/sci/ - Science & Math

Does proving ZF and ZFC equiconsistent require assuming choice in the metalogic?

Why is /mg/ dead?

Damn decent pic.

>>15498384
I want to self learn math from scratch, but I find it hard to organize a path, I'm reading a text on arithmetic right now, are discoveries still made in arithmetic, geometry and trigonometry in this day and age? I hate that I often see books like "X subject level y" or "x subject year x" it feels completely arbitrary and like it doesnt apply to me at all because Im self studying. Im thinking if I should look for a modern translation of Euclid and start from there but feels pretentious. I just want a big fat definitive book on each subject.
Thoughts and suggestions?

>>15498914
>I want to self learn math from scratch, but I find it hard to organize a path
what do you know? what do you want to know?

>are discoveries still made in arithmetic, geometry in this day and age?
yes but only with highly advanced methods

>>15498933
>what do you know? what do you want to know?
I have a high school education but there are subjects like trigonometric function that I never learned quite right but I was able to graduate still by average. I want to approach it as if i knew basically nothing. To fix any other holes I may have.

>>15498933
>>15498948
>what do you want to know?
I want to get up to calculus and statistics to an advanced level.

>>15498948
>>15498949
sorry I cant recommend any books for that
the only ressource at that level that I know is khan academy

>>15498956
Is there something like a book with all of the "rules" and concept of arithmetic, geometry, etc? With no care to provide examples or help you understand it, just, here is everything.

>>15498804
classes are out, so the 80% of the thread that's "do my homework" vanishes

I'm writing my calculus exams for next semester. What questions should I ask?

>>15498914
School level math is outdated and not how modern math is actually studied. Trigonometric functions for example are defined very differently. Geometry is completely different.

Analysis is the subject where everything in school except geometry is defined from scratch, but its focus is continuous functions. Modern algebra is the same thing as school algebra but way more general; so like instead of studying polynomials, you would study a generalised model that behaves the same as polynomials called integral domain. Number theory is the study of numbers obviously, but mostly whole numbers. Geometry is still called geometry but also closely linked to Topology which is closely linked to Analysis. And then there's discrete math, which is combinatorics, graph theory, etc.
The best place to start would be a book on proofs, then analysis.

>>15498963
Wikipedia

>>15499032
>anon wants to learn basic trig
>first study proofs, then analysis

Thanks /mg/, funny stuff.

Any book recommendations for PDEs? My background is pure maths. I've studied ODEs (local existence, global uniqueness, etc.), boundary problems (constant coefficients, Fredholm alternative, Green's operator,) and stability results (Liapunov, Poincaré-Bendixson).

I hate induction proofs, how do I show that the transitive closure of a well-founded relation is again well-founded (here well-founded is defined as "admits do well-founded induction")
I've been trying several approaches but none go anywhere

>>15499388
If it wasn't well founded then you would have an infinite descending chain of related elements which contradicts the original assumption of the relation being well founded.

>>15498933
I want to be able to do partial differential equations with ease. But I struggle with integral calculus and normal ODE

I might genuinely be stupid. How do I do this?

>>15500098

First use Green's theorem to show this is equivalent to proving the following:

Claim: given [math] X_i , Y_i \in \{+1,-1\} [/math] for [math] i \in \{1,2,\ldots,N\} [/math] ,

if we define [math] x_i = \sum_{1\leq j \leq i} X_i [/math] and [math] y_i = \sum_{1\leq j\leq i} Y_j [/math] ,

and if we suppose [math] x_N = y_N = 0 [/math] , equivalently [math] \sum_{i=1}^{N} X_i = \sum_{i=1}^{N} Y_i = 0 [/math] ,

then [math] s= \sum_{i=1}^{N} x_i Y_i [/math] is divisible by 2 iff [math] N [/math] is divisible by 4 .

(By Green's theorem, [math] s [/math] equals twice the area of the polygon enclosed by the curve obtained by starting from the origin and performing the steps [math] X_1,Y_1,X_2,Y_2,\ldots,X_N,Y_N[/math] where [math]X_i = \pm1[/math] means a displacement horizontally of length 1 in the [math] \pm [/math] direction, and similarly for [math] Y_i [/math] vertically.)
(Note a horizontal step must be followed by a vertical one, and vice-versa, due to the polygon's sides being length 1 and perpendicular. Also the total numbers of horizontal and vertical steps are the same.)

Proof of Claim:

Note N is even, which follows from [math] \sum_{i=1}^{N} X_i = 0 [/math] and each [math]X_i[/math] being ±1 .

Note modulo 2 we have [math] Y_i \equiv 1 [/math] for each i ; and , [math] x_i [/math] is even or odd according to whether i is .

Therefore, we have [math] s \equiv S \text{ mod }2 [/math] where [math] s' = \sum_{1\leq i \leq N , i \text{ odd}} 1[/math] .

Or in other words , [math] s \equiv k \text{ mod }2[/math] where k is the number of odd positive integers ≤ N .

Thus [math] s [/math] is even iff (the # of odd positive integers ≤ N) is even.

Finally, the latter holds iff N is a multiple of 4, given that N is already even.

>>15500098
Draw the shape on a sufficiently large black-and-white checkerboard. Because the boundary must execute a 90 degree turn each unit length, all squares inside the shape that are adjacent to the boundary must have the same color (see pic). Without loss of generality, take this color to be white. Let W be the number of white squares inside the shape and B the number of black squares; the total area is A = W + B. If we remove all black squares from the shape, the perimeter will increase by 4B, since all black squares were in the interior and no two shared an edge. Since the remaining shape is white squares only, the new perimeter is 4W. Thus the original perimeter is P = 4W - 4B = 4A - 8B, meaning the perimeter is a multiple of 8 iff the area is a multiple of 2.

>>15500098
One solution: First use Green's theorem to show this is equivalent to proving the following:

Claim: given [math] X_i,Y_i\in \{+1,−1\} [/math] for [math] i\in\{1,2,\ldots,N\} [/math] ,

if we define [math] x_i = \sum_{1\leq j \leq i} X_j [/math] and [math] y_i = \sum_{1\leq j \leq i} Y_j [/math] ,

and if we suppose [math] x_N = y_N = 0 [/math] ,

then [math] s := \sum_{i=1}^{N} x_iY_i [/math] is divisible by 2 iff [math] N [/math] is divisible by 4 .

(By Green's theorem, [math]s[/math] equals ± the area of the polygon enclosed by the curve obtained by starting from the origin and performing the steps [math] X_1,Y_1,X_2,Y_2,\ldots,X_N,Y_N[/math]where [math]X_i=\pm 1[/math] means a displacement horizontally of length 1 in the ± direction, and similarly for [math] Y_i [/math] vertically.)
(Note a horizontal step must be followed by a vertical one, and vice-versa, due to the polygon's sides being length 1 and perpendicular. Also the total numbers of horizontal and vertical steps are the same.)

Proof of Claim:

Note [math] N[/math] is even, which follows from [math] 0 = x_N = \sum_{i=1}^{N} X_i [/math] and each [math] X_i[/math] being ±1 .

Note [math] Y_i \equiv 1 \text{ mod }2[/math] for each [math] i [/math] ; and , [math] x_i[/math] is even or odd according to whether [math] i [/math] is .

Therefore, we have [math]s\equiv k \text{ mod }2[/math] where [math]k =\sum_{1≤i≤N,i \text{ odd}} 1 [/math] .

Or in other words , [math] s\equiv k \text{ mod } 2[/math] where k is the number of odd positive integers ≤ N .

Thus s is even iff (the # of odd positive integers ≤ N) is even.

Finally, the latter holds iff N is a multiple of 4, given that N is already even.

>>15499308
Partial differential equations Lawrence C. Evans
Or
michael taylor partial differential equations Vol I-III

>>15500411
>>15500413
Awesome, thank you both!

>>15499290
Learn to read nigger >>15498949

Points A and B belong to an n-dimensional ball in an n-dimensional Euclidean space. A and B are chosen randomly and independently of each other, have a uniform distribution. Find the mathematical expectation of the length AB. Can you write the answer without using integrals? Thank you!

Am I just retarded or there is some typo.
I can't comprehend at all how we can transform system to the last equation.

>>15500098
If every two consecutive sides are perpendicular, the shape must be a square.
Ergo, you cannot have 8n sides.

>>15500769
I got an integral *of* an elliptic integral, so you're probably fucked

>>15500891
you sure about that?

>>15500891
>If every two consecutive sides are perpendicular, the shape must be a square.
Are you sure about that?

>>15500893

>>15500893
Show me!

>>15500936
This is probably a better approach than my bullshit https://mathworld.wolfram.com/BallLinePicking.html
At first I misread the problem as the points being on the surface of an n-sphere, and even managed to find a closed form solution on my own
[math]\frac{2^{n-1} \Gamma \left(\frac{n}{2}\right)^2}{\sqrt{\pi } \Gamma \left(n-\frac{1}{2}\right)}[/math]

>>15500959
Thank you very mcuh! You just forgot R in your formula.

>>15500990
the unit ball is the only ball that matters

>>15500959
Mkay. And what about half ball, inner and surface cases?

Example of a non-compact metric space for which every infinite subset has a limit point?

>>15501202
There aren't any.

>>15501212
Didn't ask about your bitches.

>>15501202
My dick

>Non-compact
Surely not
>metric space
Oh for sure
>every infinite subset has a limit point
Although it seems infinite it has a limit point

>>15500907
fucking hell

>>15501217
Curious. I shall publish this in my upcoming paper. Look for it in the next edition of Penthouse.

>>15501213
A metric space is compact if and only if every infinite subset has a limit point.

>>15501245
I only read fine literature so post it in Penthouse Forums. We're intellectuals, after all...

>>15500959
Isn't it going to be half the surface diameter in the surface case?
Since the set of points at distance [math]d/2 + x[/math] and [math]d/2 - x[/math] are both circles with the same radius, so the probability densities should cancel out.

i have a question because i'm not sure if this makes sense in my head or not. i'm in a stats class, we're doing poisson and exponential distributions right now.
so, here's an example problem.
4 visit a restaurant in an hour on average.

the last person to visit arrived at 1:00pm. given that it's 1:45 pm, what is the probability that the next person won't arrive until 2:00pm?

easy to solve, it's p(x > 1 | x > 45/60)
= e^(-4) / e^(-4*(45/60))
= 1/e
=~ .368

but, you get the same value for if it was just
p(x > 15/60)
= e^(-4*(15/60))
=1/e
=~ .368

i can't really conceptualize why those values are equivalent because i'm pretty sure it's not generalizable to other distributions.
is it because the events are defined as being independent? like, just because you didn't get any heads on your coinflip for the past 10 runs doesn't have any bearing on the chances of getting 3 more tails?

Maybe the calculator rounds these numbers off? Maybe try to use full numbers and then get the decimal numbers like this?

>>15501398
Euclidean distance, not sphere-intrinsic distance

>>15501431
https://en.wikipedia.org/wiki/Memorylessness

>>15501431
poisson is a memoryless distribution.

>>15501708
>>15501854
i see, yes. my misunderstanding is under the heading "a common misunderstanding".
the fact that the event is binary, whether someone comes or doesn't, are independent events. and whether a new event happens is independent of any previous events, but the actual markov property, or memorylessness, of the system is what is actually displayed when you say that your past coin flips have no bearing on the 50% chance of the next one coming up heads.

just getting into ring theory. Is there any way to visualize these fucking things?

Sphericity condition in fusion 1-categories
>the snake
In fusion 2-categories
>the swallowtail
How lovely! What'd be the name for this in 3-categories? Perhaps
>the cinnamon roll

>>15498384
I failed an (undergraduate) topology test and feel like killing myself. Did some truly retarded shit on it, like intersecting a countable dense subset of X with an arbitrary subset Y of X to find a countable dense subset of Y. Couldn't finish my proof that paracompact Hausdorff spaces are regular. Fucked up a proof of the induced homomorphism of a covering map being injective (it was a simple lifting, but I lifted the wrong function). I know good mathematicians didn't experience this in undergrad. Should I just do an hero and give up mathematics for good? I don't want to keep embarrassing myself like that.

>>15501202
Compactness, sequential compactness and limit point compactness are equivalent in metric spaces

>>15502012
>visualize
in algebra imo you dont visualize as much, you have a well fitting example amd work with that
for rings that might be the integers, a matrix ring, Z[sqrt 5] or whatever has enough right properties

I got this problem fromOrganic Chemistry tutor. Can someone explain to me why I'm getting the answer sin^5x/8 + c while he and every other internet calculator is giving the answer sin^5x/5 + c? I just took u as sin^4 x instead of sinx, but shouldn't we arrive at the same result anyway?
picrel my math

>>15502531
You have to replace all x by u under integral sign.

>>15498384
Cool image.

>>15502638
I don't understand. I did replace x with u

>>15502695
With me? You have to replace all x by yourself for better understanding.

5x - 9 = 0.999...

>>15502012
scheme theory is the most profitable way of visualising commutative rings with unity, unfortunately you need some time to build up your intuition.
As >>15502368 said, you're better off thinking about "standard" examples, such as the rings of integers [math]\mathcal{O}_K [/math] of a number field, the ring of polynomials [math]R_0[t_1,\dots, t_n] [/math] over a base ring [math]R_0 [/math] easy enough (a field, or [math]\mathbb{Z} [/math]). For non-commutative rings, indeed the "standard" examples are with matrices, or group rings [math]R[G]=\bigoplus_{s\in G}R_0\cdot s [/math] (again [math]R_0 [/math] is a ring, or [math]\mathbb{Z} [/math], say) over a finite group [math]G [/math], with multiplication given by the group law of [math]G [/math].

>>15502783
it was probably clear from context, but it should read [math]R_0[G]=\bigoplus_{s\in G}R_0\cdot s [/math], with [math]R_0 [/math] a field or [math]\mathbb{Z} [/math]

>>15502730
[math]\sum_{i=1}^{\infty}2^{1-i}[/math]

>>15502730
[eqn]\sum_{i=1}^{\infty}2^{1-i}[/eqn]

>>15502730
[math] \sum_{i=1}^{\infty}2^{1-i} [/math]

>>15502332
Uhh anons... has no one else bombed like this? I need some copium....

>>15502987
I failed Spectral theory of linear operators exam. Since then I have been nolifer and hikkikomori.

>>15498804
>>15498991
the math discord unironically has much better math discussion, even if it's considerably faggier (at least there are no overt "boymoders" there)

>>15503082
Kind of based but are you serious? Did you make retarded mistakes or did you just leave too much empty? I feel like the latter is more dignified...

>>15500680
No u
>>15498948
>I want to approach it as if i knew basically nothing.

>>15503544
Yes, I am quite serious. I was depressed at that time. It was a special course, and I could not interact with basic definitions like sheaves. And what you gonna do? Can you pass your exam again?

>>15503829
>I could not interact with basic definitions like sheaves
I'm sorry but did I jump to a timeline where sheaves are basic tools in spectal theory of linear operators?

>>15502332
some people are bad in time pressure
as long as it's obvious to you how to solve all the stuff (and you can actually write out coherent arguments to justify that belief), you should be fine
For example, what should you have done for the X/Y problem?

>>15503829
No, it just tanked my grade. Quite worrying desu since I need a competitive GPA to get funding for grad school, but most importantly quite demoralizing.
>>15503891
I was getting nervous since I didn't go to class and expected the test to be about manifolds and fundamental groups but it was mostly compactness, countability axioms and properties of the covering map. I still knew enough about that to be able to complete the test, though, so that uisn't a good excuse. I guess doing well in the final exam will make me feel slightly better but I'll never be able to delete my signs of retardation off of that TAs memory, who will probably turn into a proper mathematician btw.

>>15503887
Well, one has the classic result that [math]K^0(X)[/math] can also be described as the Murray-von Neumann classes of sheaves of Fredholm operators over some Hermitian vector bundle [math]E\rightarrow X[/math].
Sheaves are not basic tools for operator algebra I'd say, unless one considers index theory "basic".

>>15502012
In the view of algebraic geometry (at least as I understand it),
rings are viewed as the algebras of ("algebraic") functions on some space (a scheme; specifically a ring R is the ring of functions on the affine scheme SpecR

>>15503887
Umm... I meant sheaves of operators. Yes, in my course and timeline it was basic thing. It was special course about it. I am sure.

>>15500872
factor of 2 missing in front of both F2(delta)alpha term

>>15498398
No, it is most easily done by defining the Gödel's constructible universe L within a model of ZF. ZF now proves the restriction of all ZFC axioms with respect to L, so that (informally spoken ) 'the substructure L is a model for ZFC', and the consistency of ZFC follows.

>>15504446
We then even obtain the interesting rrsult that ZF itself proves Cons(ZF) implies Cons(ZFC), just as ZF^- proves Cons(ZF^-) implies Cons(ZF).

Final note: in L, the generalised continuum hypothesis holds relative to L, so that Cons(ZF) implies Cons(ZFC+ CH).
One can show that Cons(ZFC) implies Cons(ZFC-CH), but this uses forcing.

Hey, I'm a first year undergrad studying programming and I know there's some holes in my mathematical knowledge. I think I got up single variable calculus, and differential equations, (though a bit spotty) but not linear algebra.
Is there some free online test or something I can take to find my weaknesses?

damn thread dead as hell, where's this discord server?

>>15504085
I am taking from this example that you're also not aware of how sheaves are useful for spectral theory specifically outside of functional analysis in general.
Btw what's up with your filenames?
>>15504131
>It was special course about it.
Oh okay, you're not saying the full course name so we can't figure out your college, that's fine.

>>15504446
I think the part that was confusing me originally was seeing statements that proving that consistent theories have models requires choice, which is only the case when you generalize beyond countable stuff.

>>15504131
Funny thing I'm the retarded anon that failed the topology test and was thinking about taking an advanced course on spectral theory next semester.... oh well

>>15504680
Exactly. By the way I am from Albania, and you?
>>15504745
Take your best, anon!

>>15504763
I'm not from Albania.

[math]2^{1}=1.999...[/math]

>>15504926
This breaks the mathfag.

>>15504926
Give me the proof.

>>15500458
Taylor is great. He has such a rich perspective

>>15505022
[math]2^1 = 2^0 \cdot 2 = 1 \cdot 2 = 2[/math]
[math]\forall \varepsilon > 0, |1.\underbrace{99\ldots9}_{\lceil1/\varepsilon\rceil{\rm~digits}} - 2| = \frac{1}{10^{\lceil1/\varepsilon\rceil}} < \frac{1}{\lceil1/\varepsilon\rceil} \leq \varepsilon[/math]
[math](1.9,1.99,1.999,\ldots) \sim (2,2,2,\ldots)[/math]
[math]1.999\ldots = [(1.9,1.99,1.999,\ldots)] = [(2,2,2,\ldots)] = 2[/math]
[math]\therefore 2^1 = 1.999\ldots[/math]

Set k\geq 1, 1<p<k, q_{1}= 2k/(p-1), q_{2} =2k/(p+1). How can I approximate a given function u \in L^q_{1} \cap W^(2,q_{2})(R^n) by smooth compactly supported functions which converge both in the L^q_{1} norm and the W^(2,q_{2})(R^n) norm ?

>>15505149
Try using Mollifiers.

>>15505179
Oh so the point is the mollifier approximates in both norms. Okay thanks

>>15502012
If you want examples with nice visuals, you could look at subrings of the complex numbers like [math]\mathbb{Z}[i][/math] or [math]\mathbb{Z}[\sqrt{-5}][/math]. Pic related is from a thread on lambdaplusjs /math/ about rediscovering some basic operations on ideals using [math]\mathbb{Z}[\sqrt{-5}][/math] as an example.

(Don't know where to ask this, I already posted it in the book thread as well)

Is there a book which studies parallel computing from a purely mathematical perspective?

>>15505631
https://cs.stackexchange.com/questions/13863/reference-book-for-parallel-computing-and-parallel-algorithms
See first answer.

>>15505146
kek

year 2023 = year 2022.999...

>>15499014
I just took my Integral calculus final and the last question for credit was "Draw a dinosaur" and it evicerated my text anxiety as I was going over the test.

I think you should ask students some practical interpretation questions as calculus is mostly taken by applied students. For example having them use a numerical method and interpreting the result.

>>15498914
Please ignore the bad advice in this thread.

Get a textbook. Openstax is a free/open source and well-made textbook series. https://openstax.org/subjects/math
Start with "algebra and trigonometry", then "precalculus", then you can do statistics or calculus books. Read the entire textbook, except sections you already know in "algebra and trigonometry" do half of the exercises in the book. Set a realistic goal for yourself and stick to it, you should be able to complete each book in 3-6 months depending on ability and time a available.

If you get stuck look up organic chemistry tutor on YouTube or ask for help here.

>>15502531
Illegal application of constant multiple rule to pull out variable x. Need to let sinx = u not sinx=u raised to the fifth. U-substitution should avoid leftover functions of x after cancelation of terms.

>>15502638
*Raised to the forth

>>15502730
False. Let x = 0

>>15506520
I typed this terribly sorry I'm phone posting from bed.
Let U = Sin (x)
Don't let U = Sin ^4 (x)
If you have an x left after replacing dx with du and cancelling the denominator under du, you have the wrong substitution.

>>15506524
5*0=0
0-9=0^-9
0^-9=0.999999999

On track for graduating with a 3.85 GPA. The average GPA of a math undergrad in the USA is 2.9. Does this mean that given a crop of math majors, I would be part of the creamy surface layer?

I want to show that these two curves do no intersect at a point with positive rational coordinates.

-x^3*y-x^2*y^2+x*y^2+x^2+3*x*y+y^2+2*x+2*y+1 = 0

-x^2*y^2-x*y^3+x^2*y+x^2+3*x*y+y^2+2*x+2*y+1 = 0

They do seem to intersect at a point in the first quadrant, but I don't think it is rational. I tried using Grobner bases but could not isolate a variable. Any ideas?

>>15507569
Forgot to mention: you get one polynomial from the other by swapping x and y.

>>15507569
If a pair (x,y) solves both equations then it also solves the difference of those equations. The difference can easily be factored to
-(1 + x + y)(x - y)xy = 0
which in the first quadrant only has roots at x=y.

Restricted to x = y the curves are
-2 x^4 + x^3 + 5x^2 + 4x + 1 = 0
which can be solved with the quartic formula.
The roots are one negative number, one irrational positive number and two non-real numbers. So there is no rational solution in the first quadrant.

How are you guys so smart?

>>15507615
We (I am) are white, now kys tranny

>>15507605
>-2 x^4 + x^3 + 5x^2 + 4x + 1 = 0
This seems to check out. Thanks!

>>15507493
Yes, nice and creamy...

Of course, grade inflation is a thing and it might
bring up questions of what a student earned is
really true. The inflation rate depends on each
college, and even accounting for that you're still
on top. Mine's a 3.55 and I'm practically tops in
my department for the graduating year.

If we have standard deviation information as well
you can see how far up from average you are--
probably 5 SD, I guess--still, great job.

>>15489323
>>15489324
>>15490701
im sorry i never replied to you, friend. life comes at you so fast, and theres so few days. thank you for your posts, ill sit down and really try to absorb them later tonight.

>>15507493
GPA is reflective of your understanding of mathematics about as much as IQ is a representation of your intellect.

>>15500413
How many hours to I have to study everyday if I want to be able to come up with this solution on my own in a year if my current level is basic proof writing?

>>15508009
>im sorry i never replied to you, friend.
my friend, don't worry about this

I've had my time to think about it, and I think there is a more intuitive way of placing everything in a single framework (I hope it actually works, because there's some ugly computations which I didn't want to do; it's strange to me that I could find no online source about it)
Let's start with an easy example: consider the integral
[eqn]\int_1^a \frac{\mathrm{d}x}{\sqrt{1-x^2}}. [/eqn]
With some work, one shows that the differential [math]\omega=\mathrm{d}x/\sqrt{1-x^2} [/math] is invariant under a rotation of the circle. So, after a change of variable, we get
[eqn]\int_1^a \frac{\mathrm{d}x}{\sqrt{1-x^2}}=\int_b^{c} \frac{\mathrm{d}x}{\sqrt{1-x^2}}, [/eqn]
with [math]c=ab-\sqrt{(1-a^2)(1-b^2)} [/math].

What is happening here? We have a plane curve [math]C:y^2=g(x) [/math], [math]g(x)=1-x^2 [/math], which also admits a group law (given by rotation) we have found a differential [math]\omega=\frac{\mathrm{d}x}{y}[/math] which is invariant under translation for this group law, and so we can compute the integral
[eqn]\int_{O\to P} \frac{\mathrm{d}x}{y}=\int_{Q\to P+Q} \frac{\mathrm{d}x}{y} [/eqn]
by making a change of variable. Here [math]O=(1,0) [/math] is the identity of the group law, don't confuse it with the origin of the plane. But here this fact is not playing a crucial role, we could have started from any other given point [math]P_0 [/math]. Also, the path [math]O\to P [/math] that we've chosen is not important, but we should be coherent about it: [math]Q\to P+Q [/math] should we that same path, traslated via [math]Q [/math]. Now we're on a circle so there aren't many paths to choose (you could go either clockwise or anti-clockwise, though), but note that all this also works when considering the *complex* solutions to that equation. Now it's topologically a surface, and so we have many different ways of going from [math]O [/math] to [math]P [/math]

1/2

>>15508374
Likewise, we can start from [math]C:y^2=g(x) [/math], [math]g(x)=(1-x^2)(1-k^2x^2) [/math], [math]\omega=\frac{\mathrm{d}x}{y} [/math], and try to compute the integral
[eqn]\int_{O\to P}\omega, [/eqn]
where [math]O=(1,0) [/math]. Again, we find that [math]C [/math] admits a group law (hopefully? I couldn't find anything online), with something like [math]P+Q=(\frac{x(P)y(Q)+x(Q)y(P)}{\sqrt{1-k^2x(P)^2x(Q)^2}},\sqrt{g(x(P+Q)}) [/math] (but I really don't like the fact that some square roots appear, since the whole reason why we introduced [math]y [/math], and considered a plane curve, was so that we could make coherent choices of square roots by having [math]y [/math] move continuously). Anyway, we also find that [math]\omega [/math] is invariant under translation for this group law (I didn't actually check this), so we can again make the change of variable
[eqn]\int_{O\to P}\omega=\int_{Q\to P+Q}\omega, [/eqn]
and so we get the sought-after formula:
[eqn]\int_{O\to P}\omega+\int_{O\to Q}\omega=\int_{Q\to P+Q}\omega+\int_{O\to Q}\omega=\int_{O\to P+Q}\omega, [/eqn]
where we compose the paths we're taken as our choice for [math]O\to P+Q [/math].

One works analogously when [math]E:y^2=g(x) [/math], where [math]g(x)=x^3-px-q [/math] is a cubic with three distinct (complex) roots. Here the group law is given by the chord and tangent method (note that we have to also consider the point at infinity, to make everything work), and the fact that [math]\omega [/math] is invariant can be checked without much of a computation, since in this case one can prove that the space of holomorphic differentials on this curve is 1-dimensional; so, if we call [math]t_Q [/math] the translation by [math]Q [/math], then we must have [math]t_Q^\ast\omega=c(Q)\omega [/math], for some [math]c(Q)\in\mathbb{C} [/math] depending algebraically on [math]Q [/math]. So, we have a morphism [math]E\to \mathbb{C} [/math], which must be constant. Since [math]c(O)=1, [/math] we conclude.

2/2

>>15499014
Word problems like the popular streetlamp and shadow problem with derivatives or the draining of a pond problem with integration helped me find ways to apply what I learned and therefore made Calc. I and II more effective to me.

>in graduate differential geometry course
>raise hand smugly
>"um teacher, when are we ever going to use this in real life?"
>get ignored, he continues writing, get no mention of detention or ISS

Is it because he is ESL?

>>15509123
Hehe could you imagine

what the fuck is a fucking t distribution?
there's like, all of 3 pages devoted to this in my textbook.
like sure i can find the degrees of freedom and then get the tscore and look it up in a little table but what the fuck does that have to do with finding probabilities?

books in higher-order logic?

Going to be starting undergrad analysis next semester. Anybody have book suggestions for self study? I learn much better on my own then in class. I have basic fundamental proof experience. I was thinking of using Understanding Analysis by Abbott but was wondering if anybody has any other suggestions or if that book is good.

>>15509991
Rudin.

>>15509850
If [math] X_1, X_2, \dots , X_n \sim \mathcal N ( \mu , \sigma ) [/math], and [math] \bar X = \frac 1n \sum_1^n X_i [/math], [math] S^2 = \frac{1}{n-1} \sum_1^n (X_i - \bar X)^2 [/math], then:
[eqn] \frac{ \bar X - \mu }{ S / \sqrt n } [/eqn]
is called the [math] t [/math]-statistic and it follows [math] t [/math]-distribution with [math] n-1 [/math] degrees of freedom.

Since it's a standard distribution with known quantiles, it allows us to infer on the unknown [math] \mu [/math]. So if substitute into the statistic the value of [math] \mu [/math] some number, and the corresponding probability of the statistic having the resulting number is very low, we can infer that [math] \mu [/math] is unlikely to be the mean.

>>15509991
Rudin.

Ive been learning about complex power series lately and I had a question. Lets say a power series [math]\sum a_nz^n[/math] has radius of convergence [math]R[/math] (for convenience lets say its finite and nonzero). Given a function [math]f:(0,\infty)\mapsto(0,\infty)[/math], is it possible to construct a new power series with coefficients [math]b_n[/math], such that it has radius of convergence [math]f(R)[/math]?
For [math]f(x)=c[/math] the answer is affirmative as we can just take [math]b_n = (1/c)^n[/math], independent of the [math]a_n[/math]. And for [math]f(x)=\sqrt{x}[/math], the answer is yes, as we can take [math]b_n = a_n^2[/math]. But I cant see how we could construct [math]b_n[/math] for e.g. [math]f(x)=\log( x+1)[/math]. Anyone have ideas regarding this?

>>15510326
[eqn]R = \frac{1}{\limsup_{n \to \infty} \sqrt[n]{|a_n|}}[/eqn]

[eqn]f(R) = \frac{1}{\limsup_{n \to \infty} \sqrt[n]{|b_n|}}[/eqn]


[eqn] \limsup_{n \to \infty} \sqrt[n]{|b_n|} = \frac{1}{f(R)} = \frac{1}{f\left(\frac{1}{\limsup_{n \to \infty} \sqrt[n]{|a_n|}} \right)}[/eqn]


This suggests the choice
[eqn]b_n = \left( \frac{1}{f\left( \frac{1}{\sqrt[n]{|a_n|}}\right)} \right)^n [/eqn]

>>15510406
That makes sense, thanks. I guess that's not the only choice because only the limiting behaviour matters, so we can take a finite number of the [math]b_n[/math] to be any arbitrary complex numbers so long as we have [math]b_n = \text{(the expression you wrote)}[/math] for all [math]n>N[/math] for some [math]N[/math].

>>15510326
The simplest solution is just [math]b_n = \bigl(\frac{R}{f(R)}\bigr)^n a_n[/math], though having to explicitly refer to the limits is kinda lame.
>>15510406
This works for all a_n iff f is left-continous and non-decreasing (and set b_n=0 whenever a_n = 0).
Counterexample: try a_n = 1 for odd n and a_n = 2^n for even n, so R=1/2. For f(x) = 1/x, b_n = 1 for odd and b_n=2^(-n) for even, so the radius is 1 instead of f(1/2)=2.

Proof: let [math]x_n = |a_n|^{-1/n},\; y_n = |b_n|^{-1/n} = f(x_n)[/math]. The two radii of convergence are the limit inferiors of x_n and y_n respectively.
The method works when
[math]f(\liminf_{n\to\infty} x_n) = \liminf_{n\to\infty} f(x_n)[/math].
If we take x_n to alternate between two values x,y, the above equation implies that [math]f(\min(x,y))=\min(f(x),f(y))[/math], i.e. f is non-decreasing everywhere.
In turn, this means there exists a non-decreasing subsequence [math]x_{\lambda(n)}[/math] whose limit is the argument of the left-hand side, such that [math]f(x_{\lambda(n)})[/math] has the right-hand side as its limit.
As [math]x_{\lambda(n)}[/math] is an arbitrary non-decreasing sequence with a limit, the equality [math]f(\lim_{n\to\infty} x_{\lambda(n)}) = \lim_{n\to\infty} f(x_{\lambda(n)})[/math] is equivalent to left-continuity.

What comes between 1/3rd and 1/2? Please help, I am retarded.

>>15510695
2/5

>>15510695
>>15510714
I will give you two dollars for a pdf explaining this

Hello Anons,
my calculus II exam is in a couple of weeks and one of the topics will be about matrices, differential equations and so on.
We haven't talked about matrices yet, but I would like to know how to solve pic related. Can an Anon help me out? thanks.

>Solve for p

>>15510750
We can't solve for p without an equation, what you've posted isn't an equation, just a (nonsense) expression

>>15510817
equals zero. Yes I figured that out, but I got it straight from a previous exam, which is why I thought it might be written that way.
There is no equals zero in the exam, I just figure it might, judging from other questions from the years before.
Thanks for any help!

>>15510695
There is an infinite amount of numbers between those 2 numbers, represented by f(x)=x in the interval ] 1/3 , 1/2 [

>>15510750
>>15510822
What I'm asking is; there must be a system to solve these type of equations. I just don't know where else to look for
They were written down by other studets from memory, which might explain the weird format.

>>15510750
How the hell do you subtract a scalar from a matrix?

>>15510843
You wing it, I guess...

Or, it could be the determinant of the matrix.

>>15510882
that would make the most sense because the way the matrix is populated makes it seem like the creator had calculating the determinant in mind

>>15510882
>>15510905
thanks anons, this might be the solution. I was looking for the determinant of the matrix. Solution is:

det(A) = ( x^6 + 3x^5 ) - 1/4 x^5
=> x^6 + x^5 * 11/4
=> x^5 * (x + 11/4)

>with a solution for - 11/4.

Seems like an easy question for the upcoming exam. Thanks a lot!

>>15510882
>>15510931
Notation-wise, it should be two vertical lines
(like the absolute value) or a "det" to the left of the
matrix--never the curved brackets itself.

>>15510269
Why Rudin? I've heard of his book but I'm just curious about why you recommended him

I'm a /sci/ tourist and I've had a desire recently to go back and actually learn geometry since I never really absorbed the material when I took the class in 7th or 8th grade.. My highest level of math I've taken is calc 1 in uni.

I sorted through some old books at my parents house and found that I have pic related 5th edition. Does it make sense to go through this book chapter by chapter to start learning? Or is it more efficient to use a resource that is not a college textbook? My goal is just to have a decent understanding of geometric concepts that will lead up into trig since I'll probably go there afterwards.

Sorry in advance if this is a common/annoying question from tourists

>>15511160
It has a concise very brief overview of all of the fun stuff in undergraduate analysis. Basically the pacing of topics is unmatched. It only gives as much information as necessary for a beginner who doesn't know what to specialise in. After that you can go for more specialised topics like analysis on manifolds, measure, functional analysis, construction of numbers, etc., which it treats very briefly but enough to provide the idea. I find that some Analysis books go too in depth in stuff I don't care for, or don't find interesting, which can be especially overwhelming as a beginner. And of course, the most important reason is its amazing exercises and proofs. His proofs are often incomplete which forces you to engage with them and deeply think about them instead of glossing it over. It's gonna take a good while before you figure out his proof regarding rational numbers and root 2.

Abbott is great if your course is a calculus course calling itself analysis. It is unrigorous. For instance, it uses trigonometric functions without defining them, and in consequence assumes their derivative without any proofs. It also is strictly real analysis, while Rudin develops the theory on general metric spaces, while specialising in real or complex where appropriate.

Since Rudin has no diagram and can be difficult to understand, you may supplement it with Pugh.

hello. Is On the study and difficulties of mathematics of Augustus De Morgan worth it from an undergrad?

What textbook for studying affine spaces and more advanced linear algebra concepts?

>>15511234
I appreciate you writing that out. You've convinced me to give Rudin a try. Right now, I'm not really studying to study for my course since it is in-between semesters, but rather I'm studying to further my understanding of math, and I would prefer a book best at doing that. Based on your description Rudin sounds more like what I'm looking for, so thank you.

>>15503103
Link to this math discord?

>>15510822
Unless there is some smarter way this is just a fifth-degree polynomial equation

I miss MJR so much

>>15511293
https://discord.gg/math

How do I learn proofs? How did you learn proofs?

Are types in type theory in any way related to sorts in mathematical logic?
I took a mathematical logic class and we briefly talked about logic with more than one sort (like, say, lines and points for a two-sorted theory of plane geometry; or theories whose models are rings together with a module over that ring).

When my programmer friends talk about type theory (which I'm not familiar with at all) it does sound a little bit similar except much more elaborate (sounds like one can create new types from old ones instead of having a fixed set of different sorts?).

Is there any connection here at all or is that analogy completely wrong?
And does anyone here know an introductory text *not* aimed at programmers or computer scientists? Maybe something aimed at people with only a mathematics or logic background?

>>15498384
How would this book compare to the Stewart Calculus book? I'm taking a 5 week calculus 1 course right now, and from what I've read on limits, I like the examples and explanations a lot more in this book. Would it be worth it to focus more on this one?

>>15511560
uuuh... that's a problems books...

>>15511560
>>15511569
Wrong book sorry, couldn't find an actual cover but it's a Russian book from the 70s which is why I was hesitant.

>>15508274
Once you know the notation it gets pretty easy. It's just passing your thoughts to writing. Everything after that it's just acquiring new knowledge and applying it.

>>15511398

>>15511476
look up per martin-lof. hott book

>>15511560
>>15511570
Rudin.

>>15511570
That doesn't look ORC'd so you will have a bad time Ctrl-F'ing. Just download whatever the sticky recommends you.

>>15512390
Rudin.

Fuck yall that don’t answer questions that are obviously not homework related

>>15509991
Strichartz

Hello I was referred here by a fellow from /wsr/
I wanted to do a school project on chaos theory because it sounds cool, so I looked it up and stumbled upon this pdf that gives an overview of it with some degree of rigor
https://web.math.ucsb.edu/~padraic/ucsb_2013_14/mathcs103_s2014/mathcs103_s2014_mlecture10.pdf
I understood everything up until this point (I think, I felt like I could follow it along) but I'm confused at what the theorem I've screenshotted is saying.

>>15513810
assuming you understand what's written in plain English:
your collection of closed intervals consists of n intervals, numbered 0 to n-1.
Applying the function to any of these intervals returns a set that has the next interval as a subset, "looping around" when you consider your final interval.
Then there exists a point on your first interval, denoted [math]x_0[/math] here, such that if you apply your function to it, and then to its output, and so on until you've applied the function n times, you will end up back at [math]x_0[/math].
Furthermore, at the kth application of the function to [math]x_0[/math], the output point will be contained in the kth interval. (Remember, the condition we started with is only that the set of outputs contains the next interval: on the face of it there is no guarantee that any given point outputs to a point within the next interval.)

>>15513742
Rudin.

I know I'm an undergrad retard, but I'm studying Calculus 1 and I need serious help with this
Where do I even start? I've never dealt with differential equations before

>>15515106
and by the way, these are practice questions and not homework if anyone gets the wrong idea. I have a final in a few days

>>15515106
A equilibrium happens when the derivative is 0.

[math]P^{BBC}[/math]

>>15511179
Algebraic geometry - hartshorne

why do we only work with one stock of variables when doing type theory instead of having an entire family of variables at our disposal (indexed by the types)?

>>15510822
I'm sorry but I don't understand what r^5 is supposed to be, how does it make sense to subtract a scalar from a matrix?

>>15516331
Other anon answered, sorry should've read on

Hey if you could answer this question I would really appreciate it:

>>15516783
>>15516783
>>15516783

>>15516784
a - b = c if b + c = a. In your example, 9 - 4 = 5 because 4 + 5 = 9. You can read this as: '9 - 4 is 5 because you need to take 5 steps from 4 to reach 9, that is, 4 + 5 = 9.'

>>15516829
What I'm saying is that if you want to know the difference between Person A who has $4 and Person B who has $9, you can count from $4 to $9 which is $5 which means Person B has $5 MORE than Person A. But you can arrive at this simply by subtracing Person A's $4 from Person B's $9 which gives you $5 too.
In other words you end up with the same answer if you count from the lower amount to the higher amount as you would if you subtracted the smaller amount from the larger amount. But why?

>>15516844
>>15516829

Oh I think I figured it out. I guess it's because addition is commutative?

4+5 = 9

So it doesn't matter if you start 4 and add 5 to get to 9 or start with 9 and take away 4 to get 5 because 4 and 5 are both components of 9?

Is that a good way of thinking about it?

>>15516844
write it as suc(suc(suc(suc(5))))

>>15504131
Well, where I'm from "special topics courses" don't exactly cover the basics. However, it does sound like an index theory course.
Could you be a dear and skim through this paper to see if anything seems familiar?
https://arxiv.org/abs/1608.04226
Thanks darling.

Not sure if this is the right place, but I am going back to school soon. I have to take a calculus readiness test and if I pass I can skip uni pre-calc. I have two months to prepare and essentially learn all of pre-calc by that time. How should I prepare?

>>15518594
look up a bunch of problems and solve em

why does analysis seem like a collection of empirical rules while algebra has theory that fits together nicely?

because analysis is just topology applied to the reals (don't hate me because I speak the truth)
algebra's rules look pretty arbitrary if you limit yourself to one particular type of structure, too

>>15518615
i dont know how to solve them

>>15518634
that explains nothing.

>>15516868
Yes.

>>15518640
go look up professor leonard on youtube

>>15518594
it depends on the effort you want to put in. if you want to sit there and actually learn pre-calc you should find a pre-calc book and go through as much of the book and exercises as you can. if you want to put in a little less effort you'd probably learn a comparable amount by watching someone's youtube lecture playlist. either of these options will probably lead you to pass the readiness test

>>15515106
this is baby stuff, you don't even the slope field to deduce that the only equilibrium T of a hot pizza is room temperature. perhaps it helps for you to understand that each mini-slope represents a normalized value of dT/dt, so a flat slope indicates T is constant (ie. equilibrium)

>>15519961
yeah I've been trying to get caught up over the few days, but I still haven't been able to figure out how to do linear approx. or euler's method

>>15519967
eulers method is easy-peasy, just add the value of the derivative times a time step. it's a first order method with no correction.
each time step is obviously 0.25 mins here.
[math]h = 0.25 \\ y_0 = T(0) = -2 \\ y_1 = y_0 + h*\frac{dT}{dt}(y_0) = -2 + 0.25 * 7= 0.875
\\ y_2 = y_1 + h*\frac{dT}{dt}(y_1) = \ ... [/math]

>>15519984
and how do I tell if it's an underestimate or an overestimate? I've seen some stuff about checking concavity from previous examples but I'm not quite sure..

>>15499014
Make them all about Covid and Trans athletes so the students can't use Chat-GPT

>>15518654
Well, because it's wrong.

Okay
I just got out of my calculus 1 final, the differential part fucked me up and I am pretty sure I got the optimization questions completely wrong since I couldn't simplify certain variables out of the way.

I still pass the course no matter what I get from the final, but I don't feel like I earned it. How do I intuitively understand maths? I've been studying whilst watching Professor Leonard and have been solving problems from textbooks but I still completely froze up today in the face of simple rates problems, like I forgot everything.
I kept confusing myself the most by rereading the problem and reinterpreting what it meant. There was a classic opt. problem where you have to minimize time spent swimming and walking, and I completely shut down when trying to place 1300-y for example, like I haven't done similar problems a hundred times for weeks

I'm trying to look for two numbers whose product is "close" to 76, and whose ratio is "close" to 281/400.

To expand on what I'm doing: I'm trying to make a phone wallpaper that's a tiling (rows and columns) of my favorite Yu-Gi-Oh! cards, and each card has dimensions 400x580 pixels (width x height), and my phone has dimensions 1080x2040 pixels.
So the cards' ratio is 20/29, while my phone is 9/20. And to get a wallpaper that almost fills my phone screen, I need to use a number of rows and columns for the wallpaper such that its ratio is as close a possible to 9:20.
So I need R (rows) and C (columns) such that (20C)/(29R) ~ 9/20, or equivalently, C/R ~ 281/400.

If I want to exactly use 76 cards, I need RC = 76, but 76 = 4*19, which is very far from the required ratio. So I could add or remove some cards. So I only need the RC ~ 76.

How would one go about this?
I could just manually try every number from 71 to 81 and look at their factors, but that's no fun.

To add to >>15520981, perhaps the best definition of "closest" is "leaves the least uncovered screen area".

>>15520981
Fuck, it's supposed to be 261, not 281.
Been writing a program for this and was working with the wrong ratio.

>>15520981
Here is the naive solution
[eqn]C \cdot R \approx 76 \\
\frac{C}{R} \approx \frac{281}{400}[/eqn]
Take the product and quotient of those equations and then take the square roots
[eqn]
C \approx \sqrt{76 \cdot \frac{281}{400}} \approx 7 \\
R \approx \sqrt{\frac{76}{\frac{281}{400}}} \approx 10

[/eqn]

>>15518594
I like this [math]{\rm cat}[/math]

I don't know what to do for this. These curvature/torsion/TNB vector problems are so tedious they make me want to kill myself. They take like 10+ minutes just to crunch the numbers and I still mess it up every time. Somebody help.

>>15521501
https://www.wolframalpha.com/input?i=curvature+of+%28x%2Cy%29+%3D+%286cos%28t%29%2C+5sin%28t%29%29+at+pi%2F2

>>15521501
well its 0 in the last coordinate so we will only have a z component in r' x r''; sin and cos make repeating cycles of length 4 as you successively differentiate with the 2nd derivative being the same as the original function with an extra -, and because we're moving at a t rate in the arguments we don't get any extra factors when we differentiate so r''= -r. intuitively r is the position vector of the ellipse, r' rotates it 90 degrees counterclockwise to give you the tangent vector, and r'' is another 90 deg counterclockwise rotation so you're back to pointing to the origin in the direction of r but antiparallel now

so r'xr'' = r' x (-r) = -(r' x r) = r x r'. now we already know these vectors are perpendicular by the picture above so |r x r'| = |r| |r'|, thus k = |r'xr''|/|r'|^3 = |r x r'|/|r'|^3 = |r|/|r'|^2. now we can plug in t, but to save even more work we remember that this ellipse is just a distorted circle (some constant * cos, some constant * sin) and at pi/2 we're actually at the x=0, y=whatever the scaling factor in the y coordinate is (where are you on the circle a quarter way through your counterclockwise revolution if you start on the positive x axis?), in this case 5 thus |r| = 5. finally r' being the tangent vector will point to the left at pi/2 (it points in the direction the curve is taking, and we're traversing counterclockwise) so it will have a 0 y component and only an x component, so -* whatever the x scaler is, in this case 6 but we're going to square this length anyways so the sign doesn't matter and so in total we get k=5/6^2.

there you go, no messy algebra computation at all

>>15521697
are there any tricks for this type of problem, on this one specifically I get a huge hunk of shit that apparently simplifies to sqrt(42) but there must be a better way

>>15520221
that's just a fancy way of say the derivative is monotonic in this case. you know what the graphs of the solutions look like given different choices of boundary conditions by looking at the provided figure. what condition corresponds to your problem? well, at t=0 the graph should be at -2, so you can match your answer to the curve that increases to 5 but becomes progressively flatter. in other words, the slope of the graph is positive but monotonically decreasing, so if you take the slope of the graph s at any point P and move some distance starting at P along the straight line of slope s, you should end up further above on the y axis than if you just sticked to the curve. the derivative at any point generates vertical movement on the curve and the generators are weakening in their upward push, so sustaining the push at a fixed point over any length of time forward should overpower and overshoots the ordinary movement along the curve and push you further above.

you can also visualize this by drawing a tangent line at 0 to the solution curve and following the line for some time t_1 to end up at p_1 (you're already above the graph now); then continuing your journey from p_1 along another straight line of slope = slope of the solution curve at t_1 and moving along this line until time t_2 to end up at p_2 (now you're even further up above the graph), etc. the error compounds over your movement along each tangent line segment. the straight line approximations all lie above the curve due to its concavity (for differentiable functions concavity = monotonically decreasing derivative, convexity = montonically increasing derivative. makes sense when you think about what the LHS and RHS mean)

>exam question asked student to solve differential equation utilizing laplace transform.
>student writes "No." and solves differential equation using power series.

Who was in the wrong here?

>>15521794
student, easily
if the exam specifies a method, they're testing you on your knowledge of and ability to use that method.

>>15521794
The student.

>>15498384
How do I get better at math

[math]N_{cock}=AIDS[/math]

>>15522689
Honestly, just do more problems

>>15498384
Hey anons, what are some advance topic atm? Like what are researchers working on?

>>15522689
These are the three M's of mathematics
More (math), Meth, and Monetary compensation
These are the things that drive a person to become good at math

>find mean of some curve on a graph
>generate function of graph by interpolating with random data points in a vandermode matrix
>just integrate the function from end to start, and divide by total range

I feel powerful.

>>15522689
read books. do problems. no shortcut to math which is why people think it's hard but it's just about time you put in.

>>15508374
>the differential [...] is invariant under a rotation of the circle
can you be a little more verbose here? are you saying that rotating the bounds of integration around the circle doesnt affect the integral?
>we could have started from any other given point
would it have to be on the circle?
>Also, the path OP that we've chosen is not important
really? does the path have to be along the circle?
>>15508397
>we find that C admits a group law (hopefully? I couldn't find anything online)
youre telling me
>Anyway, we also find that ω is invariant under translation for this group law
what kind of translation? rotating a circle is easy, but elliptic curves are harder to imagine. is it just sliding the points the same distance in the same direction along the curve?

thank you again, friend. i feel like im so close to understanding this thanks to your help.

hi im the mathematician the image is about

I'm going to graduate my math PhD with 0 publication.
How bad did I do?

What is "mathematical maturity"?

If someone is trying to brush up and relearn highschool precalc, how bad of an idea is it to just learn first year calculus instead?

>>15524301
you learn more math by seeing it in different contexts. it's not a bad idea at all

>>15524301
Not bad at all
you won't use much precalc in a normal calculus class

>>15524302
>>15524307
Thank you. When I go back to school later this year, I will be taking a calculus readiness test, which mainly involves precalc. Do you think if I were just self study first year calculus, I would be good for a precalc test?

>>15524316
depends. i think if you weren't failing high school math you will do fine

>>15524346
I was dead average in math. C+s and Bs.

>>15524316
Try to relearn some of the trigonometric stuff and refresh your algebra skills if you need to. Maybe look up other calculus readiness exams and see what topics are on them

latest

>>15524088
Only your advisor is at fault

>>15524088
Is that even possible?

>>15523454
>can you be a little more verbose here?
idk your mathematical background, but for a given smooth algebraic curve [math]X [/math] over the complex numbers one can consider the vector space of holomorphic 1-forms (I say holomorphic but you might as well consider regular forms; again, idk your background), often called [math]\Omega^1_X [/math]. Defining this properly would take me afar, but a 1-form is basically something which is locally of the form [math]\omega=f(x)\mathrm{d}x [/math], where [math]x [/math] is a local coordinate on [math]X [/math] and where [math]f [/math] is a holomorphic function on [math]X [/math]. They are designed specifically to be integrated over paths, [math]\int_\gamma \omega [/math] makes sense (again, remember that topologically [math]X [/math] will be 2-dimensional, while [math]\gamma [/math] will be 1-dimensional).
Now, one can pull back 1-forms: given a morphism [math]\phi: Y\to X [/math] between smooth algebraic curves, and a 1-form [math]\omega [/math] on [math]X [/math], one can construct a 1-from [math]f^\ast\omega [/math] on [math]Y [/math] (locally, it would look like [math]f(\phi(x))\mathrm{d}\phi(x) [/math]).
Now, in the cases that we are considering, our [math]X [/math] admits a group law, in particular for a given point [math]Q\in X [/math] we can consider the translation by [math]Q [/math] (this is the standard terminology, but in many cases it doesn't look like a translation; e.g. on the circle it is a rotation), call it [math]t_Q:X\to X,\quad P\mapsto P+Q [/math]. One says that a 1-form [math]\omega [/math] is invariant under translation if [math]t_Q^\ast\omega=\omega [/math] for all [math]Q\in X [/math].

>are you saying that rotating the bounds of integration around the circle doesnt affect the integral?
After the previous explanation, what I am using is a variant of the change of variables formula, which in our case says that [math]\int_{\phi\circ\gamma} \omega=\int_\gamma \phi^\ast\omega [/math]

1/2

>>15523454
>I am using is a variant of the change of variables formula
To elaborate, we take [math]\omega [/math] to be the invariant 1-form, [math]\phi [/math] to be the translation [math]t_Q [/math], [math]\gamma [/math] to be some path [math]O\to P [/math], and so we get [math]\int_{t_Q\circ \gamma}\omega=\omega_\gamma\omega [/math]. Note that [math]t_Q\circ \gamma [/math] is a path [math]Q\to P+Q [/math].

>would it have to be on the circle?
>really? does the path have to be along the circle?
Yes, everything that I'm doing should live inside
[math]X [/math].
And the path is important, as I said we have to be coherent about our choices. I was omitting the actual choice of our path [math]\gamma [/math] to be more in line with historical notations, but it was probably a bad idea in retrospect.

>youre telling me
Hehe you're right, that's where I got lazy and didn't want to spend too much time on this random (to me) historical problem. The point I wanted to drive across was that *if* we both take the leap of faith of believing that the lemniscate [math]C: y^2=g(x),\quad g(x)=(1-x^2)(1-k^2x^2) [/math] admits a group structure, and believing that [math]\omega=\mathrm{d}x/y [/math] is a differential invariant on it, *then* we have a nice framework justifying all of our addition formulas at once.

>what kind of translation? [...] is it just sliding the points [...]?
If you think of [math]E [/math] as a curve inside the projective plane, then [math]t_Q [/math] is a sort of slide, but a strange one at that: keep in mind that the point at infinity should end up at [math]Q [/math], so it cannot be the stiff motion that you're probably thinking of. Also, remember that we're thinking about complex solutions, and over [math]\mathbb{C} [/math] we have two descriptions of the law group: thinking of [math]E [/math] as a torus should give a better idea, see https://en.wikipedia.org/wiki/Elliptic_curve#Elliptic_curves_over_the_complex_numbers

2/2

Not even my teacher knew what's the connection between the slope of the tangent line and the area under the curve. Can someone explain this to me please.

Take a simple subtraction problem like 9 - 7

1) On a number line you can start at 9 and count 7 steps down to 2 and that's your answer.

2) Or you can start on 7 and count 2 steps up to 9 and 2 is your answer.

You get the same answer either way but in the 1st way you have to make 7 stps and in the 2nd way you only have to make 2 steps.

Why does this work?

>>15524799
>Why does this work?
because the answer was less than the subtrahend. in 9 - 1, the first way is faster, because answer is greater than the subtrahend.

>>15524804
whoops, pic unrelated

How do I prove that a set with a countable base will have a countable subcover for every open cover?

>>15524804
No I'm asking why these two different methods give the same answer.
And the second way is faster not the first.

Are all uncountable subsets of euclidean space bijective with the euclidean space?

>>15524088
I thought you were required to publish for a PhD? If you dont publish, isn't it the same as those masters programs that are coursework only? How did you do it, the only reason I wont even consider a PhD now is because of risk. If there is a zero risk and guaranteed option I'd jump on that immediately.

>>15524809
>No I'm asking why these two different methods give the same answer.
in the first way you're doing 9 - 7 = 2
in the second way you're doing 7 + 2 = 9
these equations are identical
>And the second way is faster not the first.
the first way is faster when you do 9 - 1 instead because you only count one step down, whereas if you started at 1 you'd have to count eight steps up to 9

>>15511476
You can indeed think of type theory (and higher-order logic in general) as a variant of first-order logic with infinitely many sorts. Some things you might not be used to: Sometimes first order logic assumes a nonempty domain, but types don't have to be non-empty. You can have "dependent functions" where the type of the output depends on the input. The types themselves have types; somewhat confusingly, the types of types are sometimes called "sorts." Propositions are also types, whose members are proofs of the proposition (false propositions are empty types).

I hate my thesis advisor.

What's the best textbook for Calc. 2?

what do you call it when you have an axiom that can be demonstrated to be true in the real world

>>15525454
define what you mean by an axiom in the real world.

>>15525475
If you say X is true, when X is true empirically. Would you just call it a fact instead of an axiom. If you take a set of these, what would you call it instead of an axiomatic system.

>>15525586
>true empirically
Which comes down to observation and this is inherently flawed because your sight/equipment/experiment may not be telling you everything. Axioms cannot exist in nature, facts can with the caveat "to the best of our current knowledge".

>>15498914
> should I read euclid
No. The ancient math texts that were worth reading are Euclid's elements, Appolonius of Perga's conic sections and the works of Archimedes.
You'll notice the works of Newton and other scientists of his time relied heavily on the comics. Most obviously ellipses which described the orbits of plants, but also some properties related to the reflection of light off a conic section.

I'm not sure exactly when "they" decided to modernize math , but it really lost any connection to the past and was built on its own foundation. Currently I'm writing a pdf for people like you, but in the mean time I'd suggest just doing things the standard way with the understanding modern math is disconnected from the past which is why so much of it seems to be taught arbitrarily. What makes this worse is the modern belief math is just "made up" instead of some truth that exists outside of us so the shitty modern way of teaching it can justify itself. Because if math doesn't exist outside some of us then there's no best way to teach it and so any way is ok including tranny sneed modernist methods.

>>15525885
Conics*
Planets*
Also my pdf will be called calculus for sneeds

So I found out my IQ (professionally tested) is only 123. should I abandon the idea of academia? I'm a graduate student. I obviously like math and am decent at it but if it's going to be a constant struggle to keep up with the average mathematician out there then I feel like I should probably cut my losses and just go for industry after I finish my PhD

>>15525911
PhD candidates are already above average mathematicians. You're in the 99th percentile of math knowledge. The average joe barely retains anything beyond fractions and algebra.

>>15525952
I mean mathematician as a profession, i.e. a mathematician employed at a university. I need to look at the relative competition among all math PhD students because they are the ones I will be competing against for a job

>>15525955
Well, this is easy. First, find a successful fellow PHD candidate in a similar field as you and challenge them to a math competition. Then have a professor give you both a collection of problems. Whoever solves the most wins, and the loser has to quit.

>>15498914
Don't read Euclid. If you actually want to learn synthetic geometry, get a more modern treatment like Kiselev. Following this you could work on Coxeters' Geometry Revisited.
If you aren't genuinely interested in synthetic geometry, don't do this at all. Get a precalc book or even Spivak's calculus, since a good chunk of Spivak is precalculus. Or if you really wanted, you would not miss much by going straight to Zorichs' Mathematical Analysis

>>15525040
Never heard of anywhere having a publication requirement (except some chinese universities I think)

Is there an infinite number of prime numbers with the property that no two adjacent digits are the same, and by removing any digit, the resulting number is also a prime number?

In picrelated I highlited the biggest one that someone calculated by computer.

>>15525998

I've never heard any university will give you a phd without publishing anything

>>15526065
heuristic says 1/log x probability of prime of size x
you start with number of size near x with distinct digits. approx x^a choices where a = log_10(9). it and all log x different deletions must be prime
probability (1/log x)^(log x). This is very small, much less than 1/x^a. But it's not actually directly independent so who knows

if you just want distinct adjacent digits, it should follow from generalizing Maynard "primes with restricted digits" which he won fields medal

>>15526082
show me a western university with a formal requirement (other than the dissertation itself which no one calls a "publication"), I'm pretty sure it is very common in mathematics to graduate without a publication

>>15526082
>>15526134
you have to have a thesis, it doesn't have to be published

>>15526168
>>15526134


Ok, so why not publish it then? Seems like it would only take a few minutes to submit it to a journal.

>>15525911
If you need to ask then yes, leave.

>>15526370
What's the logic here?

>>15525911
>So I found out my IQ
you failed the test, then

This video is kino for any true math frens
https://www.youtube.com/watch?v=B1J6Ou4q8vE

>>15525911
>Im a graduate student but I just found out my IQ is only 1.5 stds above the average instead of 2 should I kill myself?
Yes.

>>15525911
William James Sidis had an IQ around 275 and he worked at a post office and as an low position accountant, eventually dying depressed and alone and you think you can make it with 125? It's over. That isn't even monkey tier.

Neuron star shape obey no euclidean forms, have you noticed it?

bros, can someone tell me whats the answer to picrel is? I'm getting 0 or pi, but symbolab is telling me its pi/2

>>15526720
I'm getting negative pi/2. Just a good choice of
u-substitution should help.

>>15526765
idk what that method is called where you split the integral into two parts and substitute the positive and negative infinities with a and b. then you use arctangent formula. I'm being very vague because I can't recall the formula exactly

>>15526781
>>15526765
Cauchy's principal value?

>>15526720
[eqn]x = -e^{-t} \\
dx = e^{-t} dt [/eqn]

[eqn]\int_{- \infty}^\infty \frac{e^{-t}}{1 + e^{-2t}} dt = \int_0^\infty \frac{dx}{1 + x^2} = \arctan(\infty) - \arctan(0) = \frac{\pi}{2}[/eqn]

>>15526786
>>15526794
I'm not sure what I just did both by myself and using ChatGPT's help, but I finally got pi/2. picrel is what chatGPT did.
1/3

2/3

3/3

Be sincere, sci.
Is math the language of human hallucination? If you pay genuine attention you'll notice the very basis starts from absurd comparisons and missing aspects.

>>15526811
what is absurd about that picture? that pi is greater than sqrt 2 in a geometrically precise ratio?

>>15526827
The obsessive comparison of curves and lines.
Number sets, n-dimensions etc

>>15526837
it comes naturally to some people.

>>15524774
I think the most intuitive part is that the rate of change of the area under the curve is equal to the curve itself. If at a given point the area is increasing a lot, that means that the point is high above the axis. Or, if a given point is low, then the area is only increasing slightly. If the point is below the axis, then the area is decreasing. This is why the derivative of the integral of a function is the function itself.
https://yewtu.be/watch?v=FnJqaIESC2s

>>15526842
I'm not saying math is illogical. But just like some perfect cohesive conspiracies theories texts it lacks fidelity with reality until we get to see the whole picture.

statistics seems to be the most valuable field in the modern world

>>15525049
No the 2nd way is faster because you only count 2 steps up. The first way means counting 7 steps down. Review the image.

>>15526803
>>15526786
Are you sure it's not negative pi/2?

>>15526883
>the integral of a positive function
>negative pi/2
What do you think, anon?

>>15526883
the minus from the u sub happens at the same time as the reversal of the domain (think about what e^(-x) does) and the negatives cancel

>>15526811
isn't everything that humans have done the "language of human hallucination" in a sense?

>>15526883
>>15526898
>>15526896
Got it, it makes sense now.

>>15526811
Are you supposing mathematics or mathematical objects don't exist?
Are you assuming mathematical objects supervene on physical systems or not?
Are you aware of the historical delineations between pure and applied mathematics?
There are a plurality of views on this, and most laypeople don't even have a coherent view on what the big questions about them are.

>cant understand proof of cauchy's theorem

>>15526957
I don't know. I'm just a dreamy pythagoreanlet brainlet.

>>15526811
>Is math the language of human hallucination?
yes, for example 0 is not real, humans made it up, the whole concept of "math" is a joke. for example nobody can solve even something as simple as p=np which proves the whole concept of math is broken as fuck

Periodicity in 2n+1

I have made interesting observations in the 2*n+1 rule:

1. whenever you get a ending string of ...999 it will forever stay 9. This can be shown easily: 9*2+1=18 where you consider the 10 as a carry in the "10-ary" (plain) addition. And so an ending string of a arbitrary amount of 9ths will always stay the same. (Interestingly 9 is divisible by 3, but I digress.)


2. It is enormously rare to get an additional 9 at the end. So much so that I suspected there might be combinations in which you will never reach a second 9. So I checked. In fact this can be proven by simple calculation as above.
Let us start by calculating whether a 9 in the previous place will ever produce one the following place:
9*2+1=19
10*2+10=30
30*2+10=70
70*2+10=150 (we can cut off the leading 1 because we only care about the decimal place before that)
50*2+10=110
10*2+10=30
...
By that we can see that 1,3,5,7 will never lead to another 9 in that same decimal place (when preceded by 9). Let us now check for the other numbers:
2*2+1=5 ...
4*2+1=9
6*2+1=13 ...
8*2+1=17 ...
0*2+1=1 ...
So 1,2,3,5,6,7,8,0 will not lead to a 9 when preceded by a 9, 9 is itself already and 4 will lead to it. So the only way to get an infinite amount of 9s at the end is to produce a 4 directly in front of leading 9s. Let us now construct the number that will lead to an infinity of 9s:
We observe:
?4*2+1=49 => ?=2

?24*2+1 = 249 => ?=1

And thus
124*2+1=249
249*2+1=499
499*2+1=999
Now we extend this to 4 letters by halving 1 (=0.5) in the leading decimal:
624*2+1=1249 -> 2499 -> 4999 -> 9999
Now we quarter etc:
10000/4+1000/2+124=2500+624=3124
+100000/8 = 12500+2500+500+124=15624
+1000000/16 = 78124
+2^2*5^7 (prime factorisation) = 390624
+2^2*5^8 = 1953124
...
Now we have a formula:
[math]\sum_{k=0}^\infty 2\cdot 2\cdot 5^k[/math] This p-adic number will produce an infinite amount of ending 9s. I wonder whether it converges or anything. Can anyone maybe tell me more?

>>15527428
3. Whenever you perform 2*n+1 9...9 times, you will not only receive the string of the amount of ending 9s (according to the construction above), but before that you will receive a string that is always the same and continually getting longer. What do I mean by this? Here is a python program:

n=1
l=[]
for i in range(100000):
l.append(n)
n=2*n+1

Then
l[9]=...
l[99]=...375
l[999]=...9375
l[9999]=...09375
l[99999]=...109375
As you see in this case, there are no ending 9s and the end is periodic and growing in periodicity for every decimal power of applying the rule more. I have not yet checked this for other multiplication+addition rules. Also I wonder how this fringe periodic stability comes about. Can anyone explain this to me?

>>15527428
this is a 10-adic number (10 is not prime)
10 does not divide 5, so this doesn't actually work converge, but there are many subsequences that converge (by konig's lemma-style argument, or you can do it explicitly).
if you looked at this series 5-adically it would just be the expansion of -1 which is a fixed point of x->2x+1

>>15527430
l[k] = 2^(k+1)-1
k=10^n-1
l[10^n-1]=2^(10^n)-1
easy to see this stabilizes mod powers of 2 very fast
stabilizes mod powers of 5 by Euler's theorem
in fact this number is -1 mod 2^large and 0 mod 2^large
so chinese remainder says it is some fixed number mod 10^large, increasing number of digits all the time

>>15527430
>>15527671
also you get the same sequence from
10^(2^n)-5^(2^n)
may be

>>15527661
>>15527671
>>15527685
Thank you.

Eurler raised 13 childern and was blind the last 17 years of his life.

Are there any websites for daily math problems?

>>15527784
check your local university's website. Many of them have a "problem of the week"

Geometrically, we can interpret sine and cosine as the vertical and horizontal components of a vector, and the tangent as the slope. It's probably useless, but in this view, what are secant and cosecant?

>>15526856
This is because their textbooks are unrigorous garbage that aim to mystify pretty simple concepts and techniques to make people think their meme analysis on their worthless data is justified because of esoteric math tricks. Then these sort of tests and memes become the standard in the industry and even in science so the value goes up tremendously. In reality, either the techniques amount to do science blindly which means you cannot conclude antythings, or the assumptions are so ridiculously strong that they model jack shit and the only way of fixing it is actually investigating and thinking about the thing you want to model, i.e., doing actual science. The weirdest thing is that if you just know some basic math and just take some time into looking what the statistical tests are doing, you will realize that they are quite limited without any input and will probably just validate tendencies when it is kinda obvious they exists. But people who dare to call themselves scientists just kinda put it in a software without even looking at it and say voila my chi-squared test shows this.

>>15527910
Check wikipedia for trig functions

What is indefinite integral (cos(sqrtx))/(sqrt(cosx))??

If f(x) = 2x-1, fins the value of x such that f(f(x)) = 9

>>15528086
3

are there any measure theory books with a large number of visualizations? i've checked out a lot of books on the topic and nearly all of them are just walls of text and symbols.

>>15528778
there isn't much to visualize. it's an abstract subject

Is there a math guide to learn math from? I want to learn math but I don't know what to read. I have an O levels knowledge of math.

>>15528807
Search Math Sorcerer's videos about book recommendations or start with Khan academy.

>>15528778
Kek I think you're shooting out of your league. Measure theory is a graduate level topic, and you won't find much visualisation at this level, even in geometry books, but Stein and Shakarchi is probably your best bet for an easy introduction to measure theory.

What are some good mathematically rigorous texts on chaos theory for an undergrad

>>15529757
I added mathematically rigorous because a lot of the material I find from a cursory search is pop science junk