/mg/ - math general

[Anonymous No.11990339]

Regular polyhedra edition.
Previously: >>11975242

Are one variable polynomials with positive integer coefficients basically solved or are there any open questions in the topic? Vieta's formulas are pretty recent, being from the 1900s and all.

[Anonymous No.11990346]

>>11990339
>Elliptic curves

[Anonymous No.11990394]

>>11990346
Based

[Anonymous No.11990451]

Dropping the final version of the new chart from last thread.

[Anonymous No.11990456]

>>11990451
Stitz-Zeager is literally in every /sqt/ OP. The Book of Proof is also absent.

[Anonymous No.11990461]

>>11990339
Lads, is it just me or does this seem pretty tame for a commutative algebra exam?
https://www.maths.cam.ac.uk/postgrad/part-iii/files/pastpapers/2014/paper_1.pdf

[Anonymous No.11990464]

>>11990451

[Anonymous No.11990475]

I'm going back to school for applied math but it's been a while since I did some senior year math. What's a good little crash course I can do to refresh myself?

[Anonymous No.11990480]

>>11990346
Based

[Anonymous No.11990488]

>>11990475
Khan Academy has lots of video on highschool algebra with tests you take at the end before advancing to the next section. Main things I would focus on are factoring expressions, expanding expressions, geometric intuition of functions (analytic geometry), properties of exponents/logarithms, basic trig knowledge, and finding the roots of a polynomial.

[Anonymous No.11990529]

>>11990451
I still think Freitag and Busam is the superior complex analysis text. Shitty translation aside, it really is superb.

[Anonymous No.11990538]

>>11990339
God damn Spivak's Differential Geometry is just shitting on my life right now.

[Anonymous No.11990556]

>>11990475
Basic Mathematics - Lang

[Anonymous No.11990592]

>>11990451
>>11990529
Are any of these CA texts appropriate for an introductory level or are they grad level? I'll be taking it next semester and I'd like a good complementary text.

[Anonymous No.11990619]

>>11990451
The blue box thing is a bit fucked imo.

>spivak only covers single variable, apostol vol 2 (pictured) does multi
>algebra section has grad level books grouped with introductions
>LADR is technically meant for a second course, but whatever

[Anonymous No.11990642]

>>11990619
Well D&F serves as both an introduction and a grad level book. I mean, it's almost 1k pages.

[Anonymous No.11990671]

>>11990456
>Stitz-Zeager is literally in every /sqt/ OP.
Stitz-Zeager doesn't actually have a proper cover.
Also, /sqt/ links to the Stitz-Zeager website, not to any particular book.
>book of proof
Good point on that one.
>>11990464
Oh, yeah, gotta add that one in.
>>11990619
>apostol vol 2 (pictured) does multi
You see, I actually just used the second volume because the cover had higher resolution and I didn't notice it was the second volume.
>>algebra section has grad level books grouped with introductions
Isn't Pinter the only actual undergrad book there?
Maybe I should change the subtitle to "Books in the blue box cover roughly the same subjects, easier ones to the left."

BTW, added in Neukirch to complete the line.

[Anonymous No.11990695]

Recommend me an exersise book in group theory, if possible very difficult.

[Anonymous No.11990696]

>>11990671
Gelfand

[Anonymous No.11990702]

>>11990696
Nobody cares

[Anonymous No.11990709]

>>11990695
You only need one exercise: classify the finite simple groups. Understanding how to do that should take you long enough, especially if you do it from scratch. Come back for something harder when you're done.

[Anonymous No.11990740]

>>11990709
Quite funny
How about you eat a dick

[Anonymous No.11990749]

Good afternoon, /mg/!

https://www.pims.math.ca/scientific-event/200926-cts
>The Cascade Topology Seminar is a semi-annual conference hosted in the Pacific Northwest and Western Canada. It features topology of all kinds. This year's event will be hosted online.
By the way, is semiannual once in 6 months or once in 2 years? The first would make it literally semiannual, but the second one would also make sense if you think about annual being 1 instance/year and this would then be ½ instance/year.

[Anonymous No.11990759]

>>11990740
:(

[Anonymous No.11990819]

>>11990749
Afternoon, lad.

[Anonymous No.11990840]

>>11990695
[math]\textrm{Prove that any subgroup} \ H \ \textrm{such that ord}(H) = p^{n-1} \ \textrm{in a group} \ G \ \textrm{where ord}(G)=p^n \ \textrm{is normal in} \ G. \ p \ \textrm{is a prime number.}[/math]

[Anonymous No.11990866]

>>11990840
1. I asked for a book, not an exercise.
2. Literally just Sylow's theorem.

[Anonymous No.11990869]

>>11990840
Please stop fucking up the page formatting like this, and consider learning how to write mathematics before posting algebra exercises. A sentence should never begin with a math environment.

[Anonymous No.11990871]

>>11990840
Just show that [math] |H| = |g H g^{-1}| [/math] and that the order of H must be unique. QED

[Anonymous No.11990903]

>>11990840
there's a well known lemma that goes:
If G is group, p is smallest prime divisor of |G|, H is subgroup of G, [G:H]=p, then H is normal.

[Anonymous No.11990974]

>>11990671
hatcher

[Anonymous No.11991094]

What's the Bible of commutative algebra?

[Anonymous No.11991128]

>>11991094
Eisenbud.

[Anonymous No.11991133]

>>11990840
The result is clear if G has order p.
Otherwise, consider the normalizer N of H in G. Then N is equal to H or G.
Assume that N = H. Then H contains the (nontrivial) center Z of G. By induction hypothesis, H/Z is normal in G/Z, hence H is normal in G, a contradiction.

[Anonymous No.11991136]

>>11991128
This one?

[Anonymous No.11991141]

>>11991094
Bourbaki or Matsumura.
To actually learn commutative algebra, read Atiyah-Macdonald and Eisenbud instead

[Anonymous No.11991142]

>>11990974
No one seriously recommends Hatcher.
I recall this one time a dude told someone to read Hatcher until he noticed that Hatcher is garbage, and then to read Fuchs-Fomenko.
I recall this other time a dude mentioned Hatcher's recommendation list and made sure to mention that he didn't actually like Hatcher's book.

[Anonymous No.11991152]

>>11990749
Good afternoon fellow based Gabu-poster

[Anonymous No.11991168]

>>11991142
What's wrong with Hatcher?

[Anonymous No.11991242]

>>11991142
this is a good reclist
>>11991168
hatcher is a meme. not actually a good book. anyone who reads munkres fully should move on to fuchs fomenko. but i agree that people who only read the basic point-set part of munkres would benefit from hatcher before fuchs fomenko.

[Anonymous No.11991252]

>>11991242
Mendelson is good for basic point-set, too.

[Anonymous No.11991494]

>>11991142
"Hatcher's garbage" is just a meme spread by filtered undergrads. Despite its drawbacks, there is honestly no better textbook for an intro to algebraic topology

[Anonymous No.11991503]

>>11991494
>there is honestly no better textbook for an intro to algebraic topology
Are you forgetting that Rotman exists?

[Anonymous No.11991519]

>>11991494
Stop trying to make me add Hatcher when it's barely recommended damn it, I'm gonna have to either remove Neukirch or add Evans, Atiyah-MacDonald and Eisenbud to complete the line.

[Anonymous No.11991526]

GOAT AT book coming through.

[Anonymous No.11991529]

>>11990696
Interesting

[Anonymous No.11991533]

>>11991519
Why would you ever remove Eisenbud?

[Anonymous No.11991614]

>>11990451
What the fuck is this shill and meme list? None of these books is good.

[Anonymous No.11991628]

>>11991614
*tips fedora*

[Anonymous No.11991651]

>look up mathematician on Wikipedia
>was

[Anonymous No.11991660]

>>11991651
>Look Abel on Wikipedia
>Died (age 26)
>Look up Galois on Wikipedia
>Died (age 20)

[Anonymous No.11991676]

>>11990759
Don't care about him, he's a fool

[Anonymous No.11991689]

>>11990695
Dixon

[Anonymous No.11991692]

>>11991660
Galois was a simp.

[Anonymous No.11991786]

in compsci, they have all the complexity classes and the notions of "-hard" and "-complete"
in descriptive set theory, there are also various complexity classes and similar notion of "-hard" and "-complete"

is there any other place where the "-hard" and "-complete" naming scheme comes up? was it first conceived by compsci people or by mathematicians?

[Anonymous No.11991853]

>>11991786
Shut up whore

[Anonymous No.11991907]

why are polyhedro so cool?

[Anonymous No.11991981]

>>11990461
>Any 4 from 6
>3 hours
>Pt iii
Very tame, did majority of this at undergrad at a much inferior institution
A few of the cambridge past papers are like that, I remember having loads of tabs open going between them thinking they were exercise sheets or something

[Anonymous No.11991997]

>>11991614
>What the fuck is this shill and meme list?
It's the /sci/ shill and meme list? I thought it was obvious from the title and the description at the end.
>None of these books is good.
Impressive that you've read all of them.
Still, don't exaggerate. Fuchs-Fomenko, Cohn and Serre are good.

[Anonymous No.11992028]

>>11991981
>A few of the cambridge past papers are like that
Is there any reason for this? Like I said, for a topic such as commutative algebra that seems like a very generous exam.

[Anonymous No.11992057]

>>11991786
I don't know about those terms use in other disciplines, but the terminology goes back to at least the 40s. The notion of being complete was known by the descriptive set theorists of 1900-1910s. I think Luzin, for instance, prove that a set A is coanalytic if there is a continuous function from Baire space to the class of linear orders (represented as Polish space) so that A the pullback of the the subclass of well-orders. In modern terminology this means that the class of well-orders is is co analytic complete even if Luzin use that wording.

[Anonymous No.11992058]

>>11992057
*Luzin didn't use that wording

[Anonymous No.11992097]

>>11992028
The future of mathematics is inclusive to all backgrounds, chud.

[Anonymous No.11992116]

Can I read the Book of Proof without knowing calculus? It said something in line of "as you've already seen on calculus" and I noped out

[Anonymous No.11992123]

>>11992116
I haven't read it, but probably. There's nothing that should be in there that you need calc for, outside of maybe a few examples.

>>11991142
I'm a little sad Pugh or Abbott aren't on there, but there are literally hundreds of real analysis textbooks so whatever

[Anonymous No.11992139]

>>11992116
While you'll get more out of it if you've taken calculus, the only real prerequisite to it is a sense of 'mathematical maturity'.

[Anonymous No.11992685]

>>11990451

zero baby logic
zero baby set theory
zero baby category theory


i love how undergrads hate logic, set theory and cat theory, when math is heavily based on formal logic and models. Those people really have a naive deprecated view of maths from the middle age, which makes maths way harder to learn.

[Anonymous No.11992686]

Logarithms should be written down with reverse circumflex, so that ln(2) would be 2\/e

[Anonymous No.11992692]

[Anonymous No.11992693]

>>11990749
Good afternoon pedophile

[Anonymous No.11992695]

>>11992685
What are some good baby logic and set theory books?

[Anonymous No.11992699]

>>11990461
>Lads, is it just me or does this seem pretty tame for a commutative algebra exam?
you only have 3hs to do all this

[Anonymous No.11992750]

>>11992028
Most of the grade is in the work done before the exam as I recall

[Anonymous No.11992906]

Do you guys have cat theory book recommendations. I read a gentle introduction to category theory and it seems like it could be an interesting subject to study but obviously the book was really dumbed down

[Anonymous No.11992912]

>>11992906
Not recommending, more sparking discussion but what are your thoughts anons on "Topoi: a categorical analysis of logic"?

[Anonymous No.11992928]

>>11992906
Homotopy type theory is the real deal. Category theory is a dead end.

[Anonymous No.11992938]

A monad is a monoid in the category of endofunctors:
If you make a set with the indentity endofunctor, another endofunctor, and a morphism between the two, you get a monoid because:
A monoid is defined by a set with identity and associativity, morphisms are associative and you have identity
A morphism between endofunctors is a natural transformation
This becomes a monad because the morphism is [eqn]\mu: M \times M \to M[/eqn], it has associativity, it has identity.
Did I get that right or am I getting filtered by monads?

[Anonymous No.11992944]

>>11990339
>regular polyhedra
Biochemist here, I am trying to construct a 3d model of an icosahedral viral capsid from it's triangular face. Is there an angle of rotation around the centre I can transform the face such that repeating it 19 times will produce the entire icosahedron? How do I go about finding it?

[Anonymous No.11992954]

>>11992912
Haven’t read it but the idea of axiomizing math with category theory rather than set theory (if that’s what the book is about) seems neat. don’t know if I fully trust it though if you know what I mean.

[Anonymous No.11992980]

>>11992699
>only 3 hours
That is a lot of time for four of these questions, especially as a fourth year subject

[Anonymous No.11993070]

>>11990461
>define
>prove (aka write down the proof in the textbook)
>what does it mean that x y z
This isn't hard anon, it's just tedious. But I'd take this any day of the year in exchange for my exams. 2hrs and 7-10 non-trivial (i.e. cannot do them in your head) exercises. Maybe I'm oneupping but I'm annoyed at how these Ivy leauge unis have such low standards. No wonder women and miniroties get into and finish them

[Anonymous No.11993112]

Good morning, /mg/!

>>11992906
People like Categories for a Working Mathematician, so that would be one. I used Awodey which was also good. Also, Emily Riehl has some material which people seem to like.

>>11992938
If you want [math]M[/math] to be a monad, you need the unit [math]1\to M[/math] and the "operation" [math]M\circ M\to M[/math] which are natural transformations. The coherence conditions will correspond to the axioms of a monoid. However,
>If you make a set with the indentity endofunctor, another endofunctor, and a morphism between the two, you get a monoid because
is not true in general. You will have your natural transformations [math]M \to M\circ M\to M\circ M\circ M\to \cdots[/math], but not [math]M\circ M\to M[/math] such that the composite [math]M \to M\circ M\to M[/math] is the identity.

[Anonymous No.11993114]

>>11990346

Based

[Anonymous No.11993115]

>>11992685

can you do categorical numerical analysis?

[Anonymous No.11993119]

>>11993112
Morning, animechad

[Anonymous No.11993131]

Anyone able to imitate Tao's handwriting?

[Anonymous No.11993257]

>>11993131
>wirting in arial, like all women do

[Anonymous No.11993260]

>>11993112
>Good morning,
Good morning, pedophile

[Anonymous No.11993265]

>>11993260
Www

[Anonymous No.11993369]

>>11993131
Total dyel

[Anonymous No.11993405]

>>11992944
What do you actually want and what do you mean by "produce"? Your question doesn't seem well defined to me so far.

[Anonymous No.11993429]

>>11993405
In the same way that a straight line rotated 90 deg 3 times around a centre of symmetry makes a square, or 72 deg four times makes a pentagon, is there an xy (or xyz) rotation that when perfomed 19 times on an equilateral triangle makes an icosahedron?

[Anonymous No.11993455]

>>11992685
>undergraduates hate category theory
they love it anon

[Anonymous No.11993485]

>>11993112
Morning, lad.

[Anonymous No.11993521]

>>11993429
>In the same way that a straight line rotated 90 deg 3 times around a centre of symmetry makes a square
This isn't true.

[Anonymous No.11993595]

>>11992944
what software are you using?
perhaps this will help although it's not exactly what you're asking for: http://www.math.ubc.ca/~cass/courses/m308-03b/projects-03b/keating/projectweppage2.htm#Construction

[Anonymous No.11993615]

>>11993429
Oh, right.
There isn't. Notice from https://en.wikipedia.org/wiki/Icosahedral_symmetry that the group of symmetries is [math]A_5[/math], so every element has order 5 and none of them act transitively on the icosahedron.

[Anonymous No.11993621]

>>11993615
*order at most 5.

[Anonymous No.11993641]

>>11990339
>Regular polyhedra edition.
1. How do we know that there are only four regular star polyhedra?
2. How do we know that there are none regular Euclidean tilings with star polygons?

[Anonymous No.11993652]

What happens to geometry when the triangle inequality is strictly disobeyed
i.e. concatenation of two paths is strictly 'shorter' than the direct 'line' between two points

[Anonymous No.11993659]

>>11993641
why are the ones in OPs pic called "regular"? they have different faces

[Anonymous No.11993669]

>>11993652
well in euclidean geometry the line between two paths is equal to the concatenation of two paths in the case where the triangle inequality is ==
so youd get a contradiction i assume, or sprout an extra modified space of concatenated paths as opposed to pure paths

in non euclidean geometry though it might be different, the triangle inequality could hold == even when the concatenated paths arent lying on the direct line. in that case you would modify space but it might act strange or asymptotically at some bounds and potentially also become incoherent, unless you change the rest of the space too

[Anonymous No.11993673]

>>11993521
stop being obtuse

[Anonymous No.11993678]

>>11990451
whats on the cover of Kunze's LA book?

[Anonymous No.11993738]

>>11993429
Choose an appropriate representation and generators for the icosahedral group. You'll need more than one.

[Anonymous No.11993788]

>>11993669
I mean length is assigning a non-negative to two points and naturally we want to see linearity or semblance to euclid and cartesian coordinates
geodesics can be assumed to be the shortest path, but what happens when we assign different lengths to the cartesian product of some space with itself, only asking for L(a,b)=L(b,a)

[Anonymous No.11993802]

I'm scared to continue with mathematics. I've gotten to several variable calculus and it's not too hard if I just put some time and effort into it but everything you people do looks like it would be much harder to learn.

[Anonymous No.11993810]

>>11993802
Higher math is actually easier to some extent because more of it is justified and there's less handwavey bullshit, at least in certain areas.

[Anonymous No.11993824]

>>11993802
once you get past the index suplex notation, its not too bad
just imagine the y axis is n-1 dimensions, and x is the nth

[Anonymous No.11993838]

>>11993802
How do I unsubscribe from this blog?

[Anonymous No.11993857]

>>11993112
That makes more sense, thanks

[Anonymous No.11993863]

>>11993824
> index suplex notation
wat

[Anonymous No.11993874]

>>11993863
for me, keeping track of subscripts and superscripts was the most difficult part of it

[Anonymous No.11993890]

>A series converges if and only if the sequence of its summands converges to zero.

I love this theorem. So much simpler than in the reals.

[Anonymous No.11993899]

>>11993890
is this some surreal shit?

[Anonymous No.11993942]

>>11992695
I like
Dalen - Logic and Structure

People like
Halmos - Naive set theory?


>>11992938
Recall that [math]\bf{Set}[/math] can be equipped with a monoidal structure where units are singletons (= final objects in [math]\bf{Set}[/math],
e.g. [math]1:=\{0\}[/math]) and the product [math]\otimes[/math] can be taken to be the Cartesian product [math]M\otimes N:=M\times N[/math] (= categorical product for [math]\bf{Set}[/math]).
Here, a monoid object is a triple given by

[math]M[/math],

[math]e:1\to M[/math],

[math]*:M\times M\to M[/math].

Now a monad is given by

[math]T[/math],

[math]\eta:1_{\bf C}\xrightarrow{\bullet}T[/math],

[math]\mu:TT\xrightarrow{\bullet}T[/math]

and is also a monoid object, namely in the category of endofunctors [math]{\bf C}^{\bf C}[/math], with the monoidal product [math]\otimes[/math]
(not the categorical product) given by concatenation of functors [math]S\otimes T:= ST[/math].


Btw. I'm currently reading
https://www.cambridge.org/core/books/introduction-to-coalgebra/0D508876D20D95E17871320EADC185C6
if you want to read along. A 2012 version of it is online on the authors website

[Anonymous No.11993949]

>>11992685
>spacing after quoting a post
I'm sure you know where your opinion is going.

[Anonymous No.11993953]

>>11992695
I like
Dalen - Logic and Structure

People like
Halmos - Naive set theory?


>>11992938
Recall that [math]\bf{Set}[/math] can be equipped with a monoidal structure where units are singletons (= final objects in [math]\bf{Set}[/math],
e.g. [math] 1:=\{0\} [/math]) and the product [math]\otimes[/math] can be taken to be the Cartesian product [math]M\otimes N:=M\times N[/math] (= categorical product for [math]\bf{Set}[/math]).
Here, a monoid object is a triple given by

[math]M[/math],

[math]e:1\to M[/math],

[math]*:M\times M\to M[/math].

Now a monad is given by

[math]T[/math],

[math]\eta:1_{\bf C} \xrightarrow{\bullet} T[/math],

[math] \mu: TT \xrightarrow{\bullet} T [/math]

and is also a monoid object, namely in the category of endofunctors [math] {\bf C}^{\bf C} [/math], with the monoidal product [math] \otimes [/math]
(not the categorical product) given by concatenation of functors [math] S\otimes T:= ST [/math].


Btw. I'm currently reading
https://www.cambridge.org/core/books/introduction-to-coalgebra/0D508876D20D95E17871320EADC185C6
if you want to read along. A 2012 version of it is online on the authors website

[Anonymous No.11993995]

>>11990339
Here is Math 101
Prove below theorem using formal maths notation:
If x>= 4 then 2^x >= x^2
Let's see how many of you can prove in time.

[Anonymous No.11994009]

>>11993995
...in time for what?

[Anonymous No.11994010]

>>11993995
>>11993995
>2^x
no such thing

[Anonymous No.11994012]

>>11993899
>>11993874
what the hell are you talking about

[Anonymous No.11994013]

>>11994010
>no doubling allowed

[Anonymous No.11994015]

>>11994009
I am counting. It's been ~2 mins now.

[Anonymous No.11994018]

>>11994012
First quoted post refers to a result in surreal analysis which is much simpler than the corresponding thing for real numbers. Second quoted post is probably about index gymnastics in differential geometry.

[Anonymous No.11994019]

>>11994010
2 power x.

[Anonymous No.11994021]

>>11994015
Then I'll wait for you to answer me.
I'm not impatient.

[Anonymous No.11994043]

>>11994021
I am thinking how to do using "inductive proof"
Let k = x >= 4
Assume "If k>=4 then 2^k >= k^2" is true.
From above two statements I have to reach the inductive statement:
2^(k+1) >= (k+1)^2


I am doing this:
multiply both side by 2.
2 * 2^k >= 2* k^2
=> 2^(k+1) >= 2k^2
i am stuck here. help

[Anonymous No.11994055]

>>11994043
Use calculus.

[Anonymous No.11994068]

>>11992685
>undergrads
>hating logic, set theory, and cat theory

[Anonymous No.11994070]

>>11994043
You shouldn't solve this with induction, since it works for arbitrary real numbers.
Anyhow, if you're not the original poster, please give it some time for him to answer what's the hurry before posting a solution,

[Anonymous4 No.11994087]

Here's a lemma.
If f(a) = g(a), and f'(a) > g'(a), and f"(x) > g"(x) for all x >= a, the f(x) >= g(x) for all x >= a.
Prove.
Apply.

[Anonymous No.11994114]

>>11993899
>>11994012
I'm talking about the p-adics

[Anonymous No.11994126]

>>11994070
I am op. i posted that. the solution is written in next page of the book i m reading but i want to test myself on how can i do with knowledge i currently have.

[Anonymous No.11994129]

>>11994087
>In mathematics, informal logic and argument mapping, a lemma (plural lemmas or lemmata) is a generally minor, proven proposition which is used as a stepping stone to a larger result
You don't prove a lemma. It is established. Call it theorem.

[Anonymous No.11994139]

>>11994087
The rate of f(x) is greater than rate of g(x).
The rate of rate of f(x) is greater than rate of rate of g(x)
The rate of rate of rate of f(x) is greater than rate of rate of rate of g(x).
With above three statements it is fucking clear that f(x) will be greater than g(x) for any x >= a.

I am retarded, can't prove it using "Maths" logic notations.

[Anonymous No.11994152]

>>11993890
Based
I remember this from p-adic analysis. So much easier with the whole non archimedean norm.

[Anonymous4 No.11994166]

>>11994139
Have you seen any version of Taylor's Theorem? If not, you should google it. For example, it lets you express f(x) in terms of (x-a), f(a), f'(a), and f"(y) for some unknown y between a and x. You can then apply this to f(x)-g(x).

[Anonymous No.11994173]

>>11990339

pretty sure i prestiged through all of the levels in mw2, op

[Anonymous No.11994198]

>>11990451
Currently reading Aluffi's book, it really great

[Anonymous4 No.11994203]

>>11994166
Following myself up, alternatively you could derive f'(x) > g'(y) for x > a from f'(a) > g'(a) and f"(x) > g"(x) (for x > a) by integration, and then likewise derive f(x) > g(y) for x>a by integrating again. It seems like going backwards and forwards again, but it's so much easier to compare the second derivatives of 2^x and x^2.

[Anonymous No.11994238]

Let [math]A[/math] be a non empty set with an associative and commutative operation [math]+[/math].
Assume that, for every [math]a[/math] in [math]A[/math], the map [math]x\mapsto x+a[/math] from [math]A[/math] to [math]A[/math] is bijective.

Show that [math](A,+)[/math] is an abelian group.

[Anonymous No.11994258]

Can someone explain to me why the statement "Let eplision <0" is wrong? I say it every time I'm in math class and it causes my Professor to rage. I think it's because I'm smarter than he is

[Anonymous No.11994269]

How do you know someone else is already working on the same subject you are working?

[Anonymous4 No.11994277]

>>11994238
How does the set of positive integers under addition fit your problem? It doesn't have an additive identity element. (I think you need A to be finite.)

[Anonymous No.11994284]

>>11994277
For the positive integers the map wouldn't be bijective.

[Anonymous No.11994286]

>>11994258
kek

[Anonymous No.11994289]

>>11994238
Assume we've found the identity. Then every element has an inverse since for all a, there is some x such that x+a=e (by surjectivity of x->x+a).
Now pick some a, which we can do because A is nonempty, and by bijectivity pick the unique e_a such that e_a + a = a.
Then for any other element b, we have (e_a + b) + a = (a+ e_a) + b = b + a and by injectivity we see that e_a + b = b, thus e_a is indeed the identity element for A.

QED
>>11994277
n-> n+1 is not surjective

[Anonymous No.11994326]

>>11994198
>3
Show us the other ones anon.

[Anonymous No.11994338]

I played games and watched youtube all day instead of doing maths

[Anonymous No.11994345]

>>11994338
Based

[Anonymous No.11994350]

>>11994269
You have friends who work in your area and you go to conferences where you tell strangers about your work.

[Anonymous No.11994356]

>>11993659
I think op was inspired by a YouTube vid from Jan misali

[Anonymous4 No.11994549]

>>11994284
>>11994289
Momentarily confusing bijective with injective.

[Anonymous No.11994774]

1 + 2 + 4 + 8 + 16 + ... = -1

[Anonymous No.11994815]

Is Pinter sufficient preparation for harder books on that new chart, especially Lang?

[Anonymous No.11994859]

>>11992906
at some point it gets really obscure and you'll need motivation from other fields like al. topology or al. geometry to really have the will to continue (remember you don't do cat. theory for itself but for other theories) but for the basics you'll just need some mathematical maturity. Read Emily Riehl

[Anonymous No.11994868]

>>11994815
It'll give you a good foundation in algebra. Assuming you're also going to school and taking classes, then I would say yes.

[Anonymous No.11994914]

>>11994238
>>11994043
the absolute state of /mg/

[Anonymous No.11994921]

Why is group theory so much comfier than ring or field theory?

[Anonymous No.11994941]

>>11994914
Thanks, glad to contribute to this board's degeneracy.

[Anonymous No.11994954]

-12 -24 -36 -48 ... = 1

>>11994921
2 operations are more stressful than 1, ask any surgeon

[Anonymous No.11994976]

>>11994954
Kek

[Anonymous No.11995202]

Jesus Christ all statistics article I find online are written using word lmao. Also why do they spend like 20 pages explaining basic math concepts using words instead of just defining giving motivation for their definition and moving on? It seems highly respected statisticians can't formulate their tools using measure theory lmao.

[Anonymous No.11995274]

>>11995202
Part of mathematical maturity is having the good manners to let the statisticians have their fun in peace. We all know what it is, just leave them be.

[Anonymous No.11995305]

>>11995274
>Stats isn't mat-

[Anonymous No.11995316]

>>11995274
I mean, I wouldn't care so much if there wasn't a profound misuse of statistics in many fields. For example, there is no fucking way you can misinterpret a p value if you know how it is actually defined mathematically, but in statistics texts only a few books actually give a rigorous definitions because most of them are written for retarded psych majors. Obviously if you bullshit with words ambiguity is going to persist, the point of mathematical thinking is to avoid such things.

[Anonymous No.11995339]

>>11995305
Probability theory (i.e. using measure theory) is mathematics. Statistics is not.

[Anonymous No.11995349]

>>11995316
There aren't enough mathematicians or mathematically-literate scientists to use probability properly, so statistics acts as an easier, nonrigorous substitute. It's either that or disenfranchise everyone who can't do math, which is an especially large part of the social science community.

[Anonymous No.11995382]

>>11995305
>iCarly isn't ma-

https://www.youtube.com/watch?v=FYx_IGBX-EE

[Anonymous No.11995408]

>>11992750
I would've taken that Algebra exam this year if it hadn't been for corona. Problem sheets don't count for anything, it's 100% exam. I agree that the one posted looked quite easy.

[Anonymous No.11995435]

Today I understood how Dynkin diagrams classify semisimple Lie algebras. I had never gotten around to learning about Lie algebras but I'm very glad I did it now! The result is really satisfying, I highly recommend studying it.
I read Humphreys' "Introduction to Lie Algebras and Representation Theory". It took me about two weeks.

[Anonymous No.11995445]

>>11995382
What's the mathematical equivalent?

[Anonymous No.11995450]

>>11995435
I like Hall's "Lie Groups, Lie Algebras, and Representations: An Elementary Introduction".

[Anonymous No.11995471]

>>11995450
I had a little look at it recently,but it seems a little wordy. I never got around to studying smooth manifolds or Lie groups either, so I'm reading Warner's Foundations of Differentiable Manifolds and Lie Groups now. I get the feeling that a decent grounding in manifold theory is necessary to make sense of Lie groups properly and Hall's book doesn't offer that, even though it seems like a great text for its purpose. Also I want to understand Hodge decomposition for number theory and algebraic geometry.

[Anonymous No.11995545]

>>11990451
Why no Jacobson I and II? :(

[Anonymous No.11995621]

>>11991526

[Anonymous No.11995700]

Hey I've been stuck for a long time on this exercise pls help going to sleep now so I thank thee even though the thread might not be alive when I see a reply

Describe the sets of points in 3-space that satisfy the equation R=2cos(phi)
R, and phi denote the appropriate
spherical coordinates

Apparently the answer is a sphere centered at (0,0,1) with radius 1 and I have no idea at all where to begin or how to arrive at that,

[Anonymous No.11995752]

>>11995700
If you know the answer, then try finding the distance from (0,0,1) to an arbitrary point in that set.

[Anonymous No.11995806]

/mg/ what do you guys think is the most aesthetic branch of mathematics

[Anonymous No.11995809]

>>11995700
Project the equation into [math]\mathbb{R} ^2[/math] and think about what shape you get. Remember that [math]r = \cos (\varphi) \Rightarrow \sqrt{x^2+y^2} = \frac{x}{\sqrt{x^2+y^2}}[/math].

[Anonymous No.11995810]

>>11995806
Lie theory.

[sage in all fields No.11995812]

>>11995806
>Elliptic curves

[Anonymous No.11995814]

>>11995700
>>11995752
>>11995809
lmao is just the law of cosines you retards

[Anonymous No.11995821]

>>11995814
Yes, and?

[Anonymous No.11995841]

>>11995814
Sorry, I don't listen to hip hop.

[Anonymous No.11996112]

>>11995810
But I don't want lies. I want a true theory.

[Anonymous No.11996136]

>>11995806>>11995806
>/mg/ what do you guys think is the most aesthetic branch of mathematics

anything not bourbabki

[Anonymous No.11996137]

>>11996112
Lol.

[Anonymous No.11996280]

Math chads unite

we run it

[Anonymous No.11996668]

is there a counterpart to tropical geometry where max is used instead of min for the + operation

[Anonymous No.11996674]

>>11996668
They're exactly the same. Doesn't matter which one you use.

[Anonymous No.11996676]

>>11996668
Arctic geometry.

[Anonymous No.11996681]

>>11992906
Don't read category theory, read areas that use it like algebraic topology and algebraic geometry

[Anonymous No.11996689]

>>11996676
using polar coordinate =^)

[Anonymous No.11996696]

>tfw proving a definition doesn't depend on a choice of maps (2 maps), a choice of target, and a choice of representatives (4 choices but only 2 by symmetry)

[Anonymous No.11996701]

>>11996689
Thanks for the laugh.

[Anonymous No.11996774]

>>11994326

[Anonymous No.11997024]

>>11996774
How many do you have?

[Anonymous No.11997107]

Any number theorists in here? What are you studying/working on?

[Anonymous No.11997126]

>>11997107
number theories

[Anonymous No.11997129]

>>11997126
Do you know what number I am thinking of?

[Anonymous No.11997140]

>>11997129
no but i've got a theory

[Anonymous No.11997144]

>>11997140
I have numerous theories.

[Anonymous No.11997145]

https://en.wikipedia.org/wiki/Mathematics_of_Sudoku

[Anonymous No.11997205]

>>11997144
have you numbered them?

[Anonymous No.11997218]

>>11997205
Class Field Trip Theory
Boku want to go to Okinawa

[Anonymous No.11997219]

Is there a simple proof (i.e. one that doesn't use the full classification of finite simple groups) that there no more than two non-isomorphic finite simple groups of any order?

[Anonymous No.11997262]

>>11997140
Ok but what number I am thinking of?

[Anonymous No.11997272]

>>11997219
Yes: I am convinced that there are no more than two non-isomorphic finite simple groups of any order and I do not know of any proof to the contrary, also whilst my convictions may be wrong there are many other equivalent claims on the World Wide Web. Hence, I am pretty confident that that there are no more than two non-isomorphic finite simple groups of any order. QED.

[Anonymous No.11997276]

>>11997272
Ah yes... proof by idiocy

[Anonymous No.11997326]

>>11994356
based. as my homeboy jan misali gets more popular the base of people who learn toki pona will increase. o kama sona e ona ``la mi pona''

[Anonymous No.11997345]

>>11997219
Would be amazing if there is actually some meme duality result relating the simple groups of same cardinality.
But I don't think there is.

[Anonymous No.11997346]

>>11995349
Can you elaborate on the difference between probability and statistics? Someone once told me the difference is analogous to “looking at an animal and exploring what type of foot prints it will leave” and “looking at a foot prints and determining what kind of animal made them”.

Do you think that is a fitting analogy? Why is probability mathematically sound but statistics isn’t?

[Anonymous No.11997354]

>>11997346
>Phoneposter

[Anonymous No.11997368]

>>11997354
I was homeless for a while and my laptop is broken. Please no bully.

[Anonymous No.11997377]

>>11997346
Not that anon, but one can formulate (at least almost) all of probability using measure theory. It is thus rigorous.

[Anonymous No.11997398]

Good afternoon, /mg/!

>I am writing to announce a new site, which was created to house a collection of short research videos from junior mathematicians in all areas of pure mathematics. The goal of this site is to provide a platform for junior researchers to share their work with a broader audience, without the scheduling conflicts of an online seminar/conference. We hope that this site facilitates the spread of mathematical ideas and collaborations, especially from and between those at the start of their mathematical careers.
>You can visit our site for more information at
https://sites.google.com/view/jmra
t. email

[Anonymous No.11997415]

>>11997398
Afternoon, lad.

[Anonymous No.11997444]

>>11997346
A statistic is typically the result of a ton of probabilistic theory, which is math. Too often, stats in application and teaching is cookie-cutter and very much not math.

As in, there are many theorems about optimal testing, convergence of random variables, sequences and such. And much in the same way there are 'proofs' about models in physics, stats is very similar in that respect. It is often times math, but mostly math adjacent.

[Anonymous No.11997449]

>>11997272
>>11997276
give me a proof retards

[Anonymous No.11997452]

>>11997346
Statistics is in general just a field dedicated to the analysis of data which is obviously (for a non autist) beyond the scope of pure math. Modern statistics does have some sort of general theory that is founded in probability theory and many modern concepts in statistical inference are just that. The thing is that math is inherently abstract and general so a concrete interpretation of any concept is bound to be wrong, but you have to weight that with the fact that when learning statistics you must understand how to apply the mathematical tools you are learning, so some heuristics are sometimes inevitable.

[Anonymous No.11997507]

how fucked up is it to use a notation like this A:x

to reference x in the tuple A=(a,b,x,c)

[Anonymous No.11997617]

>>11997507

I guess it's ok since i didnt get any horrified replies yet

[Anonymous No.11997619]

>>11997507
just say x

[Anonymous No.11997626]

>>11997619

no there is another tuple B from the same set of tuples that all are (a,b,x,c)

[Anonymous No.11997632]

>over 99% of groups of order less than 2000 have order 1024

[Anonymous No.11997643]

>>11997632
There are 49,487,365,422 groups of order 1024.

[Anonymous No.11997682]

>>11997643
There are 155682086691137947272042502251643461917498835481022016 magmas of order 8, up to isomorphism.

[Anonymous No.11997796]

look at those gigachads

https://www.genealogy.math.ndsu.nodak.edu/id.php?id=18926
https://www.genealogy.math.ndsu.nodak.edu/id.php?id=49555

on the other hand, there are people who leave just fire and brimstones

> inb4 Robin COPE Hartshorne

[Anonymous No.11997838]

Any recommendations for the linguistic and semiotic (syntax, semantics) aspects behind maths?

[Anonymous No.11997841]

>>11990592
They're fine for both.

[Anonymous No.11997855]

>>11997796
>Jean-Pierre Serre has 4 students and 99 descendants.
>Robert Langlands has 8 students and 48 descendants.
How come they have so few?

[Anonymous No.11997909]

>>11997796
>Robin COPE Hartshorne

[Anonymous No.11997939]

>>11997626
the same variable x?
Or are they x_1 and x_2

[Anonymous No.11997952]

hi i made a discord servre if you frens wanna join
https://discord.gg/ctNVCA

[Anonymous No.11997956]

Why is the Langlands programme so hard to get into? It requires a lot of depth in areas as different as the infinite dimensional representation theory of Lie groups, Hodge theory, and etale cohomology. What other field requires elliptic operators and class field theory at the same time? It's frightening.

[Anonymous No.11997958]

>>11997632
>>11997643
>>11997682
Source?

[Anonymous No.11997960]

>>11997958
Aluffi pg. 82

[Anonymous No.11997977]

>>11997632
Why is this surprising?

[Anonymous No.11997999]

>>11997958
i wrote them on a napkin once but blew my nose into it

[Anonymous No.11998003]

I wonder if there's a general formula for the number of groups of order 2^n.

[Anonymous No.11998017]

https://en.wikipedia.org/wiki/Tarski_monster_group

[Anonymous No.11998022]

What’s wrong with the following ‘proof’ that symmetry and transitivity imply
reflexivity? Assume that the relation ∼ is symmetric and transitive, and argue that
a ∼ a for all a, as follows: (1) since ∼ is symmetric, a ∼ b implies b ∼ a; (2) since
a ∼ b and b ∼ a, by transitivity it follows that a ∼ a. Why doesn’t this argument
work?

[Anonymous No.11998035]

Is there a nice closed formula for

[math] \sum_{k=0}^{m-1} \dfrac{n!}{(n-k)!} [/math]

?

Mathematica sadly assumes non-natural values and returns something something EiIntegral (even if I simpify assunming m and n are ints), but there might be a nice number assuming they are ints.

[Anonymous No.11998042]

>>11998003
no, we don't know if the majority of finite groups are of order 2^n either (although that's very likely the case).

[Anonymous No.11998046]

>>11997958
http://oeis.org/A001329

[Anonymous No.11998055]

>>11998022
Let [math]A=\{0,1,2\}[/math] and let a relation be given by [math]\pi=\{(0,0),(1,1),(0,1),(1,0)\}. \ (2,2) \notin \pi \Rightarrow \pi[/math] is not an equivalence relation.

[Anonymous No.11998069]

>>11990346
he's back on top

[Anonymous No.11998085]

>>11998055
ye, for a, we don't necessarily know if a~b for some b, that is assumed without justification

[Anonymous No.11998090]

>>11998035
did you write m-1 instead of n-1 in the cool E

[Anonymous No.11998097]

>>11998090
I did.

[Anonymous No.11998111]

>>11998042
Am I missing something here? The set of finite groups is countably infinite, isn't it? And includes a countably infinite number of cyclic groups with orders not of the form 2^n. So in what sense can the majority be of that order?

[Anonymous No.11998113]

>>11998090
um actually it's called a sigma

[Anonymous No.11998124]

>>11998111
For any natural number n you let [math]f(n)[/math] be the number of groups of order [math]2^i < n[/math] over the number of groups of order smaller than [math]n[/math].
Then, if the function converges to [math]1[/math] as [math]n[/math] goes to infinity, you're golden.

[Anonymous No.11998127]

>>11998111
In terms of density. number of groups of order 2^n < k divided by number of groups of order < k is conjectured to go to 1 as k goes to infinity.

[Anonymous No.11998151]

[Anonymous No.11998167]

>>11998151
based

[Anonymous No.11998175]

>>11998151
There's nothing wrong with planar geometry.

[Anonymous No.11998186]

>>11998151
Pretty funny to see a shot at the other side.

t. someone who liked algebra and topology (and hence categories) as an undergrad

[Anonymous No.11998190]

>>11998151
I'm a problem soolver, and I'm proud of it.

[Anonymous No.11998192]

What are some good grad level texts on group theory?

[Anonymous No.11998194]

>>11998151
>Does analysis through memorization and bruteforce
Literally me, except instead of analysis it's every single field of maths.

[Anonymous No.11998196]

>>11998192
What do you need it for?

[Anonymous No.11998202]

>>11998196
Personal interest.

[Anonymous No.11998206]

What's the equivalent of Jech for group theory?

[Anonymous No.11998217]

>>11998113
the cool E will catch on, signerd

[Anonymous No.11998232]

>>11998206
That makes me curious of your characterization of Jech.

[Anonymous No.11998233]

>>11998232
something like a reference book for specialists?

[Anonymous No.11998270]

>>11998035
n!(e-1)

[Anonymous No.11998278]

>>11998233
I think group theory is too broad for that.

>>11998270
m is finite, even smaller than n, the result will be in N

[Anonymous No.11998286]

>>11998270
maybe you misread the incomplete Gamma function

[Anonymous No.11998289]

>>11998278
Well round it then if you want.

[Anonymous No.11998294]

>>11998278
What about such a book in finite group theory?
Like the tools that were developped to prove the finite simple group classification?

[Anonymous No.11998304]

>>11998278
>>11998286
Ok to be more accurate you could take
floor(n!(e-1) - 1/(n+1) )
It works perfectly for n larger than 5

[Anonymous No.11998313]

I just want a comfy group theory book, lads.

[Anonymous No.11998319]

>>11998313
Serre has one

[Anonymous No.11998354]

>open math book
>cringe
>close it
>go back to playing video games

[Anonymous No.11998356]

>>11998354
don't post here ever again.

[Anonymous No.11998363]

>>11998356
to be fair to him, 90% of math books are shit

[Anonymous No.11998367]

>>11998151
Math olympiad was a mistake.

[Anonymous No.11998370]

>>11998363
read better books. I've never read a bad book in my entire life, because I don't have enough time to waste on bad books.

[Anonymous No.11998390]

>>11998363
Name one (1) bad book from Springer or AMS.

[Anonymous No.11998397]

>>11990339
>vieta's formulas
LOL this basic fact was actually given a name

[Anonymous No.11998413]

>>11998390
Linear Algebra Done Right.

[Anonymous No.11998451]

>>11990346
Based

[Anonymous No.11998462]

>>11998390
There are GTM books which have glaring mathematical errors, it's not all perfect.

[Anonymous No.11998510]

>>11998413
Brainlet

[Anonymous No.11998813]

>end of chapter has 20 exercises
>1-10 normal
>11-20
>for each question answer questions 1-10 again but with this other assumption
>actual question count is 110
why do books do this?

[Anonymous No.11998842]

>>11998510
t. Shelly Axley.

[Anonymous No.11998851]

>>11998813
then just dont do all of the questions lol

[Anonymous No.11998867]

>>11998851
My OCD prevents me from skipping exercises.

[Anonymous No.11998878]

Tip from the coach
Rinse eyedrops has doubled my cognitive endurance
Because I thought I was getting tired from thinking when really my eyes just got dry

[Anonymous No.11998897]

>>11998294
Gorenstein - Finite groups

[Anonymous No.11999293]

Any of you currently working and not in a grad program? How do you handle self-study? How do you find people to talk math with outside of this Mongolian horticulture forum? Do you have plans to apply to a grad program? Basically, how do you cope?

[Anonymous No.11999342]

>>11999293
Have you considered going to grad school?

[Anonymous No.11999372]

>>11999342
My concerns are money and the fact that I am tied to my present location for a while. I am in the USA. I don't think I'd get into a PhD given my background / 4+ years in industry, so it'd be a Master's. A part-time degree where I am currently located is not out of the question. Full-time probably isn't an option (seems like giving up a salary would be a bad move unless it were a PhD with funding), and relocation probably isn't an option either (not single, a lot to figure out there).

So my questions are more to see if anyone in a similar position has found any sense of joy or community since I may be occupying this in-between space for a while.

[Anonymous No.11999382]

>>11999372
Nothing wrong with a master's.

[Anonymous No.11999630]

>>11999293
I'm working a full-time job and have a family to care for. There is little time for me to do math. Only in the subway while commuting and sometimes in the evening I find time to read a little bit about math or check /mg/. Still glad I chose to leave university after my master degree though and not to pursue a PhD. The atmosphere at university is too unhealthy.

[Anonymous No.11999641]

>>11999630
>The atmosphere at university is too unhealthy
Is it really that bad in the US?

[Anonymous No.11999649]

>>11999641
For women it is. She's obviously more comfortable in her home with her children.

[None !HvKi...... No.11999682]

Can somebody explain e^(pi*i)=-1 once and for all?

[Anonymous No.11999717]

>>11999682

[Anonymous No.11999770]

>>11999641
I'm in Germany. It's extremely bad here. At university you're constantly confronted with hateful feminism and genderism and there is no escape. At my former university literally every wall and every door was covered with antifa graffiti and stickers. That's what made me decide to keep intellectual pursuits as a hobby and to finish my degree as quickly as possible so I can leave this horrible place behind.

[Anonymous No.11999778]

>>11999641
Went to a commuter school for undergrad, the atmosphere was definitely progressive leaning but most students didn't care much for it.

[Anonymous No.11999794]

>>11999641
No, but I wouldn't say it's great either.

[Anonymous No.11999958]

Is there any material for learning how to use a proof assistant?

[Anonymous No.11999998]

>>11999770
>I got scared away from research because of some stickers and posters
erbärmlich

[Anonymous No.12000072]

>>11997952
?

[Anonymous No.12000101]

>>11999649
>>11999770
Some perspective from Яussia
We had around 12-15 girls on the maths programme out of 40 students total
Almost all of them went through the full 6-year course (and around 5-6 went for doctoral), so most of the tutors/professors knew what those were capable of by then. Girls were never looked down upon in terms of research topics etc., but it kinda seemed they got preferential treatment on exams from random professors who only knew them for a single semester
No political bullshit at all, obviously

Overall, it was quite a healthy climate in terms of cultural/gender/political issues. Lots of nepotism, though. And rather unappealing career choices afterwards

[Anonymous No.12000106]

>>12000101
You need to do 6 years of maths before you start a PhD in Russia?

[Anonymous No.12000109]

>>11998042
>we

[Anonymous No.12000115]

>>12000106
You finish school at around 17, then there is a 4-year bachelor programme and a 2-year master programme. Then you can apply for an aspirant programme which eventually leads you to a PhD.
So what I meant was the aspirant programme, I don't know the US nomenclature.

[Anonymous No.12000123]

in france, at 18, you have 3 years for some undergraduate study, then 2 years of masters, then you get to phd

[Anonymous No.12000138]

>>12000123
In the UK it's just 3+1. I'm starting my PhD in a couple of weeks and I feel woefully underprepared.

[Anonymous No.12000174]

>>11990451
>>11990671
>proofs books
>calculus
>lang's gebra instead of clark's gebra
>axler's abomination
>hoffman, kunze instead of shilov
disgrace of a chart

[Anonymous No.12000541]

>>11990339
Please halp me /mg/, what's the primary difference between )))calculus((( and (((calculus))?
>I am not concerned with making difficulties for you as a Jew, but only with protecting – above all – German students of the second semester from being taught differential and integral calculus by a teacher of a race quite foreign to them. I, like everyone else, do not doubt your ability to instruct suitable students of whatever origin in the purely abstract aspects of mathematics. But I know that many academic courses, especially the differential and integral calculus, have at the same time educative value, inducting the pupil not only to a conceptual world but also to a different frame of mind. But since the latter depends very substantially on the racial composition of the individual, it follows that a German student should not be allowed to be trained by a Jewish teacher.
https://en.wikipedia.org/wiki/Oswald_Teichm%C3%BCller#cite_ref-8

[Anonymous No.12000629]

>>12000541
)))calculus(((: the same calculus that was done by Euler, Gauss and Riemann. Focus on actual ideas, physical relevance and interpretation.
(((calculus))): focus on foundations, high rigor before getting to actually know the subject, disregard or indifference towards physical relevance.

[Anonymous No.12000659]

https://www.youtube.com/watch?v=e2A1Bg12bJo
This really looks like a video posted by someone who doesn't really enjoy topology much.
Here's the secret: no topologist gives a single shit about rigor and you fucking morons belong in algebra.

[Anonymous No.12000938]

>>12000138
You'll be alright, where are you going?

[Anonymous No.12001066]

>>11997024

[Anonymous No.12001116]

>>12001066
Please don't lewd Tibees.

[Anonymous No.12001163]

Do I need to use Stoke's Theorem to find what points belong to the boundary of the set of an annulus?

[Anonymous No.12001279]

>>12000659
>>11999641
zoomers are ruining mathematics. they have already ruined the internet

[Anonymous No.12001282]

What math do I need to know to move from pure to applied in industry? I can't stand the idea of growing old as a mediocre academic.

[Anonymous No.12001289]

>>12001282
Depends on what you want to do. Applied math isn't a monolith.

[Anonymous No.12001291]

>>12001289
What are the big areas though that are most useful? I heard that calculus of variations was important for engineering?

[Anonymous No.12001307]

>>12000659
Absolute rigor is the only way to BTFO the 2+2=5 crowd, who use vagueness and equivocation to trick laymen into accepting their arguments.

[Anonymous No.12001363]

>>12000123
that's if you are a retard. If you're smart, it's two years of prépa (three if you're a bit stupid, one if you're Serre) and then 4 years of ENS.

[Anonymous No.12001371]

What's the motivation for defining the trace of a matrix?

The determinant is the product of the matrix's eigenvalues, and therefore measures the factor by which the matrix stretches space. I get why it's an interesting thing to study. But why the hell should I care about the *sum* of eigenvalues?

("It's invariant under change of basis" is not a satisfactory answer, in my opinion)

[Anonymous No.12001377]

>>12001371
Trace is invariant under basis transformations, and is therefore widely used in representation and character theory.

[Anonymous No.12001588]

>>12001371
[eqn]\operatorname{tr} M = \sum_{i} M_{ii} = \sum_{i} \lambda_i[/eqn] If the fact that these two sums are equivalent is not fascinating enough to study the trace for you, then nothing will be satisfactory. Computationally it's also very useful, e.g. [math]\det e^{M} = e^{\operatorname{tr} M}[/math] and >>12001377

[Anonymous No.12001708]

>>12001371
thing is invariant -> thing might be useful
i don't think there's any deeper understanding to be found here
no one "studies the trace" just like no one "studies the chain rule", you just notice it and realize it will probably be useful
these posters >>12001377 >>12001588 have pointed out why trace is useful, but those are applications that were most likely conceived by people who already knew what's a trace of a matrix

[Anonymous No.12001899]

>>12001708
>no one "studies the trace" just like no one "studies the chain rule"
What do you mean by that, though? I mean people write books and papers about trace class operators. Or partial traces, in which case they correspond to "entangled" marginalization. So it's definately studied.

The relative change of a parametrized volume [math] \det(A) [/math] being given as

[math] \det \left( A^{-1}\,A'(t) \right) = \operatorname{tr} \left(A(t)^{-1}\,A'(t)\right) [/math]

is maybe a good intuition already.

There's this relation [math] \det(E-A)=\exp(-\sum_n\frac{1}{n}\mathrm{tr}(A^n)) [/math] that pops up occationally,

From

[math] (1-a_p) = \exp\log(1-a_p) = {\mathrm e}^{-\sum_n a_p^n/n} = \prod_n{\mathrm e}^{-a_p^n/n} [/math]

such that

[math] \prod_p^d (1-a_p) = \prod_p^d \prod_n {\mathrm e}^{-a_p^n/n} = \prod_n {\mathrm e}^{-\frac{1}{n}\sum_p^da_p^n} = {\mathrm e}^{-\sum_n\frac{1}{n}\sum_p^da_p^n} [/math]

E.g. as

[math] \zeta(s) = \prod_p^\infty\frac{1}{1-p^{-s}} \implies \zeta(s) = \exp\left(\sum_n\frac{1}{n}\sum_p^\infty p^{-s\cdot{}n}\right) [/math]

or local zeta-function's. It made me think the rationality part of the Weil conjectures aren't as far fetched as I first thought.

[Anonymous No.12001921]

>>12001899

[Anonymous No.12001944]

>>12001708
>no one "studies the chain rule"
I'm a couple weeks into calculus 1 now, doing well, already past the chain rule and beyond. Quotient rule was a joke. Product rule remains my specialty.

I ask my professor his thoughts on quantum mechanics and partial derivatives. He's impressed i know about the subject. We converse after class for some time, sharing mathematical insights; i can keep up. He tells me of great things ahead like series and laplacians. I tell him i already read about series on wikipedia. He is yet again impressed at my enthusiasm. What a joy it is to have your professor visibly brighten when he learns of your talents.

And now I sit here wondering what it must be like to be a brainlet, unable to engage your professor as an intellectual peer. All of the deep conversations you people must miss out on because you aren't able to overcome the intellectual IQ barrier that stands in the way of your academic success... it's so sad. My professor and I know each other on first name basis now, but i call him Dr. out of respect.

And yet here you brainlets sit, probably havent even made eye contact with yours out of fear that they will gauge your brainlet IQ levels.

A true shame, but just know it is because i was born special that i am special. I can't help being a genius, nor can my professor. Two of a kind is two flocks in a bush.

[Anonymous No.12002005]

>>12001944
every time

[Anonymous No.12002007]

>>>12002000

More /mg/.