/sci/ - Science & Math

Let us begin by contemplating Euler’s two-way geometric series
[eqn]S = \dotsb + x^{-2} + x^{-1} + 1 + x + x^{2} + \dotsb[/eqn]
We can “solve” it by formally applying the geometric series formula:
[eqn]S = \frac{1}{1 - x^{-1}} + \frac{1}{1 - x} - 1 = 0.[/eqn]

>yooler

>>12275396
Right. Let’s address this issue immediately. The correct pronounciation is (or is closer to) “oiler”.

But now, /mg/, I present to you, Kontsevich and Zagier’s investigation of the ring of periods… except it was published in 1775, under Euler’s name!
https://arxiv.org/abs/0711.4986

>>12275396
If any of your professors say this, you are at the wrong Uni.

Dumb question, how does a math researcher figure out if a mathematical object they made hasn't already been discovered before?
It seems incredibly easy for someone to say, invent 'Example Function', not be able to find it in the huge swath of math literature out there, prove properties of the object, only to discover 'Example Function' was published years ago by someone else under the name of 'Instance Function' and they've inadvertently plagiarized someone else's work despite having discovered it independently?

>>12275585
advisors, google scholar, etc etc.

>>12275489

>>12275393
There are no negative numbers in this sum, therefore Euler is wrong. Therefore I am smarter than Euler. QED

Consider the linear map
[math] \mathbb{R}^{m\times n} \to \mathbb{R}^{m \times l}: W \mapsto WB [/math] (where [math]B \in \mathbb{R}^{n \times l}[/math]).
Is its transpose [math] \mathbb{R}^{m \times l} \to \mathbb{R}^{m\times n}: K \mapsto KB^\top [/math]?

Similarly for
[math] \mathbb{R}^{n\times l} \to \mathbb{R}^{m \times l}: V \mapsto AV [/math] (where [math]A \in \mathbb{R}^{m \times n}[/math])
Is its transpose [math]\mathbb{R}^{m \times l} \to \mathbb{R}^{n\times l}: H \mapsto A^\top H [/math]?

>>12275236
That notes hits me hard in the feeling, when we were still cosy and sh't.

>>12275695
I am trying to derive backpropagation in a simple neural network neatly and this makes it work, but I am just guessing.

[math]
a(W) = Wx + b \\
h = f(a) \\
a(W+H) - a(W) = Hx [/math]
which is linear in H, meaning [math] H \mapsto Hx [/math] is the derivative of [math] a [/math] at (any) W.
[math]
\frac{dh}{dW} = \frac{dh}{da} \frac{da}{dW} = \frac{dh}{da} (H \mapsto Hx) \\
(\frac{dh}{dW})^\top = (\frac{da}{dW})^\top (\frac{dh}{da})^\top = (H \mapsto Hx)^\top (\frac{dh}{da})^\top = (K \mapsto Kx^{\top}) (\frac{dh}{da})^\top = (\frac{dh}{da})^\top x^{\top}
[/math]

Let [math]\triangle ABC[/math] be a triangle in a Euclidean affine plane [math]\mathbb{A}[/math] and let [math]P[/math] be any point of [math]\mathbb{A}[/math].
Prove that the three straight lines through [math]P[/math] perpendicular, respectively, to the straight lines [math]AB,AC[/math] and the median corresponding to the vertex [math]A[/math] (the straight line determined by [math]A[/math] and the midpoint [math]M[/math] of [math]BC[/math]) cut the altitude through vertex [math]A[/math] in equal line segments.

>>12275585
>>12275594 is obviously correct, but it still happens. John von Neumann rederived an entirely equivalent (though different) version of Lagrange's "Analytique Mécanique" because he was trying to develop a computer algorithm for solving physics problems, and proudly gave a talk on his method only to have von Karman have to tell him afterwards he was just reinventing Lagrange.

So it happens to the best of us.

I'm studying rectifiability of sets and measures. i'm following Mattila's book but I'm finding it very hard to understand and to have mental pictures about the lemmas and theorems. Any advice?

>>12275769
You forgot to say that if we can't solve it in ten minutes we deserve to be castrated or something like that

>>12275769
Apply Ceva's theorem.

Hey /sci/, visiting anon from /g/ here. I'm mostly a front end monkey, but I've decided I'd like to implement a loot box game as a kind of non-trivial, practical project to learn a back end language and refresh my math knowledge, elementary as it may be.

How do I go about modelling this mathematically, so that I may implement it in a program? Would this be considered a difficult problem to model? I suspect it certainly has something to do with probability theory and maybe mathematical optimization, idk. Am I way over my head? Please forgive the ignorance, I am but a lowly developer with only a cursory, mechanical knowledge of mathematics.

I suppose I could formulate the problem statement as such: a loot box could be considered as a fix-priced set of N distinct products that differ in price, of which the probability allocation distribution is non-uniformly random. Products within a box are categorized into multiple rarity classes (each product belongs to a specific class), and are allocated probabilities based on their prices. Determine the revenue-optimal combination of price and allocation probabilities for loot boxes over multiple classes of products, such that over time, revenue will always be > 0.

So, the end result would be a program that would consume a csv or a json list of products (>= 2 obviously), and would spit out a box with the optimal price and optimal probabilities mapped to each product in the box.

Any hints/ideas?

>>12275796
he got kicked out of academia for plagiarizing right?

>>12275769
Are the altitude and the vertex the same line, cause if so what if P lies on it unevenly cutting it into 2 segments

>>12275970
I think you could determine the weighed averages of the outcomes. So if you have N possibilities, multiply their likelihood of occuring by their price and sum it all up. And you want it to be so that the sum of gross profit after N times is greater than the price to open it N times. And just find different probability distributions that keep this net revenue greater than 0, by varying one parameter and changing the other(s) to compensate

>>12274855
>>12274863
Anyone?

So, let's say I have the first 1000 primes and I multiply them together and add 1. This is a prime.
Say I do the same thing but instead of adding 1 I substract 1. This would be a prime as well correct?

Shouldn't this solve the twin prime conjecture?

>>12276090
There's no way it's this simple, what am I missing here?

>>12276090
>So, let's say I have the first 1000 primes and I multiply them together and add 1. This is a prime.
Why would it be a prime?

>>12276106
Why wouldn't it be?

Gottes Hande spaltet die Erde
Die Sonne dreht sich in die Dunkelheit
Und die Nacht Licht unendlich beginnt
Holle Regen von Himmel
Die Geist die Gesellschaft wird Tod Augen, sich wie ein mude und gelfortert Masse bewegen
Tod SPRIESST aus JEDEM HERZEN
Tod bluten wie DunkelLiebe uberall
Mit jeder Verlust, wir werden nummer und heisser, fulle die Lucken
Und wir erreichen mit unsere Grab Geist Hand
Uber und unter schloss
Und die Welt ist leuchtend mit hasse und Tod, liebe und Schoen

https://www.youtube.com/watch?v=H9UIdlXhrBQ

>>12276120
Failed the Miller–Rabin test.

>>12276090
the product of the first 10 primes plus 1 is 6469693231 which isn't prime (331×571×34231), so why ought the first 1000 primes plus 1 be prime?

>>12276020

Yes, I think I can make sense of this. I appreciate the feedback anon-sama.

>>12276106
Because if you divide it by any combinations of primes, its a prime plus a fraction

>>12276150
you alright bro? feeling sick?

>>12276150
You're only considering the primes you used to construct your number. There are other primes other tahn those that are smaller than this number, these other primes are all potential divisors of it. That's what happens with the example given here.

>>12276090
here's something to think about bro:
2*3*5*7*11*13+1 is not a prime because it equals 59*509

>>12276228
Yes, I see now. Just because none of the primes you used are not its factors, doesnt mean a different prime can't be its factor

>>12276125
Speaking languages other than English outside of /int/ is a bannable offense.

>>12276287
Being a faggot is a faggot offense

>>12276150

>>12276316
When one posts a faulty reasoning it is more useful for the recipient and speaker to explain where the rationale goes wrong rather than einfach provide a counteraxmple

>>12276316
>Wolfram Alpha

can you guys give me some examples of "nice" objects
e.g
>in analysis, continuous functions are considered "nice"
>in linear algebra, linear images are "nice"
what are more like these

>>12276375
Doesn't get much nicer than commutative fields.

>>12276375
Hausdorff and compact spaces in topology
Simple groups in algebra

>>12276332
ok, p1 p2 ... pm > pm

>>12276375
Personally, I prefer "good" and "very good" objects.

>>12276375
See this short list of various rigidity results:
https://en.wikipedia.org/wiki/Rigidity_(mathematics)#Examples

>>12275695
>>12275768
Can someone help?

>>12276435
I have an answer, but this thread is too small to contain it.

>>12275695


(WB)^T = (B^T)(W^T)

>>12276649
yes, but here the map is not represented by WB
WB is a Value of that map, not the map itself

>>12276748

I fail to see the difference

For example, if you have f(x)= x^2, then you can replace all instances of f(x) with x^2

>>12276765
WB is a linear map, whose transpose is B^T W^T
F(W) = WB is a linear map of linear maps
F(W1) = W1 B
F(W2) = W2 B
What's F's transpose? (W1 B)^T or (W2 B)^T?

>>12276799
Neither, it's [math] \lambda W_1. (W_1B)^T[/math], also sometimes written as [math] W_1 \mapsto (W_1B)^T[/math], which is alpha-equivalent as an expression to, and therefore represents the exact same function as, [math] \lambda W_2. (W_2B)^T[/math]

If we can prove that 2 = 1 then we can show that any set of 2 objects in the universe contains 1 object, meaning all objects are the same as any object they could be paired with, meaning all objects are equivalent, thus the universe is all one.

But how do we prove that 2 = 1? I envision the integers as a series of points, N points on one side and K on the other, with lines/arrows connecting them. If there are no arrows that fail to land, the two sets are equal in count. Is there a space where drawing 2 lines, each to a unique point, has both lines land on the same point, a space which tweaks the definition of same and unique possibly?

>>12276840
Ok then, [math] (W \mapsto WB)^\top = (W \mapsto (WB)^\top [/math]? How come? The transpose's domain is the dual of its codomain. Here it just uses the original domain...

>be mather
>come up with math object
>oooh I just found some counterintuitive property
>Double check to make sure I didn't just come up with the empty set
>try to prove counterintuitive property
>turns out I'm just retarded and my intuition was wrong about the counterintuitive property
thanks brain for turning the last three weeks of research into unpublishable results

>>12276397
>>12276400
>>12276428
>>12276432
thanks lads

Does math have anything that resembles moving parts? I don't mean dynamic systems, those equations themselves are set and fixed. I mean equations or concepts that have any sort of varying or interactions pathways that you can see, that feel physical almost. The one example I have is the method of the solution to e^z looking like orbits.

>>12277038
take your pills schizo

>>12276090
You're confused by the proof that there are infinite primes. In that proof, you don't prove that p_1...p_n+1 is prime. You prove that either p_1...p_n+1 is prime, or there is a prime number between p_n and that number. That's enough to prove the infinitude of primes but it means that it's not always the case that a shit ton of primes plus 1 is prime

>>12277038
I'm not sure I get you
I'm assuming you're looking for more than stuff like volumes of revolution, groups and simmetries of objects, prime spirals, and such

>>12276845
>If 2=1 then...
Yes it's called explosion
>I envision (mathematical object) as...
Doesn't matter that's not how math works, make your proof from the definition of Z then we'll talk

>>12276150
What about the primes between your 1000th one and your new number?

Is there any way to mathematically represent an emotion in itself? We can describe properties of emotions through stuff like periodic functions, differential equations, randomness, etc. But is there any mathematical structures that distinctly exhibit an emotion, such as Gladness? Of course a parabola looks like a smile, but that is flat and superficial, of course positive numbers are positive, but that is only one very simple relation to an emotion. Are there any groups, spaces, topologies, or other that generate varieties of structures that interact in a glad way, for example? Or who have naturally glad operators and properties, interacting complexly as a living mind?

>>12277266
>yes it's called explosion
The idea is to see if it could be true under any circumstances
>not how math works
I'm using that model because other models might fail to correspond to reality. A rigorous way of putting would be, have two sets. If every element in one can be mapped to some other, then they are the same quantity, if one or more elements fails to land, different quantity.

>>12277150
Groups and symmetries get closer in the direction. The idea is any mathematical concept that has some notion of "life" in it, where within the realm of mathematics, you can follow anything from the logical implications of that object, to the natural sets/topologies/spaces it creates, to the operators it comes with, and it feels more like a living system than inert truth. Like how physics is mathematical but also in motion, in more ways than just having the variable "t" in their equations.

>>12277384
Maybe some neuroscience stuff? I am definitely not the least bit knowledgeable about the subject, but I imagine that a lot of brain chemical shit is represented mathematically.

>>12277392
That actually would be really cool. The algorithm that describes the time, position, and function of various brain regions in creating a happy thought might actually somehow look like a happy algorithm. In fact, e^x is a non-superficially happy function, if you think about it!

>>12277384
I understand that bijective functions are how we measure the cardinality of sets but how does that prove that 2=1

>>12277533
I literally started my post with "If 2=1"
I was asking if anyone could figure out a reasonably interesting/nontrivial system where it is true

>>12277578
>I was asking if anyone could figure out a reasonably interesting/nontrivial system where it is true
As that anon pointed out, you can simply drop the principle of explosion. This is formally dual to LEM, so the resulting system (which is called "paraconsistent logic" if I'm not mistaken) is simply intuitionistic logic with the polarities of all statements flipped.

Also
>>12277384
>Is there any way to mathematically represent an emotion in itself?
In order to decide on the adequacy of any proposed mathematical representation, you must have a list of characteristic properties of emotions that you can verify to be simulated by the model.
Chances are that once you state this list explicitly and in full, there'll already be a pre-existing mathematical model that does the job. But this is a task that requires understanding of neuroscience and/or cognitive psychology, not of mathematics (see point (1) of picrelated).

>>12277605
Principle of explosion is trivial though. It's as good as saying "everything is one therefore 2=1", but we want to find a way for 2=1 without it basically being an axiom, I was thinking of some sort of periodic, self referential type of space

>emotion
I was thinking of it from the reverse perspective. It might be hard to discern the properties of an emotion, but if we could find a structure that has the right properties, we might feel the emotion from it and know. I would posit that all emotions, being of the same dimension/category, might share a basic algorithm. If we can discern the basic algorithm that makes any element of the universe be an emotion, then it might be easier.
>pic related in context
I think a little different. I see mathematics as being parallel to physics. So all properties of anything, just as they stem from quantum fields, they also have a natural origin/placement in some mathematical space

>>12277621
>Explosion is trivial though
Yes, a paraconsistent logic is one that lets you have contradictions like 2=1 without explosion

>>12277646
Poor wording, but that's what I meant. That allowing explosion/being paraconsistent is trivial, thus isn't a valid (imo) way to say that 2=1. I want some way to find out what the ancients meant by saying All Is One. But the truth is, we observe a broken symmetry universe with separate parts. So the more likely solution is finding a way that all pieces fall out of another piece. But we have that even in trivial set theory, where all elements carry the property of being in a set, they are all united, and conversely the set is defined by all of them

>>12277621
>Principle of explosion is trivial though
It isn't, and I was just about to add to my post a caveat that LEM (its formal de Morgan dual) is classically equivalent to various other formulations (e.g. double negation rule, Peirce's law) that become inequivalent once you pass to substructural logic. This is analogous to dropping the axiom of choice from ZFC, but retaining weaker variants like countable choice, the prime ideal theorem, or dependent choice.

>if we could find a structure that has the right properties, we might feel the emotion from it and know
I'll grant that this is a historically viable strategy, it worked for Newton's model of spacetime and Turing's model of the computer. But in both cases the effectiveness of their model came from understanding their subject matter deeply enough to state their defining laws with mathematical precision, in contrast to the "I know it when I see it" approach for which mathematics is of zero utility.

>I would posit that all emotions, being of the same dimension/category, might share a basic algorithm
And my hypothesis is that insofar as you're attempting to model emotions, the existing logic of algorithms is inadequate and a fundamental revision would be needed at the logical level. As far as I'm aware, this position is standard, and your postulate is extraordinary. Do you have any supporting evidence (not necessarily empirical)?

I'm back after like a week or two, did I miss anything interesting?

Also, question. Say my third year real analysis course isn't going to go over sigma algebras to construct the Lebesgue measure, instead we're doing what's in Apostol, single variable. Is this class a trash? This is the actual third (of four) mandatory real analysis class at my uni btw. "Abstract measure theory" is a grad course, and so it's one of the fifth ones you can take (along with functional analysis and something else).

>>12277662
>allowing explosion/being paraconsistent is trivial
Holy shit it's like I'm square dancing with the special girl.
Allowing explosion doesn't mean the same thing as being paraconsistent. Paraconsistency means you have DONT have explosion. You are looking for a system that has a contradiction (2=1) and that isn't trivial. I'm telling you that what you're looking for is a paraconsistent system of logic, which allows for contradictions without trivializing every proposition

>>12277697
Are you able to email whoever teaches the measure theory course to see if you can get in as an undergrad?

>>12277686
Supporting evidence:
>neurons follow algorithms
>emotions in general follow patterns such as sin(t), and sigmoid(t)
>emotions, when meditating, feel more mechanical, nervousness and excitement share a basic sameness of nerves feeling stimulated
>highly distinct emotions such as love and hatred feel to bleed into eachother at extremes, just like how colors at high and low ends bleed into violet, suggesting similar structures reused

>>12277702
I could %100 get in, no problem, but I have other courses I am more interested in (Algebraic topology & Rings and Modules). I can do some summer reading, I'm just mostly worried that the topic is generally required for most undergrad students.

>>12277384
Pic is me reading this thread, since I've been gone

>>12277740
Not sure if I understand your class situation, if you can get in and it's required for undergrads might as well take it instead of the classes you're just interested in. That being said I sympathize with having to do something like that.
Also if you think this thread is bad you should have seen the probability shit in the last thread

Quints

>>12277756
Nope.

>>12277756
Damn it I thought that would work

How about quints?

>>12277762
It's going up like two at a time it's pointless

I'm just going to keep on making posts until I get quints, let's let some time pass while I talk about a proof:
>any non-int fraction a/b divided by an integer c is not an int
>if c is a factor of a, then you have a/c = q/b, and b can't be a factor of q because that would make it a factor of a
>if c is not a factor of a, q/b also must not divide q into an int n, because that would mean q*c*b*n=a, meaning b is a factor of a

>>12277769
I fucked that up really badly

>>12277769
Nice proof, burn How dastardly would it be if someone got 77778

>>12277773
I meant to say but not burn

I get quints, Milhouse is finally a meme

>>12277752
>it's required for undergrads
That's the one thing I'm not sure about. It's not required at MY uni, but what about literally anywhere else?

>>12277778
Let me check my uni real quick, I doubt it

>>12277769
Fixed the proof

>any non-int fraction a/b divided by an integer c is not an int
>if c is a factor of a, then you have a/b*c = q/b, and b can't be a factor of q because that would make it a factor of a
>if c is not a factor of a, q/b also must not divide q into an int n, because that would mean q=b*n, thus c*b*n=a, meaning b is a factor of a

Just finished pic related, really enjoyed it. Are there any other books that chronicle the history of a theorem or problem and the components that went into solving it?

>>12277778
Yeah it's not required for undergrads for my uni, I doubt it would be for most

>>12277784
>>12277792
ic. Ty!

Alway rember.

Just talked about the Monster group in intro group theory and woah. All y'all scizos here with secret knowledge and spiritual universe theories might be onto something, because what the hell. Why would simple, natural theorems give rise to inelegant, inexplicable results and be so fundamental about it?

>>12277815
It's understandable that complexity can arise from simple things. One way is just that something is cyclic but never returns to the origin. Like if you rotate around a circle with an irrational angle, and keep doing repeating it

>>12277725
Your reasoning, as it stands, is hopelessly circular: the first two points are blatantly begging the question, while the appeal to "feeling" in the last two lead to the infinite regress of explaining emotions in terms of themselves.
The most charitable thing I can say about this is that it could be internally consistent, but I could probably say the same of numerology and astrology.
Still, you could argue that even those systems possess some kind of artistic value, so I won't object as long as you're aware that you're not pursuing any kind of scientific truth. Though you'll probably want to understand the rules of classical logic thoroughly before attempting to break them: the vast majority of intelligent people comply with them for good reason.

>>12277846
I don't understand what you mean about begging the question in the first two. You asked for evidence that an algorithm could describe an emotion, and I showed that algorithms pertain somehow. As for the last two, the point isn't to describe emotions in terms of emotions, but to highly that I have experienced algorithms in them. The emotion tells me about itself in its own terms, yes, but that doesn't mean there can't be other terms. It's just a trailhead for discovery.

Graphs [math]\Gamma_1 , \Gamma_2[/math] are called isomorphic if there's some bijection [math]\varphi: V_1 \to V_2[/math] s.t. [math]\{u,v\} \in E_1 \Leftrightarrow \{\varphi(u), \varphi(v)\} \in E_2[/math].
Groups [math]G_1 ,G_2[/math] are called isomorphic if there's some bijection [math]\varphi: G_1 \to G_2[/math] s.t. [math]\varphi(ab) = \varphi(a) \varphi(b)[/math].

QUESTION: These definitions are similar, but not identical; Is there a way to endow graphs with a group structure such that the two definitions literally coincide?

>>12277906
Unironically category theory

>>12277868
>I don't understand what you mean about begging the question in the first two.
You're claiming to demonstrate a link between emotions and algorithms by assuming that the link exists, and then elaborating on emotions and algorithms separately. It does nothing to demonstrate the existence of the link in the first place: if I believe that no algorithm (as the term is currently understood) can express emotions, then claiming that sin(t) expresses emotions isn't going to convince me one bit.
That said, circular logic is only a sin if you're aiming for objective, universal truths. If your goal is merely to express your subjective personal notion of emotions in the language of mathematics, as it seems from
>the point isn't to describe emotions in terms of emotions, but to highly that I have experienced algorithms in them
then go ahead and do whatever you like (and discard whatever logical laws you deem inconvenient); at most I'd decide that your understanding of "emotions" would be impoverished compared to my interpretation, but this would be a subjective (non-mathematical) judgment on my end.

Though I will say that this misunderstanding could have been avoided had >>12277384 been phrased less objectively, for example something like:
>Do you feel that there are any mathematical structures that distinctly exhibit an emotion, such as Gladness? I feel that a parabola looks like a smile, but that is flat and superficial, positive numbers feel positive, but that is only one very simple relation to an emotion
which would still be farfetched in my eyes, but at least it wouldn't be pseudoscientific.

>>12277906
Model a graph with vertex set V as a reflexive symmetric binary relation on V.
Then the isomorphisms of binary relations should yield the second definition (unless I've misunderstood it), and you can get the first definition by reinterpreting the edge set as a characteristic function [math]V\times V \to \{0,1\}[/math] with the corresponding properties.

>tfw cruising through Calc II, don't even need to study.
When does math get hard?

>>12276375
any fine set
any non-finite set is a scam

>>12277984
Eventually it will, and you will hit that wall hard if you don't study.

>>12277788
I enjoyed infinitesimals by amir alexander
really nice story of the beginning of analysis

>>12275695
lads?

>>12277621
I think explosion is only trivial over N because that's a theory with a generator S that lets you reach all elements (and even that only holds in the second order formulation) and in this way all theorems.
For a theory of more complicated objects, Explosion is quite suspicious if you ask me.

I haven't given up explosion mostly because the disjunctive elemenation hinges upon it (from "A or B" and "not A" follows "B")

>>12275236
I visited Euler's grave in St Petersberg.

It was across the lane from the writers, musicians and painters. The WMP area was well tended with elaborate graves, while the scientist area was neglected and overgrown. Very sad.

How do I prove that if [math]a \in A[/math] and [math]A[/math] exists then [math]a[/math] is a set that exists?

>>12278335
Can there exist a set with elements that do not exist?
Is this some intuitionist autism again?

>>12278335
you should look up the definition of \in

Is this just a really confusing fucking book that goes all over the place, or was I just filtered by a literal meme book? I did finish it, and I even got a good grasp on the overall matrix shit, but for every subject it touches I had to actually go and look for alternative explanations and almost all the time they were more clear then the ones presented in the book.

>>12278432
It's a terrible book, I don't know why it gets recommended all the time. It jumps all over the place. And if it's suppossed to accompany the OCW lectures it's not much better because, unsurprisingly, Strang talks much like he writes. A confusing mess.

>>12278442
Thank god I'm not the only one who feels like it's just absolutely unstructured at the individual chapter level. Lectures are much the same, agreed. I feel like the only people recommending this book are the ones who never read it comprehensively. So much time lost, fuck.

>>12278432
Yes, it's a horrible engineer-tier book and Strang's lectures are just bad, even though he is very likeable person.
The main reason it's bad is that it views matrices mainly as an array of numbers (like an engineer-monkey would) instead of representations of linear maps.
Read Axler's book and watch 3blue1brown's videos.

>>12278449
Well, I need it for engineer-tier stuff, but it has to actually make sense to be usable anyway.

Thanks, I'll try. Actually watching 3B1B video on change of basis rn.

>>12278447
At times I found it unstructured at the paragraph level, hah. Sorry that happened to you. A lot of people just haphazardly recommend books, I don't know why.
>>12278461
I needed it for the same reason, had the same experience as you. The class actually recommended books like Hoffman and Kunze, but the level it was taught at was much lower. Personally I used Friedberg et al and I quite liked it. Introducing linear maps early makes things so much more intuitive.
If you're watching 3b1b's, do watch episode 3. It's the cornerstone of the whole thing

Can anyone shed some insight.
Canada fag, went to school back home in Ontario and got a 3 years B.Sc in Mathematics, graduated in 2016.
Now I'm living in Montreal and want to stay in the city. How can I further my formal Mathematics education with a 3 year degree in the city, to maybe get a Master's degree in the future? From what I understand a University won't grant me a 4 year Honours degree off 1/4 of the credits, obviously.

>>12278513
since you are done with undergrad studies, if you speak french, try getting a master 1 or master 2 in maths in france, like in paris 11 at orsay https://www.unipage.net/en/400/paris_sud_university_paris_11

https://www.jm.universite-paris-saclay.fr/sites/default/files/media/2020-05/student_handbook_paris_sud_17-18.pdf

>>12275768
Explain this in English terms

>>12278597
>not "Umm, In English Einstein!"

>>12275236
i finally proved the algebraic properties of the conjugate gradient method to myself. bloody hell it was miserable, and the ridiculous part is none of the math is hard or complicated, just tricky to spot.

>half the thread is CSfaggotry
>not even thinly veiled bc reddit infestation has suffocated the last of board culture
>literal retards asking how the back propagation algorithm works
>people can’t do basic euclidean geometry
I’m glad I chose to go into applied math desu, enjoy stewing in this disgusting carcass of a board. thanks for the textbook recommendations btw, desu!

>>12278449
It actually does introduce linear maps in one of the later chapters. I was like fuuuuck, why haven't you just started with this shit, it would be so much more intuitive.

>>12278683
>mfw I took a master in sciences instead of applied or statistics

>>12278461
Engineer stuff are respectable. The thing is engineers usually are too utilitarian and don't delve properly into math which makes them end up not understanding anything and forgetting everything after a month.
If you learn stuff rigorously and intuitively, you will never forget them.
If you want to learn math, it's best if you learn it the "math" way, regardless of your specialization.

Mech eng faggot here. Finished my bachelor this year. My university unironicaly made me hate everything related to engineering. But fortunately, I have a small chance to get into masters in mathematics/computer science in very good university, at least in my country. They don't require good gpa or previous math degree, but they have exam. Can you anons recommend me some good undegrad level books with A LOT of tricky problems? Exam will be mainly on calculus, (linear) algebra and a bit of statistics/probability theory

>>12278335
[math] a \in A [/math] is not a proposition in a set theory without atoms. Regrade your starting assumption in a closed form.

>>12278937
What about [math]X \subseteq A[/math]?

Retard reporting:
Combinational Proof

Q:

Show that 1+2+3+4+...+n = (n+1)C2

A (so far):

Let S = {1, 2, 3,..., n+1}, Note: |S|=n+1

Then, the number of 2-element subsets of S is given by (n+1)C2.

Let X denote such a subset, of form X={k, l}, Note: 1<= k < l <= n+1

k has k possible values, l has (n+1) - k possible values.

By the multiplication principle, there are k * (n+1-k) possible 2-element subsets X of S.

Idk now how to go from k * (n+1-k) to 1+2+3+4+...+n.

>>12278597
x and h are layers of neurons and you go from x to h by doing f(Wx+b) (W is called a weight matrix and b a bias vector. f is some (usually non-linear) function.
dh/dW is the sensitivity of the layer h with respect to small changes in W

>>12278994
consider n distinct elements a1, ... , an
to find all possible pairs (no repetition, order irrelevant) do the following:
We focus on a1, and find all the pairs that contain it. They're n-1 in total.
Then we focus on a2 and find all the pairs that contain it, while ignoring a1, since that would be double counting. n-2 in total.
Then we focus on a3 and do the same, while ignoring a1 and a2 since that would be double counting again. n-3
...continuing like that...
we end up at a_(n-1). we ignore the previous elements since that would be double counting. only 1 pair ({a_{n-1}, a_n}) left

Thus we counted all combinations and we found that they're (n-1) + (n-2) + (n-3) + ... + 1

One hundred thirty fourth for the Atiyah-Guillemin-Sternberg convexity theorem.

>>12279029
Ty for the response, very clear. One thing though, shouldn't we consider n+1 elements rather than n?

>>12279083
np fren
yes, should be n+1

>>12278921
Have a look at
Berkeley Problems in Mathematics by Souza, Paulo Ney, Silva, Jorge-Nuno

Can be found online (z-lib, b-ok.cc).

>>12278198
>explosion is only trivial in N
Explosion by definition means that every sentence in your language is a theorem. It makes the entire system trivial. I've been trying to tell you this but it's like you just keep using words you don't understand.

>>12279385
Posters like that should be banned imo. They're posting in bad faith if they're not underage.

>>12279385
>>12279527
The poster with the girl image was not the writer of the initial posts

>>12279385
>I've been trying to tell you this but
This was my first post in the thread

>Explosion by definition means that every sentence in your language is a theorem. It makes the entire system trivial.
You don't need to adopt the principle of explosion to derive the principle that from [math]\bot:=(0=1)[/math], every arithmetic statement follows. Because the arithmetic statements are of the form [math]n=m[/math] for particular numbers, and all those statements can be derived from [math](0=1)[/math].
Therefore, to quote myself again
>I think explosion is only trivial over N because that's a theory with a generator S that lets you reach all elements (and even that only holds in the second order formulation) and in this way all theorems. For a theory of more complicated objects, Explosion is quite suspicious if you ask me.

>>12279527
What is your concern or complain exactly?

>>12279385
>Explosion by definition means that every sentence in your language is a theorem.
Also, I now realize that you guys might talk by each other.

Explosion, or the Principle of Explosion, or ex falso quodlibet, is the axiom that from a false statement being prove, any statement can be derived.

Explosion is not the state or situation where everything is derivable. It's the name of a rule of inference, which may be expressed as
[math] (P \land \neg P) \vdash Q[/math]
for any P and Q

>>12279385
>Explosion by definition means that every sentence in your language is a theorem.
Also, I now realize that you guys might talk by each other.

Explosion, or the Principle of Explosion, or ex falso quodlibet, is the axiom that from a false statement being proven, any statement can be derived.

Explosion is not the state or situation where everything is derivable. It's the name of a rule of inference, which may be expressed as

[math] P \land \neg P \vdash Q[/math]

or

[math] \dfrac{P \land \neg P}{Q}[/math]

or simply

[math] \dfrac{\bot}{Q}[/math]

for any [math]P[/math] and [math]Q[/math].

>>12277971
>You're claiming to demonstrate a link between emotions and algorithms by assuming that the link exists, and then elaborating on emotions and algorithms separately.
How is this different than mapping physical phenomena with laws of motion? Isn't the whole idea of saying something has an algorithm, just finding an algorithm to describe it?
>if I believe that no algorithm (as the term is currently understood) can express emotions, then claiming that sin(t) expresses emotions isn't going to convince me one bit.
Is your notion that something like an emotion is less possible to communicate than something visual/physical, like observing a stopwatch timing the velocity of a falling object? I would say they exist on the same plane, you could argue that it's impossible to convey any sensation, be it visual or emotional. But I think emotions are conveyable almost as well as other sensations. If I wanted to convince you that emotions follow sin(t), I would simply ask you to pick one mood, such as your feeling of energy/tiredness, existing on one axis where tired is negative and energy is positive, and to graph it subjectively over the course of one day, for a week. You would see it would wax and wane with sun and moon rise. It is absurd to think that you and I would be unable to mutually comprehend the same notion of fatigue and energy. This doesn't describe the nature of feeling energized itself, of course, it just describes properties that govern its change. But the point was to illustrate that emotions are on some level governed by algorithms, and thus might be on a deeper level as well

>>12279608
>You don't need to adopt the principle of explosion to derive the principle that from ⊥:=(0=1), every arithmetic statement follows.
THATS LITERALLY THE PRINCIPLE OF EXPLOSION.

>>12279635
The principle of explosion is not a statements involving numbers. It's a logical principle.

In the theory of arithmetic, the principle of explosion need not be adopted, since it follows arithmetically.

The fact that you can make valid arguments such as
>if 0=1, then 0=7*0=7*1=7 and thus 4=4+0=5+7=11, i.e. from 0=1 follows 4=11
does not mean you need to adopt the principle of explosion (the logical rule of inference)

Uhmm... Guys...

>>12279655
Ok wait when you said explosion is trivial in N, did you mean that explosion causes the system to be trivial; or did you mean that the principle of explosion holds in N regardless of whether or not you're in a para consistent logic (similar to how heyting arithmetic has LEM for some statements while being intuitionistic)

>>12279738
Saying LEM holds for some statements was inexact. I actually meant that a weaker version of LEM holds for propositions without quantifiers

>>12279738
>did you mean that explosion causes the system to be trivial
If by "explosion" you mean the principle of explosion, then I certainly did not mean that the system would then become trivial. Classical logic (and even intuitionistic logic) assumes the principle of explosion and arithmetic is not trivial.

To prove
[math] \forall n \forall m((0=1)\implies (n=m)) [/math]
doesn't require the principle of explosion.

To adopt the principle of explosion in a general logic context I find debatable (although by god I haven't tried to find out how annoying logic gets if you actually do drop it)

>>12279363
Pretty interesting book anon, thanks for suggestion

>>12279777
I don't understand what you're meaning by
>classical logic assumes the principle of explosion is non trivial
I agree with you that you can prove that statement for arithmetic without needing the principle of explosion, but what do you mean by nontrivial.

broskys, I need your help.
I'm desperate because I have troubles with "tricky" problems.
I look at them and immediately give up, for example.
[math]\frac{3}{3-\frac{3}{x+3}}-\frac{3}{3+\frac{3}{x+3}}[/math]
It's just simplifying but I just pussy out.

Don't call me retard pls, my thing is medicine.

>>12279797
>classical logic assumes the principle of explosion is non trivial
Who are you quoting?

Let me quote myself, the old control+c kind of way:
>Classical logic (and even intuitionistic logic) assumes the principle of explosion and arithmetic is not trivial.

Are you trolling me?

I'm dissappointed with /mg/, all you guys talk about is newbie stuff, set theory, logic and foundations in general. No one here seems to know commutative algebra.

>>12279844
It's just a phase. But commutative algebra is a bit too broad and inaccessible to ask questions and expect thought out answers.

>>12279826
Ok never mind I understand now I've been misreading everything.

>>12279813
Just multiply the tops and bottoms, it's a matter of effort not intelligence

>>12279844
I wish my stuff was commutative.

>>12279844
Then make a post about commutative algebra yourself.

>>12279890
Maybe I'm not putting enough effort, the thing is that normally when I see a problem I think that I have to solve it in my head and in just solve it in 5 seconds without effort, that's not the way, is it?
Also, when I get stuck, should I try to use rules like a retard?

>>12279909
fucking Grammarly messing with my shit.

>>12279909
>Also, when I get stuck, should I try to use rules like a retard?
What do you think this stuff is based upon? Of course you use rules if you have those.

>>12278077
Neat! Thanks

>>12278977
>>12278335
Define "exists"

>>12279897
I tried: >>12276034

I'm not implying commutative is harder in every sense or superior to foundations etc. This is basic commutative algebra.

>>12279844
I'm mainly here to learn, what more do you want from me lmao

>>12279924
isn't there an Anki to learn all the rules? asking for a friend

>>12279844
4chan(nel) is just more suited to contrarianism and argument in general. And there's not really much that's debatable about higher math

Wtf is up with all the frogposters? Are the adhd kids shitposting while on online classes?

>>12277815
Read about monstrous moonshine. You could make a religion out of that one topic.

[math]\bullet[/math]What textbooks, papers or text did you read today?
[math]\bullet[/math]What non-textbooks did you read today?
[math]\bullet[/math]Did you write something today?
[math]\bullet[/math]Did you do some programming today?
[math]\bullet[/math]Did you build something today?
[math]\bullet[/math]Did you clean up something today?
[math]\bullet[/math]Did you plan something today?
[math]\bullet[/math]Did you manage to work off some bureaucracy/paper work today?
[math]\bullet[/math]Did you practiced any skills today? If so, which?
[math]\bullet[/math]Did you do sports/cardio/weightlifting today?
[math]\bullet[/math]What were you eating today?
[math]\bullet[/math]How much sleep did you get yesterday?

>>12280190
[math]\begin{itemize}\item benis \item in \item anus \item :::DDDDDDDDD \end{itemize}[/math]

>>1227777

>>12280201
I fugged up

>>12280203
Meant to write
>12277777

crossposter here boys, recommend me your favorite/best book on the history and development of mathematics

>>12280205
yeah it's fine anon, I catch your drift either way

>>12280207
Fuckkkk
>>12277777

>>12280210
Do you want a nice read or a historical study including the math from that time?

>>12280190
All the answers are no. I'm a shell of a man.

>>12280214
both tb h

>>12280221

>>12278077
>>12277788

I got bored, so I came to whine.
This place fucking sucks now. I don't know whose fault it is, but most of you need to fuck off.
That'll be all.

>>12277697
I am halfway through my Analysis 3 unit (year 3 UK). We finished sigma-algebras, measurability, measure spaces, integrability and he just introduced Lebesgue spaces.

Analysis 2 was purely metric spaces and some basic topology. We haven't done real analysis since year 1, but it is a formal prerequisite for both. You should do real analysis and possibly some vector spaces as well as metric spaces before measure theory imo as it uses several results from them.

https://en.wikipedia.org/wiki/Esquisse_d%27un_Programme#Extensions_of_Galois's_theory_for_groups:_Galois_groupoids,_categories_and_functors

Has anyone read it? Is it worth reading it today. I am an undergrad and have no idea what direction the cutting edge of research is going. I want a tenure so I will study something relevant.

How do you get an idea about what people are studying in general?

>>12280520
>How do you get an idea about what people are studying in general
talk to people smarter than u, go to seminars

>>12280520
>I want a tenure so I will study something relevant.
Protip: If there was a subject you could study to guarantee you get tenure, then every hopeless idiot would have tenure.

>>12280477
You suck. Go eat a dick.

If [math] A, B[/math] are submodules of [math]M[/math] such that [math]A\bigoplus B \cong M[/math], why is there an idempotent [math]g: M\to M[/math]? I've been stuck on this for the past few hours and I just can't seem to come up with the right homomorphism. Any suggestions?

>>12280527
Yes I'm doing a project this year. The meetings are quite candid. I will try and talk to my supervisor about her research and opinions on the direction mathematics will go.
>>12280545
Not gaurantee, I see many people study purely something that they 'liked' based on limited undergraduate experience with that field. It might end up being a dead end or just have not many people interested in it. I don't want to be that person.

I'd prefer pursuing fields with several active research groups which will advice policymakers on funding towards it etc..

>>12280605
Use the direct sum to compose an inclusion with the corresponding projection (in the order such that you get an endomorphism of M). This is an idempotent of End(M).

>>12280558
I’m pretty sure the anime trannies want to suck dick so your post is not very effective.

>>12280605
Identity on elements of A and 0 on elements of B (or vice versa).
Also, the identity map itself id:M->M is idempotent.

>>12280641
To be more precise, I want an idempotent [math]g: M\to M[/math] such that [math]Im(g) = A[/math]. Could you please elaborate what you mean by "compose an inclusion with the corresponding projection"?

>>12280938
Don't listen to him, he doesn't know what he's talking about.
>>12280735
is what you want
For any (a,b) in M, let f(a,b)=(a,0).
f is the idempotent map you're looking for.

>>12280994
Hmmm here's where I am confused. I thought about doing that but I have [math] M\cong A\bigoplus B[/math] so I do not necessarily think M has an underlying "ordered pair" structure. This is what has been really confusing me. I do not think I can explicitly say [math] f(a,b) = (a,0)[/math] because ordered pairs are not elements of M

>>12280994
You essentially said the same thing I already did, dumbass.

>>12280938
It's [math]g := \iota_A \circ \pi_A[/math] where [math]\iota_A : A \to A\osum B[/math] is the inclusion [math]a\mapsto (a, 0)[/math] and [math]\pi_A : A\osum B \to A[/math] is the projection [math](a, b)\mapsto a[/math].

>>12281005
M = A + B literally means that every element in M can be expressed in a unique way as a sum of an element of A and an element of B. In other words, every element of M can be described as an ordered pair of an element of A with an element of B.

>>12281005
You prove it for [math]A\oplus B[/math] in place of [math]M[/math] and then bring everything back to [math]M[/math] through the isomorphism [math]M\cong A\oplus B[/math].

In the end, your actual [math]g[/math] will be [math]\varphi^{-1} \circ g \circ \varphi[/math], where [math]\varphi[/math] is the isomorphism [math]M\overset{\sim}{\to}A\oplus B[/math].

>>12281019
Anon said A and B are submodules of M.

A previous /mg/ post had asked if every sequence [math]x_n\in l^1[/math] is also in [math]l^t[/math] for some 0 < t < 1, where we define [math]l^t := \left\{ x:\mathbb{N} \to \mathbb{R}_{ > 0} \middle| \sum_n |x_n|^t < \infty\right\}[/math]. I think the answer is "yes, and moreover it will be in every [math]t\in (0,1)[/math]", but is my proof valid? And if it is, can we make it less ugly?

Proof(?): Set t=1 and transform the sequence to [math]x_n = \exp\{ t\cdot \log(x_n)\}[/math]. Because [math]\sum_n x_n[/math] converges absolutely, we can safely expand each term [math]x_n = \sum_m \left[\log(x_n)^m/m!\right] \cdot t^m[/math] and collect the powers of t into a Taylor series [math]\sum_n x_n = M(t)[/math] with real coefficients. Since M is analytic with radius of convergence [math]\geq 1[/math], it also converges whenever [math]0 < t < 1[/math], in which case [math]M(t) < \infty[/math] which is equivalent to [math]x_n \in l^t[/math], and we are done.

This result can perhaps be made more intuitive by interpreting [math]x_n[/math] as an unnormalized probability mass function for some discrete [math]\mathbb{N}[/math]-valued random variable X, such that for some continuous function [math]m[/math] parametrized by [math]t[/math], the random-variable [math]m(X,t)[/math] is also well-defined. But not before the result is validated.

>>12281029
Yeah, but if we want to be explicit and precise, it's better to use the structure of the direct sum.
You could define the projectiion as "Let [math]\pi : M \to A[/math] be defined by [math]\pi(m)[/math] being the unique element of [math]A[/math] such that [math]m - \pi(m)\in B[/math]", but that's cumbersome and lame.

Some people are satisfied with just sounding intuitive though.

I have a retarded question, why is it that I'm able to prove a theorem, but unable to internalize it for use later?

>>12281144
The proof can be quite different from how you apply it. You gotta have a good idea of the contexts of applications.

>>12281019
Perfect. This is what I was looking for. Thank you!

>>12281050
Update: Assuming my computations are valid, the functional form of the proposed probability transformation m should be [math]m(X,t) = tX^{t-1}[/math], following the sketch
[eqn] \sum_n x_n^t = \int 1 \cdot d(x^t) = \int 1 \cdot tx^{t-1} \cdot dx = E[tX^{t-1}][/eqn]
where the integral is in the sense of Lebesgue-Stieltjes, and all equalities are up to a finite normalization constant.

In other words, the result
>if [math]x_n[/math] is a real positive sequence such that [math]\sum x_n[/math] converges absolutely, then so does [math]\sum x_n^t[/math] for every [math]0<t<1[/math]
should be mutually deducible from the (in my opinion, more apparent)
>if X is a well-defined discrete random variable whose probability distribution has support [math]\mathbb{N}[/math], then the random variable [math]tX^{t-1}[/math] is also well-defined, for any t in the unit interval

>>12281050
>>12281234
Update 2: I've found an error in my proposed proof, in that the coefficients of M are not necessarily well-defined real values. In general, convergence of [math]\sum_n x_n[/math] does not entail convergence of [math]\sum_n \log(x_n)^m/m![/math] for individual m, and in particular it already fails for m=0.

Ironically, it's the crudely sketched argument from probability that seems to be more robust, and I suspect that the proof can be salvaged by Fourier-transforming [math]x_n[/math] instead (i.e. appealing to existence of the characteristic function of X). Unfortunately my knowledge of complex analysis is sorely lacking, so the only approach I can think of is to approximate t (and hence [math]\sum x_n^t[/math]) to arbitrary precision by a rational p/q and interpolate from the p-fold convolution.

>>12276090
multiplying the first 1000 primes gives you a number which will pass other primes along the way. So yes the product-1 will not have those 1000 primes as factors but one of the other primes you passed may be factors

How am I supposed to study math if I can't stop pacing around while holding a knife to my heart?

>>12281357
Stab proof vest?

>>12281358
My mind doesn't work. "Show that there exists a bijective.... AAAAAAAAAAAAAAAAAAAAAAAAAAA". I can only see pure static redness. How do thoughts come to be. Even typing these words is majority unconscious

>>12281358
A stab proof won't do anything. The knife is just catharsis. I know suicide will fail, because reincarnation is real. Exit does not exist.

>>12281368
It sounds like you need help friend. Take a break, talk to a shrink. Maths will be there for you when you come back.

>>12281368
>>12281372
It's gonna be okay anon.

Speaking about suicide, how common is it in your uni?
Did you personally know anyone who did it?

Hey guys I was looking through my dads ammo cabinet and I found a couple of pictures of a guy with his head blown off I'm not sure what to do.

>>12281433
It looks like the pictures are from an official coroner but to me that makes even less sense

>>12281433
>>12281436
What a strange larp

>>12281453
Sorry I had to get rid of the county part

>>12281486
Damn

>>12281486
But wait. What country starts with the letters KN? And the Cubs are an american sports team.

Lol I asked my dad and he said he got it when they moved the courthouse to a newer building a couple years ago and so they auctioned off old shit. Apparently they're just too retarded to clean their shit out properly before selling it. Anyone know what laws they broke here?

>>12281504
When I say they auctioned old shit, I mean they auctioned off the cabinet

>>12281486
you didn't blot out KANSAS CITY very well. at least we don't know which one.
that's fucked up

>>12281486
another one falls to the coroner virus...

>>12275236
Can someone tell me how pic related is even possible? The term at the bottom is a rearrangement of the numerator of the exponential without the negative sign.
All variables in the bottom equation are matrices. I tried doing it by hand and figured some factorization was involved, and got through that, but I don't see how this is possible:

[math]-x^T\Sigma^{-1}\mu+\mu^T\Sigma^{-1}x=2x^T\Sigma^{-1}\mu[/math]

>>12281700

>>12275236
Can someone tell me how pic related is even possible?
From https://arxiv.org/pdf/1811.04751v1.pdf

The term at the bottom is a rearrangement of the numerator of the exponential without the negative sign.
All variables in the bottom equation are matrices. I tried doing it by hand and figured some factorization was involved, and got through that, but I don't see how this is possible:

[math]-x^T\Sigma^{-1}\mu+\mu^T\Sigma^{-1}x=-2x^T\Sigma^{-1}\mu[/math]

>>12281704
it's possible because
1) x and mu are vectors
2) sigma^-1 is assumed to be symmetric
3) x^T sigma mu is a scalar
4) transposing a (real) scalar doesn't change it

>>12281704
>>12281704
well you must have
sigma x should be the same as the transpose

and
\sigma^[-1]^transpose = sigma^-1


and sigma is
Σ covariance matrix (real,
symmetric, positive semi-definite, |Σ| ≡ det Σ > 0)

I am having a great deal of trouble understanding mathematical induction and strong induction. I understand the basic principle but applying the logic to problems in hard.

>>12282119
I'm sorry to hear about your terminal brainletism.

>>12282119
induction: show it works for the smallest case, assume it works for the nth case, then show it works for n+1 cases.
strong induction: show it works for the smallest case, assume it works for all cases up to including n, then show it works for n+1 cases.
there's gazillions of examples of regular induction. prime factorization theorem is a good example using strong induction.

>>12275695
still no response FeelsBadMan

>>12282275
Reject Induction
Return to Contradiction

Explain to me how to visualize 4D objects

>>12282293
Ask in /sqt/ lmao.

>>12282332
Say there are 2D. Now imagine what it's like for a person living there, and for you living in 3D. Now extrapolate to higher dimensions

applied or pure?

Hey /mg/!

Could you advise good introductory books on differential geometry and information theory (i.e. turing machines, language models, this stuff)

Right
>Chaos theory
>cryptography
>Vector calculus
>dynamical systems
>biomathematics
>mathematical chemistry
>division

Left
>Group theory
>statistics
>Game theory
>CS
>linear algebra
>Pythagorean theorem
>number theory
>multiplication

>>12275695
>>12282376
I found a way to do it btw.
For example for the second case:
We have that AV = [AV(1) ... AV(l)]
So, just transform V with a column vector (columns on top of each other) and multiply by a block diagona matrix containing l copies of A.
The mapping becomes
V_flat |--> diag(A, ..., A) V_flat
l*n x 1 |--> lm x ln times l*n

The transpose of that is
W_flat |--> diag(A', ..., A') W_flat
lm x 1 |--> ln x lm times lm x 1

Where W_flat corresponds to W:=[W(1), ... W(m)]
So the mapping is equivalent to
W |--> A'W, just as it was hypothesized

>>12282425
Applied is a buzzword. A dumb meme.

>>12282494
That right and left brain stuff is a completely inaccurate picture of how the brain works.

>>12282548
it's a reference to the picture retard. right being the chad and left being the virgin

>>12282579
It's kinda vague though. I thought you were talking of the political spectrum first.

>Me having to explain to a first year graduate math student why 'Linear Algebra Done Right' does them no fucking good if they haven't taken a first course in Linear Algebra and actually gained an intuition for the material for the fucking 50th time

>>12282464
>differential geometry
Do Carmo

>>12282600
How do you get to grad without taking LA?

>>12282623
Don't, Differential Geometry was made by the Analysts to try and drain federal funding from Topology.

https://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=3440E99FEA7657D58DB87323D4678DBD?doi=10.1.1.169.9076&rep=rep1&type=pdf

Can I do this after taking abstract algebra?

Today I found out that any complex number of form "a+bi" can be represented as a 2x2 matrix. I solved this simple complex number equation with linear algebra.

z = ((1+i)(3-4i)) / (2+i)

Because this equation involves division, I realized that the inverse of the matrix representing the "2+i" term could be used to transform the equation into:

(1+i)*(3-4i) * (2+i)^-1

With this trick I ended up with this result:
z = (13/5 - 9/5i)

>>12282425
The real interesting shit doesn't even happen until graduate school.

In undergraduate you're just building up a foundation of topology, combinatorics, modern algebra, and complex analysis. But in graduate school you get to take manifolds, functional analysis, algebraic topology, algebraic geometry, measure theory etc

>>12282766
>measure theory
>interesting
ebic maymay

>>12282623
yikes

>>12282790
Ok midwit.

>Engineering student
>Started expanding on my math knowledge in free time
>Go through standard exercises
>Try to solve a harder question from competitions, like those from michaelpenn's channel and such
>Fail every time
The most I can get to is like halfway, then I get stuck. Is it even possible to improve in stuff like this or I shouldn't bother? I'd really like to learn more math but some of these questions just hammer me into the ground and I feel like a complete idiot. How to improve mathematical thinking?

bros I just want to be good at it, why is it so difficult. I'm doing imaginary numbers at uni. the concept is okay but there are some difficult exercises where i'm not sure what i am doing. fuck i'm so frustrated

>>12282846
take an sophomore intro proofs class or if you're a graduate eng student take undergraduate real analysis

>>12282868
Ty, any book recommendations? Calculus feels closest to home for me, that's probably the best place to start.

>>12282858
>some difficult exercises
Such as?

>>12282885
solve equation
|z|^3 = i*z^3
i'm pretty sure you need to use exponential form so
r^3 = r^3 * e^(3i*phi) * i
solution will be for every r from 0 to infinity and then i need to find the imagnary part I think

>>12282881
For intro level proofs you don't really need a book persay.

Just read up on writing very basic proofs like proof by contrapositive, proof by contradiction, proof by induction, using cases, etc. etc.. You also need to understand set notation but that is even more simple. If you get a good underlying idea then you can understand why he begins his proofs or choose certain cases.

does anyone know what this equation might be?
found it on one of my old phones and have always wondered

>>12282982
That's a series. It goes 1 - 1/3 + 1/5 -1/7 ... forever. It's "sum" is pi/4.

>>12282623
Started with this and loved it. Back when some of your professors were grad students they learned enough Portuguese to read the original because it hadn't been translated yet. Don't hate.

Perhaps this is taboo to ask here, but what kind of maths do you need for civil/structural engineering (and engineering in general)?

>>12283026
Calculus

How does one rigorously do a surface integral of a hypersphere? Like I know for example that on R^n, [math]r^{n-1}\int_{\partial B(0,1)}f(r\omega)dS(\omega)=\int_{\partial B(0,r)}f(x)dS(x)[/math] and see intuitively why it's true. I'm a bit lost as to how this would be actually proven, or even exactly how one should think of dS.

>>12282895
z^3 = |z|^3 e^(i3θ) where θ is z's angle.
Use this in your equation to get (z=0 is a solution, we assume z!=0 from now on)
1 = i e^(i3θ)
i is equal to rotating by π/2 (or equivalenty, multiplying by e^(iπ/2)), so we get
1 = e^(i(3θ+π/2))
therefore 3θ +π/2 = some multiple of 2π
θ = (2πk - π/2)/3 for some k
Since θ is defined to be between 0 and 2π (or -π and π, I don't know what you guys use, so I 'll use [0,2π))
θ = π/2 or π+π/6 or 2π-π/6 (k=1,2,3)
as well as z=0

so it's:
the imaginary axis
and the line with angle -π/6

>>12283088
oops, it's actually:
the positive imaginary axis
the half line with angle -π/6
the half line with angle π+π/6

>>12281820
>>12281742
Thanks. Now for this one, how did they factor the polynomial-looking equation to get the last line? Do you know of a source where I can read more about this?

Given the usual dictionary order, [0,1]x[0,1] , [0,1]x[0,1) , and [0,1)x[0,1] all have the least upper bound property right?

How the fuck does the [math]C^*[/math] algebra formalism of quantum mechanics handles unbounded operators?

>>12283173
They just added and substracted the necessary terms to create the thing in the left.
There's nothing to learn, really.

>>12283263
something something rigged Hilber spaces?

donno

[math]\bullet[/math]What textbooks, papers or text did you read today?
[math]\bullet[/math]What non-textbooks did you read today?
[math]\bullet[/math]Did you write something today?
[math]\bullet[/math]Did you do some programming today?
[math]\bullet[/math]Did you build something today?
[math]\bullet[/math]Did you clean up something today?
[math]\bullet[/math]Did you plan something today?
[math]\bullet[/math]Did you manage to work off some bureaucracy/paper work today?
[math]\bullet[/math]Did you practiced any skills today? If so, which?
[math]\bullet[/math]Did you do sports/cardio/weightlifting today?
[math]\bullet[/math]What were you eating today?
[math]\bullet[/math]How much sleep did you get yesterday?

>>12283298
Rigged Hilbert spaces extend the usual theory to allow distributional arguments.

>>12282600
That's not true though, see Europe

>>12283424
>see Europe
?

>>12282766
>manifolds, functional analysis, algebraic topology, algebraic geometry, measure theory
At my school people take all of these in undergrad.

https://www.researchgate.net/publication/231906328_On_the_Fourier_transform_of_a_compact_semisimple_Lie_group

>>12283525
You didn’t do all of those in kindergarten? Get a load of this brainlet.

>>12283322
I did headstands and tried to do handstands today. Did a little topology and did a lot of suffering.

>>12283525
You could optionally take functional analysis and measure theory in my uni in undergrad

>>12283322
>slept a full 8 hours yesterday
>continued reading a random graph paper from yesterday
>finished writing down one of my proofs for my own paper
>wrote two quick project proposals for a potential colleague
>went to visit my old college bro who's now unemployed, drank lots of beer, had pizza, played co-op videogames
I've had worse days

>>12283685
I'm not trying to shitpost. I didn't take functional or differential geometry because I didn't want to, but the majority of the math autist stream takes all of those courses.
We don't have courses on pointset topology because you just learn it when you need it in those courses.

>>12282766
you don’t have to nor need to take combinatorics to complete a math degree at most universities, nor would it hinder your ability gain admission at a good graduate program. algebra and complex analysis would be much more important.

>>12282674
differential geometry is older than topology, brainet.

>>12283816
Only because the differential geometers took advantage of their arcane knowledge to manipulate the space-time manifold and travel back in time.

>>12283689
nice
>>12283782
nice

>>12283842
do not reveal the secrets of shorter than geodesic travel

>>12283842
go back

>>12283891
>anyone making jokes I don't like is redd*t

the jews

>>12283913
Kill yourself, low iq faggot.

>>12278683
what are some insights you'd give to a math undergrad?

>>12283945
Why the homophobia?

>>12283967
Homophobia is the natural response to faggot behavior.

>>12283951
think very hard about what you want from your course
do exercises
all of them
ALL of them

>>12283980
ok thanks

>>12283972
/mg/ is a nice thread, if you're going to be a rude big*t do it somewhere else.
pic-related: you

>>12283990
forgot the pic

>>12275236
Physicsfag here. Had a tangential thought while reading a paper that involved antisymmetric orthogonal matrices.

Is there a classification/understanding of Lie groups for which there exists a solution to [math]e^X = X[/math]? I suppose for simplicity consider matrix representations of Lie groups and their algebras, so that it makes sense to say that [math]X \in G[/math] and [math]X \in \mathfrak{g}[/math] simultaneously.

For instance, taking [math]G = \mathbb{R}^{+}[/math], whose Lie algebra is [math]\mathfrak{g} = \mathbb{R}[/math], there is no real number [math]x[/math] for which [math]e^x = x[/math]. But "extending" to the group of complex numbers ([math]G = \mathbb{C} \setminus \{0\}[/math], [math]\mathfrak{g} = \mathbb{C}[/math]), it turns out there are an infinite number of solutions (related by something like [math]2\pi[/math] periodicity), e.g., approximately [math]x \approx 2.06 + 7.59i[/math].

I wonder how this generalizes to beyond "1x1 matrices". What conditions are necessary and sufficient, or which classical Lie groups have such solutions? For absolute simplicity I'm considering [math]\mathrm{GL}(2, \mathbb{R})[/math] right now but I'm not sure how to go about the problem besides some sort of brute-force Taylor series approach.

>>12283995
I didn't forget the picture but I understand that joke may have been a bit too much for you.

>I didn't forget the picture but I understand that joke may have been a bit too much for you.

>>12283998
I'm not sure this question even makes sense outside of matrix groups. Without some other identification between a Lie group and its algebra it seems ill-posed.

>>12284030
go back

>>12283525
>>12283685
I really don't understand why people obsess over undergraduate coursework. Beyond the bare necesseties, i.e. at least 2 semesters algebra, 2 semesters analysis, 1 semester topology, I feel like it's not that important. You'll see all this stuff in grad school anyway.

>>12284042
What, you want more?

>>12284044
Some of the topics classes will broaden your horizons at a level where you don't need graduate algebra or measure theory to understand.

>>12284034
I suppose that's probably true. I'm just a lowly physicist so I don't really think outside of matrix groups.

>>12283298

oh god kill yourself please i'm begging you

>>12284042
he can only go back if one of you goes with him

so who's goin?

>>12283998
Given that C embeds into GL(2,R) and bigger direct sums of such, we'll quickly find infinitely many solutions.

For one, your condition on matrices implies
det(X)=exp(trace(A))

Your requirement is however much more strict than
>X∈G and X∈g simultaneously

det(X)=exp(trace(X))

>>12284044

The state of math education in universities is abhorrent. The curriculum is designed under the assumption that you are totally uninformed about math and will decide to change your major 2 years in.

If you're not in the honors program, your diploma is as good as toilet paper. If you're in the honors program (which is a completely arbitrary gatekeeping measure that does way more harm than good) you're still going to be stuck going through basic math books at turtle pace for 3-4 years.

>>12284149
wasn't expecting to engage coom mode on mg, whos that polegirl?
reverse image search gives moon runes, soviet moon runes and a million nameless reposts

>>12284179
>books
>not module specific (free(while on the course)) tailored resources
man what do your fees go towards

How do I develop mathematical intuition like this?

>>12284204

>not module specific (free(while on the course)) tailored resources

FUCK no, fuck that cancer

>>12284204
donno, saved it in 2015

>>12284207
probably by doing a lot of math. and having rich Jewish parents who know people helps too

Although I always found that quote to be sketchy

>>12284207
>>12284224
Have Ashkenazi parents really

>>12284207
It's bullshit, unironically. To answer how to develop intuition: practice and try asking and answering your own questions.

>>12284207
By fixating on the meaning of operator symbols themselves. The geometry of the paper, of the symbol.

>>12284234
No, it's cause he's hungarian. I dated a hungarian and she was like this too, not a jew

>>12284207
Have a wealthy family and lots of connections. People will fall over themselves to say you can "intuit theories" even though you know none of the details.

>>12284315
>The Rich Parents means I can spend 8 hours a day practicing mathematics without having to worry about paying for anything and gain true mastery over it Chad
vs
>Working 39 hours a week + school + other bullshit means at best you get one hour of mathematics practice a day virgin

>>12277697
Sounds kinda trashy. It not like it takes a lot of work to do things in the generality like Folland's book does.

>>12282119
>I understand the basic principle
>applying the logic to problems in hard
Contradiction. [math]\square[/math]

>>12285425
>[math]\square[/math]
But what was to be proven?

>>12285425
(1) Do you understand the axioms of Peano arithmetic, and the basic logical principles behind the validity of mathematical proof?
(2) If your answer to (1) is yes, can you apply that understanding to resolve Goldbach's conjecture?

>>12275236
I'm truly having my doubts about category theory - it seems like anti-knowledge rather then knowledge. What the hell good is ending every paragraph with the phrase "up to isomorphism: and drawing triangles.

>>12285935
You will not gain any new knowledge from studying category theory. You will, however, be able to leverage your existing knowledge better, by taking advantage of the ubiquity of isomorphisms in mathematics.