things and say, "why would you present that!?" or "why would you leave that exercise for the reader!? It is fundamentally important to understanding the chapter!!!" In my personal experience, the pedagogy linked to the book was also quite poor; it took me a few years to truly understand any of the concepts therein.

Is it that weird? @sage python

[remote sparrow]

that comment was written in 2012, but the latest edition of the text was published in 2015. axler sometimes participates on mathstackexchange, so there's a small chance those gripes could have been addressed.

[fierce hedge]

Ah, good to know

[remote sparrow]

i have a pdf and print copy of axler ¯_(ツ)_/¯

[remote sparrow]

https://www.youtube.com/playlist?list=PLMcpDl1Pr-viA25VUkHNmcUkWx9usPgyb

there's a video series to accompany the book

[slim peak]

In fact no, but actually yes 🤓

[sage python]

This is why I don't trust anyone other than me to teach linear algebra

[slim peak]

Hahaha*

The powerfulness of Real or Complex field-only approach is that you can provide very cool stuff like proof of Cayley-Hamilton via Cauchy Integrals and stuff

and still reach a full theory

While finite fields requires to go trough other representation of matrices, that are not (always, but still sometimes tbh) necessary like Similarity invariance, Jordan-blocks/reduction etc..

(up to introduce the appropriate field extension of course or even the algebraic closure)

[formal tinsel]

VMM was right

i do really like jacod and protter for probability

[midnight gulch]

Any book that explain machine learning stupidly easy?

[fierce hedge]

Andrew Ng lectures are the only thing I would categorise as stupidly easy

Otherwise ML is pretty probability -stats heavy and DL is heavy on linear algebra

[remote ginkgo]

this is so fking stupid

[placid pollen]

I checked the book a while ago, seems fine to get up to the level of highschool Math excluding calculus (if I remember right)

[timber ermine]

Hey everyone, I'm currently studying Prob and Stats, sophomore year. Could you recommend me a good book for the subject?

[hearty steppe]

Alright guys I’m gona recommend this when you finish your first chapter of your journey into dynamical systems (which currently is me looking at 12 books first on a combination of k complexity, general dynamic systems theory, and Ergodic theory before I actually get to the meat of this)

Oh boy man this stuff is explosive in terminology

https://quantumgravityresearch.org

Emergence theory

[topaz plaza]

im struggling in algebra one what book should I get so I can get better at it ? and do i don’t forget steps

so i can get advanced and start learning it better

[gentle arrow]

algebra, second edition by michael artin

ok but seriously if ur on high school algebra i just recommend you use khan academy tbh

[grand thistle]

why not lang

[gray gazelle]

Try Serge Lang’s “Algebra”

[topaz plaza]

1965 book?

[dapper root]

This was a super meme response

It’s literally just griefing

Don’t do that

[topaz plaza]

sorry didn’t know

[dapper root]

Oh no!

Sorry no I meant their recommendation

You’re totally fine, I’m just saying you should not use Lang because that’s a book for PhD students in a subject called “algebra” but not the one you want

[dapper root]

I concur with cat bread that using Khan academy is probably the best

[vague kindle]

interesting..

[topaz plaza]

I was gonna get the intro to algebra book by Richard Rusczyk

i was recommended that

[dapper root]

Yeah idk about books for that level

I just used whatever they made me do in school so I have no idea what options there are

[grave thorn]

functional analysis is infinite dimensional linear algebra

[delicate cliff]

Any book to learn calculus and functions? Are there any pre requisites?

[gentle arrow]

stewart is a pretty common one

prerequesites are

just have solid algebra/trig skills

[gray gazelle]

Spivak if you want to major in math, Stewart if you need it to do other stuff, MITOCW to Supplement, Khan when things get too difficult.

[foggy relic]

yea thats likely the best book for that

[gentle arrow]

idk if you were to do spivak

i think you would need some proofwriting skills

wait am i looking at the wrong book

i am

i was looking at calculus on manifolds

[gray gazelle]

You do, but proofs are just formal solutions and you can learn them quite quickly

[manic cape]

check out paul's online math notes

https://tutorial.math.lamar.edu/

[grave thorn]

proofs often take more than a "quick time", just saying

[gray gazelle]

Enough to begin, to be a GOOD proof-writer, yeah, it’ll take some time

[heady ember]

Just do Lee

[naive tendon]

is principia mathematica worth reading

[hazy elk]

How is this related to dynamics tho?

[remote sparrow]

Which one? If you're talking about the Russell and Whitehead one, there's a project that is modernizing the book.

[naive tendon]

yeah Russell’s

forgot another book had the same name

[remote sparrow]

https://www.principiarewrite.com/

[naive tendon]

alr thanks

[vital bane]

that book is awesome

[urban herald]

I need help with vector algebra, vector fields and integrals incorporating vectors(line integral, surface integral, etc. with their interpretations being in vectors, and also stuff like gauss divergence theorem and stone's theorem), what book or books should i get?

[urban herald]

yes i think so, i am struggling with understanding maxwell's equations and anything related to electricity and magnetism at a fundamental level, i am able to solve the problems but dont understand what terms like div and curl actually mean

[orchid mortar]

Div should be easy to get some intuition via first maxwell equation.
Curl I still don't get, but you could always treat it as a formula

[hazy elk]

You can probably just do khan academy calc 3 or maybe Griffiths (EM) ch 1 too.

Khan acad calc 3 covers the stuff in a way that you intuitively understand grad, div, curl etc

[urban herald]

thats the book i was recommended by my friend as well

i will be sure to check it out

thanks for the recommendations

[hearty steppe]

Non deterministic walks… now imagine those walks are guided through paths determined by directions on which the quasi crystal’s sharp edges are pointing as directions

The problem with Ergodic theory is it is for deterministic systems

This solves that problem

Every point is a quasi crystal structure that moves based on where those sharp node points are between each of the sides of the crystal in which it can move from one direction of the iteration in its walk to another non-deterministically

You also have objects composed of these quasi crystal structures

[hazy elk]

[hazy elk]

Are you talking about random walks on the crystal lattice wherein the probability values are dependent on the edges of the crystal?

[hearty steppe]

So imagine that the quasi crystal is a point that can move in all directions depending on its sharp node edges in terms of a nondeterministic walk. The walk might start out random but maybe there is persistent behavior that gives it some extended Ergodic behavior

Once we have persistent pathing work done in the system then we are going from nondeterministic to degrees of deterministic behavior perhaps

How does randomness become less random? When things in the system interact with eachother and cause deviations in the walks

The deviations in the walks might cause persistent behavior to happen

But do we have persistent yet nondeterministic behavior? How do we talk about that?

Maybe reasonably we can say we hit thresholds where deterministic behavior becomes unstable and leads to nondeterministic behavior when too much is happening to keep the system in an equilibrium state

Then we have emerging behavior

Degrees in which it’s nondeterministic in terms are based on initial conditions of the factors of emergence in the instance emerging behavior starts to occur

This is how I translate the usefulness of this “quantum gravity research”

And I probably saved you some time if your not ready to understand their work for yourself maybe. Or I totally made a fool of myself on math server. I’ll take both at this point

But all this is happening with one or more points simultaneously persisting in evolving state in a system in which the points themselves are these mathematically defined quasi crystal objects

Try thinking of a shrunken Hoberman sphere if that helps

Or this is a science experiment I enjoyed making. Think of an infinite amount of like these magnetic thingies but arranged in a sphere looking like crystal

And all those spikes imagine they are directional points in which the quasi crystal can walk or do movement

[hazy elk]

Ah its probably too difficult to explain meaningfully thru a discord text

[hearty steppe]

Probability values are based on our sample space where we have outcome weights for our events. Or essentially the outcome of our walks uniformly or in some way distributed where the weights total to the total possibility of one of our walk outcomes

The thing about probabilities is we have to translate them carefully because it’s a percentage value

But you can have negative probabilities too. I am not very familiar with that area to provide more comment other than you have a lot of situations where usually events cancel eachother out with “antievents “ or the complement of the events themselves

I think you can use events and antievents to explain why humans don’t act on all the thoughts they concoct in their heads

That would be a pretty cool research paper

[simple belfry]

Guys

Anyone here read a atomic habits book before

My English is so Bad 😭

[lime seal]

I am reading it . Finished just a few pages.

[hazy elk]

Probability measures are positive (at least that's what I learnt last semester in my probability course). I think you are misconstruing different things.

It's good to brainstorm and think of wierd things but I think it's important to understand the fundamental theory first.

[loud cradle]

there is a notion of signed measure, where both positive and negative measures are allowed, but I don't know if it's fruitful to apply it to probability

a signed measure is just the difference between two positive measures (see Hahn decomposition theorem)

[simple belfry]

Give me ur review for this book

[sleek python]

Negative probability (and quasiprobability distributions) are a thing in quantum mechanics apparently

https://en.m.wikipedia.org/wiki/Quasiprobability_distribution

https://en.m.wikipedia.org/wiki/Negative_probability

[hazy elk]

Oh

My bad then

[sleek python]

That seems like an interesting topic, however I cannot find purely mathematical resources that talk about it

[hearty steppe]

Look at the exchange post I referenced? I think I referenced it correctly about events and antievents?

It’s alright like I said I didn’t really dive into this stuff just kinda glanced at it

[hazy elk]

I see

Did you get around to doing brin and stuck ch 2 by any chance?

[hearty steppe]

Yea I am there I might start that tonight I don’t know. I really don’t have much of a reason to work myself that much harder this week cuz these prototypes are coming out the way I want them too 🥰

And I am a little manic driven from the climate and some lack of sleep so I should take it a little easy

[hazy elk]

I might take a look at them in 2 weeks, would be cool if we could discuss in vc

[hearty steppe]

Topological dynamics yea I been looking forward to going through that chapter. That’s not an easy book to digest.

[hearty steppe]

but honestly I need to make more headway with my k complexity books and those feel like easier reads to get through and prioritize right now

[formal tinsel]

does anyone here know of a complete solutions manual for Jacod and Protter probability essentials? I can only seem to find a partial one online

[waxen elbow]

hey, any thoughts on the open logic project ? looks like there is a lot of material

[normal reef]

Where can I er..."acquire" a solution manual to Spivak's?

[manic cape]

google

[normal reef]

yep been looking for an hour without results

[heady ember]

Idt there's one

[tacit lily]

solve it 🤓

[lime sapphire]

There is though...

[gray gazelle]

it's useful in the sense that we can consider the vector space of (Borel, regular?) measures with finite total variation. Then probability measures sit in the sphere there

Intersection of the sphere and the "positive cone"

Not quite the positive cone like on https://encyclopediaofmath.org/wiki/Positive_cone but that's how I think about it

then you can apply functional analysis to say something about the probability measures. Not sure if that's really useful but certainly something to consider

[heady ember]

Oh right that one

Oops i forgot

[grave thorn]

Signed measures aren't very important in probability

[vital bane]

i did, i just dont know if my solutions are correct

[remote slate]

what is dover books on mathematics series?

[remote sparrow]

math books published by dover. usually out-of-print works, but there are some new works published by them.

as a rule, most of them are quite affordable

[remote slate]

Ohoo

[floral agate]

Does anyone have any book recommendations that get into some higher uni level maths without loads of prerequisites?

[remote sparrow]

what have you already studied? also, while a book may have little or no formal prerequisites, it may be difficult to appreciate it without having studied its soft prerequisites.

[floral agate]

I get that