/sci/ - Science & Math

Black lives matter

why doesn't anybody post those "you should be able to do this" questions anymore they used to be fun :(

posting one myself

how is graph isomorphism not already solved in polynomial time? isn't it obviously just vertex and edge comparisons?

i've been looking through some graph theory textbook isomorphism exercises and solving them with a polynomial time algorithm that's pretty simple, and always works. i'm struggling to understand how this would require more than polynomial time

>>16113958
>how is graph isomorphism not already solved in polynomial time?
If it's so easy just write an algorithm and publish it. Wait for your Turing prize in the mail.

Do you think stuff like category theory, topos theory and hott are still going to be big in 20 years?
What is going to be next big thing in math?

>>16114008
Proof assistants and automated theorem proving are the next big trendy thing. Computers are going to become better at proving theorems than humans relatively soon. Journals will start to require formalized proofs to be submitted with papers, if papers and journals don't become basically out-moded by gigantic formal libraries of automatically verified theorems.

>>16113965
i'd first like to know if there's some constraint beyond creating an edge-preserving bijection that is required for such an algorithm. there doesn't seem to be, but my solution seems way, WAY too simple for it not to have been one of the first things tried.

for fuck's sake, i'm not even using coloring or group theory

>>16114016
Everyone goes on about the mind, intelligence and such.

....

Give it up for the heart everyone, has she not earned her time yet?

>>16114015
I have severe Lean fatigue already.

>>16114019
are you going to tell me if there's another constraint, or are you more invested in sharing your slam poetry?

>>16114038
Is it alright if I pass through the curtain and hop under the quilt with you, perv?

>>16114046
Can I ask what makes you invest so much time into derailing every conversation on /sci/? You must have some motivation. I expect you to respond with nonsense, but I'm responding with the slim hope you might give us some insight into your mind.

>>16114088
it's probably a bot.

>>16113816
Repeating something doesn’t make it true.

>>16114015
Tick tock Mochifags

>>16114015
What's the point of anything once AGI and ASI come to be?

Stats guys here?

If I have a coin that has head on both sides - does it actually affects the chances of getting it right as compared to normal coin with heads and tails?

>>16114190
Pussy.

>>16114015
They'll never AI-ify human mathematical ingenuity

>>16114218
>muh magic ghost force

>>16114233
You live on a world. Your world is 3D, isn't infinitely huge but you can't find the edge of the world. Suppose you have a giant rope. If you can tie a lasso (or a noose) around any part of your world and pull it tight, and the loop can shrink until it closes completely, no matter where or how you tie the loop, then your world is a ball.

Does anyone have any really good textbooks for self learning differential geometry? Also textbooks for differential topology.
Generally what are the prerequisites for these? Do I need to understand differential geometry before differential topology? Someone recommended me a textbook on morse theory, I should have have a general understanding on differential topology first before tackling it right?
I am a physics grad student so I am more interested in a book accessible to my level of understanding than something extremely rigorous.

>>16113926
Agreed, I missed them too. That and a light snack thinking about it.

>>16112583
Wow never heard that joke before, did you stay up all night coming up with that?

I need to re-learn mathematics in English for university (and I've got some giant holes in my studies due to sick leave). What are some good textbooks/guidebooks that contain and summarize everything you need to know in order to apply general arithmetic and geometry on a daily basis before moving on to specialist fields?

>>16114190
>>16114218
Humans still have a place in mathematics even if AI could become good enough to prove or disprove pretty much anything posed to it. It really is up to humans still to decide what is interesting enough to be proven. It's not too hard, even right now without any AI business, to generate a bunch of 'valid' propositions and their proofs in the language of dependent type theory about potentially nonsense types. And this is what 'automatic theorem proving' is good at doing, lots of nonsense valid deductions. It's still basically up to us to decide what is nonsense and what is sense. However, automated reasoning and proving will certainly be able to pick up a lot of the slack of humanity's mathematical ingenuity, and our relationship with mathematics will likely be permanently changed with these innovations.

>>16113951
Am I overthinking the answer or is there just one value z can be?

>>16113951
i'm pretty sure the inequality requires that the imaginary component of the left-hand side be zero, meaning the imaginary component of [math]z[/math] and [math]\alpha^k[/math] have to be equal.

since [math]z[/math] doesn't change for [math]k[/math], and [math]k[/math] has to hold for all integers up to [math]m[/math], [math]\alpha[/math] can't actually have a nonzero imaginary component at all - there's no way i know of to construct a complex number [math]\alpha[/math] with a nonzero imaginary component such that said imaginary component is equal for [math]\alpha^k[/math] using consecutive positive integers for [math]k[/math].

this makes it pretty easy to figure out what [math]\alpha[/math] has to be: 1.

from that, [math]z[/math] is any real number such that 0 [math]\leq[/math] [math]z[/math] [math]\leq[/math] 2

>>16114418
If you read the problem statement you would see that there is an absolute value sign on the left hand side of the equality. So you are wrong.

>>16114468
ah, you're right. well, we at least still know that range of [math]z[/math] does work for any [math]m[/math], so the answer will have to at least intersect the real number line there.

>>16114375
won't spoil for the others, but can you post your sol?
>>16114504
not true, that range, excluding 0, only works if alpha equals 1 and the premise of the question is that z should satisfy the property for any mth root of unity...perhaps i should have been clearer

>>16113958
If the graphs have n vertices, there are n! candidate bijections to check. How do you narrow it down?

>>16113951
This is just a draw a picture problem. Alpha (ill call it A) is one of the m-th roots of unity. Literally draw the unit circle in the imaginary plane, and divide it into m pie slices, where the first cut is always the line segment from 0 to 1. Each point on the circle uve divvied up is one of the A^k. If m = 1, then the only point is at (1,0), and if m=4, then the 4 points are (1,0), (0,1),(-1,0), and (0,-1).

The question asks to find z such that the distance between z and each point is less than or equal to 1, for every point. If m = 1, then the only point is (1,0), meaning z is every point within and on the circle centered at (1,0). For any point for m > 1, draw unit circles centered at each point A^k. Where can they only overlap? You should be able to do this

>>16114563
being pedantic, 1 is a root of unity, meaning it satisfies "any mth root of unity" - you seem to mean "all mth roots of unity"

you only asked for an element of the complex numbers that is a root of unity, not the set of roots of unity. that means the solutions are not a single inclusive set for z; if multiple compatible alpha exist, each provides its own valid solution. if that range works for 1, and 1 works for alpha... that satisfies every part of what you presented, but is not necessarily the only thing that does so.

you probably should have used "for all alpha" somewhere.

>>16114364
Will anyone employ you as a mathematician to do this, though?

>>16114202
Anyone?

>>16115028
Probably not but they will employ you to develop the software and the automated provers themselves. This fantasy scenario does run the risk of funneling mathematicians into 'AI devs,' which is why its probably a purely fantasy scenario. Whether or not it does get to that point I still think formal math in 10 years will be much more automated and formalized than it is now.
I've been thinking about this more, and there's an extent to which 'informal' mathematics is *already* automated in the sense of what I'm thinking of it. It's automated in that we just leave gaps in our proofs when we feel like we can. When small details are left out in our proofs, we are already automating our informal proofs as well. Maybe in this sense then, not a whole ton will change. It will just get easier to write formally in the same way that we can write informally.

how do i learn how to do geometrical proofs?
On euro high school level, I'm writing finals in 3 weeks and I havent even started studying yet
Every time I try doing one I can't come up with anything

I have to do through my ass?

>>16114700
"any" means it can be anything and not just 1 so you cannot just take it as 1, "any"="any arbitrary" mth root, similar to say, if one says let M be any metric space, prove <this property>....you cannot just assume the metric space in question is R with the euclidean metric, but yes you are right, i mean for all alpha

>>16115084
read Euclidean Geometry in Mathematical Olympiads by Evan Chen(you'll find it on libgen)...the first two chapters or so and you will absolutely love it, i know because i was in the same position as you, i absolutely hated geometry and didn't have any intuition at all

also, beware of the erratas, he documents them here <https://web.evanchen.cc/upload/geombook-errata.pdf>

you should finish the first chapter in about 3-4 days (don't get hung up on directed angles and why they work, instead, just use them and you'll see that they work and slowly understand why)

the second chapter will take longer but since you have 3 weeks, do maybe 1 or 2 problems from the exercises and not more

problems of the third chapter are significantly harder and you won't need to do them now, just reqd the first parts to learn some of the standard theorems like ceva and menelaus and you are good to go

>>16114679
> Each point on the circle uve divvied up is one of the A^k
not true, alpha has to be somewhat "special" for that to be the case but here alpha is just any mth root, actually there are [math]\phi(m)[/math] many such special alphas

>>16115110
damn that's nice
i have 3 weeks for 4 years of curriculum not just geometry so not sure if i'll be able to go through the whole thing but it's going ot be useful many thanks

Electrical engineering student fag here. What are some good books to learn calculus from? Our official literature list is just a shitty textbook that doesnt explain jack.

>>16113803
$z$

>>16113803
[Z]

>>16115209
what exactly is in your geometry syllabus? because the book i mentioned will probably not be like any other you have read, seeing that you are not into olympiads, the first chapter may as well be enough

>>16115231
[math]\mathbb{Z}[/math]

>>16114418
How do I enable mathjax?

>>16115234
How?
Is it like math stack exchange?

>>16115235
[nath] \sum_{n=0}^{\infty} n = \frac{-1}{12}[/nath]
but math instead of nath

>>16115212
Start with Inequalities (Korovkin) then move to calculus by tarasov.

>>16115240
Ok understood

>>16114612
i'm going to code this algorithm and test it on some graph pairs to make sure it actually works the way i think it does, but i never actually permute anything

>>16114358
AOPS curriculum. It's really not just for contests. There is no better books.
>>16114249
Honestly, just mathematical maturity. There is no hard prerequisite. A lot of the popular texts review the necessary results in the beginning or appendix. Some "easier" option (more or less in order of sophistication/rigor):
- Eigenchris' videos
- Baez's book
- Gravitation
- Applied Differential Geometry: A Modern Introduction
- Loring Tu's 2-books
>>16115084
AOPS
>>16115212
>Our official literature list is just a shitty textbook that doesnt explain jack.
Wrong way of thinking imo. Do more problems instead of procrastinating by looking for a better textbook.
Try a problem book. I like John Erdmann's one. It's free and legal.

>>16115104
any is more inclusive in math contexts and often implies "the set of possible examples" , generally, yes... but you used a symbol that means "is contained in." you literally have to assume the size of alpha is a complete set in the domain of that relation to get what you want the problem to say, because you never defined it as such. [math]\in[/math] by itself is just an existence check. "let M be any metric space" is a less ambiguously defining a set of metric spaces for M, but that's not what [math]\in[/math] means by itself. if you mean for all alpha, you really do have to say it (and it wouldn't hurt to say "for all M" either).

>>16114504
z works for any k, not any m.
>>16114563
I will post my solution tonight when I have time to type it up

What is the name of that guy that doesn’t like the irrational numbers and wants to do analysis with only the rationals?
He is kinda of a meme on the board.

>>16115409
Wildberger?

>>16115494
That’s him, thanks anon.

>>16113958
>>16114016
>>16115249
okay, i think i've roughly figured out what happened - i've been reading through a bunch of graph isomorphism algorithms to cover my bases for what's been tried, and i think i basically just accidentally independently stumbled into the trivial "G is a subgraph of G" case for Ullman's 1973 subgraph iso algorithm by implicitly assuming a fixed H without realizing it (which, AFAICT, does indeed make it polynomial time complex).

The first thing that truly melted my brain beyond repair was understanding what a group homomorphism is. I still get chills thinking about how much that one single thought changed my entire worldview regarding human communication and what it even means to "write down a meaningful piece of mathematics."

>A natural number is the set of all smaller natural numbers
>An ordered pair (a,b) is the set {a, {a,b}}
So 2 is the ordered pair (0,1) right? Are there any meanings or applications to this or is the first line just an interpretation of numbers and I'll have to study some jackoff "interpretation theory"?

>>16116954
>meanings
It means that a pair consists of 2 things.
>applications
You can count the 2 things using the numbers 0 and then 1, respectively.

>>16115283
>AOPS curriculum. It's really not just for contests. There is no better books.
Thanks for the tip, this might just be a lifesaver as someone who's had to resort to tired self-learning over the last few years. Never entered any contests since I was already struggling to keep up due to sick time so I'll take whatever I can get and put it in my shelves.

>>16116954
The application is for the formalization of mathematics: it's useful to know everything can be boiled down to a simple theory with few primitive objects, e.g. things like ordered pairs can be coded as certain kinds of sets. But in everyday mathematical practice such codings are not important, what matters is the structural/characteristic properties e.g. (a,b) = (c,d) iff a = b and c = d.

>30mg noopept
>still havent even touched the book
im FUCKED

>>16115212
I'm assuming you're at very basic Multivariable Calculus based on that homework problem. You need to get James Stewart's Transcendentals of Calculus. I genuinely cannot recommend this book enough, it is the most retard-friendly calculus book out there and still gives you a rigorous understanding of every theorem you will ever use in that class

>>16117207
Probably an obvious advice. But eliminate distraction as much as possible.
As in:
- no smartphone
- no youtube/stream - delete your accounts, playlists, etc.
- no video games - delete your gog and steam account, yes including the games you've padi
- no laptop - use university's computer, print things out, get physical books
- no internet - destroy your router, use university's computer, I hope you are not retarded enough to shitpost on 4chan with them
- blocking/parental control software with passwords - ask a friend/family member to set the password
Long term you can try mindfulness, therapy, CBT, meditation, etc. to control it. But for now, just do something drastic.

>>16117292
>it is the most retard-friendly calculus book out there
It keeps you retarded.
>>16115212
This is not multivariable calculus. It's literally grade school stuff.

>>16117333
you're one of those faggot kids who spends their time tryharding behind the scenes and then humblebragging as loud as you can to anybody who can hear "I'm SOOOOOOO sad, I ONLY GOT a 98.3 on that last exam! >:("

It is very obviously multivariable calculus. It is calculus, and there are many variables, and this is a great book for it. Especially cuz he is in electrical engineering. It gives you everything you could ever need to understand Greene's and Stoke's on a deep intuitive level

>>16117331
i've tried it a million times, it's impossible
sitting on my ass in front of a pc has been my life since before i turned 10
>meditation
i'll do this, it help a lot

>>16115212
pozz iz slovenije haha our math professors (chemistry in slo) have a good calc I textbook i can link. it's in slovenian, but the way it's laid out you'd understand.

Is it irrational?

>>16115669
interesting

>>16113951
posting my sol.
Suppose [math]p[/math] is the smallest positive integer(i.e, order) so that [math]\alpha^p = 1[/math].
Since [math]p \leq m[/math], [math]|z-\alpha^k| \leq 1[/math] for all [math]k \in \{1,2,\dots,p\}[/math] and [math]\alpha^k[/math] generates all the [math]p[/math]th roots of [math]1[/math] as [math]k[/math] varies from [math]1[/math] to [math]p[/math].
Now note [math]|z-\alpha^k|^2 \leq 1 \implies (z-\alpha^k)(\overline{z-\alpha^k}) \leq 1[/math].
Expand this to get [math]|z|^2-z\overline{\alpha}^k-\overline{z}\alpha^k \leq 0[/math] and then sum the inequalities as [math]k[/math] varies from [math]1[/math] to [math]p[/math].
The coefficients of [math]z[/math] and [math]\overline{z}[/math] will go to [math]0[/math] as they are both equal to sum of all the [math]p[/math]th roots and you are left with [math]p|z|^2 \leq 0 \implies z=0[/math].

think i might make an announcement thread about posting nice problems in here so that other anons can join...i am still in highschool so they won't be too technical or use heavy machinery but will definitely be interesting.
next ones will be combinatorics which is better than this ill choice of algebraic manipulation!

Any literature on the geometry of multiplication?

How long should it take someone to go from pre-algebra to calc 3, supposing they study for 2 hours/day?

>>16117331
>>16115283
>>16115233
>>16115110
i've tried reading that book but it seems to cover much more than what's relevant to me and i couldnt even figure out the easiest exercises so i gave up on it and tried to do some of the example problems for the test but i'm just staring at them for 20 minutes and i can't figure out how the fuck i'm supposed to know how to label everything in such a away that i can make it into a solution i literally cannot for the life of me figure out how to do any geometry and it's the vast majority of the test i am so fucking clueless i have 0 idea what to do at this point except just fucking pick 50 problems and memorize them and pray i'm going to get the same shit on the test I feel like there is literally no other way than just learn a certain number of these problems by heart and just recognize portions of few problems in the particular one you're doing and apply the same steps which wouldnt even be that big of a problem if my long term memory wasnt worthless i dont even know what's the point of this post i'm just sitting here foaming at my mouth because i'm stuck

>>16117733
about 50% of the test is freebies like logarithms, quadratic with variable, divisibility, limits or trig equations and the rest of the test is geometry and stereometry

>>16117733
the problems are not meant to be totally easy, like i said, the first chapter will probably be enough for you
your problem is that you are panicking when you cannot understand something quickly, because that makes you think that the same will happen in your exam
relax, you have 3 weeks still to go, that is plenty of time, 20 minutes is not much, be more patient

>>16117681
I arrived at a similar conclusion in a different way, I thought it's a fluke.
The disk [math] |z-\alpha^k|\leq 1 [/math] looked like part of the triangle
inequality [math] |z-\alpha^k|\leq |z|+|\alpha^k| [/math]. Thus, I can have
[math] |z|+|\alpha^k|=1 [/math] for each k.

Then, the system: [math] |z|+|\alpha^1|=1, |z|+|\alpha^2|=1,...,|z|+|\alpha^m|=1 [/math].
The moduli of each root of unity is 1, simplifying all to: [math] |z|=0 [/math].

>>16117681
>>16117775
Could also go for a more geometric intuition.
Note that for any given [math]\alpha^k[/math], the valid set of solutions for [math]z[/math] is going to be a unit disc shifted by [math]\alpha^k[/math]. Since your set of powers of [math]\alpha[/math] ends up as a regular polygon on the unit circle, and we're drawing a unit disc centred at each one, the only values of [math]z[/math] which work for all of them are those points of mutual intersection of the discs. In the case where we have the first root of unity (i.e. [math]\alpha=1[/math]) it's the entire unit disc centred at [math](1,0)[/math], but for any other value the sole point of intersection of them all will be the origin

Why is measure theory so fucking hard?

Everything in lecture is fairly straightforward and makes sense and then the exercises are just unbelievably difficult. Do professors just enjoy using this subject as their hazing ritual so they shove in as many brutal abstract integration proofs as they can or is there an actual purpose?

>>16117607
Yes, the sum of any two irrational numbers will be irrational.

>>16117802
>>16117775
That intuition is pretty neat. Very nice.

>>16117775
incorrect
[math]|z|+|\alpha^k| = 1[/math] is just one possibility, it may as well be less than or greater than [math]1[/math].
>>16117802
this is true but i wasn't satisfied with a geometric proof as no one ever precisely says why [math]0[/math] is the only common point, sure, its "obvious" as you can "SEE" but its not a complete proof, but yes its a good way to guess at the answer

>>16117808
why is /mg/ being flooded with retards? literally take any rational-irrational = irrational and switch the irrational over to the rhs
>>16117607
nice troll, for the ones who took the bait, this is an open question, see https://en.m.wikipedia.org/wiki/Irrational_number#Open_questions

>>16117808
>A = sqrt(2)
>B = 2-sqrt(2)
<A + B = 2

>>16117716
I've thought about this a lot for myself. I think at max it shouldn't take longer than 2 years if you are studying 2 hours a day (more like 4 hours every other day for the best results).
Pre-algebra, geometry, algebra I and II can all be knocked out in a year.
Pre-calculus isn't even really necessary, you just need a good understanding of the trig in it, so you can skip right ahead to calc in about a week or so. Calc I and II are very straightforward in their syllabus, each concept very seamlessly leads into the next (with the exception of the unit on series, but that also isn't that difficult.) but I'd still give it a full semester (not the two semesters that schools usually require)

Now, though learning all the concepts in these classes doesn't take much time, the amount of practice problems you would need to do to familiarize yourself with the tricks of the trade could add another month or so, but not more than that.
Finally, Calculus 3 is just a generalization of Calculus 2, and shouldn't take more than a month to get the whole thing down pat.
All in all, if you are a somewhat bright kid, and you aren't forced to stay in retard classes with the rest of the normies, you should have the mathematical ability of a college sophomore by the time you are leaving middle school. In fact, a family friend of mine who went to a magnet school did just that, so I'm not just blowing smoke out of my ass.

At that point, you'd be better served spending your high school years either not doing any math ever again, or just grinding number theory for some kind of Math Olympiad so you can get into MIT or Princeton

>>16117842
>>16117775
You're right in that regard. To me it seemed they matched up nicely.
I should fill in the extra cases here.

If [math] |z|+|\alpha^k|<1 [/math] for each k, then the modulus
of z becomes negative which couldn't happen by definition.
For [math] |z|+|\alpha^k|>1 [/math], the modulus of z is positive
which satisfies being in the disk of the root of unity (good) but
can simply grow without bound (bad). For each power of k
between 1 and m, and for some complex number z with positive
modulus, it can be coincident with at least one disk but not all
disks. So, equality has to be chosen to guarantee z belongs to
all disks.

>>16117806
I imagine the proofs in the lecture are roughly as difficult as the exercises, but maybe not. Post some?
The purpose is to have you learn measure theory, there's no better way than through problems.

>>16117808
sqrt2 + -sqrt 2 = 0

fuck you retards for inventing strength of materials.

>>16117681
The sum of all the pth roots is not 0 if p is equal to 1 which happens for the right value of alpha.

>>16117681
>>16118312
p=1 only when [math]\alpha=1[/math]. since the question implies that the property holds for any mth root of 1, we can just pick some other root unequal to 1(this wont be possible when m=1 as the only root is 1, but the question also implies that the property holds for all natural numbers m, so we don't have to worry about this either)

thank you for pointing this out
>>16117940
>grow without bound
wdym? the modulus of z simply needs to be bigger than 0 for the inequality to hold

>>16118365
>>16117940
Yes, you're right. I didn't want the modulus of z be so big that some z leaves the disk centered at a root of unity.

>>16117842
It at least follows pretty much immediately if [math]\alpha[/math] is a root of even order, since you'd have a unit disc about (-1,0) and another about (1,0). Obviously the only point at which they intersect is (0,0).
For odd orders, I can't be assed to work out a formal proof, but you could probably do something involving it being the circumcentre of that regular polygon formed by your roots

>>16113803

>>16118248
Sans the additive identity, this is true.

If you sum any two irrationals, the sum will either be 0 or an irrational number. You can't produce a non-zero rational number by summing two irrational numbers.

>>16118564
I don't think that's true.

We can take x = 2 - pi and suppose by contradiction that x = p/q where p and q are integers. Then pi = 2 - x = 2 - p/q is now rational, which is a contradiction. Thus, x is irrational.

Now we can see that x + pi, the sum of two irrational numbers is 2 which is rational and not the additive identity.

>>16118556
>>16118559
>>16118562
Interesting questions, anon. Where are they from?

>>16117466
>>16117331
>>16117207
I learned a neat trick recently.
If you post something like "I'm 12, how do I get jacked?" on /fit/ or something similar on other boards, you'll get permabanned for being underage.
Of course getting "unbanned" basically just mean deleting cookies and getting a new (shared) IP. But I noticed for these past few years, resetting router doesn't guarantee a new one easily.
Worth a try, if you're addicted to 4chan.
Other ways to get banned: https://4chan.org/bans

>>16113803
Doesn’t Eisenstein’s criterion claim x^2 - 1 is irreducible?

>>16118587
What prime number divides -1?

>>16118587
Nevermind I’m retarded

>>16118574
>>16118248
>>16117864
>>16117851

You are right, I was being dumb. Here's a more precise (and hopefully correct) statement.

Two numbers that are not in the same equivalence class relative to the rationals cannot sum to a rational number. As an example if you have an equivalence class
[x] = {sqrt(2) + q, q rational} and [y] = {pi + q, q rational} then any x'+y' will be irrational.

>>16118584
From my Turd World Math Book
It is quite different from general textbooks though.
It's written in a problem + commentary format, concepts follow from the problems.
I have solved each one of them, that grid path problem (1.21) is a nice one.
I am trying to clear the entrance exam of the best math college in my cuntry.

>>16118584
I can post the pdf link if you want

>>16118606
Yes, please

>>16118606
No need. I hate Pajeets.

>>16118556
>>16118559
>>16118562
>>16118584
>>16118604
>>16118606
>>16118618
>>16118632
I found the title. Educative JEE by K.D. Joshi.
I skimmed it. It reminds me of Newman's Problem Seminar.
Looks like the author is trying to pass on his problem solving skills/way of thinking. No idea if it's any good.
The problem collection is interesting though.
And the author have extra problems and solutions on his website/google drive. Not sure if it's dedication, or just looking for feedback for future books.

>>16118654
Yep, that's the book, it's a quite good one, specially if you love mathematics and problem solving.
Most students no longer like to study such texts anymore though.

>>16113926
>>16114270
Hey! I used to post quite a few of them a while ago. Once every thread I think. I stopped since I have been busy but I think that's a lie I'm telling myself. I just haven't been doing math all that much. Here's a new one. I hope you enjoy it. It's not difficult but if you want a hint or two, please ask.

>>16118869
Such an annoying way to ask a simple question. This is not math. It's more of a puzzle.

>>16118869
Gib a hint

>>16118869
Which level of math is this?

>>16118869
Pick sequence n=3: 1,2,4,6.
Split into a_i=1,2 and b_i=6,4.
W= |1-6| + |2-4|= 5+2 = 7.
Perfect square my ass.

>>16118711
I wanted to buy it. But it doesn't look like Indian bookstores do international shipping. Probably to prevent foreigners buying the international editions.
Forwarding services are too expensive imo.

The book is currently in the 3rd edition. And looks like only in hardcover.
The PDF of this book on the internet is from the 2nd edition and doesn't look the same as the one on google books. Perhaps leaked by the author himself?

>>16119355
You can buy it from here, (official website of the publisher).
Add to cart and then select shipping country.
https://www.universitiespress.com/details?id=9788173719455

>>16119338
Negro your sequence is wrong.
1,2...,2n means first 2n numbers retard

>>16119396
Learn English, ESL.

>>16119355
Or from here https://orientblackswan.com/details?id=9788173719455
Universities press is an associate company of Orient blackswan

>>16119440
COPE

>>16118869
i think the idea is to consider any two sequences and then slowly "correct" them into [math]1<2<\dots<n[/math] and [math]2n>2n-1>\dots>n+1[/math]
say 1 belongs to [math]a_n[/math], if 2 doesn't belong to [math]a_n[/math] then we shift it over to [math]a_n[/math] and shift [math]a_2[/math] over to [math]b_n[/math].
if we can show the sum remains invariant in this process, then at the end we get the sequences i wrote earlier, with the sum being [math]n^2[/math]

>>16118869
>>16119077
>>16119093
>>16119194
>>16119338
>>16119355
>>16119387
>>16119440
>>16119445
>>16119451
This sub is retarded. It's literally just a sum of odd numbers equal to squares. Redo your grade school.

>

>>16119495
what a fucking nigger retard, you are supposed to take any two arbitrary sequence, say for n=6
1<3<4 and 2<5<6
is |3-5| an odd number?
you have to transform the sequences to what i said in >>16119451 to do that

>>16119511
How can you transform them though?
1<3<4 can't become 1<2<3

big fan of the theoretical computer science, lambda calculus, intensional type theory, algebraic and categorical semantics, algebraic topology, algebraic geometry, etc.
but never bothered to learn homotopy type theory
should i?
i have a copy of that recent book on modal homotopy type theory which i may read

>>16119586
However we can indeed say that one implies other, maybe that is the key

>>16118556
>>16118559
these are very standard results(and are quite easy)
i believe there is a typo in 1.43: [math]s_0-s_1+s_2-s_3+\dots+(-1)^ns_n = 0[/math] assuming [math]S=\bigcup_{i=1}^{n}A_i[/math]
>>16118562
1.21 is the only interesting problem
i used recursion: each junction(apart from the two junctions at the edge) will have [math]j\cdot2^{k-j}[/math] people whilst the edge junctions will have [math]2^{k-j}[/math](i think)
>>16118869
this is an interesting problem(as are all the ones you post!) and i cannot dispose it with the invariant strategy (>>16119451) directly
i will definitely try again at school tomorrow tho!
>>16119194
it may be from combinatorial problems and exercises by László Lovász as that anon loves that book i think

>>16119753
The coefficient of [math]2^{k-j}[\math] will be [math]\binom{j}{i}[\math]
Hint : Pascal's triangle

>>16119785
what is [math]i[/math]?
(use /math instead of \math for latex)

>>16119806
Distance in x-direction.
Maths is beautiful.

any other examples of troll-tier counter examples?

>>16119950
In a specific field or in general?
one of my favourites relative to how simple it is is the line with two origins. you take two copies of the real number line and glue them together at every point except 0, which gives you a number of initially unintuitive counterexamples pertaining to compact sets
There is no reason to limit yourself to only two lines, of course

Could some anon please tell me how the hell I triangulate this space

>>16116954
"model theory" but you don't need that yet.

>>16119785
oof, this is so much nicer than i thought, the hard part is spotting the triangle, otherwise proving that the coefficients are binomial coefficients is just a one step induction
i am retarded to not have seen that as well as fucking up in calculation
i have never done a problem where spotting the triangle was useful so thanks!

>>16114249
for diff geo Do Carmo is great (Differential Geometry of Curves and Surfaces)

>>16114358
schaum's series of textbooks are EXCELLENT.

>>16114358
Basic Mathematics by Serge Lang.

I'm going to enroll in community college math courses

>>16117851
>le underage namefag explaining the bait
cringe as fuck

>>16113926
I got banned for posting one. They said it was homework thread. It obviously wasn’t so I’m just not doing it anymore because the jannies are faggots.

>>16117704
There’s stuff about it in The Four Pillars of Geometry, which is a Springer book. And Coxeter, I think.

>>16119950
A strictly increasing continuous function with derivative 0 almost everywhere
https://en.wikipedia.org/wiki/Minkowski%27s_question-mark_function

>>16119950
The Weierstrass function is a great one too (though it actually has some useful applications in Brownian motion and digital signal processing/digital control theory where the sampled continuous process is distorted by proper white Gaussian noise).

It's continuous everywhere yet differentiable only on a set of measure zero (meaning only at countably many isolated points with no open interval surrounding them). So it's a function which is continuous on the entire real line, but only has one countable equivalence class of singletons where it is differentiable (meaning almost nowhere).

I have come to say:
FUCK
HATCHER
this motherfucker was fine for the first chapter or two but no he's out here spending fifty fucking pages on obvious expository bullshit and then he skips the part of the proof you actually care about - that is if you don't discover he actually did a crucial step in the proof (which is important here and only here) 5 pages (ahead or behind, chosen at random) of this, or if he doesn't cite some random-ass mechanical detail from the midst of some unmarked previous proof that turns out is actually extremely important (and was, of course, halfway skipped over in favor of more blathering)
how the hell would someone who'd never seen any category theory would be able to do this shit based on the 4 pages of very brief exposition he gives (at the end, of course, of the homology chapters in the middle of the book, *after* the axioms that actually use the definitions are introduced) is as far beyond me as most of his bullshit proofs are

>>16118869
Did anyone solve this yet?

>>16115110
image the millions of Chang and Kims that waste 1000+ hours training olympiad level geometry problems like Evan Chen.
Just to end up a washed up 3rd rate mathematician like Po Shen Loh who has to earn their living by selling the competition math training scam to the next generation of gullible idiots.

>>16118869
The following proof can be extended to a proper proof by induction but I won't really bother with that. You'd have to show it's true for n=1 and probably also n=2 but these can be done on the back of one's hand so I won't bother doing it here. Spoilers ahead for anyone still working on it

Without loss of generality
[math]a_1=1[/math]
If [math]a_2=2[/math], then move on to the next step, otherwise a_2 is not 2, and must be bigger than 2. Swap the position of 2 and 3. This will not affect the total sum, and we can see this by taking cases:

Case 1: If 2 and 3 are both b_k and b_j for some k and j, then their corresponding [math]|a_k-b_k|[/math] and [math]|a_j-b_j|[/math] will each go down by 1 and up by 1 respectively (or vice versa depending on whether you make b_k=2 or whether it's 3. I'm going to make b_k=2 henceforth). I know this must be the case, because min(a_j,a_k)>=4 whereas max(b_j,b_k)<=3, so really [math]|a_k,j-b_k,j|=(a_k,j-b_k,j)[/math], which means we'd have, in our series: [math]|a_k-2|+|a_j-3|=a_k-2+a_j-3=a_k-3+a_j-2=|a_k-3|+|a_j-2|=|a_k-b_j|+|a_j-b_k|[/math].

>>16120551
Case 2: a_2=3, in which case b_n=2, it has to be 2, it can't be anything else, this is how the partitions are defined! Well, then a_n is clearly greater than 2 and 3, and b_2 is clearly greater than 3, because we know that b_2 can't be 1,3, or 2. So then:

[math]|a_2-b_2|+|a_n-b_n|=|3-b_2|+|a_n-2|=b_2-3+a_n-2=b_2-2+a_n-3=|2-b_n|+|a_n-3|[/math]

Thus we see that even in this second case, we can swap the location of 2 and 3 without changing the total sum.

So to recap what has happened, we start off with a_1 = 1. If this isn't true, then flip the b's and call them a, then flip the a's and call them b. Next step, if a_2!=2, then swap the positions of 2 and 3. This can be done without changing W. In the case where a_2=3, this will make a_2=2. Otherwise, we continue this swapping game, swapping 3 with 4. By the same arguments as before, this won't change the total sum (here is where you would need to properly write this as an inductive proof, but I'll let you imagine what that looks like because it's been a long time since I've done the 4chan latex formatting and I think I've probably fucked it up already [lmao I have]). Then we swap 5 with 6, and so on, up until we swap 2n-1 with 2n, and we therefore see that the series is equal to the series where a_1=1, a_2=2, ..., a_n=n, b_1=2n, b_2=2n-1, ..., b_n=n+1.

Showing that the resulting sum equals n^2 is then straightforward.

>>16120542
Po Shen Loh is a IMO silver medalist and a Professor at CMU, He won many scholarships and attended prestigious institutions like Caltech, Cambridge University and Princeton, He is married and has 3 children.
Meanwhile you are low IQ failed cel coomer seething on a 4chan board

>>16120555
I forgot to add that when a number jumps from the b section to the a section, you need to repeat the whole process again starting from the smallest number still in b

>>16120564
Po Shen Loh hasn't discovered anything. He's most definitely a third-rate mathematician. There's a reason why he's focused on selling the olympiad scam as opposed to researching. You're clueless

>>16120564
hit right where it hurts, huh?

>>16120567
Ah wait, no. Case 1 is dumbass territory; if a_i!=i, then you essentially jump straight to case 2. If a_i=j, then you swap j with j-1, since j-1 will be one of the b_k's. This will have the same effect as making one part go up by 1 and one part go down by 1. You can then incrementally do this until a_i=i, then you move onto a_{i+1}, and I'm pretty sure this is right now... heh how embarrassing!

>>16120551
> Swap the position of 2 and 3.
What?

>>16119093
Of course! First, I'd suggest you work out some small cases. The result is kinda clear. The second hint, and I believe this can take you right to a nice solution, is notice how one of a_i or b_i is greater than or equal to n+1. I hope this helps anon!
>>16119194
Hmm, it's from a national math olympiad for highschoolers but it's one of the easier questions. It definitely doesn't require anything advanced.
>>16119338
There seems to be a misunderstanding. 1,2,...,2n meant all the numbers from 1 to 2n, so you could split it into two sequences with n numbers. You'll get a perfect square everytime if you do this.
>>16119396
Thank you anon.
>>16119451
I think that's a good idea! I'm not sure how you'll proceed with it, I don't know if I could. But to me at least the idea seems correct, that the sum remains the same. Good luck with the rest anon. And if it doesn't work out, check out the hint I gave above.
>>16119753
>this is an interesting problem(as are all the ones you post!)
Thank you! I don't know if it is, lately I haven't been able to find problems which were both interesting and that I was capable of solving. But I appreciate the compliment, I hope we can do more problems together.
> i cannot dispose it with the invariant strategy
I should have read all the replies before writing what I did above but oh well.
Good luck with your attempt. Don't neglect school subjects though.
>it may be from combinatorial problems and exercises by László Lovász as that anon loves that book i think
I do like that book but this one is just from a national olympiad.
>>16120388
Sorry to hear that anon. I hope it doesn't happen to me.
>>16120555
Looks good. I need to leave home soon so I can't give it a proper check but I really like your idea about swapping numbers and showing how it doesn't change the sum. Thank you for answer, it's appreciated. I'll be sure to read it carefully once I'm at school.

>>16120660
>Looks good... I really like your idea about swapping numbers and showing how it doesn't change the sum. Thank you for answer, it's appreciated.

No problem, I made a slight correction to my method because I realised I wouldn't be able to swap numbers in Case 1 without betraying the order of the sequence, I fix the mistake here >>16120578 and will give a better explanation below now that I have the time to clarify.

>>16120585
Yeah I did a stupid with case 1. My fix is as follows:
We start off with a_1 = 1 (w.l.o.g.) and we want to find an equivalent sum wherein a_2=2.
Suppose a_2 = 3. Then b_n = 2. We can "swap" 2 and 3. By this I mean we make a_2=2 and b_n=3. This doesn't change the overall value of the sum, as I've explained >>16120555

On the other hand, a_2 may be neither 2 nor 3. If a_2=4, then, for some k, b_k=3 (in this case k=n-1, but it's not too important what k is) so then we 'swap' the position of 3 and 4; a_2=3 and b_k=4. This won't affect the total sum, nor will it affect the order (I show this formally below). We then swap a_2=3 with b_n=2 and get a_2=2.

In general, if a_1=1, a_2=2, ..., a_{i-1}=i-1, but a_i=j, where j is not i, then we swap the values of a_i and b_k, where b_k=j-1 (k can probably be found in terms of i but it doesn't matter). By the way the sets are ordered, it's simple enough to show that a will still be strictly increasing and b strictly decreasing (there are no integers between j and j-1).

The sum W will also be unchanged. You have to show that a_i and b_k are interchangeable, i.e.
[math] |a_i-b_i|+|a_k-b_k|=|b_k-b_i|+|a_k-a_i| [/math]

If i=k this is trivial
If i<k then a_i<b_i and a_k>b_k, so
[math] |a_i-b_i|+|a_k-b_k|=b_i-a_i+a_k-b_k=|b_i-b_k|+|a_k-a_i|=result [/math]
If i>k then a_i>b_i and a_k<b_k and, similar to before, it doesn't change W

We repeat this process from i=2 to i=n until a_i=i for each i. b will necessarily be decreasing because the order hasn't swapped. Ergo W=n^2

>>16113951
Do you have another complex number analysis problem, these are good.

>>16113803
Give this one a try everyone

Looking to do recreational maths. Where do I start? Do I just pick a subject that's interesting or do I learn something like proof theory first? I haven't touched the subject since highschool calculus so I'm lost for where to start.
>pic unrelated

>>16120920
All of the interesting math is gated behind proofs so you'll certainly want to learn that. Most first get exposed to proofs in their first analysis and linear algebra course, so just find books on these topics you like and work through them. The /sci/ wiki is great for that. Feel free to branch out to subjects you like whenever you feel like it. May even help you understand what you have to learn to reach that point

>>16118869
Let a,b be consecutive integers.
Let A be the index of a and B be the index of b.
Our goal is to show that swapping set membership of a and b does not change the sum.
For the case A=B, the term |a-b| is unchanged by swapping.
WLOG suppose A<B.
The two terms in the sum that will be affected by swapping are |a - b_A| and |a_B - b|.
Since A<B, we have a_A < a_B and b_A > b_B.
This gives:
a - b_A < a - b_B = a - b = +/- 1
and
a_B - b > a_A - b = a - b = +/- 1

So a - b_A and a_B - b have opposite signs.
Swapping a and b will have the effect of increasing/decreasing a by 1 and decreasing/increasing b by 1 (since they are consecutive).
This means a - b_A and a_B - b will both increase/decrease by 1 (note they change by the same because of the - sign on b).
Since these terms are opposite signs, one will get closer to 0 by 1 and the other will get further from 0 by 1 which means the sum will be unchanged.

You can just start with the sets of a in {1,...,n} and b in {n+1,...,2n} and just do consecutive swaps to reach any partition.
The sum remains n^2.

I know this isn't really math but it's slightly mathematical so what is a good resource to understand what a learning curve in economics is?

>>16120793
i don't you can call this "analysis", but here is another one...i don't have too many, i am going to be brushing up on my algebra so i will keep posting new stuff on /mpsg/(a general i created specifically for these kinds of problems) so you can check that out

>>16120818
gib a hint anon, i am stuck after obtaining f is odd from f(0)=0

I have nothing of value to say, I just find it funny that the three topics I decided to read at once(Munkres' Topology, Weinstock's Calculus of Variations, Chartrand's Introductory Graph Theory) happened to be the ones for make game by sheer coincidence. I picked up Munkres because I wanted to understand what the fuck the mathematical physicists were talking about, other two for shits and giggles.

>>16121342
>>16121491
Try taking x=2y, and then try f(y)=0. Since the mystery function is bounded and continuous, consider calculus techniques.

>>16120761
This is a repeat of what I said before but I really like your solution. And I've given it a through read and to me it looks quite sound. Thank you for your solution and your beautiful use of 4chan latex formatting.
>>16120953
Very nice! I'm surprised to see two solutions in this similar line of thought. Yours is quite concise and clear, I think it's a good solution. I appreciate the time and effort you spent on this, the solution I had(which I got because I had a hint that directed in a different direction) was quite different so it's good to see this alternative. So thanks! Have a nice day.

>>16121342
Let [math]\zeta = e^{i 2 \pi / 5}[/math].
Plug in [math]x = \zeta[/math], [math]x = \zeta^2[/math] and [math]x = \zeta^3[/math] to get the system
[eqn] A(1) + \zeta B(1) + \zeta^2 C(1) = 0 \\
A(1) + \zeta^2 B(1) + \zeta^4 C(1) = 0 \\
A(1) + \zeta^3 B(1) + \zeta C(1) = 0
[/eqn]
The determinant of that system is non-zero so [math]A(1) = B(1) = C(1) = 0[/math].

Hello /mg/
Where should I start if I want to start learning abstract algebra? I've taken all the way up to Differential equations at my university (engineering school), but I'm really curious about abstract algebra and would love to learn more about it.

Is there maybe a roadmap of mathematical fields (e.g calculus, diffeq, basic algebra, etc.) I should follow?

>>16122260
Start off with group theory (which doesn't require anything fancy besides a basic understanding of set theory), move into ring theory after that, and from there you should be reasonably comfortable to figure it out on your own

>>16122265
What, if any, resources would you recommend?

>>16122260
Learn topology and then go to differential geometry. You'll learn interesting algebra in context along the way.

A math degree isn't actually a bad career choice.

>>16122389
M. Sc? Maybe. PhD is a career killer.

graph theory question from CLRS regarding strongly connected components.
>Professor Bacon claims that the algorithm for strongly connected components
would be simpler if it used the original (instead of the transpose) graph in the
second depth-first search and scanned the vertices in order of increasing finishing
times. Does this simpler algorithm always produce correct results?

i can't give any formal reason for why this doesn't work and no book has addressed this.

Let's play a game.
You are given an infinite supply of arbitrary statements p_1, p_2,.. to choose from, and can form logical combinations using them. In a round, you give me three logically independent sentences: it should be impossible to derive one sentence or its negation from the other two by pure logic.
I say whether each one is true, except I lie about one of the sentences you give me.
Question: can you deduce whether p1 is true by asking me questions?

>>16122568
reproduce the algorithm here
>>16122580
yes, only need p1 p2 p3
(p1,p2,p3)
(p1,p1+p2,p1+p3)
(p1,p2,p2+p3)
and all independent mod 2 xor combinations
the allowed lies combination for false false false, false false true, and so on are all disjoint

>>16122626
>reproduce the algorithm here
an example is here.

i think i find out why. the whole reason why we transpose the graph is because it's easier to find the source of component of [math]G^{SCC}[/math] instead of the sink component and also because of all the nice lemma's that we have regarding edges going from an older component to a most recent one. For the exercise, i think we to have to simply say that a vertex with the lowest completion time doesn't have to be in a sink component.

>>16113803
Can anyone solve this?

>>16122851
I hate niggers so much

>>16123047
Which level of math is this?
Where can I find more such problems?

>>16123047
Put a(n) in the first pile. Put a(n-1),... into the second pile until the second pile has more than the first pile.
Start adding to the first until it has more than the second. etc.
Assume they are never equal until the process ends at a(1) (if they become equal you can just use induction).
If a(k) is the number that makes one pile overshoot the other, then the difference is at most a(k)-1 at that point.
Since a(k-m) >= a(k)/2^m you should always be able to overshoot or match the a(k)-1 difference since
a(k) <= 2^(k-1)
and
a(1) + a(2) + ... +a(k-1) >= a(k)*(1/2 + 1/4 + ... + 1/2^(k-1))
= a(k)*(1 - 1/2^(k-1)) >= a(k) - 2^(k-1)/2^(k-1) = a(k) - 1

>>16121342
A is the constant function of a 2nd degree polynomial. The function derived as D would have 0 change at 1.

>>16123385
> If a(k) is the number that makes one pile overshoot the other, then the difference is at most a(k)-1 at that point.
how?
> Since a(k-m) >= a(k)/2^m
what?

>>16123592
This idea that one number a(k) makes a pile overshoot the other isn't always true.
Example - 1,2,3 is one such sequence
(1) and (2,3) is a split where this logic doesn't work

>>16122260
Read Gorodentsev

>>16123047
I wonder if this can be solved by Induction

>>16113803
I am in my undergraduate degree. If I am hating my group theory and number theory classes, will I hate all of pure math as I go on? I have enjoyed my classes on analysis and advanced calculus a lot.

>>16123618
You are adding from largest to smallest.
1,2,3 would give piles:
3 then 2+1

>>16123420
>A is the constant function of a 2nd degree polynomial
What are you talking about?
One possibility is for example
[eqn]A(x) = B(x) = C(x) = x - 1 \\
D(x) = x^3 - 1[/eqn]
Another possibility is the trivial
[eqn]A(x) = B(x) = C(x) = D(x) = 0[/eqn]
Infinitely many other polynomials work too but you have to prove that for all of them [math]A(1) = 1[/math].

>>16123823
try a relatively formal/rigorous class in ODEs, a decent class in PDEs, or a rigorous treatment of numerical linear algebra (Trefethen&Bau's book is decent, if you know enough that you won't get roadblocked by section I and can jump to II and past). You could do a reading with a professor, if something like the above aren't offered at your school.
Hating both group theory and intro number theory (ah, but I repeat myself, given that these are undergrad classes) but it could just be that you've got a bad group theory prof and aren't inclined to number theory. There's lots of branches of analysis beyond the ones I mentioned, if you're more of an analyst over algebraist.

>>16123893
Thank you very much anon. I am splitting my degree between two subjects and so sadly I cannot go incredibly deep into a lot of stuff. I enjoyed my class on ODEs and Dynamical Systems. I will try to do work with PDEs next year. If I cannot, I'm sure I will have opportunities to do them in my masters. Doing a joint degree is sometimes highly painful.

>>16123882
Because the function isn't a coefficient of a x variable it has no rate of change. It is locked at a point. Given some D(x) would be a factor of all terms for a rate of change it would be 0.

>>16123919
>Because the function isn't a coefficient of a x variable it has no rate of change
It's a function of [math]x^5[/math]... just because you're not multiplying it by another [math]x[/math] doesn't mean that it has no rate of change with respect to that variable.

>>16123047
>>16123385
>>16123155
you don't need to do calculations at all desu(your idea is correct)
for others: the idea is to be greedy, this is one of those "just do it" problems as tim gowers puts it.
the source is 1957's moscow olympiad(or some competition idk whats it called)
let [math]p_n=a_n[/math] and [math]q_n=0[/math] where [math]p_i[/math] and [math]q_i[/math] are the two piles when we add [math]a_i[/math] to one of the piles
we will add the highest available element of the sequence, to the smaller pile. so now [math]p_{n-1}[/math] is still [math]a_n[/math] and [math]q_{n-1}=a_{n-1}[/math]
now you can easily see that [math]|p_{n-k}-q_{n-k}| \leq a_{n-k}[/math] for any k by the condition that [math]a_{i+1} \leq 2a_i[/math]
so at some point the difference of the piles will be [math]\leq 1[/math], but it cannot equal 1, as that would mean that the two piles sum to an odd number which is forbidden, so the piles must be equal in size

(if at any point you are able to equalize the piles before exhausting all the elements, just start again from the highest available element, noting that all the original conditions are still satisfied)

Noob question on notation: Which of the two is preferable?
"for any integer [math]m\geq1[/math]"
or
"for any [math]m\in\mathbb Z^+[/math]"
Or is it entirely a matter of writing style?

>>16123932
Both work, though if it were me I'd go with the second. It's a bit shorter and a bit cleaner, and conveys the same information.
But nobody would chide you for the former.

>>16123155
here is one other problem in a similar vein if you are interested, the source is 2018's IMO shortlist

>>16123937
Okay, thanks.

>>16123925
What exactly do you mean by being greedy?

>>16123965
https://en.wikipedia.org/wiki/Greedy_algorithm
basically, just use the biggest number you can at every possible step

job advice for a math graduate? I finished uni last year and I'm currently working as an engineer writting technical documents of things I have no knowledge of. I want a more math-loaded career but not willing to teach nor PhD. I like competitive programming, math, solving abstract problems etc. Maybe a job on the gambling industry? Suggestions?

>>16124011
Banking and Finance? That seems to be the goal of most math students I have met. It is dependant on you wanting to move to a big city and/or getting a visa to a country with a large financial sector tho and seems very stressful.

Since you like programming, (assuming by engineering you mean a conventional engineering job) what about a software engineering/development? Some sort of quant job combines both suggestions and pays well....

>>16124021
thanks anon. yeah, maybe a software engineering job will fit me. That, along with the gambling industry or even some statistics job sounds very attractive.

>>16124033
Finance is just gambling with bailouts if you loose. Good luck!

>>16113803
What's the best way to do standard deviation from a set of year brackets, e.g., a data set of age groups from 10-20 years, 20-30 years, 30-40 years, and so on? I thought I should half the integers, e.g., midway between 10 and 20 is 15. Is this the way to do it?

>>16123925
What if say while adding the number [math] a_k [/math] we realise that the pile we were earlier adding to is now greater in sum than the one we added [math] a_n [/math] at the start?

>>16124459
that's why the mod, if you read the sol it says a_i to the smaller pile at any point, anyways the mod always decreases

>>16124541
The reason I asked the question was because I am doubtful whether [math] |p_{n-k}-q_{n-k}| \leq a_{n-k}[/math] will be valid in that case

>>16124556
say the difference becomes negative first on adding [math]a_i[/math] to the smaller pile
it must have had some value [math]<a_i[/math] just before this for this to be the case, so of course, now the difference whilst negative, will still not be as negative as [math]-a_i-1[/math]

I have a degree in computer science, did Calculus I, II and III, discrete maths and I can't understand shit what you guys are talking about most of the time... suffering...
How do I fix this?

>>16124969
Did you at least do linear algebra? Please tell me they didn't let you graduate with a comp sci degree without linear algebra and ODE's.

Real answer, spend some time self-studying real analysis, abstract algebra, and some introductory topology (and the undergrad stuff you missed like LA and ODEs). That will get you a strong set of fundamentals to pursue the rest of mathematics.

>>16124976
I did get linear algebra, but man, it's been such a long time. Thanks for the tips, I will try.

I really liked discrete mathematics, I attempted reading "Concrete Math" from Knuth, but didn't go so well.

>>16124574
I understood it later yesterday, the absolute difference will decrease to 1 anyways, if sum in one of the pile increases that the other, same thing continues with the piles swapped with each other, finally we will put the remaining 1 in whichever pile required to make the sum equal.
Example - take the sequence 1,2,3,4,8
[math] \left(S_1,S_2\right) \to (8,0), (8,4), (8, 4+3), (8, 4+3+2), (8+1, 4+3+2) [/math]

>final spring quarter
>cannot fucking muster to do jack shit
>submitting incomplete homework
>quit my research
>haven't bothered to even open my books
why is this happening to me?
why am I so fucking done?
It's my last fucking quarter
I just cant stand telling myself that I need to get through one more fucking week
What the fuck am I doing with my life? Why the fuck am I studying fucking mathematics? What kind of fucking mega retard studies mathematics?
I'm fucking losing it guys
has this ever happened to you?
I don't know if I can slog through this quarter, so far I'm passing but what about finals? If I don't fucking finish I'm going to fucking end it all

>>16124556
It doesn't matter, all that matters is that you put the number into the smaller pile.
After putting [math] a_k [/math] let [math] 0 \le p_k - q_k \le a_k [/math], basically [math] p_k \gt q_k [/math]
After putting [math] a_{k-1} [/math], there can be 2 cases

1. [math] p_{k-1} \ge q_{k-1} [/math] then
[math] |p_{k-1} - q_{k-1}| = p_k - \left(q_k + a_{k-1}\right) \le a_k - a_{k-1} \le a_{k-1} [/math]

2. [math] p_{k-1} \le q_{k-1} [/math] then
[math] |p_{k-1} - q_{k-1}| = \left(q_k + a_{k-1}\right) - p_k = q_k - p_k + a_{k-1} \le a_{k-1} [/math]

Thus in both cases, [math] |p_{k-1} - q_{k-1}| \le a_{k-1} [/math]

>>16120818
>>16113803
>>16121491
This is the pdf of the answer for this problem

>>16122389
if i dont ace my exams next month (which is likely) i'll be choosing between going to a top tier (for my country) school for math or tier 2 for cs
i think i want to make big buxx more than i want to jerk off my ego so i'm hesitant about choosing math
i was thinking about going for math and then starting CS next year but i dont think i could handle both at the same time

>>16125043
If you want the "real stuff" you could jump right into baby Rudin and then Royden for integration.

Probably your best bet would be getting a refresher in the undergraduate stuff (e.g., Axler's Linear Algebra Done Right, Tenenbaum's ODE's), and then take a look at the easier side of a pure math track.

Jay Cummings has a pretty friendly book in terms of introductory real analysis, and Ross's Elementary Analysis isn't bad. Munkres' Topology is pretty approachable, and Pinter's Algebra book is pretty friendly to new people also. I like Bartle's integration book if you want to learn the easier side of Lebesgue integration without going fully into measure theory (which is a big jump in difficulty but important for some fields, like those that deal with probability and PDE's a lot).

Generally the best advice I can give is be patient with yourself and understand that it's difficult work and you will fail sometimes. That's easier said than done, but the point is to learn, not check off boxes.

Every Math Department:
>westerners research pure math
>eastern europeans research analysis and physics
>iranians research applied math
>asians research statistics and finance

>>16125730
Don't forget the metric fuck ton of Indians researching applied probability and applied statistics.

[eqn]\lim_{n\, \to\, \infty} \sum_{k = n}^\infty 1[/eqn]If you only go by the epsilon-criterion for convergence, then this diverges, right?

>>16126027
indeterminate form I'd guess

How do I know if I have a math learning disability?
When I was a kid, I was diagnosed with a math learning disability by a psychiatrist. I always struggled with math.
I dropped out of high school early on, so my mathematics knowledge is lacking. But I recently started doing Khan Academy and I think I'm doing okay... the question is, am I struggling more than a normal person, and how much will this "disability" affect me (if at all) all these years later?
Believe it or not, I really want to pursue electrical engineering, but I also don't want to spend a bunch of money on college just to find out that I physically cannot get past calculus 2.

>>16126428
You haven't really given an indication of how much you're struggling so it's hard to gauge anything. I'll say that if you're actually understanding the concepts that you're learning, then you're probably fine. Just make sure that you're understanding the topics to a decent degree and not trying to get through them as fast as possible. I tutored Calc 1 and 2 for a while and I've seen some total retards make it through. Maybe you could look into CLEP and getting Calc 1 credit through that and then taking just a Calc 2 course at a community college; if you fail and decide that you don't want to continue, you'd only be out a couple hundred dollars.

>>16126460
I don't feel like I'm fully understanding all of the topics. In some cases, I feel as though I'm just sort of memorizing rules/procedures that don't really make sense to me because "that's math and that's how it works". But I don't understand WHY they work or WHY I'm allowed to do X only in Y situation.

>>16126461
Tbf it feels like that for most up until like... trig? Even then most students just vaguely follow procedures that their teacher gives them, lol. Just keep chugging along and I'm sure you'll be fine. Even if you do have a learning disability I doubt you'd be the first to get an engineering degree.

What is the best book about groups?

My algebra lecture was held by a female professor and thus the quality was terrible. Looking at Wikipedia I notice how much I don't know about group theory. Any help?

>>16126475
Messages coming soon
I have a slight problem... I've been under attack for ages.
I am capable of doing something great
But my position is strange, keep meeting an odd ending which means I missed the target by some mode of a awkward push at a distracting moment.
I have now good aim at the enemy problem
And am installing.

In the mean time. Think reverse mechanics in psychological stance that also uses the body. To shadow calc, you must go through the body. Then it's just a matter of knowing which parts of the brain you need to use to do what you want. Firstly, you want to install a frame machine. It is like a field, but for cut-life. You must generate something to play with or you're all brain. But the .method of shadow calc should be enough, you'll find your own way

>>16126511
It's not the aim at the brain parts you're trying to achieve - it's the language You're speaking on the low. Ignore 'reverse' do what's comfortable.

>>16126475
I'm definitely not an algebra or number theory guy, but I found the chapters covering groups in Pinter's Algebra to be pretty good.

Has anyone ever see this notation?
[math]E \otimes_{S^1} E' [/math] where [math]E, E' [/math] are circle bundles. What does it mean?

what field of maths has the most awful notation?

>>16127007
Number theory wins entirely on Legendre symbols
t. number theorist

>Talk maths
>Posts theories from academia
>Mfw

I still don't understand what "completing the square" really was. Given

6x^2 + 12x + 8 = 152

You first get it to a particular form, right?
so divide by 6 and move the constant on the right

x^2 + 2x = 24

Then you imagine you have a square
x by x
2 by x
and then a square with an area of 24
Then you arrange these pieces to form a square with a hole in it? Like in this case we are missing another rectangle of size 2 by x to form a square on the left side
Then what? Do you pretend you have this missing rectangle and say this new square you formed has an area of 24? To me that doesnt give the right answer, since it would result in (x+2)^2 = 24

>>16126938
tensor the fibers

>>16127113
split the 2 by x into two 1 by x rectangles
put one vertically and one horizontally
now you are missing just 1 by 1 from (x+1) by (x+1)
so (x+1)^2=25

>>16127118
Can I just do that? What are the rules of this rearrangement game?

How do I study for exams? Throughout my whole life I have found learning math to be incredibly easy, but studying to be difficult. I usually used to self teach things and just move on when I had the hang of it to more difficult stuff. I'm now in my penultimate year of college and cannot do any memorizing of definitions as the way I'd always self teach was to just learn the work, do one or two practice questions and then move onto more advanced stuff. Now i'm stuck having to mindlessly memorize lecture notes. Any advice??

how do you learn from dense math books that follow the definitoin-lemma-theorem-corollary pattern? at the moment i'm reading at a pace of a 3-4 pages per day because i write everything down and i try to understand every sentence but it's simply not doable because i also have other things to study.
should i just read it very carefully and do the exercises? i feel like i'm not really interiorizing the stuff that's begin said to me unless i write it down (which is nonsense but i can't help it).

>>16126428
>>16126461

math specific learning disabilities don't exist (e.g. dyscalculia)
they were invented to make midwits like yourself feel better about being retarded

math is 100% logic, so if you are dumb and struggle to understand logic and how to apply rules to novel scenarios will be extremely difficult / nigh impossible

If you don't understand why algebra rules work, then prove why they work. algebra is an extension of arithmetic so if you don't understand arithmetic then algebra and its "rules" will of course seem opaque and arcane

>>16127113
>>16127123
what rules? you just make a fucking square

>>16127188
you need to be actively guessing what the point is
just grinding is like reading a mystery novel without thinking critically about how the crime could have happened, or a fanntasy novel without visualizing the scenes
bullet by bullet, sequence of events and facts, soulless
you learn and do by making a narrative, even if it's a bit wrong you'll correct it later when they say "the knife was actually in the pool room"
pretend someone is reading the book to you, and you have to stop them when you're bored and ask "what's the point?" or "is this detail really important?" only, you have to answer it yourself best you can

>>16127421
not the guy asking a question, but what's the point in being unnecessarily hostile like this?

>>16127448
i don't want people that are filtered by algebra 1, aka middle school math, shitting up this board

>>16127474
A lot of the time people are never actually taught the actual proofs and reasoning behind the methods, especially for very basic math, and if they are taught it, they were probably 13 and barely paying attention just wanting to memorize formulae for easy marks. The person who asked the question however should have probably googled it, but no need to be cruel I think.

>>16127116
But the fibers are [math]S^1[/math], not vector spaces...what does tensor mean in this context?

>>16127493
how is anything i previously said "cruel"
sorry i want people to not be retarded because figuring out "completing the square" is literally in the fucking name
anyone who passed algebra 1 should be able to figure it out with ease by exactly what i did in the picture
>they were probably 13 and barely paying attention just wanting to memorize formulae for easy marks
then fuck off of /sci/ and go back to r*ddit

>>16127188
it's natural that you do better by writing things down. it has to do with unloading cognitive load and dexterously (i.e. with the hands) engaging with material
as for studying maths, the first thing is that you miss a step missing from the definition-lemma-theorem process. the missing step is example, which in a good book will be close to definition. I will explain the purpose of example
have you ever studied euclidean geometry axiomatically? you will notice many statements are formulations of things that "look right" already before you read the proof. in fact, reading the statement itself is often enough to convince you that it's true. this happens because you already have a good mental model of the objects at play in euclidean geometry, i.e. you have already experienced straight lines and circles
(for a working mathematician, the point of doing proofs is to keep his intuitions/conjectures from becoming fact without rigorous examination)
so when you study from a textbook, you should always have a few illustrative examples of the object at hand. these examples should guide your intuition. I like to use the mathematical concept of ring as an example. for me, the core examples of rings are the ring of integers Z, the Gaussian integers Z[i], polynomials over a ring, and C(X) ring continuous real-valued functions on a topological space. each of these I can use to build intuition. for example, you can think of ideals as lattices, because in Z[i] there is a correspondence between them.
second, if you want to work fast you don't have to understand all proofs right away. writing down the important theorems and working out an example to check that it holds will give you a practical feeling for it. while reading, keep a list of theorems you want to work out the details of later. and do exercises. if you want to move fast, focus on the big ideas that characterize the subject/object at first and check details later

>>16127141
Same situation. I am second year math student, now I am teaching myself for some gay algebra exam, by watching some yugoslavia turbofolk, and learning these gay schemes and definitions in meantime just to pass the exams since academia is gay anyway.

i figured out that I need to look at math from more philosophical point of view so I am studing "algebra chapter 0" book which approaches algebra (and hence math in general) from category theory point of view. I hope that I will get some nice understanding that I hoped for before starting uni, from that.

>>16127544
Yes it is a bad situation to be in. Algebra is possibly the worst subject of them all for this sort of thing. It's just mindnumbing pedantic definitions until you get VERY deep into it. Good luck for your exams.

>>16127559
Thanks for good wishes. So what do you recommend to get the best results in understanding algebra and math? Just doing bare minimum of college stuff just get by, and in meantime learning stuff like category theory (grothendieck type of stuff even?)? Is it good approach or am I going to find out just how foolish my thinking was, some time in the future.

>>16127188
i kind of already do that but it takes a lot of time because i'm not able to concentrate deeply and before i understand the concept i think of ten other unrelated things before i actually get it. at least i'm stubborn enough that i don't filp to the next page unless i've gone through this "process".
>>16127518
i'll give you an actual scenario. i want to restudy mathematical analysis from the ground up and i'm using a book that starts with the axioms of reals and all the related consequences.
should i write down every little definition like "there exists only one number 0 in R"? otherwise i wouldn't remember a single thing, but i feel i'm just copying the whole book by hand like monastic scribes.

>>16127584
>So what do you recommend to get the best results in understanding algebra and math?
I hate to break it to you but consistency. It's something I am very awful at too. But it's by far the best way particularly with definition memory tests. Just use this as a learning experience more than anything else.

>>16127606
>But it's by far the best way particularly with definition memory tests

You are probably right on that. But I don't necessrily wish to ace one these scheme based exams.

To paraphrase what was said about Grothendieck, I feel like math is a mountain and I am walking on it, however, I wish to build a mental helicopter which will take me to the top. I wish to change paridgm (in my head obviously) of
understanding math, just as Immanuel Kant laid down the new clear understanding of reasoning in his "critique of pure reason", which made a new paridgm in philosophy

>>16127602
the real numbers are meant to bridge the gap between geometric continuity and arithmetic descreteness. let me explain

start with two distinct points (called 0 and 1) on the plane and a straight line between them. You should think of this line as the number line. Construct the square on the segment from 0 to 1, draw the diagonal from 0, and draw a circle from 0 with the diagonal as its radius. Clearly the circle intersects the original line (possibly after extending it): call this point A. The length from 0 to A is in no rational proportion to the length from 0 to 1. This means that we can endlessly mark off rational distances on the number line like 1.5, 1.4, 1.45, etc, without ever touching A. Of course the circle still intersects the line, just not at a distance from 0 that we can express with complete accuracy.

If you want arithmetic to be the foundation of your maths (maybe you don't trust geometry) and you have only the rational points, then you run into a problem as the circle seems to pass through a hole in this line. The challenge is then to have a line without "holes" without reference to geometry.
one possibility, perhaps the most natural, is interval arithmetic. Another one is the Cauchy completion formalism. Both deal with the problem of having imperfect accuracy. from this point of view, real analysis essentially becomes the study of approximation by rational numbers

>>16127734
where did you learn this?
also i'm sorry, but how is that going to help me learn mathematics from cryptic books?

>>16127188
You need to come up with your own questions/problems and (counter)examples and alternative proof.
Obviously problem book is the most efficient way to learn things. You basically offload the work I mentioned to the author, and just grind problems until you vomit. But relatively early in math curriculum, there is no such thing as problem books any more, with or without solutions. Even slightly advanced or obscure topics usually only have 1 monograph written by some asshole.

I hadn't really thought about it before, but it occurred to me about half an hour ago that you wouldn't use spherical coordinates to describe a proper sphere. Well, not without rendering the radial coordinate entirely redundant, at least.
I thought it over a bit more and I've been trying to figure out if there's any way at all to extend spherical coordinates to describe some sphere in some dimension, again without fixing the radial coordinate and thus rendering it redundant.

Am I stupid? Is it obvious and am I overthinking it?

>>16128565 (Me)
Well, I suppose that as I've phrased it you could describe a circle by just fixing one of the angular coordinates, and use a similar process for any sphere.
Let me amend my question to "without fixing any coordinates"

>>16128565
Any reasonable hypersurface is lower dimensional, so locally one of the coordinates is fixed by knowing the other [math]n-1[/math].
Not sure what the question is then, if you just don't want a "simple equation" then take a sphere not centered at the origin.

>>16128600
If we take a 2-sphere in 3-space, and a point on the 2-sphere, then to describe the point using spherical coordinates, consisting of two angular coordinates and one radial coordinate, we fix the radial coordinate to the sphere's radius - it might as well as not be there, since the point only has two degrees of freedom.
But a 3-sphere has three degrees of freedom. What I'm trying to do is to wrap my head around how, conceptually, one would use spherical coordinates to describe a point on that instead. Similarly, a point on a 4-sphere by adding another angular or radial coordinate, etc.

Though now that I think about it, I suppose it'd just be analogous to polar coordinates on a 2-sphere, so it's not really all that interesting of a concept to get stuck on.

Prove
[math]\exists C_d: |p(0)|\leq C_d\int_{-1}^{1}|p(x)|dx\ \text{for all }p \text{ of degree }d.[/math]

>>16128941
|p(0)| is seminorm, integral is norm on d+1 dimensional space of polynomials
Equivalence of Norms

Is e an t-àite seo an taigh beag agam a-nis.

>>16129007
Is that allowed with the seminorm?

>>16129105
Not him but can't you just take the two norms
[eqn] \|p\|_1 := |p(0)| + \int_{-1}^1 |p(x)| dx \\
\|p\|_2 := \int_{-1}^1 |p(x)| dx [/eqn]
By norm equivalence there is a [math]C_d[/math] with
[eqn]\|p\|_1 \leq C_d \|p\|_2[/eqn]
then
[eqn]|p(0)| \leq \|p\|_1 \leq C_d \|p \|_2[/eqn]

>>16129114
That also works.
Working with the seminorm just complicates things if you are trying to bound the seminorm from below so you still get the part of the inequality that you need in this case.

Hey /mg/cels, user of a different /sci/ general here. Where do you think you would go if the email verification system that /biz/ has right now gets introduced site wide?

I know that this Neumann series
[eqn]
\mathbf{x} = \left(I + \int_{t_0}^{t}\mathrm{d}\xi_1A(\xi_1) + \int_{t_0}^{t}\mathrm{d}\xi_1\int_{t_0}^{\xi_1}\mathrm{d}\xi_2A(\xi_1)A(\xi_2) + \cdots\right)\mathbf{x}(0)
[/eqn]
converges provided that
[eqn]
\int_{t_0}^{t}\mathrm{d}\xi \rvert\rvert A(\xi)\rvert\rvert < \infty
[/eqn]
But I have a two questions:
Integrating the norm from [math]t_0[/math] to [math]t[/math] mean that I'm analyzing the convergence up to a time [math]t[/math]?
And what is the best norm to use here?

>>16129105
>>16129125
equivalence of norm trivially means seminorm is bounded above by norm
>>16129326
yes
operator norm

>>16125718
Thanks a a lot for the tips, I am digging into it, liking it a lot so far. Also, I'm looking for more stuff computer science related. Knuth stuff is pretty good, but I imagine you come from a field more related to math given your recommendations. What about discrete math? Do you happen to have material on it? I've read and did a bunch of exercises from "Discrete Mathematics and Its Applications" by Rosen and it was pretty cool.

Kind of a very stupid "google it" question, but since people here absolutely know more than me, i guess asking for guidance isn't that bad.

I have been struggling with Calculus III in college recently and i'd like some books recommended so i can study the subject. I think i'm losing the point of studying math and that has been adding obstacles in my understanding of the subject. Tips on studying Calculus III + Differential Equations are welcome.

>>16129440
MIT OCW

>>16129440
unironically stewart textbook + looking up lectures (MIT OCW/youtube) when needed. calc 3 is about doing problem after problem after problem until you know it. same with a basic diff eq class (although diff eq is less useful than calc 3 moving forward)

also phd student here ama

I'm looking for examples of pairs [math](G, \Gamma)[/math] where [math]G[/math] is a locally compact 2nd countable group, [math]\Gamma[/math] is a lattice in [math]G[/math], and [math]\Gamma[/math] has infinite center.

Trivial example: [math]\Gamma = \mathbb{Z}^n[/math] sitting inside [math]G = \mathbb{R}^n[/math].
Less trivial example: [math]\Gamma = \begin{pmatrix}
1 & \mathbb{Z} & \mathbb{Z}\\
0 & 1 & \mathbb{Z}\\
0 & 0 & 1
\end{pmatrix}[/math] sitting inside [math]G = \begin{pmatrix}
1 & \mathbb{R} & \mathbb{R}\\
0 & 1 & \mathbb{R}\\
0 & 0 & 1
\end{pmatrix}[/math].
Other examples: go go go!

>>16130007
extend a lattice in a noncompact center of a Lie group to a full lattice

Is it possible to get onto a Pure Maths PhD with a 2:1 from a good, not great UK university?

>>16129424
>operator norm
If in a certain interval the largest eigenvalue is 1, does that mean that the series converges as long as [math]t[/math] doesn't approach infinity?

>>16130066
It's not worth it to get into hard sci/tech PhD unless you are atleast above average (preferably even better) in your field.

Is there some kind of discrete version of differential geometry that works on polygonal chains in R^N? I've got a dataset of about 17000 of these in R^5 (pedestrian routes around a national park with three variables) and I need to figure out a discrete 5d equivalent of the torsion of a curve in 3d space and then classify them according to it.

>>16130072
need a bound on whole interval, and operator norm and max of spectrum not the same

>>16130665
>need a bound on whole interval
You mean that I need to have a bounded interval?
>operator norm and max of spectrum not the same
I wasn't clear, sorry. I meant if the largest eigenvalue of [math]A^HA[/math] is 1

>>16130671
need "bound": norm <= 1 or whatever
on whole integration interval. or it could be getting infinite in part of it... bye bye

AHA eigenvalue ok if finite dimension space. In Hilbert space eigenvectors can't always exist, "Spectral theorem" wikipedia "bounded self adjoint operator" section

>>16113803
For integer n≥3 , is there always a prime p and integer k>0 such that [math] p^k < n < 2p^k [/math] ?

>>16113803
For every integer n≥3 is there a prime p such that p < n < 2p ?

>>16130737
bertrand postulate

>>16130737
Are you sure that's what you meant to write? n = p+1 is a trivial solution

>>16130759
reading glasses

>>16130767
reminder to self, don't post after drinking

>>16130737
Nvm, this is apparently just a consequence of Bertrand's postulate / Chebyshev's theorem

>>16130782
To follow up, I was looking at this problem:

>For which positive integers n is 1 + 1/2 + 1/3 + ... + 1/n itself an integer?

I think using >>16130737 one can show the answer is only n=1 .

However, does anyone know a solution that doesn't use Bertrand's postulate?

>>16130787
power of two, n/2 < 2^k <= n
sum = a/(2^(k-1)b) + 1/2^k, b is odd

>>16127007
Probably category theory.
But even "clean" notation fields looked awful back in typewriter times. Unfortunately, many niche fields rely on typewriter scans.

>>16131053
this isn't a typewriter scan retard.
I wonder how someone could be smart enough to research model theory but incapable of telling apart a typewriter scan from bland text without didactic bolding and good looking fonts
typewriter scans have the formulas written by hand, things used to be 100000x worse than picrel

why do algebraic topology papers always deliver the most minimal unambitious results ever? every math field seems to require a more contrasted imbalance between the apparent strengths of the aassumptions and the conclusion, but algebraic topology papers just list 6 pages of assumptions to deliver unbelievably niche resultd

>>16127007
Set theory.
Model theory has nice notation because set theorists deal with the infrastructure.

I'm having trouble proving that for [math]p,q \in \textbb{N} \text{ and } n_i \in \textbb{R}^2[/math] if [math]p\leq q \then \sqrt[p]{\sum n_i^p} \geq \sqrt[q]{\sum n_i^q}[/math]

God I hope I typed that right.

>>16131643
Ah fuck, I meant the naturals for p and q and the positive reals for the n

>>16131643
can assume p=1 by change variable
then it comes from convex function f(x)=x^q

>>16131675
I'm a dumbass can you elaborate?

>>16131683
0

>>16131683
xi=ni^p
q'=q/p>=1

itbecomes
(sum xi)^(q') >= sum xi^(q')

f(x)=x^(q') it becomes
f(x1+...+xn)+f(0)+...+f(0)>=f(x1)+...f(xn)
Jensen inequality for convex function

>>16129227
mathchan of course

>>16129227
back to mathstackexchange

new thread, please