Anonymous at Fri, 15 Nov 2024 06:32:53 UTC No. 16475770

Anonymous at Fri, 15 Nov 2024 06:42:39 UTC No. 16475775

Define a positive rational exponent as:
[math]x^{a+b/c} = x^a x^{b/c} = x^a\sqrt[c]{x^b}[/math]

Then we consider exponent addition:
[math]x^{a+b/c}x^{d+e/f}[/math]
First add the whole numbers, no need for proof:
[math]=x^{a+d}x^{b/c}x^{e/f}[/math]
Then get the same denominator:
[math]x^{b/c}x^{e/f}=x^{fb/fc}x^{ec/fc}[/math]
The exponents remain equal during the last step:
[math]x^{eb/ec}=({\sqrt[ec]{x}})^{eb} = (({\sqrt[ec]{x}})^{e})^b = ({\sqrt[c]{x}})^{b} [/math]
Rewrite the numerator based on [math](ab)^{1/x}=a^{1/x}b^{1/x}[/math]
[eqn]=\sqrt[fc]{x^{fb}x^{ec}}[/eqn]
And perform the final addition:
[eqn]=\sqrt[fc]{x^{fb+ec}}=x^{(fb+ec)/fc}[/eqn]
So by splitting the process into more believable steps we understand why exponent addition holds for rational powers.

Anonymous at Fri, 15 Nov 2024 06:44:08 UTC No. 16475777

Define a positive rational exponent as:
[math]x^{a+b/c} = x^a x^{b/c} = x^a\sqrt[c]{x^b}[/math]

Then we consider exponent addition:
[math]x^{a+b/c}x^{d+e/f}[/math]
First add the whole numbers, no need for proof:
[math]=x^{a+d}x^{b/c}x^{e/f}[/math]
Then get the same denominator:
[math]x^{b/c}x^{e/f}=x^{fb/fc}x^{ec/fc}[/math]
The exponents remain equal during the last step:
[math]x^{eb/ec}=({\sqrt[ec]{x}})^{eb} = (({\sqrt[ec]{x}})^{e})^b = ({\sqrt[c]{x}})^{b} [/math]
Rewrite the numerator based on [eqn](ab)^{1/x}=a^{1/x}b^{1/x}[/eqn]
[eqn]=\sqrt[fc]{x^{fb}x^{ec}}[/eqn]
And perform the final addition:
[eqn]=\sqrt[fc]{x^{fb+ec}}=x^{(fb+ec)/fc}[/eqn]
So by splitting the process into more believable steps we understand why exponent addition holds for rational powers.

🗑️ Anonymous at Fri, 15 Nov 2024 08:06:45 UTC No. 16475814

can someone prove that vectors have a morphism to topologies which is closed under transposition

Anonymous at Fri, 15 Nov 2024 08:07:53 UTC No. 16475815

>>16475814
do your own homework

🗑️ Anonymous at Fri, 15 Nov 2024 08:19:25 UTC No. 16475819

>>16475814
>>16475815

Anonymous at Fri, 15 Nov 2024 11:31:49 UTC No. 16475958

Is there a way to turn the problem of palacking 17 unit squares into a polynomial with 51 terms (17 tuples of x-coord, y-coord, angle) or is it impossible?

Anonymous at Fri, 15 Nov 2024 11:32:26 UTC No. 16475959

>>16475762
Do you guys have recommendations for textbooks on graph theory?

Anonymous at Fri, 15 Nov 2024 12:10:26 UTC No. 16475981

>>16475958
It wouldn't make sense as a degree 51 polynomial but it might make sense as a linear equation of 51 variables.

Anonymous at Fri, 15 Nov 2024 13:03:56 UTC No. 16476030

>>16475775
>>16475777
Cool! Now prove the Recusion Theorem (so that integer exponents always make sense) and the existence of nth roots (from the completeness axiom of the real numbers)

Anonymous at Fri, 15 Nov 2024 13:16:18 UTC No. 16476042

>>16475959
The standard is Dietsel's Graph Theory in GTM. It's the only graph theory textbook I have any real familiarity with, as the other graph books I've read have been focusing on applications (primarily PGM's and Markov Random Fields).

Anonymous at Fri, 15 Nov 2024 13:17:19 UTC No. 16476044

>>16476042
Diestel*

Anonymous at Fri, 15 Nov 2024 13:30:53 UTC No. 16476062

>>16475770
People always post this, but they should point to Misha Verbitsky's Problem Course in Undergraduate Mathematics too.
http://shenme.de/listki/
>This page is home to an English translation a of course that was taught by Misha Verbitsky and Dmitry Kaledin at the Independent University of Moscow in the fall of 2004 (also known under the name “Trivium”). The course was targeted for first-year students, so the prerequisites include only elementary high-school mathematics

Anonymous at Fri, 15 Nov 2024 16:14:43 UTC No. 16476352

>>>>/pol/488535575

What is written on the blackboard?

Anonymous at Fri, 15 Nov 2024 19:26:15 UTC No. 16476645

>>16476352
Idk but there’s a massive clue here if you care to look it up

Anonymous at Fri, 15 Nov 2024 21:49:17 UTC No. 16476909

How is the Suslin hypothesis independent of ZFC if it fails in L? How can a Suslin line cease to be?

Anonymous at Fri, 15 Nov 2024 22:57:20 UTC No. 16476981

>>16476062
is this as much of a meme as the chart you replied to?

Anonymous at Fri, 15 Nov 2024 23:39:49 UTC No. 16477080

>>16476981
>Misha Verbitsky
Is the same guy from the meme list
http://imperium.lenin.ru/~verbit/MATH/programma.html

Anonymous at Fri, 15 Nov 2024 23:46:42 UTC No. 16477101

>>16476042
This wasn't exactly what I was looking for. Turns out I need a textbook on *algebraic* graph theory.

Anonymous at Fri, 15 Nov 2024 23:47:14 UTC No. 16477102

>>16476909
>1. R does not have a least nor a greatest element;
>2.the order on R is dense (between any two distinct elements there is another);
> 3. the order on R is complete, in the sense that every non-empty bounded subset has a supremum > and an infimum; and
> 4. every collection of mutually disjoint non-empty open intervals in R is countable (this is the countable chain condition for the order topology of R),
If you admit more sets (go beyond L), then the suslin line may cease to satisfy 3, because there are more nonempty bounded subsets of the line.
It could also start to fail 4 because there may be more collections of mutually disjoint non-empty open intervals in R, which happen not to be countable.

Anonymous at Sat, 16 Nov 2024 00:35:24 UTC No. 16477194

What are some mathematical systems in which 45=47?

Anonymous at Sat, 16 Nov 2024 00:41:16 UTC No. 16477214

[eqn]x^2+y^2=25[/eqn]
[eqn]2x+2yy'=0[/eqn

whyyyyyyyyyyyyyyyyyyyyy

muh just take da derivative !!

Anonymous at Sat, 16 Nov 2024 00:42:38 UTC No. 16477217

[eqn]x^2+y^2=25[/eqn]
[eqn]2x+2yy'=0[/eqn]

whyyyyyyyyyyyyyyyyyyyyy

muh just take da derivative !!

Anonymous at Sat, 16 Nov 2024 00:47:07 UTC No. 16477226

>>16477217 >>16477214
Are you retardeded?
>>16477194
Arithmetic modulo 2

Jefferson01 at Sat, 16 Nov 2024 01:06:12 UTC No. 16477268

>>16477217
lolllll physics-pilled

Anonymous at Sat, 16 Nov 2024 01:36:43 UTC No. 16477322

Elliptic curves for retards/"undergraduates"?

Anonymous at Sat, 16 Nov 2024 01:37:24 UTC No. 16477324

>>16477322
Silverman and Taint (lol) or nah?

Anonymous at Sat, 16 Nov 2024 01:41:51 UTC No. 16477330

>>16477324
lmao boomer yellow books

Anonymous at Sat, 16 Nov 2024 01:48:17 UTC No. 16477344

Currently reviewing basic geometry on Khan Academy because I have terminal imposter syndrome and I'm never going to be confident in my abilities. My current neurotic obsession is that my fundamentals are mediocre so I have to systematically go through all my high school math to make sure I know it.

Anonymous at Sat, 16 Nov 2024 01:49:49 UTC No. 16477347

>>16477344
Everything in high school geometry is some kind of affine translation.

Anonymous at Sat, 16 Nov 2024 12:23:40 UTC No. 16477722

>>16477194
The field of order 1048576.

Anonymous at Sat, 16 Nov 2024 22:08:52 UTC No. 16478246

>>16477344
Don't forget gelfand

Anonymous at Sun, 17 Nov 2024 23:27:43 UTC No. 16479947

If the universe entirely* consisted of math and you are some angel being able to look at it from an outside perspective, would probability even be a coherent concept?

Anonymous at Mon, 18 Nov 2024 00:11:37 UTC No. 16479997

>>16479947
Yes since probability is just mapping binary outcomes to partitions of (0,1). It would be much less important to them than it is to us, however.

Anonymous at Mon, 18 Nov 2024 00:27:40 UTC No. 16480021

What is the difference between space and a field and R^3?

Anonymous at Mon, 18 Nov 2024 00:30:46 UTC No. 16480027

>>16479947
>>>/x/

Anonymous at Mon, 18 Nov 2024 05:13:04 UTC No. 16480226

>>16480021
>Space
Metric space? Vector space? Just "space" is a rather vague concept, just like saying "number" instead of rational number or complex number.
>Field
Two meanings, one in algebra (ring with nonzero elements invertible under multiplication), the other in vector calculus (vector field, scalar field). Returning to algebra, the geometric line has field operations (real addition and multiplication) and the geometric plane too (complex addition and multiplication). Im not sure, but maybe its not possible to define field operations for points in 3D space (quaternion operations define a 4D algebra that is not commutative under multiplication, therefore not a field)
>R^3
Is an example of vector space. Tangentially, is an example of metric space, measure space and topological space too.
>physics
R^3 is a model for physical space according to classical newtonian physics

Anonymous at Mon, 18 Nov 2024 20:09:33 UTC No. 16481171

What springer books r u guys picking up during this sale?

I'm eyeing some of Pierre Brumaud's older texts, but Im also trying to find a good text on algebraic combinatorics.

Anonymous at Tue, 19 Nov 2024 12:10:10 UTC No. 16482124

>>16481171
Damn some books are almost 75% off. I assume they're trying to get rid of some old inventory.
Anyways, I got the problem books by polya and szego. I've heard good things about them.

Anonymous at Tue, 19 Nov 2024 14:59:15 UTC No. 16482368

I posted a thread about 0^0, hoping that seeing people argue about exponentiation would better help me visualize the riemann surface of x^y and I'm really disappointed by the amount of people in the thread who are instead arguing about multiplication.
Have any of you ever gotten really good at visualising exponential surfaces?

Anonymous at Tue, 19 Nov 2024 20:18:47 UTC No. 16483061

Consider using a single cup for tea, and you never swap the cup.
Instead of completely finishing up the tea, you generally drink until some of the tea ends up cold, leaving a fraction of the tea left. Instead of dumping the tea and replacing it with new cold water, you keep that tea in there while throwing the teabag and placing a new teabag, and then filling up the rest of the cup with new boiling water for the remainder.

Concretely, with raw numbers, as an example you would drink your cup of tea until 20% remains, you will then throw the tea bag and replace it with a new, and place 80% of the cup with new boiling water, meaning 20% is tea and 80% is new boiling water.

If you do this an infinite amount of times, what will the potency of the tea end up being? For any given fraction left undrank, although it is always the same fraction X that is left undrank. Do not consider any physical factors such as evaporation

Anonymous at Tue, 19 Nov 2024 20:26:34 UTC No. 16483080

I was stuck on the logic behind implicit differentiation:
[eqn]x^3+y^3=6xy[/eqn] implies that
[eqn]3x^2+3y^2y'=6xy'+6y[/eqn]

until I rewrite the equation as
[eqn]f(x)=g(x)[/eqn]
so
[eqn]f'(x)=g'(x)[/eqn]

Anonymous at Tue, 19 Nov 2024 20:50:57 UTC No. 16483163

Are math spergs ever bad at humanities or is it always the other way around?

Anonymous at Tue, 19 Nov 2024 21:21:52 UTC No. 16483320

>>16483163
We are good at implementing algorithms in Python

Anonymous at Tue, 19 Nov 2024 21:43:26 UTC No. 16483573

>>16481171
Are they still on sale?

Anonymous at Wed, 20 Nov 2024 02:20:23 UTC No. 16485695

How do i git gud at math again after not using it except for budgeting and percentages for years? I remember exponents, probability and i can read a graph. Biz major.

🗑️ Anonymous at Wed, 20 Nov 2024 02:33:49 UTC No. 16485707

I'm like, borderline retarded, but I decided I want a phd anyway (or maybe because i'm dumb). I loved analysis and topology in undergrad but did terribly in algebra. Would any anons /here/ have some advice for/be able to tutor me through dummit and foote?

Anonymous at Wed, 20 Nov 2024 02:41:28 UTC No. 16485720

>>16483163
I'm an EE, not a math sperg, but in general most of the EE's I know don't really read or pay attention to the humanities. There's two other people in my lab (two of the only white people in the department) who also read and pay attention to the arts, but most EE grad students I know are spending all of their free time on vidya or something instead of reading/music/the arts/etc.

Anonymous at Wed, 20 Nov 2024 02:44:23 UTC No. 16485728

>>16481171
Pierre Bremaud's Fourier Analysis and Probability Theory books are both pretty good and cheap through the UTX series. I'm kind of meh on his mathematics for signal processing book, but it's good in trying to synthesize the applied Fourier analysis used in DSP with Lebesgue integral methods.

My favorite book of Bremaud's is, unfortunately, not on sale for the physical copy. His discrete probability models book is actual gold. It covers basically everything you'd need for discrete probability and even covers Markov Random Fields and a good amount of discrete source information theory. Really good stuff. As far as I'm aware that one is only available through hardcover and isn't on sale because of this though.

Anonymous at Wed, 20 Nov 2024 02:52:30 UTC No. 16485742

im trying to make a lock-in amplifier but i dont know what the fuck a Hilbert space is.
let [math]f(t)[/math] be a pulse signal of some frequency and duty cycle, and [math]g(t)[/math] be a similar pulse signal but smaller in magnitude and with a fuck ton of noise. the idea is to get rid of the noise by multiplying to two signals and integrating over some period.
1. do the integrals over one period for both functions absolutely need to be 0? it makes things easier if not.
2. do the integrals over one period of the square of both functions absolutely need to be 1? it makes things easier if not.

Anonymous at Wed, 20 Nov 2024 03:45:24 UTC No. 16485821

>>16485720
What about you? If an english professor asked you to write a short story with correct grammar that made them feel something could you do it? You have one hour.

Anonymous at Wed, 20 Nov 2024 06:52:16 UTC No. 16485976

>>16475762
x=-1/-1 =1 which maps the function to -1
x=1/1 =1 which maps the function to 1
Your setup maps one x value to two different values in the range, so you actually don’t have a function at all. 1 and -1 are coprime with all integers in case you didn’t know. Try requiring q to be a natural number.

Anonymous at Wed, 20 Nov 2024 10:09:12 UTC No. 16486072

>>16485976
Yeah, the [math] q [/math] needs to be restricted to [math] \mathbb{N} [/math]

Anonymous at Wed, 20 Nov 2024 10:35:37 UTC No. 16486099

why did he do it?

Anonymous at Wed, 20 Nov 2024 11:05:30 UTC No. 16486140

Is it possible to position a unit sphere in R^3 such that it has no rational points?

Anonymous at Wed, 20 Nov 2024 11:23:36 UTC No. 16486163

>>16486140
Consider
(x - pi)^2 + y^2 + z^2 = 1

Anonymous at Wed, 20 Nov 2024 13:40:11 UTC No. 16486329

>>16485821
I'd like to think so? I don't read as much literature as I used to in undergrad, but I still try to read at least one or two works of fiction per semester.

Recently I finished re-reading Zen and the Art of Motorcycle Maintenance (which I first read in HS). I've been going back and forth between starting Shogun and Before the Dawn. Neither are high-art literature but they aren't slop either.

Anonymous at Thu, 21 Nov 2024 00:18:04 UTC No. 16487241

>>16485728
Hm, I'll try and pick it up used.

Anonymous at Thu, 21 Nov 2024 01:14:54 UTC No. 16487312

Why isn't a flattened weierstass function (like 0.01W or sqrtW or 0.01(sqrtW)) lipschitz? It looks like it avoids the forbidden steepness regions of the plane around each point.

Anonymous at Thu, 21 Nov 2024 11:46:56 UTC No. 16487752

>>16483174

Anonymous at Thu, 21 Nov 2024 12:23:38 UTC No. 16487775

>>16487752
Lucy Calkins and her supporters have done tremendous damage to the users of this board.

Anonymous at Thu, 21 Nov 2024 14:12:39 UTC No. 16487865

>>16477101
Ah, yeah that's above my pay grade, unfortunately. In the systems side of EE we tend to pretend algebra (beyond linear algebra) doesn't really exist (which leads to some quite entertaining contortions by researchers when they do actually need group/rings or galois theory and don't want to bother learning it).

Anonymous at Thu, 21 Nov 2024 18:42:09 UTC No. 16488147

>>16483080
youre doing a derivative with respect to a variable, which you've chosen to be x. Think of y as y(x)

Anonymous at Thu, 21 Nov 2024 19:07:16 UTC No. 16488183

>>16485742
[math]
\begin{aligned}
|f|^2 & \leftrightarrow \langle f | f \rangle = \int f^2 \ dt \\
|g|^2 & \leftrightarrow \langle g | g \rangle = \int g^2 \ dt \\
f\cdot g & \leftrightarrow \langle f | g \rangle = \int fg \ dt \\
(f\cdot g)^2 \leq |f|^2 |g|^2 & \leftrightarrow \langle f | g \rangle^2 \leq \langle f | f \rangle \langle g | g \rangle < \infty
\end{aligned}
[/math]

If f is in Hilbert space, then <f|f> is finite, which means <f|g> is finite. Being forced to be equal to one just means the integral is finite. You can just do this normalization after you do all the work. You're working in the space where the integral over a period is also equal to zero, so you can prob assume that g also needs to obey this rule.

Basically, yes to both your questions, but for the latter just force it to equal 1 after you finish the work.

Anonymous at Thu, 21 Nov 2024 19:38:23 UTC No. 16488213

[eqn]x=h(u)[/eqn]
[eqn]\int{f(x)}dx=\int{g(u)}du=F(x)[/eqn]
[eqn]\frac{d}{dx}F(x)=f(x)[/eqn]
[eqn]\frac{d}{du}F(x)=g(u)=\frac{dF}{dx} \frac{dx}{du} = f(x)h'(u)=f(h(u))h'(u)[/eqn]

Substitution logixx

Anonymous at Thu, 21 Nov 2024 19:42:22 UTC No. 16488216

>>16488147
The idea is that the derivative of each side of the equation respect to x are equal if the equations are equal for all x

Anonymous at Thu, 21 Nov 2024 20:09:10 UTC No. 16488254

STEM fags can someone explain why the inside of a sphere isn't hyperbolic?

Anonymous at Thu, 21 Nov 2024 20:27:10 UTC No. 16488279

>>16488254
I mean if the inside was hollow and just the inside surface

Anonymous at Thu, 21 Nov 2024 20:27:59 UTC No. 16488282

>>16488254
What math is that from?

Anonymous at Thu, 21 Nov 2024 21:26:34 UTC No. 16488353

What is the strongest Large Cardinal axiom not known to be inconsistent with ZF alone? I saw someone mention "There exists an elementary substructure [math]V_{\kappa} \prec V[/math]". Can we do better?

Anonymous at Fri, 22 Nov 2024 02:44:04 UTC No. 16488700

What are some elementary lectures notes from the masters? For example
>Logic Lectures: Gödel's Basic Logic Course at Notre Dame
>Number Theory for Beginners - André Weil
>Notes on Introductory Combinatorics - George Pólya
>A Course in Pure Mathematics - G. H. Hardy
>Modern Algebra (based on the lectures of Emmy Noether and Emil Artin) - B. L. van der Waerden

Anonymous at Fri, 22 Nov 2024 06:37:00 UTC No. 16488883

>>16488254
Gaussian curvature doesn't change in 2 dimensions when flipping the orientation (i.e. flipping from outside the spherical surface to inside does nothing). The curvature is equal. -1*-1 = 1*1

Anonymous at Fri, 22 Nov 2024 16:04:37 UTC No. 16489271

Can someone prove this?

Anonymous at Fri, 22 Nov 2024 16:05:07 UTC No. 16489272

Anonymous at Fri, 22 Nov 2024 16:37:10 UTC No. 16489313

>>16488700
https://catalog.hathitrust.org/Record/102330536
>A freshman honors course in calculus and analytic geometry: taught at Princeton University by Emil Artin; notes by G. B. Seligman
According to Serge Lang's preface on his first edition of A First Course on Calculus
>I learned how to teach the present course from Artin, the year I wrote my Doctor's
thesis. I could not have had a better introduction to the subject

Anonymous at Fri, 22 Nov 2024 16:51:32 UTC No. 16489332

>>16476645
>Idk but there’s a massive clue here if you care to look it up

Inside the red circle:
(Ma - Marinesw 2007)
iR (superscript L) > ƐW (subscript X) on X

Alternatively, the 2007 = poot

GOOGLE LENS says the text is
Th (Ma-Marinesce 2007)

Anonymous at Fri, 22 Nov 2024 18:21:50 UTC No. 16489418

Does anyone else listen to different kinds of music when doing different kinds of math?

I listen to metal when I’m doing analysis but algebra feels like Mozart to me

Anonymous at Fri, 22 Nov 2024 19:51:22 UTC No. 16489526

>>16489271
>>16489272
Prove it supposing f is the indicator of a measurable set and then approximate.

Anonymous at Fri, 22 Nov 2024 20:19:03 UTC No. 16489570

>>16488700
>>16489313
Felix Klein's Elementary Mathematics from an Advanced Standpoint

Anonymous at Sat, 23 Nov 2024 03:08:45 UTC No. 16490125

Need some advice fellas. I've taken math up to calc 3 awhile ago, and I'm going back to school and self studying to be an EE. I need to refresh my old math knowledge but I also want to push my math knowledge a little further to get me ready for some of the more advanced EE topics. Would I be jumping the gun if I casually flipped around my old calc textbook and then went straight into an analysis textbook? Or should I start somewhere inbetween?

I'm already brushing up on my precalc

Anonymous at Sat, 23 Nov 2024 04:35:05 UTC No. 16490179

>>16488183
thanks fren <3

Anonymous at Sat, 23 Nov 2024 04:51:27 UTC No. 16490184

>>16490125
You already took some multivariate calculus in the past right? Go through a few problems in Stewart or something and see how you feel about it.

If it's way too easy, start learning some introductory analysis and linear algebra. If it's way too hard, start over at the beginning of the book and review your precalc and early calculus fundamentals before moving into an "advanced calculus" (i.e., intermediary between applied calculus and analysis) book. Either way you should be focusing on learning linear algebra, linear ODE systems and probability as soon as possible. Those are essential skills for modern systems based EE.

I'm almost done with an EE PhD, and if you can get a good grasp of the basics of analysis (up to measure, Fourier analysis, a bit of convex analysis and linear functional analysis) and linear algebra you will basically have everything you need to do phd research level systems work.

Anonymous at Sat, 23 Nov 2024 04:53:59 UTC No. 16490185

>>16489271
That notation is absolutely terrible, but the previous response is correct >>16489526

Start with indicators of measurable sets (remember, g being measurable means that the pre-image of any Borel set of measure greater than zero is also a Borel set of measure greater than zero), and approximate from below using MCT.

Anonymous at Sat, 23 Nov 2024 04:55:44 UTC No. 16490186

no one here is smart enough to solve this

Anonymous at Sat, 23 Nov 2024 05:08:57 UTC No. 16490195

>>16487312
It's because of the way that the Weierstrass functions are defined. At the limit, the cosine is oscillating back and forth between +/- 1 "infinitely quickly" meaning that the cosine part of the season is not maximal only on sets of measure zero, while also only being constant on sets of measure zero.

Anonymous at Sat, 23 Nov 2024 05:10:50 UTC No. 16490197

>>16490195
Series* not season. Late night phone posting, not even once.

Anonymous at Sat, 23 Nov 2024 05:54:24 UTC No. 16490227

Is studying math for an hour every morning a waste of time? Currently doubling back on Calc 1 and 2

Anonymous at Sat, 23 Nov 2024 08:28:21 UTC No. 16490269

Anybody like the number 326?
3 + 8 = 11
3 to 6 = [4, 5]
What else does it have?

Anonymous at Sat, 23 Nov 2024 13:16:04 UTC No. 16490408

>>16490184
>I'm almost done with an EE PhD, and if you can get a good grasp of the basics of analysis (up to measure, Fourier analysis, a bit of convex analysis and linear functional analysis) and linear algebra you will basically have everything you need to do phd research level systems work.

that's bitchin to hear. I actually have already done a linear algebra and diff eq course so that is very encouraging. Thanks for the rec's anon

Anonymous at Sat, 23 Nov 2024 19:40:58 UTC No. 16490767

I wish I would focus on math all day but I need to get my life straightened out first

Anonymous at Sat, 23 Nov 2024 20:03:38 UTC No. 16490796

you know, it makes me kinda sad how math started making sense to me once I started using LLMs to study
as in, it didn't have to be this way until just now
and that math ability isn't some deep genetic or magical trait that only "some" people have

the point is that having an AI that can rephrase things in the ways you understand them literally makes all the difference
and ultimately having your own abstractions is fine since math transcends language and notation

ty for reading mi blog

Anonymous at Sat, 23 Nov 2024 20:08:10 UTC No. 16490801

>>16490796
>math transcends language and notation
Bold statement. Proof?

Anonymous at Sun, 24 Nov 2024 18:01:35 UTC No. 16491748

Call a function f : X -> Y strongly constant iff whenever Y is inhabited then there is a y in Y s.t. f(x) = y for every x in X. Call it weakly constant iff f(x) = f(x') for all x, x' in X.
Clearly a strongly constant function is weakly constant and the converse holds with excluded middle.
Is the converse strong enough to imply excluded middle?

Anonymous at Sun, 24 Nov 2024 18:06:39 UTC No. 16491759

>>16491748
Subtle difference. How do you start to worry about this kind of formalities? Any recommended reading?

Anonymous at Mon, 25 Nov 2024 06:22:54 UTC No. 16492523

rate of change of area under f(x) = [eqn]\lim_{h \to 0}\frac{Area(x+h)-Area(x)}{h}[/eqn]
[eqn]=\lim_{h \to 0}\frac{f(x+c)*h}{h}[/eqn] with [math]0 < c < h[/math]
[eqn]=\lim_{h \to 0}f(x+c)=f(x)[/eqn]

Anonymous at Mon, 25 Nov 2024 06:51:57 UTC No. 16492546

Is it always the case that you can replace any variables with their unbiased estimators in the equation of an unbiased estimator and it will remain unbiased?

Anonymous at Mon, 25 Nov 2024 19:58:18 UTC No. 16493195

>>16491759
I'm not a constructivist or anything like that but I used to work with some of them and their brainrot tends to be infectious (I mean that in an endearing way).
This particular question mostly came to me after reading a discussion about whether [math]\varnothing\to\varnothing[/math] should be considered constant or not.
Unfortunately I don't really have any reading recommendations, sorry.

>>16491748
It seems that most people will answer with the following when asked what it means for [math]f:X\to Y[/math] to be constant:
(1) There is a [math]y\in Y[/math] such that [math]f(x)=y[/math] for every [math]x\in X[/math].

Some might come up with what I called weakly constant in my previous post, i.e.
(4) [math]f(x)=f(x')[/math] for all [math]x,x'\in X[/math].

Now [math]\varnothing\to\varnothing[/math] is constant in the sense of (4) but not constant in the sense of (1) - there is nobody in the codomain that every element in the domain could map to since the codomain is empty. So if one considers [math]\varnothing\to\varnothing[/math] to be constant then (1) isn't the right definition, even in classical mathematics.

Two adjustments could be made to (1) to deal with that edge case (the first one is what I referred to as strongly constant in my previous post):
(2) If [math]Y[/math] is inhabited then [math]f[/math] is constant in the sense of (1),
(3) If [math]X[/math] is inhabited then [math]f[/math] is constant in the sense of (1).
Both definitions consider [math]\varnothing\to\varnothing[/math] to be constant (vacuously).

Now (3) and (4) are equivalent, even constructively, so I'll stop mentioning (3) from here on out. (1) clearly implies (2) while the converse doesn't hold (even classically) due to the [math]\varnothing\to\varnothing[/math] counterexample.

(2) implies (4) constructively and with excluded middle they are even equivalent (this is what I had mentioned in my previous post).

Anonymous at Mon, 25 Nov 2024 20:29:16 UTC No. 16493232

>>16493195
Hence the only remaining question is whether the (4) -> (2) direction actually requires excluded middle, i.e. whether it doesn't hold constructively. An even stronger thing to ask would be whether that direction is even powerful enough to prove excluded middle, and apparently it is.

Letting [math]X=\{\star\mid P\}[/math] for any proposition [math]P[/math], setting [math]Y=\{0\mid P\}\cup\{1\}[/math] and letting [math]f:\{\star\mid P\}\to \{0\mid P\}\cup\{1\}[/math] be the (necessarily unique) function such that [math]f(x)=0[/math] for every [math]x\in \{\star\mid P\}[/math].
Clearly [math]f[/math] is constant in the sense of (4), so by assumption [math]f[/math] should be constant in the sense of (2). But its codomain is inhabited (since [math]1\in Y[/math]), so it by definition of (2) it should even be constant in the sense of (1).
Hence there is some [math]y\in\{0\mid P\}\cup\{1\}[/math] such that [math]f(x)=y[/math] for every [math]x\in\{\star\mid P\}[/math]. In the case where [math]y\in\{0\mid P\}[/math] we have that [math]P[/math] holds and so [math]P\lor\neg P[/math] holds too.
In the case that [math]y\in\{1\}[/math] we have that [math]y=1[/math] and I claim that [math]\neg P[/math] must hold (again implying [math]P\lor\neg P[/math], which would prove excluded middle in either case).
So aiming for a contradiction, suppose that [math]P[/math] holds. Then [math]\star\in\{\star\mid P\}[/math] and so [math]f(\star)=0[/math] by definition of [math]f[/math]. But we also know that [math]f(x)=y[/math] for every [math]x\in\{\star\mid P\}[/math] and so in particularly [math]f(\star)=y[/math], but we know that [math]y=1[/math] in the case we're currently considering, hence [math]f(\star)=1[/math]. Summing up we have that both [math]f(\star)=0[/math] and that [math]f(\star)=1[/math], hence [math]0=1[/math], contradiction.

Anonymous at Mon, 25 Nov 2024 23:51:14 UTC No. 16493479

When was the last time you really needed to use algebra

Anonymous at Tue, 26 Nov 2024 01:33:12 UTC No. 16493574

>>16492523
Very simple proof of the Fundamental Theorem of Calculus, nice work

Anonymous at Tue, 26 Nov 2024 05:10:16 UTC No. 16493747

>>16476062
Doesn't load thanks for hacking.

Anonymous at Tue, 26 Nov 2024 11:09:12 UTC No. 16493926

>>16476062
>>16493747
I don't know why it just stopped working. Too much traffic from this post? You can still access the archive https://web.archive.org/web/20240407182046/https://shenme.de/listki/

Anonymous at Tue, 26 Nov 2024 11:40:02 UTC No. 16493941

How would you explain to the curious 12 years old that infimum of the empty set is positive infinity?

Anonymous at Tue, 26 Nov 2024 11:42:46 UTC No. 16493945

>>16493941
Sketchy academia cunt

Anonymous at Tue, 26 Nov 2024 11:45:48 UTC No. 16493948

>>16493941
The infimum of a set is its greatest lower bound.
Since the empty set has no elements, any number is a lower bound. The greatest lower bound is just the greatest number, which has to be infinite since if it was finite you could just add 1 and get a greater number

Anonymous at Tue, 26 Nov 2024 11:52:55 UTC No. 16493957

>>16475770
>>16476062

Anonymous at Tue, 26 Nov 2024 11:55:17 UTC No. 16493959

>>16493957
V. Arnol'd also has a trivium but for applied math/physics students:
https://physics.montana.edu/avorontsov/teaching/problemoftheweek/

Anonymous at Tue, 26 Nov 2024 12:41:35 UTC No. 16493979

what would be your advice to someone who likes differential geometry, cryptography and set/model theory in the same intensity but doesn't like topology at all?

Anonymous at Tue, 26 Nov 2024 16:43:59 UTC No. 16494207

I heard that every rational function with only positive integer exponents had elementary antiderivatives, so I plugged one where the polynomial has nonsolvable roots into wolfram.
It gave me the result in the image, but how do I interpret wolfram's notation (circled in blue and red)?
.
I figure the red circle is just how wolfram represents nonsolvable roots but I can't find a name for this notation and every search for little omega brings up results from comp sci.
I don't know what the blue circle depicts at all. How do you have a 3-ary logarithm? There's no subscripts here to indicate that it's some kind of hypergeometric series and I don't know of any other 3-ary functions with square brackets that come up in integration.

Anonymous at Tue, 26 Nov 2024 16:45:33 UTC No. 16494209

>>16494207
I forgot the image, here it is

Anonymous at Tue, 26 Nov 2024 21:17:29 UTC No. 16494482

>>16493979
Force yourself to like topology. Actually, you don't have to force yourself, the love for topology must come to you naturallly after you study the proper authors/approaches. If you hated abstract algebra instead, much less could be done

Anonymous at Tue, 26 Nov 2024 21:48:32 UTC No. 16494533

Haven't been here for several years really.
Anything new?

Anonymous at Tue, 26 Nov 2024 22:18:39 UTC No. 16494566

>>16494533
Tom Leinster's course on structural set theory (all the constructions are isomorphism invariant or something like that) https://golem.ph.utexas.edu/category/2024/11/axiomatic_set_theory_10_cardin.html

Anonymous at Wed, 27 Nov 2024 00:32:30 UTC No. 16494714

>>16493979
von neumann didnt love topology, and he did fine. Stick to what you like

Anonymous at Wed, 27 Nov 2024 01:00:39 UTC No. 16494735

Can anyone recommend me a Calculus book from the 70's or 80's that doesn't have all those gaudy graphics found in newer undergrad textbooks? Those pop-up ad-like graphics in the margins are unnecessary and rub my 'tism the wrong way. Not actually autistic, I just think that reading through a textbook should be enjoyable, and the presence of such eyesores pisses me off.

It needs to be fairly comprehensive regarding everything you'd need for a standard undergrad Calculus course--up to the sequence of Calculus 2 is fine, but for it to include Multivariable would be preferable for the sake of maintaining consistency in explanations. Also, I'd like an inexpensive physical hardcover--otherwise I'd just resort to downloading one of the many antiquated books found in the charts frequently posted. Furthermore, I'd like for it to include a decent number of exercises of a decent variety--that is to say, problem sets that don't include 10 questions on one manner of calculation, and then move onto another; I'd prefer problem sets that cover one concept per question (or two) for the sake of efficiency. The latter is more of a particular preference for the sake of time management rather than a steadfast requirement. Lastly, it'd need to be "just on the precipice of rigour"; it'd need to be just a step (or two) above something like Stewart's; it'd need to be a book intended for Math majors, not for a general audience, something that would serve as a stepping stone for Analysis later down the line. I say "on the precipice of rigour" instead of requiring that it be rigorous since I'm self studying the topic, and I'd like the explanations to be not all that terse--just barely at the point where I can make sense of any proofs with a reasonable amount of effort.

To give context on my Mathematical background, I've read through McKeague's Trigonometry and Sullivan's College Algebra, and have taken a course on Calculus 1.

>tl;dr recommend me a semi-rigorous oldfag Calc book.

Anonymous at Wed, 27 Nov 2024 01:02:01 UTC No. 16494736

>>16494735
Having written all of that, I realize that it would have been better to just ask what book those of you who are Math majors used to learn Calculus and then look through them to see which meet my requirements, and which I can easily get my hands on for cheap.

Anonymous at Wed, 27 Nov 2024 01:58:02 UTC No. 16494755

>>16494533
Since 2017 old trip and avatar fags left and were replaced, the sticky link host died and was inadequately replaced, there's now also a shitty questions general, the mathjax implementation is completely unchanged and not updated, the average /mg/ thread has gone from 25% unanswered questions by randos to 40%, some avatarfags made their own discord and left, some avatarfags made their own imageboard (mathchan) and left, hating on category theory died out, wilderburger's youtube was deleted and ultrafinitism escaped containment.

Anonymous at Wed, 27 Nov 2024 01:59:29 UTC No. 16494757

>>16494735
>>16494736
Hava a look at these, very very terse:
>>16491764

Anonymous at Wed, 27 Nov 2024 02:10:08 UTC No. 16494761

>>16494735
Tom Apostol's Calculus is from 1967 and doesn't bombard you with modern hi-res graphics. There are some figures when necessary, but it's pretty terse and to the point for an introductory calculus text.

Anonymous at Wed, 27 Nov 2024 02:25:20 UTC No. 16494768

Wildburger was so comfy to watch,

Anonymous at Wed, 27 Nov 2024 03:06:09 UTC No. 16494816

When you project a cube to a plane figure, what is the term for all of the points of the cube that are projected to the same point of the figure? I'm talking about like the intersection of the cube and a specific ray.

Anonymous at Wed, 27 Nov 2024 03:22:46 UTC No. 16494825

>>16494816
In general terms, it's the image of that ray in the cube. The ray is the pre-image of that point, and the image of the ray is the point.

Anonymous at Wed, 27 Nov 2024 06:46:06 UTC No. 16494924

Why did nobody tell me earlier statistics can glitch money out of people

Anonymous at Wed, 27 Nov 2024 08:03:58 UTC No. 16494965

>>16494755
damn, really sad to hear desu

>>16494566
I vividly remember reading the small Leinster book in a cafe in Göttingen in 2012. I remember there's a photo of me holding the book up, with a butterfly on my cheek. Must have been Fasching 2013. Can't find it though.
I had a phase like everybody, I even programmed some Idris, I enjoyed the HoTT drama ofc. I still like topoi, but I'm not into alg. top/geom and so it never felt like that whole corner sold itself proper enough. I will come back to it when I'm old(er).

Anonymous at Wed, 27 Nov 2024 14:48:45 UTC No. 16495134

>>16494533
Alan U. Kennington had a quintuple coronary artery bypass grafting surgery earlier this year and is now working through 3000 pages of neurophysiology books http://www.topology.org/tex/conc/diary.php

Anonymous at Wed, 27 Nov 2024 17:06:28 UTC No. 16495253

>>16475762
>instagram reel on math being best subject
>engineers malding in comments
Why are they like this?

Anonymous at Wed, 27 Nov 2024 17:13:36 UTC No. 16495257

>>16495253
>inst*gr*m
You need to go back. (And don't return.)

Anonymous at Wed, 27 Nov 2024 17:14:43 UTC No. 16495258

>>16495257
I have been here since 2009

Anonymous at Wed, 27 Nov 2024 17:16:48 UTC No. 16495262

>>16495258
That explains the stench.

Anonymous at Wed, 27 Nov 2024 18:05:01 UTC No. 16495301

>>16495134
What's your take on the diff geo book?
Probably not the first on my list..

Anonymous at Wed, 27 Nov 2024 18:47:52 UTC No. 16495344

do topologists really?

Anonymous at Wed, 27 Nov 2024 22:03:02 UTC No. 16495525

>>16495344
Yes, we do. And we give "find an example where all 14 regions are different" as a homework to 1st year students.

Anonymous at Thu, 28 Nov 2024 07:20:29 UTC No. 16495982

>>16494757
First one looks to be a short pamphlet list of definitions and theorems. second one looks more like the manuscript of the book a professor I know is putting together and using to teach his class--though it's frustratingly terse and somewhat difficult to read. Also, I asked for them to be not all that terse lole.
>>16494761
I've heard of it before and seen it included in the various chart threads--most of which I assume are either memes (for the most part) or slightly overkill. From what I've heard it's comparable to Spivak's Calculus but slightly more forgiving. Unfortunately I can't get my hands on it easily nor for cheap. Pirating it is always an option, but I'd prefer to have a physical (hardcover) copy to write notes in the margins for the sake of posterity. What's a book that would be used in an honors college Calculus course with emphasis on rigour? Would Spivak work, or would that be too much of a struggle? I've seen a few copies that are (relatively) inexpensive compared to other texts which are held in high regard.

Anonymous at Thu, 28 Nov 2024 07:24:45 UTC No. 16495985

i enjoy

Anonymous at Thu, 28 Nov 2024 07:47:07 UTC No. 16495999

>>16495982
Ok, then read the preface to Serge Lang's Short Calculus and see this post:
>>16489313

Anonymous at Thu, 28 Nov 2024 10:27:33 UTC No. 16496053

The Math Sorcerer was right, old books do smell heavenly; they have a vanilla-like scent. It's only with certain types of paper however, some of them smell like ASS.

Anonymous at Thu, 28 Nov 2024 14:31:21 UTC No. 16496185

>>16496053
the ones that smell like ass are probably moldy as shit

Anonymous at Thu, 28 Nov 2024 16:16:06 UTC No. 16496268

Some quintics are solvable by radicals and some are not. I at best vaguely remember the group theory I did, but some stuff from galois lets us identify which side a given quintic falls into.
How big are the two sides? Are almost all quintics (with complex algebraic coefficients) solvable? Unsolvable? Are they both dense in the set of all quintics?

Anonymous at Thu, 28 Nov 2024 16:40:10 UTC No. 16496293

>>16495301
Good reference for beginners thinking they want complete rigour from the ground up and making them realize if that is what they really want

Anonymous at Thu, 28 Nov 2024 17:08:16 UTC No. 16496330

What is this diagram tying to show? I found it in my scratch folder dated June 9

Anonymous at Thu, 28 Nov 2024 18:49:43 UTC No. 16496435

>>16496185
I think it has to do with the type of paper. I think the type of paper modern textbooks use--the easily torn kind--are prone to smelling like ass after a while. Doesn't negate the fact that the reason for why they smell like ass is because they're prone to mold.

Older books have really nice thick, quality paper. What's the point in making students pay out the ass and handing them a shitty-made textbook?

Also, slight tangent, but I hate when people scribble in their textbooks or write nonsense in them. Notes and signatures are fine and I consider part of the history of the book, but doodles and markings are retarded.

Anonymous at Thu, 28 Nov 2024 21:26:17 UTC No. 16496550

The random variable
max(X_1, X_2)
in [0,1], sampled by generating two values in [0,1], equals that of X_1^2

Does that of min(X_1, X_2) also equal a polynomial like that?

Anonymous at Fri, 29 Nov 2024 04:48:20 UTC No. 16496875

>>16495982
My university used Apostol for "advanced calculus" (basically just calculus but with epsilon-delta proofs and some basic introduction to set theoretic argumentation). With that said, all of the homework assignments were written by the professor (as far as I can tell) and were not exercises from Apostol's book.

PDFs of it are as easily accessible. It is kind of a pain to get your hands on, but vol 1 covers all the way through a "calc 2" course in an advanced calculus sequence, alongside some linear algebra basics. Volume 2 does multivariate calculus and ODE's along with a bit more linear algebra and some probability.

Anonymous at Fri, 29 Nov 2024 04:50:57 UTC No. 16496876

>>16495982
I realized I forgot to answer your Spivak question in >>16496875

Yes, you can use Spivak. Spivak is also good, but a little harder than Apostol and (in my opinion, what do I know?) not as accessible for self study. Spivak and Apostol are basically "the standards" for a proof based calculus sequence, but Spivak usually requires a better professor for guidance.

Anonymous at Fri, 29 Nov 2024 06:47:28 UTC No. 16496933

Is there any way that .99999... doesn't equal 1

Anonymous at Fri, 29 Nov 2024 08:11:36 UTC No. 16496997

>>16496550
I assume you mean to say that the CDF of min(X_1, X_2) (call it G) is a polynomial of the CDF of X_1 (F), namely G=F^2.
This requires the rvs being iid, but not uniform per se.
Then also the CDF of min(X_1, X_2), call it H, has H=1-(1-F)^2.

Anonymous at Fri, 29 Nov 2024 09:31:22 UTC No. 16497024

>>16496933
When the base isn't 10.

Anonymous at Fri, 29 Nov 2024 11:56:31 UTC No. 16497100

>>16496933
https://arxiv.org/abs/1007.3018

Anonymous at Sat, 30 Nov 2024 08:27:37 UTC No. 16498067

"The domain of [math](f\circ g )(x)[/math] consists of the numbers [math]x[/math] in the domain of [math]g[/math] for which [math]g(x)[/math] lies in the domain of [math]f[/math]".

[math]D((f\circ g)(x))=\{x\in D(g):Range(g)\in D(f)\}[/math]
Is dat a correct way of writing out the above statement using quantifiers?

Anonymous at Sat, 30 Nov 2024 09:22:56 UTC No. 16498084

I need to speedrun group theory in about three weeks.
Should I go with Artin or Herstein? Personally speaking, I'd like to go with Artin but I'm having trouble with FOMO since Herstein is the "classic" text that everybody uses.

Anonymous at Sat, 30 Nov 2024 18:11:12 UTC No. 16498388

>>16498084
the answer is always Lang

Anonymous at Sat, 30 Nov 2024 22:43:25 UTC No. 16498645

Favorite mathematical reasoning/intro to proofs books?

Anonymous at Sat, 30 Nov 2024 23:05:29 UTC No. 16498664

>>16498645
>Textbooks
Journey into Mathematics: An Introduction to Proofs - Joseph J. Rotman
Proof, Logic, and Conjecture: The Mathematician's Toolbox - Robert S. Wolf
Alice in Numberland: A Students’ Guide to the Enjoyment of Higher Mathematics - John Baylis, Rod Haggarty
>Lecture Notes
A Primer for Logic and Proof - Holly P. Hirst and Jeffry L. Hirst
http://www.appstate.edu/~hirstjl/primer/hirst.pdf
Modicum Mathematicum: A Swath Through The Basic Language Of Abstract Math - Paolo Aluffi
https://math.hawaii.edu/~pavel/Aluffi_notes_321_Modicum.pdf
Proof, Sets, and Logic - M. Randall Holmes
https://randall-holmes.github.io/proofsetslogic.pdf
Basic Concepts of Mathematics - Elias Zakon
http://www.trillia.com/zakon1.html

Anonymous at Sat, 30 Nov 2024 23:42:31 UTC No. 16498689

>>16498645
You should find a professor at your institution that wrote his own violently context-deficient introductory proofwriting book and pay $100 so it can filter you anyways, and then stew over it for years until you've done all of the exercises and they mean nothing to you anymore and you're a sordid husk and the light has left your eyes

wait. Maybe you shouldn't do this. try the book pictured maybe.

Anonymous at Sun, 1 Dec 2024 05:50:17 UTC No. 16498919

>>16498645
Proofs and reasoning are best absorbed on-the-job in my opinion. Try picking up a linear algebra book (perhaps Axler's book) and work through that. IMO the real purpose of intro linear is to get you more used to "real" mathematics. The content itself is fuckin easy, the hard part is being precise about it.

Anonymous at Sun, 1 Dec 2024 08:20:14 UTC No. 16499010

>>16498645
Calculus by Spivak

Anonymous at Sun, 1 Dec 2024 13:09:05 UTC No. 16499165

I'm trying to wrap my head around something. Suppose we have some sequence of reals [math] x_{n}[/math] that tends to some value [math] x[/math]. Then [math] lim_{n}1_{]-\infty,x_{n}]}=1_{]-\infty,x]}[/math] almost everywhere, except at the point [math] x[/math].
Why is it the case that this is equality almost everywhere? Is it that by definition of the limit, every point of [math] 1_{]-\infty,x]} [/math] other than the point at [math] x[/math] is accounted for?

🗑️ Anonymous at Sun, 1 Dec 2024 16:45:54 UTC No. 16499318

>>16498664
Thanks for the recs anons. Morash's book gets some good reviews, and hirst & hirst's primer looks very concise, I like that.

>>16498689
>>16498919
I was akshully thinking of picking up Amann and Escher's Analysis series after brushing up on my basic calc, and it looks like chapter 1 covers a lot of the kinds of stuff those math reasoning books do, ex sets, building the real number system, etc. I think I'll go through either hirsts' notes above or the first half of marash's before moving on to the big league once I get there. Maybe touch up on my ODE's a bit. Thanks again, boys.

🗑️ Anonymous at Sun, 1 Dec 2024 16:47:46 UTC No. 16499320

>>16499318
>>16498689
meant to quote you too

Anonymous at Sun, 1 Dec 2024 16:50:57 UTC No. 16499322

>>16498664
>>16498689
Thanks for the recs anons. Morash's book gets some good reviews, and hirst & hirst's primer looks very concise, I like that.

>>16499010
>>16498919
I was akshully thinking of picking up Amann and Escher's Analysis series after brushing up on my basic calc, and it looks like chapter 1 covers a lot of the kinds of stuff those math reasoning books do, ex sets, building the real number system, etc. I think I'll go through either hirsts' notes above or the first half of marash's before moving on to the big league once I get there. Maybe touch up on my ODE's a bit. Thanks again, boys.

Anonymous at Sun, 1 Dec 2024 17:10:02 UTC No. 16499335

>open book
>do all problems at the end of each chapter
>get new book
>repeat
am i self studying right

Anonymous at Sun, 1 Dec 2024 18:20:18 UTC No. 16499385

>>16499335
Take notes

Anonymous at Mon, 2 Dec 2024 14:51:19 UTC No. 16500207

>>16500204

https://www.youtube.com/watch?v=7oWip00iXbo

What kind of math is needed to describe this for someone who took just the basics (up to Cal 3, DiffyQ, basic probabilities and linear algebra)

tl;dw: indivisible non-markovian stochastic processes

Anonymous at Mon, 2 Dec 2024 17:16:53 UTC No. 16500330

>>16500207
Start here https://arxiv.org/pdf/2302.10778
You're not going to get any of the inner workings by watching the video.

Anonymous at Tue, 3 Dec 2024 05:01:18 UTC No. 16500929

>>16498084
Lang isn't possible please shill either Artin or Herstein to me

Anonymous at Tue, 3 Dec 2024 05:21:13 UTC No. 16500947

>>16499322
> Amann & Escher

"GERMANS could be here" he thought. "I've never been in this neighborhood before. There could be GERMANS anywhere."

Anonymous at Tue, 3 Dec 2024 10:34:51 UTC No. 16501084

>>16494207
>>16494209
It's just a sum over all complex omega which are roots of the polynomial. You could derive it yourself from partial fraction decomposition (although you'd need to show the polynomial has no repeated roots before factoring into linear denominators)
"Expanded logarithm notation" is WolframAlpha autism. It doesn't seem like anyone knows what that is, even MathWorld. Or WolframAlpha itself when you copypaste the "Wolfram language plaintext output".

Anonymous at Tue, 3 Dec 2024 11:47:46 UTC No. 16501109

>>16501084
That expanded logarithm thing looks like how mathematica writes sums of logarithms. It's possible that in Wolfram autism, the expression
[math]\{log [-1+x]\}\{ \frac{1}{79} , \frac{1}{4} \}[/math]
actually means
[math] log \left( \frac{-78}{79} \right) + log \left( \frac{-3}{4} \right) [/math]
It's also possible that I've got the wrong idea. Type out the full sum into wolfram and see if it lines up with the numerical answer wolfram gives for the initial integral.

Anonymous at Wed, 4 Dec 2024 16:40:52 UTC No. 16502194

I'm 23 and I've never done research or solved and unsolved problem. I saw a high school student on twitter talking about how his paper will be published next week. Suicide fuel.

Anonymous at Wed, 4 Dec 2024 16:52:29 UTC No. 16502215

>>16502194
if it makes you feel any better, it's almost certainly a nothingburger that got blown out of proportion by people who don't understand what they're talking about, akin to the FIRST TIME ANYONE EVER PROVED PYTHAGORAS IN 2000 YEARS

Anonymous at Wed, 4 Dec 2024 18:08:16 UTC No. 16502348

>>16502194
I'm 26 and have never done research.

Anonymous at Wed, 4 Dec 2024 18:20:41 UTC No. 16502384

Why did I fail math in 3rd year?
calc 1 2 3
stats
liner algebra
numeric methods (finite difference, elements etc.)

it was all smooth going, regular wageslave work got me thr B's and A's but then
complex analysis
partial differential equarions and banach spaces and shit
real analysis
set theory shit
markov chains
optimization

I couldnt even do the fucking excercises. Immediately I got Es, Ds and fails.

Anonymous at Wed, 4 Dec 2024 18:27:55 UTC No. 16502401

>>16502384
Did you study or just wing it after skimming the book the day before?

Anonymous at Wed, 4 Dec 2024 18:43:22 UTC No. 16502436

>>16502401
I studied at the same pace as the last 2 years the first month or so but I kept failing and not understanding every god damn concept, theory and excercise

Anonymous at Wed, 4 Dec 2024 23:42:08 UTC No. 16502923

I read an edgy italian scifi comedy about an alien who comes to Earth and studies humanity and the character said a thing about math that has stuck with me. Here is my best translation:
>As many as one in two hundred humans are born with noticeably enhanced brainpower, called nerds. They are faced with a difficult problem, however, in that they are only adequate at interpreting other humans' facial expressions. Since humans can only move between their religious castes by exhibiting notably good or poor abilities at social conspiracies, however, the nerds cannot form a single governing body.
>Nerds are intelligent enough to understand that their own efforts are futile, so they all participate in the best equivalent to wireheading that their technology permits: pondering analytical formulas for measures of obscure geometric or algebraic objects, even when their society already has robust theories for effective numerical integration of all instances of algebras over monads.
Was he right? Is every analytical problem after the invention of computers just masturbation?

Anonymous at Thu, 5 Dec 2024 00:18:29 UTC No. 16502968

>>16502923
>Was he right? Is every analytical problem after the invention of computers just masturbation?
No. For an example, this problem whose solution is a triple of 80-digit numbers. which can't be found with a computer using brute force search:
https://www.quora.com/How-do-you-find-the-positive-integer-solutions-to-frac-x-y+z-+-frac-y-z+x-+-frac-z-x+y-4
If what he said was right, there would be no state secrets https://en.wikipedia.org/wiki/NSA_cryptography

Anonymous at Thu, 5 Dec 2024 03:17:55 UTC No. 16503159

>>16502923
> Is every analytical problem after the invention of computers just masturbation?

Absolutely not. Firstly, computational limitations will make it so that you're almost always better off putting some analytical muscle into getting your expressions into a more "computation friendly" formulation. Thinks like numerical stability, floating point precision limitations, and poor time-cost scaling for integration over large spaces all make it worth your while to learn and practice mathematics even in the era of computers.

Secondly, there's an almost endless supply of mathematics that is not computational in nature (and would be unlikely to be aided in meaningful ways by computers). Really anything that involves any non-trivial set mathematics will be essentially impossible to meaningfully implement on a computer. Consider, as an example, trying to do something as "simple" as a martingale filtration via realizations of a real random variable. The sigma algebra generated by your the realized samples will be quite difficult to meaningfully express computationally, and would require a serious exploration of coding/information theory to properly represent at all (let alone implement the next filtration recursion). That's a simple example of an "applied" mathematics discipline that has real world utility (calculating the minimum size event space from previous observations) that computers are essentially useless for outside of very specific circumstances.

When you need to do something more abstract (e.g., find the best smooth and bounded variation function f which minimizes some cost functional J), you're going to be completely SOL if you are relying on solely the computer to do your math for you.

Anonymous at Thu, 5 Dec 2024 12:17:43 UTC No. 16503481

>>16502194

> do nothing
> somebody else does something
> uhh guys this is suicide fuel

Anonymous at Thu, 5 Dec 2024 17:24:26 UTC No. 16503733

>first semester grad school Topology (Munkres)
>haven't done a single assignment before the day it was due
I am simply built different

Anonymous at Thu, 5 Dec 2024 18:29:29 UTC No. 16503868

>>16503733
They started with Munkres? I like Munkres but my school started with Lee's topological manifolds book and used Munkres as an "undergrad reference."

Anonymous at Thu, 5 Dec 2024 19:02:07 UTC No. 16503984

>>16503868
I wanna kill myself so bad
I live in such a shithole

Anonymous at Thu, 5 Dec 2024 21:27:16 UTC No. 16504220

I don't care if you're a PhD developing the newest computational automorphisms on stacks, if you're struggling with depression you ought to stay out of /mg/, this isn't a thread for diary posting

Anonymous at Fri, 6 Dec 2024 07:52:57 UTC No. 16504633

>>16503984
Please don't. I'm sorry you don't like where you live/are from. If you feel like you aren't getting enough because of where you're from, just take a deep breath and try to snap out of it. People can follow all sorts of life paths after all sorts of universities, and if you do good work people will barely even think about where your degree is from.

If it's just grad school depression, I understand that in spades. I've been getting my ass kicked by a problem that I probably could have solved a while ago if I just faced the music and spent time learning random matrix theory.

Anonymous at Fri, 6 Dec 2024 20:22:01 UTC No. 16505126

Approximately four years ago I saw a youtube comment where a carpenter turned programmer was saying that they need to use algebraic geometry and differential geometry to program a CNC machine to make certain wooden parts of expensive staircase handrails, because the act of making a banister profile smoothly follow a curve in three dimensions to the specifications of some clients was a complicated mathematical problem.
They went on to say that they needed the math to make the CNC software but that subsequent carpenters would be able to just input the requirements into the machine and never have to learn anything beyond precalculus.

The argument vaguely made sense but I have never heard of anything to corroborate it since. Was the whole thing made up or are there really staircases where the shapes of some of the components can't be made without university math?

Anonymous at Fri, 6 Dec 2024 20:30:11 UTC No. 16505137

>>16505126
Probably just typical programmer moment where they use big words to sound more fancy. I needed to solve a system of polynomial equations to do a thing? I did a le hecking algebraic geometry.

Anonymous at Fri, 6 Dec 2024 23:58:51 UTC No. 16505293

Are there any good proofs in linear algebra that rely on the IVT? I've only ever seen the IVT used in analysis and topology.

Anonymous at Sat, 7 Dec 2024 02:10:48 UTC No. 16505397

>>16505293
Maybe that some real matrix has eigenvalue if its characteristic polynomial is a polynomial of odd degree

Anonymous at Sat, 7 Dec 2024 02:28:19 UTC No. 16505410

>>16483163
Usually the opposite. At my uni, Math & Physics guys hung out with the Phil and Lit guys
>>16485720
CS/CE/EEs are one track minds uncultured nerds
>>16486329
>Neither are high-art literature but they aren't slop either.
Those are all slop

Anonymous at Sat, 7 Dec 2024 04:31:25 UTC No. 16505504

>>16505126
If the closed figure is embeddable in R^2, made up of finitely many algebraic curve pieces (eg lines, quartics, circle sectors) joined at angles which are algebraic multiples of pi and it does not intersect, the boundary of the solid produced by smoothly transporting it along a smooth curve in R^n is numerically computable by combining linear algebra and the newton method. You don't need anything more advanced.

Anonymous at Sat, 7 Dec 2024 04:53:10 UTC No. 16505533

>>16505504
Not true. You can construct turing machines with only the limited set of concepts you use.

Anonymous at Sat, 7 Dec 2024 05:01:56 UTC No. 16505537

>>16505410
Before the Dawn is not slop. Get out of here.

Anonymous at Sat, 7 Dec 2024 18:28:28 UTC No. 16505983

Are there any almost continuous functions that are Lebesgue integrable but not absolute (improper) Riemann integrable or vice versa?

Anonymous at Sat, 7 Dec 2024 19:08:44 UTC No. 16506011

>>16505983
for the latter, all riemann integrable -> lebesque integrable.

Anonymous at Sat, 7 Dec 2024 20:00:17 UTC No. 16506088

>>16477194
you mean "number system"?
mod 2?

Anonymous at Sat, 7 Dec 2024 20:01:48 UTC No. 16506092

>>16477217
lol wut?

Anonymous at Sun, 8 Dec 2024 00:59:23 UTC No. 16506313

Yo what are hypergeometric series? They came up on a thing I put into Wolfram but wikipedia does a bad job of explaining them.

Anonymous at Sun, 8 Dec 2024 01:34:03 UTC No. 16506328

>>16505983
Every Riemann integrable function is Lebesgue integrable. That is not true in the other direction. There are many Lebesgue integrable functions are not Riemann integrable.

Anonymous at Sun, 8 Dec 2024 03:14:40 UTC No. 16506374

Putnam Competition is over!
anyone ITT take it?
im thinking about B2, which asks about quadralaterals:
if you have quadralateral ABCD and take the diagonal AC then reflect D across the perpendicular bisector of AC (call it E) then ABCD and ABCE are called partners.
is there an infinite sequence of quadralaterals a_k s.t. a_n and a_{n+1} are partners and there does not exist any i≠j where a_i is congruent to a_j

this problem seemed too easy for me, and i fear i got it wrong. heres the sketch of my solution:
call |AB|=a |BC|=b |CD|=c and |DA|=d
note that CD and CE are congruent. and AD and AE are congruent, so each quadralateral that is a partner shares all the same side lengths a,b,c,d, albeit in a different order. the number of orderings of a,b,c,d is 4!, which is finite, so any infinite sequence must have multiples of some ordering.

Anonymous at Sun, 8 Dec 2024 09:10:42 UTC No. 16506542

>>16489313
This doesn't have any exercises.

Anonymous at Sun, 8 Dec 2024 13:00:35 UTC No. 16506609

>>16506542
Lang has exercises. Read Artin + Lang (first edition)

Anonymous at Sun, 8 Dec 2024 13:01:05 UTC No. 16506610

How do you prove that any endomorphism of a f.g. module over an arbitrary ring satisfies a monic polynomial, without using determinants and adjugate matrices?

Anonymous at Sun, 8 Dec 2024 13:31:22 UTC No. 16506627

>>16475762
in a flat circular coordinate system, can I use pi radians to do trigonometry? if so, how?
are pi radians and plain radians (already multiplied by pi) different in this regard?

Anonymous at Sun, 8 Dec 2024 14:42:45 UTC No. 16506672

>>16506609
Why first edition?

Anonymous at Sun, 8 Dec 2024 15:00:10 UTC No. 16506689

>>16506672
https://link.springer.com/book/10.1007/978-1-4613-0077-9

Anonymous at Sun, 8 Dec 2024 16:35:25 UTC No. 16506807

Hey math enthusiasts. Let I={(0,n) in R^2: -1<=n<=1} and A={(x,sinx) in R^2: x>0}. I need to prove that A union I is connected but not path connected. Any ideas? Or a counterexample would be greatly appreciated.

Anonymous at Sun, 8 Dec 2024 16:49:58 UTC No. 16506827

>>16506807
It's path connected. You made an error writing down the problem.

The correct version of the problem is solved here.
https://proofwiki.org/wiki/Closed_Topologist%27s_Sine_Curve_is_not_Path-Connected

Anonymous at Sun, 8 Dec 2024 16:52:21 UTC No. 16506833

>>16506807
As stated your set is both connected and path connected. In the definition of A, you probably meant to write (x, sin1/x) which is what I'm going to assume.
First note that both I and A are connected and path connected.
>connected
Take a continuous function X -> {0,1}. Suppose w.l.o.g. I maps to zero (I is connected so maps to the same set). Take any point x in I. By continuity, there is an open ball around x that maps to 0. But inside that ball there are points in A, thus A also maps to zero. Thus there is no surjective map X-> {0,1}.
>not path connected.
Cannot connect (0,0) with (1, sin(1)). This is because any path that starts at (0,0) stays in I, because to leave it it would have to jump too much to be continuous.

Anonymous at Sun, 8 Dec 2024 18:00:58 UTC No. 16506900

>>16506827
Why is it path connected? I mean, my prof wrote it like that, using (x,sinx). Maybe he made a mistake.

Anonymous at Sun, 8 Dec 2024 18:10:52 UTC No. 16506909

>>16506900
And why is it connected?

Anonymous at Sun, 8 Dec 2024 19:20:14 UTC No. 16506949

>>16506900
>>16506909
>Continuous functions map path-connected sets to path-connected sets
>[math]\mathbb{R}[/math] is path-connected
>[math]f(x) = (\max(0,x) ,\sin(x))[/math] is continuous and [math]f(\mathbb{R}) = A \cup I[/math]
>Path-connected sets are connected

Anonymous at Sun, 8 Dec 2024 21:03:24 UTC No. 16507063

I'm retarded. Will an Nspire CAS calculator help my limp through a calc 2 class. I roughly remember some stuff but was always kind of shit at math

Anonymous at Sun, 8 Dec 2024 22:17:42 UTC No. 16507124

>>16507063
Plotting and numerically calculating integrals can help, but it won't help you learn unless you are actually committed. Having a calculator that you can use to check definite integrals is quite nice though.

Anonymous at Mon, 9 Dec 2024 09:29:54 UTC No. 16507487

Doesn't the uncountability of Dedekind cuts on the rationals contradict the partition principle?

Anonymous at Mon, 9 Dec 2024 10:44:25 UTC No. 16507529

>>16507487
Dedekind cuts are countable. Any subset of a countable set is countable.

Anonymous at Mon, 9 Dec 2024 12:20:42 UTC No. 16507575

>>16507529
No I mean the set of all Dedekind cuts is uncountable because it constitutes the reals (though the proof relies on an alternative construction iirc).
My argument is this: Start by well-ordering the Dedekind cuts, and mapping each one to the symmetric difference of the A used in its A/B construction from the union of every predecessor under that well-ordering. By the axiom of identity, said difference must always be non-empty, and by definition it must always consist of rationals. It seems to me that by examining the resulting image, we can use the Dedekind cuts to partition the rationals into uncountably many equivalence classes.
My instinct is that this probably means the set of Dedekind cuts is probably not uncountable in the first place, but let me know if I'm misunderstanding something.

Anonymous at Mon, 9 Dec 2024 13:01:21 UTC No. 16507597

>>16507575
Following up on this, I'm not a finitist and I accept the transfinite hierarchy/Cantor's Theorem, I'm just not convinced that Dedekind cuts capture the same notion of real numbers that the powerset of the naturals does. I know it's relatively consistent with ZF without Choice that an infinite set can be partitioned into more equivalence classes than it has elements, and that Dedekind reals are only equivalent to Cauchy reals for example if you assume Countable Choice (though the fact that ZFC is equiconsistent with ZF is somewhat disturbing).

Anonymous at Mon, 9 Dec 2024 15:39:24 UTC No. 16507717

>>16507575
>>16507597
Not disturbing at all. The concepts "set", "belongs" are undefined, which mean they dont have an unique interpretation, but many, in fact infinitely many interpratations (one of them is compatible with ZFC, other with ZF+~C). Think of "position" in physics and how it has multiple interpretations as well (classical, relativity, quantum).

Anonymous at Mon, 9 Dec 2024 16:18:59 UTC No. 16507763

>>16506011
>>16506328
This is only true for proper Riemann integrable functions. There are improper Riemann integrable functions that are not Lebesgue integrable like x l--> sin x / x, but it is not absolutely Riemann integrable.

Anonymous at Mon, 9 Dec 2024 16:26:13 UTC No. 16507776

>>16507763
sin(x)/x is Lebesgue integrable. It is only discontinuous at the single point x=0, which is of measure zero on the real line.

Anonymous at Mon, 9 Dec 2024 16:29:48 UTC No. 16507780

>>16507776
It has positive and negative parts of infinite integral, which means you'd get oo - oo.

Anonymous at Mon, 9 Dec 2024 16:30:31 UTC No. 16507781

>>16507763
Ignore >>16507776

I was being dumb. Yes, you're right. It isn't Lebesgue integrable. It's been a while since I've actually used any of this stuff and I should probably brush up on the basics again.

Anonymous at Mon, 9 Dec 2024 17:06:48 UTC No. 16507821

>>16507575
What axiom of identity?

Anonymous at Mon, 9 Dec 2024 17:32:19 UTC No. 16507864

>>16507717
Oh no, I understand incompleteness just fine. What makes it disturbing is that Choice feels outright inconsistent to me, even though I understand that it holds in L so it can't be.
>>16507821
Extensionality, sorry. My idea being that the only way the image of one of the cuts could be empty is if we repeated a set.

Anonymous at Mon, 9 Dec 2024 17:43:31 UTC No. 16507881

>>16507575
>and mapping each one to the symmetric difference of the A used in its A/B construction from the union of every predecessor under that well-ordering.
Ok so in a lot of cases you just get the complement of A.
>By the axiom of identity, said difference must always be non-empty
Doesn't follow. It could be empty.
>It seems to me that by examining the resulting image, we can use the Dedekind cuts to partition the rationals into uncountably many equivalence classes.
You can't.

Anonymous at Mon, 9 Dec 2024 18:19:04 UTC No. 16507901

>>16507881
>Ok so in a lot of cases you just get the complement of A.
A is always bounded from the right, so the images must be too.
>Doesn't follow. It could be empty.
There are infinitely many rationals between any 2 reals. This is fine for other kinds of reals because rationals don't extend to infinite sequences, and I can kind of intuit almost all of the reals being "at the bottom" of the Stern-Brocot tree, but with the Dedekind reals my intuition is that even if there's no greatest element to them, there's still a greatest open neighborhood preserving the total order.

Anonymous at Mon, 9 Dec 2024 18:33:56 UTC No. 16507914

>>16507864
Does Chaitin's constant or any uncomputable number (or, farther away, any undefinable number) feel inconsistent to you?

Anonymous at Mon, 9 Dec 2024 19:21:22 UTC No. 16507972

Anonymous at Mon, 9 Dec 2024 19:21:47 UTC No. 16507973

>>16507914
No, undefinable reals make perfect sense to me. I'm somewhere between a Platonist and coherentist.

Anonymous at Mon, 9 Dec 2024 19:50:23 UTC No. 16508012

>>16503984

Anonymous at Mon, 9 Dec 2024 21:32:03 UTC No. 16508132

>>16475762
How do I get into math? I'm 30 and I just barely know what a number is, is there a book that teaches retards like me from zero?

Anonymous at Mon, 9 Dec 2024 22:07:16 UTC No. 16508148

>>16508132
https://archive.org/details/basic-mathematics-serge-lang_20240418/mode/1up

Anonymous at Mon, 9 Dec 2024 22:16:57 UTC No. 16508159

>>16508132
You don't need to know "what a number is" because it is an undefined concept. Btw, see:
>>16498664
>>16488700
>>16489043
>>16495659

Anonymous at Tue, 10 Dec 2024 00:38:41 UTC No. 16508256

Got accepted to Umich and Columbia for grad school. Applied math. I don't really give a shit about some autistic concentration in a field, I'm just here to go through the motions and into industry right after. Maybe some day teach at some SLAC.
Online rankings are misleading. Where is best for a chill young 30 year old guy like myself.

Anonymous at Tue, 10 Dec 2024 01:00:14 UTC No. 16508274

>>16508256
Bro, you sound insufferable. Are you sure you have a clue what you're getting into and you actually want to do grad school? It seems like you actually don't give a shit and are just doing it because you think it will lead to a cushy job. That's a really shit reason to do a PhD.

Anonymous at Tue, 10 Dec 2024 01:53:13 UTC No. 16508320

>>16508256
Try to figure out who your advisor would be
If they're Indian avoid at all costs

Anonymous at Tue, 10 Dec 2024 02:07:24 UTC No. 16508333

>>16508256
UMich is a great department academically, but it's not an ivy and doesn't have comparable dick length to an ivy in terms of connections/clout
The only general advantage of Ann Arbor is that (IMO) it is a much nicer place to live than New York

Other stuff comes down to a personal judgment based on supervisors and your research interests and shit like that

Anonymous at Tue, 10 Dec 2024 04:44:11 UTC No. 16508437

>>16508274
>Bro, you sound insufferable. Are you sure you have a clue what you're getting into and you actually want to do grad school?
Nothing else to do idk.
>cushy job
I'm an only child and my parents are loaded. If they died now I would make more more dividends than what I would from a normal salary. I don' even know what that means, it's just what my dad told me.

I like math and want to live at a nice school. I did my undergrad at uconn and I don't want to live in such a rural place anymore. I focused on SDE's/PDE's. I didn't do any interviews I just applied to a few schools and these two were the highest ranked ones in a place where it snows that accepted me.

Anonymous at Tue, 10 Dec 2024 18:21:47 UTC No. 16508946

>>16508437
Congrats on getting into your schools. I don't want you thinking I'm shitting on you for that. UConn is a pretty solid school for math and engineering as well, but I don't blame you for not wanting to do grad school there. One of my favorite probabilists researches there (Ido Ben-ari).

Regarding the second part, that's good for you (genuinely) and you're in a position where you really can afford to do a PhD just because you want to try. That's not a bad thing, just know that this being a good reason to start one doesn't necessarily mean it's a great reason to finish one.

Umich is a great stem school with a really strong robotics program which funds a lot of the other research they do. Columbia is just a good school all around. You'll do fine at either if you can really find a research area that motivates you enough to deal with the negative parts of grad school life. You just need to find an area you give enough of a shit about that the frustrating bits of academic research don't drive you mad. Neither are bad choices if you can find an analyst/probabilist professor whos research speaks to you.

Anonymous at Tue, 10 Dec 2024 18:33:36 UTC No. 16508959

Is this a good list?

How to Prove It by Daniel J. Velleman

Discrete Mathematics and Its Applications by Kenneth H. Rosen

Introduction to Graph Theory by Douglas B. West

Elementary Number Theory by David M. Burton


Set Theory and Logic by Robert R. Stoll

Linear Algebra Done Right by Sheldon Axler

Introduction to the Theory of Computation by Michael Sipser


Probability and Statistics by Morris H. DeGroot

Combinatorics: Topics, Techniques, Algorithms by Peter J. Cameron

Abstract Algebra by Dummit and Foote

Principles of Mathematical Analysis by Walter Rudin

Topology by James R. Munkres

---

Numerical Analysis by Richard L. Burden

Computational Linear Algebra by Trefethen and Bau

Introduction to Algorithms by Cormen et al.

Categories for the Working Mathematician by Saunders Mac Lane

Convex Optimization by Boyd and Vandenberghe

Partial Differential Equations by Lawrence C. Evans


Algebraic Topology by Allen Hatcher

Introduction to Smooth Manifolds by John M. Lee

Nonlinear Dynamics and Chaos by Steven Strogatz


Information Theory, Inference, and Learning Algorithms by David J.C. MacKay

Game Theory: An Introduction by Steven Tadelis

Stochastic Calculus for Finance by Steven E. Shreve

Anonymous at Tue, 10 Dec 2024 19:48:58 UTC No. 16509033

>>16494566
This is really good :0

Anonymous at Tue, 10 Dec 2024 19:48:59 UTC No. 16509034

>>16508959
Anyone who creates a long book list before starting to study will abandon their plan before finishing the first book.

Anonymous at Tue, 10 Dec 2024 19:50:04 UTC No. 16509035

>>16509034
Speak for yourself, anon. I'm trying to study as concurrently as I can

Anonymous at Tue, 10 Dec 2024 19:58:06 UTC No. 16509043

>>16509035
he can speak for everyone who has ever posted on this board
sitting there with a book next to you and deciding instead to plan what you're going to read 3 years and 6000 pages from now is a pretty clear sign that you like the idea of studying math a hell of a lot more than you actually like studying math

Anonymous at Tue, 10 Dec 2024 19:59:20 UTC No. 16509046

>>16509043
:<
Blah blah blah
And yet you all obsess over old images of lists

Anonymous at Tue, 10 Dec 2024 20:01:49 UTC No. 16509049

>>16509046
misha's list wouldn't have become a tenured meme if it was a reasonable thing that people actually did anon

Anonymous at Tue, 10 Dec 2024 20:08:54 UTC No. 16509057

>>16508959
If you really need "How to Prove it" you wont be able to follow Axler, Dummit, Rudin or Munkres in the next year, let alone the grad school topics like category theory, PDE or algebraic topology. If you want a taste of advacend topics, you are better off reading something like Evan Chen Infinitely Large Napkin or replacing the advanced books with frienlier ones. For example, replace Categories fo the Working Mathematician (are you a working, i. e., professional mathematician?) with Lawvere's Conceptual Mathematics. Replace Rudin with Ross' Elementary Analysis. Replace Dummit and Foote with Pinter's A Book of Abstract Algebra and so on.

Anonymous at Tue, 10 Dec 2024 21:01:01 UTC No. 16509103

>>16509057
Thx I was thinking it might be a bit out of order I'll just keep chugging

Anonymous at Wed, 11 Dec 2024 00:33:40 UTC No. 16509330

Greetings anons. Have a question regarding real analysis. If a metric space X is separable, is a subset of X separable too?

Anonymous at Wed, 11 Dec 2024 02:52:58 UTC No. 16509428

>>16509330
If X is separable, it has separable sub-spaces. Not every sub-space will be separable.

Consider the interval [0,1] and the euclidean metric. This is obviously dense in the rationals. Now consider the subset {1/2, 1/4}. This subset of [0,1] is clearly not dense in the rationals (and this has no countable dense subsets).

Anonymous at Wed, 11 Dec 2024 04:23:38 UTC No. 16509484

>>16509428
But isnt [0,1] dense in itself? Does it need to be a proper subset? Thanks in advance

Anonymous at Wed, 11 Dec 2024 04:33:24 UTC No. 16509496

>>16508959
Imo this is a horrible list. You spend too much time on trivial shit, things are out of order and the difficulty curve is wack. Here are my recommendations for what I consider core material:
If you struggle with proofs then unironically work through
>Basic Mathematics, Lang
By your list I assume you know some calculus, so read the massively underrated
>Infinitesimal Calculus, Dieudonne
For linear algebra, assuming you know matrix multiplication (if you don't then read the relevant parts of Lang), read chapters 1-6 of
>Linear algebra and applications, Lax
For abstract algebra I like chapters 1-4 of
>Basic Algebra I, Jacobson
For analysis Rudin is fine but torturous, so consider instead
>Analysis I, Tao
If you've managed to work through the books mentioned adove, you'll be well on your way to tackle any area of mathematics.
Good luck!

Anonymous at Wed, 11 Dec 2024 05:53:35 UTC No. 16509554

>>16509496
>>Infinitesimal Calculus - Jean Dieudonné
>This book treats constructive methods of real and complex analysis at the advanced calculus level and beyond. A preliminary Chapter 0 contains a brief review of concepts assumed known to the reader, such as sets, functions, real and complex numbers, continuity, derivatives, primitives, and topology of the plane.
>Chapter I, entitled ``Majorer, minorer'', deals with inequalities related to real and complex numbers, vector and matrix norms, series and integrals. Chapter II discusses methods for calculating numerical approximations to the roots of an equation f(x)=0 [...] Chapter III treats asymptotic relations with applications to series, integrals, and implicit functions. The method of Laplace and the method of stationary phase are developed in Chapter IV in connection with real integrals depending on a parameter. Chapter V discusses uniform convergence of sequences of functions and includes the Weierstrass polynomial approximation theorem.
>The theory of analytic functions is developed in Chapter VI using power series as the point of departure [...] Chapter VII continues this theme with the Cauchy theory of complex integration. Singular points and the residue calculus, as well as the local and global problems of inversion of analytic functions, are discussed in Chapter VIII. [...] Chapter IX applies complex analysis to various approximation problems. It treats contour integrals depending on a parameter [...]
>Chapter X returns to the general theory of analytic functions and discusses conformal mapping [...]
>The last five chapters of the book are devoted to differential equations. [.....] The last chapter deals with Bessel functions in the complex domain.
>Reviewer: Apostol, T. M.
The title of that book is deceiving. I think it assumes proficiency with at least the first volume of Dieudonne's Treatise of Modern Analysis, which is of grad school level

Anonymous at Wed, 11 Dec 2024 06:28:10 UTC No. 16509580

>>16509554
He states in the preface that some of the proofs can be skipped, as the students can learn them later from Foundations (they are even in small print!). This is meant explicitly as an introductory course in "hard" analysis.>>16509554
>>16509554

Anonymous at Wed, 11 Dec 2024 06:52:04 UTC No. 16509587

>>16509580
>An introductory course in hard analysis
Then the order of your list around that item doesn't make sense because Tao's is meant as an introductory course in "easy" analysis:
>>16509496

Anonymous at Wed, 11 Dec 2024 07:19:29 UTC No. 16509600

>>16509587
“Hard” as opposed to “soft”, that is, manipulating inequalities instead of topological arguments.
https://terrytao.wordpress.com/2007/05/23/soft-analysis-hard-analysis-and-the-finite-convergence-principle/
The book is available online, you can just read the preface.

Anonymous at Wed, 11 Dec 2024 12:40:29 UTC No. 16509731

/mg/bros, what are your thoughts on doing a masters? I don't have the time to do a PhD but still want to do more classes and school math, and my GI bill is still fresh and unused.
It seems the general consensus on the internet is that masters are all cash cows and waste of time and money unless you're an international student. For my case, it would only be a "waste" of time.
While there are other more lucrative degree programs, I had the same dilemma for undergrad and chose math anyways, and I immediately got hired for a 95k/yr job, so I'm not in need of a career transition or anything like that.
Oh, and I was also considering doing it online, as there aren't many schools near me.

Anonymous at Thu, 12 Dec 2024 09:24:26 UTC No. 16510512

>conjecture:

Anonymous at Thu, 12 Dec 2024 10:27:17 UTC No. 16510535

Is there a superset of combinational logic and sequential logic? Yeah the latter has states and order of operations but is there an overarching theory of logic that formalizes the distinction between the two? I don't know enough about mathematical foundations to know where to look.

Anonymous at Thu, 12 Dec 2024 17:18:28 UTC No. 16510765

>The following are equivalent:
>1. A subset [math]E\subset X[/math] of a metric space [math]X[/math] is compact iff it is closed and bounded
>2. For all [math]p\in X[/math] and [math]\rho>0[/math], finite [math]\rho[/math], [math]\overline{B_\rho}(p)[/math] is compact
>3. For some [math]p\in X[/math], every [math]\overline{B_\rho}(p)[/math] with finite radius is compact
The implications [math]1\implies 2\implies 3[/math] seem very straightforward. I'm having trouble doing [math]3\implies 1[/math].
I can assume 3 holds and [math]E[/math] is closed and bounded, and then deduce it's also compact. But I'm having difficulty doing the proof assuming that it's compact and deducing it's also closed and bounded.
Any help is appreciated.

Anonymous at Thu, 12 Dec 2024 17:48:23 UTC No. 16510791

>>16510765
A compact set in a metric space is always closed and bounded. You don't need 3 for that.

Anonymous at Thu, 12 Dec 2024 19:48:13 UTC No. 16510906

>>16510791
A compact set in a metric space is always closed, but is it bounded necessarily?

Anonymous at Thu, 12 Dec 2024 20:08:21 UTC No. 16510927

>>16510906
Yes, it's pretty easy to find an open cover of an unbounded subset with no finite subcover

Anonymous at Thu, 12 Dec 2024 20:10:02 UTC No. 16510931

>>16510906
Yes, any cover with open balls of size one has infinitely many elements.

Anonymous at Fri, 13 Dec 2024 00:44:32 UTC No. 16511193

What's the Statistics term for when causal/correlative content is "baked into" a variable that also has other stuff? Like, y=f(x), f(x)=a(x)+b(x), with only b(x) actually having the correlation and a(x) just being unrelated nonsense?

Anonymous at Fri, 13 Dec 2024 01:56:50 UTC No. 16511238

I have always struggled with understanding what trig functions really are. Like. What are they really?
We have sure

a^2+b^2=c^2
Where c =1 the hypotenuse of a unit circle

But there are no algebraic numbers for this, so I get that we use convergent infinite sums for a,b and these are the typical series for sin and cos
And we have sin^2+cos^2=1
But this is first the squares of two Taylor series, and second, in terms of rads, but we do it directly with side lengths a,b => x,y?
Maybe as something like pa^2+qb^2 and solve for all values of pa^2, qb^2 between +-sqrt2?

And but so in the special case of

2a^2=1
We can solve for a=sqrt2/2

And then use a fraction continuation and have an actual algebraic/non transcendent solution

Ofc then we want to figure out the other values, preferably without using series, which I presume is somehow impossible? Because this is fundamentally about the rate of change of a triangle whose hypotenuse is the radius of a unit circle, so these sides a,b are fundamentally being computed in terms of pi, a transcendental number?

Anonymous at Fri, 13 Dec 2024 02:21:04 UTC No. 16511259

>>16511193
I don't think your question really makes a lot of sense. In terms of functions of many random variables, you could have uncorrelated factors (meaning their covariance value would be 0), but if you are looking at a single function of a random variable x, the only way a(x) wouldn't contribute is if it is zero or near zero in magnitude at x.

Are you looking for functions of many random variables, or a complicated function of a single random variable with many different contributing sub-functions (like a mixture model)?

Anonymous at Fri, 13 Dec 2024 03:24:02 UTC No. 16511277

>>16511238
I have two books for you: The Non- Algebraic Elementary Functions - André Yandl
A Modern Intro to Math Analysis - Alessandro Fonda (Chapter 5).

Anonymous at Fri, 13 Dec 2024 08:54:11 UTC No. 16511468

I was really hoping the number of goldbach pairs per even would exceed the growth of evens at extremely large numbers
But it doesn't
:(

Anonymous at Fri, 13 Dec 2024 10:14:39 UTC No. 16511514

Anyone else hate when you're doing a test and someone lets out a loud sigh to signal to everyone how hard this test is how bad they're doing, like stfu and stop being a pussy, I'm trying to concentrate here

Anonymous at Fri, 13 Dec 2024 11:48:42 UTC No. 16511550

>>16510535
you can view the one as being contained inside the other. to try and address your wuestion: a good way of distinguishing the two could be in terms of their behavior. with the one kind of logic, I can keep pressing the same button and see output:
>True, False, True, False,…
This is not possible with just combinatory logic.

Anonymous at Fri, 13 Dec 2024 11:55:26 UTC No. 16511553

>>16510765
This depends on what definition of “compact” you are using. Is it the open-cover-finite-subcover one? In that case here is a hint. Assume A is unbounded, and make an open covfefe with no finite subcovfefe.

Super dupet hint, do not peek: you can make the open covfefe consist of balls with the same center point

Anonymous at Fri, 13 Dec 2024 13:43:38 UTC No. 16511592

>>16509330
yeah it will be. Suppose X is separable and call its countable dense subspace A. Let Y be any subspace of X; our task is to make a countable dense subspace B of Y.

For every a in A and every size 1/n, choose b_a,n to be a point in Y within distance 1/n of a; if there is no such b, that’s fine, we just don’t define any b_a,n. Let B be the set of all these b_a,n thay we did define.

Now we can prove density as follows. Start with any y in Y and any epsilon =1/n. find a in A within 1/2n of y. find b in B within 1/2n of a. This b is guaranteed to exist because we already know a possible value that could have been picked in the construction of B, namely y. Now b is within epsilon of y.

Anonymous at Fri, 13 Dec 2024 15:26:38 UTC No. 16511657

Any help with this random NT question?
x_0 = a/b coprime naturals, x_(n+1) = x_n * floor(x_n),
given x_0>0, prove/disprove that for some x_k, the sequence becomes an integer

https://math.stackexchange.com/questions/5010779/proving-a-recursive-rational-sequence-will-eventually-become-integral

lowercase sage !!IaxlA1xvEP/ at Fri, 13 Dec 2024 15:42:51 UTC No. 16511671

>>16511657
x0=3/2
x1=3/2*floor(3/2)=3/2*1=3/2
x2=3/2*floor(3/2)=3/2*1=3/2
As you can easily see, sequence doesn't have to become an integer.

Anonymous at Fri, 13 Dec 2024 15:51:42 UTC No. 16511674

Why doesn't the existence of the set with [math]\omega[/math] many parentheses around the empty set (i.e. {{{•••{}•••}}} ) contradict Regularity? It's an element of [math]V_{\omega+1}[/math] right?

Anonymous at Fri, 13 Dec 2024 15:57:23 UTC No. 16511677

>>16511592
Not to be pedantic, but I don't think it's right to say that every subset of X (under the same metric topology) will be dense. Consider X being the classic interval [0,1]. The empty set is a subset of X, and certainly isn't dense.

Anonymous at Fri, 13 Dec 2024 17:15:35 UTC No. 16511737

>>16511671
shoot, forgot to include the condition x_0>2, it's in the SE post though

Anonymous at Fri, 13 Dec 2024 17:51:32 UTC No. 16511756

>>16511674
It does (in a sense) and no, respectively. That set only exists in models with non-standard [math]\omega[/math]. Thus the descending chain looks finite "from inside the model", even though it's infinite from the outside.

Anonymous at Fri, 13 Dec 2024 17:57:13 UTC No. 16511760

You should be able to solve this

Anonymous at Fri, 13 Dec 2024 18:17:21 UTC No. 16511770

>>16511760
[eqn]\mu(\{ x \in X : |g_n(x) - g(x)| \geq \varepsilon \}) \leq \mu(\{ x \in X : |g_n(x) - f_n(x)| \geq \varepsilon/3 \}) + \mu(\{ x \in X : |f_n(x) - f(x)| \geq \varepsilon/3 \}) + \mu(\{ x \in X : |f(x) - g(x)| \geq \varepsilon/3 \}) = \mu(\{ x \in X : |f_n(x) - f(x)| \geq \varepsilon/3 \}) \to 0[/eqn]

Anonymous at Fri, 13 Dec 2024 22:49:32 UTC No. 16512064

I am reading this paper
https://arxiv.org/abs/math/0209257
Titled
"Primary Decomposition: Compatibility, Independence and Linear Growth" by Yongwei Yao. I am stuck in proof of theorem 2.2. Why does it suffice to prove that if X independent, then if for any arbitrary P in X, and P' in Ass(M), P' \subset P implies that P' is in X.
This implies X is open in Ass(M) under Zariski topology

And in next paragraph of proof of theorem 2.2, he says:
"In this case to
prove that X is stable under specialization is simply to prove that X = Ass(M)"
Why this works
I cant seem to understand this proof at all
Any help

Anonymous at Sat, 14 Dec 2024 01:05:15 UTC No. 16512193

Ever wonder what joins sums and multiplication?

X + X = X * X
1. Combine like terms:
2X = X^2
2. Rearrange the equation to get a quadratic equation:
X^2 - 2X = 0
3. Factor the equation:
X(X - 2) = 0
4. Set each factor to zero and solve for X:
X = 0
X - 2 = 0 => X = 2
Therefore, the solutions to the equation are X = 0 and X = 2.
Which equals 2^(1/1)

How about X+X+X = X*X*X ?
3X = X^3
Square root of 3 = √3, and x = -√3. = 3^(1/2)
1.73205080756888

How about X+X+X+X = X*X*X*X ?
4X = X^4
Answer cube root of 4 = ∛4 = 4^(1/3)
1.5874010519682

Seeing a pattern?
Let's try 5X=X^5
Answer: the 4th root of 5 = 5^(1/4)
1.49534878122122

6X=X^6
6^(1/5) or the fifth root of six
1.43096908110526

The equation is basically X^(1/(X-1)).

Anonymous at Sat, 14 Dec 2024 01:38:42 UTC No. 16512226

a^2 + b^2 = 1

Anonymous at Sat, 14 Dec 2024 01:45:12 UTC No. 16512230

>>16511760
Isn't this just a Markov's inequality argument using Fatou's lemma (or MCT if you really want to)? I guess you could also use a DCT argument also because of the almost everywhere equality.

Anonymous at Sat, 14 Dec 2024 02:00:32 UTC No. 16512240

>>16512226
u good

Anonymous at Sat, 14 Dec 2024 16:33:20 UTC No. 16512772

>>16511903
number theory anons how to solve this?

Anonymous at Sat, 14 Dec 2024 16:34:07 UTC No. 16512773

>>16511677
Why is that?

Anonymous at Sat, 14 Dec 2024 16:42:49 UTC No. 16512790

>>16512772
You can quickly find some upper bounds for x and y.
>x can't be bigger than 69006555934236.
>y can't be bigger than 51987524491004.

Since they are both integers and without loss of generality non-negative you only have finitely many cases to check.
69006555934236 * 51987524491004 is about 3.6 * 10^27.
Since computers can do like 10^10 operations per second it should get done quickly by brute force.

Anonymous at Sat, 14 Dec 2024 17:51:15 UTC No. 16512860

Chemist with painfully little knowledge about statistics here: how do you distinguish a very slow exponential growth from linear growth?

Anonymous at Sat, 14 Dec 2024 17:55:18 UTC No. 16512865

>>16512860
Compare graphs, and calculate O(f(n)) for the "slow exponential growth"

Anonymous at Sat, 14 Dec 2024 20:36:41 UTC No. 16513053

>>16512860
Assume it's exponential and plot the inverse exponential of your Y values. If it comes out as log of exponential = linear, then the original data is prob exponential. If it comes out looking like log of linear, then the original data is prob linear.

If you're expecting something like either Ae^(Bx) + C vs Ax+B, then do the inverse according to that instead of Ae^x vs Ax

Anonymous at Sat, 14 Dec 2024 20:46:08 UTC No. 16513070

>>16512773
Any set that is finite cannot be dense by definition. The question is whether every subset of a separable space X will be separable. If you take a finite length/cardinality subset of a dense set, it cannot be dense. One heuristic for density is the idea of "between two elements there is always another element in the set" and this cannot be true if the subset is finite.

So basically, any separable space will have separable sub-spaces (e.g , the interval [1/4, 1/2] is a separable subset of the interval [0, 1] with the standard Euclidean metric). However not every subset of your separable space need be separable because finite subsets are guaranteed to exist and finite subsets cannot be dense.

Anonymous at Sat, 14 Dec 2024 20:49:50 UTC No. 16513079

>>16513070
A finite subspace still has a countable dense subspace though

Anonymous at Sat, 14 Dec 2024 20:57:50 UTC No. 16513093

>>16513070
>Any set that is finite cannot be dense by definition.
Bullshit. Every topological space (even finite ones) is a dense subset of itself.

Anonymous at Sat, 14 Dec 2024 22:48:47 UTC No. 16513193

>>16513079
>>16513093
After looking into it some more, I'm now just confused. I thought I understood this stuff, but evidently I don't.

Anonymous at Sun, 15 Dec 2024 00:01:18 UTC No. 16513251

Just came here to say that Sovolev-spaces are cool, that's all.

Anonymous at Sun, 15 Dec 2024 00:25:27 UTC No. 16513259

>>16475762
>Thomae's function
Is the proof of the continuity for this function at [math]x\in\mathbb{R}\setminus\mathbb{Q}[/math] hard? I've just seen a very simple nonstandard analysis proof

Anonymous at Sun, 15 Dec 2024 01:11:57 UTC No. 16513282

"If 2Z and 3Z were isomorphic, then Z/2Z and Z/3Z would be isomorphic too. But |Z/2Z| = 2 and |Z/3Z| = 3, therefore 2Z and 3Z are not isomorphic."

Is this argument correct?

Anonymous at Sun, 15 Dec 2024 01:37:25 UTC No. 16513310

>>16509731
Masters are fine. I don't know who on the Internet is telling you they're not. They probably are assuming you're getting a master's in something pure vs applied.

I think genuinely people who say it's a scam were never going to get work to begin with.

My suggestion is, if your work is paying for it, look at online programs JHU, Columbia, and UPenn. All of them have very strong online programs with cohorts of people who are ACTIVELY working in industry, even through online coursework I expanded my network immensely through JHU. You quickly start to realize that the majority of people in applied industries are really bad at math (I'm talking about most data science, ML, engineering, trading, risk, etc. shops) and you quickly have a network of people in industry roles across every gamit you can think of (I was in courses and helping a guy who worked for casinos in Vegas as an example) who are GENUINELY trying to understand the math they have in front of them. The majority of them came from undergrads or other masters and realized there is more nuance to all of this than they were led to believe, but they genuinely do not have the time for a PhD and aren't going to stop their career for it.

Overall, I really recommend never stopping with university education. You don't necessarily need the courses, textbooks are enough, but if your work is paying for you to take dedicated programs online, get the certification, build out your network.

Anonymous at Sun, 15 Dec 2024 01:50:34 UTC No. 16513320

>>16512240
Show me the radical product of sinpi/257

Anonymous at Sun, 15 Dec 2024 02:39:12 UTC No. 16513350

>>16513259
>Is the proof of the continuity for this function at x∈R∖Q hard?
Nah, you just need to show that f(x) < 1/n sufficiently close to x which is not hard at all

Anonymous at Sun, 15 Dec 2024 02:40:42 UTC No. 16513351

>>16513282
No, and 2Z and 3Z are isomorphic. It's a correct argument for showing
>if A, A' are isomorphic subgroups of G then G/A is isomorphic to G/A'
is not always true

Anonymous at Sun, 15 Dec 2024 02:49:51 UTC No. 16513356

>>16513351
My bad, I was talking about RING isomorphisms. I know the case does not hold since if a homomorphism f: 2Z -> 3Z were to exist, then

f(2)^2 = f(2*2) = f(4) = f(2+2) = 2f(2)

which would imply that either f(2) = 2 (clearly impossible) or f(2) = 0, in which case f(x) = 0 for every x in 2Z (also impossible).

I'm asking if this >>16513282 argument applies. I don't know how to do a general proof.

Anonymous at Sun, 15 Dec 2024 03:50:01 UTC No. 16513398

Greetings mathematicians.

How does dividing polynomials differ from dividing integers so that I get 2 different results when I do a division?

So I divide (x^3+2)/(x^2+2) and I get the quotient x and remainder 2-2x

But if I do that integer division replacing x with 2002 (for example) I get a different quotient and remainder

Anonymous at Sun, 15 Dec 2024 04:55:51 UTC No. 16513466

sqrt2*pi/7 = pi /5

Anonymous at Sun, 15 Dec 2024 06:09:54 UTC No. 16513510

>>16513398
The condition for what counts as "fully divided" is not compatible between the two.
division algorithm for polynomials: [math]f = qg + r[/math], where [math]0 \leq deg (r) < deg (q)[/math]
division algorithm for integers: [math]a = qb + r[/math], where [math]0 \leq r < q[/math]
Now evaluation at an element is a ring morphism, so if you have [math]f = qg + r[/math] for polynomials, you will have [math]f(x) = q(x)g(x)+r(x)[/math] for any integer [math]x[/math], however it may be that the polynomial [math]f[/math] was fully divided (that is, [math]deg (r) < deg (q)[/math]), but that the integer [math]f(x)[/math] is not fully divided (we need [math]0 \leq r(x) < q(x)[/math]). your example [math]r(2002) = -4002[/math]

Anonymous at Sun, 15 Dec 2024 08:46:56 UTC No. 16513604

>>16511677
the post has 4 sets: X,A,Y are given, then B is constructed. the post did not claim that every subset is a dense subset, of course that would be a silly claim

Anonymous at Sun, 15 Dec 2024 09:06:52 UTC No. 16513606

>>16511674
No, normally that set can’t be constructed. The other anon said something about nonstandard models, but I doubt you were thinking of nonstandard models. think about how exactly you would construct this sets using the sets in V_omega, and you will see it’s not obvious (actually you can’t)
>>16513070
you are confusing two different notions of “dense”
>a subspace A of a topological space X is dense if every nonempty open set of X contains an element of A
>a strict partial order (P,<) is dense if x<z implies there is a z such that x<y<z
these are not the same thing

every topological space X is dense in X. even the empty space is dense in itself

Anonymous at Sun, 15 Dec 2024 10:22:04 UTC No. 16513625

>>16513259
Really easy.

Anonymous at Sun, 15 Dec 2024 16:52:14 UTC No. 16513851

>>16513398
Read carefully the answer of user21820 at
https://math.stackexchange.com/questions/4067510/intuition-for-polynomial-long-division?rq=1
Try to apply the same method to integer division.

Anonymous at Mon, 16 Dec 2024 01:45:30 UTC No. 16514275

>>16513356
Nah I don't think it'll be true in general, you should be able to come up with a ring R and two isomorphic ideals of R with non-isomorphic quotients

Anonymous at Mon, 16 Dec 2024 01:50:51 UTC No. 16514276

>>16513606
I can be a little dense myself sometimes. There's also the idea of a metric space being dense in which a subset A of X is dense if the closure of A is X. I think that's also different than the two you spoke about.

Anonymous at Mon, 16 Dec 2024 14:58:18 UTC No. 16514721

Is there any future for someone who hates abstract algebra?

Anonymous at Mon, 16 Dec 2024 15:03:00 UTC No. 16514727

>>16514721
Yes, topology.

Anonymous at Mon, 16 Dec 2024 15:07:06 UTC No. 16514732

>>16514721
Scientific theories come and go put we will always need people to dig ditches.
>>16514727
How do you do algebraic topology without abstract algebra?

Anonymous at Mon, 16 Dec 2024 15:18:52 UTC No. 16514738

>>16514732
If topology doesn't make you love abstract algebra via algebraic topology, you ngmi.

Anonymous at Mon, 16 Dec 2024 15:31:13 UTC No. 16514752

>>16514721
Yes, you could go into very hardcore areas of analysis. It kind of strips math of any elegance, but we do need people in such fields.

Anonymous at Mon, 16 Dec 2024 17:45:06 UTC No. 16514894

>>16490197
>Late night phone posting, not even once.
kek
waited 900 seconds to post this

Anonymous at Mon, 16 Dec 2024 18:03:22 UTC No. 16514912

>>16514894
based

Anonymous at Mon, 16 Dec 2024 20:25:49 UTC No. 16515052

where's the new bread?

Anonymous at Mon, 16 Dec 2024 20:41:28 UTC No. 16515073

>>16515052
here you go anon :)

Anonymous at Mon, 16 Dec 2024 22:08:50 UTC No. 16515150

>>16514721
Yes it is, actuary

Anonymous at Tue, 17 Dec 2024 03:35:55 UTC No. 16515466

New >>16515464

Anonymous at Tue, 17 Dec 2024 04:20:19 UTC No. 16515515

>>16500947
what mathematics should i study to destroy all germans?

Anonymous at Tue, 17 Dec 2024 10:57:03 UTC No. 16515734

>>16515515
>what mathematics should i study to destroy all germans?
Just divide them by zero.

Anonymous at Tue, 17 Dec 2024 11:43:01 UTC No. 16515767

>>16502194
What is their twitter? I am curious.