/sci/ - Science & Math

>>11733267
Rational number means integer/integer.
Rational number means polynomial/polynomial.
Meromorphic function means analytical_function/analytical_function.
Am I getting it right?

any good books on cellular automata?
most I see inspect it from the CS side, but I am trying to persue research in the math side of it
resources on functions defined in hyperbolic or graph spaces would also be neat.

>>11733307
rational number means n/m for integers n and m. think fractions
meromorphic means analytic except at some points. like 1/x or 1/((x-1)(x-3))

>>11733267
so I have a basic question about the gamma function G(a) for real positive arguments a>0.

G(a) := integral(0,inf, x^(a-1)*exp(-x) dx)

it's easy to show that
G(n+1)=n*G(n) (*)

and G(1)=1, yielding G(n+1)=n! for positive integers.

to show that G(a) exists for all a>0 it suffices to show that G(a)<inf for 0<a<1 because of (*), and that's what Im trying to do.

for 0<a<1 we have -1<a-1<0 and so I *think* I can split the integral into two parts (at x=1) and estimate those parts like this:

G(a) <= integral(0,1, x^(a-1)dx) + integral(1,inf, exp(-x)dx)

both terms are easily shown to be finite, the only question, is this step legit?
for the 1st term I use that 0<exp(-x)<1 for all positive x, so throwing that out should just make the integral larger.
similarly for the 2nd term I use that 0<x^(a-1)<1 for x>1 (remember that -1<a-1<0), so dropping it again should at most make the integral larger

is this correct? and is there an easier way of showing G(a) exists for all positive arguments?

>>11733341
I think splitting the integral into two parts it doesn't have any poles

>>11733307
>Rational number means polynomial/polynomial.
You meant 'Rational [math]function[/math] means polynomial/polynomial', right?

>>11733368
could you elaborate?

afaik G(a) shouldnt have any poles for a>0 anyway

(poles are at 0 and negative integers I think)

>>11733307
>Every meromorphic function on D can be expressed as the ratio between two holomorphic functions
https://en.wikipedia.org/wiki/Meromorphic_function

>>11733418
*flexes on your shit*

>>11733435
>how old are you
>i am ζ(s)= Σ n^-s years old
how would you respond, m*thematician.

>>11733418
THEY PLAYED US LIKE A DAMN FIDDLE!

>>11733498
To the question of how old I am? Or how would I just respond to your post? Clarify your fucking language instead of imposing your perspective.

>>11733545
DEVIL INBOUND!

>>11733498
>how would you respond, m*thematician.
Let [math]\mathfrak{m}[/math] be the maximal mathematical age. Then the answer is greater than [math]\mathfrak{m}[/math]. Furthermore, the refined Adler theorem says [math]\mathfrak{m} = 23.999\dots[/math].

>>11733557
I study hard into the mid winter night
to these black arts I dedicate my life
and no reward could ever be so sweet
oh my dark lord, when will we meet?

>>11733498
Using >>11733578's interpretation as a translation basis I would respond with [math]6 \pi[/math] then convert from radians to degrees on my skateboard and perform a flawless switch 1080 boneless while removing my pants and waggling my pants and eyebrows suggestively at speeds that cause a corona of friction burn of such intense heat that plasma begins to form and shoot out in a radial pattern on all bystanders while never breaking eye contact.

>>11733578
Pic related. I meant to get it to you sooner but you always want to be some MAGICAL FUCKING EXCEPTION TO EVERY FUCKING RULE OR OBSERVATION that the only way this was ever going to reach you was when I shot my my own cum onto my little sister's pussy, magically chant 'Anon won't be Anon for long!' 3 times as she masturbated with the frothy white mixture, and then nail the neighbor's cat onto this mongolian fertility totem for some future civilization to re-interpret the shamanic memory as a digital expression to be interpreted by some future language and individual that was neither I, Satan, or she, Little Sister.

can somebody help me with my complex analysis homework pls

Question 8. Show that for all [math]Re{s}=\frac{1}{2}[/math] the following equation holds true

[eqn]\displaystyle \sum_{n=1}^{\infty} \frac{1}{n^s} = 0[/eqn]

>>11733623
that should've read [math]\Re(s) = \frac{1}{2}[/math]

can somebody please help me with this problem from my complex analysis problem set

Question #8 Given the following infinite series, show that for [math]\left \{ s \in \mathbb{C} : Re(s) = \frac{1}{2}\right \}[/math] the convergence holds true:

[eqn]\displaystyle \sum_{n=1}^{\infty} \frac{1}{n^s} = 0 [\eqn]

can somebody please help me with this problem from my complex analysis problem set

Question #8 Given the following infinite series, show that for [math]\left \{ s \in \mathbb{C} : Re(s) = \frac{1}{2}\right \} [/math] the convergence holds true:


[eqn]\displaystyle \sum_{n=1}^{\infty} \frac{1}{n^s} = 0 [/eqn]

>>11733655
What you do is align the power drill up to the holes in the cheese slices and slowly insert the drill tip as the torque eviscerates the supple young biomatter that is attempting to present itself as an obstacle.

>>11733655
the question is false
haven't you ever heard of a p-series

>>11733307
Meromorphic means the function has at most poles, no essential singularities.
Locally they can be expressed as rational functions, but not every meromorphic function is globally a ratio.

I couldn't find it by googling, is there a name for the following problem? I'm guessing there are more general formulations.
Let there be n random variables of type 1 and 2 each, all of them independent and with uniform distribution on the d dimensional cube, of interest is a matching of variables of points of different types that has minimal length, defined as the sum of distances between the matches points.
I'm curious what general results are known about this.

>>11733783
How many apertures can you tile onto a cube? Seriously?

I haven't studied a single day during quarantine. Exams are in a little more than two weeks. Call me a retard and wish me luck.

Let [math]X[/math] be a scheme and suppose there is a [math]k[/math]-morphism [math]\text{Spec}\left(k[\epsilon]/(\epsilon^2)\right)\to X[/math]. I'm supposed to show that this gives a point [math]x\in X[/math] such that its residue field [math]k(x)[/math] is [math]k[/math], and an element of its Zariski tangent space [math]\hom(\mathfrak m/\mathfrak m^2,k)[/math], where [math]\mathfrak m\subset \mathcal O_x[/math] is the maximal ideal.

Clearly the image of the unique maximal ideal [math](\epsilon)/(\epsilon^2)[/math] of [math]\text{Spec}\left(k[\epsilon]/(\epsilon^2)\right)[/math] gets sent to some point [math]x[/math], and there is an induced local homomorphism [math]\phi:\mathcal O_x\to k[\epsilon]/(\epsilon^2)[/math], which in turn induces a map [math]\mathcal O_x/\mathfrak m = k(x)\to \left(k[\epsilon]/(\epsilon^2)\right)/\left((\epsilon)/(\epsilon^2)\right)=k[/math], since [math]\phi(\mathfrak m)\subset (\epsilon)/(\epsilon^2)[/math]. The fact that it's a [math]k[/math]-morphism then shows that [math]k(x)=k[/math] as desired.

Now, I suppose that tangent element should be obtained through [math]\phi[/math], ie, restricting [math]\phi':\mathfrak m\to k[\epsilon]/(\epsilon^2)[/math], and since [math]\phi(\mathfrak m)\subset (\epsilon)/(\epsilon^2)[/math], then [math]\phi(\mathfrak m^2)\subset (\epsilon^2)[/math], so actually [math]\phi'':\mathfrak m/\mathfrak m^2 \to k[\epsilon]/(\epsilon^2)[/math] is well defined. But now, to get a map to [math]k[/math], we have to compose with the quotienting of [math](\epsilon)/(\epsilon^2)[/math], ie we get the map [math]\mathfrak m/\mathfrak m^2 \to k[\epsilon]/(\epsilon^2)\to k[/math]. But would this not just be the zero map every time? We have that [math]\mathfrak m\mapsto (\epsilon)\mapsto 0[/math], so I fail to see how this is meaningful information. What went wrong?

>>11733819
retard

>>11733267
>pic
I've always hated that extension of the factorial function. Like, it smooths out the positive numbers nicely, then its like 'fuck the rest.' Also a little it's ugly.

>>11733819
Okay retard, I wish you luck in being able to appease the God of recordable media and its associated mediums of transcription.

>>11733829
What went wrong is you thought that any of that faggotry was meaningful in the first place beyond whatever projection it maps onto your real domain.

>>11733655
is this equivalent to the riemann hypothesis?

if so then you're neither funny nor original

>>11733785
I don't know, that's why I'm asking. Or was my description hard to undestand?

>>11733859
What do you mean there, really. I feel disagree there and it sort of makes sense.

As you say, it does it's job on the naturals and as far as the positive reals are concerned, its behaviour actually follows a unique characterization among the interpolations, namely being superconvex there
https://en.wikipedia.org/wiki/Bohr%E2%80%93Mollerup_theorem
(Bohr–Mollerup theorem)

Now if we enter nice functions on a complex domain, we have the arguably lucit fact that

https://en.wikipedia.org/wiki/Abel%E2%80%93Plana_formula
(Abel–Plana formula)

[math] \sum_{n=0}^\infty f(n)= \int_0^\infty f(x) \, dx+ \frac 1 2 f(0)+i \int_0^\infty \frac{f(i t)-f(-i t)}{e^{2\pi t}-1} \, dt [/math]

which is at least reasonable in that for integrals over functions (the first on the left) to be completed to sums (pickung up residues), you must force poles on the naturals, which is what [math] \frac{1}{e^{2\pi t}-1} [/math] does, with exp being the generic periodic function, at least if you add in the [math] \pi [/math].
The symmetrization of f and the first term being special I don't have the best intuition for, but similar things pop up along the way.

A most generic such sum is the infinite Euler product, a.k.a. the zeta function [math] \sum_{n=1}^\infty\frac{1}{n^s} = \prod_{p \text{ prime}} \frac{1}{1-p^{-s}} [/math], at least with a last product term added

[math] \Gamma(s) \zeta(s) = \int_0^\infty \frac{x ^ {s-1}}{e ^ x - 1} \, \mathrm{d}x [/math]

with a last product term is the Gamma.
In a way I feel the Gamma gives zeta convergence in a oddly straight forward "mean" kind of way

[math] \zeta(s) = \dfrac{ \int_0^\infty \frac{x ^ {s-1}}{e ^ x - 1} \, \mathrm{d}x} { \int_0^\infty \frac{x ^ {s-1}}{e ^ x - 0} \, \mathrm{d}x } [/math]

where Gamma is the denominator.
Alone, it reads

[math] \Gamma(s) = \int_0^\infty \frac{x ^ {s-1}}{e ^ x - 0} \, \mathrm{d}x [/math]

which is taken as its definition. That integrand still has poles on pi-related evenly spaced gridpoints on the real line and they somehow translate.
And, in fact, looking at the [math] \Gamma(s) \zeta(s) [/math] formula, the divergences of the Gamma function are the trival zeros of the zeta function. This is the same as cos being symmetric or Bernoulli numbers being all zero (except the first).


The harmonic series to infinity, the quint-essiential nice looking but still divergent series [math] \sum_{n=}^\infty \frac{1}{n} [/math] gives zeta it's real pole at inbetween s=0 and s=1 is s=1/2 there's various mirroring going on, such as

[math] \Gamma(s) \Gamma(1-s) =\frac{1}{\sin(\pi\ s)}\ \pi [/math]

[math] \zeta(s) = 2^s\pi^{s-1}\ \sin\left(\frac{\pi s}{2}\right)\ \Gamma(1-s)\ \zeta(1-s) [/math]

I will not say it all makes sense, but somehow it's all sort of stemming from periodicity features of exp together with homology'ish features of complex analysis. I don't know where to complain to the Gamma function for what it is.

PS first for SCV

Oh an my point about the reflection formula is that this is the "critical line" kind of variant for gamma that makes (when comparing with zeta) it being [math] \Gamma(\frac{1}{2}) = \sqrt{\pi} [/math] seems like a thankful feature.

We can nicely read it off from
[math] \Gamma(s)\Gamma(1-s) = \dfrac{1}{\sin(\pi s)} \pi [/math]
whereas zeta (despite having a similar integral representation), is a bitch there.
(Or maybe Gamma is also a bitch there and I just don't know enough about it)

>>11733926
Your description was easy to understand but I can't extend into my own reasoning a rationale as to why you would want to replicate something like a menger sponge.

>>11733970
>>11733963
>>11733961
And all this sifting and sorting through bounded regions achieves... what?

>>11733498
>mathematician is also language-let
maybe you could have spent your time learning something MORE USEFUL eh?

>>11733992
Achievement is only in everybodies mimd anyway.

Turing wrote his last paper on computing zeta zeros. Very ugly.

>>11733961
>>11733963
>>11733970
Just wanted to say thanks!

>>11734069
np.

And always remember that

[math] \int_0^\infty \dfrac{ {\mathrm d}x }{ 1 - {\mathrm e}^{2\pi\sqrt{x}} } = -\dfrac{1}{12} [/math]

>>11734066
Did he? That is actually fascinating to learn.

Achievement is primarily in the mind of the individual or group, true. I'm wondering how many intellectuals that are self-aware of their sapiosexual masturbation practices.

I guess with the insight from yesterday, that with CH as in
i : R -> w1 a bijection
we can order all of R via
x <_{w1} y := i(x) < I(y)
so that

for all (y in R).
for all (x <_{w1} y).
exists (b: N -> {u <_{w1} x)

i.e. while R is not countable, there's an order for every y such that everything up to it is actually countable.

If we e.g. say R is like w_2, then there's elements Y so late in R that things below Y can't be counted.

This actually strikes me as a quintessential character of the hypothesis.

Are there any natural objects that are by construction of the cardinality of an uncountable ordinal?
I suppose the Neumann Hierarchy isn't, since their objects (while indexed by ordinals) have sizes soley determined by the power set operation

>>11734296
https://en.m.wikipedia.org/wiki/Turing%27s_method

>>11733829
You need to compose with the map [math]a + b \epsilon \mapsto b[/math], not the augmentation.

Come to think of it, we can actually write that thing as

[math] \zeta(s) = \dfrac{ \int_0^\infty \frac{1}{e ^ x - 1} \, \mathrm{d}x^s} { \int_0^\infty \frac{1}{e ^ x - 0} \, \mathrm{d}x^s } [/math]

>>11734402
If I am an identity or value within infinity then what true worth do I or my output hold to infinity if it has access to all that is me and all that is not me and all that could ever be accessible?

>asking for a friend.

>>11734499
Why does your relation to infinity even matter?

Rain is coming.

>>11734560
even matter = worth

You just performed a polynomial reduction to my query but didn't actually change the substance of my investigation.

I want to buy some absolute essential reads from undergraduate/graduate mathematics. What should I get apart from Baby Rudin?

>>11735414
One word for you, my friend: just follow Verbitsky's list.

>>11735414
just a few that come to mind
>munkres, topology
>any one of artin or fraleigh for abstract algebra (my favourite is aluffi but the exercises are too easy. Alternatively, Jacobson, basic algebra I)
>atiyah-macdonald commutative algebra
(alternatively Bosch's algebraic geometry and commutative algebra is also a GOAT book, covers the same core topics, is modern, clearer, has good direction, and goes through some homological algebra and properties of flat modules. Has a geometric view in mind and the second half of the book does scheme theory in a very clear way, however very abstractly (more so than Hartshorne))
>do carmo, differential geometry of curves and surfaces
>Lee, smooth manifolds, or Tu, introduction to manifolds
>do carmo, riemannian geometry
>hatcher, algebraic topology (contentious pick here in /sci/, but almost every university course follows this)
>Harris, algebraic geometry, a first course, or Shafarevich, basic algebraic geometry I (I don't like either too much, but theyre classics within classical AG - if you only want to have a quick taste of AG to get motivation for scheme theory, Smith's invitation to AG or even chapter I of hartshorne is good, the former better though)

>>11735414
I think everybody should own a copy of proofs from the book

>>11733267
>this is /mg/'s discord
Holy fucking shit, please someone create another one, pic related is full of trolls

What books do I nead to read/what do I need to learn to start learning SCV?

>>11736952
Thank you for reminding me why I shouldn't use Discord.

>>11737075
The Verbitsky list & the basics of IUTeich.

>>11737075
>Complex analysis
>Differential Geometry
>Algebraic topology
Those are the very basics

while we're bookfagging, does there exist an algebraic topology book with no pictures in it?
I have to learn a bit of it eventually but I don't like geometry

>>11737704
just buy a book with pictures and cut them out

Why does iterating through input in the mergesort algorithm happen in log(n) time? Please explain like I'm retarded, I was barely smart enough to even ask that question.

I get where the O(n) comes from, just not the O(log(n)) part.

also, in relation to the above, when people say mergesorts run time is O(nlog(n)), they ALWAYS mean log_2, right? and they're just too stupid to include that very necassary component?

>>11737849
your question seems nonsensical
the algorithm takes n*log(n) operations
there isn't really any part of it that takes log(n) time

>>11737704
My diary. Also, Bourbaki could have a book on AT and that should be free of stuff for the feeble minded such as pictures.

>>11737901
Could also be e or 10 instead of 2.

>>11737956
>Bourbaki
Not a person.

>>11737901
yes, log in computer science is usually taken to be base 2, but it makes no real difference because any two bases for the logarithm are related by a constant factor; [math]\log_2(n) = \frac{\log_k n}{\log_k 2}[/math] . What's important is the logarithmic growth rate, not the exact value.

>>11733498
>"how ald are you"
>responds with a function
you must be the funny one, right?

could you beat this kid?

https://www.youtube.com/watch?v=xLT3NsDYEfc

>>11737995
Where in that post was the claim made that Bourbaki would be a person?

>>11738331
I was just correcting your ignorance, no need to get angry.

>>11737849
Merging n elements is O(n). There are log(n) layers in your recursion (because it takes log(n) halvings to reach the base case) so you have to do the merge operation log(n) times.

calculus is so simple when presented rigorously. the only thing that gets students is the bullshit roundabout presentation of trig identities that nobody saw before and have to memorize in addition to all the other little shits u have to memorize. if they just forced you to prove the identities from first principles yourself, and prove the functions themselves, there would be no need for memorization or lengthy month long sequences of plug and chugging integrals to memorize trig identities.

calculus makes me feel filthy

>>11738381
Delusional.

>>11738704
Yes, you are!

>>11737919
There are three parts to the algorithm, one of these parts takes O(log(n)) operations, the other are limited to O(n) operations. That's my understanding. This guy says it better: >>11738431.

Also, >>11738431, great explanation! Thanks.


>>11738116
Ah, thanks. I should've gathered that, a little stressed today.

What's the formula for the n-th diferential of a function from R^n to R^m?

>>11739362
[math]f^{(n)}[/math]

Ok, can I get some anons to rip this 'proof' apart? It's trying to show the runtime of mergesort is O(n*log(n)), but it doesn't feel very proof-like.

>>11740305
>but it doesn't feel very proof-like.
It's not the proof's fault if you just don't _feel_ like it's right.
It's correct and fairly clearly written.

>>11740305
It's okay, except for the claim that the number of divisions is log_2(n), that is only true if n=2^m, but it's trivial to correct that.


>doesn't feel very proof-like
It's verbose, nothing wrong with that.

>>11740305
nothing wrong with this proof
if you're looking for something else here's how I would do it:
let f(n) denote the worst-case runtime of the algorithm
clearly f(1) = 0
similar analysis shows that [math]f(n) \leq f( \lfloor{n/2}\rfloor) + f(\lceil{n/2}\rceil) + cn[/math]
now only some arithmetic remains to show that f(n) is O(n log n)

how are you guys cheating your Calculus exams

>>11740677
I hide the answers in my head

>>11739362
>What's the formula for the n-th diferential of a function from R^n to R^m?

in 1d,
[math] {d^n \over dx^n} f(g(x))=\sum \frac{n!}{m_1!\,1!^{m_1}\,m_2!\,2!^{m_2}\,\cdots\,m_n!\,n!^{m_n}}\cdot f^{(m_1+\cdots+m_n)}(g(x))\cdot \prod_{j=1}^n\left(g^{(j)}(x)\right)^{m_j} [/math]

otherwise
https://en.wikipedia.org/wiki/Fa%C3%A0_di_Bruno%27s_formula#Variations

>>11740389
thanks!

>>11740546
would the number of divisions then be ceiling(log_2(n))? and thanks!

>>11740601
this is how I started, i guess I couldn't figure out the arithmetic. I thnk I understand it better now though and might play with that idea later

>>11740677
by reading the relevant sections of apostol and knowing the general forms of the equations and being thus able to solve any stupid problem the academic throws at his student

>>11733331
CS is math.

Why do you only have to go through the square root of the maximum number to do the Sieve of Eratosthenes?

>>11741202
Cs is code monkey with basic algebra

>>11741258
Suppose there's some composite number you missed by stopping at sqrt(N). That means this number only has prime factors strictly greater than sqrt(N) (otherwise you would've crossed it out using one of the smaller primes), and since it's composite, it has at least 2 prime factors. But if you multiply two numbers both bigger than sqrt(N), you get something bigger than N. The first number you can miss is outside your range.

Had enough self-discipline today to both go for a 8km jog and finish my maths homework. Feels great.

>>11741258
Use 11 for an example:
[eqn]xy = 11 = \sqrt(11)\sqrt(11)\\
\text{Let }x = 7\text{, an integer greater than }\sqrt(11)\\
y = \frac{11}{7}\text{, a number less than }\sqrt(11)[/eqn]
Therefore, calculating up to [math]\sqrt(11)[/math] will also calculate for any possible factors above [math]\sqrt(11)[/math]

>>11741284
I'm gonna punch you.

>>11741416
BASED

calculus can be taught in a single quarter, I'm seething, 3 semesters for this baby shit. what a fucking joke

>>11741493
Not the best explanation I've seen. I'll try to give a better one.

Let's apply the sieve to the number 100. What are the proper divisors of 100?

{1, 2, 4, 5, 10, 20, 25, 50, 100}

Notice that divisors come in pairs: If you divide 100 by 50 you'll have 2, and if you divide 100 by 2 you'll have 50. So let's pair the proper divisors of 10.

{(1,100),(2,50),(4,25),(5,20),(10,10)}

As you can see, proving that one of the numbers of each tuple divides 100, implies that the other number of the tuple is also a divisor.

>>11741416
Good job, anon!

>>11741600

If your school is any good you are able to ask your professor to anticipate your tests

>>11738143
that's the point. functions are useless just like most of math. you can't buy ax+b apples.

>>11742652
Numbers are constant functions, so you're wrong.

>>11742662
>Numbers are constant functions
the absolute state of contemporary mathematics
>>11742652
>functions are useless
so what?

>>11730007
I actually just found that example on the German CH Wikipedia page.
It ends up taking the second integral of each case, thus arguing that the the assumption that this indicator function is not measurable

https://de.wikipedia.org/wiki/Kontinuumshypothese#Beispiel_aus_der_Ma%C3%9Ftheorie

what's a good book for learning Probability?

>>11742847
Kolmogorov.

>>11742713
An interesting plot twist. The Lebesgue measure seems to be allergic to both AC and CH.

>>11742847
Do not trust anyone recommending anything other than Shiryaev

Aluffi's Algebra: Chapter 0 is being sold for $44 on AMS

https://bookstore.ams.org/gsm-104

>>11742952
Aluffi is $0 on libgen.

>>11742968
Libgen's version is full of mistakes though

>>11742912
I'm untrustworthy? It hurts me to read those words, but I guess there is only a 24 year guarantee for worth...

>>11742968
Chapter $0.

>>11733267
So dedekind cuts require "threre is an rational betweeen two reals". Can "threre is an rational betweeen two reals" be proven with set theory?

>>11734328
>Are there any natural objects that are by construction of the cardinality of an uncountable ordinal?
Isn't that the whole point of Hartogs' numbers?

>>11742990
>I'm untrustworthy?
Of course, you stole my heart and you're not taking responsibility for it.

>>11742995
Dedekind cuts are set theoretic objects, it is already proved within set theory.

>>11743158
Prove there is isomorphism between real numbers and cuts i.e.every real number has exactly one unique cut

>>11743166
This is such a vague question, what representation of the real numbers do you have in mind? Dedekind cuts are defined to be the real numbers. This is just one implementation of the reals inside set theory. There are other constructions of the real numbers, like as equivalence classes of convergent sequences of rational numbers, another being decimal expansions. All these construction yield the same set since R is the only complete ordered field. I can't think a source for this off the top of my head, but I'm sure you can google it. Here is some info about the constructions. Dedekind cuts tend to be easy to define and give you quite access to the reals, though verifying the algebraic properties, in particular multiplication, can be tedious. The Cauchy sequence construction is a bit technical, with all the equivalence relations, though its advantage is that the method used generalizes to be able complete an arbitrary metric space. If you are interested in this you might check out exercises 22,23, and 24 of Rudin PMA.

>>11743258
>This is such a vague question
It is really not. I'm completely aware of Cauchy sequences and etc.
The question is: Dedekind Cuts is a one way to define REAL NUMBERS through RATIONAL NUMBERS BUT (here comes a big BUT) if we assume that there could be two numbers that have no rational number between them then these two numbers have exactly the same Dedekind Cut (by definition) and this notion becomes inherently useless.
Dedekind cuts are useful because:
1.It uses the set theory
2. There is isomorphism (Cauchy sequences have not, there are only equivalence classes)
But if "threre is a rational between two reals" property couldn't be proven with set theory (it could be easily proven with Archimedes axiom) then what's the point of Dedekind Cuts? Just honest question.
Gonna ask it everywhere I can.

>>11743303
Dedekind cuts are sets of rational numbers, if two Dedekind cuts are not the same then one is proper subset of the other, pick a rational in the set difference. There you go.

>>11743303
The rationals are dense in the reals, you can’t just assume that there are two numbers that don’t have a rational between them without violating other theorems/axioms.
>what if it couldn’t be proven with set theory
It can and is so, so why bother thinking otherwise.
Dedekind cuts are just one of many ways you can define the reals, but dedekind cuts also are very constructivist in nature and have the isomorphism you described. I guess when you ask the point of dedekind cuts, I interpret it as “why have 2 proofs of the same thing”, to which I say why not, it’s always cool to show there’s 2 ways to get to the same path.

>>11743348
>Dedekind cuts are sets of rational numbers
Yes and it's the main issue
I say there is sqrt(2) (which is not rational) and some number r2 (which is also not rational) and there no rationals numbers between them. So
>Dedekind cuts are not the same then one is proper subset of the other, pick a rational in the set difference.
there is no "rational difference" and these cuts are same.

>>11743385
>you can’t just assume that there are two numbers that don’t have a rational between them without violating other theorems/axioms.
What set theory axioms?

>>11743433
you can't pick an r2 not equal to sqrt2 that has that property, idiot, it's provable that between any 2 reals, there exists a rational, you've mentioned it yourself.

>>11743467
>idiot
Okay
>it's provable that between any 2 reals, there exists a rational
If it's hard for you to follow the discussion I can remind: prove it with set theory.

>>11743433
I'm not really sure what you mean. Let x,y be Dedekind cuts, so x={q\in Q | q <q_x} and y={q\in Q | q <q_y}. Suppose that x !=y and wlog that x is a proper subset of y, i.e. q_x \in y, now define q_0 as (q_y+q_x)/2, then q_0\in y but not in x. So q_0 is a rational between x and y, and rational.

So if I have an isometry G that is defined as counter-clockwise rotation by 90 degrees around the origin, and I apply it to a square centered at the origin, and label the corners 1 through 4, I can write e.g.
[math]
G^2 = \begin{bmatrix}
1 & 2 & 3 & 4 \\
3 & 4 & 1 & 2
\end{bmatrix}
[/math]
and similarly for the other powers of $G$. This is cyclic. It also reminds me of cyclic groups in Algebra. Is there a general name for this kind of thing? As in, a name for the branch of "things that go roundy-roundy"?

>>11743491
Oh, and I should specify I also like that you can decompose these boys when composing them just like with the cyclic groups in algebra, which also is very nice. Someone please tell me what this is called.

>>11743474
Yes. You proved that there is no rationals between two cuts. And I say there are irrationals between two cuts.

>>11743491
Group actions. The cyclic group of order 4 acts on the plane (or wherever you're defining the isometry) by corresponding applications of the isometry. This is also why you can define dihedral groups and other symmetry groups by their action on some geometrical object.

>>11743555
Where can I read more about this? Like I said, I encountered it a bit in Algebra, and I assume that's where I should look? Essentially permutations + algebra + geometry is what I'm looking for.

Holy fucking shit I'm so fucking dumb why can't I fucking understand ABSTRACT FUCKING ALGEBRA? I feel like a complete brainlet, perhaps I really am one it's so frustrating I can't understand the basic things like quocient of a set or equivalence classes why am I so fucking pathetic fuck

>>11743796
>split a set into a bunch of sets
>ask if 2 things are in the same set or not

Could someone suggest a decent resource on algebraic k theory? I have Bass's book and I've read Magurn's, but I'm curious if someone has better.

>>11743802
I think I kind get it when I read the definitions, but I can't solve any fucking exercise about it

im writing an essay which touches upon the phil of math.

historically, a lot of the "foundations" or methods of math were justified by intuition, such as infinitesimal calculus i believe. it was only until much much later that naive set theory, and ZFC took over.

did we "lose anything" by switching over to an axiomatic foundation? are there issues in ZFC that "intuitive math" doesn't have?

im just a lowly undergrad in math:

whenever i am given a computational question (i.e.: not a proof), i have absolutely no idea how to justify my sequence of actions, but somehow im pretty confident that the result after the mess of scribbles gives me the correct answer, but the more i try to confirm it to myself the less confident i get because i have no idea what i am doing

does this happen to anyone else? why does this happen? how to resolve it??

>>11742987
so is the AMS'
https://www.math.fsu.edu/~aluffi/algebraerrata.2016/Errata.html

Math textbooks have typos. This is just how the world is. Learn to deal with it.

>>11741202
CS is math related to computation.
Computation being how you can get machines to do mathematics.
Studying mathematics is not studying computation, but you can study mathematics by trying to compute formulas on a machine.
>inb4 cs is more pure math
complete lie if you know anything about computation. I heard this last time and I'm still salty about it.

>>11743843
What do you mean by intuitive math?

>>11743853
>why does this happen?
you don't understand the underlying theory
>how to resolve it??
learn the underlying theory

>>11743860
math you can see in your head easily.
But the anon doesn't realise that really smart people can see most math as intuitive.

>>11743864
I had a class with a set theorist who could easily see all the proofs as a picture in his head. To keep himself sharp, he would only give himself an outline and force himself to do most of the thinking on the spot. He would mumble to himself for a brief bit, then draw a nice picture and mumble to himself "Yes, that's right." Then he would write out the detailed symbolic proof like nothing happened; it was funny.

>>11743867
it is definitely trainable, but some people will be able to get there quicker.
It's impressive when you see someone doing hard problems in their head.

what math will pill me on dual polygons like a cube and octahedron
. I've been fucking around finding angles and lengths and and wow there's a lot and it's hard to categorize,i see graph theory from the wiki but what about analysis? some of this I can use a bit of vector/matrix algebra, but I'd like something a bit more general for finding related volumes

>>11743992
Have you heard of "geometry".

>>11743858
>CS is math related to computation.
Not really, I doubt anybody would call numerical analysis computer science.

>Studying mathematics is not studying computation
What are you on about?
So much mathematics has been developed for the sake of computation, both in applied and pure mathematics.

>complete lie
Define pure math

>>11742898
Foundations of Probability, I assume?

>>11743664
I think dihedral groups are discussed in most Discrete Math classes, probably start there if you're unfamiliar.

What's the RIGHT formal way of defining complex numbers rather than just saying "lmao let's just call \sqrt(-1) i and pretend [math]i i = -1[/math]

/gmmg/

>>11743148
Hmm, that is a valid point. But what if I am keeping your heart as a hostage so that you will grow big and strong on your quest to reclaim it? Then I can say "I raised that boy" when you have trained yourself to the psycho-physical peak.

>>11743796
Approach from some nice examples. If you want a general equivalence relation, take all humans and say x and y are related if they live in the same country. What would be the equivalence classes then? For quotient groups, try approaching first from the direction of integers mod n. Then you can generalise it to more complicated cases and eventually reach the point where you are taking the quotients of kernels over images. Can you give an example of a problem most problematic? I believe in you, querido.

>>11743807
I learned the basics from Swan's book on the subject, although it is typewriter stuff. I don't know if it is good all the way through, but the beginning was well-written. This one: https://www.springer.com/gp/book/9783540042457

>>11743853
You need to bang your head against the material until you notice how the situation you are in is suitable for evoking theorem 6.66 etc. Then you apply its claim to move one step forward and recheck your situation. It's obviously not necessarily the only way to proceed. This post is getting algebraic, so let's continue with that. Say you were told to prove that the semidirect product [math]\mathbb{Z}/3\mathbb{Z} \rtimes \mathbb{Z}/2\mathbb{Z}[/math] is isomorphic to the symmetric group of 3 letters. You could construct an explicit isomorphism, or you could use the fact that there are only 2 isomorphism classes for groups of order 6, one for abelian and one for non-abelian groups. Then you would simply note that neither the group in question nor the symmetric group is abelian, from which you would conclude the result. Care to share an example of where you would struggle?

>>11744325
My favorite is as an extension field to the reals. You can do the same with say sqrt(2) which is also a favorite the wild burger.

>>11744358
>extension field to the reals
Undergrad here. Care to explain this a bit more or link source?

>>11744283
Yep. It's old but still sufficient at least for undergrad stuff. Spasibo dyadya Andrei.

>>11744325
[math]\mathbb{R}[x]/\langle x^2+1\rangle[/math].

Someone please suggest a solid rigorous but more introductory probability and statistics text. I have a very basic background in analysis and a decent understanding of linear algebra but the current text I'm using is nearly worthless. Not having to learn measure theory at least for now until I have a better background in analysis would be preferable but I don't mind having to do proofs and have already taken Calc I-III with a good deal of self-teaching from more advanced texts outside of class.

>>11744365
https://en.wikipedia.org/wiki/Field_extension
In the most basic way you introduce a new element to a field and some identity that describes its behavior, and then consider the field generated by it and the original field.
For example, you can think of it as introducing a new element [math]x[/math] without any sort of additional rule leading to polynomials and power series.

>>11742990
>Chapter $0
nice

gmmg

bros...

>>11744528
[math]\mathscr{GMMG}[/math]

>>11744563
...?

>>11744394
+1

Math undergrad is going fine. Currently working on a publishable project and I have a workshop to attend soon. Personal life is utterly shit though. I have fucking issues man

>>11744623
I don't think cats have their mouth open when they smile.

>>11744643
Do you even lift?

>>11744652
Do you even know what characteristic classes are?

>>11744669
Revolution is
not possible when
the internets attention
span has been reduced
to blips. That's true for
anti-capitalism as well as
mathematics and physics.

bros... i have been doing maths today... for the third day in a row

>>11744680
Listen, I'm really good at math and my sense of self-worth comes from being good at math, I don't need to lift. However, my personal life is shit and can't be fixed.

>>11744698
Lift.
It's important.

And don't make shitting homotopy jokes now.

>>11744697
Have you lifted today? If not, plan it. It's important.

>>11744643
It's a good way to put stuff under the rug, but then you just have to keep pushing further and further to avoid all the repressed stuff. By good, I mean it works in that sense, not good-good.

>>11744652
Yeah, that is true. My granny had two, the older of which liked apricot baby food. They didn't like to be touched, though, because she got them from an animal shelter and the animals there can have pretty shady pasts. Then there was this one guy who had a really nice cat who liked to be touched and stuff. The owner lived the sloth-like NEET life of taking a lot of naps, so if I didn't get any simultaneous sleep, I'd just crawl on the floor playing with his cat. I even had the nose connection with her, but he was full of scratch marks. Good times.

>>11744697
Very good, anon! You actually inspired me to try not to drink myself to sleep every night.

>>11744680
Did you intentionally give the Nick Land vibes with this?

>>11744698
>However, my personal life is shit and can't be fixed.
You know the /sqt/ reply. What have you tried?

>>11744723
not intentionally, no

>>11744715
Why should I lift? I already run like 4-5 miles at least 5 days a week. Cardio is certainly better for cognition. Lifting is really boring for me and most guys that lift are dumb.

>>11744723
>You know the /sqt/ reply. What have you tried?
I haven't tried anything. I don't know what to try. I'm pretty fit and I go to a good university but I still can't have a good social life since I focus too much on school. Right now I'm at home because of summer break, though, and I've literally not had any social interaction.

>>11744764
Not him

>I still can't have a good social life since I focus too much on school
Unless you are literally spending 14 hours a day on mathematics this is not true.

I have to say that, even though I am somewhat anti-social myself, having friends is really important.
And sitting alone at home for weeks on end really can't do you any good.

>>11744764
So you are basically a workaholic whose efficiency is hindered by the fact that you suffer inside and then you have to put more work in to keep up with the goals you set yourself, and this creates a vicious circle of perpetually worsening conditions, or am I wrong? Do you have any uni comrades who you could do something with online? Could you imagine trying to find people sufficiently close to your location online? That cat guy I mentioned, I got to know him over the internet and he happened to live in the neighbouring city, so it sort of escalated it when the both of us seemed to enjoy each other's virtual company things. Suppose you managed to do the same, then you would an online+offline buddy or maybe even a few, and you could stay in touch online after you go back to the uni (assuming it's not where you are now). Even better: if it is, then you can even see them during the terms! I'm not saying it will work, but I am saying it could be worth trying.

amigos... I forgot to do maths today...

can i get job with a Master's degree in applied Mahtematics

>>11733883
No, it’s just false.

>>11744627
Im going to fucking kill you faggot

If you be a woman, find a man before 24

>>11738592
>trig identities
Maybe you aren't cut out to be a mathematician. It's like learning a combo to a video game.

>>11744352
>What would be the equivalence classes then?
One person from each country?

>>11745502
nan

>>11744394
> solid and rigorous
> no measure theory
Sorry but just learn measure theory if you want to do probability theory rigorously(its really not that hard), otherwise just get on with your life and do it the bullshit way.

>>11745616
one equivalence class is the set of all people living in the same country
if Bob is a person, then the equivalence class containing Bob is denoted [Bob], and remember that it's a set:
[Bob] = {all people living in the same country as Bob}
Bob is called a representant of this equivalence class

>Dear Professor X,
>I am writing you to ask if I can write my bachelor's thesis under your guidance.
>I would be glad to receive a topic suggestion.
>I am mainly interested to write about a topic in [area of math].
>best regards, anon
is this okay?

>>11746234
lol

>>11746352
is that a yes or no?

>>11745819

>>11746234
I would:
- Mention a particular course you were especially interested in
- A tiny bit of flattery "I really enjoyed X subject (that you taught) so I would like ... " will not hurt, especially if it is sincere
- It is really bad style to start every sentence with the same word, you do not have to write like you are autistic, even if you are.

>>11746234
I agree that the I I I structure is bad prose.
You can also skim his arxiv and write that you know that Magneto works on so and so and you're interested in that.

>>11746754
>you do not have to write like you are autistic
I think reading greentext all the time fried my brain

>>11746858
>I agree that the I I I structure is bad prose.
It only looks like bad prose because of the greentext making it look like he's starting 5 sentences in a row with the word I.
To say what he wants to say without a bunch of Is would be way more awkward than just using pronouns in their natural place. If he were to rewrite it in normal English I don't think it would look weird.

>>11747086
I disagree, but I also don't care enough for a discussion.

>>11747293
>kraut who posts garbled crappy English on a daily basis
>entitled to opinions about English style

>>11743843
yes we did

>>11747465
This is unrelated to what anon is asking.

i can't find a proof for this, can someone help?

>>11747465
>This is an extended text of the address at the discussion on teaching of mathematics in _Palais de Découverte in Paris_
I always thought this was just an article, I never noticed before that it's actually a speech he gave in fucking Paris
what a madman

>>11745860
Told ya I'm a fucking retard, I'm finished

>>11745819
>>11746434
Ok.

Any books you particularly like? I have Billingsley, Cramer, and Fristedt and Gray’s book.

>>11747395
I'm neither a "kraut", nor is it up for debate that starting sentences the same way repeatedly is bad style. Neither is this insight tied to the particular language. Why do you even attack me on it, I responded in the least combative way possible. It's a good tip, everybody take it if they like or not.

>>11747497
did you try actually doing the computation?

>>11747637
a little bit, i don't see how the determinant is relevant

>>11747564
>I'm neither a "kraut"
non-krauts don't name their screencaps bildschirmfotos
>nor is it up for debate that starting sentences the same way repeatedly is bad style
This is true, but this is also exactly what I posted.
Perhaps if you were fluent enough in English to be telling people how to write you would have understood this.

>>11747676
determinant nonzero implies that such h, k can be found

>>11747687
Donno. I know my English is more than sloppy but I'd think that aspect of writing, and this approach to systematically just chain blocks of information in an email to a person, disregarding prose, is not tied to the particular language. If would jump you in any language. The line breaks in the greentext make it worse, but I also notice this otherwise.

But hey, I wasn't the one pointing it out in the first place. Writing a nice text takes effort and I'm not saying he should change it if he feels the rewrite would be worse. E.g. if the text gets doctored via multiple Ctrl-V's and sent off with errors.

Semi-related, here's something fun
https://youtu.be/A8zWWp0akUU

This is good too
https://youtu.be/2yzMUs3badc

And I have no emotional tie to Otto v. Bismarck and his states or his Germany, whether my laptop screencaps as Bildschirmfoto.

>>11747688
it's not obvious how the determinant would ensure that

Remember to stay hydrated, /mg/!

>>11745616
They would be basically the countries. Next attempt. Person x is equivalent to person y if their names start with the same letter. What are the equivalence classes?

>>11746234
I remember you. Let me try to help you with another broken and garbled message:
Dear Professor X,

I am writing to you to ask if I could have you supervise my bachelor's thesis. My main topics of interest lie in [area], and I would be glad to receive suggestions of possible topics to write it on.

Best regards,
the cutely uncertain anon who should get a head pat

bros...

>>11748798
please kill yourself

>>11749175
what is it bro

>>11749227
One day.

>>11748798
are you a girl (female) or a girl (male)?

>>11749542
Not science or maths.

>>11747908
>whether my laptop screencaps as Bildschirmfoto.
But your windows is in German, thus you most likely speak German and there is a reasonable probability that you have ties to Germany.

>And I have no emotional tie to Otto v. Bismarck and his states or his Germany
"His" Germany died over 100 years ago?

https://www.youtube.com/watch?v=5_r3V9S0Qzo

newton btfo

can anyone prove this pls? >>11747497

>>11748148
It is.

>>11749659
prove me then, should be easy if it's obvious

>>11749657
If you are >>11748148, then you need to brush up extensively on you linear algebra knowledge before continuing to think about differential equations.
Since you apparently do not know basic conditions for when a 2x2 system is solvable...

>>11749663
Determinant non zero means system is solvable.
How do you not know this?

>>11749682
it's still solvable even if zero because further down from the screenshot it assumes a case where det = 0 and b1 != 0 and gives info how to solve but im not there yet

>>11749682
ok i remembered that if a whole row or column of a determinant is zero it efferctively becomes not a square but rectangle so 1 variable will not have a proper solution.

>>11749721
>of a determinant
*matrix then it's determinant will be 0

>>11748798
>what are the equivalence classes
Not the person you were asking but just to try my luck. Letters?
Also what are equivalence classes used for?
I am new to math.

>>11744273
It sounds like you agree with me.
numerical analysis is something you do in both computer science and mathematics

>developed for the sake of computation
At that point it's no longer pure mathematics. But you can use computation and use a variation of the idea, to assist in gaining a better understanding of the idea.

>pure mathematics
the mathematical ideas separate from any application.

>>11749785
>It sounds like you agree with me.
I would put our differences at deviation our views on mathematics.

>At that point it's no longer pure mathematics.
My counter point would be the euclidean algorithm.

>the mathematical ideas separate from any application.
Name a single one.

>>11749692
>>11749721
For fucks sake, read up on your linear algebra.

To make sure you understand this:
A nxn system has a unique solution iff the associated determinant is nonzero.

>>11749843
i know fucking autist, if it's 0 it has infinitely real solutions,

>>11749872
lol, how many solutions does x+y = 0; x+y = 1 have?

>>11749872
>i know fucking autist
>immediately followed by wrong statement

>>11749872
>if it's 0 it has infinitely real solutions,
No, not at all.
Take:
1*y+0x=1
1*y+0x=2
The determinant of that system is very obviously zero, yet, just as obvious, there is NOT A SINGLE solutions.

PLEASE brush up on your linear algebra, it is pointless to study something when you have serious issues with the assumed prerequisites.

>>11749872
You gotta study linear algebra my friend.

>>11749893
>>11749890
>>11749894
>>11749911
my statement is assuming the system is otherwise consistent, so then if there exists 1 solution and the determinant is 0 does that mean there are infinitely many solutions?

>>11749916
yes

>>11749783
Letters indeed. If you want a more mathsy example, so be it. Do you know modular arithmetic? Choose some positive integer [math]n[/math]. We can then define [math][a]_n = \{ b\in \mathbb{Z}\ |\ b \equiv a\text{ mod }n\}[/math] (congruence mod [math]n[/math] is an equivalence relation). These sets are the equivalence classes of the congruence relation. Next, notice that if [math]a\equiv b, c \equiv d \text{ mod }n[/math], then [math]a+c \equiv b+d \text{ mod }n[/math]. It follows that we have an addition for the equivalence classes: [math][a]_n + [c]_n = [a+c]_n[/math], and this actually leads to a group structure (and gives the group [math]\mathbb{Z}/ n\mathbb{Z}[/math]). This idea generalises to all quotient groups.

>>11749916
Again. If you knew the tinies bit of linear algebra you would not be asking this question.

There is nothing wrong with not knowing something, but you should be aware that there is a serious gap in your knowledge that is hurting you.

>>11749943
I'm finished, thanks for your help, but I can't understand algebra, it's over for me, algebra has oficially retired me from math, I'm so ashamed right now

>>11744738
Lift so you can fight me off when I try to rape you twink.

>>11750069
Post physique.

>>11744738

start lifting naked at home, it changed my life

>>11749973
i am not a mathematician, i only try to understand something once and when i do i will most likely never read it again.

>>11750069
>what is a gun

>>11750047

abstract algebra isn't meant for humans to understand, just linear algebra is enough

>>11750047
Instead of retiring, maybe you could tell what you are struggling with and maybe you will end up becoming less troubled afterwards.

>>11750109
Then we must transcend humanity.

>>11750090
>i will most likely never read it again.
Guess what! The time has come, you now need to use your previous knowledge and are faced with the fact that you forget basically everything.
This is a fucking retarded attitude and especially clinging to it when it just bit you in the ass seems ridiculous.

You don't know the prerequisites for a course, I honestly couldn't care less what you remember and what not. But clearly you are having serious issues with the material which are certainly to some extent caused by you not understanding the prerequisites. If you want to pass the course, you obviously have to do something.
So, I am giving you the advice to go back to linear algebra in order not to get stuck on extremely simple issues again and again.

>>11750119
>maybe you could tell what you are struggling with
I can't solve anything in pic related, I looked up 1.2's solution and I understood perfectly but I couldn't come up with something like that at all, and that's just the first chapter, talking about simple concepts, if I'm struggling with these I can already imagine I won't even understand shit when I read about rings, groups, fields etc

>>11749943
Nice!
I don't really know modular arithmetic but I will save this to take a look later. Many thanks.
>>11750047
I don't know algebra anon but I am sure you can do it by spending more time on it with focus on what you don't understand and seeking examples alongside definitions. Well, I hope anyway.
>>11750109
Pic related

>>11750142
>I looked up 1.2's solution and I understood perfectly
You don't know that. Wait a few hours and reproduce the proof on your own. See if it matches with the given solution, what's different. That's the only way to know if you actually understood it.

>>11750142
If you understood it perfectly, you can share it with us, right? Alternatively, you can answer these questions:
(1) If [math]a\in S[/math], why is it the case that [math][a]\neq \emptyset[/math]?
(2) If [math]c\in [a]\cap [b][/math], what can you say about [math]a, b[/math]?
(3) What can you say about [math]\bigcup\limits_{a\in S} [a][/math] when we notice that [math]S = \bigcup\limits_{a \in S} \{ a\}[/math]?
Also, notice how 1.2 and 1.3 combine to give you this nice result: there is a bijective correspondence between the equivalence relations and partitions of a given set. For 1.3, what would you say about the following idea? Let [math]\mathscr{P} = \{ P_i\}[/math] be a partition of [math]S[/math]. Define [math]a\sim b[/math] if the elements are in the same partitioning set. I suggest you give this a try and maybe write it out here and we can bul... check it.

>>11750148
No problemo. Modular arithmetic is sort of like looking at the clock. It doesn't matter if you use 12 or 24 hours, as 25=1 anyway mod the number of hours. Hopefully your motivation rubs onto the struggling anon.

>>11750136
>>11749973
>>11749922
>>11749911
>>11749843
>>11749682
>>11749659
>>11749676
>>11747688
>>11747637
>>11747497

you gaslighting niggers i derived it, could've just told me to use the fact that dy/dx = d(v + k)/d(u + h) = dv/du

i wasted basically half a day on this retarded shit

>>11750075
>>11750097
Seething twinks.

>>11733267
I'm trying to find a short (half a page or so) proof for the spectral theorem (for unbounded operators, but even a general one for bounded ones will do) in the spectral measure form that uses the Gelfand-Naimark theorem, but everything I find are long as fuck ramblings. Can a kind mathanon sauce me or at least give an outline of a quick proof?

>>11746234
hijacking but here goes
>had to write a thesis to graduate this semester
>asked a prof from a related department (super nice, gives easy As and extensions) who I've taken classes with before and asked recs for grad school from to be my advisor
>Man agrees, we spend the semester meeting weekly and I keep updating him on research
>Never really type up my shit, so as time goes on he recommends a couple of times that I write something up and send it to him before too long as deadline approaching
>say I'll send stuff but never really happens
>virus so everyone kicked out
>Ohfuckhowwillifinishthis.jpg
>have to do a thesis presentation a few weeks before end of semester
>he again asks me to send a thesis draft and presentation outline (twice) about ten days before the presentation
>I can't do this in time lmao
>Ask department if thesis has to be turned before presentation, dep. theses' coordinator says daijobu if no turn in before but he'll need grades from my advisor end of sem
>epicface.jpg, inform advisor
>First doc I send him is 2 days before presentation
>it's 6 pages
>send presentation outline an hour before presentation
>he didn't end up catching my email and sees the presentation the first time when I'm actually presenting
>nervous as fuck but goes well
>he emails after saying it was good, says he looks forward to receiving the full written thesis soon

wait 4 newpost

>>11750286
by spectral theorem, i assume you mean "every normal operator can be written as [math]\int z E(dz)[/math] for some spectral measure E"
i know a proof but it's long and doesn't use gelfand naimark at all
are you saying there should be a fast proof using gelfand-naimark?

>>11750222
Sugoi! You're awesome, senpai! You realized the answer to the question without even seeing the solution, which is all about those three topics you wrote for me, amazing, I can't come up with that, my mind gets blank when I read the exercise, but your intelligence motivated me a little, it's not that hard, right? I just need to put more work into it, right? I'll read chapter 2 and try doing all exercises by myself without looking up the solutions.

she looks like she knows some funny math small penis jokes

>>11750348
contd.
>2 days before grades due
>10 pages, not a single chapter complete
>Send email basically saying welp virus fucked me not sure I can do this lol sorry kill me pls
>he personally emails dep coordinator who tells him dep fine with receiving writeup later and just needs a grade in time
>advisor says okerino I'll give your grade but let's wrap this up by day X (about 5 days of extra time)
>man gives A
>Thank him and kiss his feet for being jesus christ
>tfw I graduate with cum laude without finishing thesis
>Fastforward a few meetings later to day X
>Thesis now 20 pages, send him updated version
>Prof says he's super impressed at quality of work and thesis looking monster, makes small suggestions and asks me to try and send updated thing the next day
Now this is where it's fucked up
It's been 10 days since day X and I haven't contacted him at all
My thesis is now 40 pages and nearly complete (been that way since X+3 days actually) but I'm totally fucking stuck on ironing out a couple of lemmas in a chapter. 0 progress in a week.

I don't know what's fucking wrong with me, I could probably just delete the problem parts and turn it in and no one would say anything but here I am self-sabotaging and literally destroying my reputation with my prof. I guess my perfectionism OCD's running loose now that a hard deadline's no longer pushing a gun against my temple and I'm trying to polish it in case I have to link the thesis in the future for jobs/possible grad school transfer (I didn't exactly get into a top place).

We were emailing and meeting every other day from end of sem of to day X but he hasn't sent a single email since our last contact. My dep coordinator CC'd an email a week ago to about ten kids (not trash alone??) from the graduating class requesting we send our written thesis for dep. records but I've just pushed it off. How big of a scum do you think I look like right now? How do I redeem myself? The guilt and pain is pure suffering.

>>11750398
I can't even imagine the lack of self-awareness it takes to type something like this, read it, and not immediately blow your brains out

>>11750399
looks like a wide brain

>>11750408
Eeeeee? Nani? W-what did I do wrong?

>>11750441
talked like a retard who thinks children's cartoons are real, no biggie bro

>>11750441
What is it with /mg/ and anime?

>>11750407
Just send it to him and explain your autism. Both the full and the thesis without autism.

>>11750441
He's attacking your weeb typing style, not the substance of your post. Don't listen to him, I find your posting style mildly entertaining <3

>>11750398
It's not that hard. You just need to bang your head against the wall until it breaks. And it is not your head. It is the wall. What will be hard is your head, and your mind will be sharper than a razor blade. Effort is the key to understanding this (and any other) stuff. Even I managed to learn these things, so I am sure you will too. I believe in you. By the way, if you did 1.4, what did you get?

>>11750452
I think it's cute.

>>11750463
You think it's cute because you are also an adult man pretending to be a little cartoon girl. This does not make it any less pathetic.

>>11750469
>man
I’m not a “man”.

>>11750474
yes, you are.

>>11747525
>>11746434
>>11745819
I hate you fucking people so much fuck you.

>>11750253
huh that looks like this

>>11750469
Or maybe because I had a nice day. I went outside and saw some dogs and one of them came to me to be scratched. It's the small things that can make a day good.

>>11750483
Dumb question: how do you get inverted colours on a pdf?

>>11750492
>how do you get inverted colours on a pdf?
i use sumatrapdf and there is a config.txt that can invert them

>>11733267
Alg geo friends, I keep seeing the spectrum of a ring R defined as simply the representable functor [math]\hom(R,-)[/math] with the claim that the more geometric interpretations can be derived from this. I'm not an algebraic geometer, and I only know of the prime ideal definition. How do I reconcile these two ideas?

Much thanks in advance

>>11750495
Thanks!

>>11750463
>By the way, if you did 1.4, what did you get?
The number of partitions on {1,2,3}, 5.
>I think it's cute.
wwwwww, o kawaii koto

>>11750446
But my thesis is not complete anon
I think I'm going to die

>>11750069
But what if I don't to stop you?

>>11750501
Foxit has it too!

>>11750507
Yup, and that's also the number of eq rels. No retirement for you, yet. Not until THE AGE.

>>11750522
No dying until the age of 24.

>>11750534
>No dying until the age of 24.
I've lost count of the number of all-nighters I've pulled off in the last month trying to get this thing complete. I've been proving every statement I make in full detail but this one section is really fucking me with me because I realize I'll have to write another 5-10 pages to rigorously provide the entire definitions-lemmas setup before I can present the result I want to use and move on. I'm so utterly exhausted and I don't know if I can do that hence the limbo
I need someone to come here and finish these last few pages or just kill me

>>11750553
I know that feel. It's funny how it starts growing in the beginning when you make arguments in the end and then it hits you: I DID NOT PROVE THIS FACT! Then you try to prove it, things go smoothly and your argument, that clever little trick you wrote smiling, was not proved before. Care to share an example of something you are uncertain whether to prove or not?
PS. If you are under 24 years, you don't have my permission to die.

>>11750589
Or does it grow from the beginning rather than in? Anyway, additions to the first sections.

How to ensure that Bernoulli numbers ([math]B_n[/math]) and Bell numbers (also [math]B_n[/math]) won't get mixed up?

zoomers are so fucked hooooooooooooly shit. it's all up to us now, millennials. somehow we always knew it would be like this, much as we tried to think it don't

>>11750619
use function notation for Bell numbers. either B(n) or Bell(n)

>>11750627
If first Bernoulli number is equal to -1/2 then mark them [math]B^{-}_n[/math]. If it's equal to +1/2, then mark them [math]B^{+}_n[/math]

>>11750671
Oops, meant to comment here:

>>11750619

https://sites.google.com/view/nialltaggartmath/oats
>Kirsten Wickelgren (Duke University)
>Title: There are 160,839<1> + 160,650<-1> 3-planes in a 7-dimensional cubic hypersurface
>Abstract: The expression in the title is a bilinear form and it comes from an Euler number in A1-algebraic topology. Such Euler numbers can be constructed with Hochschild homology, self-duality of Koszul complexes, pushforwards in SL_c oriented cohomology theories, and sums of local degrees. We show an integrality result for A1-Euler numbers and apply this to the enumeration of d-planes in complete intersections. Classically such counts are valid over C and sometimes extended to the real numbers, but A1-homotopy theory allows one to perform counts over a large class of fields, and records information about the solutions in bilinear form. The example in the title then follows from work of Finashin--Kharlamov. This is joint work with Tom Bachmann.
In 15 hours.

Nighty night~

>>11750348
>>11750407
at least where i live, the truth is: no one really cares about what's written in your bachelor or master thesis
in the rare situation when someone proves new and interesting results in their thesis, the advisor should suggest making this into a separate article and sending it to a journal

>>11750727
good night math general

>>11750627
What do you mean? In what way am I fucked? :(

what math is this?

>>11750780
looks like inter-universal triple barnett integration

>>11740305
The only issues I see with this are missing \log.

>>11744325
I've always thought the ordered pairs definition was neat, where each complex number was defined as an element of [math]\mathbb{R}^2[/math], redefining complex addition and multiplication as new operations.

>>11750589
I'm not going to go into a lot of detail since my situation seems pretty revealing as it is and I don't want to get any more fucked that I already am, but I hope you get the feeling with this long story cut short example: I'm talking about domain issues of densely defined operators in this one section and I give an example related to locally compact second countable Hausdorff spaces (because it happens to be relevant to what I've been talking about at that point) and then find that I have to prove the space of compactly supported continuous functions are dense in L^p on such a space wrt a Radon (or more generally locally finite) measure for my example to be rigorously be justified
I could just post on stackexchange but I have been honest with all my work so I push through the tears. Been questioning why the fuck I decided to this in the first place every night

>>11750744
But anon when I was a sophomore and was doing summer work with a prof, I was handed a couple of bachelor theses to read and write my notes off
At my uni, honors student theses go into the university library for everyone to access. My advisor had actually recommended my theses for university honors (based off the research work I'd showed him during the semester) and I ended up graduating with that title. I'm not the only one in my dep. for whom this happened so it's nothing special, but because of that, my work is will be publicly accessible too, and I'm terribly afraid everyone will laugh and wonder how such a trash thesis got in there

I'm too stupid to prove anything of value but if my thesis can at least help explain a topic to some student who decides to read it I'd be happy. Just skipping over parts would defeat the whole purpose of writing a thesis and giving my own explanations/proofs/intuition on everything instead of just referencing the 50+ papers/textbooks I read that already talk about the same thing

Is there a nice way to do congruence proofs for stuff in the 2d plane? Most of it is blatantly obvious, but proofs are often tedious to write. For example, suppose you wanted to show that two squares whose sides have equal measure are congruent, it's incredibly easy to see this (Just move them so one of the sides overlap, and then either flip it or rotate it). Actually writing out the proof is complete ass though.

Honest question, does the IUT have anything to do with Several Complex Variables? I saw an anon posting about it somewhere.

>>11750785
what do I need to know to do.. that?

Came across those notes

https://mcgreevy.physics.ucsd.edu/

https://mcgreevy.physics.ucsd.edu/f18/2018F-217-lectures.pdf
>The Renormalization Group
>Fall 2018, Lecturer: McGreevy

looks fun

>>11750985
There's nothing fun about math, it's an honest work like any other else

>>11750992
Work can be (should be) fun?

Actually I don't make my money doing pure math, so I'm not sure what I'm doing.

>>11750997
Work that is fun is called 'hobby'. Real work is by definition an activity which you must do in order to sustain your household.

>>11751007
Rich people can't do math, then?

>>11751007

I make a living solving triple integrals

dy/dx = dy/d(u + const) * du/dx

this is true right?

>>11751322
oh no, that means that const = 0, so this is wrong then again >>11750253

>>11750492
Okular, Evince, Zathura and I am sure many more pdf/document viewers have it by default. For evince and okular you select it at the top, there should be an option in the dropdown menus related to view.
For zathura I think it was CTRL+r that does the trick.

>>11750253
Nobody is gaslighting you. The issue is with you lacking the linear algebra knowledge...

>>11751564
the screenshot might not be complete information because there is further info and it's worded a little ambiguously. is there enough info in the screenshot to be able to solve for y(abstractly speaking)?

>>11751614
You mean in >>11747497?
The whole point is that it becomes a homogeneous equation, which can be solved by means of integration.
Obviously the integral need not result in an analytic expression again.

>>11750901
>move them so one of the sides overlap, and then either flip it or rotate it
isn't that already a proof?

>>11751642
i can't find a way how substituting x = u + h, y = v + k, that will get it in the form y' = f(y/x)

>>11750497
I'm not sure but it sounds like a really bad case of category theory autism. recall that you can equip a spectrum of a ring with a certain structure sheaf and get a locally ringed space. a locally ringed space that is isomorphic to one like above is called an affine scheme.
now we remember that the category of affine schemes is equivalent to the opposite category of rings. now by applying the yoneda embedding we can identify R with hom(R, -).
in short:
[math]affScheme \simeq Ring^{op} \hookrightarrow Set^{Ring} [/math]
[math](Spec_R, \mathcal{O}_R) \leftrightsquigarrow R \leftrightsquigarrow hom(R,-) [/math]

G M
M G

>>11750805
I can't say I'd know much anything about that stuff, but it sounds like something not to skip.

>>11750843
>I'm too stupid to prove anything of value
You don't know that yet. It is likely that we all are, but the only way to know it for sure is to give up.

>>11751437
Thanks. I also found a nightmode in Foxit.

>>11747497
This is garbage notation followed by even more garbage posts. [math]h,k[/math] are CONSTANTS by which you're shifting the VARIABLES [math]x,y[/math]. The determinant being non-zero means the linear transformation associated with that matrix is invertible. That is, if [math]T:\mathbb{R}^2\rightarrow \mathbb{R}^2[/math] is a linear map which sends [math](1,0)[/math] to [math](a_1,a_2)[/math] and [math](0,1)[/math] to [math](b_1,b_2)[/math] (recall that linear maps are uniquely determined by what they do to the basis of their domain), then [math]T[/math] is injective and surjective.
In particular, for a given [math](-c_1,-c_2)\in \mathbb{R}^2[/math], there exists a point [math](h,k)[/math] in the domain so that [math]T((h,k))=(-c_1,-c_2)[/math] by surjectivity. Normally I would ask you to write this in full, but given how much shit you've had to deal with, I'll show you myself: [eqn](-c_1,-c_2)=T((h,k))=T(h(1,0)+k(0,1))=hT(1,0)+kT(0,1)=h(a_1,a_2)+k(b_1,b_2)=(a_1h+b_1k,a_2h+b_2 k).[/eqn] Equating coordinates exactly gives you the last two relations in your posted image.
The whole purpose of doing this was to "absorb" [math]c_1,c_2[/math] into the variables: [eqn]\frac{a_1x+b_1y+c_1}{a_2x+b_2y+c_2}=\frac{(a_1u+b_1v)+(a_1h+b_1k+c_1)}{(a_2u+b_2v)+(a_2h+b_2k+c_2)}=\frac{a_1u+b_1v}{a_2u+b_2v}.[/eqn]
Now, remembering that
>[math]h,k[/math] are CONSTANTS
you can see that [eqn]y'=\frac{dv}{dx}+\frac{dk}{dx}=\frac{dv}{dx}=\frac{dv}{du}\frac{du}{dx}=\frac{dv}{du}\left(\frac{dx}{dx}-\frac{dh}{dx}\right)=\frac{dv}{du}\left(1-0\right)=\frac{dv}{du}.[/eqn]
To actually find [math]h,k[/math] given values for [math]a_1,b_1,c_1,a_2,b_2,c_2[/math], you can invert the matrix of [math]T[/math] that is provided to obtain the matrix for the inverse transformation [math]T^{-1}[/math] and right-multiply it by [math](c_1,c_2)[/math] in column vector form. You should know how to do this. (Equivalently, solve the simultaneous equations directly by hand - it's only a 2x2 system.)

>>11747497
This is garbage notation followed by even more garbage posts. [math]h,k[/math] are CONSTANTS by which you're shifting the VARIABLES [math]x,y[/math]. The determinant being non-zero means the linear transformation associated with that matrix is invertible. That is, if [math]T:\mathbb{R}^2\rightarrow \mathbb{R}^2[/math] is a linear map which sends [math](1,0)[/math] to [math](a_1,a_2)[/math] and [math](0,1)[/math] to [math](b_1,b_2)[/math] (recall that linear maps are uniquely determined by what they do to the basis of their domain), then [math]T[/math] is injective and surjective.
In particular, for a given [math](-c_1,-c_2)\in \mathbb{R}^2[/math], there exists a point [math](h,k)[/math] in the domain so that [math]T((h,k))=(-c_1,-c_2)[/math] by surjectivity. Normally I would ask you to write this in full, but given how much shit you've had to deal with, I'll show you myself: [eqn](-c_1,-c_2)=T((h,k))=T(h(1,0)+k(0,1))=hT(1,0)+kT(0,1)=h(a_1,a_2)+k(b_1,b_2)=(a_1h+b_1k,a_2h+b_2 k).[/eqn] Equating coordinates exactly gives you the last two relations in your posted image.
The whole purpose of doing this was to "absorb" [math]c_1,c_2[/math] into the variables: [eqn]\frac{a_1x+b_1y+c_1}{a_2x+b_2y+c_2}=\frac{(a_1u+b_1v)+(a_1h+b_1k+c_1)}{(a_2u+b_2v)+(a_2h+b_2k+c_2)}=\frac{a_1u+b_1v}{a_2u+b_2v}.[/eqn]
Now, remembering that
>[math]h,k[/math] are CONSTANTS
you can see that [eqn]y'=\frac{dv}{dx}+\frac{dk}{dx}=\frac{dv}{dx}=\frac{dv}{du}\frac{du}{dx}=\frac{dv}{du}\left(\frac{dx}{dx}-\frac{dh}{dx}\right)=\frac{dv}{du}\left(1-0\right)=\frac{dv}{du}.[/eqn]
To actually find [math]h,k[/math] given values for [math]a_1,b_1,c_1,a_2,b_2,c_2 [/math], you can invert the matrix of [math]T[/math] that is provided to obtain the matrix for the inverse transformation [math]T^{-1}[/math] and right-mutliply it by [math](c_1,c_2)[/math] in column vector form. You should know how to do this. (Equivalently, solve the simultaneous equations directly by hand - it's only a 2x2 system.)

>>11747519
did you at least understand it ?

Im trying to find the continous version of adjacency matrix for the euclidian plane

its simple for a discrete infinite chessboard. You can build up the graph from a tensorproduct of two linear graphs, thus the chessboard Adjacency matrix is the kronecker product of the two linear adjacency matrices
Or one can get the value directly from the indices
Mx[x1, x2] = 1 if |x1-x2|==1
Mxy[x1, y1, x2, y2] = 1 if (|x1-x2|=1 XOR |y1-y2|=1)

This can be done for a 4-connected or an 8-connected neighborhood, but how to do the jump from discrete to continuous, where one can move 1 step in any direction?
I got some good numerical results, by scaling everything up x5 or x20 and use a pixellized circle of radius 5 or 20 for the neighborhood, but that doesnt seem right, and is definitely not scalable (kills my CPU and takes hours)

>>11751744
>I can't say I'd know much anything about that stuff, but it sounds like something not to skip.
>You don't know that yet. It is likely that we all are, but the only way to know it for sure is to give up.
Thank you for the encouragement anon. I just got yelled at by my parents again because I was supposed to help them with their jobs after graduation but I'm "crazy" and a "sham" because I'm still not done with my work even though I should supposedly be getting my diploma in the mail any day now. I guess it's time to sleep for 2 hours yet another day and then slap myself awake to get back to grinding. I pray with all my might that I finish this today.

>>11751811
>I pray with all my might that I finish this today
Give yourself time till maybe 4 pm, and then go ask on Stackexchange if it you are stuck. If things are as bad as you make them look like with your posts, now is not the time for high idealism. It's like being chased by a bear and carrying a rifle as a vegan. Because of veganism, you don't want to fire at the bear, but if you don't then it will eat you. You are the vegan, the problem is the bear and your principles are the veganism. Good luck, anon.

Bros....................
I studied for 24.1 hours......
I have forgotten everything.....

>>11751922
just revise the material now bro...

just got absolutely destroyed on my exam in algebraic number theory... fuck corona

>>11733267
Can someone please tell me where the hell the x-2 comes from..
Sorry for bad quality picture I am on my phone

>>11751951
just multiply the first fraction by (x-1) and it should be clear

> All serious mathematics is rigorous!!!!
> Gotta have rigor!!!!
> OMG I bet you don't even know Coq!!!!

>>11751960
people criticising rigour usually haven't reached the maturity required to abandon it yet

>>11751954
I end up with x-1, not 2..

>>11751974
Then you have one copy of the e-thing without x.

was it autism?

>>11751951
>>11751974
That confused me as well anon. But here is how.
Sorry for the low quality pic as well, don't know how to use LaTeX yet.

>>11751996
Wow..
I can't believe that took me half an hour to notice..
Thank you so much!

>>11752007
No problem anon. We all make mistakes like that. Sometimes it is really hard to spot it until someone tells you and then it becomes crystal clear. I experienced it multiple times.

bros... i have been doing maths today... for the fourth day in a row

>>11752057
dude... at this rate...

>>11751773
Yes.

bros... i finished maths...

would a decent bongland uni likely take someone who graduated with a 2:1(3 years ago) as a masters student
I want to study maths full time again despite the debt, waging isnt fun and i dont want to end up as an old raisin trying to learn about homotopy groups rubbing elbows with kids young enough to be my children

>>11752323
Go for it, man, the heart moves where the heart wills.

>>11752058
slow... and steady... wins the race

>>11752323
whip your dick out and try
if you're really going to do it, it makes no difference whether it's "likely" or not

>>11752323
I started my phd at 22, against some people of 25+, so yeah

>>11752347
what is up with this "24 year old" meme

>>11752398
there was(is) a faggot here who was constantly spamming whiny "crying anime girl in rain pic" blogposts about how his life was over and it was too late for him to make it in math because he was 24

he went at it so hard that he turned himself into a running gag

>>11752347
>>11752393
>>11752396
cheers lads

>>11752406
kek
>his name? evariste galois

>>11752398
>>11752406
I think it was after I posted my life story here
>>/sci/thread/S11685818#p11686028
>tfw nothing changed after all this time
Feels bad man

>>11752396
I started mine at 24, ironically enough
the postdoc working for my advisor is fresh out of his PhD and he's turning 40 soon

I'd say the majority of people do go straight from undergrad to grad school with only a summer in-between, but it's not that big of a majority. It's incredibly common to take a year or a couple years to do who the fuck cares after undergrad, and most programs have a couple of students in their 30s too.

>>11752419
fuck off whore

>>11752141
good

*inhales blunt*

dude what about Conway's game of life on the nodes of an undirected graph?

you could enumerate all the possible starting configurations and see what happens

>>11743843
>did we "lose anything" by switching over to an axiomatic foundation?
"Intuitive" maths are just memes, our feelings towards certain proofs or patterns are not necessarily correlated to actual mathematical properties, that's why we formalize our theorems, i.e, to have the certainty of not making mistakes or assumptions, that is the reason why proof-checking algorithms exist.

Also the foundations of maths give birth to a totally new investigation area. Things like transcendental numbers have their origin in problems regarding set theory. Measure theory and topology are other examples. Zorn's Lemma, Hausdorff Maximal Principle and others are the result of certain axioms which we understand better by having a clear basis of what we are allowed or not to use.

>>11747465
>Mathematics is a part of physics.
Into the trash it goes

>>11752582
>Things like transcendental numbers have their origin in problems regarding set theory.
I don't think this is true. Transcendence theory started with special constants like e and pi. People were discussing the transcendence of individual numbers decades before formal set theory even existed.

>>11751990
No, it was based.

>>11752618
based on what?

>>11752582
>>11743843
>"Intuitive" maths are just memes, our feelings towards certain proofs or patterns are not necessarily correlated to actual mathematical properties, that's why we formalize our theorems, i.e, to have the certainty of not making mistakes or assumptions, that is the reason why proof-checking algorithms exist.
I think the ultimate realization has to be that mathematics needs to be investigated intuitively and then written down rigorously.
So many proofs are so obviously based on geometric intuition, but yet are actually fully rigorous.

Neither "intuition" nor "rigour" make sense on its own. Both have to work together.

>>11752625
Based on seething Anlytishits mad that continuity implies no differentiability at all, although ""intuitively"" it should.

I mean, you even have something like Lusin which gets you from measurable to "continous in a lot of places", yet still you can have a continous, but nowhere differentiable function.

>>11752625
'based' is a /pol/ slang to describe an action that's praiseworthy in the extreme right political spectrum

>>11752647
>'based' is a /pol/ slang to describe an action that's praiseworthy in the extreme right political spectrum
lol

>>11752647
You can certainly say Blade Runner 2049 is based without being for free speech etc.

>>11751768
thank you

I need help or I'll lose my mind.

According to this post >>11748184 positive derivative at x doesn't imply that function is increasing at x. I've done my research and it's true. There's even a function differentiable everywhere and monotonic nowhere. What I want to know is where is the hole in the following reasoning (which must be wrong).

Suppose [math]f'(x) > 0[/math]. This is to say [math]\lim_{h \to 0}\frac{f(x+h)-f(x)}{h} = L > 0[/math]. That means there exists [math]\delta > 0[/math] such that [math]|h| < \delta[/math] implies [math]\frac{f(x+h)-f(x)}{h} > 0[/math], for example pick [math]\epsilon = \frac{L}{2}[/math]. Then for [math]h > 0[/math] we have [math]f(x+h) > f(x)[/math] and for [math]h < 0[/math] we have [math]f(x+h) < f(x)[/math] which means [math]f[/math] is increasing on [math](x-\delta,x+\delta)[/math].

>>11752748
okay, I've been thinking about this for hours and as soon as I hit send I realize where's the problem.
I've only showed that f is "increasing" for pair of points a < x < b. if x < a < b, I don't know that f(a) < f(b).

>>11752748
How is monotonic and increasing defined, on a smooth domain?

>>11752625

>>11752595
When Cantor posted his proof that algebraic numbers were enumerable, therefore transcendental numbers were equipotent to the reals a lot of the criticism was that "he didn't posted any example of a single transcendental number". It took a while (between 20 to 30 years) figuring out that e and pi were transcendental

>>11752761
its a totally ordered manifold, obviously

i am stupid
i understand proofs by induction (i think) but what does "suppose inductively" actually means?

>>11752837
That's not true though, according to wikipedia
https://en.wikipedia.org/wiki/Transcendental_number#History

>>11752902
That phrase means that you are invoking the induction hypothesis.

>>11752748
apostols calculus, anon, u need it in ur life

>>11753103