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/sci/ - Science & Math

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Types of mathematicians edition
Formerly >>13576443
Talk maths.
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>>13617361
Perhaps it's just my lack of experience in mathematics, but I don't buy his distinction between J and S type mathematicians. I remember reading somewhere about the work of Serre, Cartan and other frenchmen in algebraic topology which was initially regarded by the international community as extremely abstract and abstruce, the french topology, but after one or two years they were regarded as standard techniques, not any more abstract or difficult than other stuff. I think this is true to some degree for all new mathematics. Given the Bieberbach's hostility towards jews, it's not surprising that he viewed their contributions to Hilbert's theory of formal logic as an intellectual performance. I don't exactly know which jew logicians he was referring to in the period 1900-1934, he doesn't specify someone explicitly. The only one that comes to mind was a german jew Leopold Lowenheim, whose contributions to the field are indispensable and I don't think the style of his mathematics is viewed any differently than the contributions of goyim like Godel, or even nazis like Gentzen.
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>>13617515
its just nazi cope, dont take it seriously.
>>
I used to be very good at math and took a few college courses like Linear Algebra, and now that I'm free from a lot of restrictions in life, I'm now able to pursue it again. I have a lot of holes in my foundations and I kind of stumbled through the college classes, leaving me stunted.
Is this a good track to go through? Would you recommend changing anything?
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>>13617621
Just pick up a book and start reading.
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>>13617643
I'm going through Discrete Mathematics with Applications by Epp right now, I'm about halfway through. Im just wondering if this track is reasonable.
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>What is your study/work routine?
I'm just starting out, my current routine is 30 minutes in the morning and 30 minutes at night
>What habits do you do with your study time?
For example, I always put on tea and drink that while listening to no-lyrics liquid drum n bass.

I'm interesting in seeing what other anons do. Feel free to give more info.
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>>13617361
Nazi copetards fuck off
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Which Millenium Prize problem do you think is going to be solved next?
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>>13618871
The Birch and Swinnerton-Dyer conjecture
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What does it mean by 'describe the domain of \phi_4(U_{14})' of (\phi_1\circ \phi_4)? It makes no sense why \phi_4(U_{14}) is the domain of (\phi_1\circ \phi_4).
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>>13619636
I forgot to post the picture.
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>>13619636
>>13619758
It is essentially defining $U_{14}$ to just mean $U_1\cap U_4.$ It's fairly straightforward to see that the domain of $\phi_1\circ\phi_4^{-1}$ is just $\{(x,z)\in\mathbb{R}^2:x>0\land x^2+z^2\leq1\}.$
The reason this is the domain of $\phi_1\circ\phi_4^{-1}$ is because you need $\phi_4^{-1}$ to map the domain into $U_1.$
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>>13618991
Why? I believe the RH is gonna get solved relatively soon.
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>>13619791
Hmm, it's not clear to me x^2 + z^2 \leq 1. Could you please explain that? Maybe it's something really obvious but all we know is that x>0 and y < 0.
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>>13619949
Every point on $U_4$ is of the form $(x,y,z)$ such that $x^2+y^2+z^2=1,$ and $y<0.$
$\phi_4$ is the homeomorphism that maps such an $(x,y,z) \mapsto (x,z),$ and we have $x^2+z^2=1-y^2<1$ (sorry, I should have written $x^2+z^2<1,$ not $\leq1$ in my previous post).
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Nice fictional number system you got there. It would be shame if someone destroyed it.
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I want to impress my engineering boyfriend by getting better at math. I've never been confident in it, and really want to change that. I feel a lot more motivated to do so since I live with someone who already analyzes the world in patterns and equations and whatnot. Do you think you could lead me to some helpful instruction websites maybe starting from algebra 1?
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>>13620796
>my engineering boyfriend
Forgot your anime pic tranny
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>>13620796
You've already asked this question before. What was wrong with the responses you got the last time?
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>>13618851
There's not a single nazi copetard in this thread. y u mad bro
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forgive me if I seem ignorant about the way numbers work (Have only had through Alg 2 so far). My dad just pointed out to me that e^pi√(-1) = -1. I'm thoroughly amazed and confused as to how three irrational numbers found throughout the universe can come together to make -1. Any explanation/links to videos about this would be helpful
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>>13617361
can you recommend me a book series that covers all of the math topics in depth
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>>13622046
Just follow the /sci/ charts
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>>13621730
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What are the polynomials foe animal bodies and movement?
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>>13620796
Impress him by sucking his dick. If you really want to get into the "math" he does just let him explain it to you. Then you can identify and work on what you are missing, the fundamentals that you don't have.
t. engineer
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Anon, do you think there's a difference between how a mathematician and "a man who must know math" (for example, an engineer) should study?

Personally, I want to enter a field where I need a solid command of stats, graphs and linear algebra, but do not need to work on math research.
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Hello mathchads, mathlet here. Someone dared me to solve this problem and if I do, he will donate to a charity. Here is the problem:

There is a country called X. Its currency is called C.
You have been sent back 40 years. You have all the internal economic data of the X nation, but you don't have any foreign investments data. There are 5000 people who have a 1 million dollars (7 million Cs) in liquid assets, and ask you for advice to maximize their money for next 40 years. You have data on all the prices of possible assets in the next 40 years, inflation rates, currency exchange rates, and every other possible economic data.

0. Prove that this problem can be solved.
1. Solve the problem.

Honestly this looks like a optimization problem to me. Remember that it has to be solved in 5000 dimensional space.
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Why would I ever need calculus as a computer scientist?
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>>13619636
>>13619758
>>13619791
What are you guys talking about? What is this and what area is it in?
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>>13622665
Manifolds 101. You didn't cover this in high school?
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>>13622590
>calculus
To make a physics game engine or any non-trivial simulation.
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>>13622669
Where can I read an explanation based on real world context?
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Why are maths undergrads so arrogant? They tend to act like all-knowing demigods, while they have no more significant knowledge than physics or (quality) CS fags.
Like once you are doing a PhD, you might be heading somewhere, but the PhD students don't tend to be such pricks.
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>>13622961
300k starting
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>>13620796
>>13617621
>>13617643
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Is herstein substantially harder than fraleigh? I'm a brainlet but also a poorfag
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>>13623091
libgen.
Pick a book and start reading. It doesn't matter. They're all the same. Math is math. If you don't like it switch it.
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I am incredibly biased because I study math, but I am constantly in awe of the seemingly endless volume of math that needs an absolutely insane amount of prerequisites to even partially understand, even for things that were known over one hundred years ago. If I had to guess, the only other academic fields that I could imagine that could possibly compare to math depth-wise would be physics or philosophy.

Do you agree that math is the "deepest" field, or is that notion too naive?
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>this exercise is so easy even an undergrad could do it
>I can't do it
bros...
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>>13622961
mathfags understand the deep underlying theory, and therefore tend to have a better grasp of physics and computer science than the average physics or CS undergrad without ever having to directly learn it.
Often will be the case where a single lecture in a math class will cover 75% of an entire CS or physics course, but in a much more succinct and easy-to-understand fashion.
Obviously this doesn't apply for post-grad, nor the top of the physics and CS classes (who interestingly enough, tend to be well-versed in math).
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>>13623162
Physics is inherently deeper than math, but isn't taught so for various reasons.
Philosophy is part of the humanities, all of which are broader than math, but none of which are deeper than math.
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>>13623567
>mathfags understand the deep underlying theory
No they really don't after a couple of years of undergrad. I know people with a math title that did theses on algebraic curves and have close to no idea about physics or algorithmics/optimization topics.. and frankly, the algebraic curves are easy to pick up for a physicsfag aswell.
Honestly, you just proved my point a bit. All mathfags think they have this universal magic knowledge, while others have plenty of that too and don't have their heads stuck up their asses.
>>

Hello mathbros, you study all of this in your degree don't you? Why don't you all become quants?
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Perpendicular lines are lines whose the sum of their slope = -1. A horizontal line parallel to the x axis has a slope of 0, while a vertical line parallel to the y axis has a slop of undefined/infinity.

They are perpendicular, as the angle they make is 90 degrees. Does this mean 0 * undefined = -1?
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>>13624685
No. You either say a line zero slope and the other an undefined slope are perpendicular because they must be the axes, or you rotate the axes and all the zeros vanish.
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>>13624666
>only begin complex anal and rings in third year
you do realise thats just textbook advertising space to fill out the webstie
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Are there any z besides (2k-1)(ipi) where e^z is minus 1?
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There is no smallest real number r>0 according to my analysis text, but what about non-standard analysis?
Is there a smallest r>e, where e is 0 + any infinitesimal? Is this analogous to the first transfinite ordinal, but in the other direction?
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>>13624775
Shouldn't that make math anons even more likely to go into quant?
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>>13620161
looks like wildberger has been hitting mathdoctorbob's gym
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>>13625040
no it wouldnt
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>>13623567
>and therefore tend to have a better grasp of physics and computer science than the average physics or CS undergrad without ever having to directly learn it.
this is a non-sequitur and it's also blatantly false
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A night i was thinking about squaring numbers...and then discovered that the difference between two numbers squared (if the difference between them when they aren't squared is 1)is equal to two times the first minus one or two times the second plus one...so...if x-y=1 then x2-y2=2x-1 or 2y+1(they are the same number)...then thinking about it...i discovered that 2x-1 and 2y+1 are equal to x+y...so if x-y=1 then x2-y2=x+y...has this law been discovered yet or am i the first to discover it?
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>>13625344
This is discovered by most smart children when they're about 6 or 7.
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>>13617361
So Professor Bieberbach was wrapped up in the ideology of his time and it made his scientific publications muddled and cringeworthy. I think progressive mathematicians and scientists of today should take note and try not to fall in to the same trap.
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The wording of this task is a bit strange in my language so if you could help me with this and how you came up with the answer, I would be grateful anons.
And I am 13 hours into my math and pedagogy tasks and I am getting exhausted, so the numbers are hazy atm.

"In a class are 2/6 girls. 5/8 of the boys play football. Of the boys who do not play football, 1/3 are in the Scout. How much of the whole class consists of boys who are in the Scout?"
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>>13625512
>"In a class are 2/6 girls. 5/8 of the boys play football. Of the boys who do not play football, 1/3 are in the Scout. How much of the whole class consists of boys who are in the Scout?"

We don't know because we don't know how many of the boys who play football are in the Scout.
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>>13625512
Not enough information. We also need to know how many boys are in the scout of those who do play football. But we can say it's at least 3/8*1/3*4/6
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>>13625515
>>13625516
Yeah that's what I guessed myself, but the teacher said the solution was a fraction so.
Two of my group mates are stumbled too.

The task in Norwegian if it makes it easier, somehow.

" I en klasse er 2/6 jenter. 5/8 av guttene spiller fotball. Av guttene som ikke spiller fotball, er 1/3 med i Speideren. Hvor stor del av hele klassen består av gutter som er med i Speideren?"
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>>13617361
>>13625509
I do think there are two different types of mathematics being done, which may divide roughly into the two categories Professor Bieberbach was groping for. However they are based in the type of mathematics and who pursues them, namely:

One type of mathematics is about solving unsolved problems.

The other type of mathematics, becoming more existent in the 20th century, is the type of mathematics that argues over which definitions that other mathematicians should use.

For examples of the latter, consider all the different equivalent systems for the foundations of mathematics. Set theory, Goedel numbering, lambda calculus, turing machines, category theory... it's an argument about what symbols should be, but doesn't actually solve any unsolved problems...

And don't get me started on the trannies who want to replace pi with tau for uneducated reasons...
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>>13625522
>I en klasse er 2/6 jenter. 5/8 av guttene spiller fotball. Av guttene som ikke spiller fotball, er 1/3 med i Speideren. Hvor stor del av hele klassen består av gutter som er med i Speideren?
It's only a complete question if your teacher assumes that people who play football cannot also be in Scout
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>>13625552
So you're basically saying It's a trick question?
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>>13625566
It's either:
1. a trick question to get you to realize or be fooled by the fact an unspecified fraction of boys can play football and be in scout at the same time
or
2. poorly written because it doesn't explicitly state an assumed exclusion of boys playing football and being in scout at the same time
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>>13625344
(x+1)^2 = x^2 + 2x + 1
or if you want to complicate it = x^2 + x + (x+1)
i invite you to look up pythagorean triples and play with those ideas

>>13625512
>>13625566
you can likely assume that scouts dont play football
4/6 of the class are boys, then go from there
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>>13625512
>>13625566
Tell your teacher to go fuck him/herself for such a bullshit question. Such uneducated people should not hold the responsibility of one's education.
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>>13625595
>>13625612
>>13625606
Yeah thanks lads. Talked with my group mates and we agree that this is a trick question, since there is too little information to go on about.

If anyone has an answer though except "There is too little information given, can the boys play football and be in the scouts at the same time? Unless we have that information, the question is unsolveable." I would appreciate that too.
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>>13625702
>>13625595

>If anyone has an answer though except "There is too little information given, can the boys play football and be in the scouts at the same time? Unless we have that information, the question is unsolveable." I would appreciate that too.
>>2. poorly written because it doesn't explicitly state an assumed exclusion of boys playing football and being in scout at the same time
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>>13625512
if we assume it is not a trick question and assume that no footballers are also scouts, and also that 4/6 in the class are boys, then:
>1/8 of the boys are scout
>1/8 of 4/6 of the class are male scout
>1/8 of 4/6 is 0.5/6 or 1/12
probably wrong since im a remedial mathlet but thats what i thought of
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The only problem I seem to be having is with algebra and its basic rules. I want to get a degree in math and I know that I wont be able to get by with out knowing them.

I got a book called : "No BS guide to math and physics" and its pretty good. What other books or online MOOCs would you recommend me read/watch. My end goal is to become a Mater at algebra and trigonometry :D
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What's a good way to learn combinatorics? Bona's A Walk Through Combinatorics is filtering me hard. What are some good supplements for that book?
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>>13625924
just work through that one hungarian book
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>>13625929
Which one?
I mostly struggled with applying the pigeonhole principle on the first part of the book btw. The rest were so and so.
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at school i went halfway thru calc2, i remember volumes and taylor series, last summer i did the first 1/3 of apostol calc
what’s the best way to finish calculus so i can begin diffq/linear/analysis should i just go chronologically through khan? i want to start analysis and statistics, i’m comfortable with the basic handfuls of derivatives and integrals, but i’m unfamiliar with multivariable and while i followed the trig proofs they’re not intuitive
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>>13625879
>The only problem I seem to be having is with algebra and its basic rules. I want to get a degree in math and I know that I wont be able to get by with out knowing them.
>I got a book called : "No BS guide to math and physics" and its pretty good. What other books or online MOOCs would you recommend me read/watch. My end goal is to become a Mater at algebra and trigonometry :D

If you can recite Euclid's Elements by heart then you will be a true master of the type of algebra and trigonometry taught in high school
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Any of you met John Conway?
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if i take multiples of 3
3,6,9,12
and add any digits length >1
12 =1+2=3
what is this? there are a lot of patterns
someone said these don’t repeat to infinity because the strong law of large numbers
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>>13626477
who
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>>13626648
Get out

>>13626597
You came across modular arithmetic.
I'm not gonna go through the full proof, but basically you are reducing mod 9.
When you write down a number like 1234, it actually means 1(1000)+2(100)+3(10)+4(1), that's what our notation means.
Replacing it with 1+2+3+4 means you are removing 1(999), 2(99), and 3(9).
1(999)+2(99)+3(99) is clearly a multiple of 9, and you're subtracting it from the original number.
You keep doing this process, and you're basically doing division (with remainder) by 9, since that is just repeated subtraction.
What you end up with is a number that has exactly one digit left, so between 1 and 9, which is the remainder after dividing by 9. (except the remainder for the case of 9 ought to be 0, but that can't happen with the way we're doing things here, since we're adding positive numbers)
You don't have to limit yourself to multiples of 3. You can take any natural number, and doing this process will give you the remainder after dividing by 9 (again, except in the case whn the number is divisible by 9, in which case you'll get 9 instead of 0).

Try it.
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Is anyone here actually smart and knows math, or are you all fucking morons?
I'm trying to solve a project euler question and want hints.
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Please ignore my post >>13626683
I took a look around the board
You're all clowns

>>13624996
No. Those are all the solutions.

>>13624685
The PRODUCT of the slopes is -1, not the sum.

>>13622483
What the fuck does this even mean

>>13619758
>>13619791
>>13617515
You're the only ones who aren't clowns here
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>>13626720
Make a thread on the problem and enjoy the show :)
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>>13626683
thought i solved 754 in seconds - misread 1000000007 as 1000007 lol
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>>13626750
You should be familiar by now; it's their favorite prime.
Did you end up solving it?
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>>13626750
754 is a classic "switch the sum order", except it's a product.
g(i) = prod(k, k = 1 to i, gcd(k,i) = 1)
And then G(n) = prod(g(i), i = 1 to n)

How many i's is 1 coprime to, up to n? All of them.
How many i's is 2 coprime to, up to n? About half of them.
How many i's is 2 coprime to, up to n? About 2 thirds of them.
How many i's is k coprime to, up to n? About phi(k)/k of them.
Calculating phi(k) for up to a hundred million is not a big deal. The big deal is that "about" that I included.
If k does not divide n, then there's some remaining amount unaccounted for, and that's the crux of the question.
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Behold our generation's Ramanujan.
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What are some good books on Ramsey Theory?
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Does it make sense that 1 is the only number whose value is equal to its number of digits?
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>>13627822
Yes, but it's an artifact of our base 10 representation.
Let B be your base number (in our case, 10).
The number of digits in a natural number n is ceil(log_B(n+1)).
You want n to equal its number of digits, meaning n = ceil(log_B(n+1)).
The LHS is linear, while the RHS is logarithmic, so the LHS will grow too quickly. This means any solution to the equation will be a small number. And the bigger the base, the less the solutions (because then the RHS would be even smaller).

You can test out the first few numbers and see 1 is the only solution in base 10.
However, if you go to base 2, then 1 and 2 are both solutions.
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After literally fighting Stewart at every turn, I've decided to skip his Calculus proofs altogether and just write down all the formulas and learn how to plug and chug them into the problems.

Obviously, I'm very disappointed that I've been reduced from trying to learn math on a conceptual level to just memorizing theorems, but Stewart's Calculus book is just so poorly done (and given the time constraints of a college math class), you just have to adopt a plug and chug mentality and abandon any further attempts to understand the proofs side of things.

I'm incredibly frustrated and angry that Stewart's proofs are far too difficult for the beginner Calculus student to grasp. I've spent literally hours just trying to piece together his logic and thinking. He throws in complex graphs with markings on it out of nowhere, and he refers to past theorems and equations from 4 chapters back without so much as an annotation for reference.
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>>13627965
Could you post something like that so that I can help you with it?

Also: Understanding the proofs is not the point of a calculus class. You ARE meant to "plug and chug". You need to first take a proofs course to understand how logic flows, and then take an undergrad analysis course to go over calculus again, but rigorously. You're getting ahead of yourself.
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>wasted a whole year not doing any math
>almost 23
I am literally never going to make it
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Is this equation true?

(x+y)/(z+y) - (x/z) > 0

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>>13628105
Sorry, forgot pic
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>>13628105
already false for z=x
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I’m doing a take home test and I swear this is the easiest question I’ve seen, unless I’m a dumbass and doing it wrong. Here it is:

Given the function below, find f(7).

f(x)=-x2 - 6x

Do I just plug in the 7?
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>>13628175
Yes. Be confident when you are doing math. When you are wrong, it will be a substantial learning experience and really identifies what you do and do not know.
>>
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What's a common sense description of $\Sigma_2^0$ in the arithmetical hierarchy?
(Not how the predicates are formally defined, but what they can be understood to represent)
>>
My TA is autistic about proper formatting and simplifying. There is a double integral in my homework equal to $\frac{-31}{16}$. They've marked that as wrong, saying I should have written $\frac{1}{16}-2$.
What the fuck? Is there some kind of rule about not allowing negative fractions? This is bullshit.
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>>13629661
nah, sounds more like schizo or TOC faggness. Unless you're using it to prove anything else that requires the $\frac{a}{b}-n$ format thing.
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>>13629667
>TOC
I meant OCD.
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>>13629667
hisp detected
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>>13629661
You've done nothing wrong.
>>
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>>13617361
In my work, I have encountered a pretty interesting problem that I just don't know how to attack, maybe you find it interesting as well:

>You have two time series, that is, 2 structures each with 2 even length vectors. one of the vectors represents a time axis and the other some variable measured throughout time. Example: myWeight = {v: [90, 89, 90.2], t: [10/08/21, 11/08/21, 12,08,21]}

>Now the spacing between the input timestamps (sampling period) can be completely arbitrary for both series.

>The time series can both have completely different lengths as well. For example one can be 3 observations long whilst the other ist 80 observations long.

>They can also have different start and end timestamps.

>Now the problem is given such two super arbitrary time series, perform linear interpolation on both until they have matching sampling periods and matching array lengths. Needless to say all initial samples (both time and and measured variable ) must remain unchanged in value and the relative distance between all timestamps must remain the same.

How the fuck do I do this in such a way that wouldn't take forever for a computer to solve?
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>>13629884
>until they have matching sampling periods
I just realized this after typing the response below: By "periods", do you mean the time between the first and last observations, or do you mean the all the times between consecutive observations? What I'm doing below is the second one.
Here's what I typed:
This seems like the easier part. Instead of dates, assume we're using another unit that's easier to work with, like minutes or hours or days.
You want a sort of "gcd" for the periods of both time series (together).
For example, if the time between 5 observations is 1.5, 2.7, 0.9, and 1.2 (four periods for five observations), then the "gcd" between them is 0.3, so you want to have an observation every 0.3 units of time.
Gcd is only defined for integers, but not the rationals. So the way to do it multiply your times by a common number D to clear the denominators, which would probably be a power of 10.
Once you do that, you take the usual gcd G of the new times, then divide all the times back by D, as well as G.
Now you go to the earliest observation and interpolate every G/D units of time.

For the other parts of your question, it might be impossible, as theymay include extrapolation rather than interpolation.
If one series ends earlier than the other (say the first ends before the second), then it seems you want to extend the observations of the first until they reach they reach where the second reached, so they end at the same time. This means extrapolation.
>>
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>>13630063
Seems like you're working mod 9, but using 9 as a representative of the class [0] = [9] rather than 0.
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>>13617361
Am I retarded or are Geometric Algebra and Geometric Calculus better than regular Algebra and Calculus?
I recently got into it because it seemed like a fun idea to explore
but now im realizing you get things like complex number to fall out of GA for free and without added complexity.
why aren't more people ranting and raving over this?
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>>13630131
>are Geometric Algebra and Geometric Calculus better than regular Algebra and Calculus
Just about every field of math is better than high school algebra and calculus.
>you get things like complex number to fall out of GA for free and without added complexity
What do you mean?
The way I know how it's constructed is that you take the ring $\mathbb{R}\left[x\right]$ and quotient out by the ideal generated by $\left(x^{2}+1\right)$.
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>>13630088
So it's higher math and fully understanding this stuff is years away?
Mod arithmetic alone isn't too difficult for me to get, just wrap around the modulo
But when will I know how it all fits together?
It's just sort of choosing, every number, every other number, every 3rd, etc
Abstract algebra?
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>>13630157
Unfortunately you fucked up the Latex so you're a moron and I won't read your post
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>>13630182
>not reading quality abstract algebra because of a stick up your ass
sure bud
>>
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Ladies and gentlemen: Quality abstract algebra
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>>13630178
What do you mean "how it all fits together"?
You learn about modular arithmetic in either number theory or abstract algebra, and it's important in both.
Number theory would seem like an easier course, though.
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>>13630202
Why are you so mad?
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>>13630206
I'm just having a giggle over here, no need to get so offended.
>>
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>>13630206
If you're not mad at the world then you're not paying attention
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>>13630157
>\left[x\right]
Why did you even bother? It looks almost the same.
$\mathbb{R}[x]$
$\mathbb{R}\left[ x \right]$
>>
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>>13630229
Because this is what it shows.
>>
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>>13630240
Works on my machine
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>>13630213
I will never understand the mind of a person who barges into a thread so mad that he starts cussing everybody out, then pretends to have done none of that.
Sure, you're not mad.
>>
>>13630240
>that R's font
What ancient browser are you even using?
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>>13630243
Famous CS words
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>>13630250
Firefox 90.0.2
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>>13630203
do other mods have the same patterns? or the same iterative pattern mod9 has i think it’s an n choose?
and idr rings and fields but yes arithmetic on a field i think is a ring, or smth, and the mod is a property of the number line
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>>13630283
You basically wrote down the multiplication table mod 9.
I don't get most of your comment, though.
>>
>>13630259
Sasuga Mozilla.
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>>13629884
If you have 3 and then 4 how would you add an interpolation without changing relative distance
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>>13630293
oh i see, i think it seems my expectation was correct these have the same pattern
i’d ask for books but i think i know what to do, i haven’t finished calculus so i don’t have the vocab to communicate wtf i’m thinking
>>
>>13630313
Try starting with a proofs book so you can learn the general math language and how to communicate with mathematicians. Then try a number theory book.

But most importantly: play around. Explore modular arithmetic, and you'll learn new things and get a better feel for what's going on.
Exercise: Try raising the numbers 0 to 9 to the power of 5, and look at the last digit. Notice anything? ;) Then try raising instead to 9, then instead to 13.
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>>13630327
so it goes
1,2,3,4,5,6,7,8,9
and then 2 selects from that list
2,4,6,8,1,3,5,7,9
which
1
>2
3
>4
5
>6
every other
3 is every 3rd
which like of course, we’re adding increments of 3 as we multiply so, yes
>the sky is blue
>>
>>13630157
>What do you mean?
I should have put it this way
>It generalizes complex arithmetic to spaces of arbitrary dimension
or maybe this way
>The subalgebra G2+ is isomorphic to the complex numbers C.
>>
>>13630355
I said "raise to the power".
0^5 = 0
1^5 = 1
2^5 = 32
3^5 = 243
4^5 = 1024
5^5 = 3125
Look at the last digits.
>>
>>13630367
oh yes yes ok
so please tell me more about the 5 here, reflection on the number line?
>>
>>13630394
There is something called "Fermat's little theorem", or Euler's generalization of it.
Basically says if you have a modulus m (in our case m = 10) and a number A coprime to it (so 1, 3, 7, 9), then A^phi(m) mod m = 1.
phi(m) is how many numbers from 1 to m are coprime to m.
In the case of m = 10, we know there are four, which I listed above: 1, 3, 7, and 9.
So A^4 mod 10 = 1 for these numbers.
So if you multiply by A once more, you get A^5 mod 10 = A, meaning if you raise A to the power of 5, you get the same last digit you had.
That explains it for the case of these four numbers. But what about the other six? (0, 2, 4, 5, 6, 8)
Well, that is a little bit more involved, and requires explaining the Chinese Remainder Theorem.
>>
>>13630367
Yeah, numbers are cheeky like that.
Check this trick out:
6^n, n>0; will always end in 6
76^n, n>0; will always end in 76
376^n, n>0; will always end in 376
9376^n, n>0; will always end in 9376
09376^n, n>0; will always end in 09376
109376^n, n>0; will always end in 109376
etc...
>>
>>13630570
You posted 9376 twice.
>>
I'm proving that $\sin u$ is transcendental for any algebraic $u,$ using Lindemann-Weierstrass, and I feel like the proof I've found is too ugly and inefficient.

What I've done is first, of course, reduce the problem to showing that $y=e^{iu}-e^{-iu}$ is transcendental. Then, I show that for any polynomial $f(x)$ over the algebraic numbers, $f(y)$ can be written as a sum of integer powers of $e^{iu},$ i.e. [eqn] f(y)=\sum_{k=-n}^na_ke^{iku}. [/eqn] Recall now that by the alternate form of the Lindemann-Weierstrass theorem, since for each integer $k,$ $iku$ is a unique algebraic number, then the $e^{iku}$ are all linearly independent. Hence, if $f(y)=0,$ then by the linear independence of the $e^{iku},$ we see that all the $a_k$ are zero. Therefore $y$ and hence $\sin u$ are transcendental.

Does anyone have a quicker, more elegant proof?
>>
>>13630587
It's ugly to skip digits.
>>
>>13630570
You basically found an idempotent in the ring of 10-adic numbers, meaning a number which when you square, you get itself back.
There is another one that starts with 5 (from the right), which you can find here: https://www.physicsforums.com/threads/interesting-property-of-idempotent-10-adic-number.877983/

Note how when you add the 6-digit number 109376 to the 6-digit number 890625 in the link, you get 1000001. Not a coincidence ;)

High-level explanation:
The ring of 10-adics is a product of the 2-adics and the 5-adics. Each of these has two idempotents, which are 0 and 1 (as with every ring).
But then the product ring will have four idempotents: 0 = (0,0), 1 = (1,1), ...109376 = (0,1), and ...890625 = (1,0). I put the ellipses on the left because in the p-adics, the digis go infinitely to the LEFT.

You can do operations on them component-wise, and so ...109376 + ...890625 = (0,1) + (1,0) = (1,1) = 1, hence why we got that 1000001 earlier (the more digits you have, the closer it will be to 1 in the p-adic 10-adic topology, so 100000000001 is even closer to 1 than 1000001, because higher powers of 10 are small). In other words, if we call the number you found x, then the number in the link is just 1-x.

And if you multiply, you get ...109376 * ...890625 = (0,1) * (1,0) = (0,0) = 0. Indeed, when you multiply the two 6-digit numbers, you get 97413000000, which is approximately zero in the 10-adic topology (because of all the zeros on the right).

Keep playing around with math :)
>>
>>13630591
That looks like the elegant proof. What are you on?
>>
>Complex analysis
>Partial Differential Equations
>>
Given four points on a plane, how to check whether their convex hull is triangle or tetragon? (BTW, why is it tri-ANGLE, yet tetra-GON?)
>>
>>13617361
Based, I’ve been looking all over for this article. Bieberbach was fucking right damn it.
>>
>>13630943
Use an algorithm to calculate the convex hull.
List has length 3? Triangle. Length 4? Tetragon.
>>
>>13630716
Oh, I just thought there's surely got to be something really quick and obvious that I was missing.
I also refined it quite a bit when I wrote it compared to how I found it, so in my head it seemed more complicated.
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>>13630943
if one of the points lies within the triangle formed by the other 3 points then hull is is triangle

>(BTW, why is it tri-ANGLE, yet tetra-GON?)
maybe because greeks where obsesed with angles before the sides of the shape
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>>13630943
or maybe if the shape formed by connecting each point is convex then the hull is tetragon, otherwise is triangle
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>>13630995
...triangles are convex...
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>>13631012
ah rats, yeah, my bad
>>
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>>13631012
no wait, I actually meant this.
If conecting points gives you a convex shape then hull is tetragon
If not convex the hull triangle
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>>13631019
>If not convex the hull triangle
bullshit
>>
>>13631045
well, sorry for trying to help you with your hs homework, you picky asshole.
>>
>>13629553
>What's a common sense description of Σ02 in the arithmetical hierarchy?
1. Σ01 for an oracle machine that can solve the halting problem
2. there is a computer that takes x and says 010110101010 for a long time but eventually says only 111… iff x is in the set (and keeps saying both 0 and 1 otherwise
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>>13631073
3. typical Σ02 complete problem: Does turing machine number n ever stop giving output? (doesn’t have to halt)
>>
>>13629927
>If one series ends earlier than the other (say the first ends before the second), then it seems you want to extend the observations of the first until they reach they reach where the second reached, so they end at the same time. This means extrapolation
No, I probably explained myself incorrectly.
I just want to extend the number of samples until they both have matching number of samples and matching sampling periods (yes, time between consecutive samples)
And yeah I use dates for example purposes the time unit is usually milliseconds
>>
>>13631161
Ah shit I now see that if the.series have different durations then matching amount of samples does mean extrapolation.
>>
>>13631169
>>13631161
If they DO start and end at the same time, you can still do what I described and get exactly what you wanted.
>>
>>13631073
>>13631045
So are you saying Sigma 2 is like Sigma 1 except the former is a particular machine and the second is for all?

Since Sigma 2 starts with an existential quantifier, I don't understand why it would be "does Turing machine n" (index of the machine).
Existential statements sounds like it should be the problem of finding a particular machine.
>>
Reminder that self-proclaimed ML guru Siraj Raval plagiarized a quantum computing paper and thought he wouldn't get caught if he changed "complex Hilbert space" to "complicated Hilbert space", "logic gate" to "logic door", and "we" to "I".
>>
Trying to solve Project Euler 644: https://projecteuler.net/problem=644
Honestly don't think it's hard. Just know Sprague-Grundy numbers and you're good to go.
Don't really see why it's a 90% difficulty question and only 144 people solved it. I'd say 70% max.
>>
>>13631664
>Just know Sprague-Grundy numbers
>>
>>13631352
>So are you saying Sigma 2 is like Sigma 1 except the former is a particular machine and the second is for all?
no. whereas Sigma 1 is like “this set is computably enumerable”, Sigma 2 is like “this set is computably enumerable provided you have access to the halting problem”
this family of equivalences is called Post’s Theorem
>Since Sigma 2 starts with an existential quantifier, I don't understand why it would be "does Turing machine n" (index of the machine).
a Sigma2 complete problem is a set A satisfying
$n \in A \iff \exists x \forall y \phi(n,x,y)$
where phi is a computable yes-no question (or, to some ppl, a formula in the language called bounded arithmetic). for those 2 and 3 the phi could be something like
$x\leq y Rightarrow (\mathrm{the~machine~doesn’t~output~0~between~times~}x,y$
>Existential statements sounds like it should be the problem of finding a particular machine.
ya sometimes but alot of the time the machine index is the input n, and then you’re looking for either a time (as above) or an input value that does something special or what have you

a typical Pi2 complete problem: Does turning machine #n always halt on every input?
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>>13631678
>a Sigma2 complete problem is a set A satisfying
whops I mean just any Sigma2 problem, not all of them are complete
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>>13631676
That's why I said "70% max".
Look at this:
Requires SG numbers and at 45% difficulty: https://projecteuler.net/problem=509
Requires SG numbers and at 55%: https://projecteuler.net/problem=550

It's a really simple concept, and I'll show you:

Definitions:
Impartial games are games where there are 2 players taking turns, the game is deterministic, is guaranteed to end, and does not distinguish players (e.g. crewmates and imposters).
The mex function takes a set of natural numbers and returns the smallest natural number not in the set. E.g. mex({0,1,2,4,5}) = 3, mex({}) = 0.

Each game state has a Sprague-Grundy number assigned to it, which is defined recursively: Get the game state's successor states (the game states you can immeditely move into), put their SG numbers in a set, and take their mex. That's the SG value for your current state. No need for a base case for the recursion ;)

If your game can be split into two games A and B, meaning on your turn you can pick one of the two games and only play a move in that game, then you can also easily calculate the SG number. Take the SG value for A and the SG value for B and xor them. That's the value.

That's all you need to know.
>>
>>13631709
>>13631676
One last thing: States with an SG value of 0 are losing states. Everything else is a winning state.
>>
>>13631709
>>13631710
Great. You know a trick to simplify the question, but its obvious from the answer rate almost everyone attempting the question does not. Without that trick and trying to solve it brute force computationally, like much of that site requires, is much much harder.
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>>13631729
>trying to solve it brute force computationally
>project euler
Maybe you should check out leetcode?
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>>13631760
They say explicitly it themselves

> Project Euler is a series of challenging mathematical/computer programming problems that will require more than just mathematical insights to solve. Although mathematics will help you arrive at elegant and efficient methods, the use of a computer and programming skills will be required to solve most problems.

You know the efficient method. It's a 90% difficulty because at least 90% of people do not.
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>>13631729
>You know a trick to simplify the question
It's not a "trick"; it's literally the only way to tackle questions about impartial games.
>>
>>13631678
I think I got it, thx
>>
>>13617621
Nice choice
>>
What are courses that are (sometimes) not core courses, but you'd want them to be?

For me:
>Topology
>Number Theory
>>
>>13633523
What third tier shithole do you attend where topology is not a core course?
>>
>>13634142
They only recently made it a core course.
When it was an elective, they rarely offered it.
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>>13634142
hes clearly talking about undergraduate, and lets be honest its a good reason its not because at that level all the topology you need is taught in real analysis.
>inb4 my school had it required
liar. even if it was,
>(sometimes)
>>13633523
Differential geometry in R^2 and R^3. Its a good culminating course for calculus and a good transition to the techniques of analysis and topology in geometry. Sometimes people go straight into manifolds which is fine but a more visual class would be illuminating for many.
>>
>>13634166
lmao, differential geometry was also a core course at my uni (topology and differential geometry were combined in the one course.
>>
>>13634166
>liar
Rude.
>>
How can I use Rouche's theorem to count the roots in the unit circle?
>>
>>13635239
carefully
>>
>>13635239
look up (or derive) Jensen's formula
>>
what is the integral of sin^2(x/2)dx?
>>
>>13635409
sin^2(x/2)=(1-cos(x))/2, so the antiderivative is (x-sin(x))/2. Haven't you heard of google, you retard?
>>
>>13635477
how is sin^2(x/2)=(1-cos(x))/2? I get that 1-cos^2(x/2)=sin^2(x/2)
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>>13635499
https://themathpage.com/aTrig/sum-proof.htm
It's one of the shittest proofs, and is just one of the many reasons why trigonometry is the worst part of all of math.
>>
>>13635499
cos(2y) = cos(y)^2 - sin(y)^2 (double angle formula)
cos(y)^2 + sin(y)^2 = 1
Get rid of cos(y)^2 in the fist formula using the second, solve for sin(y)^2, then substitute y = x/2.
>>
>>13635524
There's a simpler, more natural proof. Let A_x be the transformation of the plane which rotates it counterclockwise by x radians. Then A_x has the matrix
[ cosx -sinx
sinx cosx ]
Now multiply A_x by A_y, read the first column to get the formulas for cos(x+y) and sin(x+y)
>>
>>13635845
Shit, that is really nice, thanks! I don't know why I hadn't seen that before.
>>
>>13635936
Do you know about the exp proof? e^ix = cos x + isinx. So
e^ix e^iy = e^i(x+y) = cos(x+y) + isin(x+y) = (cos x + isinx)(cosy + isiny)= (cosxcosy - sinxsiny) + (cosxsiny + sinxcosy)i.
This is arguably simpler but it's less elementary and perhaps harder to motivate. It relies on analysis which is required to prove e^ix = cosx + isinx, which is quite hard for someone who's never seen it before.
On the other hand, the matrix proof relies on only the following assumptions:
1. Rotation transformations are linear. This is justified by noting that it maps parallelograms to parallelograms (corresponding to f(x+y) = f(x)+f(y)) and that it commutes with scaling (corresponding to f(cx) = cf(x) for c real).The latter is justified by the fact that rotations preserve lengths.
2. Composition of linear maps (in our sitation rotation transformations) corresponds to multiplication of their matrices. This is easy to show directly.
3. The matrix of rotation through angle x counterclockwise around the origin is as shown before. The first column is just the definition of cosx, sinx. The second column relies on the formulas for sin(x+ pi/2) and cos(x+pi/2), which are again simple to justify by drawing a diagram.
>>
>>13636013
>Do you know about the exp proof?
Yeah, I do recall seeing that one. I did want to avoid it in the above post though.
Even though it's essentially the same, I prefer the matrix proof to the exponential proof.
>>
thanks to sci teaching myself undergrad math is ez
but what about higher math?
what if i knew and understood undergrad math, but those youtube lectures only go so far
at what point do i run out of things to teach myself?
do i need to suffer an undergrad if i want to study higher maths?
>>
>>13636714
>but what about higher math?
Pick up a textbook and start reading.
>at what point do i run out of things to teach myself?
Never.
>do i need to suffer an undergrad if i want to study higher maths?
You don't.
>>
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I have two fairly fundamental questions about abstract algebra.
The first is: why do we only deal with structures that have one operation (groups) or two (rings/fields)? It's clear that numbers also have exponentiation and you can define superexponentiation and so on, so why don't we study algebraic structures with arbitary numbers of operations?
My first thought about the above question is that it's because exponentiation is just repeated multiplication in any ring/field, but there are various examples (e.g. complex numbers) where multiplication is not repeated addition. But... why is this? Why aren't there interesting structures with three or more operations?
The second question is: is there any inherent relationship between the two operations in a ring/field? Obviously they have to distribute over each other. But does this imply anything interesting? Does anyone have an example of a ring where calculating a multiplication doesn't involve anything to do with addition?
>>
>>13637025
Isn't there only addition? Subt is adding in the opposite direction (negative)
>>
>>13637025
>Why aren't there interesting structures with three or more operations?
There are. Vector spaces have addition, scalar multiplication, technically negation is an operation, and if your vector space is normed you have that operation, and if it's an inner product space then you have an inner product.
Nothing stopping you from learning stuff with more operations, and no one is saying you should only study structures with only a few operations.
But the point of studying groups/rings/fields is that they're general just enough that you can see them all over the place and can apply the techniques you learn in a lot of situations, yet have enough structure to be interesting.

>is there any inherent relationship between the two operations in a ring/field?
No. Usually, when you define a structure with multiple operations, you want the operations to interact (preferrably nicely). You can define two structures on a set, but if they don't interact in meaningful ways, what's the point of studying them together? You can study them separately.
>Does anyone have an example of a ring where calculating a multiplication doesn't involve anything to do with addition?
For a ring R, take the ring R[x,y]. Then multiplying x and y is just xy.
>>
>>13636714
YouTube lectures can run out, but boks never do.
>>
>>13637137
Good luck defining multiplication in terms of repeated addition in Z[i] or any other nontrivial ring.
>>13637025
Because exponentiation is stepping into the land of analysis. But it's used in algebra too, for example you can take p-adic completions of number fields and do exponentiation and logarithms in them by looking at the power series sum x^n/n!.
>>
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>>13637165
I'm talking about binary operations. Yes negation is a unary operation but it just maps an element to its inverse under addition so it doesn't contain any information that isn't already contained in the binary operation of addition itself.
Vector spaces / modules are defined on top of a field / ring, and I'm asking about the underlying structure itself (the field / ring).
I might have phrased it badly. You can consider a set with one binary operation which is a group (assuming the operation satisfies the axioms). You can also consider a set with two binary operations to get a ring or a field (again, assuming the operation satisfies the relevant axioms). Both of the these are studied widely. I've never heard of anyone studying a set with three binary operations. Is there an obvious reason why these aren't interesting objects?

>>13637180
Sorry I'm being dim here but I don't follow what you mean by 'exponentiation is stepping into the land of analysis'.
To state the obvious: we can forget about addition for the time being and consider real numbers with multiplication and exponentiation, and we'll get a perfectly serviceable (if artificial and redundant) field. So there's nothing magic about exponentiation itself that makes itself un-algebraic. It seems to me like it's the interplay from having all three operations together in play at the same time - addition, multiplication, and exponentiation - that's interesting. But we just ignore that when we start studying groups and rings.
>>
>>13637248
>I've never heard of anyone studying a set with three binary operations. Is there an obvious reason why these aren't interesting objects?
https://en.wikipedia.org/wiki/Algebraic_structure#One_set_with_operations
>>
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>>13637308
The only examples in that list with more than two binary operations are lattices with meet/join. These are abstractions of the max/min operation, and they're not really what I'm asking about.
I'm asking if there's an abstract structure consisting of a set with three binary operations which are analogous to addition, multiplication, and exponentiation of numbers. Lattices are about ordering. The part of the real number line they capture is the order. It isn't the exponentiation operation.
Since it doesn't look like this is a thing that people study, I'm guessing people have tried this out and not found anything interesting. Exponentiation in a ring or field is only ever dicussed as repeating the multiplication operation in that ring/field. But since the multiplication operation is frequently defined not in terms of addition, I'm wondering why I never see a set equipped with a third binary operation (which we'll call exponentiation), which lives alongside the two which make it into a ring/field, which is not defined in terms of multiplication.
>>
>>13637416
https://en.wikipedia.org/wiki/Ring_(mathematics)#Rings_with_extra_structure
>>
>>13627473
also looking for books on infinite combinatorics
>>
>>13638240
Why do you seem to think this is a hub for experts?
This is 4chan. It's full of morons.
Go to mathoverflow, or hell, even Reddit.
>>
I have $n$ distinct singular points of a curve $f$ of degree $d$ lying on a line $l$. I'm trying to prove that if $2m > d$, then $l$ is a factor of $f$. I really have no idea how to approach this. I really don't see how the inequality $2m > d$ helps me here. Any help?
>>
>>13638370
is it because all 2m are even?
>>
>>13637416
Exponentiation is really more of a function than a binary operation, morally speaking.
>>
>>13638554
Binary operations are functions.
>>
>>13638643
The point he's trying to make is that the base and exponent may come from different sets, rather than the same set (like with binary operations).
For example, if you're exponentiation x^y mod p, p a prime, then x comes from Z/pZ, but y comes from Z/(p-1)Z (or Z).
So exponentiation is not a binary operation in general.
>>
>>13638370
Maybe? I do know the fact that if $l$ meets more than $d$ times, then $l$ is a factor of $f$. I just haven't completely figured out how to use it.
>>
>>13638659
Oh, I see, thanks.
>>
>>13638281
You have to go back.
>>
is there a standard algorithm to find a particular solution of any riccati equation?
>>
>>13639806
mmm the ricotta equation
>>
bros... i forgot.... to do maths today.......
>>
>>13640363
there is still time
quick use cylindrical coordinates to parameterise a solid tetrahedron
>>
is it reasonably common to self study no degree and become an actuarial fellow?
>>
>>13639797
Back to the past
Samurai Jack

Anyways, you're a fucking moron if you think this is a place to get *actually* intelligent answers.
>>
when do the undergrads get back
>>
Proving that a metric obeys the triangle inequality. I can do it by contradiction but I'd like to do it directly - any (general) tips for doing so?

Would rather not post the question
>>
>>13641199
sorry i meant to post this in /sqt/ but I had both threads open
>>
>>13641124
Middleschoolbros, when will high schoolers come back to /mg/ so we can bask in their amazing knowledge of calculus and matrices?
>>
>>13641199
>any (general) tips for doing so
>Would rather not post the question
Then I can't help you with anything more specific.
>>
>>13641199
do calculations on 3 arbitrary points using properties of the metric
thats about as good as it gets
>>
>>13641199
I'm trying to recall any "this is a metric" proofs that explicitly work by contradiction and I can't recall any besides the length space one.
But I don't think you can do that one without contradiction.
>>
>>13641199
Any metric automatically satisfies the triangle inequality by definition.

Unless you're saying you have a function $d: X\times X \to X$ and you want to show it is a metric by showing it satisfies the triangle inequality.
>>
>>13641417
Why is it that only mathematicians use the phrase by definition correctly? Everyone else uses it and when asked by what definition they are stumped.
>>
>>13641417
Not him, but yeah, obviously.
You a priori know it's a metric, but you're asked to prove it.
>>
>>13641441
>redefine a phrase for formal use in mathematics
>wtf why noone else use it right
hmmm
>>
>>13641441
No idea. Probably because mathematicians use strict definitions more than any other discipline

>>13641465
Yea you're right. I just read it too quickly. I also cannot recall any time I have ever proven the triangle inequality for a metric with contradiction. In general, fix a $x,y,z\in X$. You want to show $d(x,y) \leq d(x,z) + d(z,y)$. If there is another metric hidden inside of $d$ then you want to take advantage of that. For instance, if you want to show that the $d_1(f,g) = \int |f(t) - g(t)|$ satisfies the triangle inequality, then you should use the fact that the absolute value already satisfies the triangle inequality and so $d_1(f,g) = \int |f(t) - g(t)| = \int |f(t) - h(t) + h(t) - g(t)| \leq \int |f(t) - h(t)| + \int |h(t) - g(t)| = d_1(f,h) + d_1(h,g)$.
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I've been stuck on the following problem for the past few hours. There is a canonical map $\phi: T_p\mathbb{R}^n \to mathbb{R}^n$ by $\sum_{i=1}^n a^i \frac{\partial}{\partial x^i} |_p \mapsto (a^1, ..., a^n)$. If $L: \mathbb{R}^n \to \mathbb{R}^m$ is a linear map, then I want to show that $\phi \circ L^* = L\circ \phi$. On the RHS, we have $L(\phi(\sum_{i=1}^n a^i \frac{\partial}{\partial x^i}||_p)) = L((a^1, ..., a^n))$. I can't seem to figure out the LHS. We have $\phi(L_*(\sum_{i=1}^n a^i \frac{\partial}{\partial x^i}|_p)) = \phi(\sum_{i=1}^n a^i L_*(\frac{\partial}{\partial x^i})) = \phi(\sum_{i=1}^n a^i \sum_{k=1}^m a_i^k \frac{\partial}{\partial y^k}|_{L(p)})$. I don't really know where to take it from here.
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>>13637416
You don't study/define an algebraic structure just cause
You study it because it arose time and time again in many places
You haven't seen algebraic structures with 3 binary operations? Probably because they never arose anywhere.
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>>13641824
Idk wtf this is, but did you try evaluating at the basis elements to see if they match?
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>>13641824
How's $L^*$ defined again?
$L^* = \phi^{-1} \circ L \circ \phi$?
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>>13641918
In general, if $F: M\to N$ is a $C^\infty$ map between two manifolds, then $F_*: T_pM\to T_{F(p)}N$ is a linear map of tangent spaces defined as follows: Let $X_p\in T_p$ and $f\in C_{F(p)}^\infty$, then $(F_*(X_p))f = X_p(f\circ F)$. When F is a map between two Euclidean spaces, then its differential $F_*$ is represented by the Jacobian matrix.
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>>13641936
What does Jacobian mean? I saw it contrasted to something called "Hessian" in this artificial neural networks book.

idk what this stuff means though. I barely know basic math. pls no judge.
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>>13642135
https://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant

You should definitely learn at least calculus and linear algebra
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>>13626720
>You're all clowns
honk honk smart man
*chucks a pie filled with human feces in your face*
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>>13642148
Okay. Will do. What does Hessian mean? I recently read about Cramer's law. I forgot what it is for though.
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>>13642151
The Hessian of a function f is the matrix whose (i,j)th entry is the partial derivative $\frac{\partial^2 f}{\partial x_i \partial x_j}$
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>>13642135
You know how a single-input-single-output function f(x) has a derivative? (Ignoring all the technicalities)
Well if you have a multi-input-multi-output function, then you can also define its derivative, and that derivative is a matrix, called the Jacobian.
The Hessian would be its second derivative.

(Lots of details omitted)
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>>13642189
Oh. Well okay.
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>>13641251
I like middleschool girls. Especially blondes.
>>
lets say you have a back and forth with someone in 4chan
then you decide to talk in discord
how do you verify that its actually him/her?
i feel like this is not possible because every observer have basically the same amount of information
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>>13642624
Correct.
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>>13642624
unless they were tripfagging you cant really be certain
>/her
no she wont show you her tits
>>
Is there a $\omega -1$? What is it?
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>>13643678
There isn't an ordinal whose successor is w. In other words, w is not a successor ordinal. So no.
But for ordinals a, b, b<=a, you could define a-b to be the unique ordinal c such that b+c = a. In that case, w-1 = w.
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Studying for first calc 1 test and am totally lost
The text just has the slope formula
(y2-y1) / (x2-x1)
I'm not sure how to use that for these questions.
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>>13644068
how do professors of calculus cope with the fact nothing they ever do will be as good as khan does it?
they even say go, do khan, do it
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>>13644113
huh?? 4khan?
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>>13644068
I wish I could just take the exams and not be allowed to attend the lectures and just pay a fee for the tests. Instead I have to shill out 2k just for this one math class
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>>13644222
no you must first watch the lecture where 30% of time is dedicated to useless administrative concerns
you must then watch khan academy to learn something, do khan practice problems to learn, and then do your actual school homework for certification of learning
btw the exams have retarded problems where a calculator is required, and are designed to trick you because we train thinkers here
oh yeah i’m the teacher but you’re the only whose responsible to learn the material, and also learn all this new remote learning software that i just don’t have time to futz with so i’ll waste your time while i learn it
no i don’t answer questions, just go watch khan
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>>13644222
for calculus you can use khan and then take AP exams for 1/2
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>>13643713
So $1+\omega = \omega = \omega -1 \neq \omega+1$? I thought infinite ordinals commuted under addition?
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>>13644321
Yes. They don't.
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>>13641883
Literally just consider basic arithmetic of integers. You have addition, multiplication, and exponentiation. Those are three binary operations defined over one set.
I hate this fucking place so much.

For /ants/ anon: this falls apart if you start to consider complex numbers because of the difficulties involved in defining exponentation with a complex number as the exponent (the solution is nonunique).
In general as you go from integers -> rationals -> reals -> complex numbers -> quaternions -> octonions, you first gain then lose structure, with the sweet spot being somewhere around rationals or reals depending on your views on infinity.
Go from reals to complex numbers and you lose the ordering and also exponents.
Go from complex numbers to quarternions and you lose commutativity.
Go from quaternions to octonions and you lose associativity which is pretty fatal.
The question you should probably zero in on is why complex exponentiation falls apart and you need to find an explanation that works for you. And think about why it isn't commutative over the reals, unlike addition and multiplication.

>>13638659
In most cases exponentiation is a binary operation. It just isn't commutative in general.
In general exponentiation has to do with spaces of morphisms and their sizes. The broadest definition is category theory where A^B is the object that encodes all morphisms from B->A. e.g. in the category of sets, A^B is the set of functions from B->A, and the notation here comes from the usual exponentiation of numbers because there are a^b functions from a set of cardinality b into a set of cardinality a. These exponential objects are present in cartesian categories and tend to make things nice. The more structure the objects have, the less likely you are to have exponential objects.
All of this is rather dodging the fundamental question posed by /ants/ anon.
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>>13644068
this is calc 1?
this is for 13-14 year olds in the uk
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>>13644068
Part (a): P = (15, 250) and Q is the five other points in the table.
(x1, y1) = P, so x1 = 15 and y1 = 250. x2 and y2 are the coordinates of whichever point you're currently using for Q.
Use the slop formula to calculate the slopes of the secants.

Part (b): The slopes of the secants approximate the slope of the tangent at P. They want you to get a good approximation of the slope of the tangent. Take the secants of the two nearest points (which you calculated in part (a)), and take their average.

Part (c): Plot the points in the table. Try to make out what the shape is. Find out what the slop at P is.
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>>13645816
>Literally just consider basic arithmetic of integers. You have addition, multiplication, and exponentiation. Those are three binary operations defined over one set.
Yeah, but how often does this structure arise that you have to abstract and generalize it and give it a name?
He and everyone else already know about the integers already, you pretentious dumbass.

>All of this is rather dodging the fundamental question posed by /ants/ anon.
He doesn't even know what wants to ask. Every time you answer him, he says "no not like that" and modifies the question. Stop trying to act like YOU even know what he wants to ask.
>>
Is it possible to "count" in a non-integer base as we do with numbers in integer bases? What would that look like?
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>>13646245
shapes
i’d go line, cross, triangle, square, etc
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>>13646245
Messy, probably.
But I think algebraic integer bases would probably be nicer. See the examples here: https://en.wikipedia.org/wiki/Golden_ratio_base
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Is type theory a tough to get into or a dead field? Should I start thinking of arternatives for postdoc? I live in a French speaking cunt bordering France.
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>>13644113
High schooler, go away.
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>>13646932
>Is type theory a tough to get into or a dead field?
>Should I start thinking of arternatives for postdoc?
If you’re far enough along in a type theory phd to be asking the second question, you probably can answer the first question better than anyone else here
>>
Wtf are homology and cohomology groups? (outside of topology)
>>
>sqrt2 cannot be expressed as a sum of rational numbers
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>>13647459
What am I looking at?
>>
>>13642135
>>
shapes

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>>13647417
Basically, they count holes.
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>>13647459
What's that pic supposed to prove?
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>>13647996
How tf does a group have holes?
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>>13647417
>(outside of topology)
they measure exactness of (co)chain complexes.
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>>13648074
>(co)chain complexes
What are they conceptually? Like how would one look at these topological notions and find something there to be abstracted and applied to general groups, without anything being related to topology?
I can read the definitions and go through them again and again, but who could even come up with such a definition?
Cohomology groups seem ubiquitous in algebra and I don't have any idea what they mean, and I feel like I'm missing out on a lot.
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>>13648065
They don't, they count the holes of your complex.
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>>13648091
learn algebraic topology
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>>13648159
https://en.wikipedia.org/wiki/Group_cohomology
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>>13648161
No
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>>13648167
Original post referred to cohomology groups, not group cohomology.
There isn't really an intuition for what (co)homology is in general, asking what cohomology is is like asking what a space is, it doesn't make sense.
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>>13648170
but there's where the answer to your question lies
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>>13648203
No
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>>13647337
I'm finishing undergrad, but I'm reasonably convinced that I can do the PhD, in maybe not at a cream of the crop, but at least a relevant school to my field of interest.
I'm just asking how's the research after that. I know that INRIA affiliated people shit out interesting papers regularly, but I'm wary of, for one, the amount of competition for it, since all the CS grads are also considered for the position, for two, that it's actually a niche field that relatively few people actually care about, and that it's going to die out if the interest dries up.
I asked some people I know, and the opinion used to be "nobody outside CS cares", but now it seems "gradually more people are becoming interested". Granted, the person who said that doesn't work in the field, but his friend does, so I dunno.
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>>13648376
Dropping in on your discussion, and idk if type theory is much the same as homotopy type theory, but apparently the US Department of Defense is interested in the latter: https://aperiodical.com/2014/05/make-math-%C2%AC-war-american-military-invests-in-homotopy-type-theory/
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>>13648422
Sure, HoTT is a hot meme, even M\$ is working on it, I'm just wondering whether there's something an outsider can't see that'll make it go the way of the Meme Learning bubble.
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>>13648453
Microshaft Research has been funding weirdo programming language and type theory research for decades now.
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>>13648461
But they're not like Goolag in that they don't regularly pick something up just to kill it off, there's usually some results like Z3 at the end.
The French guy I was talking about who works in the field, I don't know what exactly he does, but he said Lean is not bad at all. I thought it was some of the usual NIH Microshart from what people said, but if someone who's not a undergrad overdosing on memes says it's alright, then I'm not inclined to believe it's just a passion project.
Now, will I have to be a slave to Kevin Buzzard translating tedious proofs 24/7, or is it still not an oversaturated field? More interested in the situation in Europe.
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>>13647459
>sqrt2 cannot be expressed as a sum of rational numbers
Yes. The sum of two rational numbers will always be rational. Therefore the square root of 2, being an irrational number, cannot be. expressed as the sum of two rational numbers.
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>>13649399
>proof by cases
>first case is a subset of the second case
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>>13649406
You must be fun at parties.
>>
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I'm looking for any function as follow
- $f:\mathbb{R}^{+}\times\mathbb{R}^{+} \to\mathbb{R}^{+}$
- $f(x,y)=f(y,x)$
- $y=0 \Leftrightarrow \frac{\partial f}{\partial x} \approx e^{x}$
- $y=x \Leftrightarrow \frac{\partial f}{\partial x} \approx 1$

I'm retarded. sorry if it doesn't make much sense
>>
Is Ebbinghaus' "Mathematical Logic" any good? It's not on the /sci/ wiki list for Logic books, but it's pretty recent so i wouldn't be surpirsed if it wasn't on there.

Anybody have experience with this book, either in a course or self-study?
>>
how does one get good at the maths :(
>>
>>13649593
it doesn't
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>>13650300
he(she) is looking for a function from 2 positive real numbers to a positive real number (and 0) such that

a) it's symmetric wrt $y, x$
b) At $y = 0$ the partial wrt $x$ is $e^x$
c) Similarly partial wrt $x$ is 1 if $y=x$

>>13650277
you don't; nobody is *really* good at math, they just larp that they are in order to convince you that they are "smart" by showing you a piece of paper issued by an institution specializing in self-gratification of know-it-all assholes, emptying grandparent's retirement funds, and ruining people's credit scores.
>>
>>13650332
>(she)
also
>studying in america
>>
>>13649593
Did you try just solving the series for enough terms to work with?
Assume the function has the form $f(x, y) = \sum _{i, j}a_{i, j} x^i y^j$. By $f(x, y) = f(y, x)$ you know that $a_{i, j} = a_{j, i}$. $y=0 \Leftrightarrow \frac{\partial f}{\partial x} \approx e^{x}$ tells you that $a_{i, 0} = \frac{1}{i!}$. Considering $g(x) = f(x, x) = \sum_{n} \sum_{i + j} a_{i, j}x^n = \sum_n a_n x^n$. You want $\frac{\partial g}{\partial x} = 2$, so you can compute, for example, $0 = a_2 = a_{0, 2} + a_{1, 1} + a_{0, 2} = 1/2 + a_{1, 1} + 1/2$, hence $a_{1, 1} = -1$, and so on and so forth.
>>
i have not a clue on how to do my physics homework about vector addition (and subtraction). it says a line is going 20m @ 90° and then 30m @ 60°. im supposed to find the length of the start to the end. i put it into a vector calculator and it tells me the length is 39m. i put it into a triangle calculator it tells me the length is 26.5m. i dont know which one is right or how to do it on my own. my professor was out today and he emailed us a worksheet and told us to try it out.
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>>13650538
nvm i figured it out. it was because i used 30° and not 30° + 90° for the cosine law. lel
>>
Is there any relationship involving [eqn]f^{-1}(U \times V)[/eqn] and [eqn]f^{-1}(U) \times f^{-1}(V)[/eqn] ?
Whe. Do you have the equality between them?
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>>13650600
This doesn't make sense.
What is the codomain of f?
You have three different sets for the codomain.
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>>13650607
fine, smartass
Let [eqn]f: A \rightarrow X \times Y [/eqn], [eqn]U \subset X[/eqn] and [eqn]V \subset Y[/eqn] such that [eqn]U \times V \subset f( X \times Y )[/eqn].
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>>13650615
I'm being serious; I'm not trying to be a pedant for the sake of it. This does not make sense.

In your initial question >>13650600, let's say U and V were the real numbers R. You're asking about the preimage of RxR and the product of the preimages of R. f can't both give you a single real number AND a pair of real numbers. It doesn't make sense.
Moreover, the preimage of UxV lives in A, but the second set lives in AxA, so they can never be equal.

(In your second post, I think you meant to say U×V⊂X×Y in the last line)
>>
Is university level math something that you can understand if you power through and dedicate hours to it, or is it possible to be "just too stupid" for it?

I'm going to university for the first time next week and I'm terrified I'll be too stupid for it and won't make it.
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>>13650819
Only if youre honest with yourself and understand the amount of work ahead of you, and actually DO that said work, youll make it.
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>>13644068
If you can't do algebra 1 material, get the fuck out of calculus you midwit
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>>13645962
It's algebra 1 content, not calculus. Sad
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>>13650819
>Is university level math something that you can understand if you power through and dedicate hours to it
Yes
>I'm terrified I'll be too stupid for it and won't make it
They already expect you to be stupid, and they'll teach you accordingly. The math you'll take at the beginning will be a review of stuff you took in highschool.

Chill. You'll be fine.
>>
>>13650600
>>13650615
lmao what a brainlet
>>
Do any of you use your skills to btfo brainlets at poker?
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>>13651608
Stop watching dumb Hollywood movies to form your idea of what a mathematician is.
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>>13651792
>>13651792