/sci/ - Science & Math

SLE curve is neat

Still mostly interested on getting a hold on the provable range of things in CZF and polishing Wikipedia articles, and trying to survey various notions of the reals (in particular the small conservative ones) but on the more straight and hands-on side, I'm coming back to pic related today.
My Conway semiring question in the last thread didn't get a reply, I think. It was fascinating mating for me last week to discover there's a corpus of texts around the algebraic characterization of that star operation that at least for dense rings ends up as X mapsto 1/(1-X).
I first tool interest in this object as a thing on its own when I saw the relatively broad range of spaces in which the geometric series gives a formula of powers of operators (can also sometimes be used to compute solutions of differential equations where X is d/dt And their inverse are integrals) as well as all those number theoretical local Zeta functions formulas where it pops up.
In the theory of semi-rings you ca. e.g. use it to reason about existence of paths in graphs, which powers of the adjacenct matrix tells you, which is also a wonderful formula.

When I find cool for formulas in pic related I may try and squeeze them into this instructive virtual machine Python setup I cooked up with recently.

Whats the most I would have to know to work through concrete mathematics and understand it? Would finishing stewart's calculus first be enough?

y^1/3(y^2/3+y^5/3)
I have been spending hours trying to figure this one out
how does it simplify to y^2+y

I understand the classical argument from calculus of variations for showing that the shortest path between two points is the straight line. But this minimization kinda only works for picewise C^1 curves. How can someone proof it actually is a minimum over general continuous curves?

>>10913240
Exponents add

>>10913068
>Isn't it only true if J is symmetric?
No, you're not transposing your matrices correctly when computing the gradient of [math](1/2)f^T f[/math].
If that's confusing, write [math]f\cdot f[/math] as [math]\sum_i f_i^2[/math]. Then, compute the j-th derivative of this summation for all j to get [math]\sum_i 2 \frac{\partial f_i}{\partial x_j} f_i[/math]. This proves that the j-th entry of [math]\nabla (f\cdot f)[/math] is equal to the j-th entry of [math]2 J^T f[/math]. Finally, divide everything by 2.

>>10913229
Don't be condescending. Explain things in a way they can understand.

https://en.wikipedia.org/wiki/Mathematical_universe_hypothesis

>>10913240
It doesn't.

>>10913241
By Sobolev embedding [math]W^{n,1} \hookrightarrow C^0[/math], solutions to weak EL [math]\int_\Omega dx \delta F = 0[/math] achieved in [math]W^{n,1}[/math] is automatically achieved in [math]C^0[/math]. In addition, the solution is unique (uniqueness of weak limit [math]u_n \harpoon u[/math] in [math]W^{n,1|[/math]). This means that once you've found a solution, you can stop looking.

For the problem of geodesics, the kernel [math]F \propto \sqrt{-g}[/math], where [math]g[/math] is the metric tensor on [math]\Omega[/math]. For sufficiently "nice" spaces [math]\Omega[/math] or smooth manifolds ([math]n \neq 3,4[/math]), [math]F[/math] is Caratheodory and hence the weak EL is [math]W^{n,1}[/math]-solvable.

>>10913378
Shieett
[math]u_n \rightharpoonup u[/math] in [math]W^{n,1}[/math]

I like /mg/ and it's in theory naively a good way to spend your time (by answering). But in reality it's one of the cases where the anonymity really is a hindrance:
You don't know how much the asker cares, if he's genuinely interested or just lazy, whether he or other people will engage and in fact you don't even know if your investing time and engaging and posting a solution will even get a reply..

>>10913388
I once asked a question here and some anon gave me a counter example matrix and I still appreciate it to this day.

>>10913402
unrelated to what I said, but good for you

>>10913372
call my textbook company "Cengage Learning" and call them retards pls

>>10913241
definition of curve length for arbitrary continuous curves is:
- choose a finite set of points of curve, they create segments, take the sum of lengths of segments
- length of curve := supremum over all possible choices of intermediate points
the C^1 case obviously includes "piecewise linear curves". So you're taking supremum over some numbers, all of which are not smaller than the minimum you already obtained with a piecewise C^1 curve.
>>10913378
you learned so much advanced stuff and yet can only regurgitate hard words without thinking for yourself

>>10913388
>You don't know how much the asker cares, if he's genuinely interested or just lazy, whether he or other people will engage and in fact you don't even know if your investing time and engaging and posting a solution will even get a reply..
As in real life

>>10913212
>Would finishing stewart's calculus first be enough?
Why don't you try it and find out?

>>10913417
Kinda, but in a reading club or uni class the change of people responding is naturally higher. O

I have finally proven how many ways there are to choose rows of the Sierpinski matrix of order n such that the rows form a system of linear equations with 0 degrees of freedom.

It turns out valid selections from [math]S_n[/math] are in bijection with full binary trees of depth n.

So, there are A002577 of them.

>>10913132
Pls someone give brief on series. Why is it important and what is the basic 101 of it?

What are some examples where induction is actually invalid as a proof approach instead of being "not elegant enough"?

>>10913212
High school algebra, just a lot combinatorics problems.

https://www.springer.com/gp/book/9783319138435
Almost same topics,good problems and rigor but very smooth learning curve.

>>10913590
>What are some examples where induction is actually invalid as a proof approach instead of being "not elegant enough"?
Any theorem that doesn't involve a collection of statements that can be indexed by the natural numbers.

>>10913309
You should take some lessons before opening your mouth

> Check your work you look like an idiot right now

>>10913378
You are wrong too wtf guys

What’s the use of ordinary differential equations?

You get to find the function from knowledge of function’s relation to its derivative.

>>10913649
Wrong.

>>10913634
>>10913638
>>10913654

>>10913591
How many words of length n can be constructed from the alphabet {a,b} such that no word has two adjacent a’s?

>>10913591
Why soany combinatorics books ha e graph theory shoehorned in?
Is it because of computer scientists?

>>10913442
Yes, but you have to worry about a large amount of interpersonal rigmarole, which is distracting. The tl;dr phenomenon is marginalized only in appearance.

>>10913663
Wrong

>>10913654
Wrong

>>10913608
Wrooong

>>10913591
God, this is embarrassing

>>10913514
Please tell me you are joking

>>10913640
Gaaah

>>10913413
Seriously?

>>10913459
This isn't even a thing

>>10913659
LOL

>>10913142
Gay

>>10913132
Asshole

>>10913199
I'm not even reading that

>>10913240
Ultra gay

>>10913387
I bet you thought this was hilarious

>>10913294
Imagine living on the same planet of this idout fuck fuck

>>10913678
Hello, I love you. Won't you tell me your name.

>>10913665
Yes

N=nP
Which means that the function of N is actually equal to an infinite number of P variables

While covering the differential in multicalculus variable the professor left this proposition as an exercise: "let [math]f : \mathbb{RR}^n \mapstop \mathbb{RR}^m [/math] be a function. f differentiable at some point [math] x_0 [/math] iff all components of f are differentiable"
The point that my reasoning is a bit speculative is this: if I've [math] \sqrt{a^2_1 + \cdots + a^2_m} = o|\mathbf{x} - \mathbf{x_0}| [/math] then: 1) the norm of the vector [math]x[/math] must be comparable as in speed of infinitesimal to x, 2) all components must [math]a^2_i = o(x)[/math]

How wrong is this reasoning? I need this argument so that I can later use the triagular inequality to split the whole square root (as it's a norm of a vector) and then use 2) in the limit case to prove my point as the a terms are a lazy way to write the components of f.

>>10913700
f(a,b) is just the sum of the square root of a and all real numbers of b

>>10913700
mistake, is just [math]a_i = o(x)[/math]

>>10913700
I'm floored how fucking dumb are you really though? Literally go back to school and take some lessons before opening your mouth.

>>10913711
Be more constructive

>>10913716
Die

>>10913725
nice argument, it fixes my proof
go circlejerk on some maniford now

>>10913731
Neck yourself bud

>>10913413
>you learned so much advanced stuff and yet can only regurgitate hard words without thinking for yourself
His proof is correct, lad.
Yours is nicer, tho, even if proving uniqueness from it is tricky as fuck.

>>10913700
Are you stupid? The sum of all values of a is just equal to the absolute value of x- the initial x multiplied by the a

>>10913142
tfw this one equation has produced more fields medals than entire fields of math

>>10913372
>>10913388
>>10913406
>>10913459
>>10913634
>>10913638
>>10913654
>>10913668
>>10913678
>>10913686
>>10913695
>>10913711
>>10913725
>>10913736
moronic posts, stop ruining the thread for fun

>>10913142
>>10913916
Did you reply to yourself? I'm asking because I never heard of that or seen the Wikipedia page. Where or since when is that a popular subject?

>>10913700
1. Fix a basis [math]\{e_i\}[/math] of [math]\mathbb{R}^m[/math] and define coordinate functions [math]e_i: \mathbb{R}^m\rightarrow \mathbb{R}[/math] sending [math]x \mapsto x_i[/math].
2. Prove the smoothness of [math]e_i[/math] and [math]f_i = e_i \circ f[/math].
3. ???
4. Viola.

>>10914115
>calling the dual basis the same as the basis
what a moronic post, the "functions" should have superscripts. and what is x_i?

>>10914115
He said iff.
Which does remind me I have jack all idea what definition of smooth he`s using, since he`s trying to prove the one I`m used to. There`s
>that really autistic one in differential topology
But I`ll guess that`s not the case.

>>10914129
Ah, he said differentiable, not smooth.
Fuck if I know what he's asking.

>>10913240
It really doesn't

>>10914129
>`

show this integral [math]\int_{0}^{\frac{\pi}{2}} \frac{1}{1+tan^{\sqrt{2}}x} dx=\frac{\pi}{4}[/math] is convergent
Can i just say that since the area of function is defined for the interval then it cannot be divergent? Since if we evaluate a function between some interval [a,b] to be divergent then the area of the function between interval [a,b] would stretch toward infinity?

>>10914312
Yeah if you accept the pi/4 result, then it's not divergent by definition.
I expect they just want you to prove the result.

>>10914312
I think what you’re trying to say ( in easiest terms) is the maximum of the divergent or convergent integral in which you need to find the whole area? In other words, just solve the integral and plot it

>>10914385
Didn’t finish plot it at the maximums you find by doing any of the divergence tests

>>10914379
is it even possible to solve this integral without using imaginary number?

>>10914395
Yes if you don’t know your basic integrals for tangent, then you can’t solve it

>>10914397
I dont understand how [math] tan^{\sqrt{2}}x [/math] has anything to do with basic trigonometric integrals. I assumed you solve it by substitution

>>10914420
There is an identity in there look again

>>10914422
u making me trip ass, even if i took out [math]tan^{\sqrt{2}}x = tan^{\sqrt{-2}}x\cdot tan^{2}x[/math] to get the secant doesnt get me anywhere, same for [math]tan^{\sqrt{2}}x = \frac{sin^{\sqrt{2}}x}{cos^{\sqrt{2}}x}[/math]

>>10914312
i mean the term in the in the integral is positive and bounded above by 1 for all x from 0 to pi/2, so of course the integral converges and is bounded by pi/2.

Why the fuck are the reals considered uncountable? It's easy enough to do

>>10914815
if you claim to me that you have an enumeration of the reals, i can find a real that is not on your list. this contradicts your claim.

>>10914856
My method can bootstrap any number into itself

>>10914863
Would you mind elaborating on what your post means in a manner comprehensible to people lacking schizoid personality traits?

>>10914887
I'm still refining it. I'm probably wrong, but it's at least worth consideration. I'm going to post it once I get it somewhat rigorous.

Is he right about determinant or just imagination?
https://www.youtube.com/watch?v=Ip3X9LOh2dk

I'm trying to learn multivariate spline interpolation but I can't find a textbook that covers it. Should I just use Wikipedia and some random papers on it?

>>10914312
okay so I ended up with this approach.
Since I cannot evaluate the integral, I shall prove its convergence by evaluating 2 functions that bind my function from above and below. So i'll use [math] f_{1}(x)=\frac{1}{1+tan^{2}x} [/math] and [math] f_{2}(x)=\frac{1}{1+tan(x)} [/math] Since [math] tan^{2}x<tan^{\sqrt{2}}x<tan\,x [/math] when [math] 0\leq{x}<\frac{\pi}{4} [/math] and [math] tan^{2}x>tan^{\sqrt{2}}x>tan\,x [/math] when [math] \frac{\pi}{4} <x\leq{\frac{\pi}{2}} [/math] where [math] x = \frac{\pi}{4} [/math] is trivial since points cannot have an area. So if we do the the integral of both function between the intervals [math] [0,\frac{\pi}{2} [/math] we can prove that the initial integral is convergent between said intervals.

Would this be a good approach to the problem?

>>10914106
The SDE was first seriously studied around 1999 or so. People found out you could model loop erased random walks with SLE, which gives them applications to 2d percolation in statistical physics. They have conformal invariance properties. You can uniquely characterize the family by these properties. Then people pushed these connections to the 2D Ising model. The fields medals came in 2006 and 2010.

>>10914312
try any exponent, not only sqrt2. Try 99, pi, 666/665. You will have the same result.

>>10914900
Hahaha.
Everybody laugh at the retard

>>10915054
Conjecture:
[eqn]\int_{0}^{\frac{\pi}{2}} \frac{1}{1+tan^{a}x} dx =\frac{\pi}{4}[/eqn] for every [math]a \in \mathbb{R}[/math]

>>10915083
the function is inversely proportional from the interval [math] [0,\frac{\pi}{4}] [/math] to [math] [\frac{\pi}{4},\frac{\pi}{2}] [/math] so everything cancels out and will always give us an average of [math] \frac{\pi}{4} [/math]

How do I prove (1)

>>10915210

Thank you anon who believed in me that I could pass the RIMS entrance exam! Only like 3 people including you thought I could do it.

What's the largest piece of math you guys have ever come across?

>>10915454
my highschool textbook was pretty large if u ask me

>>10915460
>>10915454
I meant *hardest sorry.

>>10915285
Absolutely great job, lad. Knew you could do it.

>>10915042
You work in that field?

>>10915454
It's a bit of a philosophical question what "hard" out to mean, especially since there's different amounts of people in every field. Many people succeed and many people fail in calculus, so it's hard to compare it to e.g.
https://en.wikipedia.org/wiki/Hodge_theory
or
https://en.wikipedia.org/wiki/Arithmetical_hierarchy
that view people even get into and there's comparably few resources. I'd spontaneously cite those two as answers, since I know my fair share of math but those are hard to get started with.

complete newbie here taking my first calculus course. It's been years since I did math.
the course book is "Calculus" by Adams and Essex. Anyone familiar with it? Seems "meh" based off reviews.

I've got these books but I don't know which one might be better for the course:
>Calculus - Apostol.pdf
>Calculus - Spivak.pdf
>Calculus - Strang.pdf
>Elementary Calculus, An infinitesimal Approach - Keisler.pdf
>Calculus, Made Easy - Thompson .pdf

Exam questions are things like:
>R is the limited area of the 1st quadrant above the curve y=x^2 and below the curve y=8-x^2. Calculate the volume of the body generated when R rotates around the Y-axis.
>The half-life of C14 is [so-and-so]. The amount of C14 in a dead organism at a current time y(t) satisfies a differential-equation of the shape y' = k*y for some constant k. A piece of bone contains 60% of the original C14. How old?
>The ellipse E is provided as solutions to the equation 5x^3 + 5y^2 = 10. Decide an equation for K that tangents E in point Q = 1, sqrt(3)/2).
>Integrate(x^5 ln x dx), for 1:e.
>Let P(x) be a 2nd deg Taylor polynomial around x=0 to the function f(x) = e^x. Use P(x) to give an approximation to sqrt(e). Decide if the error in the approximation is greater or lesser than 0.02.
>f(x) = (x^3) / (2x^2 - 1) ; Draw the graph y = f(x) and make it clear where the function increases and decreases, what local maxima/minima the function has, where the slope is zero, and its asymptotes.
(not all the numbers are correct, I just want to give you an idea of the level)

What book or resource should I use for the above?
I'd like to understand the course contents and then go beyond.

As for number theory: would you recommend Andrews, Hardy, or Niven for an introduction?

>>10915888
>an infinitesimal approach
Absolutely would not recommend.
Use Spivak, Apostol or Strang.
For number theory, use Weil`s An approach to number theory through history. Classical number theory hasn`t changed anyway.

>>10913388
Consider your shitposting as some sort of sand mandalas

>>10913402
What counterexample?

>>10913388
In 2012, someone on /sci/ told me the best way to prepare for STEP 2 and 3 was to work through Spivak’s Calculus.
I didn’t get into Cambridge

Can anyone give me a good ‘pathway’ to rigorous PDEs?

I’m guessing a good knowledge of harmonic analysis is useful, but is it necessary?

>>10916016
Just learn multivariable calculus, linear algebra, complex analysis, ODEs, then jump into Cohn`s Measure theory, follow up with Lax`s functional analysis, Tu`s differential topology, then Hormander`s set on linear partial differential operators.

>>10916000
One in which a non diagonalizable matrix when exponentiated becomes diagonal implies nilpotent. And the hint afterwards after about jordan form being useful for these problems proved immensely useful. If you were that anon thanks again!

>>10915042
>applications to 2d percolation in statistical physics
Please tell me more.

When you use a calculator to compute a value for an inverse trig function like cos^-1 that is outside of the domain, what are the values that the TI type calculators are giving you? If you put arccos(2) it will return (0, 1.3169.....). What is this value supposed to be? Why doesn't it just say "error domain" like it sometimes does with other things?

>>10916402
>Yukari_smile3
How many do you have? And who did take time to turn touhou into a manga?

>>10916025
>`
What Ctrl-AltGr-BlNum path does it take to do the quirky apostrophes?

>>10916025
Me likey Tu, do you mean the Differential Forms in Algebraic Topology?
I have his other two, but not that one.

I've seen some bad reviews for that Lax book, also it's very expensive.
Any other functional analysis texts you'd recommend?

>>10916610
>differential forms in algebraic topology
No, the intro to manifolds one.
I`ve seen people recommend Riesz Sz-nagy, but absolutely not the foggiest whether it`s good or not.
>>10916557
No idea.
I input my computer`s password with the wrong keyboard configuration, so sometimes I forget to fix it after logging in.

>>10915454
Motives

>>10916634
Yeah, that one looks good actually. Cheap as well.
I'll check the Lax one out of my library anyway, I should at least give it a look.
Fanks

>>10916557
It's a backtick ``
They're normally on the same key as tilde ~~

I can't stand emailing professors

I'm an undergrad and need to be punished by wearing a cock cage

>>10916676
I have no tilde on my keyboard :C

Should I get a math minor as a Biochemistry major?

I want to switch major from economics to math is this a wise choice

also I suck at math but I am depressed and interested in what will lead me to most mental stress and misery while being in uni

I'm taking a course in differential geometry next term but I plan on taking the prerequisite manifolds course in parallel, I pretty much know 60% of the content already so it should be an easy course to ace if I am gonna be seeing the same thing but harder in DG. I was wondering if I should dump the manifolds course and take a course in Lie Groups instead and pick up everything as I go along with the harder examples I'll see in DG and Lie.

Are Lie Groups hard if you have solid analysis experience but aren't so good at algebra (except linear which I find the proofs a lot easier)?

Do any anons think it's a useful/important course to know for someone mainly interested in dynamical systems? I was also thinking it might look good flexibility wise in PhD applications since atm my course choices make it look like I'm crap at algebra (which I am, but don't want people to know).

Or is Lie Groups too niche and not a suitable course for algebraically inclined that I would be tanking my grades for not much gain.

Thanks frens

>>10917057
It's really hard explaining how fundamentally wrong everything in this post is.

>>10917102
I mean I took a course in analytic number theory with dodgy foundations in number theory and a term before complex analysis, so I know what I'd be getting into, but I like to take calculated risks, not slow motion car crashes.

>>10916491
> When you use a calculator to compute a value for an inverse trig function like cos^-1 that is outside of the domain, what are the values that the TI type calculators are giving you? If you put arccos(2) it will return (0, 1.3169.....). What is this value supposed to be?
(Real part, imaginary part). acos(2)=i*log(2+√3)=1.31696i.

Ok lads, I think I understand the basics of distribution theory, but I can't seem to really connect to how they are used in the usual intuitive way. My main problem right now is, if you consider this Poisson equation:
[math]\Delta\phi=\rho[/math]
But in a distribucional way, I don't see how to generally define something like
[math]\int \phi(x)\rho(x)dx[/math], which is important in many applications. This because the potential isn't necesarilly a Schwartz function, so for singular delta-like distributions, it doesn't seem to be well defined. i mean, the greens function for the Laplacian isn't even integrable over the whole space. However, it is consistent, for example, to apply that equation for system of particles whose distributions are deltas, and naively just evaluate the potential at the position of the other charge to get to the correct result. Is this something not suitable for distribution theory? Or is there a way to expand the domain?.

Random question. In a well shuffled deck, what are the odds of any card value appearing at least twice in a row?

>>10917607
>important in many applications
What do you mean? Distribution theory gives you tools to turn badly posed problems into well posed problems. It's similar to how extending the real domain to the complex domain makes it easier to analyse and construct results about polynomials, or how working over reals helps you understand continuous functions, even if you're only interested in their rational values.
Is there any particular reason why you feel the need to work over the entirety of R^n instead of restricting to the compact sets you're actually working over?

>>10917610
> In a well shuffled deck, what are the odds of any card value appearing at least twice in a row?
Around 98%. It's one minus the probability of the contrary, i.e. of each card being different from its predecessor, which is approximately 1-(12/13)^51 = ~0.983.

>>10917610
Isn't this just [math] \frac{4}{52} \cdot \frac{3}{51} [/math] ?

>>10917683
huh. Cool

>>10917643
Don't really see your point. Distributions are perfectly well defined over Schwartz functions over all R^n. And the you can show rigorously that the fundamental solution to the Laplacian goes as ~1/r which is a distribution as a Cauchy principal value. But the integral I mentioned appears in many places as the energy that is in the distribution in whatever model you are using. My question may be more precise, and it's that in many cases, the "delta function" is used to justify many integrals over functions that are not test functions. They are distributions, but it seems that there are a lot of problems with defining the product of two arbitrary distributions, and it gets worse with singular ones.

>>10917683
Empirical evidence suggests ~95.4%.

>>10917713
To clarify, it goes as 1/r^k with k depending on the dimension put it's 1 in R^3. By definition, you cannot "apply" the delta function to that harmonic function.

>>10917057
Read this, He introduce basic modern algebra and build, lie groups and lie algebra, He create a whole book several years later but without Basic modern algebra intro.
https://arxiv.org/abs/math-ph/0005032

Is injective function always a linear function in two variable?
If so, Why didn't any book state this ever?

>>10917930
No it isn't. I have no idea how you got to that conclusion.

>>10917930
how the heck would that be the case

>>10917607
Are you referring to the second forcing term in the weaker form [math]S[\phi] = \int_\Omega (\frac{1}{2}|\nabla \phi|^2 - \phi \rho)[/math] of the Poisson equation? Remember that convolution is well-defined on distributions, so if you have [math]\rho = \sum_i q_i \delta(x-x_i)[/math] the second term just reads [math]\sum_i q_i (\phi\ast \delta)(x_i)[/math]. In other words, there is no need to identify the second term with the evaluation [math]\langle\phi,\rho\rangle[/math], there is another multiplicative structure on distributions you can use.
Also, the massless Green function [math]G = \nabla^{-2}[/math] is well-defined away from [math]0[/math], so its domain of ill-definition has Lebesgue measure zero. It's in general a distribution, and one wouldn't expect it to be integrable in the conventional sense.

>>10918056
libtards don't realize that heck is far more folksy and endearing than HEYALL FOWKES will ever be. sad!

>>10918060
Kek

>>10918090
are you mentally ill? fuck off.

>>10918060
I'm referring to that extactly (I'm actually with a non relativistic EM lagrangian density but I'm trying it first with electrostatics). I understand that convolutions can be defined between elements of the Schwartz space and distributions, but not between distributions themselves in general. And the problem is that the electric potential is not even locally integrable and it's defined as a distribution as a C.P.V. which from the examples I've looked, are the distributions that are difficult to give a multiplication structure.

>>10918092
Fuck you too anon

>>10918096
>but not between distributions themselves in general
Well, I guess that depends on what you mean by "in general". It works as long as one of the distributions belongs to [math]\mathcal{S}'_c[/math], i.e. has compact support.
https://ncatlab.org/nlab/show/convolution+product+of+distributions
There is also a way to loosen the regularity conditions so that [math]S[/math] takes distributions in [math]L_\text{loc}[/math], namely those that have evaluates to [math]\int_\omega dx \phi f [/math] on test functions [math]f[/math] and all [math]\omega \subset\subset \Omega[/math] compactly contained in [math]\Omega[/math]. Fixing [math]\omega[/math] lets you work with [math]\phi \in \mathcal{S}'_c[/math]; of course the problem then becomes finding [math]\phi[/math] that optimizes [math]S[/math] for all [math]\omega\subset\subset\Omega[/math].

Again, because of Sobolev embedding (and the fact that [math]\mathcal{S}'[/math] is fuck-huge compared to the usual "nice" Sobolev spaces like [math]H^1[/math]), if you can find optimizers without distribution theory then you won't find any new ones with it. You'd probably need to look at something non-linear like KdV in order to see the full strength of distribution theory.

>>10918111
I'm still new to this topic so I don't know if your post also answered this more precise question. Is there a way to at least define the convolution between Fundamental solutions to constant coefficient linear pdes and the dirac delta? In other words, can it be justified to think of applying the dirac delta to the fundamental solution of the Laplacian?

>>10918111
>yukarishit linking nLab
like pottery

>>10916610
I like the Stein and Shakarchi 3rd and 4th volumes - they cover a wide range of topics fairly well. Chapters 4 and 5 go over basics of functional. Meanwhile the fourth volume will go over it plus lots of applied topics. Highly recommend if you're interested in going deep.

If you want more cursory overview, Nochtergale and Hunter's text is quite good.

>>10918111
How can someone talk like this and post anime pics? I just don't get it.

>>10918295
Touhou isn't anime.

>>10918295
you need to go back

>>10918295
I know right

>>10918310
It's worse than that.

does "word line parametrized by proper time" mean anything else than "curve parametrized by arc length" ?
goddamn physicists

>>10918358
World line implies it's timelike, i.e. in the direction where the metric has the opposing sign (conventionally, this means it's negative). The "length" is a time duration.
It's a "curve parametrized by arc length". But from all the examples where physicist might have unhelpful diverging nomenclature to mathematicans, this sure isn't a good example.
Also, fuck you.

>>10918295
I someones wonder if these guys are copying and pasting from mathoverflow. No way people who post here are this smart right? >>10918111
>>10918060
>>10914115
>>10913700
>>10913378

>>10918657
To the extent that others find it helpful (and I think that's the case) if he'd copy from MO he'd still roughly need to grasp it.
I'd like to know if the anime posters are one guy of if there are generally several, and also whether it was always the same guy.
I've been on and off posting on /sci/ since 2008, so I'm sure there's a few more PhD's around. The site is definitely more social and fun than reddit of SO/SE. On that note, let me point to a call to read into Artin:
>>10918608

>>10918654
thanks. emphasis was not on physicists using different terminology but on me not knowing.

Have any of you read any good math-related fiction? If so, what?

>>10918675
All is forgiven. Join the Artin reading group.

>>10918683
Alice in Wonderland?
Also, pic related.

>>10918683

the flatland movie was kind of cute but meh

>>10918694
>>10918667
Why would you try to get math grad students to read an undergrad text on algebra?
>>10918358
Lorentzian geometry is some weird fuckery.
Which does remind me that libgen has extremely few books on the subject.

Does anyone want to either read a first course on modular forms or problems in algebraic number theory with me? I think there are some people in this thread interested in them.

>>10918667
Tbh reddit math feels like a circlejerk. They are good people I am sure, but they seem to love praising each other.

>>10918901
they're mathematicians, of course they're circlejerking

>>10918901
I never had a reddit account in the first place, but from my occational getting there via google, it didn't hit me that there are different people on reddit than on 4chan.

>>10918791
I doubt there's just enough grad students interested in the same topic (and I like going over fundamentals and reflecting)

>>10918129
The Laplacian is elliptic and the wave front set of the fundamental solution is empty. More generally than what Yukari said, you can take the convolution when the wave front sets don't intersect (this is somewhere in Hormander).

>>10918657
>>10918295
The best people in math are so much better than you imagine. Also, Yukari is one person, there's no reason to think otherwise.

>>10919048
>the laplacian is elliptic
>water is wet
>unbounded operators are unstable
>algebraic geometry is dogshit

I have few very genuine question.
How did we discover matrix?
How did we discover vector?
What lead us to those two?
What was discovered first?
Who the operations on them formulated and how they are attached to the concepts of geometric change? Like transformation.
Do vector and matrix belong to same set?
Can a set have two different types of elements?
I know set of real positive numbers, set of natural numbers but never knew set of consisting of heterogeneous elements like set of natural numbers and real positive numbers.
Wtf man.

>>10919079
>never knew set of consisting of heterogeneous elements like set of natural numbers and real positive numbers.
[math]\mathbb{N} \cup \mathbb{R}_{>0}[/math]

>>10919079
>how did we discover vector
no one discovered a vector, it's literally "let's put numbers in a sequence"

>>10918295
Take this opportunity to learn the very valuable lesson that being smart and/or well-educated has absolutely zero correlation whatsoever with not being a faggot.

>>10919181
But I thought smart people were sophisticated and drank wine and listened to Bach and watched art films?

>>10919184
No, that`s faggots.

Can someone explain sentence one?
Is author asking to multiply A by determinant of A?
What is the domain and codomain of the map mentioned?

>>10913132
how hard is discrete math in undergrad and is it a useful course

>>10919390
Nah.
We set A as the matrix with a and b as its collumn vectors, and B is a matrix acting linearly through matrix multiplication.
We also note that det AB = det A det B.
Finally, we leave finishing the explanation as an exercise to another poster.

>>10919396
>how hard is discrete math in undergrad
depends mostly on who's teaching the course at your uni

>>10919442
A chinese Ph.D. who I can barely understand

>>10919445
it's a question you should ask your friends at the uni, we can't give you an answer
whether a course is hard or easy depends 99% on your abilities and expectations of the lecturer

>>10919390
Any [math]2 \times 2 [/math] matrix [math]A[/math] determines a linear map [math]A \colon \mathbb{R}^2 \to \mathbb{R}^2[/math] (I hope you know this, otherwise you can stop reading). Take your favorite planar shape and denote its area by [math]V[/math]. The mapping [math]A[/math] moves and distorts this shape, denote by [math]V_A[/math] its new area. The relationship between these two areas is [math]V_A = \det(A)\cdot V[/math]. Therefore determinant is the factor by which [math]A[/math] scales volumes. In particular, if our chosen shape is a unit square, its image in [math]A[/math] is some parallelogram, and since [math]V=1[/math], the volume of this parallelogram is precisely [math]\det A[/math].

>>10919396
>how hard is discrete math in undergrad
Depends on what your uni means by "discrete math". In some places it's a respectable combinatorics/graph theory course under a different name. Combinatorics can be hard if you aren't used to the problems, or if you're a brainlet.
At other unis "discrete math" is a freebie course that is a mixture of shitty set theory, shitty logic, shitty combinatorics and shitty number theory basically intended to be an introduction to proof course.

>is it a useful course
Probably yes. Combinatorial arguments pop up quite frequently in most parts of algebra, so it's useful to be familiar with the basics of them. It's also fundamentally necessary if you care about computer science on a level higher than codemonkeying.

>>10919445
Chinese you can't understand are usually fine, because FOB Chinese profs all just copypaste the textbook onto the board.
It's only pajeets with horrendous accents that you should try to avoid.

I passed the Pre-Algebra exam on Khan Academy.

>>10919048
>Yukari is one person, there's no reason to think otherwise.
>he's never pretended to be Yukarifag

https://youtu.be/rHsuesTdFLM

Having some fun with basic function types

What properties do Kravchuk polynomials have and in what context do they arise?

The wikipedia page for these is just plain weird. Especially the generating function (it's a sequence of polynomials?)

>>10919469
>volume of parallelogram?
you mean area?

>>10919543
Please don't.

>>10913678
B A S E D

A

S

E

D

>>10913240
It does retards. Y^2/3 is the same as the cube root of y^2. So inside the brackets you get (cube root of y^2 * cube root y^5) and all that if multiplied by cube√y. So if you expand it it's cube√(y^3) * cube√(y^6), which is y + y^2.

I know its cyclic because [math]81=3^4[/math] but i dont know how to find the generator without just bashing my head into the wall by taking the powers of all the elements, is there a nice way to do this? the group is all units coprime to 81

>>10918657
it doesn't take much to be yukariposter, just a phd or at most a postdoc and a lot of vocabulary

>>10919065
very based post

>>10920845
2-volume.

My thesis advisor offered me a PhD after graduating from undergrad in February. I'm unsure if I can really manage it without a masters, but he knows his shit and thinks I'm ready. Should I be fine if I have strong analysis and geometry basis? At least at an undergrad level.

>>10921034
in america typically you do a phd in mathematics without a masters
if you're not in america i am less certain

>>10918896
Wake me up when you get to TMFs.

>>10921086
>You should not do your PhD at your undergrad institution
I've never understood why this is the case. Is there a reason? It's not even relevant to me, but it is to friends.

>>10921093
>>10921086
It's abroad the PhD, my advisors just has connection with the institute.

I want to get a PhD. I'm an undergrad sophomore and I have always enjoyed learning about math for fun, but I feel that I need to start thinking about my career this year.

what should I do?

>>10921245
kys

>>10921245
Get serious. Start going through textbooks. Push up how much you are learning.

You're lucky, since you have much more freedom in what you study and can find things that you enjoy most. You won't always have this freedom so don't waste it.

>>10918295
he's a tranny

>>10921314
>>10921245

And papers. And mathoverflow.
Don't stop learning.

>>10921245
Take it a step at a time and do undergrad as best you can

What is the IQ required to get accepted into a Princeton Math PhD program?

Thoughts?

>>10921323
>he's a tranny
Yukari is a "she".

>>10915210
>>10915215
What have you tried?

>>10921386
Begin non-white or Jew, just check last years photos of PhD students

>>10921386
>>10921521
Do people apply for pure Math PhDs anymore?

Waiting for results of oral exam, they come out in a week and a half. I don’t know what to do to pass the time. I cannot concentrate on anything until they come out.

>>10921386
High enough to understand Rick and Morty

>>10921386
I applied to Princeton and UCB straight out of undergrad. Excellent grades, a few papers and decent recommendations. Got rejected pretty fast. Not sure who they're looking for.

>>10921738
>Excellent grades, a few papers and decent recommendations. Got rejected pretty fast. Not sure who they're looking for.
Straight A+'s, more than a few papers, and recommendations from Fields medalists.

Is there any well-written statistics book that goes into detail about the theory behind ANOVA and LSD?

>>10922072
Probably one of the books written by Daniela Witten. Or you can go to the websites of the reputable schools' statistics departments and take a look at their recommended readings.

I want to check if the improper integral [math]\int_0^{infinity} e^{-x^2}[/math] is convergent or not. My first step is comparing it to [math]e^{-x}[/math] which is obviously convergent, but is that mathematically sound to say? I'd like to compare the function to [math]x^{a}[/math] to have a sounder argument but how do I do that?

>>10922340
Well, just split into the integral from 0 to 1 and the integral from 1 to infinity. The first part is convergent because the function is bounded above and the interval is finite length. You can use your argument for the second part since e^(-x) bounds e^(-x^2) for x > 1.

>>10922340
>>>/sci/sqt

>>10921346
>>10921377

I mean, how do I guarentee I'll have the option to do a PhD

It's easy for me to spend time doing math that ultimately goes nowhere but I haven't ever done research

How/when did you find your main field of study? Nothing calls me with certainty.

>>10922357
Something called me with certainty.

>>10922363
Which was?
[math]youfucker.[/math]

>>10922357
All my fun professors were algebraic geometers. I did really well in geometry and algebra classes. I did piss poor in analysis classes. The decision was obvious: Graph Theory.

>>10922367
PDEs. Took a class in early undergrad and knew pretty quickly that I wanted to go further with it. Then I did some some research and got a pretty cool result, and I've never felt anything else tug me away since.
Something about it just feels right in a way no other math can.

>>10913132
Artin's Algebra is so rigorous that I am having hard time going through single page.
If I finish this book, will I become god at computer graphics since it use a lot of linear transformations?

>>10922395
What? No. You will benefit from the theory but if you want to do computer graphics you'll also need to read about computer graphics.

>>10922356
You can't guarantee acceptance into a good/top PhD program. You have to essentially convince the school that you're good enough to be funded to do math. Can you do that?

>>10922356
>how do I guarentee I'll have the option to do a PhD
be the best in your year
take the putnam and score high
publish papers

>>10922413
What is considered high?

>>10922401
But it has transformations which is in abstract algebra some group or field. I don't know what these things are but having a thorough background about transformation would give me more confidence to execute things.
What is transformation in abstract algebra sense though?

>>10922395
If you have a hard time going through a single page it means it might be out of your level and you might benefit from reading another book before it.

>>10922434
You need linear transformations and it's what you learn in the book. But the book of Artin is the the theory of linear transformations what a youtube video history lesson is to studying history. (Best example I could come up with on the fly. Point being it's a vast topic.)

I try to push the "reading club" (unless in the next 3 days nobody else is posting) and I happen to work in computer vision. Pic related is a standard text that covers the various relevant transformations in the field early on and readings such as Artin will indeed by very helpful to understand that book

>>10922560
Ah I see it's actually online upon a google search

http://cvrs.whu.edu.cn/downloads/ebooks/Multiple%20View%20Geometry%20in%20Computer%20Vision%20(Second%20Edition).pdf

Anybody have a decent Dover-type book for a retarded engineer on calculus of variations? I'm trying to understand these papers

https://pubs.acs.org/doi/10.1021/acs.iecr.9b01694

https://pubs.acs.org/doi/abs/10.1021/acs.iecr.5b04277

how do I express it when I want to apply the same function twice on a variable
f º f?

>>10922711
[math]f^2 (x)[/math], just avoid it in huge equations because your eyes trick you and you mistake it for [math]f(x)^2[/math].

how do I figure out where is good to apply for PhD in my country?

my country is the UK so any suggestions (beyond oxbridge) would be welcomed

>>10922798
Ask faculty, students, whomever. Use the internet.
If you're too dumb to figure it out, you're too dumb to get in anyway.

>>10920963
Watch me.
>>10922340
I apply the residue theorem to give a textbook solution, but I phrase it in terms of de Rham cohomology and Hodge theory.

If I want to do stuff with type theory and formal systems should I be looking at maths programs or CS programs?

>>10923751
CS

Why didn't Sobolev win the Fields?

>>10923824

maybe if you want to learn java and a bunch of useless "design patterns". go with a math major and take some of the better cs classes a la carte if your uni will let you

I can't be the only one who's noticed these russians who refer to their education as 'different', with regards to calculus. Is this true? What soviet magic did they use?

ok /mg/ help with this, I've wasted at least 70 hours and have gotten nowhere

I limit the numbers to 1, 2, and 3
This is three variables and with 8 solutions for addition.

none(0),1,2,3,1+2,1+3,2+3,1+2+3
so naturally our range is 0 - 6
This is a combination problem. Now how would you access these 8 solutions?

Anyways I'm wondering because polynomials can give us many solutions but one equation, ie x^4 has 4 roots but some might be imaginary

So how to link all the possible summations of three numbers but obtainable by one equation.

>>10923751
Apart from a fringe fraction within algebraic topology and as well as set theory, I don't think mathematicans even know type theory exist. It's certainly on no math university curriculum I've ever seen.

Is matroid theory interesting?
If so, what's a teaser why it is?

Anyways I was working with Pascal's triangle and thought somehow you could use fibonacci numbers to pinpoint to solutions by going through layered pascals.

"Fibonacci numbers are strongly related to thegolden ratio:Binet's formulaexpresses thenth Fibonacci number in terms ofnand the golden ratio, and implies that the ratio of two consecutive Fibonacci numbers tends to the golden ratio asnincreases."

Anyways I believe nature tends to imitate this golden ratio as it can be linked to simplicity of pascals triangle.

>>10924954
https://en.wikipedia.org/wiki/Representation_theory_of_SU(2)

>>10924991
thanks

>>10924938
>Now how would you access these 8 solutions?
don't know what you mean by access

>So how to link all the possible summations
don't know what you mean by link

Did I fuck this substitution up?
[math]\int_0^1(2+x)\sqrt{1+x}dx ,\enspace u=\sqrt{1+x},\enspace dx=2udu,\enspace =>\enspace \int_{-1}^0(u^2+1)2u^2du[/math]

>>10924954

So you be sayin that the universe is math n shiet?

>>10925173
Lol I just found out today the guy who tutored me maths got a Third from Cambridge.

>>10925193
>tutoring

>>10923824
>>10924950
Ok, thanks friends. I shall go look into CS programs.

>>10924953
Matroids are combinatorial objects that generalize the idea of linear independence in vector spaces. There are numerous equivalent definitions (one is to define a matroid in terms of "cycles" motivated by graphs which also carry a natural matroid structure).

I personally thing it's quite dull, but there are some interesting results like the whitney planarity criterion: https://en.wikipedia.org/wiki/Whitney%27s_planarity_criterion

>>10925246
think*

>>10913132
>left derived functors are right Kan extensions
>right derived functors are left Kan extensions
Whose idea was this?

>just got to first order differential equations
Is it normal to feel like youre “missing something” entering this class?

stuck in this low IQ prison for the rest of my life ...

>>10924355
>Why didn't Sobolev win the Fields?
Why should he have won?

>>10924953
https://arxiv.org/pdf/1908.09820.pdf

>>10913132
[math]\textlit{Is Rn a vector space with operator "+" and scalar multiplication"}[/math]

>>10913700
Adding two differentiable functions gives a differentiable function

>>10926620
Why shouldn't he have?

>>10926995
>Why shouldn't he have?
The burden of proof is on you.

>>10927227
What proof? This is a matter of opinion. Sobolev's ideas shaped a large portion of modern mathematics.

>>10922878
>Use the internet
You mean like I'm doing?

>>10926665
I computed a few phi^4th Fayman diagrams in my time, but I'm afraid that article needs a motivation on its own

Can a big brain help me with pic related? I'm stuck on i) and generally have a poor understanding of problems where I need to prove or show something (I'm an engineer! :^)). Any hints & tips are all appreciated.

>>10927778
Like, re-read your textbook carefully, think about the proprieties of convolutions and laplace transforms you saw.
Try doing another easier/simpler exercises. If you can't solve some problem them there's ankther, easier problem, you also can't solve.
Your question is pretty standard stuff, so if you can't see a way foward you probably didn't grasp something that came before.
You might want to read some other text like Lamar's math notes or something if your textbook alone doesn't help.

>>10927843
Thanks for the response.

It is the final task on the assignment I have left, and the other ones have been ok, though with help from the internet. It's not like it's completely alien to me, but I'm not confident in my ability and understanding of the concepts to write them down in good faith, so I have trouble starting a path I deem reasonable when I do problems like these. I've reread my textbook a few times without much success (my professor has also been complaining of it).
Also been a solid 18 months since last time I had a math class.

>>10921386
Admissions to the highest-level PhD programs (roughly top 5-10 in the States) like Princeton is pretty much luck. They receive 5x as many perfect or near-perfect applications as they can admit, and there's really no rhyme or reason to who gets in and who doesn't at this point.
Unless you have an unusual personal connection with one of the faculty who is willing to pull strings to have you as a student, all you can do as a top candidate is apply to all the Ivies and cross your fingers that you get into one.

>>10927863
Isn't the Princeton Math department the best in the world? (At least for pure Math) I would imagine you need stellar publications and recommendations from well-known mathematicians to have a shot at that.

>>10927888
>best math department in the world
No such thing.

>>10927778
A few questions:
Is there a way to rewrite the laplace transform of f in terms of integrals of length T instead of one of length infinity? Remember your basic integral rules.
How can you use periodicity after you do this?
Is there something you can factor out? Take a look back at where you're trying to get and see what you think.

>>10927778
>engineer
Refer to >>>/sci/sqt

>>10927940
Sorry, I'll post the right place next time.

>>10927932
Thank you for posting in this format, was helpful to collect my thoughts by answering your questions.

Considering the group -> groupoid relationship, has anyone ever actually bothered to define an "algebroid" as an algebra where sum and multiplication aren't necessarily defined, i.e. the "algebroid" of all matrices over a ring?

>>10928051
What do you get you wouldn't get by considering the relevant groupoids?

>>10928098
I have the impression that it has absolutely kino topologies that aren't coming to me.
Plus, the adjoint map. And quivers, potentially.
Also, it doesn't have inverses, since it isn't a groupoid. It's the "set of all linear transformations on a set of Hilbert spaces, with partially defined sum, composition, the adjoint map and multiplication by a scalar."

is
| 0 0 |
| 0 0 |
| 0 0 |

in row echelon form, reduced row echelon or neither? why? I read once that such a 2x2 matrix is reduced low echelon

>>10928260
It's in both trivially.

>>10928277
Why

>>10928311
row echelon:
>all nonzero rows (rows with at least one nonzero element) are above any rows of all zeroes (all zero rows, if any, belong at the bottom of the matrix),
satisfied

>the leading coefficient (the first nonzero number from the left, also called the pivot) of a nonzero row is always strictly to the right of the leading coefficient of the row above it (some texts add the condition that the leading coefficient must be 1)
yup

reduced row echelon:
>It is in row echelon form
check
>The leading entry in each nonzero row is a 1 (called a leading 1)
vacuous
>Each column containing a leading 1 has zeros everywhere else.
also vacuous

>>10928325
Thx!

| 3 6 -3 -2|
| 0 -2 3 -2|
| 6 6 3 4 |

Solve this.
I got
| 1 0 2 0|
| 0 1 -3/2 0|
| 0 0 0 1|

(No solutions)

>>10921386
Basically this >>10927863 among the grad students I know at some top unis even they don't know how they got in. There's always way more people qualified to enter the program than there are spots. Hell, it's gotten to the point where people who could've succeeded in top 5 programs instead go to top 25 programs. Still great, mind you, but not the at the peak.

>>10914128
finite dimensional real space is self dual and equal to it's bidual in the simplest imaginable way. stop being a pedantic shithead.

>>10921996
it was over before it even began, unless you've been trained by a top mathematician to become a top mathematician yourself you will almost certainly not become one

>>10928496
calling the basis vectors same as the dual basis vectors is just stupid

>>10928051
Groupoids aren't really studied as algebraic objects per se. From an algebraic point of view, partial operations on a set are equivalent to total operations with an extra unary operation.

>>10928519
Nah.
I calling both math [0 ... 1 ... 0]' and [0 ... 1 ... 0] e_i is fine.

how does "e_i" transform when you change the basis then ?

>>10928623
>>10928594

>>10919079
Your teacher must really suck ass, because you are asking the right questions, yet you haven't the answers.

Watch this:
https://www.youtube.com/watch?v=fNk_zzaMoSs&list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab

>Can a set have two different types of elements?
A set can have anything.
"Set" is actually the only thing not rigorously defined in mathematics. It's just a collection of stuff. Nothing stops {!,@,1,, p} from being a set for example.

>set of consisting of heterogeneous elements like set of natural numbers and real positive numbers.
The union of the set of the naturals and the set of the reals is just the set of the reals, since the naturals are a subset of the reals.

>>10928757
>links yt video while critiquing an instructor you’ve never met

How does it come the solution to this picture are those points? How am I supposed to know that I have to take point in the middle of each part and add the half of given value to each?

>>10928867
Wavy lines don't mean equality.

>>10928889
I meant to say how do you get those points marked with red lined according to the formula above.

What branch of maths are estimation question like "approximately how many birds are there on the planet?", and can you recommend a book for that?

>>10928867
Gaussian quadrature chooses it's points as the zeros of some family of orthogonal polynomials.
Given it's just named as Gaussian quadrature, I guess it's using Legendre polynomials.

Look for/to derivations of the method in any half decent numerical analysis book

>>10928960
But you are not allowed to use any help to solve it, that's the issue. You need to get those points just by knowing that one approximation that is listed.

>>10928953
Extremal Combinatorics and related stuff like Ramsey Theory, extremal graph theory, etc.
Pick some of the mainstream Combinatorics books (Lovasz, Diestel, Stanley, etc) and try it out.

>>10928953
Numerical approximation methods are pretty close. You get n areas with m mount of birds, add some weights and go on.

>>10928953
Statistics.
In statistics, you take data that come from something unknown, and then you try to infer stuff about that unknown

For example, given the weight of some people from a country, you might want to try to estimate what is the average weight of the total population.
Then you might try to analyze how sure you are about your estimation, etc.

To learn statistics, you need to learn calculus and probability theory first.

>>10928953
not really math, but look up fermi problems

>>10928494
Cambridge (arguably comparable or better than Princeton) is actually not THAT difficult to get into if you went to high school in the UK - admission rate is like 1 in 5. From there you need a distinction in Part III to be elligible for their PhD program.

>>10928953

orithnology

>>10928973

bitch please

I have a general linear algebra question:
Is every consistent system of linear equations solvable using only the elementary row operations (Gauss-Jordan elimination)? Why or why not?
This isn't some challenge I'm genuinely curious.

>>10919079
In the beginning, there were not literal variables, no algebra, there were only numbers and arithmetic. Although the arabs started to use symbols, it was very hard for them, even for negative and irrational numbers, being accepted. Only in the 17th century there was a conmon ground from which mathematicias could work. Once there was algebaric equations, linear equations as well as cuadratic, cubic and quartic equations was widely known and used, for example, in astronomy. In particular linear systems of equations. On the other hand the determinants were used before the notion of matrices, as ways of knowing if a linear system has unique solution, for example by Leibniz. After a while of knowing determinants, only in the 19th century, there was understood the notion of matrix. And of a vector, is was clear only at the verge of 19th century. Yes, there was the notion of points and coordinates, earlier, in the 17th century with Descartes, and many notions related to vectors, but it was only until the 20th century that all these concepts were clarified.

>>10929053
Is satisfactory a passing grade for step 3?

>>10929137
Yes. Read the wikipedia on rref.

>>10928965
Have you tried computing the quadrature? Like split the interval like the image shows, shift and scale it to the [-1,1] and use the points if the quadrature above the one you want.
Then see where the points you used land on the original interval

In high school we just learned math to pass a test at the end of the six years of high school.
It was mostly just rote learning and drilling problems into your head so you could say "this is a simultaneous equation, here's how we get the answer" instead of more like "it's a simultaneous equation, why am I doing what I'm doing to solve it"

All that matters was getting the answer and showing you did the steps.

So how the fuck do I change that to the point I'm learning the whys and hows of what I'm doing instead of "it just makes sense, just use the formulas". Anything past basic algebra I'm fucked on. I can follow along just fine but I'd like to actually understand why I'm doing what I'm doing. This is mostly just out of interest since I have around 6-7 hours of free time at work and outside of basic variables for algebra, I don't realy understand how high school math actually works.

>I need to prove statement A
>but this is easy
>it obviously follows from B
>but I haven't proved B either
>turns out it is equivalent to A
gets me every time

What do my fellow american mathematicians think of pauls notes as a resource for calc? I always recommend it to my students because I think it often has more in depth explanations than the books we use. (The discrimination against non americans is because I know you guys learn calc in a more rigorous way)

How did you guys decide on which research area to focus on?
I've been browsing all the research groups of my new uni (starting postgrad soon) and I cannot make up my mind.

I'm thinking maybe I could do Algebraic Topology, but I'm afraid of consequences later on. Cause it's not applicable to real world jobs.

>>10929765
One of my profs offered me a chance to do a paid research project in undergrad. No way in hell I was going to turn down several thousand dollars so I accepted and I ended up enjoying the field so much I stuck with it in grad school.

This is one of those things that comes to very, very few people in a rational manner. Just meet as many people as you can, expose yourself to as much math as you can and worry about planning the long-term future as little as your brain will allow you to. Eventually something will fall onto your head.

Is there a better way to learn proofs than jumping head first into Spivak/Apostol?

>>10925173
should be [eqn]\int_{1}^{\sqrt{2}}(u^2+1)2u^2du[/eqn]

Is there a text on ODE's that doesn't make many assumptions on the primitive differential operator used. I.e. basically develops a theory of ODE's mostly just based on the chain rule and such?

>>10930080
Actually I'm convinced that basic linear algebra and abstract algebra (group theory), has much easier proofs than analysis. But the best way is just jump into something, the only way to learn proofs is by doing proofs, there's no going around it.

>>10930295
Can you recommend a rigorous beginner text on linear algebra?

>>10930329
https://www.amazon.com/Linear-Algebra-Undergraduate-Texts-Mathematics/dp/0387964126

>>10914900
spoiler: it's wrong

>>10930457
it's gonna start with something like pic related

What is the absolute trace polynomial for [math]\mathbb{F}_{3^3}/\mathbb{F}_3[/math]?

Isn't it [math]P(x)=x^9+x^3+x[/math]?

Doesn't that mean that P(1) = 3 which isn't in [math]\mathbb{F}_3[/math]?

new thread >>10930547

>>10929765
There's jobs (probably not many) in data science with algebraic topology.