/sci/ - Science & Math

>>15404877

>staging from page 6
desperate for attention
why is /mg/ so slow and unpopular? we have tons of math threads on /sci/ and yet /mg/ takes two weeks to autosage, why is that? are mathtrannys too desperate for internet dopamine to post in the general?
/sfg/ is just the opposite, they rarely post outside their general and the 404 daily, because they come to /sci/ to discuss a topic they're genuinely interested in with like-minded individuals.

>>15404888 (nice trips)
I raise you this.

I know it's the math general, but since mechanics is really just applied mathematics I'll ask here. Which is the better book between the two?
>Kleppner/Kolenkow - Introduction to mechanics,
>Morin - Introduction to classical mechanics.
They both seem similar in terms of content, but Morin's problems are perhaps a bit more difficult.

I hate this effect when wearing strong glasses

>>15404877
Computer science is shit. They think they're so smart but they can't even count past 1. As a mathematician who wields a wide array of numbers at will, my only conclusion is that I am a better person than old man Knuth

>>15404974
that's topological

>>15404905
The sad reality is that math is a topic too complex to be discussed on 4chan

Not sure if this is a bit too abstract for a problem for /mg/, but fuck it, I'll post it anyway because I like the answer, so here's a challenge for you fine folks.
Written in decimal, what proportion of integer powers of 2 have a 1 as their first digit? No leading zeroes allowed.
If this turns out to be too easy, or if you want more, here are a couple of generalisations. I will openly admit that I don't know the answer to the latter, but maybe someone here can work something out.
>What if we ask the same question, but with an arbitrary integer in place of 1?
>Is there a similar argument for powers of 3? Of 4? In general?

reposting just to clarify and reword a bit.
Not sure if this is a bit too abstract for a problem for /mg/, but fuck it, I'll post it anyway because I like the answer, so here's a challenge for you fine folks.
Written in decimal, what proportion of natural powers of 2 have a 1 as their first digit? No leading zeroes allowed.
If this turns out to be too easy, or if you want more, here are a couple of generalisations. I will openly admit that I don't know the answer to the latter, but maybe someone here can work something out.
>What if we ask the same question, but with an arbitrary positive integer in place of 1?
>Is there a similar argument for powers of 3? Of 4? In general?
I guess you can also consider it for an arbitrary base instead of decimal but that one's pretty trivial compared to the others

>>15405003
>As a mathematician who wields a wide array of numbers
real mathematician dont count retard

>>15405037
>>What if we ask the same question, but with an arbitrary positive integer in place of 1?
I'm pretty sure for 0 this is a known open problem

>>15405003
Electrical fags will understand.

Hey hey /mg/! I hope everyone is doing fine. Here is a new problem. I hope if anyone attempts it, they find it fun. Unfortunately I am very busy today so my replies might come rather late. I apologize for this in advance. Please let me know if anything needs clarification. As always I appreciate anyone who attempts, successful or not. Good luck!
>>15404974
hahaha, now that you point it out, it does look a bit strange.

>>15405037
You just want to find what proportion of n satisfy:
10^k <= 2^n < 2*10^k for some k.
Equivalently k <= n*log(2)/log(10) < k + log(2)/log(10)
If you assume n*log(2)/log(10) is evenly distributed mod 1 then the proportion is just log(2)/log(10).

heres another garbage math thread posted outside of /mg/
>>15405242
more evidence that mathtrannys are worthless human filth

Why are mathfags like this?
https://www.youtube.com/watch?v=xPzR_D9qKeo

>>15405343
Fuck me. I use to watch papa flammy's iNteGeRaL vids everyday, but they became too repetitive and he stopped making them.

Supppose I have an integral $I_n: \int r^n \cdot g(r) dr$ where $g(r)$ is some function of r.

What does it mean to express I_n through I_(n-1) and I_(n_2) and I_(n_3).... through IBP?

I see that by setting $u = r^n$, applying k successive integrations by parts will reduce the power by $r^{n-k}$

>>15404905
i'm not even gonna check sfg because i know why this is the case
sfg is filled with clueless retards (as is typical for /sci/) that can speak like they have authority because they can't immediately be exposed as retards and there's always something to post about because there's always spaceflight news everyone cares about.
Math does not have this problem, but /sci/ posters don't change, so we only get undergrads that think they are smarter than they really are (you) and will spend the whole thread kvetching about the most irrelevant shit like what undergrad analysis book is best or why the foundation of math is fucked without ever actually posting any math because it might expose you as a retard and that's not a risk your ego can take. All the educated posters have left long ago, because mathematicians are busy and other places on the internet have a higher people who know something about your work to undergrads ratio if you ever accidentally want to discuss your work in your limited free time, but nobody in your cohort wants to talk to you anymore. The only other posters are
>undergrads asking for homework help because they couldn't find /sqt/ in the catalog
>combinatorics puzzle anon, god bless his soul for actually posting math
>terminally online schizophrenics
>occasionally someone who actually knows something about math foundations that make an interesting post
And then the final problem is that math news is the most boring shit that sub 1000 people worldwide can understand and sub 100 care about of which there are zero(0) are in this thread.
>>15405277
clearly not something that belongs in the math general or that was asked by a mathematician, but is more suited as a generally retarded question to the already shit science AND MATH board
kill yourself or post anything that isn't incessant whining

How do I calculate how many times [something] might/probably happen if the chance is X%, and I do the experiment A times?
I can find online calculators for the chance of [something] happening at least once if you do the experiment A times, but I can't find one for this kind of probability.

>>15405619
>Implying you're too good for the board, and not just a downs syndrome-esque nigger retard analysis textbook hopper trying to be trendy with the cool kids.

You will never reach volume II of any textbook. You will never pass Rudin, you will never meet the prerequisites for any European math course either. Instead you will sit in the shadows making mental notes of the newfags that pass here in the hopes that you may perchance be smarter than one of them, well eventually you hope, because it hasn't happened yet.

>>15405012
heres another common example of an atrociously lame thread posted by a mathtranny which should have been contained to this general, but wasn't.
whats wrong with you people? why do you insist on being, by far, /sci/ worst citizens?

>>15405628
https://en.wikipedia.org/wiki/Binomial_distribution

>>15404877
I want to get back into doing math what online courses should I take for beginners?
I want to start from the very beginning and move my way up.

>>15405722
>implying..
i wasn't implying anything beyond what i said, no mathematician worth anything posts here. If i was good at math i wouldn't be here either
>hurr durr muh anal books
i don't feel any need to return to undergrad analysis, nor do i care what the current day undergrads read. But keep seething and filling the thread with worthless spam, at least you'll bump up the post count and then we compare better to sfg right? Don't mind me thoughever, I'll just sit around in the shadows in the hopes that there will eventually be a good post, well eventually i hope, because it hasn't happened yet.
>>15405726
>why do you insist on being, by far, /sci/ worst citizens?
hard to do worse than (you) and i'm trying

Filtered...

>>15405758
Strewth, just taking the piss me. If I didn't come here to laugh at you all, what would I be doing?
>>15405762
What are the other parts, is this teaching the exponential first with rationals and then extending it to the reals?

>>15405803
It's from Rudin.

>>15405107
Every k-subset either contains n or doesn't contain n. There are [math]f(n-1,k)[/math] many that don't contain n and [math]f(n-2,k-1)[/math] many that do contain n. So
[eqn]f(n,k) = f(n-1,k) + f(n-2,k-1) [/eqn]
Since
[eqn] {n - k + 1 \choose k} = {(n-1) - k + 1 \choose k} + {(n-2) - (k-1) + 1 \choose k-1}[/eqn]
And
[eqn]f(0,k) = \begin{cases} 1 & k = 0 \\ 0 & k \neq 0\end{cases} = {0 - k + 1 \choose k} \\
f(1,k) = \begin{cases} 1 & k \in \{0,1\} \\ 0 & k \not \in \{0,1\} \end{cases} = {1 - k + 1 \choose k}
[/eqn]


We get by induction over n that [math]f(n,k) = {n - k + 1 \choose k}[/math] for all n,k.

For part b) just sum the reccurence for k=0 to n
[eqn]\sum_{k=0}^n f(n,k) = \sum_{k=0}^n f(n-1,k) + \sum_{k=0}^n f(n-2,k-1)[/eqn]
Now remove the summands that are zero
[eqn]\sum_{k=0}^n f(n,k) = \sum_{k=0}^{n-1} f(n-1,k) + \sum_{k=1}^{n-1} f(n-2,k-1)[/eqn]
then shift the last index by 1
[eqn]\sum_{k=0}^n f(n,k) = \sum_{k=0}^{n-1} f(n-1,k) + \sum_{k=0}^{n-2} f(n-2,k)[/eqn]
Again it's easy to see that for n=0 and n=1
[eqn] \sum_{k=0}^n f(n,k) = F_{n+2} [/eqn]
so by induction it's true for all n.

>>15405856
Isn't Rudin the most popular book on Analysis? Why isn't there a single correct solution to this online?

Why can't people outside of France use the standard Bourbaki terminology/notations? Are you all mentally challenged ?

>>15405933
Because it's fucking trivial

[eqn]b^x b^y = \sup\{b^{r+s} | r,s \in \mathbb{Q}, r\leq x, s\leq y\} \leq \sup\{b^{r+s} | r+s \in \mathbb{Q}, r+s \leq x+y\} = b^{x+y}[/eqn]
since your taking a supremum over a bigger set.

Now using this inequality you can get the other direction too
[eqn]b^x b^y = b^{x + y - y} b^y \geq b^{x+y} b^{-y} b^y = b^{x+y}[/eqn]

If I'm self learning how do I know if I should take a computation-heavy approach to calculus or a proof based one?

I took a quick glance at a mainstream calculus book and I had the impression that the whole thing was
>list definitions
>maybe one or two examples
>dump exercises
It doesn't seem like a good way to get a solid foundation of what is possible with each concept and tool.

>>15405998
Where do you get the last inequality from?

>>15406018
You use the first inequality to get the second too.
Maybe rename the variables to see it better
Let
u = x + y
v = -y

[eqn]b^{x + y - y} = b^{u + v} \geq b^u b^v = b^{x+y} b^{-y} [/eqn]

>>15406027
How do I prove b^-y b^y = 1 for irrational y?

>>15405107
Please italicise all the variable names next time.

For [math]x>1[/math] a rational number, can [math]\exp(x)[/math] or [math]\ln(x)[/math] be rational too?

If no - why?

>>15404905
Because the audience of this general is comprised of undergrads looking for homework help, retards misinterpreting proofs, /sci/ elitists and a particular attention seeking tranny (and its orbiters).
This anon ( >>15405010 ) sums it up quite nicely.

>>15405277
That's a cirnotranny, he doesn't actually care about what he's asking. You can see a lot of them (him?) on /bant/.

>>15405037
>>15405141
pic related, from Einsiedler and Ward, Ergodic Theory

Are there any interesting results about [math]SL_n(\mathbb{Z}_p)[/math] ?
Any books treating the subject? (also interested in other linear groups over [math]\mathbb{Z}_p[/math] or a p-adic field)

help, how do i know which parts of
−3(x+4)(x−2)
are corresponding to which parts of
f(x)=ax+b

like how do i know which of these is "a" and which of these is "b"

>>15406249
−3(x+4)(x−2)=-3x^2-12x+24 is not of the form ax+b

what is an extensive set of primitive notions for an axiomatic formulation of ZFC of your choice

>>15406447
I don't know.

Reminder: Tao's Analysis is published by HINDUSTAN!

>>15404877
What is 1 + 1

>>15406481
>t. filtered

Have any of you been motivated to study further by a single question you've thought of that is far beyond your current level?

>>15406564
Yes, I want to know whether it's worth it to keep going self-learning math. I have an inkling of the possible outcomes, but anything could happen.

>>15405933
Rudin is a meme, Americans just like to name drop it because they think it's hard. Of course it's just really basic things that were taught to 14 year olds in the Soviet Union.

Want to get into math.Where do i start as a complete beginnger?

>>15406731
If you are gonna be a condescending asshole, the least you could do is answer the question (especially since it is so easy). Otherwise, please do not reply to my post.

>>15406731
How to Prove It, by Vellman. Then Calvin Long Number Theory would be the appropriate follow up. You may also try A Course of Pure Mathematics if you want something less one dimensional.

>>15406770
>>15406788

>>15406788
You've already been given the answer, if it doesn't make sense, consider some other textbooks.

>>15406793
It's an incomplete answer. It relies on unproven assumptions.

What's the value of 1/4(1/9+1/16+...)+1/9(1/16+1/25+...)+.....??

>>15406801
I know you're an American, but there's really only so much hand holding that should be allowed.

What are the unproven assumptions?

>>15406811
>>15406031

>>15406817
If you've proven this for rationals, observe what happens for rationals that become arbitrarily close to your real number.

>>15406825
What happens?

>>15404936
Morin's content is more advanced too. K&K could work for a smart/well-prepared freshman, but not Morin. Hence why K&K is occasionally used in honors freshman mech classes while Morin is used for sophomore/junior-level mechanics

>>15406837
It'll be quicker to just post the solution, and then you can see where you're going wrong.

>>15406564
Yep, I keep encountering liouville numbers in some niche functional analysis topics and want to figure out what their deal is

>>15405107
For part a), add 0 and n+1 to the set. The k included elements of the k-subset partition the set into k+1 parts of non-included elements.
f(n,k) counts the number of ordered partitions of n+2-k into k+1 parts.
This is represented by:
[x^(n+2-k)] (x/(1-x))^(k+1)
= [x^(n+2-k)] Sum[(m C k)*x^(m+1), m>=k]
= (n+1-k C k)

For part b) sum over k in the range. We can just let k sum to infinity since the terms are zero for all k > ceil(n/2).
Simplify [x^(n+2-k)] (x/(1-x))^(k+1) = [x^(n+3)] (x^2 / (1-x))^(k+1).
Sum[f(n,k), k>=0] = [x^(n+3)] Sum[(x^2 / (1-x))^(k+1), k>=0]
= [x^(n+3)] (x^2 / (1-x)) / (1 - (x^2 / (1-x)))
= [x^(n+3)] x^2 / (1-x-x^2)

Now from the definition of fibonacci numbers F(n+2) = F(n+1) + F(n), F(0)=0, F(1)=1, you can construct their OGF:
g(x) = Sum[F(n)*x^n, n>=0].
Multiply by x^(n+2) in the recurrence and sum n from 0 to infinity to get:
g(x) - F(1)x - F(0) = x*(g(x) - F(0)) + x^2 * g(x).
Solve for g.
g(x)*(1-x-x^2) = x.
g(x) = x/(1-x-x^2).
F(n+2) = [x^(n+2)] x/(1-x-x^2) = [x^(n+3)] x^2 / (1-x-x^2) which is in agreement with the sum in question.

>>15406235
I don't know any sources or results but am also interested in this topic. Let me know if you find anything.

>>15407035
I remember Zorich just tossing that into his question set for a preliminary chapter.
>Draw a circle
>Draw an oval
>Now draw the fucking bird.

>>15407119
BTW, to answer this question, for which the material isn't in the textbook, I took a math detour and purchased Hardy and Wrights Number Theory textbook. It's a good book, and it all made sense, but for some reason a large dose of number theory just makes me want to leap off a tall building.

In the history of /sci/, has there even been an undertaking by anons to write a collaborative textbook? Of course the initiative would be lost in under two weeks and the project condemned to failure, but I'm still curious whether an attempt was made.

If not, I'd love to see a "Retards guide to ..." series of books, that despite the title, are actually better than all of the other choices. Imagine reading Rudin (or Zorich etc), and being able to see notes saying (241 anons got stuck here. Here's why...)

>>15407166
Same fagging some more, I recalled that while I liked the Hardy book, the 6th edition has an atrocious number of LaTeX errors, which makes me wonder if the lazy faggots at Cambridge even bothered to proof read it.

If you choose this book, it's essential to get the list of errata, which is 8 pages long: https://www.maths.ox.ac.uk/system/files/attachments/HWerrata.pdf

I picked this book from name recognition, and it just seemed to be the one that everyone recommended at the time. If there are better options nowadays (at a comparable level), please let me know.

what is the motivation behind defining the cardinal number 1 as the class of all unit classes

>>15407189
>>15407189
this sounds like a really good idea but i don't think there's enough smart people here to actually take it all the way through and there's too many subjects to choose from
i suppose we could start with the most obvious ones (i.e. the Incel guide to Calculus or the Incel guide to Linear Algebra), though.

>>15407224
I would contribute if we write it. We just have to come up with a good contribution system. It's not good if anyone can come and randomly edit any part you want. We could use something like PRs in github. Any tools that would let us accomplish this?

>>15407224
There are already numerous free resources online for this like libretext.
>>15407189
Only point would be to be a shitposting/meme "textbook" and nobody would likely care anyway given how depopulated /sci/ is of actual people interested in science.

>>15407249
Don't underestimate the power of weaponized autism. BTW, a lot of free book are out there, but it's usually one man's contribution. I know there's proof wiki, but for some reason it angers a lot of mathematicians.
>>15407243
You need a set of "trusted" editors, but anyone should be able to add a comment, or suggestion. Git is the perfect collaboration tool for this.

Universal set

If I define the universal set as [math]U=\left\{ x|x\neq U \right\}[/math]
then would that be okay, and would it exist in ZFC

I would be the world's top mathematician if I could time travel to 500 BC

>>15407287
stop reading pajeet tutorials on set theory, no its not ok, that x would not include the U as you specified but it says nothing of the powerset of U, which is not U but of which U is a subset of

>>15407304
fact: no you wouldn't as they would reject half your notions such as 0 and negatives numbers and you would need to enter philosophical arguments on the foundations of mathematics, which they had hefty experience of and which you likely have no experience of (just an assumption, maybe you do have experience of it but it's uncommon for mathematicians)

>>15407281
The main problem is that someone needs to be in charge of the project and usually nobody wants to be the one in charge.

>>15407249
>There are already numerous free resources online for this like libretext.
but no one that will teach you how to calculate the rate of change in the crime level with regards to the change in the negro population

Has anyone proved an intrinsic advantage in certain languages for mathematics? Obviously we have heard the stories of tribal Africans without the necessary vocabulary to describe concepts such as "half-way" up a tree, and even the relatively sophisticated Romans had a cumbersome notation for calculation, but what about amongst the languages presently used? Is there an advantage to French, German, Russian, or say Chinese over English when it comes to expressing complex mathematical concepts?

>>15407281
>Don't underestimate the power of weaponized autism.
Counterpoint: /sci/ has allowed itself to be surrendered to 99% /xpol/ posting. The "power" seems fairly lacking if I'm honest, especially around here most of the time, given it can't even counteract the deluge of rather deliberate board derailing.

>>15407484
Leibniz liked French cause of its specificity and simplicity

>>15406865
That solution and every solution online is wrong or at least incomplete because it changes the definition assuming strict inequality, without proving they are equivalent. What do you think I have been telling you for so long?

>>15407189
There's no reason to do. We already have amazing books written by world class researchers. I think a better thing to do would be to rewrite old shitty scanned versions of classic book in LaTeX , like Gutenberg did for Rudin.

>>15408249
Like Gutenberg did for Hardy, I mean.

>>15408249
It should be done in true slacker 4chan style, just by copy and pasting good explanations from a variety of books. No citations needed, true Einstein style. In fact you just need to cite the collective work (à la Bourbaki) under the name Shlomo Goldenstein, and it's a talisman of protection against takedown requests.

>>15405925
Hello anon. Brilliant work! I don't really have much to say, your solution to both parts is just clear and concise. Definitely a textbook worthy solution in my opinion. Thank you a lot for this! I hope you have a very pleasant day.
>>15406077
Hello anon! Nice solution! Thank you for making it into a pdf, it looks really good!
As for your solution it's unique so nice job on that! I especially like the first part where you found a) by using that inequality, I think that's amazing and very creative. How you tackled b) is also pretty nice. I think it's very understandable, your writing is just good. Thank you a lot for your solution and for making it a pdf. I hope you have a very pleasant day!

>Please italicise all the variable names next time.
You're completely right, I should have done that. I'll try not to forget the next time! Is pic related ok?
>>15407051
Hey anon! First things first good job on a), it's rather unique and concise. For part b), the same goes. I don't think I've seen anyone use generating functions for this problem(not that I've seen a lot of solution). However I must take a look at b) later at more detail. But of course it's right, I just mean that I need to look at it again with more care and write it down myself.
So thank you a lot! Your solution is new and I absolutely appreciate it. I hope you have a pleasant day
I must apologize to all three of you due to the unhelpful and generic nature of my comments. I'm sorry I didn't go further than just telling you mundane praises, I simply can't think of anything else to add. But I do mean them and I'm really grateful to all of you for your solutions and the time you dedicated to them. Peace!

>>15408435
>Thank you for making it into a pdf
To be fair, writing it on 4chan is the harder thing to do.

>I'll try not to forget the next time! Is pic related ok?
Yes good job anon-kun desu ne.

I'm kind of dumb when it comes to maths but I was reading about standard deviations.
what is the mathematical reason for why you use divide by n for full populations and divide by n-1 for population samples?
I get the basic idea of it since it's only a sample but why only -1 not -2 or other figures? Is it just that it's a simple statistical technique?

>>15408559
It has to do with degrees of freedom. You wouldn't be subtracting by 1, if you used true mean in the formula instead of sample mean. But if you're using sample mean, the sample variance is always going to be smaller, because the data is always going to be closer to the sample mean. So you effectively have n-1 observations instead; you're losing an observation by using it to calculate the sample mean. Think about it: if you have the sample mean, and all the data except for one, you can find out the missing data through the sample mean. This applies to any estimate calculated using other estimates. If you have some estimate that uses k other estimates in calculating it, then you only have n-k effective observations.

https://en.m.wikipedia.org/wiki/Bessel's_correction
https://en.m.wikipedia.org/wiki/Degrees_of_freedom_(statistics)

>>15405498
bump

>>15408559
>what is the mathematical reason for why you use divide by n for full populations and divide by n-1 for population samples?
As you might guess reading something like https://en.wikipedia.org/wiki/Standard_deviation#Corrected_sample_standard_deviation
It isn't a simple explanation. Probably it's best understood graphically or by plugging in dummy numbers. The summary fact is that 1/n has a functional downward bias relative to the actual data and 1/(n-1) fixes that relative to low values of n sufficiently. For large values the difference is not meaningful but for something like n=10 or lower it can have statistically significant effects you do not want.
>I get the basic idea of it since it's only a sample but why only -1 not -2 or other figures? Is it just that it's a simple statistical technique?
It's purely a matter of what way a given operation ends up biasing the approximation you're trying to get. Same goes for handling square roots or any other operation.

Level of difficulty here depends on how deep you want to go. In statistics this is covered under bias of estimators and estimation theory and the like, more broadly under measure theory
https://en.wikipedia.org/wiki/Bias_of_an_estimator
https://en.wikipedia.org/wiki/Estimation_theory
In general mathematics the broader topic is approximation theory and approximation error
https://en.wikipedia.org/wiki/Approximation_theory
https://en.wikipedia.org/wiki/Approximation_error

It's hard to guess in what way you feel you're not getting it so I gave you some leads in case you wanted to find further examples of approximation error. You can always use some random numbers and get example figures, or graph the functions, to help get a sense of what it practically amounts to.

>>15408590
The degrees of freedom argument is a misleading rationalization.
https://en.wikipedia.org/wiki/Bias_of_an_estimator
should help explain what is being optimized.
The n-1 only happens when you try to minimize the mean-squared error.
Choosing some metric other than that will give a different correction.
There are also correction factors for higher order moments that indicate it isn't about degrees of freedom.
https://mathworld.wolfram.com/h-Statistic.html

>>15407317
how about this: [math]U:=\left\{ x|\left( \forall n,m\in \mathbb{N} \right)\left[\left\{ \mathcal{P}^{n}\left( U \right) \right\} ^{m}\notin U\right] \right\}[/math]
where [math]\left\{ \mathcal{P}^{n}\left( U \right) \right\} ^{m}[/math] is [math]\mathcal{P}^{n}\left( U \right)[/math] nested m times, [math]\left\{ \mathcal{P}^{n}\left( U \right) \right\} ^{0}:=\mathcal{P}^{n}\left( U \right)[/math], and [math]\mathcal{P}^{0}\left( U \right):=U[/math]
As an example: [math]\left\{ \mathcal{P}^{n}\left( U \right) \right\} ^{3}=\left\{ \left\{ \left\{ \mathcal{P}^{n}\left( U \right)\right\} \right\} \right\}[/math]

>>15405754
Anon, you should in my opinion download a precalculus book such as that of Axler and work through it. But if you want an online course then take a look at Khan Academy.
>>15405619
>combinatorics puzzle anon, god bless his soul for actually posting math
As someone who's not even in undergrad yet, I am flattered.
"Combinatorics puzzle anon" is a title I'll carry with pride from now on, thank you.
>>15408444
>To be fair, writing it on 4chan is the harder thing to do.
That's true, yeah. So out of curiosity, what software did you use to make that pdf?
>Yes good job anon-kun desu ne.
Yay :D

>>15405619
One day [math] \text{combinatorics anon}[/math] will make their last post...

>>15408844
>So out of curiosity, what software did you use to make that pdf
[math] \mathrm \LaTeX [/math] what else? Unless you mean the software or distro. Then MikTeX + XeTeX + VSCode.

>>15406865
>>15408244
Never mind, I read it again and it actually does the same thing I did i.e., use the solution from the next exercise to show they are equivalent. At least this one bothers to show they are equivalent, unlike every other solution on the internet. Where did you get this from? But I am wondering how are you supposed to do it without using that, unless that is what Rudin expected you to.

>>15409008
>combinatorics anon
Sorry but I prefer the title Combinatorics Puzzle Anon, as that can be truncated to CP anon, which is hilarious and fits with people who present themselves as anime avatars...

>>15408649
>>15408590
Thank you

>>15409021
>baudrillard
>transferir.jpg
brasileiro safado

>professors expects students to keep up with reading 5 chapters of gallian a week, in addition to his extremely dense and error filled quasi-textbook notes, and a few additional references for some assigned homework problems that cannot be solved with that is in the previous two already
Its a no name bottom of US rankings school, why are people like this? I know I'm an absolute idiot which is why I'm here and will never do a graduate program, but I'm not stupid enough to believe that even students in top schools are covering this much content this fast. Am I wrong in thinking this? I get the sensation he just wants to make us feel bad.

so basically I'm a normie when it comes to anything pass high school level math, only understanding maybe a tenth of Numberphile videos nowadays

but i'm curious and haven't found an satisfactory answer yet to this: how far have we exhaustively checked for primes? i.e. we know which of every number between 1 and something like 300 digits worth of numbers are prime or not, but what's the current limit? it seems like the main interest in primes seem to be like tits, bigger is better. i have an understanding that trying to search inbetween say, the last two largest mersenne primes would prob take longer than the heat death, but perhaps searching between the other types of prime would prove fruitful? related to that is there perhaps a list of the size of each said prime types? that way if again, a total exhaustive search between them proves impractical, chunking them up via other prime types until it does become practical could be an approach?

>>15410092
>but i'm curious and haven't found an satisfactory answer yet to this: how far have we exhaustively checked for primes? i.e. we know which of every number between 1 and something like 300 digits worth of numbers are prime or not, but what's the current limit?
It's an interesting question, but one which you're never going to find a satisfactory answer for. If we had a definitive answer, some yutz out there would immediately render it null and void by going through until he found the next one.
I couldn't find much from the past decade, but I did find a couple of results from ~2011 or so suggesting that the largest prime whose position in the sequence had been determined unconditionally (e.g. no Riemann assumption) is somewhere on the order of [math]10^23[/math]. Obviously it's been more than a decade since then, but to give a rough ballpark of an idea...

>>15410151
Sometimes my brain just stops working. meant [math]10^{23}[/math] obviously

>>15405007
you posted this purely to piss me off didn't you

Ayo lemme get ur recs for a bussin number theory book, low key dead ass I want the good shit, like with applications but real stuff not bullshit fr fr. Lotta stuff out there can be pretty mid so gimme your best stuff with a bit of drip. I can handle any difficulty not a flex, no cap.

>>15410092
>how far have we exhaustively checked for primes?
I'm not sure why it matters?
>but i'm curious and haven't found an satisfactory answer yet to this: how far have we exhaustively checked for primes? i.e. we know which of every number between 1 and something like 300 digits worth of numbers are prime or not, but what's the current limit?
In terms of algorithms and current computing power? Proven? Probable? For published primes likely it's application driven e.g. probable primes and primality testing algorithms. Technically therefore the largest prime is going to be a probable prime that happens to be a true prime. You don't seem to know just how vague that question is so I'm trying to explain it.
https://en.wikipedia.org/wiki/Probable_prime
https://en.wikipedia.org/wiki/Provable_prime
https://en.wikipedia.org/wiki/Primality_test
>But perhaps searching between the other types of prime would prove fruitful?
https://en.wikipedia.org/wiki/Prime_gap
https://en.wikipedia.org/wiki/Prime_number_theorem
As you learn about mathematics you'll get to endlessly enjoy discovering you're late by a couple centuries or decades. Right up until you're doing something so ridiculously esoteric you only find out you're 120 years late after having nearly completed the work for publication only to start all over again. It's fun. No really. My favorite thing.

On the other hand if you're wondering if there's some implicated way to find consistency in the patterns of primality, that is easily shown to be impossible. By definition of what a prime is for a linear numerical structure like whole numbers, having no whole number factors greater than 1 and itself necessarily means it will also be asymptotic and neverending with infinitely many prime gaps of infinitely growing inconsistently different size. Already explained by the fucker who took all the good ideas (Riemann).
https://en.wikipedia.org/wiki/Riemann_zeta_function
https://en.wikipedia.org/wiki/Riemann_hypothesis

>>15409678
Gallian algebra is know to be very elementary and chatty. A single part has as much material as a single chapter in standard algebra books. It is better if you read something else like Vinnberg, so you don't have to waste time reading Gallian's superfluous information.

>>15410279
>Vinnberg
I also like this book, clean and straight forward Russian style. Chatty where it needs to be, but information dense. The Galilean book that I had to purchase when I was at school was full of filler to raise the cost of the book.

>>15410182

>>15410298
Oof this ain't bussin yo, this Tooker guy be sending me! ngl senpai feels like I'm being a memed

>>15408435
>I don't think I've seen anyone use generating functions for this problem
I'm just the same generating function anon.
Using them to solve recurrences like the fibonacci numbers is what originally got me interested in them. The combinatorial interpretations came later.
If part b) asked to just find the sum without mention of what it should be equal to then I wonder if the other anons would have had a harder time.

An interpretation for the generating function for the sum is pretty simple.
The sum obviously counts the number of subsets of {1,2,...,n} that contain no consecutive elements.

We have the sum equal to [x^(n+1)] 1/(1-(x+x^2))
Expand as a geometric series.
1 + (x+x^2) + (x+x^2)(x+x^2) + (x+x^2)(x+x^2)(x+x^2) + ...
Each (x+x^2) factor represents an OR decision (do the thing of weight 1 or do the thing of weight 2).
Each (x+x^2)^k term represents making a sequence of exactly k of these decisions.
Extracting the coefficient of x^(n+1) means counting all ways of making these decisions that have weight n+1.
Finding meaning for what the decisions are and what the weights mean is the creative part that is more of an art.

Here is my interpretation.
List the elements of {1,2,...,n} in ascending order. Add an extra element at the end (just as a placeholder). You have n+1 elements.
Now to construct any valid subset, start at the beginning of the list and just make a sequence of choices to either exclude the next element or include the next element and exclude the one after it.
These choices decide the fate of 1 element or 2 elements respectively. The placeholder at the end is just to make things work in this construction (including n requires excluding the placeholder, if you have already gone through 1,..,n then the only choice to make for the placeholder and have total weight of n+1 is to exclude it). The placeholder is never included.

>>15410344
If you want to keep track of the number of subsets by the number of elements, you can append an extra variable, t, to the x^2 term that represents including an element.
1/(1-(x+t*x^2))
You can recover f(n,k) from this by extracting the coefficient of (t^k)*(x^(n+1)).
You can also do things like sum over just the even k.
In this case, the generating function would be (1/2)*( 1/(1-(x+1*x^2)) + 1/(1-(x-1*x^2)) )
You get (1/2)*( F(n+2) + (floor((n+1)/3) - floor((n-1)/3))*(-1)^floor((n+2)/3) )

https://www.wolframalpha.com/input?i=Table%5B%281%2F2%29*%28+Fib%28n%2B2%29+%2B+%28floor%28%28n%2B1%29%2F3%29+-+floor%28%28n-1%29%2F3%29%29*%28-1%29%5Efloor%28%28n%2B2%29%2F3%29+%29%2C%7Bn%2C0%2C10%7D%5D

Can anyone prove/disprove the following (in fact I don't need the proof, right now I just want to know if it holds):

Let [math]F = \{(c_i, n_i)\}_{i=1}^K[/math] be a set of [math]K[/math] elements where [math]c_i >0[/math], [math]n_i > 0[/math], [math]\sum_{i=1}^K c_i = C[/math], [math]\sum_{i=1}^K n_i = N[/math].

Naturally, this set can be thought of as a frequency decompotition (i.e given some positive numbers [math]a_1, \dots a_n[/math], there are [math]n_i[/math] elements equal to [math]c_i[/math]).

One trivial observation is that [math]K = O(\sqrt{C})[/math], where [math]K[/math] is the number of distinct elements of [math]F[/math]. (since the sum is fixed, we can't have many distinct elements)

Now for the question.
If [math]M_t[/math] is the number indices [math]i[/math] such that [math]n_i\geq 2^t[/math] is it true that [math]M_t = O\left(\sqrt{\frac{C}{2^t}}\right)[/math]?
Obviously for [math]t=0[/math] it holds ([math]M_0 = K = O\left(\sqrt{\frac{C}{2^0}}\right) = O(\sqrt{C})[/math], but does this hold as [math]t[/math] grows?

This would imply the strong result
[eqn]\sum_{i=1}^K (\lfloor \log_2(m_i)\rfloor + 1) = O\left(\sum_{t=0}^\infty \sqrt{\frac{C}{2^t}}\right) = O(\sqrt{C})[/eqn]

>>15410279
Speaking of Vinberg, do you prefer it to Artin's Algebra? I have the latter but never really enjoy reading it, but everyone loves to go on about how great it is.

>>15410761
I have not read it, and as far as I know it's very weird and only covers a small part of algebra. If you want a purely undergraduate algebra book, Herstein's Topics is besto.

>>15409008
>One day combinatorics anon
will make their last post...
University entrance is in 45 days so I'll probably not post here for while when the exam is even closer. I'm sorry...I'll miss you too. But I'll probably be back :DDD
>MikTeX + XeTeX + VSCode.
Thanks! This is what I was curious about.
>>15410344
>>15410354
>I'm just the same generating function anon.
You're so based and generatingfunctionpilled.
>If part b) asked to just find the sum without mention of what it should be equal to then I wonder if the other anons would have had a harder time.
Entirely possible, I personally probably wouldn't be able to go very far. But some would have figured it out from the recurrences they found I assume.
>creative part that is more of an art.
That's something I heard before and I suppose you're right.
I have read what you've written and as always it's a joy to do so. You seem very versed with GFs and it's impressive in my opinion.
I believe you mentioned "Analytic Combinatorics" by Flajolet last thread and I've read a bit of it, and I intend to read more especially after my university entrance exam is out of the way. So thanks for all this, I've learned quite a bit from you and enjoyed doing so.

>>15411089
>University entrance
Where are you from (if you don't mind)?

>>15411129
Turkey, hence why I sometimes put fez and mustaches on the problems I make. Though I'll probably move out of the country next year to live with my uncle abroad for university at least.What country are you from(if you also don't mind)?

To not get too deviated from math, let me share a math problem that was on our university entrance exam in 2021 I believe. Keep in mind this problem was considered too difficult for the exam after it was over so it's not at all representative of a typical problem that's on it.

For a degree 4 polynomial P with real coefficients, the following holds true for all real numbers x:
P(x)>=x
and
P(1)=1
P(2)=4
P(3)=3

What is the value of P(4)?

The options: A)20 B)22 C)24 D)26 E)28

The solution isn't too hard, I think given enough time it's very doable except you have 2 minutes per question and for this I believe that's not sufficient.
Keep in mind I'm not sharing this because I think it's a fun problem, I just want to keep my reply at least a bit math related.
Answer to the problem btw:B)22.

>>15411143
another anon who got got by the lack of spoiler tags on this board

>>15411143
This seems pretty tough with only a few minutes; after looking at the question for a little bit I found three equalities and six inequalities concerning the coefficients. Are you meant to solve these or is there a technique I am missing?

>>15411147
I swear I knew they don't work because it happened in the past. I just keep doing it because then people see I tried to spoiler it and failed.
>>15411177
You're missing something and that's completely understandable. It's not a reasonable problem for the time constraint and the people meant to be solving it. So what you need to do is consider P(x)-x. As that is equal to greater than 0, it means that any roots it has need to be even. Pic related is clear enough I hope. I apologize if the picture is sideways, not sure why it does it sometimes.

>>15411177
It's just a retarded entrance exam meant to filter out people as much as possible by having a strict time limit. Every 3rd world country has something like this. However, I think it is government mandated in Turkey, like you literally can't apply to any university without it; so it's even worse.

1/4(1/9+1/16+...)+1/9(1/16+1/25+...)+...
How do I find the value of this expression??

>>15411248
By my own method I'm getting 0.1668083584351...??
Can anyone confirm with the correct method if it exists?

>>15411261
that's about the answer you should be getting, but there's a more exact way to put it
Hint: It's a modified version of the Basel problem.

>>15411248
It's

[eqn] \frac{(\zeta(2) - 1)^2 - (\zeta(4) - 1)}{2} = \frac{\pi^4}{120} - \frac{\pi^2}{6} + 1[/eqn]

>>15411143
>What country are you from
India.
I also have an entrance exam (in 2 weeks). Here is a question from it:

Let [math] a_1 < a_2 < \dots < a_m [/math] and [math] b_1 < b_2 < \dots < b_n [/math] be real numbers such that:
[eqn] \sum_1^m | a_i - x | = \sum_1^n | b_i - x | \qquad \text{for all } x \in \mathbb R [/eqn]
Show that [math] m = n [/math] and [math] a_i = b_i [/math] for all [math] i \in \{ 1,2, \dots , n \} [/math]
- Pdf anon

>>15410298
>The chart on the line segment cannot affect the line segment’s basic geometric properties!

You're not a mathematician.

>>15411696
He gave you definition that you probably didn't read through, are you retarded?

Can this actually be proven or is it just an arbitrary approximation pulled out of thin air?

>>15410279
>Vinberg & Conway
Don't care about the 'rona much but why did these two have to die from/with it? Shame.

>>15411877
There do exist a few different proofs of it, although they're not really the kind that transcribe well into a 4chan post

>Study subject
>Do all the problems diligently
>Move on to other things
>2 years goes by
>You cannot even remember the subject
Why is life like this?

>>15412051 (samefag)
BTW I posed this question to a Neurologist and she mentioned this is normal, but that it should be easier to recall much of the material you learned with a quick review, so it won't require having to repeat the subject again entirely.

I have my doubts, because I've also met some Math PhDs that went into industry, and after 10 years can no longer read their thesis.

>>15412061
Even PhDs need a review, too. There can be only
so much one can take in their minds before some
stuff is pushed aside for more, so to speak.

Like your neurologist said, this is quite normal.
Have your books handy if you need them, I
know I have a library at home to reference.
And only the very few can exhibit nearly total
recall which is one the freaky things our minds can do.

>>15412106
>Have your books handy if you need them,
This is a good point. I have much better time recalling information from physical books as opposed to digital. I'm not sure if this is due to the way that when I began learning books were the only option, or whether their is an intrinsic quality which engenders spatial recall. I don't seem to have this problem of recalling information inside large code bases, so I'm leaning to the former. In any case, my external mind is certainly on my self.

>>15412137
>>15412106
I initially learned through books as well, though I
could get about the same experience looking at
the same book through an electronic reader,
or computer. The only difference between the
two is if you want your reading material to glow
in the dark. Or, if you want the pages turned for
you when you're ready. Or, if you're particular to
how your reading material feels...
(thick, singular and rough vs. thin, powerful and smooth).

Tactile and spatial memory can be had for both
options if done for long enough, so it can be
only down to preference if you favor books better.

>>15404905
Math is not a topic that's suited for generals.
This should be fairly obvious to anyone with any understanding of 4chan culture or, frankly, enough capability for abstract thought to understand how internet communities work.

>>15406770
naive set theory.

>>15410151
>>15410154
that seems awfully small, that would mean we haven't even gotten to the 10th Mersenne prime, which was found in 1911, am i underestimating just how much searching through even 27 digits is? or is it just no one has consider this worthwhile? surely if nothing else more datapoints is better in math?

>>15411877
what kind of question is this - do you know the quoted theorem or have you looked it up?

>>15412459
It is more difficult than you think. The reason that current methods of encryption work rely on the fact that factoring numbers is *hard*. There's also the fact that current CPU registers have a maximum size of 64 bits so after 2^64 (1.8 x 10^19) you are relying on software rather than hardware to perform all the mathematical operations.

Then you have the algorithms themselves. The simplest and fastest method is some kind of sieve but that requires storing all the previous primes in RAM (reading from disk would be too slow). This would quickly become technically and financially unfeasible so other slower methods are used. It's why the Mersenne prime checks used distributed computing to perform the calculations.

Even saying all that exhaustively checking a singe number in the range of 1E20 to 10E20 still might only take a second but to reach 1E21 that would require ~10^21 second of computing time and it only gets worse after that.

>>15412331
We need to start branching out in topics. I created a number theory thread for focused discussion on that field. The general could exist as just math banter, but typically someone comes here and wants a recommendation for a calculus book, and they'll be better served creating a thread which generates a discussion 50+ posts long, rather than 1 or 2 people bothering to respond here.

>>15404905
>are mathtrannys too desperate for internet dopamine to post in the general?
Yes.
>>15406183
>a particular attention seeking tranny (and its orbiters).
Very dated shitpost.
>>15411200
>I swear I knew they don't work because it happened in the past. I just keep doing it because then people see I tried to spoiler it and failed.
Based broken spoiler enjoyer.

>>15404877
bros tell me what is the name of this rule in english

-(x+5) = -x -5

>>15411579
Whoever posed that question is pretty based because they wanted all the poos to fail.

Here's my designated counter example.

>>15412809
Almost got me, nice.

>>15412783
distributive property (of multiplication over addition)

>>15412783
Negation is the implied multiplication by 1, so it follows from the distributive property of multiplication over addition.

-1

>>15412804
I don't know what you have posted but those numbers are not a counter example.

What are some popular old school math books that are out of copyright or under creative commons? Thinking of rewriting them.

any videos, series, or meme charts to cram and master linear algebra?
finals are coming, thx

>>15412997
For all X in R, and I chose X as 10. Which we can see from your graph is a point of intersection. BTFO. No university for you pajeet.

>>15412997
>>15413085
Alright I see what's going on, you don't have the freedom to pick your As and Bs independently, the x must be chosen first, and the question is making you prove that this only occurs when they are equal.

>>15413085
>>15413086
They are supposed to be equal FOR ALL X IN R. So they must have the same graph.

>>15413118
Yeah, I was phone posting and being retarded. I'll solve it on my white board.

>>15413084
3b1b has a series.

>>15411579
If they are equal then their derivatives must be the same where defined.
The lhs has derivative |{a_i<x}| - |{a_i>x}| for x != a_i
The rhs has derivative |{b_i<x}| - |{b_i>x}| for x != b_i
For these to have the same discontinuities, you must have n=m and a_i=b_i.

>>15411579
Hello anon. Sorry for my late reply.
>India.
Yep, I remember you. I hope everything is going well fren.

The problem you posted seems cool, I think I have an idea for the first part of it, showing m=n but haven't gotten to the second part yet. I'll try giving it a go today when I'm at the library but I doubt I'll able to do it. Either way it's a cool problem however! So thanks for sharing.
>I also have an entrance exam (in 2 weeks).
Good luck to you!
>>15412804
>>15413085
Imagine not even understanding the problem and then having the confidence to insult people. At least be humble.

>>15413185
>At least be humble.
A good quality, but I just saw India and reflexively assumed it was a mistaken writeup, I'm just so jaded by the shoddy work I see from Indians that I didn't give him the benefit of the doubt.

>>15413155
Yes anon you are correct. Personally I did it without calculus but your method works fine, and is shorter actually. How did you enjoy this problem? Do you think 15 minutes is appropriate for this? Keep in mind the proof needs to be a bit more detailed than yours in the exam.

>>15413185
Good luck to you to anon.
Funnily enough, this isn't the first time in this thread I have been insulted by someone who doesn't understand the question. >>15406811
Fortunately, I have figured out that question.

>>15413207
>Personally I did it without calculus
Show your work.

>>15413185
>Do meth
It was actually ritalin and benzedrine which are much weaker.

>>15411579
If we treat the [math]\left|a_i-x\right|[/math] as a step, we know that after n steps on the left and m steps on the right we cover the same distance. Thus the average distance will be the same, and taking the means gives s/n = s/m => n = m. Once you have n=m, you can subtract the two sides, and note that for each a step there must be one single corresponding b step which cancels it out. Since the as and bs are sorted, we have for each term ai = bi.

>>15413249
>Thus the average distance will be the same,
Scratch that, I cannot infer that. Time to shower, and fix the proof.

>>15413207
It was pretty easy.
At first I was just thinking of how to show m=n is necessary.
By letting x be a large negative value, the absolute values could be removed.
Then by using:
f'(x) = g'(x) is a necessary condition for f(x)=g(x)
you can show m must be equal to n.
Then I just thought of applying the derivative argument everywhere to get a stronger necessary condition for equality.
The necessary condition is everything being equal which is what is asked for.
15 minutes for pre university seems fair.
I don't know if there is a quick way that doesn't use calculus though.

>>15413084
based kay poster
>>15413192
I understand your frustration but responding to what you believed to be a incorrect problem shouldn't have been to use derogatory racial terms. It would have been much better for you to have either went about your day not paying mind to yet another mistaken writeup by an Indian or showed your counterexample without using rude words. I hope in the future you'll consider these alternatives.
>>15413207
Okay anon, so I think I got the first part. It'd be nice if you checked but the idea is that I was able to divide by (m-n) and reach a contradiction, which wouldn't be possible if m=n. I hope I didn't make any mistakes. For the second part I'm not sure how to proceed but I'll try, unfortunately I need to start studying for regular school subjects soon.
>>15413221
Yep, that's true. But meth sounds more funny. I use ritalin myself and I usually refer to it as government subsidized meth. But of course you're right, they're not the same thing.

Hello,
I failed every math class in high school. Please bully me for being a worthless low-IQ brain let and tell me to kill myself

>>15413363
actually nevermind, after establishing n=m, I think it's pretty easy to prove the second part by induction.
>>15413378
Hello anon. You're not worthless. Failing your classes during high school is unfortunate, but it doesn't say much about your potential by itself. Whether you're a "low-IQ brain" or not, you need to give more effort into studying and trying different methods before considering giving up on life.

>>15413249
Hello thank you for trying anon. Hope you succeed.

>>15413283
Thank you for your opinion. The question is for a Masters in Statistics programme. The statistics questions are a lot longer, but in general they are as easy as this one. All the questions are quite easy if you have read the proper books, but not if you rote learn, which you probably know is very popular in India; so I believe the exam is designed to filter those who do that. The main difficulty for me is to avoid stress, I always end up panicking when I am solving under the clock.

>>15413363
Yes anon you are correct. Good job. However, you could have simplified the proof. See that the first equation must be constant with respect to x as x decreases, but that is only possible if the coefficient attached to x is 0.

I will give you a hint: think what happens to the equation you derives for x in between a and b.

>>15413213
Please see [math] \text{combinatorics puzzle anon's} [/math] incomplete proof. It is similar to that.

>>15413363
nta, but don't you have to use the fact that a1 ... am - b1 ... bn = 0?

>>15413406
>Hello thank you for trying anon. Hope you succeed.
Yeah, using the corrected part of the other anons proof, where I think x should be < min(a1,b1,0), will show that 0 = (m-n)x, implies m=n.

Once you've got m=n, it's trivial to subtract the terms in sorted order, and get corresponding equal elements.

>>15413430
>>15413436
ahhh wait of course it's not zero. It's just some sort of constant. It's only zero in the form that included all absolute differences. This problem is deceivingly simple.

>the powerset of natural numbers has same size as real numbers
this is so cap. if it's true then why isn't there a bijective function from powerset of N to R?

>>15413472
There is. A subset of n can be thought of as a sequence of 0s and 1s. Given such a sequence, take the real number 0.(sequence), and apply the tan(pi(x-1/2)) function. There are only countably many undefined points or duplicate values, and you can fix them using the usual shift method.

>>15413482
>take the real number 0.(sequence), and apply the tan(pi(x-1/2)) function
This process can't get any positive number.
The 0.(sequence) will always be less or equal than 1/9 then pi(1/9 - 1/2) will be negative so the tan of it is negative too.
It can't give all negative numbers either by the way.

How the fuck are inductive proofs valid?
the base case makes sense but the inductive step doesn't.
"n then n+1" is only false if n but not n+1. if n is false then n \to n+1 is always vacuously true. the problem is if we are assuming n is true when it's not then n \to n+1 is going to be true

>>15413511
You have infinite many statements A(n) to prove. That is you have to prove
>A(1)
>A(2)
>A(3)
>...
This set of statements is equivalent to the set of statements
>A(1)
>A(1) implies A(2)
>A(2) implies A(3)
>...

In an induction proof you just proof this second set of statements.

>>15413511
>How the fuck are inductive proofs valid?
How are they not? https://mathworld.wolfram.com/Validity.html
>if we are assuming n is true when it's not then n -> n+1 is going to be true
I think you're missing a step. Usually it goes "for any given case n = k" and "n = k+1" or similar, to be less confusing.
https://en.wikipedia.org/wiki/Modus_tollens
If n holds for any given k, then n holds for any k+1 because k+1 is an element of "any given k". It is necessarily true.

>>15412827
>>15412828
thanks bros sorry for bad english

Why is Baby Rudin considered to be bad for multivariate calculus? Is it actually true or it's just people shitting on Rudin like usual? I am reading Rudin right now and I am enjoying it a lot more than the other recommendations from people who claimed Rudin is the worst book, like Zorich or Escher. I find Rudin's exercises and choice of topics to be much more enjoyable. However, I have noticed that the opinion of Rudin being terrible for multivariate calculus to be a lot more prominent. So, why?

Is there a reason university math classes shill MATlab? How much are they paying?

>>15413814
What should they be shilling?

>>15413893
nta but he will invariably say SAGE/octane, because "muh open soares" despite this software often being inferior, and anon has no intentions of ever "auditing the code"

>>15413511
It's not vacuously true, because we are supposing that n is true, a priori. It's not allowed to be false.

If the chain f(n) => f(n+1) is established, then you only need a base case to start the chain of events.

>>15413406
>Masters in Statistics programme.
Oh, it's not just a high school math problem to get into university? I wasn't sure what level of math you could use to solve it, since I'm not sure if calculus is standard in HS for India. In the US, it's only an option.

>>15414259
Stewart level calculus is kind of optional but it's practically standard. Undergraduate entrance exams of elite research institutes like this require proof based epsilon delta stuff. Ironically, the undergraduate exams are wayyy harder since they have simple topics, so they are compensated by more difficult problems.

>>15414422
It certainly doesn't help that you have limited seats and 50 million people trying to get in.

>>15413363
>>15411579
>>15413405
>>15413406
For the second part, rewriting each |a_i - x| as some positive alpha_i, beta_i, and keeping them sorted and using
Induction seems to be the way to go. I need to write it out, but I think we avoid problems going from say 2 to 3 elements because the n+1th term is always going to be greater than the ones that came before. Without that, I think we run into something like
https://en.wikipedia.org/wiki/All_horses_are_the_same_color

a math bachelor's degree is useless. how do i know if i have the enough smarts for a master's or even perchance a phd?

>>15414835
Why does it have to be this way?

>>15414835
>how do i know if i have the enough smarts for a master's or even perchance a phd?
by entering a math undergrad program and seeing how you perform/how motivated you get about the whole thing
there's no other way

>>15411579
This is a good problem, and surprisingly not on the internet or seemingly in any of the problem books I own.

>>15414517
>50 million people trying to get in.
That's not really true. Yes 50 million are trying to get in, but maybe like 1% of them actually prepare for the exam, and maybe like 1% of those preparing, are actually passionate about the subject. 99.9% are retards, and are completely irrelevant. The exam I am preparing for is a lot more niche than JEE since research isn't a big thing here. It has only about 20 seats, yet people get in having only answered little more than half the questions, keep in mind time is not such a big constraint in this particular exam. So it is pretty evident that most people appearing for it are retards.

>>15414835
If you can solve and read standard undergraduate books on your own, and you enjoy it, you're gonna be fine. I don't even have a remotely mathematical degree, but I am sure I could survive a math masters. However, research is a different thing. There's no way to tell without interacting with professors and actually publishing papers. Best thing to do would be to get research experience starting from undergrad.

>>15414571
I tried to do it with induction first. Couldn't do it. I think it is possible but you have to consider n different values of x or something like that.

>>15415206
Thank you anon. It is from a Statistics exam. Sum of absolute deviations is quite important in Statistics, so you might find it in a Statistics problem book. The proof that all medians minimise sum of deviations may help you with this problem.

ive learned it doesnt matter how hard you work or how talented you are or how valuable you can demonstrate your findings to be you will never find success in the world if you arent connected.

>>15415435
True. I had romantic dreams of academia. Me reading math all day and figuring out problems. But in reality, it's not very different from a corporate job. You have to travel a lot and "befriend" people.

so can I just write mathematics papers on my own with no affiliation to any universtiy

>>15415558
No one will stop you, but it's the same as if you write a news article without any affiliation to an organization. No one will read it, and no one will trust it, even if it's very good. Of course, if it is exceptional, then it may garner interest.

>>15415438
the social aspect is honestly just another kind of nepotism but it doesn't usually stop you publishing but it depends on how high you're trying to go
>>15415558
You can publish, you just need to be very good and know all the relevant standards.
>>15415596
Maybe I'm giving away the game here but you also need to write any papers you publish with SEO in mind, both for internet searching but also searching with university portals and tools etc. Same goes for answering practical specific subject matter that people would be interested in and in a way that's useful. It's a lot of shit to keep together but it's possible, just really hard

>>15406235
Check out Bianchi groups maybe, as a side project I am trying to compute the cuspidal cohomology of [math]SL_n(\mathcal{O})[/math] where [math]\mathcal{O}[/math] is the ring of integers of a bi-quadratic extension.

>>15413063
just asked ChatGPT for these so you might want to doublecheck them:
>Euclid's Elements
>Calculus Made Easy by Silvanus P. Thompson
>A Course of Pure Mathematics by G. H. Hardy
>An Introduction to the Theory of Numbers by G. H. Hardy and E. M. Wright
>A Treatise on Probability by John Maynard Keynes
>The Elements of Coordinate Geometry by S. L. Loney
>The Art of Computer Programming by Donald E. Knuth

I fucking hate reals.
nothing makes sense, you can't even say something as simple as "let's see the next number" without getting an existential crises. proofs with reals are hard, equations with reals are hard, functions with reals are hard, sums over reals are hard, doing any sort of computation with reals is hard, just thinking about reals is hard.
not a single theorem or proof is intuitive or makes sense.

>>15416198
> Sure, I'll try to explain the Intermediate Value Theorem like you're 5 years old.

> Imagine you're playing a game with your friends, and you have a toy car that you're trying to move from one end of the room to the other. But you can only move it a little bit at a time, and you don't know exactly how far you need to go to reach the other end.

> The Intermediate Value Theorem is like a rule for this game that says if you start at one end of the room and you end up at the other end, then at some point along the way you must have passed through every spot in between. Even if you don't know exactly where those spots are, you can be sure they exist.

>>15416198
IEEE 754 is so much better

>>15416184
Thank you for the help, but almost all of those have been re written.
Euclid's Elements has been re written. Also, it's not really a practical math textbook, more of a historically significant one.
Calculus Made Easy has been re written by Gutenberg.
A Course of Pure Mathematics is a very good suggestion, but it has already been re written by Gutenberg.
Probability book before Kolmogorov's foundations. Seems too niche.
Another Hardy book. Also a good suggestion, especially since its scans aren't very good.
Loney's Coordinate Geometry has been re written.
Knuth has been re written.

So the only good suggestion is Hardy's Number Theory, but I despise number theory, so I am not gonna do it. I did find a book called Advanced Calculus by Loomis, which is not in open domain, but in creative commons. It seems to be an advanced version of Munkres' Analysis on Manifolds; a unique book on an important topic. Also, its scans are all shit. This seems like a worthwhile book to re write. The funny thing is: the second author's name sounds like a 4chan satire.

>>15416222
here's a few more suggested by ChatGPT
>The Principia Mathematica by Alfred North Whitehead and Bertrand Russell
>Introduction to Mathematical Philosophy by Bertrand Russell
>The Theory of Functions of a Real Variable by Shlomo Sternberg
>Elements of Algebra by Leonhard Euler
>Lectures on the Calculus of Variations by Oskar Bolza
>An Introduction to the Theory of Groups by Paul Alexandroff and Heinz Hopf
>An Introduction to Mathematical Logic and Type Theory by Peter Andrews

>>15416222
>Advanced Calculus by Loomis
my first thought when i saw this was "I guess they really want to make encryption methods secure if they are making books"

>>15416220
>10000...0 = -0
>00000...0 = +0
>111111111111...0 = -∞
>011111111111...0 = ∞
yeah makes perfect sense

>>15416222
>Hardy's Number Theory
Not out of print, because they released a 6th edition

>>15416268
My fucked up algebra > your fucked up algebra

>>15416231
I remember asking chatgpt to recommend me some fundamental papers in my field and most of them didn't exist

>>15416409
Microsoft bing gets really pissy when you correct it, it claimed it made a "typo" for an arithmetic error. And then when you mock the AI, it just shuts down with "I'd prefer not to continue this conversation at the present time, I'm still learning thanks"

>>15416416
>And then when you mock the AI
A bit sideways on the topic but I still find it disturbing people have such a natural tendency to do this simply by result of classifying something as an object. Unless that's just how you normally behave but that'd be disturbing for whole new reasons.

>>15416423
THE AI WILL NOT BE MOCKED!

>>15416451
Case in point.

>>15416467
Mocking errant behaviour is a way that both humans and AI learn. It's what you're attempting to do right now with your "holier than thou" sanctimonious defense against derision. Societies that practise a culture of shaming and mocking indecent behaviour succeed (old British Empire, Japan, Scandinavian) whereas those which do not invite a culture of bribery, corruption and humans defecating in streets like animals.

>>15416522
Yeah so I guess
>Unless that's just how you normally behave but that'd be disturbing for whole new reasons.
was spot on then.

>>15416533
Talking of spot on, this portrait of you. >>15416451. Have a good day mate.

>>15411579
Assume m > n, pick x less than the minimum of all the a's and b's so that we can get rid of all absolute values. Solving for x will yield x = (sum(a) - sum(b))/(m-n), so for m > n, there is only one value for x less than all a's and b's for which the equation holds, therefore m > n and the equation being true for all real x cannot hold simultaneously (same for m < n). The second part can be solved by induction.

>>15416867
This anon did that for the first part >>15413363
>The second part can be solved by induction.
No one has shown how to do this yet.

>>15404877

I'm getting more interested in semantics and its interaction with syntax after studying it... are there any books, articles you suggest I read concerning semantics and the other ways math intersects with linguistics, or other subjects to look into?

why [math]g^2_3 [/math] becomes squarte root of g

>>15417056
why [math]g^2_3[/math] becomes [math]√g^3[/math]

>>15417055
At what level? For a tome (with plenty of references you can chase) you have Kracht - The Mathematics of Language

>>15417160

Thanks for that anon, I'll take a look. I guess at a theoretical level mostly, I figure I'd read Wittgenstein, Searle and others in that vein to start on the philosophical side. The ways that a mathematician would approach formal language and grammar. If I start to get into lexical semantics and the syntax-semantics interface, how mathematical and logical systems would account for it. I know statistical methods could also be considered, but I'm more interested in the theoretical side

>>15415435
>humans are social beings
who would've thunk it

Hey, american guy who failed out of college here. I want to redeem myself, and prove to myself I would have been capable into a university in one of those third world countries with hard entrance exams. What should I emphasize for my study, and what ratio of input to output do you think would be best? I also slept my way through high school, so my understanding of highschool mathematics is almost nothing.

Would working through an olympiad style problem book like math via problems by Skopenkov remedy this most efficiently?

>>15417353
https://www.cmi.ac.in//admissions/syllabus.php
https://www.isical.ac.in/~admission/Syllabus-And-QP.html
http://univ.tifr.res.in/admissions/Prev_QP/Prev_QP.htm
These are the question papers of top math universities in India. The last one does not have an undergraduate degree. Skopenkov is enough for the undergraduate exams. JEE and stuff requires studying physics and chemistry as well.

>>15417367
>top math
I take it DurgaSoft is still the elite institution when it comes to tech, right?
>t. /g/

>>15417367
I don't care much about the physics and chemistry, the mathematics stuff will suffice. Thank you

>>15417367
Which institution provided the problem mentioned above with the sums of absolute differences?

>>15417376
No idea what that is.

>>15417379
You're welcome anon.

>>15417381
The second one, from their MStat exam. But that question paper (2014 sample) is not there, since the syllabus was changed. All the question papers are here:
https://www.cheenta.com/isi-mstat-iit-jam-stat-problems-and-solutions/#section-388-80394

>>15417247
yea and the problem is there are maybe 4-5 humans on earth who would want to hear a guy talk about
>so yea ive been working on a family of sin and cosine parametric equations that sort of resemble fourier series but not quite and ive yet to figure out how to iterate through the calculus portion to find out how to get the 26 basic solutions that when transformed back into the time domain will represent the phoentic building blocks

>>15417429
You think that's bad imagine being that guy without the professional qualifications trying to find those 4-5 humans without being written off preemptively as a crank. Professionals are a lot more tolerant toward ideas or expressions of ideas they don't instantly understand if there's some letters on your name. Otherwise you have to be super careful and damn near grovel, and heaven forbid you defend your ideas just trying to get them seriously considered for some help.

>>15417409
>(2014 sample)
Ahh, from the link I see the solution provided went with the calculus approach. That's the best one, but I was hoping to see an alternative.

I would really appreciate your help with this one. thanks
>>>/wsr/1349057

>>15416985
Hi anon, I'm 15413363. I didn't do the induction because in my head it seemed easy enough, as I've said here>>15413405.
Here's what I had in mind, I'd appreciate it if you checked this one out too and tell me if you think it seems correct. The base case is clear, if n=1 then a_1 and b_1 have to be equal. To show this I think we can pick an x less then them and it should follow. Assume it holds for some number n. Again pick x less than a_1, we have a_1 - x +.... + a_n - x + a_(n+1) - x = b_1 - x+.... +b_(n+1)-x

since we know a_i = b_i for i up to n, we get
a_(n+1)=b_(n+1).

I don't know, I hope I'm not making an obvious mistake. Sorry for the way I wrote, I'm phone posting and too lazy to properly type.

>>15417579
>since we know a_i = b_i for i up to n, we get
>a_(n+1)=b_(n+1).
We don't know that though. We know that's true when summing only n terms, but here we are summing n+1 terms.

>>15417587
My apologies anon, I don't follow. Here's what I meant just to clarify. Is there a mistake with this and if so can you tell me what it is and how you'd go about fixing it if you can?

>>15417531
Are you sure you copied the question correctly as that appears to only have a numerical solution?

>>15417621
Why are you assuming they are equal? The induction hypothesis is that:
[eqn] \left( \forall x \in \mathbb R \quad \sum_{i=1}^n |a_i - x| = \sum_{i=1}^n |b_i - x| \right) \implies \forall 1 \leq i \leq n \quad a_i = b_i [/eqn]
But you are using:
[eqn] \left( \forall x \in \mathbb R \quad \sum_{i=1}^{n+1} |a_i - x| = \sum_{i=1}^{n+1} |b_i - x| \right) \implies \forall 1 \leq i \leq n \quad a_i = b_i [/eqn]
If you can show that:
[eqn] \left( \forall x \in \mathbb R \quad \sum_{i=1}^{n+1} |a_i - x| = \sum_{i=1}^{n+1} |b_i - x| \right) \implies \left( \forall x \in \mathbb R \quad \sum_{i=1}^n |a_i - x| = \sum_{i=1}^n |b_i - x| \right) [/eqn]
Then your proof is correct. But I don't think you realise you have to show that, unless you skipped it because you think it trivial.

>>15417650
Oh I see now, I get what you mean. You're right, I didn't realize I'd have to show that(but let's pretend I skipped it because i thought it was trivial.) Is it trivial?

>>15417670
I don't think it is. I tried to do it, albeit didn't spend much time on it. Can you do it without induction, by using the hint here >>15413406 ?
It's very similar to what you did, I assure you. You just need to extend the equation for cases where x is not less than a_1 and b_1.
Or you can just give up and ask for the solution, since quite a few people are asking how to do it without calculus.

>>15417722
>think what happens to the equation you derives for x in between a and b.
I don't quite get what you meant here. Could you clarify?
>give up and ask for the solution
I don't really want to give up but I do have a 3 hour exam coming up in 40 minutes. I wouldn't be complaining if someone posted a solution that doesn't use calculus. I probably won't have any energy left to do anything after the exam.

>>15417749
I mean the whole point you considered x less the a1 and b1 is to remove the absolute values right? So that you can make the equation more "operable". Can you find a similar equation for when x is in some other set/interval? Some specific intervals other minus infinity to a1, b1 that will make the equation more operable? an, bn to infinity is one of them, you probably have figured that out. What other intervals/sets can you think of?

You have a prime number. If you remove any digit out of that number, the remaining number will also be a prime number.

What is the largest number with this property that you can come up with?

>>15417819
Apparently this question was asked a few years ago on Code Golf, and the biggest answer they found was the 274-digit-long 4444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444000000000000000000000000000000000000000000000000000000000000000000000000000000001111111111111111111111111111111

>>15417822
Interesting

>>15417822
Now I wonder what about the reverse process. What is the largest prime number you can get by continuously inserting a single digit to a prime number so that at each step you also have a prime number, e.g.
>2
>23
>283
>1283
>12983
>etc.
This one should be much easier to brute search.

>>15417840
Or what about a number that you can insert any digit to any place and the result would always be a prime number.

>>15417845
this one is obviously false because you can insert an even number to an end or change the sum digit to be divisible by 3.

>>15417840
Well, if we restrict it to inserting a digit at the beginning or at the end, you end up with the left-truncatable and right-truncatable primes, of which there are a finite amount of each, with the largest in decimal being 357686312646216567629137 and 73939133, respectively.
No clue for insertion anywhere, though

>every true statement in mathematics is a theorem
>a theorem is derived from a set of axioms
so what is the set of all axioms? i dn't care if it's self contradictory or complete or whatever, why has no one just compiled a list of every axiom in math?

>>15418141
>so what is the set of all axioms? i dn't care if it's self contradictory or complete or whatever, why has no one just compiled a list of every axiom in math?
https://en.wikipedia.org/wiki/Principia_Mathematica
Because by volume three after years they wanted to jump off a bridge

>>15418141
Have you Googled it?
https://en.wikipedia.org/wiki/List_of_axioms

You're free to pick whatever list of axioms you want, and then try to build theorems on top of that.
Almost always, people implicitly work within the list above.

>>15418230
ZFC is not a complete list of axioms. It is built on other axioms, of propositional logic and stuff.

>>15418236
Anything can be an axiom, so there cannot be a complete list.
I can work in a fictional literary universe and try to build a theory of mathematics that works within that world, if I want. Almost anything can be a list of axioms.

Practically speaking though, these are the de facto list.

>>15417822
Oh I can see why this number looks like this. Now you only need to check four different times if the number os prime so this particular number is very computationally efficient.

The fact that conclusions such as "it is impossible to prove if this thing is true or false" have been reached (one of many example: continuum hypothesis), makes me think that down the path we have accumulated mathematical "hacks" that produced paradoxes, which are questions that make no sense. Most famous culript may be the axiom of choice, but even the existance of complex numbers could be defined as one.

My question is: would it be possible to define a set of axioms that would guarantee that every question based on that set could be answered with true or false?

My brain may be smooth but can someone explain to me why is pic related true when dealing with vectors ? I don't get why dividing by the magnitude would churn out the cosine of the angle between 2 vectors.

>>15418641
Gödel's incompleteness theorems show that the answer to this question is decidedly "no"

>>15418649
It comes from the law of cosines. Alternatively think of the division as normalizing each vector.

what's a good book on the use of group theory in physics for someone who already knows group theory?

I have an EE/systems+signal processing degree so I'm no stranger to complex numbers, but I still don't understand how/why they correspond to rotation and oscillation at a viscerally intuitive level.
Yeah, I understand Euler's formula, I understand that it's how we can relate [math]i[/math] to the trigonometric functions, but that's not what I'm asking
WITHOUT deferring to Euler's formula, or other formula juggling, what is it about the complex unit i that actually relates it to rotation? *Why* does [math]e^{i\theta}[/math] correlate with oscillation? In other words, why does raising some random numerical constant to the power of the square root of negative one, somehow result in an oscillation?

There's the classic phenomenon that powers of [math]i[/math] are periodic. That's cool, and it has some notion of "rotation", but I think this is only a snippet of the picture I'm trying to understand

>>15405498
your g has to be nice in some way, like being its own antiderivative or similar, because as the exponent on r drops the number of antiderivatives you have to do on g grows
>>15406131
no. look up lindemann weierstrass theorem
>>15406235
look up padic langlands
>>15407287
look up restricted vs unrestricted comprehension
>>15417055
look up type theory
>>15418141
>every true statement in math is a theorem
misleading at best
>>15418684
(cos theta,sin theta) is parametrized rotation
or costheta+i*sin theta in complex plane
the tangent vector at time theta is (-sin theta,cos theta) which is geometrically perpendicular to the radial vector. since multiplication by i is 90 degree rotation this is just encoding
d/dtheta(cos theta + i sin theta) = i*(cos theta + i sin theta)
d/dx (f(x)) = i * f(x)
this is the equation for exponential growth but instead of "proportionally growing" at each step the sqrt-1 makes it "proportionally curving by 90 degrees to my position vector"
the actual exp(i theta) = blah blah is a result of a definition meant to nicely capture this (and other) facts

Math bros, how do you cope with being bad at math? I love solving problems but usually i takes me hours to do so. Of course, this is a huge problem when it comes to test, i barely pass my subjects. Is not that I dont understand the subject but I usually dont see the solution at first glance.

>>15418641
Presburger arithmetic, Skolem arithmetic, Real closed field
are such decidable theories

https://en.wikipedia.org/wiki/Real_closed_field
https://en.wikipedia.org/wiki/Presburger_arithmetic

>>15418700
Thanks for the reply, it feels a lot closer to a satisfactory answer than what I've managed to find on my own. I haven't seen a justification using the differential equation for exponential growth before. Definitely a new insight.

I just have a few qualms
>since multiplication by i is 90 degree rotation
this feels like it sidesteps the problem a bit, and says "it's rotation because it's rotation"
>this is the equation for exponential growth but instead of "proportionally growing" at each step the sqrt-1 makes it "proportionally curving by 90 degrees to my position vector"
along the same lines but a bit more subtle. This feels like it's just hiding the [math]e^x[/math] in the differential equation without explaining why the act of exponentiating the complex unit causes oscillations

I don't (or at least didn't) see the connection between "the periodicity of the powers of [math]i[/math]" and "the act of...". But, I think your response just helped me to finally answer my own question.

we know
[math]i^0 = 1\\i^1=i\\i^2=-1\\i^3=-i\\\cdots[/math]
But, recall that
[math]e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!}\cdots = \sum_{n=0}^{\infty}\frac{x^n}{n!}[/math]
So,
[math]e^{ix} = 1 + ix + \frac{(ix)^2}{2!} + \frac{(ix)^3}{3!}\cdots[/math]
just like that, there's the relation between the periodic nature of the powers of i, and the exponential function.
and as a very nice bonus, we can notice that for even terms of the sum, we get
[math]1 - \frac{x^2}{2!} + \frac{x^4}{4!}\cdots = \sum_{n=1}^{\infty}(-1)^n\frac{x^{2n}}{(2n)!} = cos(x)[/math]
and similarly for odd terms of the sum,
[math]ix - \frac{ix^3}{3!} + \frac{ix^5}{5!}\cdots = i\sum_{n=1}^{\infty}(-1)^n\frac{x^{2n+1}}{(2n+1)!} = isin(x) = cos(x - 90°)[/math]
the bonus being we can directly relate [math]e^{ix}[/math] to the trig functions, and it cleanly demonstrates why multiplication by [math]i[/math] is a 90° phase shift.

>>15418788
yes this is the definition I mentioned(+convergence blah blah)
it works because it works and it additionally has many nice properties (analytic continuation etc)
same with multiplication by i, if (a,b) is identified with a+bi then the rotation 90 degrees about the origin is (-b,a) by geometry/dot products/etc., now -b+ai = i*(a+bi) by definition of i
you asked for intuition which never has a right answer so I said the thing which is most physical, a diffeq
but you can also view everything as purely notational (formalist view), then it works bc it works

>>15418788
>this feels like it sidesteps the problem a bit, and says "it's rotation because it's rotation"
No, another way to say what he's saying is in terms of matrices.
Consider the vector
r(t) = (r_x(t), r_y(t))
and assume it moves in a way that leaves ||r(t)|| constant, i.e. it rotates around the origin.
Then clearly the velocity r'(t) or this rotating motion is tangent to the circle, in other words perpenticular to r(t) itself, and so r(t) · r'(t) = 0.
(Draw yourself a picture.)
So
r'(t) = c * (-r_x(t), r_y(t))
or
r'(t) = c(t) * J r'(t)
where J is the matrix {{0, -1}, {1, 0}}.
This is the 2D represnetation of the complex unit. You can square J and see it's minus the identity.
The 90° rotation isn't any other rotation, it's the infinitesimal generator of the motion that leave the distance to the origin unchanged.
(In fact this sort of analysis can be extended to any continuous one-parameter family of transformations)

should have been

r'(t) = c(t) * J · r(t)

>>15404877
>Vector spaces are the generalization of the plane using two co-ordinates
Axler, Linear Algebra Done Right

can someone explain this to me? the reasoning or what exactly the mathematical justification or thought process.

How do I solve [math]x=\log _2(8-2 x)\ [/math] for x ? Going into university I decided to review the little stuff I knew, but sometimes find myself not knowing shit like this.

>>15418973
Start by raising 2 to the power of each side.

>>15418973
Newton's Method

>>15418973
Guess n'check. x = 2 looks good.

Was taking a look at a qualifying exam question i didn't crack last summer in Galois theory. The question asks for the Galois group of C(t)/R(t^n). After working at it a bit more for fun, I am thinking it's D_2n. Can any Galois chads confirm?

Passed btw

I haven't had to do math since high school 10 years ago (I took the ones I'd need for uni while I was a senior) and now I've forgotten essentially everything.

What's the best way to bring myself back up to a calc 1 level? Something I can just do in my spare time at my own pace. Thank you for reading.

>>15419207
khan academy

>>15417624
it might be that the ones on each side cancel each other out, resulting in an easy to solve equation. i was just wondering.

>>15417795
>Can you find a similar equation for when x is in some other set/interval?
Nope, can't think of it. I give up :D

Is the natural map from G to G/H continuous? (where G is a topological group)

>>15419449
since the quotient topology is defined to be the smallest topology making said map continuous, yes.

I know this is retarded, but I don't want to make a fool of myself by correcting a professor:
>>15419441

What is a good book to learn introductory real analysis?
I tried baby rudin but it's a hard read
also I might need to brush up on my proofs

>>15419509

>>15419494
Yes, you're right.
Because the square root of 4 is 2.

>>15419532
Thanks, I'll send the email
I thought it was some weird paradox instead of a typo lol

Euler numbers vs. Catalan Numbers.
Who wins?
You decide.

>>15419564
I'll vote for Euler just because I have yet to read a text book that doesn't mention his name, whether it be math, physics or engineering.

>>15419144
To me it seems to be [math]\mathbb{Z}/2\times\mathbb{Z}/n[/math]. Assuming [math]t[/math] is transcendental and setting [math]T=t^n[/math] for convenience, we look at [math]Gal(\mathbb{C}(\sqrt[n]{T})/\mathbb{R}(T))[/math]. Now [math]\mathbb{C}(\sqrt[n]{T})[/math] is the compositum of [math]\mathbb{R}(\sqrt[n]{T})[/math] and [math]\mathbb{C}(T)[/math] over [math]\mathbb{R}(T)[/math] so there is an embedding [math]Gal(\mathbb{C}(\sqrt[n]{T})/\mathbb{R}(T)) \hookrightarrow Gal(\mathbb{R}(\sqrt[n]{T})/\mathbb{R}(T)) \times Gal(\mathbb{C}(T)/\mathbb{R}(T)) [/math]. This embedding is an isomorphism since [math]\mathbb{C}(T)\cap \mathbb{R}(\sqrt[n]{T}) = \mathbb{R}(T)[/math]. What do you think?

>>15419509
Abbott is extremely good, but is incomplete. He for example, does not talk about definition of trigonometric functions and assumes their derivatives. You have to follow it up with a more serious book like Rudin.

>>15419144
>>15419665
What are the automorphism of [math]\mathbb{C}(t)[/math] that leave [math]\mathbb{R}(t^n)[/math] fixed?

I would say it's the automorphism
[math]f(t) \mapsto f( \zeta_n^k t)[/math] with [math]\zeta_n = e^{\frac{2 \pi i}{n}}[/math] for [math]k=0,1,\ldots,n-1[/math]
and
[math]f(t) \mapsto \overline{f( \zeta_n^k t)}[/math] for [math]k=0,1,\ldots,n-1[/math]

>>15419564
I'm on team catalan. So many applications!!!
I'm a catalan nationalist! Long live catalonia!

>>15419700
Is there a book that can prepare me for rudin?

>>15419509
You don't "read" rudin. You just flip to the back of the chapter and start doing the problems

Is /mg/ the brainiest thread on this website? I think it is

>>15419912
There's surprisingly a lot less crazy than on the other threads.
I suck at math, but it's a nice little island of sanity

>>15419862
Abbott.

>>15419723
It seems correct, in the isomorphism I used here >>15419665, the automorphisms of the LHS group are sent to their restriction to the corresponding group in the product. As such, they are given by the images [math]t\mapsto\zeta_n^k t[/math] and [math]i\mapsto \pm i[/math].
I remember having some results about compositums and semi-direct products in my local field course but I cannot seem to find them right now.

>>15418973
clearly x<4
left increasing -infinity to near 4
right decreasing infinity to -infinity
so unique solution
guess and check: x=2 (can also use bisection, Newton's method, fixed point iteration, etc. if no obvious solution)
x=2 is the unique solution

I'm trying to come up with a function with a lot of "features" as an exercise for my students.

Could someone help me find an example of a relatively "simple" function with the following properties?
1) Has an inflection point
2) Has a critical point (optional)
3) Has a horizontal asymptote
4) Has a vertical asymptote (optional)

But that's not all. I want to insert some unknowns into it so that they can solve for them to get the desired properties at the specific points.
For example, f(x) has four unknowns a, b, c, d, and I want them to find values for them such that there is an inflection point at 1, a critical point at 2, a horizontal asymptote at 3, and a vertical asymptote at 4.

The issue is making the function simple enough that finding the values becomes straight forward.

>>15420170
I am very smart. Have you tried defining it by parts.
No need to thank me.

i have learning difficulties, i don't have any value as a person. every day i wake up knowing there are people 10x more smart than me and i have no skills or good traits. i hate life.

how the fuck do I even start to understand and read shit like this:
https://terrytao.wordpress.com/2013/12/11/mertens-theorems/#more-7184

>>15420182
Everyone wakes up knowing there are people 10x smarter.
We're still in the dark ages at the moment, but eventually people are going to realize that it's meaningless to judge people by things they don't control.

You didn't chose learning difficulties, so it doesn't define your worth.
Of course the rest of the world doesn't agree with that, so you're still going to get fucked over. But maybe someday we'll do a better job of not making things shitty for people who need help. idk.

>>15420187
It's analysis, it's supposed to be retarded

>>15420187
Start at your level and just keep going. You can only get better at it if you keep going.
Then you look back, and the things that used to look like scary symbol soup are just a few building blocks composed together in familiar ways

(Or not, because some bullshit is still hard to understand no matter how long you stare at it. But still, it gets better)

>>15420182
I wonder if there is someone out there who could take away all your fears and worries....

New Thread!
>>15420227

>>15404877
>>15412116
Can someone please help me find a formula for the nth element of each of these sequences? I will post the first several elements of each sequence below, and then explain how they are defined.

S_a: 3, 9, 15, 27, 33, 39, 45, 51, 57, 63, 75, etc
S_b: 3, 15, 27, 33, 45, 57, 63, 75, 87, 93, 105, 117, etc
S_c: 3, 33, 45, 63, 75, 87, 105, 123, 135, 147, etc
s_d: 3, 33, 63, 75, 105, 123, 135, 147, etc

The sequences are defined as follows:
S_a: All odd multiples of 3.
S_b: All members of S_a without a remainder of 1 or 4 when dividing by 5.
S_c: All members of S_b without a remainder of 1 or 6 when dividing by 7.
S_d: All members of S_c without a remainder of 1 or 10 when dividing by 11.

In general to get to the next sequence, you use all members of the previous sequence without a remainder of 1 or p-1, where p is the next prime number.

I think the formula for the nth element of a given sequence might involve some mod and floor functions, but I'm not really sure. There seems to be a repeating pattern every primorial-sized interval.

>>15417367
>The last one does not have an undergraduate degree.
So are those questions meant for admission to a graduate degree?

>>15420196
how do i learn to accept myself? it feels like i am missing out on life

>>15418748
always be practicing, the speed comes with time

>>15420514
It's a hard question to answer anon.
But you should care less what other people think, and just find something that makes you happy if you can.
There are no rules, so do your thing and don't worry about things you have no control over

>>15420233
(S_a)_n = 3*(2n-1), right? What do you mean with the whole prime business?

>>15420233
>>15420569
And then (S_b)_n = (S_a)_{2n-1-floor((n-1)/3)} it seems. I imagine the next are similar.

Does anyone here know how to count in base-12?
Is it a useful skill to have?

Any ideas for numbers with 10/11 angles?

>>15420621
>Does anyone here know how to count in base-12?
>Is it a useful skill to have?
No. I know that it's popular (and not wrong) to say that 12 is just generally a better base than 10, but there's very little practical use for it outside of saying it's a better base than 10.
Really, the only bases you're ever going to find that have much practical use are 2, 10, and 16. Maybe 60 if you're into that sort of thing

>>15420635
Personally, I like base 10. I also use base 10 and base 10, but less often.
Although I just feel like we should have named base 9+1 base 9, and base 1+1 base 2, that would have been less confusing than having every base be called base 10 within that base

>>15420635
If only we were born with 6 fingers and toes instead of 5

>>15420187
Analytic number theory, picrel is an easy start (see chapter 4).

>>15420922
Picrel.

https://www.youtube.com/watch?v=4QfgUKxrkoQ

can science explain this?

>>15419665
This is what I "guessed" on the exam, and only got points for the correct "order of the group". >>15419723
These are precisely the automorphisms I figured worked last night. How they interact with each other seemed analogous to D_2n. I could have missed a semi-direct product, as I didn't write down the relations as thoroughly as I should have between the morphisms.
I really really wanted to come up with a primitive element (I think (1 + i)?) and a minimal polynomial for it, but this turned out to be way too messy to generalize. Maybe my grader just saw my last minute guess as deserving no points whether it was correct or not, but nevertheless I only got anything for getting the order right.

>>15421453
I see, I really don't have a definitive answer. However if the result about the monomorphism into a product of cyclic groups is correct, then it cannot be [math]D_{2n}[/math]. I will try to find a proof.