/mg/ - maths general

[Anonymous No.11512441]

Previously >>11502149
Actual pictures of mathematicians edition.

Talk maths.

[Anonymous No.11512490]

I have n convex sets in the euclidean plane, I take the first and second set, A and B respectively, and rotate and translate them over each other such that their intersection is maximized, now their UNION is a new set,let's call this new set (AB), we take the third set C and rotate and translate it such that it's intersection with set (AB) is maximized, this new set we call (ABC), we take the fourth set and do the same process with the previously generated set (ABC) to get set (ABCD), and we do this process
with all n sets, and end up with the set(ABC...n) which within itself contains all sets from A to n, the question is is the set (ABC... n) the same as the set that has minimum area but also contains all sets from a to n? PicRelated is an explicit example-

[Anonymous No.11512500]

Asked in the last one.
How many irreducible reps are there of the complex Lie group/Lie algebra E6?
Is it 51840 since that's the dimension of its Weyl group?

I can't even seem to find a resource to learn the theory.

[Anonymous No.11512572]

>>11512500
Lie groups/algebras don't have a finite set of irreducible representations. It's odd that you somehow got to asking questions about E6 without working through the representation theory of sl2 first.
Irreducible rep'ns of complex (semi)simple lie algebras are classified by the theorem of the highest weight; irreducibles are in bijection with dominant integral weights (of which there are infinitely many, since they're just N-linear combinations of a basis).

[Anonymous No.11512670]

>>11512490
what an ugly problem
the answer is probably not
try pic related

[Anonymous No.11512698]

>>11512572
I'm not sure what you mean, the compact complex Lie group of E6 doesn't have a finite set of irr reps? It's a finite group, so it has a finite number of conjugacy classes

[Anonymous No.11512735]

>>11512441
What do you guys do when people ask you "what are you going to do with math?" Also how do I combat "muh when are we going to use this?"

[Anonymous No.11512745]

>>11512698
>It's a finite group

[Anonymous No.11512746]

>>11512670
your picture is a valid counterexample :( , guess I'll try a new way to make the set with minimum area which contains all other sets from A to n. Got any ideas?

[Anonymous No.11512751]

>>11512746
>is a valid counterexample
Are you sure?

[Anonymous No.11512756]

>>11512745
Am I retarded? Then what is the atlas project doing, what would then the character table of the real split group of E8 be?

[Anonymous No.11512856]

[Anonymous No.11512882]

[Anonymous No.11512891]

[math]\color{#781b86}{3.}\color{#751e8a}{14}\color{#72218d}{15}\color{#702591}{92}\color{#6d2894}{65}\color{#6a2b98}{35}\color{#672e9b}{89}\color{#64329f}{79}\color{#6135a2}{32}\color{#5f38a6}{38}\color{#5c3baa}{46}\color{#593ead}{26}\color{#5642b1}{43}\color{#5345b4}{38}\color{#5048b8}{32}\color{#4e4bbb}{79}\color{#4b4fbf}{50}\color{#4852c3}{28}\color{#4555c6}{84}\color{#4258ca}{19}\color{#3f5bcd}{71}\color{#405fcd}{69}\color{#4364c9}{39}\color{#4668c4}{93}\color{#496dc0}{75}\color{#4c71bc}{10}\color{#4f76b8}{58}\color{#527ab3}{20}\color{#557faf}{97}\color{#5883ab}{49}\color{#5b88a7}{44}\color{#5e8ca2}{59}\color{#61919e}{23}\color{#64959a}{07}\color{#679a96}{81}\color{#6a9e91}{64}\color{#6da38d}{06}\color{#70a789}{28}\color{#73ac85}{62}\color{#76b080}{08}\color{#79b57c}{99}\color{#7bb779}{86}\color{#7eb777}{28}\color{#81b875}{03}\color{#83b872}{48}\color{#86b870}{25}\color{#89b96e}{34}\color{#8bb96c}{21}\color{#8eb96a}{17}\color{#91b967}{06}\color{#93ba65}{79}\color{#96ba63}{82}\color{#99ba61}{14}\color{#9cbb5f}{80}\color{#9ebb5c}{86}\color{#a1bb5a}{51}\color{#a4bc58}{32}\color{#a6bc56}{82}\color{#a9bc54}{30}\color{#acbc51}{66}\color{#aebd4f}{47}\color{#b1bd4d}{09}\color{#b3bc4c}{38}\color{#b6bb4b}{44}\color{#b8ba4a}{60}\color{#bab849}{95}\color{#bcb748}{50}\color{#bfb648}{58}\color{#c1b547}{22}\color{#c3b446}{31}\color{#c5b345}{72}\color{#c8b244}{53}\color{#cab043}{59}\color{#ccaf42}{40}\color{#ceae41}{81}\color{#d1ad40}{28}\color{#d3ac3f}{48}\color{#d5ab3f}{11}\color{#d7a93e}{17}\color{#daa83d}{45}\color{#dca73c}{02}\color{#dea63b}{84}\color{#dea03a}{10}\color{#de9938}{27}\color{#dd9337}{01}\color{#dd8c36}{93}\color{#dd8635}{85}\color{#dd7f33}{21}\color{#dd7932}{10}\color{#dc7231}{55}\color{#dc6c30}{59}\color{#dc652e}{64}\color{#dc5f2d}{46}\color{#dc582c}{22}\color{#db522b}{94}\color{#db4b29}{89}\color{#db4528}{54}\color{#db3e27}{93}\color{#db3825}{03}\color{#da3124}{81}\color{#da2b23}{96 \text{...}}[/math]

[Anonymous No.11512910]

>>11512751
pretty sure. consider the isosceles triangle ABC inscribed in the circle, it is impossible to fully put triangle ABC inside the semicircle with diameter DB. Rotate triangle ABC around the point B so that DB and AB overlap. because C rotated on a circle with a radius larger than the semicircle, now point C lies outside the semicircle. I know this doesn't test all the possibilities but it's pretty obvious

[Anonymous No.11512943]

>>11512756
Why do you read "character table of real split form" and see "representation of compact complex lie group"? They're very, very far from the same thing. Lie groups are manifolds; if they're finite, they're extremely uninteresting. All the usual suspects are infinite, and have infinitely many irreducibles.
The real form of a complex lie algebra is a real lie algebra that complexifies to it. "Split" means it corresponds to a noncompact real lie group, whereas the compact real form corresponds to a compact real lie group.

You can't build a literal character table for a lie group/algebra, since there are infinitely many irreducibles. What you _can_ do is find a uniform method for computing a character; the Weyl character formula does this in the complex case. Even though there are infinitely many representations of a given lie algebra, the coefficients in the formula to go from representation -> character are exactly the same for all of them, since they depend only on the Weyl group. All you have to do is compute that set of coefficients once, and you can evaluate it anywhere you want.
You should think of the "character table of split E8" as something like a monstrously complicated Weyl character formula. With split E8, not all characters can be computed with only one formula, but you can partition your irreducibles up into finitely many blocks so that every representation in each block uses the same set of coefficients. The table tells you what the coefficients in the formulas are.

Some of this may be slightly inaccurate, so take a small grain of salt here (I do not specialize in this).

[Anonymous No.11512946]

>>11512910
Are you super duper sure the maximization for A and B doesn't look like this, or like some other shape which still admits inclusion into the disk?

[Anonymous No.11512949]

>>11512946
Forgot my image.

[Anonymous No.11513002]

>>11512745
I mean [math]technically[/math] up to Morita equivalence the irreps of Clifford groups are classified by their ranks up to Bott periodicity but anon obv wasn't aware of that.

[Anonymous No.11513162]

>>11512441
Oi, piece of shit, it was my time to create the thread.

[Anonymous No.11513205]

>>11512943
Thank you
>>11513002
Correct

[Anonymous No.11513241]

How much time do you guys spend on the internet? How much time do you guys spend studying?

Me: almost all day on the internet, I only get motivated to study when the tests are near, I wanna know if there are more like me here

[Anonymous No.11513256]

>>11513241
>How much time do you guys spend on the internet?
all day
>How much time do you guys spend studying?
not much, i do the mandatory coursework and study for exams but nothing more

[Anonymous No.11513273]

>>11513162
I'll wait for you to make them if you quit it with the off-topic images.
I mean, nothing wrong with memes, but at least make them topical.
>>11513241
Basically, I spend most of the day alternating between studying and skimping out on studying by shitposting, reading manga, watching videos on youtube and making and drinking coffee.
Sometimes I get tired and just don't study anything for a day or two.

[Anonymous No.11513283]

Threadly reminder to work with physicists.

[Anonymous No.11513291]

>>11513241
I study hard for maybe 2-ish hours earling in the morning. 3 on a good day.
The rest of the time I'm online, but I work some of the time I'm online too. It's not as effective, but you can still do math while watching crap on youtube.

[Anonymous No.11513325]

>>11512946
Not super duper sure, but quite sure, don't know what to try anymore, picrelated

[Anonymous No.11513337]

Ayo boys, philosophy chad here.

Are there any exercise books you guys could recommend so I can develop my algebra 1 skills? I am aware that's an embarrassing request, but my mathematics illiteracy is holding me back in my philosophy of science studies.

[Anonymous No.11513340]

>>11513337
Gelfand/Shen
or Lang's Basic Mathematics if you want to get memed

[Anonymous No.11513349]

>>11513337
Herstein's Topics in Algebra should be good, I used it for my Algebra 1 class

[Anonymous No.11513354]

>>11513283

[Anonymous No.11513362]

>>11513325
there is only one way to put the halfcircle into a circle (up to rotation)
there is only one way to put this triangle into a circle, up to rotation
so if you assume "set(AB) can be still covered by the circle C" you have only one degree of freedom, just rotating the triangle around
so you can just put a parameter [math]\alpha[/math] for this angle and compute the minimum of a one-variable function
finally, compare to the configuration on the left of your picture

[Anonymous No.11513366]

>>11513337
Art of Problem Solving - Pre-Algebra (It's mostly not just pre-algebra but it's very good to read this too)

Art of Problem Solving - Introduction to Algebra

[Anonymous No.11513375]

>>11513273
>I'll wait for you to make them if you quit it with the off-topic images.
Ok, deal, you can make the one after the next.

[Anonymous No.11513380]

>>11513366
>>11513340
>>11513349
Thanks, fellas. I appreciate it a lot.

[Anonymous No.11513383]

>>11513349
Herstein is algebra II.

[Anonymous No.11513452]

>>11513362
well yeah, but from the right picture it is pretty obvious that whichever way you rotate it it will never be as small as the tiny area on the left, and the max area included in the semicircle will be at most 3/4 of the area of the triangle. So it doesn't work I guess.

[Anonymous No.11513460]

>>11513452
What if it's like this tho?

[Anonymous No.11513502]

>>11513460
the other post said there's only one way to put an that (inscribed)triangle into a circle, up to rotation. will check tomorrow

[Anonymous No.11513860]

I like maths, I like cute animals, I like peace and quiet. I like the quarantine.

[Anonymous No.11514045]

>>11513860
>Have 6 girls ready to fuck
>Can't because quarantined
I fucking hate it.

[Anonymous No.11514057]

>>11513273
R u me?

[Anonymous No.11514067]

>>11512441
what is tropical geometry about and what is it's motivation?

[Anonymous No.11514071]

>>11513337
elements of modern algebra by linda gilbert is a nice one

[Anonymous No.11514072]

>>11514067
it's like algebraic geometry, but it's hot and sweaty and there's a lot of bugs

[Anonymous No.11514112]

>>11514072
Algebraic Geometry is the european superior continent

Tropical geometry is the south american peasant

Based

[Anonymous No.11514442]

Greetings,
One of the required classes for my masters states:
>The student must have a solid understanding of linear algebra, calculus, ordinary differential equations, and Fourier theory...
It's an engineering analysis course. What textbooks would you recommend for preparing for this? The most math I've taken so far was calc III & diffeq like... two years ago. I would just audit a bunch of math courses, but the math department at my uni is iffy. Plus the credits would still go towards my total registered amount and I'd have to pay for them.

Thanks in advance.

t. stupid mechE student

[Anonymous No.11514610]

/mg/pill me on topos

[Anonymous No.11514612]

>>11514610
muh gomboc

[Anonymous No.11514616]

How do I get better at forming differential equations from a paragraph of information. I was always aces at forming an algebraic formula from word problems but with differential equations the language used just confuses me.

Also, with something like
>a raindrop evaporates at a rate relative to is surface area
How tf do I get (dV/dt) = -kV^(2/3) (where k is constant)? I knew it would be dV/dt and I know the volume and surface area of a sphere, but otherwise I have no idea.

[Anonymous No.11514619]

For my final year of undergraduate, I have to pick either a paper on Math Logic or Hilbert Spaces do get a math major. Which should I do and why?

[Anonymous No.11514626]

>>11512735
What math is it? Just say physics, finance or engineering.

[Anonymous No.11514770]

>>11513283
cringe

[Anonymous No.11514776]

>>11514610
Go for it if you are interested in categorical logic, pointless topology or the mixture of those two. Just remember that abelian toposes are pretty much the most well-behaved categories you can have.

[Anonymous No.11514987]

>>11513002
You're extremely fucking annoying.

[Anonymous No.11514989]

>>11513241
Studying is for smart people, so I just waste my time some way or other.

[Anonymous No.11515014]

>>11514045
Unvirgins are not welcome here.

[Anonymous No.11515019]

>>11514045
Thanks for the reminder anon.
I'm trying to prove Riemann Hypothesis to distract myself from my terrible situation.

[Anonymous No.11515025]

>>11514619
Hilbert spaces are very useful and interesting, and easy. Math logic can be immensely autistic, so if you're not sure you like the topic you should avoid it. Besides that, what do you want to do after your undergrad? If a Master's, on which area?

[Anonymous No.11515075]

>>11515025
I've done a discrete math/math logic precursor course to the 300 level math logic course I'm considering which I really enjoyed, but can't see the use.

Im considering doing a masters in nuclear physics, quantum physics (I'm doing a double math/physic major) or math masters of some kind... Idk just where ever life takes me. I don't know much about Hilbert spaces, cant they be quite useful for quantum physics?

[Anonymous No.11515080]

>>11515075
>I don't know much about Hilbert spaces, cant they be quite useful for quantum physics?
Kek, obvious bait.

[Anonymous No.11515146]

>>11514442
You can probably just sit in on the lectures without registering anything, no?
Anyway, I recommend MIT OpenCourseWare 18.06SC Linear Algebra. The lecture videos are by Gilbert Strang and they're very good. Should be all you'd need.
I'm sure the most Fourier theory you'll need is to read a tiny bit about fourier series and fourier transforms, and the basic properties of each. Spend a day on each one.

[Anonymous No.11515156]

>>11514616
It's just algebra. Your variable is volume, you want to express surface area in terms of volume. What are the formulas for surface area and volume of a sphere? Can you solve for radius in terms of volume, and then plug the result into your surface area equation?
I suppose the coefficients don't matter since it says "rate relative" instead of the exact rate.
Now why the negative sign? Cause you want evaporation, i.e. decreasing volume.

[Anonymous No.11515160]

>>11515075
Hilbert spaces are quite literally the foundation of all quantum physics. Do that.

[Anonymous No.11515166]

>>11515146
Don't "just spend a day on each one", anon. Spend as much as you may need.

[Anonymous No.11515173]

>>11515166
Of course. I just meant he doesn't have to go deep into the theory of fourier series convergence, sampling theorems, poisson summation stuff, etc. All he needs is definitely "this is the definition, this is the inversion formula, this is what it does to derivatives, shifts, scaling, and products, hey look you can use it to solve the heat equation"
that's it

[Anonymous No.11515486]

unrelated to math really, but I am an UG about to graduate in a semester, and at my school I can take some first year grad classes for credit. I work at a catholic school, and my principal knows im doing a graduate class, but I have reminded her multiple times that I am an UG. She seems to forget and advertises me as a graduate student in mathematics at our local university(to make me seem like a well qualified tutor for the kids) to other faculty and parents. I literally just don't feel like sending an email again to remind her cause it seems nitpicky. If I'm not lying about my academic credentials, but my boss is, can I get in trouble?

[Anonymous No.11515547]

>>11515486
You're much more likely to fuck up your situation by ceaselessly trying to correct your boss than you are by somebody finding out you're secretly not quite a grad student.
It's quite unlikely anyone is ever going to know anyway; not even the most bored of helicopter moms are going to start calling around your department doing background checks on the tutor provided by their son's school.

[Anonymous No.11515723]

Why are Eigenvectors and Eigenvalues important and why should I give a fuck about learning them?

[Anonymous No.11515741]

>>11515723
many reasons
one of them is they let you decompose matrix into a nice form called jordan form

[Anonymous No.11515787]

>>11515547
alright, thank you lad that makes sense.

[Anonymous No.11515792]

>>11515723
They permit a method for calculating an nth power matrix easily

[Anonymous No.11515898]

>>11514067
Tropical mathematics redefines the rules of arithmetic by replacing addition with taking a maximum, and by replacing multiplication with addition. After briefly discussing a tropical version of linear algebra, we study polynomials build with these new operations. These equations define piecewise-linear geometric objects called tropical varieties. We explore these tropical varieties in two and three dimensions, building up discrete tools for studying them and determining their geometric properties. We then discuss the relationship between tropical geometry and algebraic geometry, which considers shapes defined by usual polynomial equations.

https://arxiv.org/pdf/1908.07012.pdf

A lot optimization,computation or math problems can be defined as Max or Min problem

[Anonymous No.11515970]

>>11515723
the most immediate answer:

eigenvalues and eigenvectors solve the following two problems. given a linear transformation written as a matrix, how do we determine what does it do geometrically? given a linear transformation described by the geometry, how do we write it in matrix?

long answer:
just keep doing math and you will see that they appear fucking everywhere

[Anonymous No.11516188]

>>11515723
https://m.youtube.com/watch?v=8F
The matrix of a linear transformation with a basis of eigenvectors is structurally revealing

[Anonymous No.11516201]

>>11515146
cool, thanks

[Anonymous No.11516263]

>>11516188
It seems you goofed off some of the video ID in the URL.

[Anonymous No.11516279]

>>11516188
>>11516263
Yes
https://m.youtube.com/watch?v=8F0gdO643Tc

[Anonymous No.11516410]

>>11510404
Could you explain a bit more where this identity comes from please?

[Anonymous No.11516556]

>>11516410
Just expand the polynomial.

[Anonymous No.11516558]

>>11516556
i asked where it came from, not why is it true

[Anonymous No.11516573]

>>11513337
I'm back again.

Can one of you poindexters explain to me why -a + -a = -a

Yet

-a × -a = a

Why does multiplying two negatives get a positive?

Who invented this algebra shit anyway.

[Anonymous No.11516583]

>>11516573
I miss middle school math.

[Anonymous No.11516594]

>>11516583
I worked it out. I'm retarded.

[Anonymous No.11516622]

>>11514610
Nonetheless one should learn the language of topos

[Anonymous No.11516663]

>>11516410
You know, lad. The norm. Numberino times conjugaterino.
[math]\overline{ \omega } = \omega ^2[/math].

[Anonymous No.11516671]

>>11516663
yes but that expression on the left is a multiplication of three terms, while when i think of norm i think of two terms

[Anonymous No.11516676]

>>11516671
I also don't know where the first term comes from.
Second two are literally the norm, tho, since [math]\omega ^4 = \omega \times \omega^3 = \omega [/math].
Presumably just compute the norm normally and notice that stuff cancels out if you multiply by (a+b+c).

[Anonymous No.11516681]

>>11516676
Well, but that doesn't really tell me where it comes from other than a clever identity that happened to be helpful here - I want to know if there is a more systematic way of handling that type of question. And in fact, the a+b+c term is pretty significant, since it's what gives us the "0" on the left hand side of the equation to give us the result

[Anonymous No.11516744]

Is there a neat and tidy definition of category which doesn't appeal to objects?

[Anonymous No.11516851]

>>11516744
You can do it all with just the arrows. You replace an object $C$ with a "0-arrow" $c$ such that, whenever defined, the composites $f\circ c, c\circ g$ are merely $f$ and $g$, respectively. Not a big surprise that this $c$ will turn out to be the identity of $C$.

[Anonymous No.11516853]

>>11516851
Oops, imagine [(/)math] instead of $.

[Anonymous No.11516875]

>>11516851
Yah, but that just tells me how to remove the objects from a given category. It doesn't give me axioms which guarantee I have a category (in the usual sense) from just the arrows.

[Anonymous No.11516936]

>>11513283
Thanks

[Anonymous No.11517047]

Can you guys aolve IMO problems?

Can you guys solve Putnam Problems?

I'm an average-IQ person, so I'm always insecure about whether or not I can train hard enough in order to solve them one day.

[Anonymous No.11517621]

>>11517047
my scores at the putnam have varied between 40 and 80
not sure if that qualifies as "being able to solve them"

[Anonymous No.11517665]

>>11516744

[Anonymous No.11517766]

>>11517621
>I vary between top 5% and top .5% dunno how good I am tho xp
kill yourself animenigger

[Anonymous No.11517793]

>>11515898
>tfw can't get past the drawing a tropical curve step
"just draw your triangles differently bro lmao"

[Anonymous No.11517944]

Can someone please explain to me how you actually calculate global and local truncation error of a finite difference method? I'm not asking for an explanation of the terms, I'm asking how to *calculate* global and local truncation error for a specific FD method, as in what is the general procedure.

[Anonymous No.11518278]

>>11517665
That works, thanks.

[Anonymous No.11518476]

>>11516744
partial monoid

[Anonymous No.11518498]

>>11518476
>partial monoid
The one defined here?
http://math.chapman.edu/~jipsen/structures/doku.php/partial_monoids
http://math.chapman.edu/~jipsen/structures/doku.php/partial_semigroups

Do the axioms actually guarantee that it's the morphisms of some category?
Specifically, can you prove that, if [math]ab \neq *[/math] and [math]ac \neq *[/math], then [math]db \neq *[/math] if and only if [math]dc \neq *[/math], which roughly translates to the possibility of tracing back a codomain for [math]b[/math] and [math]c[/math]?

[Anonymous No.11518504]

>>11512441
I'm an undergrad in math and it worries me that I literally don't understand wtf is being discussed on mathoverflow. Does this mean math isn't for me?

[Anonymous No.11518511]

>>11518504
Why do you expect that you should be able as an undergrad to understand a question so complicated that one of the best mathematicians in the world couldn't figure it out and decided to ask for help?

[Anonymous No.11518538]

>>11518504
Yes, this is basic symplectic geometry. I learned it in the second year of my undergrad.
Go read Hofer's book on Symplectic Capacities or something.

[Anonymous No.11518543]

>>11518498
I'm too lazy at the moment to think about it, but I've found this overflow question
https://mathoverflow.net/questions/168724/can-a-categorys-partial-monoid-of-arrows-be-completed-to-a-total-monoid

[Anonymous No.11518631]

>>11512891
3.14159265358979323846264338327950288419716939937510582097494459230781640628620899862803482534211706798214808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493038196...

[Anonymous No.11518646]

>>11512441
yeah I don't know what 1+1 is guys please help. It's for homework.

[Anonymous No.11518656]

>>11513241
all day on the internet here and only study near exams. If i were to rate possibly 7/10 since the internet was my tool for everything and because I hate that I study before an exam

[Anonymous No.11518671]

How would you write a polynomial whose variables are a function and it's partial derivatives up to an order using multi-index notation?

[Anonymous No.11518673]

>>11512441
20 years old and want to be proficient at math again. Started on Khan academy to revise on known concepts and to be 100% clear on them but is it a good start to do so?

Would like to know what sort of roadmap I should have since I'd like to be able to know more than what I had learnt about maths, potentially university level topics (i know there is a img general for books related), as well as, higher tiered concepts not relatively close to univeristy maths, as a means of a test of understanding. It really is one of my favourite subjects but due to bad life choices and silly mistakes, I have figured that this year onwards is my only chance at redemption.

[Anonymous No.11518687]

>>11518646
it's the most beautiful equation in maths and involves euler's constant. 1 + 1 = e - (e-2) + 1 + i^2

[Anonymous No.11518692]

>>11518671
Like this [math]\Sigma _{i, ~ \alpha} a_{i, ~ \alpha} ( D^{ \alpha} f )^i[/math]?

[Anonymous No.11518703]

>>11518692
>mfw I see people use \Sigma for summation
Also we perhaps don't want the [math]i[/math] index there since then [math]\sum_\alpha a_\alpha(x)D^\alpha[/math] ceases to be a linear operator on [math]f[/math], whence much of diff top/OA no longer apply. In particular I don't know how one would define the symbol of such an operator.

[Anonymous No.11518704]

>>11518538
>tfw there's people far ahead of you by their second year of undergrad
But it's about how much you yourself progress, r-right guys? Haha, I mean, it's crazy to compare myself to this guy haha. S-Surely I don't have to think too much of it.

[Anonymous No.11518742]

>>11518703
That would be nice, but he said that the variables are the function and the derivatives, so we unfortunately probably aren't working with the polynomial ring of differential operators.

[Anonymous No.11518745]

>>11518742
You might be right, but that is definitely weird since not much can be done for those objects.

[Anonymous No.11518749]

nothing is comfier than waking up at noon with the rest of the day ahead of you to write proofs and explore mathematics.

[Anonymous No.11518871]

>>11518704
he's memeing, many mathematicians won't even know what is symplectic geometry if it doesn't intersect with their research interests

[Anonymous No.11518964]

https://strawpoll.com/fz4g5wxa/r
the absolute state of this board gentlemen

[Anonymous No.11518981]

>>11518964
>implying that matters
Behold a poll truly capable of gauging this board's quality.
https://www.strawpoll.me/19664399

[Anonymous No.11519095]

why are there so many chimps on this board? every other thread is 86iq garbage

is /sci/ not moderated?

[Anonymous No.11519100]

>>11518981
>no Kahler nor hyperKahler
Won't be voting

[Anonymous No.11519154]

>>11519100
>no anon, you need to spend the entire week adding in web geometry, systolic geometry, Bieberbach groups and whatever specific object it is I like instead of hoping I can place it in a broad subfield myself
Just vote for complex geometry, symplectic geometry or Riemannian geometry lmao.

[Anonymous No.11519245]

>>11519100
>>11519100
You don't even know what a kähler manifold is you pseud

[Anonymous No.11519261]

>>11518981
You forgot hyperbolic geometry, algebraic geometry and arithmetic geometry

[Anonymous No.11519298]

Why are pure maths people so fixated on one-upping each other in terms of wanting to seem like the smartest person in any given maths conversation? It's like they get off on explaining something in the most abstruse and useless way possible. The worst part is after they've formulated their point in the most abstract and obscure form possible, they'll claim that that's the most natural way to think about it for them, which of course is a lie meant to impress their equally insecure mathematician friends. You can visibly see them cum in the pants when they do this.

Why do so many mathematicians do this? Is it because they're insecure about maths being the only facet of their life in which they're able to express competence? It's like jocks trying to out-alpha each other.

I know this of course doesn't apply to everyone, but I would argue this is the case for a lot of mathematicians.

[Anonymous No.11519340]

I am taking Trigonometry and this week we are discussing Lissajous figures. I have to write a 500-word summary along with graphs, and for the research, I have to find two interesting facts about them. Apart from searching online and finding the usual answers (which I still have to source), I figured I would come to /sci/ and see if there are any unique perspectives in regards to Lissajous that are not online. I looked through the sticky and couldn’t find anything related.

[Anonymous No.11519342]

>>11519298
I'm pretty sure it's a general human trait, much independent of the subject.

[Anonymous No.11519364]

>>11519340
reminds me of
https://youtu.be/4gibcRfp4zA

[Anonymous No.11519384]

>>11519298
not all math people are like this, but a lot are
sadly I am guilty of doing this too although I taught to stop myself mostly
>Is it because they're insecure about maths being the only facet of their life in which they're able to express competence?
yes

[Anonymous No.11519385]

>>11519364
That's pretty amazing from what I watched. Thank you, Anon.

[Anonymous No.11519445]

When converting complex numbers from rectangular to polar form, do you need to change the angle only when the form is not contained in the first quadrant?

[Anonymous No.11519462]

>>11519445
>form
>change angle
?

c=|c|exp(i*arg(c))
where I'm sure there's a nice formula for arg, involving arctan or whatnot

[Anonymous No.11519480]

>>11519462
What I mean is, say I have the polar form of:
r = -5
theta = -45
-5<-45 degrees

Do I leave it as that, or do I do 180 + (-45) = 135 and rewrite it as
-5>135,
because it is contained originally in the third quadrant.

[Anonymous No.11519501]

>>11519480
is r the radius?

just take a positive radius and add 360 to the angle in case it's not in the range [0, 360)

[Anonymous No.11519610]

>>11513380

If you care about results over prestige, use Schaums, and Dummies books are surprisingly good too. I personally after being out of high school for years wanted quick and dirty results for algebra used this book:

Practical Algebra: A Self-Teaching Guide, Second Edition 2nd Edition by Peter H. Selby (Author)

And it was fucking awesome. Low brow books, books for "idiots" are great learning sources because they ironically DONT spoonfeed you generalizations like the "smart people" books do, so YOU have to generalize from the concrete examples and exercises they give you BY YOURSELF. Always amuses me when I think about it but yeah. Have at it. Experiment, have fun, solve problems.

>>11518504

Set theory is pretentious cancer and you are sensible for being immediately repelled by it. Set theoretic notation literally adds ZERO new/novel information to whatever you are trying to say. It is completely redundant. How do I know this? ZFC is literally just predicate logic with membership "operator", i.e. set theory just forces you to talk about math in terms of predicate logic with a membership operator. That's it. It doesn't help you solve problems, it doesn't give you any extra intuition ABOUT how to solve a problem, it literally is just extra notational weight to add on for the illusion of rigor. If anything it decreases one's intuition and makes solving problems harder because you become more focused on description and making your notation look neat and pretty than actually figuring out how to solve the fucking problem. It's high order faggotry for people who are bad at math but want to sound good at it. All the "great" set theorists were essentially lawyers and logicians- verbally oriented faggots. It's the reason why physicists in the past 70 years have become better mathematicians than most math majors, and why the only good mathematicians coming out of math departments these days are students who study operations research.

[Anonymous No.11519622]

I made an argument that proves mathematically the existence of a God.

>Proposition I: Mathematics is the structure of the universe
This is true because mathematics is essentially the language in which physics amd chemistry use to explain the processes of the universe
>Proposition II: All mathematics stems from theorems, which must be proven either valid or invalid
This is, again, an intrinsic aspect of mathematics itself
>Proposition III: All proofs must have beginnings, those beginnings are the axioms, which can't be proven, just accepted as true
So, essentially, all math stems from axioms, it didn't come out of nowhere, all math came from a set of axioms which gave birth to several theorems and so on

Proof for the existence of a God:
From proposition I we can assume that mathematics is everything in the universe, if mathematics is everything then math was also the beginning, but from proposition III we know that all math begins with a set of axioms, but axioms are just absolute truths, therefore the beginning of the universe is an absolute truth, therefore the beginning of the universe can't be something like proposition III, therefore the beginning must be something that is absolute, a being that can't be from the universe itself, since the universe is just a case of proposition II, therefore there is a being, a god, who always existed and created everything, an axiom who started everything.

[Anonymous No.11519625]

>>11519610
>Self-Teaching
>Set theory is pretentious cancer
lol

[Anonymous No.11519633]

>>11519622
Ah yes, a rewording of the watchmaker argument
>not science or math

[Anonymous No.11519634]

>>11519622
>therefore the beginning of the universe can't be something like proposition III
I meant proposition II

[Anonymous No.11519783]

>>11512441
Can anyone give me a hint on this proof? I am trying to prove that the sum of all the terms of the nth row of Pascal’s triangle is equal to [math]2^{n}[/math]. I wrote a proof using the binomial theorem and considering 2 as [math](1+1)^{n}[/math], but I get the feeling that there is probably a good way to prove it using induction and I wanted to practice that, but I am not able to figure it out.

[Anonymous No.11519791]

>>11519783
>dude how many subsets does a set with n elements have
Literally that simple.
>inb4 how do I count subsets
Consider functions [math]S \rightarrow \{ 0, 1 \}[/math] or whatever.

[Anonymous No.11520045]

>>11519783
Induction is a kind of aids way to do this (either the binomial theorem or the direct combinatorial argument is much better) but if you really wanted to you'd use pascal's identity:

[eqn]\sum_{k=0}^{n+1}{n+1\choose k} = 1 + (\sum_{k=1}^{n}{n \choose k}+{n \choose k-1})+1[/eqn]

[Anonymous No.11520053]

>>11519625

That's it? Wow great comeback.

[Anonymous No.11520092]

>>11519783
First try to prove:

[math]\binom{n+1}{k} =(1 +\frac{k}{n+1-k})\binom{n}{k}[/math]

[Anonymous No.11520108]

>>11519783
First prove that

[eqn]\binom{n+1}{k}=(1+\frac{k}{n+1-k})\binom{n}{k}[/eqn]

[Anonymous No.11520649]

>>11519622
axioms are strings of symbol that a computing machine parses to output theorems. they are not "true", just accepted by some machine. yes, mathematics and its axioms are structures of the universe existing in thought or symbol, but it is not physics itself, we are just matching up symbols with observations. if you take two objects and stack them, they are not "adding." you are describing them additively. for a thought experiment, you could pour one glass of water into another, and you could either say youve added the volumes or that 1+1=1, where two solo cups return to one when mixed. a physical system is describable in any way, as long as our mind can perceive a connection between a description and the system

[A͞n̕onymous No.11520877]

Hey guys, how's it going? What did my professor mean by this and what does this tell us about disease spreading?

[An͟onymo҉͟us͟ No.11520893]

I know this is a difficult time but can we please keep /mg/ and /sqt/ math related! Thanks.
The virus is nothing to worry about!

[Anonymous No.11520900]

>>11520893
please stay isolated in your threads don't spread the virus!

[Anon͡y̢mo̴̧us No.11520927]

>>11520900
No.

[Anonymous No.11521287]

>>11516573
The negative of a number is such that you get 0 when you add it. -x is defined by

x + -x = 0

What is -(-x) then? It would be the number such that

-x + -(-x) = 0

but we already know that -x + x = 0, so therefore -(-x) = x.

[Anonymous No.11521294]

>>11515075
>I don't know much about Hilbert spaces, cant they be quite useful for quantum physics?
funny anon

[Anonymous No.11521307]

>>11517047
I don't solve problems. I don't care. Why? Because I'm not a mathematician.

[Anonymous No.11521314]

M always comes before D

its 1

8 / 2(2+2)
8/ 2(4)
8 / 8
1

[Anonymous No.11521315]

>>11518704
Almost no one knows symplectic geometry. That said there are definitely people far ahead of you so don't get too relaxed.

[Anonymous No.11521329]

>>11519298
All people are like this. You're singling out math for whatever reason.
>It's like jocks trying to out-alpha each other
Yes exactly. It seems you know the answer already.

[Anonymous No.11521335]

>>11520053
>That's it? Wow great comeback.
lol

[Anonymous No.11521352]

>>11521287
isn't this circular?

[Anonymous No.11521353]

>>11513337
>philosophy chad

Lel just switch to math. It has more actual results about transcendental ideals than philosophy and you will discover that philosophers suck dick at careful arguments.

[Anonymous No.11521361]

>>11521352
No. Think about it. How would you define a negative of a number? It's the number you have to add to get 0. Then it's clear why the negative of a negative is a positive, because what you have to add to the negative to get 0 is just the positive.

[Anonymous No.11521381]

>>11521361
couldn't you just say if x + -x = 0 then multiply by -1
-x + -(-x) = 0 so
-(-x) = x

[Anonymous No.11521402]

>>11513337
You should learn CT instead, it's philosophy with cute symbols!

[Anonymous No.11521412]

>>11521381
That works too if you use -1 * x = -x. You can prove this using the definition I gave with the distributive property of addition x + -1 * x = (1 + -1) x = 0 x = 0 so -1 * x = -x.

[Anonymous No.11521500]

>>11521412
very cool!

[Anonymous No.11521546]

>>11521352
take natural numbers with addition (which we take for granted) and multiplication defined as repeated addition. this naive arithmetic satisfies some properties such as the distributive law. now you make this into integers by introducing additive inverses. then you make this into rationals by adding multiplicative inverses. but each time you add new numbers, you need to say how does the addition and multiplication work for them. you can define -7 + 2 =: 100 for all I care. BUT there's always only ONE way to do this so that you stay consistent with the rules of the naive arithmetics that you started with.

[Anonymous No.11521596]

>>11519622
>This is true because mathematics is essentially the language in which physics amd chemistry use to explain the processes of the universe
There is no chemical or physical law that is known to be true. In fact historically every single physics theory turned out to be false (see e.g. Newton's laws, various models of atoms, etc.). There simply is no evidence for this claim.

>This is, again, an intrinsic aspect of mathematics itself
False. In fact, this is provably false. See CH as an example.

>So, essentially, all math stems from axioms, it didn't come out of nowhere, all math came from a set of axioms which gave birth to several theorems and so on
True.

So 2/3 of your propositions are either false or without evidence...

[Ań́o̴̴n͘͝ymo̡u̷s̶͢ No.11521609]

e=mc2 + 2 - 3 = COOOOF COOF COOOF COOOF!!!

[Anonymous No.11521639]

>>11521596
>So 2/3 of your propositions are either false or without evidence...
That's how it goes with religions.

[A̶nơnym͡ous͞ No.11521647]

>>11521639
>That's how it goes with religions.
Or with nearly anything else, like science.

[A͡n͟on͞ym̸ous No.11521670]

What are the most autistic fields of maths? I already know about categories, sheaves, topoï, etc... I want something as autistic as that if possible.

[Anonymous No.11521691]

>>11521670
How about the topopluralogy which is the study of the plural of "topos"?

[Anonymous No.11521907]

>>11521670
set theory/formal logic is as autistic as it gets
the autism value of heavily abstract categorical shit like topoi isn't bad, but it's offset by the pretentiousness.
logic is pure distilled autism without anything to dilute it

[A̕͢҉n̡͡on̶̕ym͞oùs͠ No.11521912]

>>11521670
model theory

[A̵n͏o͜ny҉҉m͏͞oú̴s͠ No.11521914]

>>11520893
you don't seem so well anon. are you sure you don't want to take the day off ?

[A͞͡ņ͡o̷ņ͟ymo̸u̵͞s No.11521991]

how are you coping with the virus guys

[Anonymous No.11522003]

>>11521991
Have we found out already how it's contracted?
If I were to write a simulation, I guess it's just sticking and writing in threads with infected guys

[Anonymous No.11522063]

>>11521907
model theory is just algebraic geometry with extra steps tho

[Anonymous No.11522126]

Is there a way to prove the formula [math]\int_0^t x^n= \frac{t^{n+1}}{n+1}[/math] in general without appealing to the FTOC or Bernoulli numbers? Specifically using Riemann/Darboux integration.

I've tried it and eventually reached the point where one gets [eqn]\lim_{m\to\infty} \frac{t^{n+1}}{m^{n+1}}\sum^m_{i=1}i^n[/eqn]

Evaluating for small [math]n[/math] gives the expected answer using the known formulas for sums of powers, but is there a more clever way?

[Anony̶̛͞m̷͘͟͜ous̴͘͢ No.11522216]

>>11522126
You can do an induction, but for that you would need a product rule, I think, which of course involves the Fundamental theorem of calculus...

[Anonymous No.11522280]

Oh dear. /mg/ appears to be flying close to the sun today.

[Anonymous No.11522282]

>>11522278
.

[Anonymous No.11522283]

>>>/pol/251391231

/pol/ has a huge cure thread, from what I've seen they have a script to prevent infection.

[Anonymous No.11522284]

>>11521315
I don't care anymore, I just want to surrender.

[Anonymous No.11522304]

>>11522248
15 minutes have now passed since this clean post, noting observations.

[Anonymous No.11522305]

Infected fucking shits please stay 6ft away from me.

[Anonymou͢s̶ No.11522306]

>>11522126
Have you tried using different approximations of the area, i.e. trapezoidal?
Ultimately, you just need to figure out how to make shit cancel out.

[An͜onymou҉s No.11522319]

>>11521670
>most autistic
Anything with "derived" in the name.

[Anonymous No.11522322]

>>11512441
>that picrel
is it that guy from "blues brothers"

[Anonymous No.11522433]

I wonder if there's a factor where you can only transmit it if you don't show symptoms.
Maybe people with the quarantine symbol are sick, but can't get others sick

[A̵͞no̵ń͞y͝͝moư͘s̀ No.11522434]

>>11522319
Can you provide an example please?

[An̶on҉y͘m̢o͘͝us No.11522474]

>>11522434
Derived differential geometry.

[Anonymous No.11522520]

>>11522126
You don't need to know anything about Bernoulli numbers here; every term with a Bernoulli number in front of it has degree less than n+1 and is just going to die in the limit anyway. All you need to prove is that [math]\sum_{i=1}^mi^n = \frac{m^{n+1}}{n+1}+(lower degree stuff)[/math]

and this you prove this inductively by expanding out [math](m+1)^{n+1} = \sum_{i=1}^m(i+1)^{n+1}-\sum_{i=1}^mi^{n+1}[/math]

[A҉n̡o͏n͘y͡m͘o̧͜u͏s̀ No.11522525]

>>11522474
jesus christ, is there any reason to do differential geometry like that besides autism ? I mean, does this acomplish anything, does it lead to interesting results which would be hard or impossible to prove in less abstract setting ?

[Anonymous No.11522541]

>>11522525
Dude replacing 200 pages of math with 1100 pages of commutative diagrams is the Right Way To Think About It™ who cares if it accomplishes anything new

[Ánon̛y̷mo̕̕ù̴̸̡s͠ No.11522577]

>>11522525
There are usually three reasons to abstract things further:
>there are interesting constructions you can do on set of objects which doesn't return an element of your set of objects, but still returns an object that shares many features with it, so you abstract things until your categories close off and your functors become endofunctors
>the new object is genuinely simpler to define and intuit than the previous one
>genuine autism
I'm pretty sure DDG is a case of the last one.
But sometimes, cases of the last one become cases of the first one, so you might want to check back on it in a few decades.

[A̷n҉òn҉͢y̡m̀oús͟҉ No.11522747]

>>11521670
Higher Topos Theory

[A̕n̶͢o̡̡͞n͜y͏mǫ͟us No.11522758]

>>11522541
*coof*

[Anonymous No.11523006]

>>11521670
I'd say set theory, logic, model theory just like the other anones said.
although my opinion on CT is that it isn't autistic - but just too pretensions for its own good

[Anonymous No.11523155]

>>11513337
We meet again
I recommend A-plus notes for Algebra, it covers everything you'd need to know plus more

Also you said eastern metaphysicians didn't understand that base realtiy is pain, does the Buddha not realize this?

[Anò̶n͜͜͝y͏ḿou̶s̨͜ No.11523493]

Rotations - logarithm edition

https://youtu.be/gBMdTSXhYsY

[A̡͞͠͡no̸͝͝ǹ͡y̷mous No.11523998]

Is there an elegant way to generate list {x, f[x], g[x], f[f[x]], f[g[x]], g[f[x]], g[g[x]], f[f[f[x]]],...,g[g[g[x]]]} in Mathematica?

[Anonymous No.11524332]

>>11522520
True, but you would need to know a priori what that sum looks like

[Anonymous No.11524502]

yo nikolaj why doesn't ron maimon ever hang out with us

[Anonymous No.11524980]

>>11518704
If you do a solid course in real analysis (in R^n) and algebra by your junior year/start of senior and get in one or two graduate classes for senior year you are doing just fine.

You aren't on your way to starting a harvard Phd, but you can reasonably apply for phds or masters at state schools.

[Anonymous No.11524985]

>>11520649
>axioms are strings of symbol that a computing machine parses to output theorems

No. This just one formalization to reason about them.

Every predicate is true or false, unless you don't believe in excluded middle. Any predicate which you can accept as a starting point for deduction is an axiom.

[Anonymous No.11525013]

>>11520877

IDK how to do math on chan.

integral of f over rectangular region is described
by nested sums of i and j. The base of a piece has area 1/n^2 and a height of f(i/n, j/n).
Plug into exp. Exp(a + b) = (exp a)(exp b).
So sums become products. Apply an exp definition to get the right form.

[Án̴ón̛y̶m̸̀o̸͘us̶ No.11525020]

>>11525013
Press the tex button on the top left of the reply box

[Anonymous No.11525025]

>>11518703
I fucked you're Touhou

[Anonymous No.11525031]

>>11525020
thanks

[Ano͢n͟ymoù͟s No.11525243]

>>11519340
Yes.
Look up the book by VI Arnold "Mathematical Methods of Classical Mechanics". He has a great chapter on lissajous figures!

[Anonymous No.11525329]

>>11512441
Is there any prime number [math]p[/math] larger than 3 so that [math]p^p+2[/math] is also a prime number? I got stuck on this problem.

[Anonymous No.11525340]

>>11525329
No. Use Fermat.

[Anonymous No.11525341]

Was watching a NJW video recently and he said the Fundamental Theorem of Arithmetic doesn't work for big numbers since we can't write down there prime factorizations. In fact it seems to me he's claiming most basic number theoretic properties break down on really big numbers. Is this true or is he just being a crackpot?

[Anonymous No.11525348]

>>11525340
Brainfart. Forget.

[Anonymous No.11525354]

I'm learning topology and I don't understand one thing.

Let [math]V[/math] be a normed space for simplicity, and let [math]C[/math] be some set in [math]V[/math]. For some continuous linear functional [math]f: V \to \mathbb{R}[/math], define

[math]\mu = \inf_{x \in C} f(x)[/math].

Now, this means that there is a sequence of [math]\{x_n\}[/math] in [math]C[/math] such that [math]f(x_n)[/math] converges to [math]\mu[/math]... but in what topology? Is this predetermined? Do we fix the topology at some stage? I'm lost about this.

[Anonymous No.11525358]

>>11525354
If there is no mention of a topology on [math]\mathbb{R}[/math], then it is the usual one. Note that the convergence is happening in that space.

[Anonymous No.11525366]

>>11525358
Right, but also [math]x_n[/math] are converging to some point in the space [math]V[/math], is that not correct? I was curious about that convergence.

[Anonymous No.11525380]

>>11525366
Judging by your post, there is no reason to assume that. The image of that sequence converges, but that doesn't imply anything on the original one. A stupid example: [math]a_n=n[/math] but [math]f(a_n)=1[/math] for all [math]n\in\mathbb{N}[/math].

[Anonymous No.11525381]

>>11525366
No, not necessarily.

[Anonymous No.11525387]

>>11525380
>>11525381
Okay, thanks. To expand: say that the set [math]D = \{ y \in V : f(y) = \mu \}[/math] is non-empty. Can I conclude that [math]\{x_n\}[/math] converges to some [math]y \in D[/math] if [math]f(x_n)[/math] converges to [math]\mu[/math]?

[Anonymous No.11525391]

>>11525387
No. Take the zero function.

[Anonymous No.11525395]

>>11525391
I just meant some non-pathological example, although I'm not quite sure what that would be. Say that the closure of [math]C[/math] intersects [math]D[/math] at only one point, do we get convergence to that point?

[Anonymous No.11525431]

>>11525395
>Say that the closure of CC intersects DD at only one point, do we get convergence to that point?
I think this is true. you can deduce that distance between x_n and D goes to zero in any case.

[Anonymous No.11525478]

Does anyone know anything about the regularity of the obvious function
[math]sort: \mathbb{R}^d \longrightarrow
\mathbb{R}^d.[/math]
Is it differentiable?

[Anonymous No.11525489]

>>11525478
Assuming you mean the function that takes in a sequence (x_1, x_2, x_3, x_4) and returns the sequence in nondecreasing order,
It's continuous.

[Anonymous No.11525496]

>>11525478
>is it differentiable
No, consider the partial derivative along x at the point (0,0,0,0)
right limit of sort(t,0,0,0)/t= (0,0,0,1) but the left limit is (1,0,0,0)

[Anonymous No.11525504]

>>11525478
Adding to it: the total derivative should be something like
[math] D_sort(x) = P_{\sigma(x)} [/math] where [math]\sigma(x)[/math] is the sorting permutation and [math]P_{sigma(x)}[/math] is the matrix that performs this permutation (or the inverse idk)

[Anonymous No.11525510]

>>11525478
>>11525504
Adding to this, it's differentiable at and only at the points away from the closed set {(x_1, x_2, x_3, x_4): for some i!=j, x_i=x_j}. The derivative depends on which component of this open set you are. It's constant on the components and is a permutation matrix.

[Anonymous No.11525512]

>>11525341
>Is this true or is he just being a crackpot?
It is just his view on mathematics.
He believes that if it is physically impossible to calculate something then it is meaningless to talk about it.

>In fact it seems to me he's claiming most basic number theoretic properties break down on really big numbers.
He believes the natural numbers are finite, of course that HAS to break things like prone factorizations...

[Anonymous No.11525519]

>>11525510
neato
I'm trying to explicitly differentiate the Steiner symmetrization as a function that takes a polygon with n points to a polygon with 2n-2 points. It involves sorting the points by their x-components.

[Anonymous No.11525531]

>>11525354
You can rewrite the infimum as inf {f(x) | x in C}, you see that the ONLY topology that matters is the topology on R, since this is just an infimum of a subset of R.
You don't really have to consider any properties of V, it doesn't need to have any topology at all, so you can't really expect to get any knowledge on the sequence of preimages.

[Anonymous No.11525574]

>>11516573
-a + a = -a + a
-a + - a = 0 -a - a
3 * 2 = 0 +2 + 2 + 2
3*-2 = 0 + -2 + - 2 + - 2
-3*-2 = 0 - - 2 - - 2 - - 2

[Anonymous No.11525834]

Take a norm-closed subset [math]X[/math] of a hilbert space [math]\mathcal{H}[/math]. We get a set of trace-class operators by defining [math]\{ xx^* : x \in X \rset[/math]. Is this set norm-closed in the space of trace-class operators?

[Anonymous No.11525835]

Take a norm-closed subset [math]X[/math] of a hilbert space [math]\mathcal{H}[/math]. We get a set of trace-class operators by defining [math]\{ xx^* : x \in X \}[/math]. Is this set norm-closed in the space of trace-class operators?

[Anonymous No.11526153]

>>11523998
There's various "Nest" functions, but for this one I'd generate all binary strings and then to a small loop.

>>11524502
I know that he was on 4chan for the truther stuff at one point, but I don't take him for a big memes consumer. Not sure.
Last time I've emailed with him was years ago - where does he currently hang out, if at all?

On a different note, Todd Trimble is going live now with a talk

https://youtu.be/gcYMrzQZilM

>MIT Categories Seminar LIVE - Todd Trimble: Geometry of regular relational calculus

[Anonymous No.11526209]

>>11526153
>but for this one I'd generate all binary strings
no wait, I'm stupid, what's better is probably, in pseudocode,

r = {x}; while(true): r = map(f, r) join map(g, r);

and dump the inbetweens

[Anonymous No.11526348]

>>11525835
is "[math]\{xx^* : x \in X\}[/math] is subset of trace class operators" an assumption? what do you mean by trace class, because the way I know it the problem doesn't make sense

[Anonymous No.11526413]

>>11520877
anons, I don't think this is correct, unless they mean an approximate.

I can get down to the products, and one could approximate exp(x) by 1 +x, but I don't see why the other terms should go away.

[Anonymous No.11526415]

>>11526413
pic related

[Anonymous No.11526416]

Hi can you reply to this
>>11526409
with "this", "/thread", "agreed" and so on, thanks.

[Anonymous No.11526419]

>>11526416
no

[Anonymous No.11526429]

>>11526348
I mean, the set of projection operators [math]\{ p_x : p_x(\cdot) = \langle \cdot, x \rangle x, \; x \in X \}[/math]

[Anonymous No.11526461]

>>11526429
>this nigga

[Anonymous No.11526521]

>>11526413
>>11526415
I find a bunch of formulas like this, mostly because in one dimension it relates to the Dyson formula (there the one dimension is time)
https://ncatlab.org/nlab/show/Dyson+formula
I imagine this thing pops up if you deal with field theories and you got some x's to integrate over.

Could it be that you're off by one an you get a boundary term via
[math]\int_0^1 F(x) dx = \sum_{i=0}^{n-1} G(i) = \sum_{i=1}^n G(i) + G(0) - G(n) [/math]
?

[Anonymous No.11526557]

I find a bunch of formulas in my notes, mostly because it relates to the Dyson formula
https://ncatlab.org/nlab/show/Dyson+formula
In that context, it's one dimensional with time, but for field theories in RxR^3 such integrals might pop up.

Could is be that you're off by one, leading to a canceling boundary term via
[math] \int_0^1 F(x) dx = \sum_{i=0}^{n-1} G(i) = \sum_{i=1}^n G(i) - (G(n) - G(0)) [/math]

It's interesting, can you try a simple monomial or rational function and take the limit on wolframalpha or so?

[Anonymous No.11526608]

>>11526415
the form in the question does not have an exponential inside the double product, and it also has a +1

[Anonymous No.11526615]

>>11526415
I can see that the exponential could be approximated by a 1+ x expression, but dont see the equality

[Anonymous No.11526646]

>>11526209
I made following code:
RESULT = {{x}}; Table[AppendTo[RESULT, Flatten[{Map[f, RESULT[[-1]]], Map[g, RESULT[[-1]]], Map[h, RESULT[[-1]]]}]], 3];
Flatten[RESULT] gives
{x, f[x], g[x], h[x], f[f[x]], f[g[x]], f[h[x]], g[f[x]], g[g[x]], g[h[x]], h[f[x]], h[g[x]], h[h[x]], f[f[f[x]]], f[f[g[x]]], f[f[h[x]]], f[g[f[x]]], f[g[g[x]]], f[g[h[x]]], f[h[f[x]]], f[h[g[x]]], f[h[h[x]]], g[f[f[x]]], g[f[g[x]]], g[f[h[x]]], g[g[f[x]]], g[g[g[x]]], g[g[h[x]]], g[h[f[x]]], g[h[g[x]]], g[h[h[x]]], h[f[f[x]]], h[f[g[x]]], h[f[h[x]]], h[g[f[x]]], h[g[g[x]]], h[g[h[x]]], h[h[f[x]]], h[h[g[x]]], h[h[h[x]]]}

[Anonymous No.11526737]

>>11526557
good ideas. I am not familiar with the dyson formula.

I'll try a concrete f in wolfram as you suggest.

[Anonymous No.11527555]

If I’m a pretty bright guy, can I read Lang or Jacobson as my first exposure to algebra? I had to learn basic group theory for Van Kampens theorem but that was on a need to know basis.

[Anonymous No.11527588]

>>11525341
He is mentally ill.

[Anonymous No.11527619]

I can lift one grain of sand.
If I can lift n grains of sand, I clearly can lift n+1 grains of sand.
Thus, by induction, I can lift an entire beach.

[Anonymous No.11528391]

>>11527619
Peano axioms BTFO!!!!!!!!!!!!

[Anonymous No.11528397]

>>11525243
Very cruel.

[Anonymous No.11528409]

>>11527619
I really like this. This is one of my favorite mathematical paradoxes.

[Anonymous No.11528713]

>Nature: Mathematical proof that rocked number theory will be published
http://archive.is/vdO7J

>After an eight-year struggle, embattled Japanese mathematician Shinichi Mochizuki has finally received some validation. His 600-page proof of the abc conjecture, one of the biggest open problems in number theory, has been accepted for publication.

>Acceptance of the work in Publications of the Research Institute for Mathematical Sciences (RIMS) — a journal of which Mochizuki is chief editor, published by the institute where he works at Kyoto University — is the latest development in a long and acrimonious controversy over the mathematicians' proof.

>Two other RIMS mathematicians, Masaki Kashiwara and Akio Tamagawa, announced in Japanese the publication at a 3 April press conference in Kyoto. The paper “will have a big impact”, said Kashiwara. When asked how Mochizuki reacted to news of the paper's acceptance, Kashiwara said, “I think he was relieved."

>At the press conference, Tamagawa said the solution itself had not changed in response to Scholze and Stix's criticism. There are some comments about it that will also be published in the manuscript, but no fundamental change, said Tamagawa.

[Anonymous No.11528839]

>>11528713
how analytics is his theory, really?

[Anonymous No.11528851]

>>11528839
>how analytics is his theory, really?
What do you mean?

[Anonymous No.11529032]

>>11528713
>Mochizuki is literally publishing a 600 page book in a journal where he's chief editor

[Anonymous No.11529254]

>>11513241
>internet
a lot
>study
i study on my own for fun pretty much all day

[Anonymous No.11529351]

Why do multiple Julia fractals look like one giant Mandelbrot fractal?

[Anonymous No.11529364]

>>11529351
Because Mandelbrot did with computers what Julia tried to do by hand.

[Anonymous No.11529892]

Would someone recommend me book about Calculus written c. 1920? (In English, German or French)

[Anonymous No.11529898]

>>11529892
Why not Rudin?

[Anonymous No.11529903]

>>11529898
>Why not Rudin?
Rudin is a meme.

[Anonymous No.11529906]

>>11529903
Only if you're a moron.

[Anonymous No.11529907]

>>11526429
it's closed: if [math]p_{x_n} \rightarrow p_x[/math] then
[math]\langle x, x_n \rangle x_n = p_{x_n}(x) \rightarrow p_x(x) = \langle x, x \rangle x[/math]
now write [math]x_n = a_n x + y_n[/math] where [math]\langle x, y_n \rangle = 0[/math] and you'll get that [math]x_n \rightarrow x[/math]

[Anonymous No.11529911]

>>11529898
Science is a process. I'd like to see how Calculus looked like 100 years ago.

[Anonymous No.11529966]

>>11529892
Calculus Made Easy by Thompson. Published 1910 originally. It's actually better than many modern books. Actually tells you why they call it calculus.

[Anonymous No.11529973]

>>11529906
>Only if you're a moron.
If you're a moron or if you're not a moron.*

[Anonymous No.11529991]

When I was learning proofs, I recall my professor telling me that
[math] P(n-1) \Rightarrow P(n) [/math]
was a better or more robust or more educated way to do induction proofs than
[math] P(n) \Rightarrow P(n+1) [/math]

but I don't remember how he justified that. Can anybody provide input, re: how that might be the case?

[Anonymous No.11529995]

>>11529991
There's no difference between the two.

[Anonymous No.11529999]

>>11529991
Your professor was insane. Not uncommon. Many of my professors were as well.

[Anonymous No.11530013]

>>11529991
Usually my natural response would also be that the former is weird, but I happened to write a Wikipedia article a few months ago where I found doing it like this helped me making a point.
I found the former makes stating some statements more natural, because you get P(n) as conclusion, but as far as the proof praxis goes, plugging in n-1 to derive at something will usually be more awkward.

[Anonymous No.11530047]

>>11529991
your professor is an autist

[OP No.11530069]

Hey anons, I've started what I hope is a reading-club type thread >>11530059, dedicated to field and galois theory

[Anonymous No.11530097]

>>11512441
Physics friend here learning some set theory. I'm reading Halmos' "Naive Set Theory" and I don't entirely follow something he says:
A relation in a set is an equivalence relation if it is reflexive, symmetric, and transitive. ... the largest equivalence relation in [math]X[/math] is [math]X \times X[/math].
No matter how I think about it I don't see how [math]X \times X[/math] is an equivalence relation like this. Can anyone offer some understanding?
Thanks nerds!

[Anonymous No.11530107]

>>11530097
Anon, a relation on [math]X[/math] is literally a subset [math]R[/math] of [math]X \times X[/math], where we write [math]aRb \leftrightarrow (a, b) \in R[/math].
Ask this shit in >>>/sci/sqt/

[Anonymous No.11530116]

>>11530097
Kill yourself, brainlet. You will never be smart.

[Anonymous No.11530117]

>>11530097
>I don't see how X×XX×X is an equivalence relation like this.
Everything is in relation with everything.

[Anonymous No.11530118]

>>11530107
I understand that, what I don't get is how it's an equivalence relation

I wasn't sure about posting this in here compared to /sqt/ but I had a scan through and decided it was fine

[Anonymous No.11530124]

>>11530118
This is literally kindergarten mathematics, moron. Gtfo of this thread.

[Anonymous No.11530130]

>>11530118
>what I don't get is how it's an equivalence relation
The three axioms hold trivially. Literally just check them. I don't understand what there is not to get. Which one of the three axioms are you incapable of verifying?

[LMC !!6bFXHDKL67m No.11530152]

[math] \color{red}{\heartsuit} [/math]

[Anonymous No.11530156]

>>11530118
It's the relation where everything is in relation to everything. LITERALLY look at the axioms, you don't even need to think, since the conclusion holds trivially.

[Anonymous No.11530167]

>>11530130
I dunno, probably my issue is with relations in general more than this specific example

[Anonymous No.11530169]

>>11530118
>I had a scan through and decided it was fine
Please don't depress me like that.

[Anonymous No.11530173]

>>11530167
If your issue is not with the particular question but the subject itself then you shouldn't ask for help with the question.

This is seriously trivial and immediate from the definition.
Try re-reading the chapter and asking in /sqt/ if you don't understand something.

[Anonymous No.11530205]

>>11512441
if you consider all the irrational numbers between 0 and 1, and presume that they are regular - in that all the 10 digits appear in their decimal expansion with equal probability, then

What's the difference between one irrational number and any other?

[Anonymous No.11530222]

>>11530205
What the fuck is this question?

[Anonymous No.11530223]

>>11530205
I'm pretty certain that not every irrational number is normal. (Take normal number. remove all 0s from decimal expansion.)

[Anonymous No.11530239]

>>11530223
For me, it's the irrational number which goes n 0s, one 1, n+1 zeroes, etc.

[Anonymous No.11530394]

>>11530205
>assume false thing
>then ask a gibberish question
great job there my lad

[Anonymous No.11531392]

[math]2^bump[/math]

[Anonymous No.11531418]

>>11512891
woah

[Anonymous No.11531443]

>>11531392
To make [math]2^{bump}[/math] write 2^{bump}.

[Anonymous No.11531597]

Has anyone here used this book before? How is one meant to learn the subject without exercises? Should I use a secondary text for exercises?

[Anonymous No.11531600]

>>11531597
Read Ireland&Rosen

[Anonymous No.11531607]

>>11531600
Thanks. In lock down and have nothing but time

[Anonymous No.11531805]

Let [math]v_1, \dots v_k[/math] be vectors in [math]\mathbb{R}^2[/math], such that [math]||v_i|| \leq 1[/math] for each i, and [math]\sum_{i=1}^k v_i = 0[/math]. Here [math]||\cdot||[/math] denotes the Euclidean norm.
Let's define [math]f(v_1, \dots v_k) := \max_{1\leq j \leq k} ||\sum_{i=1}^j v_i||[/math].
Prove: there exists a constant [math]C > 0[/math], such that for any sequence of vectors [math]v_1, \dots v_k[/math] as above, there exists a permutation [math]v_1', \dots v_k'[/math] such that [math]f(v_1', \dots v_k') \leq C[/math].

[Anonymous No.11531812]

>>11531443
[math] 2^{bump}[/math]

[Anonymous No.11531821]

>>11531597
For se illegal reasons I have this book, and I think I like it, but it's just so thick. Other books on number theory jump more quickly into aspects of general theories (ring theory tools, etc.) while that on is more long wondered, I felt.
As to your question, I agree that exercises are important, but actually just trying to understand the proofs in honesty will have you occupied as well.

>>11530069
(bump)! for your thread

[Anonymous No.11532016]

>>11531821
>Other books on number theory jump more quickly into aspects of general theories (ring theory tools, etc.) while that on is more long wondered, I felt.
These things are mostly just indications of the book's age. The first edition of Hardy and Wright was published in 1938; I doubt English students in the 30s would not have been expected to know very much abstract algebra, if any at all, whereas today it's just basic bedrock knowledge so authors are comfortable using it freely.

And I doubt Hardy would even have known Bourbaki existed at all yet, and their influence on writing mathematics was decades from really taking hold. Pre-Bourbaki math texts tend to be much more focused on verbal rather than symbolic explanations of things, which can make them feel wordy and rambling to people who grew up with the modern style.

[Anonymous No.11532040]

>>11531805
Set [math]C = k[/math].
The remainder is trivial and left as an exercise to the reader.
Unless [math]C[/math] needs to be independent of k, in which case 2 probably works, but the proof sucks.

[Anonymous No.11532044]

>>11532040
>2
My bad, 1 works and is optimal.
Just run the whole "If the sum until k is negative, set a positive number for k+1, and if it's positive, drop a negative". Trivially the modulo is always smaller than 1, and you can show that you can in fact choose negative/positive elements from the sum being zero.

[Anonymous No.11532045]

>>11532040
yes I wanted C to be independent of k

[Anonymous No.11532047]

>>11532044
>if the sum is negative
we are talking about R^2 vectors dude

[Anonymous No.11532054]

>>11532047
Irrelevant tbqh. If summing any element to the sum until k increases the norm, then the total sum cannot equal zero.

[Anonymous No.11532059]

FRIENDS
I need help for my bachelor thesis please
https://stackoverflow.com/questions/61001535/how-to-compute-co-occurence-probability-from-a-co-occurence-matrix

It's a statistics/probability question first and programming - python question second.

[Anonymous No.11532062]

>>11532054
Take 3 unit vectors each at 120 degree angles with each other. The bound cannot be 1.

[Anonymous No.11532068]

>>11532062
>sum two of them
>negative of the third one
>norm is trivially one
Nice counterexample.

[Anonymous No.11532071]

>>11532059
I gave you an upvote to be at least positive.
W.r.t. the question, I'd go for a simple 2 or 3 dimensional example to validate your guess of what's going on and rule out what the formulas can't be.

There's also Math and Stats SE btw.
I don't know why you tie your question up with programming at all - seems like once you know the statistics of it, the python would be trivial to implement.

[Anonymous No.11532083]

>>11532054
C = 1 doesn't work, try for example
[math]v_1 = v_2 = \dots = v_{10} = (\sqrt{1 - 1/400}, 1/20)[/math],
[math]v_{11} = v_{12} = \dots = v_{20} = (-\sqrt{1 - 1/400}, 1/20)[/math],
[math]v_{21} = (0, -1)[/math].

[Anonymous No.11532084]

>>11532071
thanks for the reply, I'm a brainlet on a university for brainlets.

the problem (for me) is, the co-occurence matrix is something completely new to me that I've just copied and pasted from other stackoverflow question. as such, I don't really understand if what I'm doing with it is correct on the basic level.
thank you also for the SE maths tip, I will post it there.

[Anonymous No.11532100]

>>11532083
Fair enough.
You do know that you could run the counterexample with only five vectors, tho, right?

[Anonymous No.11532101]

>>11532100
What would that look like?

[Anonymous No.11532124]

I think we can all relate.

[Anonymous No.11532127]

>>11532124
No. In fact it has been quite a long time since I had a written exam.

[Anonymous No.11532129]

>>11532100
[eqn]v_1 = v_2 = (\sqrt{1- 1/25}, 1/5)[/eqn]
[eqn]v_3=v_4 = (- \sqrt{1-1/25}, 1/5)[/eqn]
[eqn]v_5 = (0, -1)[/eqn]

In other words, the exact same thing with 20 swapped by 5.

[Anonymous No.11532132]

>>11532129
Wait, fuck, might have made a mistake somewhere.

[Anonymous No.11532196]

>>11532124
I wish I could, as then I'd be a girl.

[Anonymous No.11532577]

>>11532196
same