/sci/ - Science & Math

My math book says that almost all periodic functions are sinusoidal. Is that true? If so, why? The entire idea of the unit circle and right triangles and how they're related seems so fascinating, it seems like i'm missing something important. I wish I was more intelligent, it's like i'm right on the brink of something incredible, but too dumb to grasp it.

>>9515240
What's an example of a periodic function that isn't sinusoidal?

>>9515240
>I wish I was more intelligent
were*
>>9515245
I don't know

>>9515245
>What's an example of a periodic function that isn't sinusoidal?
floor and ceiling

>>9515240
https://en.wikipedia.org/wiki/Fourier_series

>>9515254
that's not continuous though

>>9515254
Those aren't periodic though.

>>9515262
>that's not continuous though
Why would it need to be?

>>9515282
>Those aren't periodic though.
I meant frac(x) of course

>>9515283
so can do calculus on them

>>9515285
Ah. It has a fourier series expansion for non-integer x though.

>>9515240
>dea of the unit circle and right triangles and how they're related

nothing mystical about it
1) a^2+b^2=c^2
2) create a situation where c=1 at all times

since sin = a/c and cos = b/c,
now that c=1, you get sin and cos
straight from a and b

>>9515260
https://www.youtube.com/watch?v=spUNpyF58BY

>>9515240
>almost all periodic functions are sinusoidal
define "periodic function," and what the hell does "almost all" mean? i'm not sure i buy this. if you're saying that certain functions can be decomposed as linear combinations of sinusoidal functions, then sure

>>9515349
>create a situation where c=1 at all times
Can you explain this a bit more?
>>9515363
Apologies, it says that "almost all" periodic functions can be _described_ using sine and cosine, I misread that misinterpreted it as "almost all" periodic functions _are_ sinusoidal.

>>9515260
>fourier series
>has more than four terms

>>9515394
what do think the "unit" in 'unit circle' means?

>>9515407
lightyears

>>9515419
're ya go, you just reinvented relativity

>>9515245
greatest common denominator with one argument fixed.
[math]\gcd(a, b) = \gcd(a+b, b)[/math]

>>9515240
Infinite polynomials

Is this the "engineering general"? Why does it say "talk maths" then? My browser seems to be malfunctioning.

>9516444
it's you that's malfunctioning, weebshitoid degenerate subhumANO

>>9516516
You don't mean that.

Does anyone know a source for the proofs of at least some of the cases that [math]\mathbb{Z}[\sqrt{D}], D < 0[/math] is a PID?
Wikipedia gives the complete list of negative integers it works for, but I can't find proofs for any of them other than the Gaussian integers.

Picked up a book on constructivist analysis, gonna be MAXIMUM comfy learning this

>>9515187
Do graduate classes (in math) usually have exams?

One of mine straight up doesn't, and in the other the professor hasn't really made up his mind.

>>9516641
>Does anyone know a source for the proofs of at least some of the cases that Z[D−−√],D<0 is a PID?
You might have an easier time finding proof that Q(sqrt(D)) (for D = 2 or 3 mod 4) has class number 1 (which implies Z[sqrt(D)] is a PID):

https://en.wikipedia.org/wiki/Stark%E2%80%93Heegner_theorem

Can posting this be an evidence of autism? "Homology groups of a Pokeball".
https://math.stackexchange.com/q/2576957/434968

What's so special about Feller processes? Why do we care about the Feller property and don't just simply deal with Markov processes only?

>>9517151
>Why do we care about the Feller property
We don't care about engineering properties around here.

>>9515240
All analytic periodic functions are sinusoidal, which is why modular forms have a q-expansion with q=e^2ipiz

How does one go about showing that all projective transformations of [math]\mathbb{F}P^{n}[/math] take the form
[eqn] \mathbf{y} \mapsto \frac{ \mathbf{c}+D\mathbf{y} }{ a + \mathbf{b}\cdot \mathbf{y} }[/eqn]
in affine coordinates [math](y_{1},\dots, y_{n})\longleftrightarrow [1 : y_{1} :\dots : y_{n}] [/math]

>>9517151
>Brownian motion and the Poisson process are examples of Feller processes. More generally, every Lévy process is a Feller process.
>Bessel processes are Feller processes.
>Solutions to stochastic differential equations with Lipschitz continuous coefficients are Feller processes.
>Every Feller process satisfies the strong Markov property.

not sure if this is the right place but what are the best books on measure theoretic probability theory for someone who has only a bit of familiarity with functional analysis and the idea of measure?

>>9518214
The Holy Bible.

>>9516800
Bishop?

Is it possible for brainlet to love math and hate physics?
They like to say that physics in close relationship with math and I must love(or hate) both.

>>9518538
For a brainlet, yes.

>>9518542
Physcuckery is really terrible.

>>9518565
Better to take pride in ignorance then.

>>9518214
Check out Probability and Measure by Billingsley, the book introduces measure theory so you dont need to know it

>>9515070
>We tried to hire a mathematician who actually had expertise in this and failed. Which is actually sort of sad, because if we actually did we're pretty sure they could solve this entire field of engineering we're in, greatly reducing the need for peons like us.

Probably. What field are you in?

What differential geometry books take a geometrically meaningful approach (think curves as opposed to derivations), but don't make you go into coordinate hell?

>>9518709
>>9515070

Also, all you need to do is give him (yes, him) a few months to learn it. Learning domain-specific applications is easy for a mathematician.

>>9518720
any differential geometry for physicists book

>>9518736
nah I want something mathematically rigorous.

>>9518709
>What field are you in?
I'm curious as well, I can't really imagine any kind of practical (in the sense of money-making business) application of category theory besides maybe in computer science.

Any good books (or other resources) to learn calculus and linear algebra, Discrete math, Proof techniques ?

>>9518775
>Calculus
Spivak
>Linear Algebra
Hoffman & Kunze
>Discrete math
don't
>Proof techniques
Daniel Velleman's How to Prove It and Solow's how to do read and do proofs

Hammock's Book of Proof is also good if you got the time

>>9518829
>don't
why ? I need it.
I can't afford a good school that will teach me.

>>9518829
I forgot to say thanks.
Sorry

>>9518832
what do you need it for, anyway?

>>9518832
Not that guy, but my impression is that discrete math is generally a hodgepodge of a bunch of important topics that should be studied individually.

Why are math textbooks always full of type errors? Particularly differential geometry books. Sick of this shit.

>>9518839
>what do you need it for, anyway?
Genetically engineer cat girls.
Computer Science.

Course Content:
>Mathematical vocabulary
>Basic techniques from discrete mathematics
>Integers and number theory
>Graph theory
>Generating functions
>Recurrence relations
>some stuff with primes
looks like you're right.

>>9518867
this >>9518844
but if you're not interested just grab Schaum's discrete math

>>9518775
start with the greeks

>>9518947
euclid: unrigorous, you can prove all triangles are isosceles with his axioms
apollonius: who the fuck cares about them conics
archimedes: anything this brainlet came up with is doable with integration

>>9518956
archimedes INVENTED integration nigga

>>9519107
that was yakub, though

>>9518956
> Draws impossible picture
> "Euclid isn't rigorous"

>>9518775

Maths books for three lives.

https://mathblog.com/mathematics-books/

>>9518720
doCarmo's Riemannian Geometry

>>9518720
Try Barrett O'Neill's "Semi-Riemannian Geometry With Applications to Relativity".

>>9518538
>physics in close relationship with math Retardedly wrong. The physics threads are over at >>>/toy/.

>>9519253
that's specifically what I had in mind when I said "coordinate hell"

How do cloud storage algorithms work?

What's preventing someone from constantly uploading junk data and filling up the provider's storage space?

>>9519522
Just use derivations. They are more universal anyway.

>>9519606
Actually revolting.

>>9519609
Explain what a derivation is intuitively

>>9519661
A linear map satisfying the Leibniz rule. The real intuition behind the algebraic definition of the tangent space comes more from the cotangent space. Pretty much the following...

"If [math]{\mathcal{O}_{X,p}}[/math] is the local ring of a variety/scheme/manifold at a point p and [math] \mathfrak{m}_p [/math]its maximal ideal, then the cotangent space of [math]X[/math] at p is [math]{\mathfrak{m}_p}/{\mathfrak{m}_p}^2[/math]. We also have [math]\operatorname{Hom} \left( {{\mathfrak{m}_p}/{\mathfrak{m}_p}^2,k} \right) \cong \operatorname{Der} \left( {{\mathcal{O}_{X,p}},k} \right)[/math] where k is the residue field at p.


Consider the local ring [math]R = C_{{\mathbb{R}^n},0}^\infty [/math]. Then [math]R/{\mathfrak{m}^2}[/math] splits canonically as [math]\mathbb{R} \oplus \mathfrak{m}/{\mathfrak{m}^2}[/math]. So consider [math]f \in R[/math] and look at the Taylor expansion [math]f\left( {{x_1},...,{x_n}} \right) = f\left( 0 \right) + {\sum {\left. {\frac{{\partial f}}{{\partial {x_i}}}} \right|} _0}{x_i} + r\left( {{x_1},...,{x_n}} \right)[/math].

See [math]f\left( 0 \right)[/math] lies in [math]\mathbb{R}[/math], the remainder [math]r\left( {{x_1},...,{x_n}} \right)[/math] lies in [math]{\mathfrak{m}^2}[/math], and we are left with the differential [math]\operatorname{d} f = {\sum {\left. {\frac{{\partial f}}{{\partial {x_i}}}} \right|} _0}{dx_i}[/math] in our cotangent space (where the differential map is viewed as the natural projection [math]\operatorname{d} :R \to \mathfrak{m}/{\mathfrak{m}^2}[/math] under this splitting)."

>>9519661
A linear map satisfying the Leibniz rule. The real intuition behind the algebraic definition of the tangent space comes more from the cotangent space. Pretty much the following...

"If [math]{\mathcal{O}_{X,p}}[/math] is the local ring of a variety/scheme/manifold at a point p and [math] \mathfrak{m}_p [/math]its maximal ideal, then the cotangent space of [math]X[/math] at p is [math]{\mathfrak{m}_p}/{\mathfrak{m}_p}^2[/math]. We also have [math]\operatorname{Hom} \left( {{\mathfrak{m}_p}/{\mathfrak{m}_p}^2,k} \right) \cong \operatorname{Der} \left( {{\mathcal{O}_{X,p}},k} \right)[/math] where k is the residue field at p.


Consider the local ring [math]R = C_{{\mathbb{R}^n},0}^\infty [/math]. Then [math]R/{\mathfrak{m}^2}[/math] splits canonically as [math]\mathbb{R} \oplus \mathfrak{m}/{\mathfrak{m}^2}[/math]. So consider [math]f \in R[/math] and look at the Taylor expansion [math]f\left( {{x_1},...,{x_n}} \right) = f\left( 0 \right) + {\sum {\left. {\frac{{\partial f}}{{\partial {x_i}}}} \right|} _0}{x_i} + r\left( {{x_1},...,{x_n}} \right)[/math].

See [math]f\left( 0 \right)[/math] lies in [math]\mathbb{R}[/math], the remainder [math]r\left( {{x_1},...,{x_n}} \right)[/math] lies in [math]{\mathfrak{m}^2}[/math], and we are left with the differential [math]\operatorname{d} f = {\sum {\left. {\frac{{\partial f}}{{\partial {x_i}}}} \right|} _0}{dx_i}[/math] in our cotangent space (where the differential map is viewed as the natural projection [math] \operatorname{d} :R \to \mathfrak{m}/{\mathfrak{m}^2} [/math] under this splitting)."

They won the Nobel Prize for discovering dark energy in 2011 and I solved the anomaly in 2009. Did anyone find an error yet? Fuck no.

>Modified Spacetime Geometry Addresses Dark Energy, Penrose's Entropy Dilemma, Baryon Asymmetry, Inflation and Matter Anisotropy
>http://vixra.org/abs/1302.0022
>old timey guy's illegible cursive is fine and we like him but fuck your MS Word bullshit

>>9519810
>MS Word
>using software that isn't Free As In Freedom

>>9519810
All that paper does is state nonsense. It doesn't even attempt to prove it.

>>9519772
I see lots of symbols here but no intuition. Manifolds are supposed to embody geometry and there is no geometry in what you're talking about.

>>9519854
>geometry
Please define this.

>>9519854
Tangent vectors are supposed to represent some type of infinitesimal displacement. What I explained shows this algebraic stuff clearly captures that notion.

>>9519835
That's to be expected. It is a physics paper after all.

When did you realized you will never learn all the maths?

I can't understand a concept unless I have a visual to hang it on. Can I ever become a decent mathematician?

>>9519937
Depends on what you mean by "visual".

>>9519937
You would do well in geometry but algebra might be difficult.

>mr mochi ball edition

'tis the season
https://www.youtube.com/watch?v=LqPj5cA-1gw

I'm trying to teach myself stats and I'm a dipshit and I'm wondering about working with different random variables and their PDFs
Say I have a random variable X with a distribution f(x). I also have another random variable N = tan(X). Generally speaking, if I want to know the distribution of N, it would simply be f(n) = tan(f(x)), right?

>>9520010
Work with the cumulative functiom and apply chain rule

>>9519968
>geometry
What's that?

Has anyone here reached a state of quasi-enlightenment where nothing even angers you anymore?

>>9520052
My secret is, I'm always angry.

>>9520052
Are you sure you're not just depressed?

>>9520015
Is this what you mean:

X is a rv with CDF fx(x) and N = tan(X)
fn(x) = P(N < x)
fn(x) = P(tan(X) < x)
fn(x) = P(X< atan(x))
fn(x) = fx(atan(x))

and then to find the pdf I take the derivative d/dx(fn(x)) = p(x) / (x^2+1)?

>>9520052
>confusing apathy with enlightenment

>>9520056
I'm not weak enough for that anymore.

>>9520074
I'm pretty sure "apathy" means something different, anon. And I said "quasi-enlightenment", not the ascended version of it.

>>9520071
https://stats.stackexchange.com/questions/239588/derivation-of-change-of-variables-of-a-probability-density-function

>>9518171
Combine rigid motions with inversions.
>>9519772
>local ring variety/scheme/manifold
In particular your example is most concretely realized as a sheave of germs on a Riemannian manifold. Sheaves on paracompact manifolds are the essence of Cousin's theorem for the existence of global sections.
>>9519854
>but no intuition
What do you mean? It's plenty intuitive.

9520157
>concretely
Nobody gives a fuck about this here, proceed to >>>/toy/.

>>9518538
Yes, applied math is like a professional sport for brainlets.

>>9520187
>Nobody gives a fuck
Do you need to swear?

Let π:C[x,y]→C[x,y]/(x2+y2+1), and for z=(z1,z2)∈C2, define ϕz:C[t]→C[x,y] by t↦z1x+z2y. Let π⋆ and ϕ⋆ denote the induced spectral maps πz⋆:A2C→A1C, and ϕ⋆:V(x2+y2+1)→A2C. For what values of z is ψz=πz⋆∘ϕ⋆ a finite and surjective morphism?

Pls help ((

Let [math] \pi: \mathbf{C}[x,y] \to \mathbf{C}[x, y]/(x^2 + y^2 + 1) [/math], and for [math] z = (z_1, z_2) \in \mathbf{C}^2 [/math], define [math] \phi^z: \mathbf{C}[t] \to \mathbf{C}[x, y] [/math] by [math] t \mapsto z_1x + z_2y [/math]. Let [math] \pi_\star [/math] and [math] \phi_\star [/math] denote the induced spectral maps [math] \pi^z_\star: \mathbf{A}^2_{\mathbf{C}} \to \mathbf{A}^1_{\mathbf{C}} [/math], and [math] \phi_\star: V(x^2 + y^2 + 1) \to \mathbf{A}^2_{\mathbf{C}} [/math] . For what values of [math] z [/math] is [math] \psi^z = \pi^z_\star \circ \phi_\star [/math] a finite and surjective morphism?

Pls help

>>9520215
Why the fuckphobia?

>>9520262
Since everything here is affine and [math] (x^2 + y^2 + 1) [/math] is a prime ideal (since it is generated by an irreducible poly), I know that for the finiteness part, it suffices to find all [math] z [/math] such that [math] \pi \circ \phi^z [/math] is an integral morphism. I was really struggling to make progress here and would love a few hints!

Anyone know what [math]\mathbb{F}[/math] stands for?

>>9520873
>Anyone know what F stands for?
Context? Could be a field.

>>9520873
if youre reading axler's LA done right it means either [math]\mathbb{R}[/math] or [math]\mathbb{C}[/math]

>>9520873
As >>9520889 says, it stands for [math]\mathbb{F}\text{iction}[/math].

>>9520157
>In particular your example is most concretely realized as a sheave of germs on a Riemannian manifold

The local ring is the stalk of the structure sheaf at that point. In the classical case of varieties and manifolds, this is the ring of germs of functions at that point.

>>9521161
Obviously, but he was asking for intuition, and classical theories are the most helpful for that.

daddy bullies me because i like to practice math, he tells me to stop being a nerd and get a gf

>>9519937
You will become more familiar with totally abstract objects the more you work with them. A mathematician takes visual examples and determines patterns such that an analogous proof of a totally abstract object can convince you,

For a double variable function defined in (a,b), if its partial derivatives exist and are continuous in (a,b), but do NOT match, is the function still derivable? Or do they have to match?

>>9521491
Limits are unique lmao.

>>9521494

>>9521491
What is your question, exactly? You are wondering whether a function [math]f(x,y)[/math] is differentiable if [math]\frac{\partial{f}}{\partial{x}}\neq{\frac{\partial{f}}{\partial{y}}[/math]?

>>9521491
What is your question, exactly? You are wondering whether a function [math]f(x,y)[/math] is differentiable if [math]\frac{\partial{f}}{\partial{x}}\neq{\frac{\partial{f}}{\partial{y}}}[/math]?

If I wanted to be involved in mathematics research, what would the expected prerequisites be? I get that this obviously depends on the field interested but what would the common path for someone interested be?

>>9521526
Yeah

>>9521533
Yes, it will still be differentiable. The partial derivatives don't need to be the same.

>>9521533
>>9521526
In a give point (a,b) to be precise

>>9521538
Yep, still differentiable.

>>9521537
Thanks a lot, anon!!

>>9521540
Alrighty, thanks! Here's another (You). You more than deserve it

i want to pursue a career in astrophysics, i have a huge passion for the universe. im aware its really math heavy, and im average with math. are these sort of fields only for the mathematically gifted or can averages like me get as good as the gifted?

> math heavy
>astrophysics

Should I take brief calculus as a stepping stone to taking calculus w/ analytic geometry? Would it just be a waste of time?

>Brief Calc
Introduction to the theory, techniques and applications of the differential and integral calculus of functions with problems related to business, life, and the social sciences.

>Calc w/ Analytic Geometry
Limits, continuity, differential and integral calculus of functions of one variable.

>>9521559
astrophysics is very math heavy.

>>9521568
I'm not sure, but, based on the provided descriptions of the courses, I am inclined to say that Calculus with Analytic Geometry will be taught at a higher level than Brief Calc. Brief Calc comes off as one's first exposure to calculus, whereas the course with analytic geometry will probably assume some familiarity with calculus and will approach things more abstractly.

If you've never taken a calculus course before, I'd suggest Brief Calc, but I'd need more information to be certain.

>>9515187
Does this estimate have a specific name?
[math]{\left\| \operatorname{A} \right\|_{p,1}} \leqslant C\left( {{{\left\| \operatorname{A} \right\|}_p} + {{\left\| {\operatorname{d} \operatorname{A} } \right\|}_p} + {{\left\| {\operatorname{d} ^*\operatorname{A} } \right\|}_p}} \right)
[/math]

Help? the answer is a constant that's for sure

>>9521828
Zero.

Write

[math] f(x) = -\frac {1}{x^2} f(1/x) = \frac {d}{dx} \int_a^{1/x} f(y) dy[/math]

Then work from there

guys, please help me. i'm a complete retard, and i'm bad at math, but i'm trying to improve myself. i'm not asking for a solution, but i would like to understand how to solve this problem myself:

i can do an activity with an 18% success rate. i get 4 attempts. what are the odds that i succeed at least once?

again, don't feed me the answer, but please help me wrap my brain around how to get the answer myself. i don't want to be stupid anymore.

>>9521849
Thank you senpai!

>>9521852
Common tool used in probability is the complement: The probability of succeeding at least once, is the same as 1 minus the probability of failing every time.

>>9521861
Which book is this?

>>9521852
1) If you do the activity one time, what are the odds of you succeeding?
2) If you have 100 people who do the activity, 18 will succeed and 82 will fail. The 82 ppl who failed get to try again. How many will succeed? This number is how many people out of 100 will succeed after 2 attempts. Do you see the pattern?

>>9521876
Just a question bank by Mcgraw hill. Contains high school level questions. Pic related.

Let [math]C[/math] be the cantor set
For any point [math]x \in C[/math], there is a set [math]A_x[/math] - dense subset of [0, 1]
Does there always exist a continuous function [math]f:C \rightarrow [0, 1][/math], such that [math]f(x) \in A_x[/math] for all [math]x \in C[/math] ?

>>9521435
>classical theories are the most helpful for that.
Perhaps for brainlet physishits.

>>9521494
>they works only in Hausdorff spaces

>>9521530
depends. on the applied side there's not many requirements in general. the deeper a field goes, the more books you'd have to read beforehand.

>>9522853
>on the applied side
He said "mathematics".

>>9522817
>Does there always exist a continuous function f:C→[0,1], such that f(x)∈Ax for all x∈C ?
This is vacuously true since you assumed the existence of "the Cantor set".

I need to make a presentation on my math master's project - what software is good for this?

>>9521550
>>9521559
It seems like you're lost. The physics threads are over at >>>/toy/.

>>9522854
>He said "mathematics".
I'm not a "he".

>>9523144
It's pronounced, "My Lady, Faggots."

それは、「マイ・レディ、Fagots。」、発音されます

so I know the FDC theorem states that I can find one of the anti derivatives of the function or a specific one if Im given a bounded interval.

Its easy to find the antiderv if Im given the function, but my text book only states it as f(x)

So I have this: integral f(x)dx = 10 between 0 to 4. It then asks me to evaluate the integral of f(2x)dx between 0 to 2. (the answer is 5). (pic related)

Now I would have gotten the result long ago if I was given the goddamn function and not just f(x). I imagine its related to the first theorem. (which Ive done many exercises but I was given a function and I just had to plug in x or do some variable substitution.

Btw is my description of the FTC-Theorem right or am I wrong? Thanks.

>>9523333
>Now I would have gotten the result long ago if I was given the goddamn function and not just f(x)

that's the point... this holds for any function f(x) via a simple substitution

https://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus#First_part

Are there any resources out there to help make studying this book not so terrible?

>>9523765
Stewart.

>>9515187
Nobody can provide clear explanation of corollary 3.12, which is essential to validity of IUTeich. He has to either explain it, or make smaller leaps so other people can follow it.

>>9523770
The standard calculus text? I already studied with that when taking the earlier calculus courses

>>9523793
>Nobody can provide clear explanation of corollary 3.12, which is essential to validity of IUTeich. He has to either explain it, or make smaller leaps so other people can follow it.
Which step is unclear?

>>9524110
Where he claims that every ring has a non-empty spectrum.

Is it possible to show in a Euclidean domain that 0%a = 0 for all a without using a valuation function which has the property "For all nonzero a and b in R, f(a) ≤ f(ab)"?

I'm trying to implement euclidean domains in a theorem proving language but the problem I am having is that I can't construct a valuation satisfying that property without using what amounts to the axiom of choice, which makes the division algorithm it satisfies uncomputable.

>>9524369
The valuation function is literally part of the definition of a Euclidean domain. Perhaps you need a slightly different definition.

I think my advisor is becoming senile. He no longer seem to remember anything at all and he keeps getting confused by simple arguments. I mean, last week was probably the fifth time I explained our joint paper to him. Even worse is that he will get irrationally angry at me every time there's something he doesn't understand. He also gets angry if I start explaining something he actually do remembers, so our meetings just keep getting more and more ridiculous.

What do I do?

>>9524791
Kill and eat him to gain his power.

>>9524796
I think I'd prefer not gaining the power of senility.

Need sanity check. positive factors/divisors = 1 to n right?

For a homework question I need to find the common divisors of 16 and 48. The solution set I'm looking at doesn't list 16 as a common divisor, but it should be right? It's the gCD* ffs.

>>9524868
>The solution set I'm looking at doesn't list 16 as a common divisor, but it should be right?
Yes.

Zero is both positive and negative.

>>9515245
Literally give up on this board if you don't know

Brainlet here. I'm currently working through an introductory book on abstract algebra (covering basics of group theory, ring theory, etc.). Will I be able to get into some basic topology after this?

>>9525255
>Will I be able to get into some basic topology after this?
You don't need abstract algebra for basic topology.

>>9524738
I want to use a valuation function, but not require it to have the property "For all nonzero a and b in R, f(a) ≤ f(ab)"

The valuation function isn't defined to require that property by most authors. The reason for this is you can show that given that you have a valuation function you can define a valuation function with that property in terms of it.
The problem is that said new valuation function acts as a valuation function for some division algorithm on the euclidean domain, but not necessarily the same one (at least as per the proof in this: https://www.jstor.org/stable/2316324).). The new valuation function requires the axiom of choice and is thus non-computable. The new division algorithm which we can define in terms of it cannot be explicitly defined either.

>>9524369
It's not possible.
As a counterexample consider the integers with the usual definition of division except 0/1 = 1 and 0%1 = -1 and the valuation which takes 0 to 0, -1 to 1, and everything else to their absolute value plus one.

Are you aware that Mochi has basically failed? You can forget about him

>>9521530
You would need up to a Master's education in mathematics, in the field you wanted to work on. Then, you need access to a LOT of research papers related to the specific problem you wanted to work on. Not including high school, but including college, you're looking at essentially 6-8 years of work.

t. grad student

Essentially a bachelors gets you up to date everything up to about 150 years ago. A masters gets you up to date with math until like the 1950s. Reading research papers will get you up to modern day math, and from there, you can actually try solving a research problem.

So yeah, best of luck to you

>>9525255
nah man

just go read some topology bro

>>9525255
topology doesnt need algebra, if anything it needs some real analysis. However proper topology done nowadays is algebraic topology so you'll need it if you go down that path

>>9525452
The workload was to be expected. I can't see myself wanting to do anything else just yet so that's my current path. Thanks for the thorough reply. Time to get to work.

How would I simplify a’bd + a’bc + bcd’ to a’bd + bcd’ ?
I can sort of guess at it because everything in (a'bc) appears in a'bd and bcd', but I can't get it through an actual process.

>>9526506
a’bc = a'bcd + a'bcd'
a'bd + a'bcd = a'bd
bcd' + a'bcd' = bcd'

>>9525469
Eh, if you want to do anything remotely interesting or geometric in topology you need to know some algebra. Point set topology is fucking miserable.

>>9526879
yeah but point set topology is what you need for any geometry field. Algebraic topology is miserable anyways:
>muh spheres
>muh knots

is it common to run out of short-term memory space while trying to evaluate integrals or am I just a brainlet

>>9527682
>evaluate integrals
Engineers are known to be brainlets.

>>9527718
I have never met a useful mathematician

>>9525469
>>9525255
>>9526879
it never hurts to get good at point-set topology, even though you only need the basics for algebraic topology. I self studies topology, I just downloaded Munkres thinking I would learn about knots etc., never was I more wrong in my life. but I forced myself through that stuff and now I have zero regrets. I understand algebraic topology on a much deeper level (no more handwaving about "obvious" continuity of something etc.) and my analysis skills improved immensely.

>>9527740
Theyre really good at auto-fellatio

>>9527750
??? you claim to have learned algebraic topology WITHOUT knowing CONTINUITY???

lol

LOL

HAHAHAHAHAHAHAHAHAHAHAHAHA

>>9527763
"The Knot Book" by Colin Adams seems to be fairly elementary and you can do a lot of knot theory combinatorially. And Algebraic Topology: An Introduction by Massey is still fairly elementary and might be possible with only the analysis definition of continuity (epsilon-delta) or even the intuitive high-school understanding.

>>9527740
>useful
This is a meaningless notion.

>>9527851
I think you can do pretty much all of algebraic topology without every invoking any homological algebra. All you need is a good knowledge of real and complex analysis. Makes you wonder if the subject should be renamed.

>>9527949
This pretty much. A full classification of topological spaces up to homeomorphism only requires some set theory and point-set topology. If you need more, you're probably a brainlet.

Can someone shine a bit of light on what the symmetric bilinear forms have to do with the Hessian?

https://www.youtube.com/watch?v=z5lo0uNQL7I
i love her so much it hurts

>>9527763
I don't

>>9527682
depends on the size, thats why we invented paper!

>>9528126
The Hessian induces a symmetric bilinear form [math]({\bf x},{\bf y})_f \equiv x_i \frac{\partial^2 f}{\partial x_i \partial y_j}y_j[/math] which tells you the hyperbolicity of [math]f[/math].

>>9528450
This isn't even well-defined.

>>9526797
Thanks, I still have zero clue what I'm doing for any of these

>>9528190
What's she on? She seems too calm as if she's smoked 3 fat blunts before every video, she's definetly a stoner girl

>>9527949
The point of algebraic topology is classify things up to homotopy.

(Co)homology only exists because computing homotopy is hard. i.e. Usual singular homology is a much more computable "homology theory" than homotopy.

So of you abandon homology, you will be able to compute very little. And when you can actual do computations of homotopy, they usually require serious machinery like spectral sequences.

Mochizuki reminds me of Mark in his eyes

>>9529013
true mochi should review some video games on youtube

>>9529010
I'm pretty sure that was a joke.

What happens if we assume ACC instead of AC?

>>9529006
shes just a naturally kind and calm person
i want to do maths problems with her and have her smile at me as i explain why shes wrong

>>9529106
we get absurdities like IUT

>>9529106
They are equivalent, since both are equivalent to the empty set having an element.

>>9529179
>the empty set
No such thing.

>>9529168
How so?

My brain keeps degrading and I keep getting duller and duller. My interests are all gone and I am just a hollow shell with no intelligence or creativity. The meds stop working after a couple of weeks and then the flat affect and delusions come back. I just want to die /sci/. I am going to have to drop out soon.

>>9529400
>I am just a hollow shell with no intelligence or creativity
You can't lose what you never had in the first place.

What exactly is a "formal" difference?

For example, in topological K-theory, the 0th K group's elements are taken to be formal differences of vector bundles over the same base space.

Are these formal differences meaningless and just a useful construction, or do they actually encode something?

>>9529554
Its just the group law on the Grothendieck Group.

i.e. [A]-[B]=[C] if 0-->A-->B-->C-->0 is a SES

>>9529590
Yes but does E-E' mean anything topologically other than being part of a group?

3rd row of equations. First one. How did that one come about? I'm confused and stuck.

>>9530670
use the formula for sum of first n numbers

>>9530671
I meant the equation below where it says "then, on the average, it takes". The bottom equations. And then the first one. Where did it come from.

>>9530696
>I meant the equation below where it says "then, on the average, it takes"
Yes, use the formula for the sum of first n numbers.

>>9530696
looks like the expected value

What's a good textbook for affine and projective geometry?

What's the error in me thinking:

[math]A[/math] countable, so can write it as [math][a_0, ..., a_n, ...][/math].
[math]<_A[/math] an order, so wlog [math]a_0 <_A a_1 <_A ... <_A a_n <_A ... [/math].
Then [math]a_i \mapsto i [/math] is the required map and [math]S = \mathbb{N}[/math].

It seems a bit too good to be true

>>9530866
You just wlog'ed the whole problem away :) You don't know that the ordering has a least element, like Z for example. In general it won't line up with the ordering on N.

This is proven by a "back and forth argument".

>>9530889
Oops. Just looked up that method and thanks

>>9515245
y=const. obviously

If I assume deg(p(x)) = 1 for a polynomial p(x) in F, a field, doesn't this mean p(x) can have indeterminate x^0 or x^1?

how do i stop doing shit like pic related

>>9531249
i wish i knew, i do the same exact shit on tests. making some retarded fuckup somewhere that even a 5th grader wouldn't make

also, you have very neat handwriting... are you a girl, anon?

Does this make sense?

What is the most beautiful and creative area of mathematics that can also make me decent money?

>>9531466
probability is pretty neat and you can wrangle that into a career in the financial sector

>>9531156
yes

>>9531249
I've been doing the same my whole fucking life. Just accept it I guess, it's not like your multiplication accuracy means anything about you as a mathematician. I space out quite often and do stupid shit like that too.

>>9531466
>beautiful and creative
>makes money

pick one

Honestly though your best bet might be in software development, but you'd have to look for some really advanced start-up idea or work for Microsoft or something.

I've been trying to do this one problem for three and a half hours and I haven't made any progress.
How would I get (a' +b')(a' + d)(a + c') from this?

>>9532297
has your instructor not taught you what a karnaugh map is?

>>9532297
If you're gonna spend three and a half hours on it just do a K-map and be done in a few minutes.

Get an expression for all of the minterms where the expression is false, and then DeMorgan's Law will convert that to product of sums naturally.

>>9532305
>>9532311
No.
How would I do it with a K map?

>>9531249
gb2reddit

>>9532315
Write the K-map for the function, if you don't know how to do that, just wiki or youtube it, it's easy. Use the values for which it is false to write it into POS. Alternatively, you could just write the truth table, and for every value that it's 0 write the expression and combine them for POS form.

>>9531466
Polymathmatics.

>>9515187
Ok I know there some topologists in here.

How tf do I do this?

"5. Consider the space Y from HW 1.

a) Show that the 0-cell has an open neighborhood C homeomorphic to the cone on a torus. Hint: The preimage of C in the two simplices before identifications is a disjoint union of 8 cones, each being a cone on a triangle. By inspection (or by an Euler characteristic argument), the 8 triangles glue together to form a torus.

b) Let M=Y-C. Show that M is an orientable manifold with boundary a torus.

c) By the "half lives, half dies" theorem the kernel K of H_1(bd M)-> H_1(M) has rank 1. Draw a curve in Y generating K. Hint: Note that M/bd M is homeomorphic to Y."

(This was HW that was due today, so not cheating.)

>>9532379
>Ok I know there some topologists in here.
>How tf do I do this?
>"5. Consider the space Y from HW 1.
>a) Show that the 0-cell has an open neighborhood C homeomorphic to the cone on a torus. Hint: The preimage of C in the two simplices before identifications is a disjoint union of 8 cones, each being a cone on a triangle. By inspection (or by an Euler characteristic argument), the 8 triangles glue together to form a torus.
>b) Let M=Y-C. Show that M is an orientable manifold with boundary a torus.
>c) By the "half lives, half dies" theorem the kernel K of H_1(bd M)-> H_1(M) has rank 1. Draw a curve in Y generating K. Hint: Note that M/bd M is homeomorphic to Y."
>(This was HW that was due today, so not cheating.)
What have you tried?

>>9532384
For part (a) I just kind of drew a picture. For part (b) I bullshitted an argument. Didn't even try (c).

>>9532311
>>9532364
>>9532305
You guys are the best

My intuition tells me the answer to this question is 9 but my professor says it's higher

Am misunderstanding something here

>>9532660
>My intuition tells me the answer to this question is 9 but my professor says it's higher
>Am misunderstanding something here
Yes, in mathematics you use reasoning, not intuition.

>>9532660
>Am misunderstanding something here
Do you know what a duel is?

>>9532675
actually not really
im not anglo or french

>>9532676
>actually not really
>im not anglo or french
You should look it up then.

>>9532676
It's like sex but with cocks instead of vaginas.

>>9532683
well I don't really know what a dueling contest is or how it works

Is it supposed to be single-elimination?

Why are all programming languages in English?

>>9532700
Americans dominate the world

>>9532674
both, ideally

>>9532315
http://www.scienceheap.com/?1vbx

>>9531744
>>9532123
>>9532700
The >>>/g/hetto/ is over there.

>>9532660
Well there are 9 rounds, but the number of duels would be:
1st round: 256
+
2nd: 256/2
+
.
.
.

Any math kino?

https://www.youtube.com/watch?v=6ybd5rbQ5rU

>>9533276
You seem lost. This isn't the >>>/g/hetto/.

How do I prove that if I have a square ABCD with P a point outside of the square such that the angle <APB=π2 and we call M and M the intersections ofAB with PD,PC respectively, then MN2=AM×BN

Hi guys, I'm in undergrad math and just started proof based lin alg. While I am understanding the concepts of lin alg well, I am struggling extremely with actually proving the theorems, so much that I am considering switching to actuary science.

Can I be successful in maths and suck at proofs?

>>9535442
>proof based lin alg
There is no other lin alg.
>struggling extremely with actually proving the theorem
>understanding the concepts of lin alg well
No, you aren't understanding them well. And you won't until you realize that.
>Can I be successful in maths and suck at proofs?
No, but you don't know yet if you truly suck at them. The way linear algebra is usually taught is a clusterfuck.

>>9535442
>While I am understanding the concepts of lin alg well, I am struggling extremely with actually proving the theorems
I refuse to believe that anyone could type out this sentence with a straight face.

>>9527926
we caught one boys!

>>9517094
Yeah I'd say he has some autism

Hi guys, I'm stuck on a question on differences of squares :
Q(a,b,c,d,e) = ab+ac-bc+2be-cd+2de

I managed to go to:
[math]
Q = (a-\frac{1}{2}b)^2 + (a-\frac{1}{2}c)^2 - (b-\frac{1}{2}c)^2 +2*(b-\frac{1}{2}e)^2 - (c-\frac{1}{2}d)^2 + 2*(d-\frac{1}{2}e)^2 -2a^2 + \frac{5}{4}b^2 + c^2 +\frac{7}4{}d^2 -e^2
[/math]

How the fuck am I supposed to get rid of the last 5 ?

>>9524791
This actually sounds kinda sad and I'm still an undergrad, so feel free to ignore me, but have you tried talking with him? If not, he may just have some mental decline and is smart enough to recognize it is happening, but also kind of at a loss of how to deal with it, thereby lashing out. This could all be bullshit, I'm just trying to give a suggestion. Hope it all works out.

>>9536491
>How the fuck am I supposed to get rid of the last 5 ?
Please refrain from swearing in this thread.

I suck at proofs. How can I get better? :-/

>>9536848
>How can I get better?
Work harder.

Does mathematics research face any sort of reproducibility crisis similar to psychology?

>>9536899
Not on the same scale but proofs are found to be flawed sometimes (watch Voevodsky's talk on why he went into foundations)

So um I am sure this question has been asked a million times but how do you know if a Math major is right for you? I would like to try out "pure" maths but I would probably end up in applied or else my dad would probably stop paying my tuition. I haven't gotten below an A in any of my math courses. The highest level of math I have taken so far has only been differential equations. I just read my textbook if I have to, go to lecture, and make sure I understand everything and avoid memorization at all costs unless I am cramming due to procrastination (which has only happen a very small number of times). Can you guys describe how you were when you were taking lower level math courses like I am? How did you know you would do well in your major?

>>9535449
>>9535447
I understand how things relate to eachother. I.e det(a) = 0 means that A isn't invertible. But I don't know how to PROVE things, like proving the rank nullity theorem. It makes sense that nullity + rank = n, since if there aren't any free variables, nullity will be 0 and that means you have nothing but pivot 1's so rank = n. But I don't know how to actually PROVE that.

>>9521849
wait, is f actually zero on a whole domain?

I'm a first year maths major taking physics classes and they're so boring. When I try to give a description of what I'm studying it still sounds cool (muh "discovering and comprehending the universe") but it's just not fun. Should I drop it and take CS classes or stick with physics since physics is a good application of maths?

>>9536833
fuck off faggot

>>9538218
>faggot
Why the homophobia?

>>9538214
Ask in a physics thread over at >>>/toy/. And read the subject before posting next time.

>>9537748
>but how do you know if a Math major is right for you?
Pick up an actual book from a field which interests you and start studying.
>The highest level of math I have taken so far has only been differential equations
So you haven't even seen any math yet?

>>9538323
>So you haven't even seen any math yet?
Yes that is what I was saying. I probably wouldn't be asking this question if I had already been introduced to higher level math. I am not sure what the point of this comment was.

>>9538487
That's like saying that the highest level of math you've taken so far was organic chemistry or a course in cooking, neither is math.
>I probably wouldn't be asking this question if I had already been introduced to higher level math
You should be learning that on your own instead of waiting for someone to introduce it to you, which might not even happen in undergrad depending on your school.

Guys, can you recommend me any other authors similar to Terence Tao? I mean, I'm searching for those who present material (like proofs, key ideas, big picture, new definitions) in a clear and rigorous way
Undergraduate stuff prefered, but anything is appreciated

>>9538894
Milnor, Halmos

The answer is .643. .08/12 = .667 isn't the answer, and the logic on my notes is that this is due to it compounding monthly.

Any ideas?

>>9539998
>>>/sci/sqt/

>>9539998
Think about this and what the question means. Just because you have two numbers (.08 and 12) doesn't mean that just any formula involving these two will work.
So what is this asking ? Start with an account at the bank satisfying the condition with an initial balance of P > 0, and let m be the monthly interest rate.
Then after a month, the balance in your account is (1+m)P. After two months, it is (1+m)^2P, etc.
After a year, your balance is (1+m)^12 P.
Hence, your annual yield is (1+m)^12 - 1, and you want it to be .08.
Solving for m, you rapidly get m = (1.08)^{1/12} - 1 = .00643... ~ .643%

http://www.kurims.kyoto-u.ac.jp/~motizuki/2018-02-02%20Tan%20---%20Introduction%20to%20inter-universal%20Teichmuller%20theory%20(slides).pdf
>slides for lectures on IUTeich by Fucheng Tan.

>>9540658
is this a proof?

>>9515187
What the fuck? Shouldn't this sequence converge to 32/8 = 4 since the dominant terms are n^5 and the others will tend to zero? How is this sequence not converging?

How my other grad applicant niggas doing? Any responses yet from PhD programs?

>>9540658
So is this just theatre or is it actual evidence that some people actually do understand IUT?

>>9540728
disgusting

>>9541279
>So is this just theatre
No, it's Hodge theatre.

>>9538323
>Pick up an actual book from a field which interests you and start studying.

Terrible advice, most math books are not good to study directly from.
Take a proof-based math course to get a better idea.

>>9540728
it's a joke

welcome to post-modern mathematics

>>9540908
did you check that the limit was to infinity and not to 0?

>>9523765

>>9515187
Critical Error #1: It does not submit to '1' to the exclusion of all other numbers.
Critical Error #2: Presuming that mathematics will have any impact on reality until someone with actual reputation and resources will get involved/invest.
Critical Error #3: 運転楽しさ | Fahrspaß

>>9542336
ching choing, hoi ting wing bing

Explain this

6
4
2
0
8
6
4
2
0

4
1
8
5
2
9
6
3
0
7

1
2
3
5
6
7
0
1
3
6
9
2
5
8

1
4
7
0
3
6
9
2
5
8

>>9541866
>Terrible advice
It worked for me and any non-brainlet I know. Maybe he is one of us as well.
>proof-based math course
Entire post disregarded. It's clear that you are engineer scum if you have to add in "proof-based".

>>9542531
This is why America isn't going to beat China. Not because of 'same random internet comment', but because Chinese people take their reading/writing more seriously so even if I am posting on a quantum computer, it will be processed by 'future Chinese scientists' because I declared them the winner by virtue of 'honor' and my daughter.

The China that did disrespected women will exist no more, to the point only 'female' can be truly Chinese. The men I will happily discard.

Are there countably or uncountably many infinities?

>>9542844
Uncountably + counted notes + numerator - SPEED OF LIGHT

>>9542844
The empty set is provably countable.

>>9542844
There is no set of cardinalities.

>>9542897
This is irrelevant to his question.

>>9542900
No because cardinalities are only defined for sets.

>>9542900
>This is irrelevant to his question.
I'm not a "his".

>>9542906
It can be shown in some non-standard models of ZF that every proper class is a set.

>>9542907
Then you're a 'Him.' A himster.

,>>9542913
Okey. You could take out the axiom of regularity and you are done, but in every workable system, there is no set if all cardinal numbers.

What's a good book that gets me up to speed on basic representation theory? Just general theory

i am retarded so please explain how you double integrate this shit? iirc its as if you put dp to both sides and put the integral symbol, then integrate? does it become ln(p) p dV/dp = a first?

>>9515187
this guy looks like he eats people

>Attempting to solve integral
>Solution gets more and more messy every step
>Margins of the page slowly run out of space to house a single term

How do I know if I screwed up or if the answer is supposed to be that way?

>>9532660
For a winner to be determined all people except for one must be eliminated. Each duel eliminates exactly one person, therefore 511 duels must take place before we have a winner.

>>9535442
>proofs
Proofs aren't leaving, so you need to become very comfortable with them. Every class from here on out will be proof centric.

For this, doing proofs will help, as will understanding what it is you are trying to prove. Perhaps you can post examples of what you're struggling with?

Some resources I'd suggest are Book of Proof / How to Prove It for developing proof strategies and they are good problem repositories. For Linear Algebra, you want to read your text well, of course, but maybe pairing it with the book "Linear Algebra: An Introduction to Abstract Mathematics" by Robert Valenza will help, as the proofs in that book are deceptively simple. Linear Algebra Done Right is another popular

3B1B:
>https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab

Lectures by Gilbert Strang, Sheldon Axler, and Wilderberger, respectively:
>https://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/video-lectures/
>https://www.youtube.com/playlist?list=PLGAnmvB9m7zOBVCZBUUmSinFV0wEir2Vw
>https://www.youtube.com/watch?v=yAb12PWrhV0&list=PLIljB45xT85BhzJ-oWNug1YtUjfWp1qAp

Additionally you'll want to work on your problem solving and learning abilities in general, and for that there's a whole host of separate texts I could recommend, but a few worth naming might be How to Solve It, Discourse on The Method, The Art and Craft of Problem Solving, How to Study as a Mathematics Major, Learning How to Learn / A Mind for Numbers, etc..

>>9542657
you're an idiot

>>9523765
>https://ocw.mit.edu/courses/mathematics/18-024-multivariable-calculus-with-theory-spring-2011/index.htm

Might help, might not. Gl