Anonymous at Sun, 7 Jul 2024 09:43:01 UTC No. 16271294

Trying to see if I can generalize the notion of the spectrum of a commutative ring.
A ring is a monoid object in the category of abelian groups, so we should look at monoid objects in any monoidal category with a terminal object (maybe we want it to be an abelian category as well). I'm not sure if we need commutativity, but if so, then we look at commutative monoid objects.

Given any prime ideal [math] \mathfrak{p} [/math] of a commutative ring [math] R, [/math] we can localize at the [math] \mathfrak{p} [/math] and then mod out by the unique maximal ideal of that local ring to get a map [math] R\to k_{\mathfrak{p}} [/math] where [math] k_{\mathfrak{p}} [/math] is a field. Conversely, given any morphism from [math] R [/math] to a field, we can recover a prime ideal which are all the elements sent to [math] 0. [/math]
Hence, prime ideals exactly correspond to morphisms into a field (i.e. simple objects in the category of commutative rings).

Therefore, we can define the spectrum of a (commutative) monoid object of a monoidal (abelian) category as the morphisms to simple objects. I guess we might as well consider the spectra of all such objects, and so we consider the collection of morphisms from monoid objects to simple monoid objects. I think this view allows us to compare spectra of two rings, but it may be a little loose.

Next, we want to place a topology on this spectrum; I guess we try to mimic the Zariski site, and so we need the notion of inverting a single element. I guess the key is to note that in the setting of commutative rings, if [math] a\in R [/math] and [math] R\to k [/math] where [math] a^{-1}\in k [/math] for some field [math] k, [/math] then this map factors through [math] R\to R[a^{-1}]. [/math] In particular, for any prime ideal [math] \mathfrak{p} [/math] not containing [math] a, [/math] then the corresponding map [math] R\to k_{\mathfrak{p}} [/math] factors through [math] R\to R[a^{-1}]. [/math]

Anonymous at Sun, 7 Jul 2024 09:44:02 UTC No. 16271296

>>16271294
Now we only really care about prime elements [math] a\in R [/math] since if [math] a=bc [/math] then [math] b^{-1}=c(bc)^{-1}\in R[a^{-1}] [/math] and similarly [math] c^{-1}\in R[a^{-1}] [/math]
Now obviously each prime element [math] a\in R [/math] corresponds to a (minimal) prime ideal, viz. the one generated by [math] a. [/math]

Here's where I'm a little unsure, but I think we want to consider all the prime ideals which don't contain the ideal generated by [math] a. [/math] Since they don't contain this ideal, they don't contain [math] a [/math] and so all the corresponding morphisms [math] R\to k_{\mathfrak{p}}\ni a^{-1} [/math] factor through [math] R\to R[a^{-1}]. [/math] I want to say that this factorization satisfies a universal property so that [math] R[a^{-1}] [/math] is some colimit.

The difficulty seems to be with choosing which prime ideals we want to consider; in particular identifying a minimal prime ideal. Actually, it is pretty easy to see that if [math] \mathfrak{p}\subset\mathfrak{q} [/math] then [math] k_{\mathfrak{p}}\supset k_{\mathfrak{q}}. [/math] A prime ideal [math] \mathfrak{p} [/math] is minimal then for any field extension [math] k_{\mathfrak{p}}\hookrightarrow k, [/math] any map [math] R\to k [/math] factors through [math] R\to k_{\mathfrak{p}}. [/math]

So I guess we want to consider in some sense a "maximal" object [math] k_a [/math] in the category of simple monoid objects, and this will correspond to all the ideals of an element [math] a\in R. [/math] We should then take the category of all simple monoid objects which are not subobjects of [math] k_a, [/math] and the obvious functor from this category to the category of objects under [math] R. [/math] The colimit of this functor will then give our desired object [math] R[a^{-1}]. [/math]

Finally, we mimic the Zariski site, taking the topology on the opposite category of monoid objects to have base consisting of isomorphisms and maps [math] R[a^{-1}]\to R. [/math]

Anonymous at Sun, 7 Jul 2024 09:49:48 UTC No. 16271303

>>16271294
>>>/x/

Anonymous at Sun, 7 Jul 2024 09:51:25 UTC No. 16271304

>>16271294
>>16271296
In some sense, I think this seems nice, since it really exhibits the important role that simple objects play, which is something that is possibly overlooked.
On the other hand, it feels kind of clunky, having to jump to the category of simple objects and find the maximal such ones. I think there might also be problems if you try to take injections as opposed to honest-to-god embeddings.

Anonymous at Sun, 7 Jul 2024 09:53:50 UTC No. 16271305

>>16271303
Sorry, I'll try to stay on topic.
I'm really struggling on my calculus homework. My teacher keeps telling me that I can't just plug in the variable when taking the limit. I tried explaining to him that it works everytime to get the correct answer, but the retard just won't listen.

Anonymous at Sun, 7 Jul 2024 10:11:34 UTC No. 16271322

>>16271294
>>16271296
interesting idea. I don't see why you are trying to recover the spectrum as a set though. It is probably easier to just focus on recovering the site straight away, maybe by generalizing some universal property, e.g. spec is adjoint to [math]\Gamma[/math]. Then you can play around with what categories you have your functors between.

Anonymous at Sun, 7 Jul 2024 10:46:32 UTC No. 16271347

>>16271296
>>16271294
Mhm. Does this take into account that 1+1 equals 1? No... Well.... pity.

Anonymous at Sun, 7 Jul 2024 11:18:57 UTC No. 16271373

>>16271237
Hey I'm kind of a brainlet. Does anyone know a simple or neat example of adjunctions arising from monads (especially in programming contexts)?

Anonymous at Sun, 7 Jul 2024 13:18:49 UTC No. 16271457

Does there exist a space-filling rhombic disphenoid?

Anonymous at Sun, 7 Jul 2024 19:12:27 UTC No. 16271804

bump

Anonymous at Mon, 8 Jul 2024 15:32:39 UTC No. 16272869

I wonder where all the analysis to go. The number theory and algebraic people are way more active in /mg/

Anonymous at Mon, 8 Jul 2024 15:35:00 UTC No. 16272871

>>16272869
*analysists

Anonymous at Mon, 8 Jul 2024 15:42:11 UTC No. 16272875

>>16272871
Mouf. Now.

Anonymous at Mon, 8 Jul 2024 15:51:03 UTC No. 16272885

>>16271237
Why are proofs of spectral theorem and the likes so darn complicated in books? I have seen proofs using induction, minimal polynomials etc. However, I discovered a simple proof using orthogonality of eigenvectors of symmetric which gives way to such a nice enlightening geometry. I just simply don't understand the reason for all this convoluted proofs.

Anonymous at Mon, 8 Jul 2024 16:22:42 UTC No. 16272908

>>16272885
Which spectral theorem are you referring to?
> I just simply don't understand the reason for all this convoluted proofs.
Which proof in particular confuses you? Ask specific questions and we may be able to help.

Anonymous at Mon, 8 Jul 2024 16:34:23 UTC No. 16272933

If you have a fraction so that the denominator is the numerator backwards, do you ever get an integer?

Anonymous at Mon, 8 Jul 2024 16:36:24 UTC No. 16272936

>>16272933
Yes. Providing an example is left as a (trivial) exercise for the reader.

Anonymous at Mon, 8 Jul 2024 16:39:55 UTC No. 16272939

>>16272933
many such examples
infinitely many in fact

Anonymous at Mon, 8 Jul 2024 16:59:04 UTC No. 16272963

>>16272933
1/1

Anonymous at Mon, 8 Jul 2024 16:59:07 UTC No. 16272965

>>16272933
In base-1, all of them

Anonymous at Mon, 8 Jul 2024 16:59:49 UTC No. 16272966

>>16272933
90/09 = 10

Anonymous at Mon, 8 Jul 2024 17:15:41 UTC No. 16272984

>>16272966
Is there an example that is multiple digit and doesn’t end in 0?

Anonymous at Mon, 8 Jul 2024 17:27:04 UTC No. 16273004

>>16272984
There are infinitely many.

Anonymous at Mon, 8 Jul 2024 17:32:36 UTC No. 16273008

>>16272933
lrn2code
the obvious ones are palindromes and palindromes with any amount of zeros appended, i wondered if there were others and one minute of writing code produced others: 8712

Anonymous at Mon, 8 Jul 2024 17:33:07 UTC No. 16273009

>>16273004
Have any examples been explicitly constructed?

Anonymous at Mon, 8 Jul 2024 17:33:39 UTC No. 16273010

>>16273009
Yes.
The shortest I know of is only 4 digits long. Very easy to find.

Anonymous at Mon, 8 Jul 2024 17:38:09 UTC No. 16273015

>>16273010
I wonder how quickly the number of examples below N grow as a function of N. Is it log, like prime numbers? Or worse? Or better?

Anonymous at Mon, 8 Jul 2024 17:38:44 UTC No. 16273017

>>16273015
Who cares? It's base dependent. Boring!

Anonymous at Mon, 8 Jul 2024 17:38:52 UTC No. 16273019

>>16273008
Interestingly, 8799912 (or any amount of 9s) works.

Anonymous at Mon, 8 Jul 2024 21:52:29 UTC No. 16273296

>>16272966
Doesn't count. You can't start a number with a zero

Anonymous at Tue, 9 Jul 2024 00:15:03 UTC No. 16273455

>>16273017
Is the asymptotic rate of growth base dependent though?

raspberry pie at Tue, 9 Jul 2024 00:16:35 UTC No. 16273456

>>16273004
*There are infinity.
Saying 'infinitely many' is a bloated and cluttered way of just saying what we already have a word for: Infinity. Anyone that ever uses the phrase 'infinitely many' is a turbo autist.

Anonymous at Tue, 9 Jul 2024 00:19:52 UTC No. 16273459

>>16273456
You are wrong.

Anonymous at Tue, 9 Jul 2024 01:07:45 UTC No. 16273530

>>16272933
10[__99999...__]89 works

For various reasons, the quotient and the first integer of the denominator can only be (2,6), (4,8), and (9,9).

We assume the quotient of 1 and 5 aren't allowed if you can't do 0's

raspberry pie at Tue, 9 Jul 2024 01:10:28 UTC No. 16273534

>>16273459
Wrong. You are wrong.

Anonymous at Tue, 9 Jul 2024 01:15:18 UTC No. 16273537

>>16273530
(2,6) do not work, so it's just (4,8) and (9,9)

Anonymous at Tue, 9 Jul 2024 01:29:38 UTC No. 16273551

Anonymous at Tue, 9 Jul 2024 01:46:28 UTC No. 16273574

>>16273530
>>16273537
Any combination of [1089..1089...1089] works actually, and with the 99999... in the middle. I wasn't too thorough but that might be it for the quotient of 9. I'm not gonna do 4, im done.

Anonymous at Tue, 9 Jul 2024 04:13:31 UTC No. 16273676

Anyone here cracked with Sn rep theory? I'm reading Vershik and Okunkov's paper on their approach and trying to apply it to Khovanov's Heisenberg category, but I'm still getting my feet wet so it seems.

Anonymous at Tue, 9 Jul 2024 05:45:36 UTC No. 16273761

>the reader will enjoy proving
ok I'm done
fuck you book

Anonymous at Tue, 9 Jul 2024 05:50:46 UTC No. 16273763

>>16273761
Authors who do this are based. Math should absolutely retain its high barrier to entry, and not pander to the common minds.
Just look in any grad department, and the quality of the students is disappointing to say the least. It's not coincidence that this shift in quality coincides with the push to make math more inclusive and "user-friendly".

Anonymous at Tue, 9 Jul 2024 05:53:12 UTC No. 16273764

>>16273763
Hard agree.
The book very clearly defined its target audience, >>16273761 was not in it, and he acts like it's somehow the author's loss

Anonymous at Tue, 9 Jul 2024 06:11:12 UTC No. 16273778

>>16271305
Let f(x)=1 for all x ≠ 0 and f(0)=1. Now your method gives the wrong answer.

Anonymous at Tue, 9 Jul 2024 07:06:30 UTC No. 16273822

How did ya'll get better at maths?

Anonymous at Tue, 9 Jul 2024 07:24:35 UTC No. 16273833

>>16273822
We molested the fabric of the universe until it told us more about itself.

Anonymous at Tue, 9 Jul 2024 07:36:28 UTC No. 16273849

>>16273833
That's physics, not math! For math you have to molest the fabric of the multiverse of all logically possible universes.

Anonymous at Tue, 9 Jul 2024 08:41:14 UTC No. 16273892

Are there any number theorists on here? How are you supposed to raise a number to a p-adic number? Nobody I've talked to knows how to do it. The only clue I have is that the resulting value should only depend on the residue class of the numbers mod 4

Anonymous at Tue, 9 Jul 2024 08:48:36 UTC No. 16273896

>>16273892
Generate a p-adic system
Define each point as a subset of a set of other points
>by definition, each point is generate by a number of points raised to another set of points

I actually can only do truth tables, so sorry if my answer doesn't help!

Anonymous at Tue, 9 Jul 2024 08:51:17 UTC No. 16273898

>>16273892
https://math.stackexchange.com/questions/436984/raising-a-rational-integer-to-a-p-adic-power
might provide you some insight you'll find helpful

Anonymous at Tue, 9 Jul 2024 09:00:17 UTC No. 16273907

>>16273896
>tfw you just make yourself right by definition instead of doing math
>truth tables are op

Anonymous at Tue, 9 Jul 2024 09:08:22 UTC No. 16273912

>>16273892
aren't they just units in z2, not p-adic numbers?

Anonymous at Tue, 9 Jul 2024 09:08:49 UTC No. 16273914

Is studying Gelfrand's books and Lang's Basic Mathematics really necessary to get into other fields? I can understand Stewart's Precalculus book just fine and I haven't done those

Anonymous at Tue, 9 Jul 2024 09:09:23 UTC No. 16273916

>>16273914
Lang is a meme

Anonymous at Tue, 9 Jul 2024 09:11:34 UTC No. 16273917

>>16273916
Well what do you recommend for the basics? I find Gelfand dull and obtuse

Anonymous at Tue, 9 Jul 2024 09:17:38 UTC No. 16273921

>>16273912
an invertible p-adic number is a p-adic number
>>16273898
thanks I might be able to get this to work

Anonymous at Tue, 9 Jul 2024 17:23:11 UTC No. 16274438

hello math, I posted this in /sqt/ but haven't got a good enough response yet so I'm posting it again here.
math/cs/stats folks itt, I'm sort of an average joe, midwit, autodidact ig and I wish to learn analysis and discrete math, but I only know pre calc algebra. I've got some questions regarding this.
For calculus and analysis:
>Should I just complete calc1,2,3 b4 doing analysis or can I go straight ahead?
>What are the best texts from a theory & problem solving perspective for learning calc 1,2,3 and analysis? I did search a little bit and found baby rudin and apostol being recommended for analysis. I was also able to find some not so often recommended texts like Courant's calculus(for analysis?), Keisler's elementary calculus, thomas' calculus, etc.. What do?
>Any good place for finding problem sets & lecture notes that may really help in improving my grasp of applications based calculus and analysis? What about books like schaum series, as in do they have a nice variety of problems? How about previous year GRE papers, putnam, etc.?
For discrete math:
>Again, what books do I use from theory & problem solving perspective? CL Liu? Kenneth Rosen? Don knuth's concrete math? something else?
>where do I find good problem sets & lecture notes that are really useful and again, is schaum's series, previous year GRE papers useful?
>Do I have to spent incredible amounts of time trying to study topics like combinatorics & number theory independently if my aim is to do machine learning, theoretical comp sci like algorithms, cryptography, etc.?

Anonymous at Tue, 9 Jul 2024 17:30:10 UTC No. 16274451

>>16274438
>>Should I just complete calc1,2,3 b4 doing analysis or can I go straight ahead?
about that: it depends on the book. You have books like Amann Escher, Zorich, Apostol which do both simultaneously while Rudin expects prior knowledge in Calc.

Anonymous at Tue, 9 Jul 2024 20:22:24 UTC No. 16274938

>>16274438
just do calc 1 up to and including the fundamental theorem of calculus, then jump into analysis
reason: calc 2,3 are only needed for engineers. All relevant theorems from calc 2,3 are contained in multivariate analysis courses/books or as preliminaries to differential geometry

Anonymous at Tue, 9 Jul 2024 20:25:18 UTC No. 16274946

>>16274438
Look up Evan Chen's notes for Math 55b, Honors Analysis. It has most of the interesting analysis

Anonymous at Tue, 9 Jul 2024 20:30:39 UTC No. 16274951

Are quadratic polynomials over the p-adics classified? Or, say, cubics in two or three or so variables?

Anonymous at Tue, 9 Jul 2024 20:33:08 UTC No. 16274956

>>16274438
read a different book on each topic: brezis for func ana, ahlfors for complex ana, gelfand for harmonic ana, bogachev for measure theory, etc. Books that do all these topics at once are incredibly shallow and you won't get any wiser

Anonymous at Tue, 9 Jul 2024 22:10:04 UTC No. 16275109

>>16274946
>Evan Chen
legit tranny or was this pic some prank? kek
>>16274451
>Amann escher, zorich apostol which do both simultaneously
you mean single & multi variable calc or were you trying to mean that it has a lot of computation related stuff like tricks for solving integrals. the kind of stuff commonly seen in calc 1 or 2?
>>16274938
what texts can I use if my prime goal is just calculus 1,2,3 and analysis of single variables? I ask this because I may or may not have time for analysis and was expecting calculus texts good for applications and problems.
>>16274956
ok

Anonymous at Wed, 10 Jul 2024 05:12:14 UTC No. 16275596

>>16275109
>you mean single & multi variable calc
yes but also analysis. They have everything a calc book would have but treat the theory rigorously

Anonymous at Wed, 10 Jul 2024 05:21:49 UTC No. 16275601

>>16274438
You'll want to learn up to Calc V. In Calc I one learns how to differentiate functions of one variable. In Calc II one learns how to integrate them. In Calc III one does both with finitely many variables. The natural continuation would be Calc IV: differentiation of functions on infinite-dimensional spaces. Finally, Calc V is integration on infinite-dimensional spaces (essentially the theory of Gaussian Feynman path/functional integrals).

Anonymous at Wed, 10 Jul 2024 10:59:36 UTC No. 16275829

How the fuck do you prove the equalizer existence thing in singular cohomology with points? I have functions f,g from A to X and an equalizer of f,g (up to homotopy) h: X to Z. Given a cohomology class in X that pulls back to the same class under f and g, i want to prove that it's a pullback of a class on Z. All cohomology pointed. I've tried defining it at a singular cochain level but didnt get anywhere.
This is in an attempt to prove that the pointed singular cohomology functors are homotopy functors (this is one of the two properties).

Anonymous at Wed, 10 Jul 2024 12:55:50 UTC No. 16275943

Should I go for a PhD in theoretical CS?
I have a math degree, getting excelent marks on mathematical subjects was no issue, but it turned out my real talent is hidden in algorithms design. Should I pursue it, or go for e.g. algebraic geometry?
I'm asking because I'm not sure how rewarding TCS will be in the end.

Anonymous at Wed, 10 Jul 2024 12:58:22 UTC No. 16275947

>>16273761
>not trying to prove all the results in the book yourself first anyway

Anonymous at Wed, 10 Jul 2024 13:04:59 UTC No. 16275958

>>16273917
I'm working through Basic Mathematics right now and I think it's pretty good. It can be a bit of a slog in the beginning because of the precise language he uses but you get into the flow and really helps you develop a deeper understand of why things work the way they do.

Anonymous at Wed, 10 Jul 2024 13:15:48 UTC No. 16275975

>>16275943
>algorithms design
Such as?

Anonymous at Wed, 10 Jul 2024 13:23:26 UTC No. 16275989

>>16275958
you'l want to get used to that language. Basically all math is written in a precise manner

Anonymous at Wed, 10 Jul 2024 13:51:22 UTC No. 16276006

>>16275109
no evan chen actually trannied out

Anonymous at Wed, 10 Jul 2024 13:52:46 UTC No. 16276008

>>16276006
sad. Was he a manlet who got race & heightpilled?

Anonymous at Wed, 10 Jul 2024 13:53:39 UTC No. 16276009

>>16276006
>>16275109
Please dont use that word. It's uncalled for.

Anonymous at Wed, 10 Jul 2024 14:03:31 UTC No. 16276016

>>16276009
didn't mean to sound rude anon, but it breaks my heart to see talented people getting filtered from the genepool because they just so happened to be physically unattractive.

Anonymous at Wed, 10 Jul 2024 14:41:29 UTC No. 16276053

>>16275989
That's why I decided to persevere

Anonymous at Wed, 10 Jul 2024 16:16:09 UTC No. 16276122

>>16275829
Now that I'm on my computer let me typeset this a bit better.
In the category of path connected nondegenerately based spaces we are given
[math]f_0, f_1 :A\to X[/math]
[math]g: X \to Z[/math] s.t. [math]g \circ f_0 \simeq g \circ f_1 [/math]
For all spaces [math]Z'[/math] and maps [math]h: X \to Z'[/math], if [math]h \circ f_0 \simeq h \circ f_1[/math],
then [math]\exists k:Z \to Z', h \simeq k \circ g[/math].
Let [math]z \in H^q(X, x_0, G)[/math] be such that [math]f_0^*(z) = f_1^*(z) \in H^q(A, a_0, G)[/math].
I want to prove that
[math]\exists w \in H^q(Z, z_0, G), g^*(w) = z[/math]
H is singular cohomology, G is an arbitrary abelian group.

🗑️ Anonymous at Wed, 10 Jul 2024 17:58:46 UTC No. 16276256

>>16272908
>Which spectral theorem are you referring to?
Of matrices.
>Which proof in particular confuses you?
None of them confuse me and I mentioned which proofs.

Anonymous at Wed, 10 Jul 2024 18:05:50 UTC No. 16276265

>>16272908
>Which spectral theorem are you referring to?
Of matrices.
>Which proof in particular confuses you?
None of them confuse me and I mentioned which proofs.

Anonymous at Wed, 10 Jul 2024 20:12:46 UTC No. 16276436

>>16276265
>Of matrices
normal, real, complex operators? Just be specific. When I hear someone talk about a "general" spectral theorem for linear operators over a field, I assume they're referring to the decomposition of a vector space in generalized eigenspaces and a non-spectral part. This decomposition can look wildly differently depending on the operator

Anonymous at Wed, 10 Jul 2024 20:27:35 UTC No. 16276472

>>16275109
bro just start reading and find out yourself, youre wasting your time here

Anonymous at Wed, 10 Jul 2024 21:09:12 UTC No. 16276529

>>16276436
Matrices over IR with Euclidean inner product.

Anonymous at Wed, 10 Jul 2024 21:12:50 UTC No. 16276536

>>16276122
There's a supposed answer to this on Math SE but it's wrong https://math.stackexchange.com/questions/2342851/proof-that-cohomology-functor-is-a-homotopy-functor
It depends on the specific topology of the equalizer by decomposing it and applying Maier Vietoris sequences. The decomposition doesn't make any sense (perhaps they misunderstood the question). Also it's not given that the equalizer in question is unique and equal to the one in the construction.
For example, an equalizer of the two elements
[math]f_0 = [id], f_1 = 2[id] \in \pi^1(S^1, p) = [S^1, p; S^1;p][/math] can be any space, since the map that's asserted to exist is explicitly allowed not to be unique.
Thus if [math](Z, z_0)[/math] is any space, [math]f: (S^1, p) \to (Z, z_0)[/math], [math]f(x) = z_0[/math] satisfies the condition, since if
[math]g:(S^1,p) \to (Z', z_0')[/math] is any map, then
[math][g \circ(id)] = [g] \in \pi^1(Z', z_0')[/math] and
[math][g \circ(2id)] = 2[g][/math]
Thus [math]g \circ f_0 \simeq g \circ f_1 \implies [g] = 0[/math]
and any map [math]h: Z \to Z' [/math] will do.

I know there are people ITT who claim to have read Spanier and liked it. If you're one of them, please explain, because I'm getting filtered.

Anonymous at Wed, 10 Jul 2024 21:15:48 UTC No. 16276542

>>16276006
damn most asian bois make good femboys but he is NOT one of them

Anonymous at Wed, 10 Jul 2024 21:25:48 UTC No. 16276559

>>16276536
Nevermind, by the property we pull the cocycle in the standard construction to the random equalizer and the pull it back to the space X, which equals the original cocycle by commutativity. So in fact it does suffices to consider the standard construction.

Anonymous at Wed, 10 Jul 2024 21:55:21 UTC No. 16276614

>>16276559
Yup it works out, but not the way that the person wrote in the answer (idk wtf he was thinking) and also using a bit of a different construction than the one Spanier provides.
[math]H^q(Z, z_0) \to_m H^q(A) \oplus H^q(X, z_0) \to_n H^q(A) \oplus H^q(A)[/math]
Where
[math]m(z) = (z|_{A\times\{1/2\}},z|_X)[/math] and
[math]n(a,x) = (a - f_0^*(x), a - f_1^*(x))[/math]
Which is pretty much exactly what we need. However, to get this decomposition we have to use a different equalizer than the one defined in the book. In the book, Spanier collapses the interval
[math]\{z_0\}\times I[/math] but in our construction we don't, so that we can take a nice slice of [math]A[/math] in the middle of our wrapped cylinder. That's why some of the cohomology groups are pointed and others are not.

Anonymous at Wed, 10 Jul 2024 21:57:01 UTC No. 16276615

>>16276614
Actually it's not that we can take a nice slice of [math]A[/math] but so that we can take its neighborhood without intersecting the space X. Because if we do, then we also intersect a neighborhood of X which is problematic to deal with, and we can no longer nicely deformation retract onto spaces X and A in our Maier Vietoris decomposition.

Anonymous at Wed, 10 Jul 2024 22:03:22 UTC No. 16276618

>>16276529
>>16276265
The spectral theorem for symmetric/hermitian matrices is essentially algebraic because the hypotheses are algebraic. You can relate the result to geometric ideas but at some point the key of the proof will depend on an algebraic idea. The approach suggested by your image is to use the quadratic form to show the existence of the maximum or minimum eigenvalue, but how do you justify a priori without the spectral theorem that you can see the quadratic form as an ellipse? Well ignoring that we know that the maximum is achieved irregardless of this let's say by [math]X_1[/math] now, how can we conclude that it is actually an eigenvector? Here is the issue you can basically say [math](AX_1-\lambda_1 X_1,X_1)=0[/math] and then why is it that this implies that it is an eigenvector, that is, [math]X_1\in Ker(A-\lambda_1 I)[\math]? Well here is the part where all boils down to the algebra since this is an if and only if for symmetric semi-definite matrices and I don't see any way to make sense of this part without an algebraic argument. Actually this if and only if is usually justified using the spectral theorem but well.

Anonymous at Wed, 10 Jul 2024 22:03:45 UTC No. 16276619

It's over.

Anonymous at Thu, 11 Jul 2024 00:29:26 UTC No. 16276751

>>16276619
No it's not the act of computation is symmetrical.

Anonymous at Thu, 11 Jul 2024 01:30:03 UTC No. 16276788

Can someone explain the structure sheaf and canonical bundle to me? I don't get it at all.

Anonymous at Thu, 11 Jul 2024 01:48:07 UTC No. 16276803

>>16276788
The way I think of the structure sheaf (I assume you're referring to this iteration of it) of a suitable variety is the set of the "allowed" rational functions on an open subset U. If you take an open ball on a surface for instance, you cannot have rational functions that assume zeros of the bottom polynomial inside that set allowed in your set of "allowed" (regular) functions. So this creates a picture where when you restrict your set more and more up to a single point (this is the stalk), you end up with a ton of rational functions, and it's actually the same as localizing away from the prime ideal that point represents in the coordinate ring. But when you look at the entire variety, you're not allowed basically ANY rational functions at all. This creates a picture that resembles a sheaf of wheat, if you're able to picture it. If you pinch the bottom of the sheaf of wheat, you get a big plumage on top, i.e. lots of allowed regular functions, and if you make the plumage at the bottom of the sheaf very large, you get only a small number of regular functions. This particular sheaf is one we get "for free" and so we use it a lot.

The key here is that in general the structure sheaf tracks what rational (again we use the term regular) functions you are allowed to have, i.e. no zeros of the bottom polynomial, in any particular open subset of the variety.

As for the canonical line bundle, you have to be familiar with the cotangent bundle first. In that case it's very similar to the volume form of a manifold, if you know what that is. It's actually something you can think of pictorially quite effectively if you consider 3d space as a motivating example.

Hope that helps, i'm kinda retarded

Anonymous at Thu, 11 Jul 2024 03:57:34 UTC No. 16276875

>>16275975
New and faster algorithms for well known graph problems. I'm not comfortable doxing myself by sharing my most cited paper.

Anonymous at Thu, 11 Jul 2024 05:31:56 UTC No. 16276935

>>16275109
>I may or may not have time for analysis
but you sure do have time for shitposting on /sci/

🗑️ Anonymous at Thu, 11 Jul 2024 07:07:35 UTC No. 16276965

>>16276618

Anonymous at Thu, 11 Jul 2024 09:31:16 UTC No. 16277067

>>16276618

Anonymous at Thu, 11 Jul 2024 12:28:28 UTC No. 16277234

>>16277067
Read Lang's Algebra, it's so much better

Anonymous at Thu, 11 Jul 2024 12:31:00 UTC No. 16277237

>>16277234
Kill yourself.

Anonymous at Thu, 11 Jul 2024 13:25:23 UTC No. 16277283

>>16275943
consider specializing in computational aspects of algebraic geometry. have a look at Gröbner basis theory, for example

Anonymous at Thu, 11 Jul 2024 15:16:46 UTC No. 16277386

>>16277237
not that anon but Lang's section on spectral decomposition is genuinely a nice read. Great exercises too

Anonymous at Thu, 11 Jul 2024 15:18:31 UTC No. 16277390

Chapter 10 of this book (spectral theory for unbounded operators) is owning me hard, anyone got some good sources for learning about it? I think Grandpa Rudin Covers the spectral theorem for unbounded operators towards the end, is it worth giving it a shot?

Anonymous at Thu, 11 Jul 2024 16:10:36 UTC No. 16277460

>>16271296
>>16271294
This is your brain on commutative algebra

Anonymous at Thu, 11 Jul 2024 18:46:48 UTC No. 16277657

>>16277390
I think it is pointless without knowing some measure theory. https://www.mat.univie.ac.at/~gerald/ftp/book-schroe/schroe.pdf
Here are some notes wich are more foccused but the writting is sometimes quite shit and has some errors.

Anonymous at Thu, 11 Jul 2024 19:31:12 UTC No. 16277734

>>16275601
>The natural continuation would be Calc IV: differentiation of functions on infinite-dimensional spaces. Finally, Calc V is integration on infinite-dimensional spaces (essentially the theory of Gaussian Feynman path/functional integrals).
is that graduate level stuff?

Anonymous at Thu, 11 Jul 2024 19:52:40 UTC No. 16277772

>>16277734
late undergrad/grad and nobody calls it "calc IV/V"

Anonymous at Thu, 11 Jul 2024 20:03:42 UTC No. 16277784

hi /math/

how do you take notes for future reference in a clean and efficient manner? pen and paper? digitally with OneNote or other note-taking app? pen and paper and then LaTeX + pdf? going through a book right now and I'd like to keep my solutions as reference

Anonymous at Thu, 11 Jul 2024 20:32:04 UTC No. 16277818

>>16277784
I usually sketch a solution on paper and then type it up in latex. I like this because when writing a proof out in detail I often catch mistakes I made before.

Anonymous at Thu, 11 Jul 2024 20:40:17 UTC No. 16277827

I'm reading through this book about Differential Geometry, where a surface at a 3D point p can be parametrized by a function X that maps parts of R^2 to the surface around p. The author is going through a proof that doing a coordinate change from one parametrization to another is diffeomorphic. The proof is shown in the pic, and my abbreviated version is in the bottom in red. In my short version, I don't talk about neighborhoods - it's more a gist of the proof.

My issue is, he goes on to say that in that proof, it is vital that X^{-1} and Y^{-1} are continuous, and honestly, I'm not seeing it. At the top, he does say that (X^{-1} o Y) is a homeomorphism, and that the rest of the proof is to show it is also a diffeomorphism. But what if X^{-1} isn't continuous? I don't see how the proof fails. The Inverse Function Theorem (IFT) concerns the differentiability of X and Y, not that their inverses are continuous.

Like, does it have something to do with the neighborhoods of X(q)? I wish he just explicitly mentioned the importance during the proof instead of later on

Anonymous at Thu, 11 Jul 2024 20:42:17 UTC No. 16277832

>>16277784
I just use pen and paper
TeXing notes is way too much fucking work unless you intend to hand them out to other people and I don't own a tablet

For a long time I didn't take any notes at all but eventually you reach the point where people start telling you things that aren't written down anywhere and I kept forgetting them

Anonymous at Thu, 11 Jul 2024 20:57:16 UTC No. 16277854

>>16277772
what is called then?

Anonymous at Thu, 11 Jul 2024 21:15:35 UTC No. 16277871

>>16277657
I do know measure theory, not sure what makes you think I don't.

Anonymous at Fri, 12 Jul 2024 05:06:32 UTC No. 16278244

>>16273778
this is just the constant function 1. The signum function defined by f(x)=-1 for x<0, f(x)=0 for x=0 and f(x)=1 for x>0 would show the failure of anon's method.

Anonymous at Fri, 12 Jul 2024 05:11:26 UTC No. 16278248

>>16273456
Infinity is a more elusive concept than its use in this particular context

Anonymous at Fri, 12 Jul 2024 05:12:56 UTC No. 16278249

>>16272933
let d1=...=dn. Then the number d1...dn/dn...d1 is an integer.

Anonymous at Fri, 12 Jul 2024 05:33:10 UTC No. 16278270

>>16277871
Lmao I was just stating it because it is a requirement. I don't know who the fuck are you or what you actually know.

Anonymous at Fri, 12 Jul 2024 06:15:19 UTC No. 16278328

>>16276875
Cool, what type of methods do you use?

Anonymous at Fri, 12 Jul 2024 07:45:16 UTC No. 16278414

>>16277827
by chain rule, the local inverse to a C^k function is C^k. you care about the inverses being continuous because otherwise the change of coordinates might not be a homeomorphism, which would just be weird: the underlying space remains the same, so a change of local coordinates should retain the local geometric structure (i.e. be a C^k isomorphism AKA diffeomorphism)

Anonymous at Fri, 12 Jul 2024 08:41:55 UTC No. 16278469

I love mathematics but I am stupid :(

Anonymous at Fri, 12 Jul 2024 08:48:44 UTC No. 16278472

>>16278469
gotchu covered pal

Anonymous at Fri, 12 Jul 2024 08:52:33 UTC No. 16278474

>>16278472
based helpful anon this looks great thank you

🗑️ Barkon Approved Post at Fri, 12 Jul 2024 08:53:41 UTC No. 16278475

>>16278474
You were had.

Anonymous at Fri, 12 Jul 2024 08:54:58 UTC No. 16278477

>>16278475
I told you I was stupid :(

Anonymous at Fri, 12 Jul 2024 11:52:46 UTC No. 16278614

>>16278469
https://sheafification.com/the-fast-track/

Anonymous at Fri, 12 Jul 2024 11:53:47 UTC No. 16278616

>>16277854
Functional Analysis and Quantum Field Theory. Zinn-Justin's book handles functional integrals rather thoroughly

Anonymous at Fri, 12 Jul 2024 16:04:09 UTC No. 16278913

Do we have any serious algebraists/number theorists (preferably ~PhD-level) who would be willing to tutor me in local class field theory for free?

Anonymous at Fri, 12 Jul 2024 18:13:12 UTC No. 16279060

>>16276615
>>16276614
>>16276559
>>16276536
This doesn't work because the uncollapsed space is not an equalizer, because the inclusion maps are not point-homotopic.
At this point I'm at a loss as to how to proceed. If anyone who has a clue about elementary algebraic topology would like to help, please do.

Anonymous at Fri, 12 Jul 2024 18:14:14 UTC No. 16279062

>>16278913
Are you self-studying? What's your skill level?

Anonymous at Fri, 12 Jul 2024 18:22:33 UTC No. 16279071

>>16279062
My goal is to learn localCFT from this book https://link.springer.com/book/10.1007/978-0-387-72488-1
The author follows the approach of Hasse-Noether via CSAs and the Hasse invariant. I've self-studied all of the other relevant chapters last year
>Absolute values on fields (completion, Hensel's lemma, extending AVs)
>Local fields (classification, ramification)
>Semisimplicity and Artin-Wedderburn theory
>CSAs and the Brauer group
>Brauer group of a local field
but the actual chapter on CFT kicked my ass, so I dropped it, now I want to come back

I'm a a first-year master's, reasonably well-versed in algebra and number theory

Anonymous at Fri, 12 Jul 2024 18:38:30 UTC No. 16279102

>>16279071
Too bad nobody here actually knows any math.

Anonymous at Fri, 12 Jul 2024 18:58:59 UTC No. 16279137

>>16279102
I figure as much, but I posted on the off-chance. Screw you for making me list out my credentials if you weren't planning on helping

Anonymous at Fri, 12 Jul 2024 19:01:57 UTC No. 16279140

So imagine you're starting at (0,0) and you started running in a straight line to any location on this grid, I want to know where you were when you crossed a 1 or -1 on either axis (the grey dotted box in my picture).

Example, if you ran straight to (4,4), you would've crossed 1 at (1,1). If you ran straight to (-4,0), you would've crossed 1 at (-1, 0).

How do you deduce this mathematically in the simplest way possible? Say I ran to (3.23, 7.4), where was I when I crossed a 1 on either axis?

Anonymous at Fri, 12 Jul 2024 19:02:59 UTC No. 16279143

I didn't save the picture with the grey dotted box, goddamn it.

Anonymous at Fri, 12 Jul 2024 19:12:43 UTC No. 16279162

What book would you recommend for module theory, about direct sums, free, projective and injective modules in particular?
Keep in mind that I'm dumb.

Anonymous at Fri, 12 Jul 2024 19:26:06 UTC No. 16279188

>>16279140
dude this is like high school math desu , you shouldn't be having much trouble with it unless you are in high school. make the equation of the line from the two points (0,0) and (3.23 , 7.4) slope= 7.7/3.23 , y-intercept is zero . So equation is y= 7.7/3.23 x , and now pug in x=1 and x=-1 you get y=2.38 and y=-2.38 , so the points are (1,2.38) and (-1 ,2.38) for y-axis . the points are (0.42,1) and (.0.42,-1) .

Anonymous at Fri, 12 Jul 2024 20:06:26 UTC No. 16279263

>>16279162
Matsumura Commutative Ring theory

Anonymous at Fri, 12 Jul 2024 20:08:40 UTC No. 16279267

>>16279162
Bro, direct sums are basic shit. Projective/injective modules are also basic, the only real "challenge" is proving every module has an injective resolution. That said, I have 2 options for you
1) The relevant chapters of Jacobson's Basic Algebra 1&2: in BA1 he introduces modules and their direct sums, and proves the decomposition theorem for modules over PID, the rest he does in BA2. Jacobson's writing is absolutely impeccable, he gets across everything you need to know in a very clear manner.
2) Weintraub's "Algebra -- An Approach through Module Theory": does what it says on the tin, he introduces modules right after groups and does all "basic" algebra through their lens. The book definitely stands out from others in the field and is pretty well written, but it has nothing on Jacobson imo

Don't listen to this guy >>16279263, he's likely trolling (I haven't read Matsumura myself, but I don't expect you'd need to read a text of this level to learn about modules)

Anonymous at Fri, 12 Jul 2024 20:14:23 UTC No. 16279276

>>16279162
this is what I learned from

Anonymous at Fri, 12 Jul 2024 20:31:30 UTC No. 16279297

>>16278414
But where in the proof is this even required? The Jacobian dF is nonzero in the neighborhood of q which leads to F^{-1} to be in C^{\inf} in that neighborhood. Doesn't that automatically mean that x^{-1} is in C^0? I don't see why it is vital beforehand that x^{-1} is continuous.

Anonymous at Fri, 12 Jul 2024 20:32:21 UTC No. 16279299

>>16279162
>>16279267
>Bro, direct sums are basic shit
In fact, I'll lay them out for you right now. I assume that you know what a ring is, what a (left) module over a ring is, and what a module homomorphism is -- that is all you're going to need. In the following let [math]R[/math] be a fixed ring, we're considering (left) modules over it (I won't be saying "[math]R[/math]-module" every time). Module homomorphism are referred to simply as morphisms.

First, let [math]M_i[/math] be a collection of modules indexed by some set [math]I[/math], then their (external) direct sum [math]\bigoplus_iM_i[/math] is defined as the subset of the cartesian product [math]\bigtimes_iM_i[/math] of all those tuples [math](x_i)_{i\in I}[/math] such that [math]x_i=0[/math] for almost all [math]i\in I[/math] (i.e. every tuple has at only finitely many non-zero components). Check that this set, with the operations defined component-wise, is a module. Check that for all [math]j\in I[/math] the map [math]\iota_j:M_j\to\bigotimes_iM_i[/math], where [math]x\in M_i[/math] is taken to the tuple whose every component is zero except for the [math]j[/math]-th one, which is [math]x[/math], is an injective morphism.

The (external) direct sum has the following (two-part) universal property
1) Given a module [math]N[/math] and collection of morphisms [math]f_i:M_i\to N[/math], there exists a *unique* morphism [math]f:\bigoplus_iM_i\to N[/math] such that [math]f\circ\iota_i=f_i[/math] for all [math]i\in I[/math]
2) The direct sum is the unique module with this property, i.e. if [math]M'[/math] is another module with this property, then there is a unique isomorphism [math]\bigotimes_iM_i\to M'[/math].
Prove this.

(If you know category theory, this means that the direct sum is the coproduct in the category of modules)

Anonymous at Fri, 12 Jul 2024 20:35:30 UTC No. 16279303

>>16279299
I got testicular torsion from reading your post due to the direct sum twisting into a tensor product.

Anonymous at Fri, 12 Jul 2024 20:41:15 UTC No. 16279308

>>16279299
Now for internal direct sums: if [math]M[/math] is a fixed module and [math]M_i[/math] are its submodules, then their sum [math]\sum_iM_i[/math] is defined as the subset of all finite sums [math]x_{i_1}+\cdots+x_{i_n}[/math] with [math]x_{i_r}\in M_{i_r}[/math]. Check that
1) This is a submodule.
2) This is the smallest submodule containing all the [math]M_i[/math].

The following are equivalent for the submodules [math]M_i[/math]:
i) Whenever [math]x_{i_1}+\cdots+x_{i_n}=0[/math] for some [math]x_{i_r}\in M_{i_r}[/math], then [math]x_{i_1}=\cdots=x_{i_n}=0[/math].
ii) For all [math]j\in I[/math] holds [math]M_j\cap\sum_{i\neq j}M_i=\{0\}[/math].
Prove this.

If any of the above equivalent conditions hold, we say that the submodules [math]M_i[/math] are independent. If [math]M=\sum_iM_i[/math] and the [math]M_i[/math] are independent, then [math]M[/math] is said to be the internal direct sum of the [math]M_i[/math]. Prove that [math]M[/math] is the internal sum of the [math]M_i[/math] if and only if [math]M[/math] is isomorphic to [math]\bigoplus_iM_i[/math].

>>16279303
Kek, my bad

Anonymous at Fri, 12 Jul 2024 20:47:03 UTC No. 16279315

>>16279299
>>16279308
Finally, concerning free modules and direct sums: a (possibly infinite) subset [math]\{x_i:i\in I\}[/math] of a module [math]M[/math] is called
1) Linearly independent, if *finite* non-trivial linear combinations of its elements are non-zero, i.e. [math]a_1x_{i_1}+\cdots+a_nx_{i_n}=0[/math] implies [math]a_1=\cdots=a_n=0[/math].
2) Spanning, if [math]M=\sum_iRx_i[/math], i.e. every [math]x\in M[/math] is equal to a linear combination of its elements.
A basis of a module is a linearly independent spanning subset and a free module is one that has a basis.

Prove that [math]M[/math] is free if and only if there exists an index set [math]I[/math] such that [math]M\cong\oplus_{i\in I}R[/math], i.e. [math]M[/math] is isomorphic to the [math]I[/math]-indexed direct sum of [math]R[/math] with itself.

And you're pretty much done, that is all you need to know about direct sums up until you start learning about projective modules and exact sequences.

Anonymous at Fri, 12 Jul 2024 23:28:12 UTC No. 16279545

>>16279267
Seconding for Jacobson. Absolutely outstanding resource, as he writes in a concrete way that you'd want for a first encounter of algebra, while still getting across key concepts that display the bigger picture down the line.

Anonymous at Sat, 13 Jul 2024 13:04:27 UTC No. 16280216

I was thinking about a solution to pic related and came up with a way to generalize it.
Take any nonzero commutative ring R and
[math]f_1(x_1), f_2(x_2), \ldots, f_n(x_n) \in R' = R[x_1,...,x_n][/math]
Prove that there is no set of polynomials [math]a_i(x_1,\ldots, x_n) \in R'[/math] s.t.
[math]\sum_{i=1}^n a_i(x_1,\ldots, x_n)f_i(x_i) = 1[/math].
Seems like there should be an elementary, combinatorial way to prove this but I am unable to without extending the coefficient ring (after which it becomes trivial). Would anyone like to have a go at it?

Anonymous at Sat, 13 Jul 2024 13:45:54 UTC No. 16280239

>>16280216
Oh I forgot to note that f_i are all supposed to have a positive degree.

Anonymous at Sat, 13 Jul 2024 18:24:17 UTC No. 16280448

>>16280216
>>16280239
Oh yeah another condition is that for each f_i(x), not all the coefficients of x^i for i>0 are nilpotent, so that f_i(x) is not a unit in R[x]

Anonymous at Sat, 13 Jul 2024 18:37:23 UTC No. 16280479

Why are there no good textbooks on proof theory?

Anonymous at Sat, 13 Jul 2024 23:58:18 UTC No. 16280812

A few years back I saw a video maybe on numberphile where some mathematician wrote a book where he came up with some new axioms/system of writing math and he developed it. I can't seem to find it, does anyone know what it is?

Anonymous at Sun, 14 Jul 2024 09:18:54 UTC No. 16281154

As someone who's an absolute brainlet that learns from seeing other people solve problems are there any good videos on linear programming/the basics of graph theory in relation to Operations Research?

Anonymous at Sun, 14 Jul 2024 09:23:00 UTC No. 16281160

>>16280812
If you're thinking of law of forms it isn't really what it's described as

Anonymous at Sun, 14 Jul 2024 10:43:16 UTC No. 16281233

I study math because I want anime to be real. What's your reason for studying math?

Anonymous at Sun, 14 Jul 2024 11:44:48 UTC No. 16281270

I'm trying kind of a "backwards proof" for the kernel representation of the fractional brownian motion, using pic related and trying to calculate the variance of (1) using the properties of the distribution of the Wiener integral, and hoping to find [math] t^{2H} [/math], but I'm not seeing it, I've tried substituting by [math] x=\frac{u-s}{t-s} [/math] but I'm not seeing anything interesting, if you guys could give me some pointers (or just tell me that it was a shitty idea from the get go) , I'd appreciate it.

>>16277784
I use goodnotes on my ipad, it has handwriting recognition, so it's useful when I want to quickly look up a theorem or proof.
I was at some point interested in typing notes in latex using vim snippets by following this tutorial https://castel.dev/post/lecture-notes-1/ , but it proved to be too much of a hassle just for making my notes just a bit cleaner looking

Anonymous at Sun, 14 Jul 2024 12:30:43 UTC No. 16281316

How do we respond?

Anonymous at Sun, 14 Jul 2024 14:17:32 UTC No. 16281369

Currently writing my thesis and I have a (not strictly mathematical) question:
If I have a single line of text between the end of a proof and the start of the next theorem, should that single line of paragraph be indented? Or should I use \noindent to push it towards the left?

Anonymous at Sun, 14 Jul 2024 14:34:39 UTC No. 16281380

>>16281160
No bro it's a video of a guy who wrote a book about all the derivations that came from his system, and it was cutesy and shit with lines and dots or something. It was some casual thing he did to begin with but then derived some more interesting things and put them all in a book and he was describing it from the start of the system. It was an actual mathematician dude, and it was on numberphile or some other channel like that.

🗑️ Anonymous at Sun, 14 Jul 2024 15:11:52 UTC No. 16281402

>>16281270
I've also tried substituting by [math] x = \frac{u}{s} to obtain an incomplete beta function, but still no idea

Anonymous at Sun, 14 Jul 2024 15:14:12 UTC No. 16281404

>>16281270
I've also tried to substitute by [math]x=u/s [/math] in the inner integral, and got an incomplete beta function, but still no idea on how to go from here

Anonymous at Sun, 14 Jul 2024 15:35:35 UTC No. 16281416

>>16281369
All paragraphs should be indented, no matter how small
If that feels awkward it may be a sign that there is a better way to format what you're writing. If it's a comment on one of the theorems, or how the proofs relate to each other, it should probably go inside a remark

Anonymous at Sun, 14 Jul 2024 18:01:23 UTC No. 16281520

>>16279545
>>16279315
Nta but nice explanations thanks. If you feel like explaining more of module concepts I would read it (but I am not asking you to do free teaching labor of course).

Anonymous at Sun, 14 Jul 2024 19:26:32 UTC No. 16281578

>>16281270
If I'm not mistaken this is done in lemma 3.1 in https://link.springer.com/article/10.1023/A:1008634027843.
There's some different scaling with their constant [math]V_H[/math] and your constant [math]c_H[/math], but it appears to be the same.
Also, it's probably very difficult to directly evaluate your integral (at least, I can't think of something nice and the article doesn't do it this way either), so it might not be a very fruitful idea.

Anonymous at Sun, 14 Jul 2024 20:52:01 UTC No. 16281620

>>16281578
gotcha, yeah I was starting to feel that maybe I was being a little overconfident, thanks for the link

Anonymous at Sun, 14 Jul 2024 21:51:01 UTC No. 16281661

>>16281416
I see, thanks.

Anonymous at Mon, 15 Jul 2024 09:52:15 UTC No. 16282192

>>16281369
You should never use formatting commands in the main document (including commands like [math] \texttt{\mathbf} [/math] and [math] \texttt{\vspace} [/math]). All formatting should be automated by the preamble. Otherwise, you'd be going against the point of [math] \mathrm \LaTeX [/math]. Equations are sometimes an exception since it is hard to automate their formatting.

Anonymous at Mon, 15 Jul 2024 10:25:54 UTC No. 16282217

>>16282192
Good point.

Anonymous at Mon, 15 Jul 2024 10:34:01 UTC No. 16282223

>>16282192
Actually, now that you mention that, I do actually have one more question to pose: should I make it a habit of leaving a blank line in the TeX file before and after a [math]\texttt{\begin{align}}[/math] and [math]\texttt{\end{align}}[/math], respectively? Or should whatever text is before or after not have a blank space in between? Because I think for the case of the [math]\texttt{itemize}[/math] environment, for example, leaving a space actually indents the following paragraph but not doing so has the same effect as [math]\texttt{\noindent}[/math].

Anonymous at Mon, 15 Jul 2024 12:45:28 UTC No. 16282327

>>16282223
I don't think you should leave a space. In all use-cases I can think of, an equation is connected with the surrounding text. Leaving a blank line would start a new paragraph. Sometimes an equation ends a paragraph, in which case you should end the equation with a [math] \texttt{\mathpunct.} [/math] and if the situation calls for it, [math] \texttt{\qedhere} [/math]. Then, leave a blank line. Leaving a blank line after itemize is fine, but typically not before. You can always leave a commented blank line if you think the code looks nicer that way.

Anonymous at Mon, 15 Jul 2024 12:47:47 UTC No. 16282331

can anyone here help me with a problem I'm stuck on from Basic Mathematics by Lang.
I'm trying to teach myself so getting stuck is terrible.
The prompt is to rationalize the numerator and remove any square root. it's in the chapter on real numbers. I took a picture to show the start, the textbook answer and the answer I got. is the textbook answer simply mistaken??? the parenthesis don't make any sense to me because without them I think i would be right.

Anonymous at Mon, 15 Jul 2024 13:04:44 UTC No. 16282341

>>16282331
No, you're correct. It's a misprint.

Anonymous at Mon, 15 Jul 2024 13:06:18 UTC No. 16282343

>>16282341
THANK U

Anonymous at Mon, 15 Jul 2024 13:12:32 UTC No. 16282348

>>16282331
What a weird exercise. There's basically never a reason to rationalize the numerator if it's going to make the denominator irrational. Then again, Lang's books are a meme for a reason.

bibi !!ZEC9/+ffJN3 at Mon, 15 Jul 2024 13:15:22 UTC No. 16282350

>>16282348
thank you. I'm trying to teach myself math so I just used textbook chart guides on /lit/ to find where to start and some of them listed this textbook but now im wondering if I should be spending my time with a better resource

Anonymous at Mon, 15 Jul 2024 13:21:46 UTC No. 16282353

>>16282350
You say no. Fag.

bibi !!ZEC9/+ffJN3 at Mon, 15 Jul 2024 13:24:59 UTC No. 16282354

>>16282353
what does this mean

Anonymous at Mon, 15 Jul 2024 13:26:06 UTC No. 16282357

>>16282354
His drinksees. OMFG YOU ARE SOOOO DOOPID, KYAD. physical stuff is not heaven faggy kid.

Anonymous at Mon, 15 Jul 2024 13:27:38 UTC No. 16282359

>>16282354
Fart in my mouf now

Anonymous at Mon, 15 Jul 2024 13:32:21 UTC No. 16282362

>>16275601
>The natural continuation
What about, you know, doing calculus on manifolds, or learning actual measure theory? That seems the most "natural" continuation to me

Anonymous at Mon, 15 Jul 2024 13:46:44 UTC No. 16282373

>>16282350
Lang's books are more of a 4chan meme than a real recommendation. Given what you were studying in >>16282331 It looks like you're trying to complete a precalculus curriculum. I suggest following the AP Precalculus curriculum, any of the study guides will be suitable for learning (e.g. Princeton Review, Barron's, Shuam's Outlines, etc.) and then you can use practice exams to self-assess how well you know the material. Here's the site from college board for AP Precalculus. https://apcentral.collegeboard.org/courses/ap-precalculus/course

bibi !!ZEC9/+ffJN3 at Mon, 15 Jul 2024 13:51:04 UTC No. 16282377

>>16282373
thank you!! yes I will check all this out. My ultimate goal is to understand enough to code 3d graphics impressively using trigonometry and stuff

Anonymous at Mon, 15 Jul 2024 13:52:05 UTC No. 16282378

>>16282373
no, Lang isn't a meme. Reading long ass books on precalc is.

Anonymous at Mon, 15 Jul 2024 14:15:47 UTC No. 16282398

Started reading about constructive math recently to see what's all the fuss about, I don't think I'm going back bros, the arguments for it are just too strong

Anonymous at Mon, 15 Jul 2024 14:17:11 UTC No. 16282401

>>16282398
like?

Anonymous at Mon, 15 Jul 2024 14:18:49 UTC No. 16282404

>>16282377
http://pastebin.com/sy2MbenC

Anonymous at Mon, 15 Jul 2024 14:19:12 UTC No. 16282406

You may be mathematicians, but can you solve the TORMENT on POKEMON EMERALD?

You know when they ask you interview questions and you have to select words to make a phrase? That stuff is a secret hard cryptology game which can be worked out by studying the game. If you succeed, a secret room opens where you can fight all previous Pokémon.

Anonymous at Mon, 15 Jul 2024 14:20:13 UTC No. 16282408

>>16282331
It seems learning math isn't really making you intelligent because if you were intelligent you would have just tested it for different values instead of coming here and asking. I think you should just not bother.

Anonymous at Mon, 15 Jul 2024 14:22:03 UTC No. 16282411

>>16282401
Computational meaning for all existence proofs, that's like, so kosher
Also intuitionist logic just makes sense

bibi !!ZEC9/+ffJN3 at Mon, 15 Jul 2024 14:22:12 UTC No. 16282412

>>16282408
that's a good idea, next time I'll try that first! i was getting too confuzzled

Anonymous at Mon, 15 Jul 2024 14:23:04 UTC No. 16282415

>>16282331
you can always plot simple shit like this on desmos. You're right btw. Also focus on finishing Lang and starting with Linear Algebra soon (e.g. Shilov or Gorodentsev's book) if you care about graphics design.

Anonymous at Mon, 15 Jul 2024 14:23:32 UTC No. 16282416

>>16282412
Don't let that nigga discourage you, I know him personally and he is a certified BITCH!

Anonymous at Mon, 15 Jul 2024 14:29:43 UTC No. 16282423

>>16282350
these charts and the /sci/ wiki usually have you work through 300 proof, precalc, calc, trig, euclidean geometry, and olympiad problem books before doing any remotely serious math. Just try reading a book on the exact thing you want to learn and go back from there. You'll have a much better picture of what you actually have to learn.

Anonymous at Mon, 15 Jul 2024 14:29:51 UTC No. 16282425

>>16282327
Thanks, that's very insightful. I will admit that I'm not too familiar with the usage of [math]\texttt{\mathpunct}[/math], but I can read up on it.

Anonymous at Mon, 15 Jul 2024 14:31:40 UTC No. 16282426

>>16282401
>>16282398
>>16282411
Despite being being a PDE/FA guy myself, I admit I too have been heavily influenced by constructivism, particularly the strain called Ultrafinitism. If you haven't read this essay yet, you're missing out: "Real" Analysis is a Degenerate Case of Discrete Analysis
https://sites.math.rutgers.edu/~zeilberg/mamarim/mamarimPDF/real.pdf

Anonymous at Mon, 15 Jul 2024 14:47:34 UTC No. 16282442

>>16282331
Are you a girl?

bibi !!ZEC9/+ffJN3 at Mon, 15 Jul 2024 14:56:46 UTC No. 16282448

>>16282442
I'm an adult woman, but yes, still a girl

🗑️ Anonymous at Mon, 15 Jul 2024 15:13:10 UTC No. 16282468

Are there any continuous functions on a connected set that do no have the limit
[eqn] \lim_{h \to 0} \left| \frac{ f(x + h) - f(x) }{h } \right| \mathpunct, [/eqn]
and if so, do these functions have a name?

>>16282415
From one meme book to another.
>if you care about graphics design.
This is just the cherry on top.

Anonymous at Mon, 15 Jul 2024 15:17:22 UTC No. 16282475

>>16282415
From one meme book to another.
>if you care about graphics design.
This is just the cherry on top.

Anonymous at Mon, 15 Jul 2024 15:31:24 UTC No. 16282491

>>16282448
ywnbaw

Anonymous at Mon, 15 Jul 2024 16:09:04 UTC No. 16282531

any chance the author of these pictures still posts here? (1/2)

Anonymous at Mon, 15 Jul 2024 16:10:06 UTC No. 16282532

>>16282531
he said he'd compile his notes in a book/pdf but haven't heard from him yet :(
(2/2)

Anonymous at Mon, 15 Jul 2024 16:45:29 UTC No. 16282562

>>16282448
Do you have a boyfriend?

Anonymous at Mon, 15 Jul 2024 16:50:44 UTC No. 16282567

>>16282562
What does it matter? Trannies are always willing to add one more male to their harem of beta male orbiters

Anonymous at Mon, 15 Jul 2024 17:01:54 UTC No. 16282578

>>16282567
>>>/pol/ nazi trash

bibi !!ZEC9/+ffJN3 at Mon, 15 Jul 2024 17:02:52 UTC No. 16282581

>>16282562
No I do not math anon

Anonymous at Mon, 15 Jul 2024 17:19:05 UTC No. 16282598

>>16282578
>nazi
"nazi" is what trannies call normal people, anon

Anonymous at Mon, 15 Jul 2024 18:00:29 UTC No. 16282635

I have a rank-one matrix [math]M = xx^T[/math].
I fix some basis and replace the diagonal entries of M with ones.
Is there some trick to compute the eigenvalues of the resulting matrix easily?

Anonymous at Mon, 15 Jul 2024 18:08:21 UTC No. 16282644

>>16282426
This is not really much like constructivism. It doesn't use intuitionistic logic or anything.

Anonymous at Mon, 15 Jul 2024 18:20:03 UTC No. 16282660

never ask a mathematician what is 843 times 2349 or what is the definition of a set

Anonymous at Mon, 15 Jul 2024 18:34:07 UTC No. 16282673

>>16282426
>zeilberger
Love this based jew like you wouldn't believe. But I don't really agree with his position. There's always been a tension bteween concepts and calculations in math. I think classical mathematicans push it too far to one side by not caring at all about how meaningful their existence statements are. They only care about consistency(can't even prove that!) and to me this is not enough and it's very dangerous, with the risk of turning mathematics into a mere language game. On the other hand, finitists push it too far into the other direction: only focusing on what is "realistic", you risk turning math into a mere cookbook, whrere you really don't understand what's going on. I think constructivism in the sense of bishop is the best approach so far (it has been expanded upon since he dropped his banger "constructive analysis", where you get the best of both worlds imo.
>ultrafinitism
Another point on this: while I do have a certain respect for the sentiment (as things get bigger, the less meaningful they are), it's in my view an informalizable aspect. Suppose there's an absolute hard computational limit: You would merely put an asterisk in from of every established result saying "ermmm only valid till this big ass number!!", which is kinda unimportant. Then we would have to realize that ACTUAL computational limits are relative to the context you are working with, so even this theoretical absolute limit would be useless because of that too.

B at Mon, 15 Jul 2024 18:35:58 UTC No. 16282676

>>16282673
No. You will be paid off and repaired. Yes. It's a problem. But it's not too much of a problem nor is it difficult for me to fix. The extra life was done this way. Morals on me

Anonymous at Mon, 15 Jul 2024 18:42:20 UTC No. 16282682

>>16282676
what in the goddamn...

B at Mon, 15 Jul 2024 18:45:00 UTC No. 16282684

>>16282682
It's all legal and financial faggot. Get with the times.

B at Mon, 15 Jul 2024 18:46:55 UTC No. 16282686

>>16282682
It's a fixed fee and service order, or hell. Take your meds

B at Mon, 15 Jul 2024 18:47:56 UTC No. 16282688

>>16282686
& Lesser hell. Like a jester. Involved in transact 1 of the former statement.

Anonymous at Mon, 15 Jul 2024 18:51:09 UTC No. 16282693

>>16282682
You should really just filter out any namefags, at best they're midwits, at worst they're nonsensical

Anonymous at Mon, 15 Jul 2024 18:53:10 UTC No. 16282698

>>16282475
>t. doesn't read books

B at Mon, 15 Jul 2024 18:53:46 UTC No. 16282699

>>16282693
MOUF. Now fag.

Anonymous at Mon, 15 Jul 2024 18:56:01 UTC No. 16282700

Some OC

Anonymous at Mon, 15 Jul 2024 19:09:59 UTC No. 16282717

>>16282700
variant

Anonymous at Mon, 15 Jul 2024 19:15:57 UTC No. 16282723

How would you explain to normies the features of “mathematical understanding” that you just don’t get with a “casual understanding” of something like statistics or ML for example

Anonymous at Mon, 15 Jul 2024 19:19:18 UTC No. 16282728

>>16282723
Ask them to explain what they understand in detail, and hit them with "so you don't actually understand it. That's the difference between you and a mathematician." when they inevitably stumble.

Anonymous at Mon, 15 Jul 2024 19:25:14 UTC No. 16282735

>>16282682
Kek

Anonymous at Mon, 15 Jul 2024 19:30:45 UTC No. 16282741

>>16282676
>>16282684
>>16282686
>>16282688

Anonymous at Mon, 15 Jul 2024 19:31:33 UTC No. 16282742

>>16282723
For ML there is no difference. It's probably the least mathematically rigorous field imaginable that still technically uses "math"

Anonymous at Mon, 15 Jul 2024 19:39:44 UTC No. 16282744

>>16282742
I explained poorly, I don’t mean math vs ML, I mean the way a mathematician understands ML vs how a normie understands ML at a very shallow level.

Anonymous at Mon, 15 Jul 2024 19:40:31 UTC No. 16282745

>>16282728
Yes trying to explain it to them without making them feel stupid (because they pay me 250k a year)

Anonymous at Mon, 15 Jul 2024 19:48:18 UTC No. 16282754

>>16282744
Still very little difference. ML is mostly just matrix multiplys, approximating gradient descent, and the intuition for that is simple.
The truth is even the mathematician only understands things at a shallow level. Nobody knows why architecture X is slightly better than architecture Y for some task, or why xavier uniform initialization is sometimes better than normal initialization. The only real advantage to being a mathematician is being able to understand notation a bit easier

Anonymous at Mon, 15 Jul 2024 19:54:53 UTC No. 16282761

>>16282331
c z e c h i a
z
e
c
h
i
a

Anyway as the other anon said focus on finishing one book, there's no need to hunt for the "perfect resource" especially with precalculus, just focus on getting the intuitive idea behind as much of the covered material as possible. You won't be usually doing operations like rationalizing the numerator but doing such exercises mainly helps with getting mileage and understanding the underlying concepts better. If you want to do 3D graphics then linear algebra will be a good next step, take a look at Gilbert Strang's book "Linear Algebra and Its Applications" and the accompanying course (leftures are on YouTube) are touted as one of the holy grails of computational linear algebra resourcesnso take a look at those. Really depending on how far you're into Lang's boom you can probably try one of the linear algebra resources and see how if goes, then go back to Lang as needed if it's too much or whatever.

Anonymous at Mon, 15 Jul 2024 20:28:21 UTC No. 16282787

Am I overthinking or is the last part of 15 quite complicated to prove?
I did it by first reducing all the terms in the relation to ones of the form (a - mu_ij(a), by using the fact that the connecting maps are homomorphisms. Then I reduceng to the case where the only term in the finite sum of relations is nonzero, is the term with the greatest index, by taking an upper bound j of all the indices and replacing m with m - (m - mu_{ij}(m)).
Then I reduced m into two parts. One which shows up as a_ij term in the sum, which I showed that eventually maps to zero. The other which is a sum of terms mu_{ji}(b_j). To show that this term equals zero, I minimized the number of terms in the sum and essentially used graph reduction to show all the corresponding indices are mutually incomparable, then, that they do not connect to anything lower, and then finally I got that they're all zero.

Seems like a lot of work for something that seems so trivial.

Anonymous at Mon, 15 Jul 2024 20:29:59 UTC No. 16282789

>>16282761
>c z e c h i a
>z
>e
>c
>h
>i
>a
Can someone explain why I keep seeing this (usually with london instead of czechia)? What does it mean? Is it all the same schizo spamming it?

Anonymous at Mon, 15 Jul 2024 21:06:20 UTC No. 16282832

>>16282723
From my experience, mathematicians don't understand shit about statistics. They just want to do math. It's people who actually work with data that know how statistics actually work.

Anonymous at Mon, 15 Jul 2024 21:07:41 UTC No. 16282835

>>16282789
It means the poster hopes the woman is in the said country.

Anonymous at Mon, 15 Jul 2024 21:38:39 UTC No. 16282861

>>16282835
How in the world do you know that? :o

Anonymous at Mon, 15 Jul 2024 21:43:19 UTC No. 16282865

How do I know if I know math? I have a serious imposter complex regarding my abilities, despite having a BSc in mathematics.

Anonymous at Mon, 15 Jul 2024 22:32:13 UTC No. 16282932

>>16282789
>>16282861
it's an old /fit/ meme

Anonymous at Mon, 15 Jul 2024 22:52:06 UTC No. 16282961

>>16282865
Math is possibly the easiest field to test your abilities in. Here's a few ways to test your ability
1. Go through a serious math book and do all the exercises (examples: Hartshorne, Spanier)
2. Do the same except prove all the theorems yourself instead of reading them.
3. Do the same for papers.
4. Solve some open problems and publish your proof in reputable journals.

Anonymous at Tue, 16 Jul 2024 00:42:56 UTC No. 16283042

>>16282754
> Nobody knows why architecture X is slightly better than architecture Y for some task
Plenty of people do. The only reason it isn’t emphasized today is that it’s easier to train data science monkeys to iterate through a bunch of models programmatically than to actually sit and explain to them which family of models is likely to work best on each type of problem

Anonymous at Tue, 16 Jul 2024 01:13:07 UTC No. 16283058

>>16283042
>Plenty of people do.
Then prove it. Write a mathematical proof demonstrating that some architecture is the best at a certain task. Until that becomes common place, ml research will remain a meme
>iterate through a bunch of models programmatically
That basically is the current state of ml research right now, unless things have changed in the last few years. Take a pre-existing model, use a new pooling/transfer function or architecture or whatever, and publish a paper if it produces slightly better results. It's just code monkeys throwing shit at the wall

Anonymous at Tue, 16 Jul 2024 02:45:04 UTC No. 16283114

>>16271237
Was ist die Unterschied zwischen deutsche Physik und andere Physik?

Anonymous at Tue, 16 Jul 2024 04:21:29 UTC No. 16283155

How correct is the following "proof"? Can it be improved? How original is it?
-------------------------------------
Natural numbers can store an arbitrary amount of information, while real numbers can store an infinite amount of information.
The first one is pretty obvious. To encode a sequence of 1s and 0s into a natural, you can pick the convention of choosing the natural whose binary representation always starts with a 1 and only then is equal the binary sequence that you want to represent, to avoid the issues of leading zeros getting truncated. So if you can convert your information to a sequence of ones and zeros (which, afaik, that is roughly* what is meant as "information" is mathematically) they you can pack as much information as you want into a natural.
The second one is a bit more subtle. Reals can contain an infinite amount of information because of the way they are defined. Real numbers can be defined by infinite Cauchy sequences. The terms of this infinite Cauchy sequence do not need to be computable. They just need to be arbitrary rational numbers such that for any ε there exists an α that guarantees |xᵢ-xⱼ|<ε given i,j>α.
If any two Cauchy sequences converge to the same real number, we say that they are both equal, so we cannot rely on the exact terms of the sequence to encode information. But we can use the convergence requirement to our advantage.
Let μ=1/2ʰ.
Let l(ₑ,ₖ)=2kμ+eμ k∈ℕ, e∈{0,1}
For any real number r, we define bₕ as follows:
bₕ=1 if there exists a k∈ℕ, β∈ℕ such that l(ₒ,ₖ)<xₚ<l(1,ₖ) for all p>β, and bₕ=1 if there exists a k∈ℕ, β∈ℕ such that l(ₒ,ₖ)+μ<xₚ<l(1,ₖ)+μ for all p>β.
In this way we can construct a number such that bₕ is either 0 or 1 depending on the terms of the Cauchy sequence used to define it up to the β term, for any natural number h. Thus we have proven that we can encode an infinite amount of binary digits in a single real number.

Anonymous at Tue, 16 Jul 2024 04:22:30 UTC No. 16283157

* The reason it may not be exactly that, is you can have a trit of information (choosing one out of three elements) which stores an amount of information that is strictly between 1 and 2 bits. But you can define a convention, such as storing the base at the beginning of the sequence and then adding redundancy such that some sub-sequences of the binary sequence are considered equivalent.

Anonymous at Tue, 16 Jul 2024 10:04:58 UTC No. 16283346

>>16283058
The problem is that mathematical proofs take time and there’s not enough people competent enough to do it and not enough funding to get those people to dedicate their time to it. There’s tens of millions of ML ants now that can import scikitlearn and run a loop over a bunch of models they don’t understand for some problem no one cares about than there are qualified mathematicians to make a proof about the absolute best model to train on some problem that no one cares about.

Anonymous at Tue, 16 Jul 2024 10:10:27 UTC No. 16283357

>>16283155
I didn’t read all of it but basically yes both statements are true. That’s why storing large numbers or arbitrarily precise numbers requires an increasing amount of computer memory. Anyone who wants to have fun can open a Python console and input 10 ** (10 ** 10) and have fun watch their computer crawl to a halt. Watch your system resources as this happens too

Anonymous at Tue, 16 Jul 2024 12:34:59 UTC No. 16283438

I'm looking for a PDF of this book in particular:

https://www.amazon.com/Lectures-Modern-Mathematics-T-Saaty/dp/0471748269

I've looked online, but I couldn't find a PDF in the usual places.

Anonymous at Tue, 16 Jul 2024 12:45:12 UTC No. 16283443

>>16283438
That's funny, I found a pdf from the usual place immediately.

Anonymous at Tue, 16 Jul 2024 12:45:17 UTC No. 16283444

>>16283438
https://archive.org/details/lecturesonmodern0000saat_m3k0/page/n5/mode/1up

Anonymous at Tue, 16 Jul 2024 13:51:14 UTC No. 16283493

I'm going insane. A while ago I saw a video about a statistician(?) giving a lecture about conspiracy theories and the math behind them. I think one example he used was how ancient sites being on special Ley Lines isn't special, because there are just so many ancient sites so by pure probability some will fall on a line. I think he then demonstrated this by showing how a bunch of Tesco stores are also on Ley Lines or something.
Does anyone know who I'm talking about, I can't find the video

Anonymous at Tue, 16 Jul 2024 13:56:46 UTC No. 16283496

>>16283493
Oh and I think another example he used, is how you can pick any random number sequence and correlate it to words in the bible and make it spell almost anything you want to

Anonymous at Tue, 16 Jul 2024 14:11:56 UTC No. 16283510

>>16283493
>>16283496
https://www.youtube.com/watch?v=sf5OrthVRPA
Nvm found it

Anonymous at Wed, 17 Jul 2024 02:51:09 UTC No. 16284452

>>16282761
Strang is pure garbage and nobody should ever be subjected to this

Anonymous at Wed, 17 Jul 2024 07:51:10 UTC No. 16284631

>>16283444
It's unavailable though. Nothing on libgen or scihub

Anonymous at Wed, 17 Jul 2024 08:38:30 UTC No. 16284647

>>16284631
What do you mean unavailable , it’s right there on the page, just read it

Anonymous at Wed, 17 Jul 2024 09:05:40 UTC No. 16284661

Whats the most advanced kind of math that you would use in a business context?

I think something like stochastic calculus maybe.

Anonymous at Wed, 17 Jul 2024 09:55:17 UTC No. 16284694

>>16284661
Depends what business you’re in. Can’t tell you how many times I’ve been asked to solve reformulations of the halting problem

Anonymous at Wed, 17 Jul 2024 10:22:46 UTC No. 16284708

>>16284452
Why
I've read that his is a computationally focused approach etc. and to be fair I think I've seen some people share views similar to yours now that I think about it?

Anonymous at Wed, 17 Jul 2024 12:36:44 UTC No. 16284860

>>16284708
Not him, but Strang never gives intuition for his results and is mostly just blind algorithms, but to be fair most pure math books don't give either; they'll use purely algebraic shit like minimal polynomials to prove purely geometric results like spectral decomposition. I have found Halmos's proofs to be quite enlightening despite being a very terse book. Surprisingly, Lang also has a few geometric proofs and is very close to my preferred proof of spectral decomposition, but his could be better (it's Lang after all). If you want to apply linear algebra, geometric intuition is quite important because a lot of the results based on linear algebra are motivated by geometry. For instance, Principal Component Analysis is completely motivated by elliptical shape of Normal contours, but most statistics book will just blindly apply eigenvectors for no apparent reason. That said, most people hate Strang because they think they are real badasses for wanting all math to be soulless le heckin Bourbaki style.

Anonymous at Wed, 17 Jul 2024 14:56:42 UTC No. 16285093

>>16284860
Ah, thanks for the detailed answer. I have never used any of Strang's materials, was just going off what I'd seen recommended online. Am working through Axler's book right now and really enjoying it, I especially like the exercises and him including a photo of himself and his cat.

Anonymous at Wed, 17 Jul 2024 15:26:36 UTC No. 16285145

>>16285093
Shartxler is also garbage.

Anonymous at Wed, 17 Jul 2024 16:54:08 UTC No. 16285226

>>16285145
>>16285093
Yes, I especially dislike Axler because his entire book's philosophy is that determinants are unintuitive and hence, should be avoid yet he FUCKING uses polynomials to prove spectral decomposition which is way more unintuitive than determinants. Axler is just a knock-off Halmos.

Anonymous at Wed, 17 Jul 2024 17:58:46 UTC No. 16285303

Behold, the [math] \textbf{Barnett-Tooker-Wildberger conjecture}[/math]

Let

,,[math]\qquad \hat\zeta_b(s) = \sum_{k=1}^{\hat\infty - b} n^{-s}`[/math]

then

,,[math]\qquad \hat\zeta_{e + \pi + \sqrt 2}(-1) = \frac{e^{i\pi}}{11 + 0.999\dots}[/math]

Anonymous at Wed, 17 Jul 2024 18:01:30 UTC No. 16285308

>>16285093
Check out Shilov. Great for a more geometric perspective that doesn't have the shortcomings these other books have. It also does some tensor algebra which many books skip despite being essential

Anonymous at Wed, 17 Jul 2024 18:11:35 UTC No. 16285333

>>16285145
>>16285226
I'm liking Axler so far but I'm not too far in yet.
>>16285308
Thank you for the recommendation.

I'm going to keep with Axler for the time being because I don't want to keep jumping between books, do you guys think I could go to Shilov, Halmos or some other book afterwards and read the relevant parts to get the full picture? I'm also going through Aluffi M's "Algebra: Chapter 0" and he seems to cover a lot of linear algebra using modules apparently I've read? But I don't know as I'm not in those parts yet.

Anonymous at Wed, 17 Jul 2024 19:45:23 UTC No. 16285500

What's a good resource for brushing up on integral / calculus knowledge? It's coming up for a probability exam and I feel like brushing up on general knowledge in that area would be nice because it wasn't really covered in the past. Stuff like splitting integrals is kind of giving me a headache. I know the basics but not the finer details / tricks, some of which look like magic to me.

Anonymous at Wed, 17 Jul 2024 20:22:16 UTC No. 16285522

>>16285333
Aluffi is fine but Lang has more shit

Anonymous at Wed, 17 Jul 2024 20:29:17 UTC No. 16285527

>>16285333
great idea imo. Both lang and aluffi are a bit outdated on algebra so make sure to supplement Hotta's book on D-Modules

Anonymous at Wed, 17 Jul 2024 23:59:45 UTC No. 16285773

>>16282635
idk, i feel like doing a qr decomp is the straightforward way. You're turning a rank 1 matrix into a possible rank n. Maybe instead of starting off with the Identity matrix for qr decomp, the first row could be set as the original basis x. But that aint much of a trick other than maybe slightly faster convergence.

Anonymous at Thu, 18 Jul 2024 00:44:32 UTC No. 16285820

>>16285522
>>16285527
Thanks for the answers. My plan was to go with Atiyah MacDonald and "An Invitation to General Algebra and Universal Constructions" by George Bergman after Aluffi, I guess after that I can look at categories and sheaves, then sheaves on Manifolds and then D-modules? but that is probably very far off in the future still

Anonymous at Thu, 18 Jul 2024 08:21:59 UTC No. 16286295

bump to beat the soijak spam

Anonymous at Thu, 18 Jul 2024 08:26:13 UTC No. 16286312

Is this the only general standing?

Anonymous at Thu, 18 Jul 2024 08:40:09 UTC No. 16286367

>>16285333
Halmos does Axler's shtick (linear algebra from a functional analyst's perspective) way better.

Anonymous at Thu, 18 Jul 2024 08:44:51 UTC No. 16286386

>>16286312

Anonymous at Thu, 18 Jul 2024 09:19:20 UTC No. 16286522

bump

Anonymous at Thu, 18 Jul 2024 10:05:11 UTC No. 16286713

up

🗑️ visit soygem.party at Thu, 18 Jul 2024 10:06:10 UTC No. 16286718

>>16271237
If yall are so good at math, solve this then:

Anonymous at Thu, 18 Jul 2024 10:27:51 UTC No. 16286798

>>16286718
Millenium problem#8

Anonymous at Thu, 18 Jul 2024 10:29:33 UTC No. 16286804

>>16286798
millennium
the 'jak cobjecture

Anonymous at Thu, 18 Jul 2024 10:32:20 UTC No. 16286808

>>16286718
zero

🗑️ Anonymous at Thu, 18 Jul 2024 10:38:54 UTC No. 16286827

>/mg/

>Deutſche Mathematik edition
>Talk maths, formerly >>16240472

Anonymous at Thu, 18 Jul 2024 11:53:15 UTC No. 16287039

up

Anonymous at Thu, 18 Jul 2024 12:28:21 UTC No. 16287095

glad that was deleted
If hate funny riddles in my /math general/

Anonymous at Thu, 18 Jul 2024 12:36:05 UTC No. 16287100

>>16286718
narrowed it down to two possibilities

Anonymous at Thu, 18 Jul 2024 12:42:09 UTC No. 16287105

>>16287100
I think the "there is no way for me to know"s can be taken into account as well, I got just one solution then.

Anonymous at Thu, 18 Jul 2024 12:43:43 UTC No. 16287106

>>16287105
which solution? I'm not sure how to use "there is no way for me to know" as a clue

Anonymous at Thu, 18 Jul 2024 12:52:36 UTC No. 16287112

>>16287106
BASEDJAK SPOILER
[spoiler]See if you can use the knowledge of how many ears there are in total and of the jaks being able to deduce that also (they have more information than you)[/spoiler]

Anonymous at Thu, 18 Jul 2024 13:08:43 UTC No. 16287128

>>16287112
SPOILERS
So Short 2 Tall 0 Mute 0 Fat 2 is a consistent answer based on the numbers that must be true and it gets around the "I can't know" people by calling them liars. So that is one valid solution.

Now I want to think out loud how to exclude the Short 2 Tall 0 Mute 2 Fat 0 solution.
In this case the Fat one is telling the truth that he doesn't know how many ears the mute one has. But he does know he has 0 ears. What are the possibilities he thinks there are?
Could he think there are 6 ears i.e. everyone else has ears but him? Well no because that would Short a liar meaning the total is less that 6 ears, so Fat doesn't think that case. So he must think at least one of the others doesn't have ears. Thus the max ears he thinks is 4 and so he knows Short is a liar and so Short has ears.
Could he think there are only 2 ears? Well in this case that would make Tall a liar, having 2 ears. But he also knows short has 2 ears, which would make the total 4 contradicting the total only having 2 ears.
Therefore Fat now knows there are 4 ears, with Short having 2 of them. He also know Tall is telling the truth and therefore has no ears. And recalling Fat knows his own 0 ears this means that Fat must know that Mute has 2 Ears in this scenario, which contradicts the idea of Fat telling the truth about him not knowing how many ears Mute has. Therefore Short 2 Tall 0 Mute 2 Fat 0 can be excluded.

Whew.

Anonymous at Thu, 18 Jul 2024 13:26:37 UTC No. 16287154

>>16287128
EARJAK SPOILERS DISCUSSION
Nice yes! I hope we get more 'jak riddles in the future
I think you could also assume that fat, having access to all the info you did, can run the same program that you have and obtain the ear count from there

Anonymous at Thu, 18 Jul 2024 13:49:52 UTC No. 16287182

>>16285820
not that anon but if you want to learn about sheaves check out kashiwara

Anonymous at Thu, 18 Jul 2024 14:18:19 UTC No. 16287230

>>16287154
SPOILERS
Oh right that makes sense, he runs that program, knows how many ears he has, and therefore knows how many ears Mute has, thus making him a liar, making it clear which of the two program choices it is. nice

Anonymous at Thu, 18 Jul 2024 17:10:03 UTC No. 16287448

>open book
>first theorem after a section is false

Anonymous at Thu, 18 Jul 2024 17:11:10 UTC No. 16287450

>>16287448
retard

Anonymous at Thu, 18 Jul 2024 17:12:56 UTC No. 16287452

>>16287450
The set of fixed points of a continuous function only has to be closed if X is T_2.
It's not true for every topological space.

Anonymous at Thu, 18 Jul 2024 17:52:22 UTC No. 16287499

>>16287448
What book?

Anonymous at Thu, 18 Jul 2024 17:54:31 UTC No. 16287503

>>16287452
nta but R is Hausdorff, so {0} is closed, and thus g^-1({0}) is closed too.

Anonymous at Thu, 18 Jul 2024 18:11:35 UTC No. 16287521

>>16287503
There's two parts of this problem, first one is false, second one is true.
>>16287499
M.A. Armstrong, Basic Topology

Anonymous at Thu, 18 Jul 2024 18:27:19 UTC No. 16287528

>>16287521
mystery solved

Anonymous at Thu, 18 Jul 2024 18:43:48 UTC No. 16287545

Anonymous at Fri, 19 Jul 2024 02:42:43 UTC No. 16287998

>tfw became an alcoholic and can't finish my PhD thesis

it's over for me bros.

Anonymous at Fri, 19 Jul 2024 07:13:07 UTC No. 16288169

>>16287998
How does being an alcoholic prevent you from finishing your phd thesis?

Anonymous at Fri, 19 Jul 2024 07:30:13 UTC No. 16288174

>>16288169
I'd imagine you need a lot of fully functioning brain cells to do a math phd

Anonymous at Fri, 19 Jul 2024 10:30:52 UTC No. 16288254

>>16287182
thank you anon, I was meaning to check out his books, by categories and sheaves I meant his book with that name
But that is still far off for me

Anonymous at Fri, 19 Jul 2024 11:57:12 UTC No. 16288300

>>16287998
In the month before my dissertation defense I only slept 4 hours every two days.

Anonymous at Fri, 19 Jul 2024 16:43:28 UTC No. 16288540

How much better are the job prospects for an Applied Math Ph.D thab in pure? I did a bachelor's and master's in pure math but was thinking moving into applied would be a better bet in gaining employment while still getting to do something interesting. In particular I'm looking to go into numerical methods of PDE and FEM.

pic unrelated.

Anonymous at Fri, 19 Jul 2024 17:01:54 UTC No. 16288554

>>16288540
They’re literally the same prospects.
No one is hiring you for your specialization anyway, they’re hiring you because they can trust you with highly technical topics

Anonymous at Fri, 19 Jul 2024 17:12:39 UTC No. 16288569

>>16271237
Testing tex
[math]\frac{4}{4}[/math]
[eqn]
\sin^2(x) + \cos^2(x) = 1
\tan^2(x) + 1 = \sec^2(x)
[/eqn]

Anonymous at Fri, 19 Jul 2024 19:40:46 UTC No. 16288739

>>16288569
[math]{\rm \TeX}[/math]ting

Anonymous at Fri, 19 Jul 2024 21:15:59 UTC No. 16288869

Hello faggots
https://www.youtube.com/watch?v=VGmv-dq0YVY

Anonymous at Fri, 19 Jul 2024 21:20:22 UTC No. 16288878

>>16288869
More people should be interviewing these outstanding mathematicians.
I love this podcast and Math Life balance.
Any other podcasts?

Anonymous at Fri, 19 Jul 2024 23:51:57 UTC No. 16289029

Does anyone know what the Analysis sequence at the graduate level is like at Northwestern University? What it emphasizes, how hard it is, etc.?

Anonymous at Sat, 20 Jul 2024 00:10:08 UTC No. 16289041

>>16288878
>>16288869
Honestly I can’t watch happy mathematicians. When I was doing my undergrad and then my PhD people were cordial and even friendly but I never saw anyone being goofy or silly. I fucking hate faggots who can do math and be carefree. There is nothing carefree about mathematics. You should be doing it because you are compelled by your drive to make sense of the world and find meaning in life, not because it’s some fucking game to you.

Anonymous at Sat, 20 Jul 2024 05:08:28 UTC No. 16289262

>>16289041
Real mathematicians kill themselves, everyone else is just a tourist

Anonymous at Sat, 20 Jul 2024 07:16:26 UTC No. 16289330

>>16289041
>When I was doing my undergrad and then my PhD people were cordial and even friendly but I never saw anyone being goofy or silly.
I thought this was gonna lead into a complaint about people not being silly and happy but instead you take this fucking stance lol?

Anonymous at Sat, 20 Jul 2024 07:49:22 UTC No. 16289348

>>16286367
I very much enjoyed going through Halmos' Linear Algebra Problem Book. Any other books from him that are recommended?

Anonymous at Sat, 20 Jul 2024 07:57:15 UTC No. 16289359

>>16274438
Bumping for discrete math part recs

Anonymous at Sat, 20 Jul 2024 08:07:41 UTC No. 16289363

>>16289348
"Finite-dimensional vector spaces" is the book I was referring to. Most of his books aren't really in my field so I can't speak much about them. I know people like his book on basic set theory.

Anonymous at Sat, 20 Jul 2024 11:49:43 UTC No. 16289456

How do I git gud
I'm doing "Problems in real analysis: Advanced calculus on the real axis" and I'm struggling hard on the introductory problems
How long will it take until I do these problems with ease?

Anonymous at Sat, 20 Jul 2024 12:13:48 UTC No. 16289464

>>16289456
Give me an example of a problem that you've struggled with for the longest but managed to solve. Also say how long it took you to solve it. Then I'll be able to answer your question.

Anonymous at Sat, 20 Jul 2024 12:52:55 UTC No. 16289494

>>16289464
Here's one (the very first exercise) that took me 5 hours:
Prove that for real number [math]p\geq 0[/math] we have [eqn]\lim_{n\to \infty} \frac{(1^{1^p} \cdot 2^{2^p} \dots n^{n^p})^{1/n^{p+1}}}{n^{1/(p+1)}} = e^{-1/(p+1)^2}[/eqn]
Hint: Use the mean value theorem (it also shows the particular case of p=2 which I've omitted, and nowhere in my proof did I use the mean value theorem)

Anonymous at Sat, 20 Jul 2024 13:00:25 UTC No. 16289502

>>16289363
Looked at the books he's published and I'll give the measure theory book a try. Always wanted a deeper insight into it

Anonymous at Sat, 20 Jul 2024 13:09:27 UTC No. 16289506

>>16289041
cringe,
try-hard, larper etc

Anonymous at Sat, 20 Jul 2024 13:48:53 UTC No. 16289553

>>16289494
Ok I just went through the exercise. I probably wouldn't have gotten it without the hint but with the hint it was relatively easy. Have you tried working out the case p=2 using the hint?
Also I would like to mention that it's impressive that you've spent 5 hours on this problem and managed to solve it without the hint. Would you like to share your proof?

Anonymous at Sat, 20 Jul 2024 13:49:48 UTC No. 16289555

>>16289262
Don't forget this guy

Anonymous at Sat, 20 Jul 2024 14:10:40 UTC No. 16289578

>>16289494
>>16289553
>>16289456
Not any of the anons involved but I'll chime in anyways.
I just did the exercise posted, without the hint or looking at any special cases.
Moreover, I don't think there's any 'tricks' in how I did it.
I'll try my best to explain the thought process, maybe that's useful.
>there's a lot of powers and such, let's take logs and simplify
[math]\frac{1}{n^{p+1}}\sum\limits_{j=1}^n j^p \log j - \frac{\log n}{p+1}[/math]
>simplify the sum because there's multiple powers with p. also, if I do this I get a term [math]\frac1n[/math] and a sum, which may help in taking the limit (think integrals / laws of large numbers / ...)
[math]\frac{1}{n}\sum\limits_{j=1}^n (\frac j n)^p \log j - \frac{\log n}{p+1}[/math]
> it starts to look like the definition of a riemann integral; but the log has j, not j/n. Let's fix that
[math]\frac{1}{n}\sum\limits_{j=1}^n (\frac j n)^p \log (j/n) + \frac{1}{n}\sum\limits_{j=1}^n (\frac j n)^p \log (n) - \frac{\log n}{p+1}[/math]
>the first term becomes an integral (in the limit; sloppy), and the second seems simplifyable
[math]\int\limits_0^1 x^p \log(x)d x + \frac{\log n}{n^{p+1}}\sum\limits_{j=1}^n j^p - \frac{\log n}{p+1}[/math]
>I know/remember faulhaber's formula / how to expand sums of powers of integers, also integration by parts
[math]\frac{x^{p+1}}{p+1}\log x|_0^1-\int\limits_0^1 \frac{x^p}{p+1} d x + \frac{\log n}{n^{p+1}}(\frac{n^{p+1}}{p+1}+O(n^p)) - \frac{\log n}{p+1}[/math]
> the first term is 0 and the last terms cancel
[math]-\frac{1}{(p+1)^2}[/math]

This took ~15 minutes, and I've been doing math seriously for ~three years now, for all the context that may give.
Not trying to show off, but I think there's value in approaching stuff almost algorithmically, there's a lot of ideas that re-appear across exercises, and recognising where they are applicable has only gotten better with a lot of practice, for me.

Anonymous at Sat, 20 Jul 2024 14:12:42 UTC No. 16289582

>>16289553
My proof is pretty similar to the one described here: https://math.stackexchange.com/questions/2018793/lim-limits-n-to-infty-frac-big11p22p-cdots-nnp-big1-n/2018986#2018986
I couldn't for the life of me figure out how the case p=2 or how the MVT yields a satisfactory result. The "geometric" approach is much more intuitive, for me at least. Care to explain how you managed to prove it with the MVT?

Anonymous at Sat, 20 Jul 2024 14:39:40 UTC No. 16289615

New thread >>16289614

🗑️ Anonymous at Sat, 20 Jul 2024 15:41:52 UTC No. 16289711

>>16289464
>Give me an example of a problem that you've struggled with for the longest but managed to solve. Also say how long it took you to solve it.
How to modify and augment the Lotka Volterra predator-prey population dynamical system to include another species that’s poisonous to the predator and that the prey tries to be a Bayesian mimic of without the whole ecosystem collapsing. Took me about 8 months.

Anonymous at Sat, 20 Jul 2024 15:43:52 UTC No. 16289715

>>16289262
> Real mathematicians kill themselves, everyone else is just a tourist
While not exactly what I was saying, I still think this is a lot closer to the truth than the opposite

Anonymous at Sat, 20 Jul 2024 15:44:45 UTC No. 16289716

>>16289330
What’s wrong with the stance. If you think math is something to laugh about then it doesn’t mean enough to you and you should get out of the way for someone who reveres it

Anonymous at Sat, 20 Jul 2024 15:56:29 UTC No. 16289732

>>16289615
We’re on page 4 faggot

Anonymous at Sat, 20 Jul 2024 17:42:19 UTC No. 16289841

>>16289615
Still here

Anonymous at Sat, 20 Jul 2024 19:27:11 UTC No. 16289956

>>16289732
>>16289841
It's reached the bump limit though.

Anonymous at Sat, 20 Jul 2024 19:30:55 UTC No. 16289964

>>16289956
It’s still going to take days to fall off. Traffic is so slow on /sci/ during the summer and a premature new general cuts off what little conversation there was

Anonymous at Sun, 21 Jul 2024 08:17:07 UTC No. 16290616

>>16289041
what an autistic thing to say.