The title says it all. Post your mathematical problems or thoughts and have people discuss them with you.
No college advice, no textbook recs.
>>7909437
>tfw want to be good at math and other sciences
>I'm shit at it
oh well
>>7909437
What you imagine this thread to be:
>advanced university level math questions
>informative answers everyone can learn from
>inspirational discussion
What it actually will be:
>muh calculus homework
>everyone ignores the very few advanced questions
Welcome to /sci/.
>>7909442
Yeah, I have started some before and this is exactly what happened but who knows..
>>7909439
The shitposting has already started.
Here's something cute I've been wondering about:
We have formulas for [math]\sum_{i=1}^n x^a[/math] for any [math]a\in\N[/math]. Can you find a similar formula for [math]1^1+2^2+3^3+\cdots+n^n[/math]?
Using linear combinations of terms in the sequence of such sums I found that [math](1^1+2^2+3^3)2^3=4^4[/math], for what it's worth. Can a geometric argument be given as it can for the sum of powers, perhaps using cubes in n-space? Can calculus somehow be made to work for this in the same way it does for the sum of powers when when [math]x^x[/math] is infinitely differentiable?
>>7909442
>everyone ignores the very few advanced questions
you know why this happens? Because the morons who post it don't bother defining what they're talking about.
There could be 10 anons who studied math but not in english who could help you, or who could provide ideas, but the person asking the question almost never bothers defining the terms and notations.
>>7909459
Obviously I meant to write [math]a\in\mathbb{N}[/math] where [math]\mathbb{N}[/math] indiciates the natural numbers.
>>7909459
>Can you find a similar formula for
I can't fucking solve basic algebra, anon
>>7909473
Thanks, your contribution really justifies the time I spent typing up that question.
>>7909459
looking into this now.
I'll go shop for some food and keep looking after.
>>7909459
This is interesting. I'll look into it at the airport today and post what I have tonight
Bump
Does anyone have a simple proof for the fact that the number of monic, squarefree polynomials of degree n over [math]\mathbb F_q[/math] is [math]q^n -q^{n-1}[/math] ? I have a fun proof of the fact that it is asymptotically equivalent to it but I think there should be a simple and direct proof of the first identity by counting (considering how simple it is).
>>7910641
Square free reminds me of the mobius function and that form you have written reminds me of the euler phi function, the number of numbers relatively prime to a number. So if q is prime and n is an exponent greater than 1, the Dirichlet convolution is with N(n)=n
[latex] (\mu \star N)(q^k) = \varphi(q^k) = q^k - q^{k-1} [/latex]
>>7910684
Oh shit I had not thought of that ! Funny enough, my proof of the asymptotic estimate actually used some sort of Mobius function and convolution but for some reason I didn't see this.
Thanks, I'll look into it !
>>7910748
Yeah honestly I am just pattern matching I have no idea how to prove this I just thought it seemed the same somehow. Also testing latex [math]\mu(n)[/math]
>>7909459
Somewhat interesting.
I found that if
[math]\displaystyle a(n)=\sum _{i=1} ^{n} i^i \\ \mathbb{then} \\
\frac {a(n+1)} {a(n)} = en+O(e/2) [/math]
But that's all I've gotten so far.
What is d in d/dx? You cancel it out right? Then it becomes x? :S
>>7910826
You probably misread his post.
Generating functions look like [math] x^1+x^2+x^3...[/math] e.g. [math] 2^1+2^2+2^3...[/math], he asked about [math] 1^1+2^2+3^3...[/math].
I have proof that 0/0 = ∞
first, we set up an algebraic equation
y = x/0
(y can be any number, we will be figuring out what x is)
we need to multiply by 0 on both sides to make x by itself
0(y) = (x/0)0
0y is 0, and (x/0)0 is x, since multiplying a fraction by its denominator gives you the numerator
0 = x
so
0/0 = y
since y can be any number, and 0/0 = y, then 0/0 is any number, meaning its ∞
>>7910842
my bad. might still be fruitful though
>>7910866
Wouldn't (x/0)0 just equal 0?
I don't think you thought this one through.
>>7910886
no, because
>(x/0)0 is x, since multiplying a fraction by its denominator gives you the numerator
you did read it fully, right?
>>7910889
Yes, but any number multiplied by 0 results in 0. Are you smarter than a 5th grader?
>>7910891
Why I'm quite smart, especially if I managed to solve the 0/0 debate.
>>7909437 Is there a decent model for the distribution of prime numbers? Does the distribution have a pattern?
>>7910907
Can you clarify?
>>7910866
>>7910886
>>7910889
>>7910891
>>7910895
>>7910898
Time to learn limits.
So a limit is used when you have functions of indeterminate forms, e.g. 0/0, infty/infty, 0*infty, etc. You use a limit to see what the [italics] ratio [/italics] is approaching.
For example, [math]\lim_{ x\rightarrow \infty} 1/x = 0 [/math] or even [math]\lim_{x\rightarrow 2} x^2 = 4 [/math] even though that is defined there.
However there's a clear problem, what if a function is approaching something differently depending on which side you look at? An example would be 1/x near 0, it's at negative infinity and infinity.
Here's how you solve that simple issue: [math]\displaystyle \lim_{x\rightarrow 0^+}1/x=\infty \ \mathbb{and} \ \lim_{x\rightarrow 0^-}1/x=-\infty [/math]
Ok, so what would [math] \lim_{x\rightarrow 0} 1/x [/math] be? Undefined, since the limit from the left and the limit from the right don't match up.
Neato, so what about holes in graphs, like in polynomials? [math] \lim_{x\rightarrow 1} \frac {x^2-1} {(x-1)(x+2)} [/math]? well if you take the limit from both sides (or cancel terms which is only allowed for limits), you'll see that it's 2/3. Even though I'm dividing by 0 on top and bottom, the ratio is approaching a number, which is 2/3.
>>7910932
shut the fuck up
Does the 'coherent' in coherent homotopies have anything to do with the 'coherent' in coherent sheaves?
>>7910932
Why don't we just make up a new number, say [math]\alpha[/math], which is a solution to [math]0x=1[/math]?
>>7910941
https://en.wikipedia.org/wiki/Wheel_theory
Any fun math books that incorporates games (like chess)?
How to calculate the volume traced by a unit circle that rotates x rad while translating a distance d?
>>7910969
Arc length s = radius r * angle theta, in radians.
For unit circle, that's actually the definition of a radian.
Things get uglier if r changes as you do the sweep, though.
>>7910951
That doesn't answer my question. It just dicks me around by defining / to be a unary operation. I am asking for a multiplicative inverse of 0.
>>7910978
If you have such a unary operation consider the well-defined expression 1/0. Certainly we have 0(1/0)=1, which is what you asked for.
>he doesn't know that 1+1=0 and he's been lied to all along
https://www.youtube.com/watch?v=ICv_0ln_yzw
>>7910834
retard
How exotic can pullbacks be depending on the category that permits them? Are they basically just always going to turn out to be well-behaved product-esque objects?
Math is fucking hard.
>>7910975
How to calculate the volume?
r is constant.
>>7909437
Why is a point on the edge of a set of say the open unit circle in the real number system considered near that set?
Nearness requires that the of the "area" around the point is contained in the set, so I understand that if the point is on the edge of the set -even if not within the set itself- that some of its area will be contained in the set, but most of the area around the point will be in the opposite direction away from the set's domain, so why is it considered near if the area around the point is not entirely within the set?
>>7910988
or consider the limit of x/x as x approaches 0
>>7911328
>well-behaved product-esque objects?
It's a "well-behaved product-esque object" by definition: The universal property of it involves projections (p1 and p2 in your image).
However, some points (and "=" denotes isomorphic):
- Keep the notion of
https://en.wikipedia.org/wiki/Concrete_category
in mind. If a category is concrete, then I guess you can argue that all objects defined in this and that way are like their setty analogue.
The standard counterexample, the homotopy cateogry, still has setty objects. The same is true for categories and their equivalences. If you go along these lines, maybe you're already at the model categories spiel. I can't tell you counterexamples here, but note this:
- Even in Set, I think it's not very healthy to force and think of the object P as subset of X x Y.
If consider Y is initial in Set, e.g. Y={0}. This is the case for the pullback in terms of which the subobject classifier Z={0,1} in Set is characterized. Then, as X x {0} = X, the pullback P is some subset of X x {0} = X, i.e. something smaller than X. It's still true that then an element u in P is technically a pair <x,0>, but thinking this way is almost evil (in the technical sense of the word).
- Depending on where you come from, I'd on the other end keep the cases in mind where you're supposed to think geometrically. Where instead Z is big and X is "in" Z, such as with pullbacks of fibre bundles in geometry. X->Z maps the place X into Z, the map Y->Z is from a space like a vector space to some place Z, and then the pullback P, is "X-indexed subsets" Y, i.e. the maps P->X are pulled back from Y->Z along X->Y.
- If you're only in Set and you really speak of sets as such and not something modeled via sets, then I'd say the proper way to think of the pullback is as solution set to equations:
The universal property that the diagram commutes says
f . p1 = g . p2
and if p1 and p2 are really just projections, then this says <x,y> in P is solution to
f(x)=g(y).
>>7909459
I found that the terms I tried to compute this could be written as the sum of two squares.
1 1
2 5 =1^2+2^2
3 32 = 4^2+4^2
4 288 = 3^2*2^5 = 12^2+12^2
5 3413 = 7^2 + 58^2
6 50069 = 62^2+215^2
7 873612 = ???
wolframalpha doesn't give anything for the last one. Maybe my hypothesis is wrong
>>7911644
Well a point that is in the boundary of a set but not in the set itself it is as "near" to the set as point can be without actually belonging to the set. If I understand you correctly the description of "nearness" you are trying to give is a topological one. It says that EVERY open neighborhood of a boundary point (or any limit point) of a set cointains some points of the set, which means no matter how small you choose your neighborhood, part of it will always overlap with the set, and I think thats pretty much as near as you can get to a set. Sure maybe the rest of the neighborhood is a million times "bigger" (whatever that means in this context) but that doesnt exactly say anything about how near the point is to the set. The crux is that the neighborhood here can be choosen arbitray.
If we have an actual way to measure distance like a metric, then this concept of nearness gets much more concrete. We can define the distance between a set [math] M [/math]and a point [math] x [/math] as [math] dist(x,M):=\inf_{y\in M}(d(x,y)) [/math] where [math] d [/math] is a metric. The the distance between any boundary point of a set and this set is zero, which is as close is it gets.
>>7911678
I think it's hard because not even the integral of n^n has a nice expression
>>7911677
Thanks for the well-written response, gives me lots to think about
What conditions do a,b and c have to meet to solve that equation when all variables are positive(a,b,c,x,y,z,s1,s2,s3)?
How can I solve this? Help /sci/bros
So I'm just getting into infinite polynomials (Taylor series etc). What kind of functions *can't* be estimated using these? Any functions have interesting results after summing enough terms?
I only learned about these a couple days ago so I know nothing.
>>7910866
>axioms
>>7912441
y + z - ax > 0
x +z - by > 0
x + y - cz > 0
>>7912446
First of all always call them power series instead of infinite polynomials, polynomial strictly refers to a finite number of terms.
The only data required to write down a Taylor series is each of its derivatives at a specific point, so the function must be smooth (infinitely differntiable) at that point.
I don't know what kind of interesting results you're looking for but summing enough terms just gets you closer to the function you started with (there is an error term associated to Taylor series that bounds the difference)
Is there a nice way to show that $E_n = \frac{1}{C}e^{k2^n}$ will take a big more than log(N) iterations to converge to 0?
>>7912464
Whoops [math]E_n = \frac{1}{C}e^{k2^n}[/math]
>>7912446
Every function that is in the space of infinitely differentiable functions can be approximated with a taylor polynomial. However, there is a radius of convergence that is important to consider.
Interesting properties: using taylor polynomials, you can calculate the value of infinite sums more easily by noticing similarities to the taylor polynomials of familiar functions. Additionally, using taylor polynomials, you can integrate e^(x^2), though you get a series result.
>>7912446
Interestingly, a whole bunch of functions cannot be approximated by taylor series (in the sense that the series does not converge to the value of the function).
The first and most obvious reason why a function might not be equal to its taylor series is if it is not infinitely differentiable. The second most obvious reason why this might not work is if the Taylor series does not converge at the point you are considering: For example, consider the function [math]f: x \mapsto \frac{1}{1+x^2}[/math]. The Taylor series for this function is [math]\displaystyle \sum_{k \ge 0} (-1)^k x^{2k}[/math] and does not converge if [math]|x| \ge 1[/math] even though f is defined everywhere on R.
Finally, you have cases where the function is infinitely differentiable and the Taylor series converge but the function simply is not equal to the sum of its Taylor series.
For example, if you consider the function [math]f: x \mapsto e^{-1/x^2}[/math] extended to R by setting f(0)=0. Then f is infinitely differentiable (well-known but not so easy exercise) and we have [math]f^{(n)}(0) = 0[/math] for each n. In particular, its Taylor series at 0 is defined everywhere and is the zero function. Therefore, f cannot equal its Taylor series anywhere else but at 0 since it never vanishes outside of 0.
This is all to show you that infinite Taylor series are things to be *very* careful with.
That being said, Taylor *expansions* (that is truncated Taylor series) are extremely useful and are one of the main tools for working with differentiable functions, as Taylor's theorems tell you that any continuously differentiable function is approximated by its Taylor expansion and gives you an estimate of the error (the various formulations give you several expressions for the error term, each more or less useful depending on the context).
can someone solve this?
>>7912458
the conditions need to be in terms of a,b,c only, for example, for example, something like a*(b-1)*(c+2)>1
>>7912565
I don't have any suggestion for a solution, but here's an obvious observation: a has to be even, because otherwise you have odd numbers divisible by even numbers.
>>7909437
>mfw 6 years of math studies have turned me into a fat slob like OP picture.
>>7912426
Where do you come from, i.e. why do you learn it and with what background?
>>7912623
>you will also never slay pussy like him
that nigga is OD lit. it's a cropped out pic. in the original one, he's crouched down between two fine chicks at some event.
What is my "goal: in getting good at real math and where do I start? Should I just know how to chug and plug shit on the fly? Memorize formulas? Or should I aim to have a fluid understanding of what certain things mean?
Where do I start? Fug.
>>7912727
Well, where are you at? One of my favorite things when I was an undergrad was trying to isolate a perfect transcendental "why" from the proofs I read. Not always achievable but good practice.
>>7912727
Ok so math has essentially 3 main fields from which everything else comes from.
Your goal is to gain a foundation of each of these 3 fields and then when you find one you like particularly, study it further and further until you reach a point where you can contribute via research.
These three fields are Analysis, Algebra, and Topology.
I know this isn't a textbook recommendation thread, but for starting you, you can read analysis books. Rudin's "Principles of Mathematical Analysis" is a classic and very good. However, there are many, many other good options as well. For algebra, I know Artin's book "Algebra" to be very concise and an excellent introduction. For topology, Munkres book "Topology." I'm lumping geometry here in with topology, but as you get more advanced, difference fields obviously interact. You might study algebraic topology or differential geometry or functional analysis or whatever but to start, just try to gain a solid foundation in each of these 3 fields.
Once you have that, you'll start to see the bigger picture of how all math comes together and you can start going deeper.
>>7912697
learned about them for the first time in algebraic geometry but have only really learned the categorical definition and basics on what they do in the category of quasiprojective algebraic sets, I was just wondering if these universal properties give anything strange in some extreme categories
>>7912727
Solve problems. That's mostly the greatest advice I could give you.
Mathematics is not a spectator sport. You really have to get in there and do it. Even professors are like that.
>>7909437
>general
>>7912755
excellent post what a contribution oh my lord
>>7912758
Request threads are against the rules.
>>7912760
Are you on the wrong website or something?
This is a thread about math.
>>7912768
There is no math in the OP. It is a request thread.
>>7912770
Yeah, a request for discussion about math you moron.
>>7910965
Mathamatical carnival by martin gardner. Good read my neighbour found it in a charity shop in the 90s its pretty cool http://www.amazon.co.uk/Mathematical-Carnival-Lightning-Calculators-Dimension/dp/0394494067
>>7912711
>fine chicks
its at a magic the gathering convention, and he's crouched down next to the ass cracks of fat dudes
>>7912752
Okay, if you're coming from a particular mathematical theory (lots of semantic ideas), maybe also have a look at the abstract definition of a limit in general.
The pullback (like the more special products, or monomorphisms) is a special case of a limit.
The limit is a special (complicated) case of a terminal morphism, also known as universal property.
Terminal morphism (from U to [math]X[/math]) always means that loads of information U is squeezed into a single object U(A), that corresponds with your object of interest [math]X[/math] via the terminal morphism phi:U(A)->X.
In the case of a limit, [math]X[/math] is a functor from some category that's merely consisting of a graph, U(A) is another certain functor and phi natural transformation.
In the case of the pullback, you have a graph involving 3 objects (a,b and c) and 2 arrows (i and j):
(a) --(i)--> (c) <--(j)-- (b)
[math]X[/math] maps this graph to the graph in your pic
X --f--> Z <--g-- X
and U(A) maps this graph to another such graph
U --u--> V <--w-- W
and phi:U(A)->X is the spider web connecting the two
U --u--> V <--w-- W
| | |
X --f--> Z <--g-- X
And the joke now is that U(A) is the diagonal functor that maps a,b and c actually to the same object P.
So you actually deal with
P --id--> P <--id-- P
| | |
X --f--> Z <--g-- X
and the pullback P is the one object incarnation of that which naturally embeds into the whole graph
X --f--> Z <--g-- X
The naturality is exactly the claim that the vertical arrows commute.
I hope this wasn't too confusing.
Btw. you can also define limits without reference to the diagonal functor by introducing cones, which are the above natural transformations in disguise.
https://en.wikipedia.org/wiki/Cone_%28category_theory%29#Universal_cones
the | | | was supposed to look like pic related
Btw. the Q in your picture is the U(Y) in >>7913006.
The terminal morphism spiel exactly says that U(A) (which maps to P in your pic) is the thing that's closest to X. Everything else that's connecting to X factors thought phi (the projections, in the case of the pullback).
How should I go about this question? Not even sure where to start tbqh
>>7910834
Leibniz notation is intuitive but not literal.
>>7913021
> [math]\Phi(\frac {k-\mu} {\sigma} )[/math] and not [math]\Phi\left(\frac {k-\mu} {\sigma} \right)[/math]
Absolutely fucking disgusting.
>>7910969
Please answer.
>>7913006
Sorry for the slow reply, I had to look up limits, graphs and cones (these seem like different cones than the ones in homological algebra, i.e. mapping cones?) but I think I understand what your point is though with all the data included via U (or U(A)). I need to get more used to these universal properties, probably skim through Leinster/Aluffi some more.
Given two finitely presented modules [math]M,N[/math] over a semilocal (i.e. finitely many maximal ideals) [math]R[/math] such that [math]M_P \cong N_P[/math] for all maximal ideals [math]P[/math], I want to show that [math]M \cong N[/math]. I know that each isomorphism [math]\alpha_P : M_P \to N_P[/math] will give rise to some map [math]\phi_P : M\to N[/math] (this works something like \alpha_P \left( \frac {m} {u} \right) = \frac {\phi_P(m)} {u}[/math]), and I'd like to take a well chosen linear combination of these maps to get my isomorphism. Any thoughts?
>>7914583
I'm so used to having \phi redefined to \varphi. \phi is so much uglier.
The part I fucked up is [math] \alpha_P \left( \frac {m} {u} \right) = \frac {\phi_P(m)} {u}[/math]
>>7914570
U mapping form some object to some very ugly (imho) functor indeed makes it weird and that's why I avoided speaking it that part.
I'm not sure if it's worth viewing the pullback this way at all, I just wanted to give the raw perspective. Also you don't "need" cones, they are just a natural transformation in disuse, however they are the direct way to speak about limits. Limits and colimits are very well explained in Simmons:
http://www.cs.man.ac.uk/~hsimmons/zCATS.pdf
>>7911678
I don't think 873612 can be written as a sum of squares.
If we don't make any more progress on this problem in this thread I'll repost it in a week or something.
>>7910933
awesome response, literally lol'd.
How do you find the derivative of e^((-x)^2)?
>>7916811
every number can be written as a sum of at least 4 squares
>>7916909
Exactly four squares, I believe.
>>7916909
I meant the sum of exactly two squares.
>>7912751
>These three fields are Analysis, Algebra, and Topology.
>What is geometry
>>7916942
Geometry/topology are grouped together. cf. any departments faculty interests page.
>>7916945
Could've at least recommended some book for babbys first algebraic geometry
>>7916954
That's grouped under algebra. You wouldn't read an algebraic geometry book before an algebra book, anyways. That would be fruitless.
>>7912623
Weeaboos are a burden on society. Get out of this thread.
>>7916913
yeah, i meant that, though some can be written as a sum of 1, 2, and 3, without including zeros.
Help?
>>7917555
>can't do the most basic of calculus problems
>these are the people that post on /sci/
>>7917555
80% of mathematical discussion on /sci/ is retarded pre-calculus homework questions.
We're trying to have one thread without it, please go to the stupid questions thread if you want someone to take you seriously.
>>7917555
im geoscience and last time i had maths was 2 years ago and i could still solve that shit
nigga pls
>>7916893
2xe^x^2
1) (-x)^2=x^2
2) chain rule
Are there any good video lectures/course on rep. theory?
>>7917555
The gradient is a scalar??
>>7916893
[math] y=e^{-x^2} \implies \ln (y) = -x^2 \implies \frac { 1 } { y } y' = -2x \implies y'= -2x e^{-x^2} [/math]
>>7909437
What's this theorem (or something similar) called?
[eqn] \lim_{x\to c} f(x) = L \Rightarrow \forall g: \lim_{x\to c} g(x) = 0 : f(x) = L + g(x) [/eqn]
I would call it false. I'm also not completely sure what you're trying to write with these colons -- just use English sentences, please.
>>7917892
Sorry.
If f converges to L at c, there is a function g (or every function g) such that g converges to zero at c such that for all x: f(x) = L + g(x)
>>7917930
why not just g(x)=f(x)-L??
>>7917930
Oh, I see. You mean to say there exists ([math]\exists[/math]) rather than for all ([math]\forall[/math]).
If f is the identity and g is 0 everywhere, then you have a counterexample to your "for all" statement.
>>7917878
That theorem is false. I can come up with some counter examples fairly quickly.
>>7912751
How do you choose a particular field to study. I'm in third year wanting to grad studies but I can't decided whether I want to focus on Analysis, Algebra, Topology, or Geometry. They all interest me equally.
>>7917939
?
but... L = L + 0...
Context time.
From Rudin, what's the NAME of the theorem that proves the existence of the function u
>>7917956
I shouldn't have to spell this out. Let [math]f(x)=x, g(x)=0[/math] for all [math]x \in \mathbb R[/math]. Notice [math]\lim_{x \to 0} f(x) = 0 = \lim_{c \to 0} g(x)[/math], but [math]f(x) \ne 0 + g(x)[/math] for ANY [math]x \ne 0[/math].
>>7917555
Why must ever math thread here have one of you faggots. Pay attention in class next time.
>>7909442
Grad student lurking
Will participate, unless the topic is a class non-math majors might take. I don't want to get baited into talking to an engineer by accident
>>7913027
It's funny to ask if it's a fraction on here though. You get everybody triggered. Thank god we use Leibniz notation over that horror called Newton's notation
>>7917978
MechE here, what is 1 + 1? Help me please. I'm taking my introduction to basic arithmetic class this year and it's fucking hard. It's taking away precious time from me and my bf!! :(
>>7917949
You don't have to choose anytime soon. Those are all very different and broad fields, however. You've taken classes in each field and enjoyed them all completely equally? I honestly find that hard to believe.
>>7910906
They're really pretty. Type in prime number gif on google, should look like a flashing grid. Just pick up a number theory book and read some prime number stuff, it's better to expose yourself that way if you're curious about t.
>>7917970
Thanks....I've been messing this up for a bit.
So...existence is okay?
>>7917956
He is not using a theorem there.
Let
u(t) = (f(t)-f(x))/(t-x) - f'(x)
Then by definition of derivative, u(t)->0.
>>7918000
see
>>7917938
It's hardly much of a statement, and has nothing to do with the red box in >>7917956
This is literally the definition of the derivative -- [math]f'(x)[/math] differs from [math]\frac {f(t) - f(x)} {t - x}[/math] by some function [math]u(t)[/math] whose limit goes to 0 as t goes to x.
>>7918009
Perfect thank you.
That's exactly what I needed
>>7918032
You're welcome.
>>7909437
[math]f(a,t)[/math] is the probability of dying at age during the year t. (Each year has its own probability of the age distribution).
It follows that
[math] \sum_{a=0}^{\infty }f(a,t)=1 [/math].
If a person was born at [math] t=t_b[/math], then the probability of living to the age A is
[math]P(A) = \prod_{a=0}^{A}(1-f(a,t_{b}+a))[/math].
Does
[math] \sum_{A=0}^{\infty}P(A)=1 [/math]?
If, or if not, why?
>>7918111
I meant:
[math] P(A) = \prod_{a=0}^{A} ( 1-f(a,t_{b}+a) ) [/math]
>>7917555
>What's a ou8 6 and wo w oo f uS theeu?
so how do you actually get good at mathematics?
The mathe in my engineering major is fucking me over constantly.
>>7911328
the points of fibred products is to be products and taking sub''entities''.
the fibred product is a subset of the biproduct though, since you have the condition that everything agrees in Z.
this condition is what keeps track of the re-indexing, when pulling back. this is why pullbacks are better than products constructively.
>>7914570
>I need to get more used to these universal properties, probably skim through Leinster/Aluffi some more.
implying these books are not the best
>>7918493
sit down and read through the math until you understand it
>>7918493
Read a lot of math, think about the concepts a lot, do a lot of problems.
Can someone explain what a fucking monad is in simple terms?
>>7917842
I think mathdoctorbob has videos on representation theory
Soft request : I'm a grad stud in proba theory/combinatorics.
For some reason I've never had a real course on vector/tensor algebra. I've got standard knowledge of algebra (groups/rings/modules/linalg/quad forms/representation theory).
Can anyone provide a good intro video on this topic ? Foreign languages if necessary
>>7909437
Couldn't find the sqt so goona post this here. I'm slogging through rudin chapter 2 and I don't understand how he can say that G forms an open cover of K1 here. If G is in the complement of Kalpha then how can it cover K1 if although no point of K1 belongs to every Kalpha it could still belong to some Kalpha.
>>7919312
>Couldn't find the sqt so goona post this here
You're too stupid to find the stupid questions thread.
I find that pretty amusing
>>7919331
Sure, on my phone at work during lunch hour. Couldn't find it.
>>7919333
>hurr how do I use the search bar
Anyways, for any [math]x \in K_1[/math], there exists [math]\alpha[/math] for which [math]x \not\in K_{\alpha}[/math]. Then [math]x \in K_{\alpha}^{c} = G_{\alpha}[/math]. There is no trick whatsoever; it is basically just the way the G's are defined.
What is everyone working on? I like writing up expository notes and just came across a proof of quadratic reciprocity in an old notebook, so I'm going to type up a nice little paper on that today.
>>7919344
Ah so it's just for some Gs not all Gs per point. He could have been better about that. Thanks man!
>>7909442
I will forever be nostalgic of the topos theory thread that flourished once.
>>7919395
Mind linking it?
Is it on the archive?
>>7919426
Stop.
>>7919395
>once
when? weeks or months ago?
Today I was reading about self-similar graphs, i.e. graphs where the distribution of number x neighbors of a vertex follows Pareto law [math]x^c[/math] for some exponent c, and I discovered the meme-expansion around [math]x=1[/math] given by
[math]f(x) = x^{F(x)} [/math]
with
[math] \left( (x-1)^{-1} + \dfrac{1}{2} - \dfrac{1}{12} (x-1) \right) \, \log \, f(x) [/math]
>>7919518
weeks, a month tops
>>7919518
the last expression is supposed to be F(x), and btw. the f(x) is essentially arbitrary, it just werks
Played around with a divergent infinite product. Took care of divergence pretty well and it turned out to be (2Pi)^2 *DedekindEta^4, even got the correct weight. The only thing missing is the dedekind sum. Can anon explain to me where the fuck the dedekind sum comes from?
>>7910969
This question is TOO difficult for the integration wizards of /sci/.
I've asked this on multiple occasions, no one can answer. If someone really wants to answer but isn't satisfied with the rigour of the question, I'll post details later on.
To me this is a very interesting problem, I hope someone else finds it interesting too.
>>7919716
Someone on this forum once calculated the trace of that kind of movement on this forum. It's more difficult than it looks. Here it is in the archive:
warosu.org/sci/thread/S7337389#p7343837
I's need help wid did
>>7920297
Now that's fairly complicated.
Maybe there's a simpler way of approaching the question about traced volume?
>>7909437
>mfw Grand Prix Richmond Crackstyle was 2 years ago
http://imgur.com/a/SjcgE
>>7917989
lol
>>7917989
>>7917555
what's a oud b aup wow do fad theim?
>>7920583
>why does the FDA have super low p-values for significance (alpha)?
It's to reduce false positives. You don't want homeopathy's placebo to suddenly infect the pharmaceutical industry.
Hello sci sorry to post a retard level question but I'm programming something and I'm stumped on this math question (I am terrible at maths) that I can't afford to lose any more time in, so I'm turning to asking internet strangers for help.
Say I have the coordinates of a point on a cartesian plane A who's center is 0,0. I want to find out the cartesian coordinates of that point in a plane that would have a center of for example 325,325.
How would you call that problem ? I've been googling "change of cartesian plan" and any other variation I could think of but I couldn't find anything that related to what I actually do, just conversions to polar or to cylindric coords or some other such.
Alternatively if you have the answer I'll gladly take it.
Anybody has a good recommendation on a book about differential equations? Both ordinary and partial, with focus on computer tools to solve and plot solutions (preferably matlab or octave).
>>7923448
x' = x - x0
y' = y - y0
Both x0 and y0 are 325 in the example you mentioned.
>>7910929
i'm not that anon, but I did this is a math class recently. here it is:
https://en.wikipedia.org/wiki/Prime_number_theorem
![720px-Expinvsqlau_SVG.svg[1].png](https://i.imgur.com/vhZMeaT.png "720px-Expinvsqlau_SVG.svg[1].png")
>>7912446
For functions of a complex variable:
Def: An analytic function is a function whose derivative exists in some open set on the Complex Plane. (This is like an open interval on the number line.)
Def: A radius of convergence is a literal circle on the complex plane, into which a certain sequence or series may converge.
A Taylor series will properly approximate any Analytic Function within a radius of convergence.
For functions which have singularities (such as rational expressions or functions composed with rational expressions) then if we have a finite number of singularities on the plane, then we can use a Laurent series to approximate outside of the radius of convergence of the Taylor series.
Picture related: Using the Laurent series to compute only the Real Valued part of the function [math]\frac{1}{e^{x^{2}}}[/math]
>>7912563
Just an addition: some weird examples of infinite differentiable functions with divergent taylor series in any neighborhood of any point can be construct.
See http://math.stackexchange.com/questions/675555/show-that-the-iterated-lnn-of-tetrationx-n-is-nowhere-analytic
I'm reading about simplicial homology from Hatcher. I can't seem to get the damn signs right for the boundary map on 2-simplices. For the torus, I find that, using a counterclockwise orientation, you should get that [math]\del_2 (U) = -a-b+c[/math] and [math]\del_2(L)=-a+b-c[/math]. Hatcher says they should both be equal to [math]a+b-c[/math]. I'm having similar problems with each surface, so something seems to not be clicking.
>>7924996
Forgot the picture, and meant to type [math]\partial_2 (U) = -a-b+c[/math] and [math]\partial_2 (L) = a+b-c[/math]. What a disaster of a post.
>>7924996
Your answer looks correct to me
>>7925189
>I think you are just switching orientation when you shouldn't be
I agree, and it bothers me that I can't seem to pinpoint how to determine the correct orientation. Seemingly it should have something to do with the way the face folds towards the back when you glue everything. It makes no difference in the homology groups, so it doesn't matter I guess.
ayy yo. engineer here. taking heat transfer this semester and my professor is kicking my ass with MATLAB coding. My Numerical Methods teacher was more about proofs and shit than actually applying what we learned in a computerized environment.
what are some good numerical analysis books that integrate MATLAB ?
>>7925197
yeah i would ask someone else im too rusty
maybe post on stackexchange if no one here does
>tfw too retarded to factorize
I'm gonna get raped on Calculus.
Anyway, how do you do trigonometric functions with only pen and paper? It is possible?
>>7925198
Google "Ozge Ozcakir Ohio State". Her webpage has a link to the book we used for numerical analysis. It's 1600 pages long, and I didn't really use it much because she was a good teacher.
It might be what you're looking for. The book is titled "Learning Matlab, Problem Solving, and Numerical Analysis Through Examples".
>>7925198
Do you need somebody to tell you how to turn algorithms into code?
>>7909459
>[math]\sum_{i=1}^n x^a[/math]
Is basicaly just [math]n x^a[/math] since there's no [math]i[/math] in [math]x^a[/math].
What did you mean?
>>7925265
i don't know how to code at all. thats the problem. everything up until now had this huge emphasis on analytical solutions.
>>7917555
Jesus Christ anon, improve your handwriting.
>>7925228
Everything you need to know is [math]\cos^2 + \sin^2 = 1[/math], a few formulas like [math]\sin\left(\frac{\pi}{2}-x\right) = \cos x[/math] or [math]\cos(\pi+x) = -\cos x[/math] that you can read on the unit circle, a few well-known values (basically at [math]\frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{2}, \pi[/math]) and the addition formulas.
I'm sure you can learn that in a weekend
>>7910684 corrected
[math](\mu \star N)(q^k) = \varphi(q^k) = q^k - q^{k-1}[/math]
>>7910684
>the euler phi function, the number of numbers relatively prime to a number
Wouldn't that always output infinity?
>>7916989
Why the hell are you on 4chan?
>>7925377
I'm not that guy but as there are infinitely prime numbers, yes, you are correct, and good on you for pointing that out.
However if you study a little number theory (and as you might expect from context), the function that that person was referring to, which is also known as the "totient" function, is defined for some number as the number of natural numbers /less than/ a given natural number which are coprime (another way of writing relatively prime) to that number. Consult wiki or a book on number theory to clarify whether zero is included as a natural for these purposes (should be moot I would think) and/or my top-of-my-head definition needs to be modified to /less than or equal to/.
>>7925402
>(should be moot I would think)
Correct. 0 is not coprime to any number: given n, n divides both n and 0.
>>7925377
He just 'misspoke' a bit, I would expect. But of course that doesn't fly in a math lecture or an argument, which is why I've complimented you on the nitpick.
But it is true that even most very competent undergrads can fill in the context-blank of what the totient function really is.
>>7916989
2D > 3DPD
[math]\mathbb{R}^3[/math] is bad bad bad!
Okay, autists, if your "beyond calculus" problems are so fucking superior, why don't you answer a simple trigonometry problem?
All of what I'm about to say is in degrees. If a hyperbola can be polynomially morphed, stretched, transformed, et cetera, can you modify a g(z)=(cos(z))/(sin (z)) equation so that every z value would be the same as it would in a g(z)=arccot(z) equation, without directly influencing the power of the functions into negative one, or replacing z with f(z). The equation must begin with f(z)= and must not have an inverse function within the equation. This is what they teach in schools now. Good luck.
Oh, and values in the +-, -+ or -- quadrants do not matter.
>>7910932
Are there cases in which the pairs of limits are not [math]-\infty[/math] and [math]+\infty[/math]?
>>7925438
The answer cannot be 1 or =/=, btw.
>>7925197
the orientation needs to match up along common edges
>>7925481
That makes a lot of sense, thanks! This also clears up an issue I was having picturing the boundary map on 3-simplices.
>>7925228
[eqn]\sin{x} = \frac{e^{-ix} - e^{ix}}{2i} \\ \cos{x} = \frac{e^{ix} + e^{-ix}}{2} \\° \tan{x} = \frac{\sin{x}}{\cos{x}} \\ \sec{x} = \frac{1}{\cos{x}} \\ \csc{x} = \frac{1}{\sin{x}} \\ \cot{x} = \frac{1}{\tan{x}} [/eqn]
Now just chain rule and quotient rule the fuck out of them, you're welcome.
>>7909459
I have a formula. But you need to know sums up to n-1 to use it.
Baby's first summation
[eqn]
\begin{aligned}
\sum_{n=15}^{150} \left(4n+1\right) &= \sum_{n=15}^{150} 4n + \sum_{n=15}^{150} 1 \\
&= 4\left(\sum_{n=1}^{150} n - \sum_{n=1}^{14} n \right) + \left(\sum_{n=15}^{150} 1 - \sum_{n=1}^{14} 1 \right) \\
&= 4\left(\frac{n(n+1)}{2} - \frac{n(n+1)}{2} \right) + \left(1n - 1n \right) \\
&= 4\left(\frac{150(150+1)}{2} - \frac{14(14+1)}{2} \right) + \left[1(150) - 1(14) \right] \\
&= 4(11,325 - 105) + (150 - 14) \\
&= 44880 + 136 \\
&= 45,016
\end{aligned}
[/eqn]
>>7925997
You can solve DiffEQ's with me, senpai!
What location statistic should I use for a discrete distribution that looks somewhat like [0, 0, 0, 0, 1, 1, 0, 0, 0]?
>>7926017
[eqn]
\begin{bmatrix}
0 & 0 & 0 \\
0 & 1 & 1 \\
0 & 0 & 0 \\
\end{bmatrix}
[/eqn]
Does it look like the matrice above?
>>7926025
No, I was trying to imply just an array of values tbqh. If you're trying to represent some sort of graph with that matrix, then no since it's kind of centred on the x-axis (assume [0, ...] corresponds to (0,0) , (1,0)) etc.
>>7925833
I made a mistake, oops. [eqn]
\begin{aligned} \sum_{n=15}^{150} \left(4n+1\right) &= \sum_{n=15}^{150} 4n + \sum_{n=15}^{150} 1 \\ &= 4\left(\sum_{n=1}^{150} n - \sum_{n=1}^{14} n \right) + \left(\sum_{n=1}^{150} 1 - \sum_{n=1}^{14} 1 \right) \\ &= 4\left(\frac{n(n+1)}{2} - \frac{n(n+1)}{2} \right) + \left(1n - 1n \right) \\ &= 4\left(\frac{150(150+1)}{2} - \frac{14(14+1)}{2} \right) + \left[1(150) - 1(14) \right] \\ &= 4(11,325 - 105) + (150 - 14) \\ &= 44880 + 136 \\ &= 45,016 \end{aligned}
[/eqn]
How much math does someone need to know before they can start working on proofs?
Or rather, what does someone need to know in order to start writing proofs? I have been just skimming Spivak's Calculus, and a lot of the questions at the end ask for proving something. I'm not really sure how to give a prove.
I am a Calculus 1 student and so far find the material really easy, and was hoping for something a little more interesting, but this seems beyond me.
>>7926219
>How much math does someone need to know before they can start working on proofs?
If you want to prove something to be true, you need logic or else it'll just appear as a "hurr x is true cause i said so after plugging n chugging the shit above i wrote".
>>7926162
you have a line with formal n, instead of numbers
>>7926225
Okay, cool. How does one go about learning mathematical logic to the point of being able to prove things?
>>7926233
I'm not fixing the statement anymore because trigger happy janitors are bound to come. In the end, the result should still be correct.
>>7926243
Read a bunch of proofs and start learning. Either that or pick up a book on proofs, maybe "How to Prove It."
At least give an attempt on some easy yet important proofs like the infinite number of primes or the irrationality of the square root of two.
>>7926219
>How much math does someone need to know before they can start working on proofs?
Literally none. Euclid proof-based geometry was most's first introduction to math in the 19th century.
>>7926243
set theory
I tried making a separate thread for this the other day, and got no answers. I'll see if it'll get any proper answers here...
So... with hyperoperations, can I represent super roots using fractional powers and knuth arrows? I know I can use something like... n↑↑2 to mean n^n, but would n↑↑0.5 be equivalent to ssqrt(n)?
>>7926243
you don't need to know any logic. A proof is just a common sense argument that uses what has already been established.
>>7926550
>A proof is just a common sense argument that uses what has already been established.
false
>naturally good at math thanks to high functioning autism
>still think it's boring as shit
>>7912472
You cannot approximate every infinitely differentiable function at every point you wish, just think of exp(-1/x^2) (the derivate ist alwas polynomial(1/x) * exp(-1/x^2) and thus, its always 0 as x goes to 0). What you are looking for are analytic functions.
>>7926619
>its always 0
And this can be considered a good approximation in neighborhood small enough.
>>7926619
>the derivate ist alwas polynomial(1/x) * exp(-1/x^2) and thus, its always 0 as x goes to 0
I'm missing why this is an issue, considering that the original function is not even defined at 0. No good can come from approximating around 0.
>>7926692
It's defined as 0 for [math]x \leq 0[/math] and [math]exp(\frac{-1}{x^2})[/math] for [math]x > 0[/math].
>>7926640
This is BS. With the same argument, we could approximate any continuous function by a constant function "in a neighborhood small enough".
What we mean in saying that a function can be approximated by a Taylor polynomial is that the power series definied by the Taylor coefficients does not converge to the original function. This means that we can approximate it in any neighborhood with arbitrary precisio if we just take enough terms. Seriously, just look up the definition of analytic functions.
>>7926831
Oops I meant "does converge to the original function".
>Thank you for your application to the University of Michigan REU program at the Mathematics Department. We were impressed with your file and feel you are qualified for an REU. Unfortunately, we have not been able to find a mentor and funding for you. For this reason, we are unable to offer you a position in our program this summer.
end my life
>>7926550
Search for the definition of logic, you will change your stance after.
I've got a good one for you guys
2+2 + 2+2 = ?
>>7926971
[eqn](2+2) + (2+2) = (2+2)(2+2) = (2+2)^2[/eqn]
I am trying to understand how Eicher-Shimura implies that
[math]X_1(N)(\textbf{F}_q) = q+1[/math]
whenever there is only the 0 cusp form of weight 2 level N.
I think Eicher-Shimura implies that the global zeta function of [math]X_1(N)[/math] is the Mellin transform of some cusp form f. Since there is none, this implies that the zeta function is 1??? From this, L functions must be 1. Hence the trace of X_1(N) must be zero. In other words, [eqn]#X_1(N)(\textbf{F}_{q^n}})=q^n+1[/eqn]
Anyone know how to solve a two dimensional recurrence? This comes out of the ballot problem.
[math]A[/math] receives [math]n[/math] votes and [math]B[/math] receives [math]m[/math] votes with [math]n > m[/math]. Then, the probability that when the votes are drawn that [math]A[/math] is never behind is given by [eqn]Q_{n, m} = \frac{n}{n + m}Q_{n - 1, m} + \frac{m}{n + m}Q_{n, m - 1}[/eqn]
I'm not sure how to use this to get a closed form expression for [math]Q_{n, m}[/math] just in terms of [math]n[/math] and [math] m[/math].
>>7910641
There are q^n monic polynomials of degree n, so you have to show that there are q^(n-1) polynomials that are squarefree.
Is that what you're trying? How far have you got?
>>7926978
Dunno, but an observation:
[math]Q_{n,m} = \frac{n}{n+m}Q_{n-1,m} + \frac{m}{n+m}Q_{n,m-1}
= \frac{n}{n+m}\(\frac{n-1}{n+m-1}Q_{n-2,m} + \frac{m}{n+m-1}Q_{n-1,m-1}\) + \frac{m}{n+m}\(\frac{n}{n+m-1}Q_{n-1,m-1} + \frac{m-1}{n+m-1}Q_{n,m-2}\)
= \frac{2nm}{(n+m)(n+m-1)}Q_{n-1,m-1} + \frac{n(n-1)}{(n+m)(n+m-1)}Q_{n-2,m} + \frac{m(m-1)}{(n+m)(n+m-1)}Q_{n,m-2}[/math]
I definitely see some factorials forming. This recurrence has a binomial flavor to it. I'm gonna check a textbook and see if there are existing techniques to handle this, rather than just flailing around with algebra.
>>7927039
Let's try that again...
[math]Q_{n,m} = \frac{n}{n+m}Q_{n-1,m} + \frac{m}{n+m}Q_{n,m-1} = \frac{n}{n+m}\(\frac{n-1}{n+m-1}Q_{n-2,m} + \frac{m}{n+m-1}Q_{n-1,m-1}\) + \frac{m}{n+m}\(\frac{n}{n+m-1}Q_{n-1,m-1} + \frac{m-1}{n+m-1}Q_{n,m-2}\) = \frac{2nm}{(n+m)(n+m-1)}Q_{n-1,m-1} + \frac{n(n-1)}{(n+m)(n+m-1)}Q_{n-2,m} + \frac{m(m-1)}{(n+m)(n+m-1)}Q_{n,m-2}[/math]
>>7927042
[eqn]Q_{n,m} = \frac{n}{n+m} Q_{n-1,m} + \frac{m}{n+m} Q_{n,m-1} = \frac{n}{n+m} \left(\frac{n-1}{n+m-1} Q_{n-2,m} + \frac{m}{n+m-1} Q_{n-1,m-1} \right) + \frac{m}{n+m} \left(\frac{n}{n+m-1} Q_{n-1,m-1} + \frac{m-1}{n+m-1} Q_{n,m-2} \right) = \frac{2nm}{(n+m)(n+m-1)} Q_{n-1,m-1} + \frac{n(n-1)}{(n+m)(n+m-1)} Q_{n-2,m} + \frac{m(m-1)}{(n+m)(n+m-1)} Q_{n,m-2}[/eqn]
>>7926946
I didn't get into that REU either a few years back, and it had no negative impact on my career. Remember that REUs are primarily meant for minorities and people at schools without researchers. Find someone at your school to work with.
>>7910842
You can still use generating functions in situations like this, usually. However, an approach using generating functions leads to a dead end.
>>7926978
[math]nQ_{n,m} + mQ_{n,m} = nQ_{n-1,m} + mQ_{n,m-1}[/math]
Generating function [math]f(x,y) = \sum_{n=0,m=0} Q_{n,m}x^ny^m[/math]
[math]xf'_x = \sum_{n=0,m=0} nQ_{n,m}x^ny^m[/math]
[math]yf'_y = \sum_{n=0,m=0} mQ_{n,m}x^ny^m[/math]
[math]x(xf)'_x = \sum_{n=0,m=0} (n+1)Q_{n,m}x^{n+1}y^m[/math]
[math]y(yf)'_y = \sum_{n=0,m=0} (m+1)Q_{n,m}x^ny^{m+1}[/math]
[math]xf'_x + yf'_y - x(xf)'_x - y(yf)'_y = [/math]
[math] = \sum_{n=0,m=0} (nQ_{n,m} + mQ_{n,m} - nQ_{n-1,m} + mQ_{n,m-1}) = 0[/math]
You get equation [math](x+y)f + (x^2-x)f'_x + (y^2-y)f'_y =0[/math]
I think you may try to find answer of the form [math]f(x,y)=e^{r(x,y)}[/math],
where rational function [math]r(x,y)= \frac{p(x,y)}{g(x,y)},deg( p) = 1, deg(g) = 2[/math].
The proofs in this fucking book is literally the worst pieces of shits I have ever read.
>>7912565
A={2,4}
>>7927440
Aww that's sad. Spivak's calculus on manifolds is no exquisite work of art either. I don't know what to recommend.
How many parts are there in 3d space cut by 10 planes? Why?
>>7927440
Please, you must not be versed in proofs.
>>7909437
does the square root of cosec^2(x) = cosec(x)?
where's that giant picture of a bunch of math books recommended by /sci/
>>7929929
Check the archives.
>>7929819
[eqn] \sqrt{ \operatorname{csc}^2 x } = \lvert \operatorname{csc} x \rvert [/eqn]
>>7929929
All you need pham
>>7929819
[eqn]\sqrt{x^2} = x[/eqn]
Therefore, [math]\sqrt{ \csc^2{x} } = \csc{x}[/math]
>>7930189
>sqrt(x^2) = x
Pleb detected
>>7930210
[eqn]\sqrt{x^2} = -(- \sqrt{x^2} ) = |x|[/eqn] Happy?
>>7930210
show me an x for which that equality fails to hold
>>7927440
end yourself that was my favorite undergrad textbook
>>7930296
Step 1. Go fuck yourself
Step 2. Get off the internet
what do you math ppl think of mod 3 as logic instead of mod 2, what are the difference to binary logic?
pic related, add operator
>>7930339
You're upset and I don't understand why. Most of the proofs I remember being particularly instructive, and it made transitioning to graduate level geometry particularly easy. I don't really like the exercises for the most part though.
how smart do i have to be to do research in differential topology
>>7930364
literally asking the same question.
what interests you in this field ?
