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/sci/ - Science & Math

Previous : >>10167486
Grillet's Abstract Algebra is an absolute goldmine edition.
Talk maths.
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Any recommended books for Elementary/Intermediate Algebra? Geometry? Fuck Khan Academy.
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>>10185490
>Elementary/Intermediate Algebra
Herstein, Rotman or Gallian.
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>>10185490
Try "algebra" by lang, i think that's appropriate for your level
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>>10185517
>>10185519
I keep hearing of Gelfand Algebra? No good?
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>>10185522
No idea.
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Help me out here /sci/

I have proven that if integer z is square and (z^2-3z+3) is square then z must be 1.

I want to show that if 1/z is square and (1/z^2 - 3/z + 3) is square then 1/z must be 1. Square here means rational m/n with m and n both square.

This is true. I have proven it by a really ugly separate way using elliptic curves. How can I, in a nicer fashion, get from the second case to the first? Thanks.
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>>10185629
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>>10185629
If z, 1/z and (z^2 -3z+3) are square, then I'd imagine
z 1/z=(z^2 -3z +3)(1/z^2 -3z +3) forces (1/z^2 -3/z+3) to be square.
Alternatively, set x=1/z.
Then again I know fuck all what you're talking about, and your definition of square is circular.
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>>10185648
I briefly defined what it meant for a rational number to be square. That is, it is m/n with m and n square integers in the regular sense.

z 1/z does not equal (z^2-3z+3)(1/z^2-3z+3) and (1/z^2-3/z+3) is not the reciprocal of (z^2-3z+3). If it was then the problem would be trivial.
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Show there is no entire function such that $F(x)=1-exp(2i\pi/x)$ for $1\leq x\leq 2$
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>>10185629
If z is square, by the definition given, 1/z is square.
Substituting 1/z by x, x is square and (x^2-3x+3) imply x is one, which implies x=1=1/z, so z=1.
>>10185655
He never said it was the reciprocal.
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>>10185667
The statement
"z is square and (z^2-3z+3) is square implies z is 1"
has only been proven for integers.

1/z is not an integer
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I cant do math right now, can someone help me.
Whats 2,090 divided by 2 with a difference of 100?
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>>10185658
An entire function is equal to it's Taylor series.
Calculate the Taylor series around the point a=3/2. Then show that this series doesn't converge if z goes to 0.
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>>10185648
>>10185667
To clarify.

The following has been proven:
"Let z be a square integer, with (z^2-3z+3) a square integer, then z=1."

What I want to prove:
"Let z be a square integer, with (1/z^2-3/z+3) a rational number with square numerator and square denominator, then z=1"
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>>10185459
Daily reminder that if you're not getting a PhD in algebraic geometry you are probably a very dull and uninteresting person
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I don't understand the concept of dummy variables in multiple regression. How can your model be accurate if you randomly assign values to different levels?
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>>10185809
They aren't randomly assigned. You assign a dummy variable when you know something is up. For instance, if you were trying to study health outcomes over a period of time which includes a terrorist attack then you need a dummy variable for that attack so the model doesn't ascribe the sudden change to other means.
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>>10185714
hmm okay. My expansion about x=3/2 is pretty ugly, looking at it when z=0 doesnt easily show it to not converge.
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>>10185754
Algebraic geometry is solved. We already have grobner bases.

Also inb4 the anime autists post a bunch of smug anime girls claiming otherwise. Schemes are pure intellectual wankery.
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>>10185754
>alg*bra
>geom*try
>their intersection
if it's not analysis, it's not math. merely tautological masturbation.
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>>10185629
It's been a while since I did any arithmetic but I'll try the second now
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>>10186946
what a lovely handwriting
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>>10185754
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>>10186846
>telling me im retarded for trying to solve every equation and integrate every function i pull out of my ass
You don't know how lucky you are. Stay mad kiddo
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>>10185490
"Algebra" by Artin
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Can you help me /sci/? What are the components of model? And how are these components reflected on the cumulative hierarchy $V_{\alpha}$?

According to Jech a model for a given language $\mathfrak{L}$ (wtf does that mean?) is composed of a universe $A$ and an interpretation function $I$. I take it the universe is the set $V_{\alpha}$ itself. But what is the interpretation function? Is it the relation $\in$?

Also, wikipedia and other books seem to have different definitions of models.

Help me bros.
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>>10186946
Thank you! This is the bit I'd proven.
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>>10186846
>merely tautological masturbation.
All math is tautological masturbation you pleb
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>>10187837
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What is the derivative of a complex valued function? We know the derivative of a real valued function is the slope of the line tangent or rate of change, but what does it actually mean to take a complex derivative? I read it has something to do with rotation?
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>>10188122
For a real valued function $f$ the derivative at a point $a$ tells you the slope of the affine function $g(x) = f'(a) (x-a) + f(a)$ that best approximates $f$ in a small neibhoorhood around $a$.

For complex valued functions it's the exact same.
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>>10188122
f'(x)~=(f(x+Δx)-f(x))/Δx
=> f(x+Δx)-f(x) ~= f'(x)*Δx
IOW, a differentiable function f looks like an affine transformation in a small neighbourhood.

Multiplication by a complex number corresponds to rotation and uniform scaling, so f'(z)*Δz is just Δz transformed by rotation and uniform scaling. In particular, the argument (angle) of f'(z)*Δz depends only upon the argument of Δz and not its magnitude, and the magnitude of f'(z)*Δz depends only upon the magnitude of Δz and not its argument. IOW, the mapping is conformal (preserves shape).
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>>10185490
I’m a complete newfag to /sci/, why the aversion to khan academy?
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Is this the correct interpretation of a convolution operator with a measure? I'm a little shaky on how they define mu, so I'm not sure if this is accurate.
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>>10188097
Could you recommend me one straight and to the point? Do you know one that covers the cumulative hierarchy? I'm a CS major and I'm really struggling with this.
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>>10188210
It's explanations are absolute trash tier.
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>>10187837
>I take it the universe is the set $V_{\alpha}$ itself.
It does not have to be, it can be any set provided with an interpretation function.
Let's see a concrete example:
Start with the language $\mathfrak L = \{\cdot, 1\}$ consisting of a binary function symbol $\cdot$ and a constant symbol $1$.
Then, a model $\mathfrak A$ of $\mathfrak L$ is going to be a set $A$ provided with a distinguished binary function $\cdot_{\mathfrak A}$ and a distinguished constant $1_{\mathfrak A}$.
The interpretation function is simply the assignation $\cdot \mapsto \cdot_{\mathfrak A}; \quad 1 \mapsto 1_{\mathfrak A}$.
A good reference for this is eg. Marker's "Model Theory: An Introduction".
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I have a question about the P-series test for convergence.

Is a series still a p-series if you have 1/(constant)+n^p or 1/(constant)*n^p etc? The picture does a much better job of explaining the question.
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>>10189262
Firstly, you need to go from n=1 not n=0 or else you are dividing by 0.

With adding K, note that the terms are smaller than without K so will of course also converge.

With multiplying by c, note that this is the same as multiplying the whole sum by 1/c, so also converges.
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>>10189262
None of those are p-series. But:
$\frac{1}{n^p+C}< \frac{1}{n^p}$, so it also converges for positive C. Proof for negative C follows the same line, but is a bit of a hassle.
$\sum \frac{1}{cn^p}= \frac{1}{c} \sum \frac{1}{n^p}$.
>>10189275
> n^0=0
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>>10189278
It is 0^p not n^0
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>>10189275
>>10189278

thanks! So basically, no matter what constant you add/multiply it with, it will not change its convergence/divergence because, if it was converging before adding/multiplying a constant, than adding/multiplying that by a constant can only give a finite number, right?
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>>10189284
Yep
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>>10189288
cool, thanks!
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>>10189284
Be wary of what constant K you add though. If you add -n^p for some n then the series isn't even defined. Anything else though and it is convergent.
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>>10189301
oh, makes sense. Thanks for the heads up.
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>>10188396
You need to start with metalogic, Enderton's book intro math logic is best. Then you can start to understand the cumulative hierarchy. V is a model of the usual set theory, ZFC, if you don't know what models are or how they work you're in a car with no gas.
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bros what went wrong
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>>10189805
>tfw there aren't enough smug anime girls in the world to properly express my opinion of this guy
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final exam on thursday and I haven't started studying yet.
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what's the intuition behind an ideal of a ring?
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>>10190004
It's like a subspace but not really.
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>>10190004
The idea comes from Dedekind. In these so called Dedekind domains, of which number fields are a key example, we have unique factorisation of ideals into prime ideals (whence "ideal" comes from: they are "numbers" that behave ideally), therefore getting a massive generalisation of the fundamental theorem of arithmetic for general number rings, which is extremely helpful in solving Diophantine equations. The motivation behind the aforementioned comes from Kummer and specifically and famously Lamé's false proof of Fermat's last theorem, where he accidentally assumes unique factorisation in these rings of integers.

Later came the idea of these objects that are to rings what normal subgroups are to groups, although it was still incomplete, since ideals cannot be subrings and don't behave as well as in groups. But of course, that's where modules come to the rescue, and naturally, the only submodules of a ring are its ideals.
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>>10188122
Notice that a 1-forms on $\mathbb{R}^2$ is closed iff its components satisfy the Cauchy-Riemann equations, hence exterior derivative on 1-forms is equivalent to the complex derivative. Since $d$ manifests as the rot operator (the 2D analogue of the 3D curl operator) on 1-forms, the complex derivative is exactly rot. The fact that holomorphicity is equivalent to form-closure is used to prove the $\partial$-Poincare lemma in Dolbeault complexes.
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I started reading Topology via Logic by Vickers and I'm totally blown away by how clear it is. I'm finally understanding what a frame is and what the deal is with geometric logic. I'm getting close to the definition of a locale and I'm just fucking thrilled!
>mfw I read the definition for a frame and saw how it corresponded to topology

Can we just take a minute to appreciate textbooks that are clear as fuck?
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>>10190004
Groups have subgroups.
Rings have ideals.
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>>10190640
Problem with clear textbooks is when you stop reading it and taking notes/completing the proofs and start to do the exercises and you don't understand shit
Also usually it becomes very hard at one point
Landau-Lifshitz is a prime example of that
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>>10191565
I remember reading an anecdote about Feynman's Caltech lectures (that spawned the book) talking about exactly this problem.
Students complained everything made perfect sense in lectures and they would go home and have no idea how to do the problem sets, because his explanations during lecture were so good they didn't have to think about stuff themselves.
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>>10188415
I didnt use khan for explanations, just having the practice problems was nice, but i honestly think you shouldnt need explanations for the stuff on khan unless you get all the way past their calc 1 section
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>>10191565
>>10191597
What.
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>>10191559
*rings have subrings
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>>10191565
This is why you do exercises and proofs as you're reading. It's also a very good idea to try to pause and try to preemptively come up with conjectures and proofs once you see a definition before you continue reading and see the theorems the book contains. Anyone reading a graduate level text on mathematics without any accompanying lectures/courses knows to do these things.
That said, what you're describing is a real phenomenon that isn't talked about very much in the context of education. More precisely, information that is easy to process (for a variety of reasons not limited to exposition and formalism but also including things like organization, typesetting, visuals, etc..) is said to have more processing fluency. As a side effect, when someone is consuming information of this type they typically have more positive feelings about it, are more likely to believe it, and are more likely to feel that the writer is intelligent. Unfortunately, in some cases this can cause people to believe things that are false or to believe they have a deep understanding of something when really they only saw one clear example. There isn't much research on this and most of it is in the context of writing questionnaires for people (e.g. simply changing how a questionnaire is typeset without altering any of the material can affect people's responses). Unfortunately there isn't much writing about it in the context of mathematics but it's something to be aware of.
>Landau-Lifshitz is a prime example of that
Google says this is a physics book. I'm not interested in physics (hence why I'm in /mg/) but I get the sense that it's easier to gaslight yourself into believing you understand things in physics because the math is so sparse and handwaved and there is so much over-reliance on real world intuition.
>>10191597
>more physicsposting

I'm gonna go back to reading my book.
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>>10191893
Don't be all pretentious now, LL is a physics book but it's one of the more rigorous ones.
I kinda agree with what you're saying, however.
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Is there anyway that I can get Math World or something like that in my terminal without a text based web browser? sdcv doesn't seem to have any math dictionaries. I just want a quick reference db in the bash terminal.
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>>10191634
Ideals are subrngs.
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>>10189805
i have a friend who had qiaochu as a ta a year back or so, and he said that the guy was really excited about spreading interest and love for math. too bad.
we had some unfortunate shit with grad students here last spring though so that could have impacted his passion i suppose.
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>>10191559
corrected: groups have subgroups, some of which are normal subgroups
rings have subrings, some of which are ideals
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>>10192480
Subrings aren't necessarily ideals.
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>>10192513
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>>10192501
ideals are almost never subrings
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>>10192530
This is true.
t. >>10192480
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>>10192480
>unironically talking about "rngs" instead of just accepting that rings don't necessarily need a unit like a normal human being
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>>10192552
RNG is an acronym of Random Number Generator. I thought maybe you would like to know that.
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>>10192523
>>10192556
>thinking I know about that cs mumbo jumbo
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>>10192552
Cry some more, faggot.

>>10192559
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>>10191565
>LL
>clear
LMFAO LL is clear only if you've studied symplectic geometry prior.
>>10191597
Anyone who couldn't learn from Feynman's lectures is beyond saving.
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does anyone know what he means by the top definition? Like, what are the elements? Is that the symbol for elements in a product, or disjoint union, or just elements of the stalk in general? I can't seem to make it out given that the only other definition I found on the internet was that thing that doesn't look too similar
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>adviser replied to emails at 1:30am, 4am, 6am, all while having a meeting the next day at 9am
>always looks well rested
what the fuck? Is uberman is a meme a meme?
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>>10193068
pic with "proof"

he also replies at 4pm, 6pm, 8pm 10pm, so it's not like he sleeps in the afternoons

this guy is a fucking machine
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>>10193027
A section $f\in \mathcal{F}(U)$ over a stalk $\mathcal{F}_x$ for $x\in U$ is by definition a collection of sections over stalks $f: U\rightarrow \coprod_{x\in U}\mathcal{F}_x$. The intuition for the sheaf of sections over a manifold/topological space is useful here: the section $s: U \rightarrow E$ of a fibre bundle over $U$ is by definition a collection of sections over fibres $s: U\rightarrow \coprod_{x\in U} F = U \times F$ by a local trivialization (in the case of Seifert fibre bundles, for instance, the equality $\coprod_{x\in U}F = U \times F$ no longer holds for all $U$).
Evaluating the section $f$ at a point $x\in U$ results in an element of the stalk $\mathcal{F}_x$, and the compatibility condition tells you that the sections are now continuous.
For instance, suppose we have a cover $\{U\}$ of the manifold $M$, then the compatibility condition states that a refinement $\{V\}$ exists such that sections $s_U$ on the cover $\{U\}$ restricts to sections $s_V$ on the cover $\{V\}$; in particular they restrict to the intersection $U\cap U'$ of two open sets.
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>>10193169
you managed to completely not answer my question. the answer was the product btw, thanks to ncatlab
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>>10193181
**thanks to stacks project
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>>10193181
Huh?
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>>10193190
My question was what did he mean by this. It turned out to be an element of the product of stalks $\prod_{p\in U}\mathcal F_p$.

Also I don't know what you mean by a section over a stalk
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>>10193199
It means $\{f_p \in \mathcal{F}_p\mid p\in U\}$; the parenthesis usually means that the set is ordered but idk how he'd order $f_p$.
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>>10193208
Well, it's ordered in the sense of a product. As I said, I found the solution here https://stacks.math.columbia.edu/tag/007X
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>>10193213
>it's ordered in the sense of a product
Huh?
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>>10193220
you know what i mean, like cartesian products aren't ordered per se but they are taken over an indexing set. Here the indexing set are the points of the open set U
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>>10193068
>>10193089
Some people can operate with very little sleep, kind of crazy really.
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>>10193229
kinda jelly ngl, sleep is killing my mental gains
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Fucking heck, this Ravi Vakil book is killing me. There are no proofs in this book, and 80% of the text is exercises, 18% definitions, 2% examples.
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>>10193237
>Fucking heck
Do you need to swear?
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>>10193246
yeah, once in a while, you just wanna have some smooth cruising in a book, but no, every definition is followed by 5 relatively easy exercises that you have to do because there are no theorems/propositions and you have no idea of how the theory is going to develop without doing them, or their importance.
>>
anyone here understand enflo's counterexample to the approximation property?
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>>10193311
All I know is that Banach spaces can be fucking terrible. I don't understand the funky constructions, myself.
>>
Any north american anons applied / considered applying for PhDs in europe? The more I read about how american grad students live in what's basically poverty the less appealing the prospect seems.
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>>10193370
PhDs in europe are very (very) different from USA PhDs.
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>>10193370
>The more I read about how american grad students live in what's basically poverty the less appealing the prospect seems.
I never understood where the destitute grad student meme comes from.
Any half-respectable institution will give you at the very minimum 18-20k a year in funding. Sure, that's objectively dogshit wages, especially considering you're probably working 60-80 hours a week, but you have to be a complete mongoloid to not be able to live easily on 20k a year.
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>>10193384

>>10193414
Source? I've heard people throw figures more around 15k around, which after rent and shit may be basically nothing.
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>>10193456
>Source?
The FAQ page of most graduate schools will give you some indication of how funding works.
I checked two random unis for a sample: Michigan says their base GA support is 10.5k per semester, and Washington says they pay their greenhorn TAs about \$2200 a month (your salary goes up as you get experience). I don't go to either of these places but this is about in line with what I make.
I never even saw anything as low as 15k when I was shopping for programs. This would only be acceptable if you're in fucking Arkansas or something and the cost of living is pennies.
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i wanna learn financial math, i'd imagine it'd involve some calculus and probability, what's the best way to get into it?
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>>10193414
you’re literally a slave and if you fuck up even a little you’re homeless, you can’t go out, would have to get multiple roommates to live in a decent city, no chance of living with gf and good luck attracting mid-late 20’s women. Its awful, im dreading it
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>>10193542
You're basically correct although the probability involved can get very high-octane depending on what exactly you mean by "financial math".
The basics (the boring shit where you calculate dividends and bond prices and annuities and stuff) is very elementary, you barely even need calculus honestly.
If you wan to do it seriously you will need to learn stochastic calculus, which although pretty cool is not exactly baby math.

Also, it's worth remembering that if you intend to make a career out of this you will either need to learn to code well or learn to be chad.
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>>10193552
>no chance of living with gf and good luck attracting mid-late 20’s women
>wanting to deal with women
one mans trash anothers treasure, i guess
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>>10193552
here's a novel idea dipshit
don't do a phd then
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>>10193554
>The basics (the boring shit where you calculate dividends and bond prices and annuities and stuff) is very elementary, you barely even need calculus honestly.
yeah that seems pretty straight forward and intuitive

>If you wan to do it seriously you will need to learn stochastic calculus, which although pretty cool is not exactly baby math.
that's the stuff i was thinking about

>Also, it's worth remembering that if you intend to make a career out of this you will either need to learn to code well or learn to be chad.
i have no intention of learning to code, why is that needed? i was thinking about making it a career in the future but while i've got a good job now and i wanted to learn it for [spoiler]fun[/spoiler]
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>>10193558
>i have no intention of learning to code, why is that needed?
I'm not sure if you think quants sit in their offices working out stock prices on a pad of paper all day or something.
It's 2018. Whatever you do is going to be implemented via computer models. In all likelihood the computer is going to be doing the actual trading too.
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>>10193564
yeah you're right, i didn't want to put more time into another field

as far as the maths goes, any book recommendations?
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>>10193580
I can't really recommend you something specific to read, sorry. I learned the bits I know from a patchwork of courses I took in undergrad.
I think the standard reference for the field is Hull but that's like 800 pages long. Generally when you're self-studying you want to shoot for the shortest book available.
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>>10185490
Lang's Algebra, Revised Third Edition
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>>10192498
So what happened last spring?
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>>10193589
Thanks mate
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>>10193602
a few suicide attempts, then someone to whom i believe qiaochu was close actually went through with it. at least publicly, due to math-related unhappiness. i knew the guy personally, he was a wonderful person. i believe they may have been officemates or something. given the date of the quora post and the fact that i know qiaochu was at the school just a year back, i don't think it's a long shot to assume that this turmoil killed his passion.
honestly, i think anyone who knew the dude kind of got turned off from going into hard research. obviously not everyone is going to have the same experience but the first attempt was so surprising and the subsequent success was just way too sad. i'm sure it's played an implicit role in my recent "interest in branching out to possible industry paths" after undergrad. i.e., looking for a path to escape academia.
at the same time, some of us are more resilient than others, and obviously many many people make it through academia. it's unfortunate that events external to qiaochu may have played a role in him moving elsewhere.
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>>10190004
You literally define an ideal I so that the quotient ring R/I has the structure that you want. So the intuition behind the definition is really just to make sure that quotient rings 'work'.
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You are given set A = {a1, a2, ...., a11}. How many sorted pairs (X, Y) are there such that X ⊂ A, Y ⊂ A? The number of elements in sets |X| = 8, |Y | = 7 i |X ∩ Y | = 5

how do i solve this?
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>>10193885
You are given set A = {a1, a2, ...., a11}. How many sorted pairs (X, Y) are there such that X ⊂ A, Y ⊂ A? The number of elements in sets is |X| = 8, |Y | = 7 and |X ∩ Y | = 5 **
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Hey /mg/!

I recently found a very useful lemma which I want to use in my masters thesis.
Unfortunately I only found it on stackexchange and can't seem to find this exact lemma to cite from any actual paper/book.
https://math.stackexchange.com/questions/642487/conditions-for-global-invertibility-of-a-function

It looks like it could be included in some calculus book around the inverse function theorem, but I wasn't able to pin it down any further.

Has anyone seen this before and is able to tell me where it's from?
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>>10192670
>LMFAO LL is clear only if you've studied symplectic geometry prior.
That's wrong, good calculus, knowing what a manifold is and a working knowledge of basic mechanics is enough for the tome I.
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>>10194158
you can cite it from stackexchange. There's a button that says 'cite' somewhere in the answer and it gives you already the latex code for any of the big bibliography options
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So I was revising my complex analysis with Lipman Bers, and I saw an exercise that asked you to show that the exponential function is transcedental (which means that there's no non-constant polynomial F(z, exp(z))=0).
So wrote down the format of the polynomials, passed the polynomial of the exponential to the other side, and wrote down that I can differentiate one side n+1 times before constant zeroes, but can differentiate the other however many I want without getting zeroes, so that the left has to have infinite values.
Does this count or am I doing something really stupid without noticing?
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>>10193208
>>10193169
>>10193027
thinking back, there's an obvious correspondence between a function $f:U\to \coprod_{p\in U}\mathcal F_p$ and an element of the product $\prod_{p\in U} (f_p,U_p) \in\prod_{p\in U}\mathcal F_p$
>>
>>10195087
Sheafs seem like next-level mathematics to me, but maybe I'm just a brainlet
>>
>>10195098
well, particularly that type of correspondence works for any set, not just sheafs
>>
>>10193714
Oh wow. I mean, I myself have never become depressed due to academia, but you can tell that people put up a face when you talk to them here. It makes me wonder how many of the students feel like they do not belong, or have 'imposters syndrome'. I think that a key issue is that people become overworked and thus learn at a slower rate, hating themselves for it. There is not a clear answer to avoid this problem other than the basic advise of knowing yourself at a fundamental level.
>>
>>10189805
Well, haven't you ever felt like this ?
Tbh, the more I attend seminars, the more this feeling is creeping up on me. You spend years studying something, writing papers that nobody will read, give talks that nobody will understand and that will only ever scratch the surface of what you do anyway. And if you work in academia, your whole life is going to be like this.
The loneliness is crushing. Research is nothing like undergrad. You will likely never have a deep math conversation with a colleague again because, even if you are in the same team (eg. algebra or probability), the depth of specialization is such that it's like you don't speak the same language, even in the same area.
Moreover, the "publish or perish" mentality is so pervasive that it is hard to take a serious look outside of your very narrow field, if only because it is so time-consuming.
Nah, I definitely see where he is coming from.
>>
thoughts on
Sets, Logic and Categories by Cameron?
I already saw naive set theory and Logic but just want a short refresher.
>>
>>10185459
Prove all finite models are saturated.
>>
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>>10189826
2 finals next tuesday, gf's dog is dying and i'm working 2 jobs who want me at all times
>>
>>10195172
Not that guy but it's not uncommon at the undergrad level either. Several friends who graduated from a math bachelors and didn't go onto grad school all talked about having imposters syndrome throughout their undergrad. Several of them were actually really good and did really well in their classes (some had almost straight A's). Now that I've graduated I've spoken to several other students still in their undergrad and I kind of get the sense that most students feel that way at least during some portion of their undergrad or in the context of some area of mathematics. Even students who do research and math camps and are wrecking the curve for everyone else in their classes have said that they're terrified that their performance could all just be a fluke and they actually don't know anything.
A friend of mine in the math department (undergrad) recently attempted suicide and afterwards several of us talked about it and it seems that suicidal ideation isn't all that uncommon either. I don't know how much of it can be chalked up to the difficulty of the math program and how much of it is just all the other stuff going on in student's lives.
>>
>>10195466
>Prove all finite models are saturated.
What have you tried?
>>
Help a brainlet out. I finished calc 1 and still don't understand continuity. I know what the professors expect as answers to the problems, but I don't get it.
In a piecewise function for example, it's easy to tell whether that point is continuous or not. But let's say you're given a random function, say a polynomial, how do you prove it's continuous everywhere?
I mean it's obvious, but I know enough about that that "it's obvious" doesn't constitute as a proof.

Let's make it even simpler
>prove that f(x) = x^2 is continuous

I feel like there was some serious handwaving in my calc class. How can we just assume that a function is continuous because it looks continuous?

I'm not even sure if I'm framing the question right.
>>
>>10195716
1. Choose an $\epsilon$ for your error margin.
2. Take $f(x + \Delta x)-f(x)$ and expand it out and regroup everything in terms of powers of $\Delta x$. Note, there is no zeroth power "constant" term. This is true for any polynomial, and it's very important.
3. Take the absolute value, and expand the right side with the triangle inequality for each power of $\Delta x$ to get an upper bound on $|f(x + \Delta x)-f(x)|$.
4. We want to find a $\delta$ such that $|\Delta x| \lt \delta$ implies $f(x + \Delta x)-f(x) \lt \epsilon$. But we don't have to choose the biggest possible $\delta$, so let's only pick ones less than one. This now gives us that $|\Delta x|^n \lt |\Delta x|$ for any positive integer n. Use this to bound your $\Delta x$ to get an upper bound on $|f(x + \Delta x)-f(x)|$ again so that every term is just some numbers times $|\Delta x|$.
5. It should be straightforward from here. For any x, you get $|f(x + \Delta x)-f(x)| \leq K |\Delta x|$ for some K. Just find out how large you can get $|\Delta x|$ while still being within your $\epsilon$ bound.

Maybe not the best way, but it works, more or less. Now we know polynomials are continuous.
>>
>>10195588
lol it's real fucking nice just getting to enjoy feeling good about myself and my place in math
like damn i care about how my grades are but i can't imagine feeling like i don't belong in math given how consistently high they are and how well i'm able to talk about math with friends and work through arguments off the top of my head
i think math is one of those things where you really just can't see yourself spending your life doing anything else, and since that's the case for me i don't see how i'd feel like i didn't belong. i feel like i was fucking chosen for this shit or something, like it's out of my control.
must be strange to not feel that way all the time and to still go through the shit that a mathematician has to endure. obviously i'm not in grad school or anything yet so the worst is yet to come, and i'm probably dunning krugering the shit out of this, but at least moving from calculations to proofs was unnoticeably smooth. basically instant.
>>
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From chapter 4 in Probability: A comprehensive course by Klenke
I've stared at this for so long I have no idea why I can't get it, I've been able to get the other exercises. If you define the partial sum F_N(t) and calculate the Lebesgue integral from 0 to infinity and let N go to infinity, we get divergence and so the integral of that sum is infinite (not that we should expect it to be finite) but it goes to infinity at O(log N) and my intuition is that if the integrand was finite, we would not expect such a slow divergence but I don't know how to use that.
>>
>>10185522
Good for a smart child learning the material first time, preferably with a tutor. For a person who had already went through school once and just wants to self-study/relearn the material? It will be annoying as fuck, even if you remember virtually nothing. Gelfand throws these tangentially-related math puzzles at you all the time without explaining anything and expects you to just *get it* and use the same reasoning in the future chapters. But if you've already passed algebra in high school, you already have *some* intuition, just from doing a shitload problems in class. You'll probably be better served just doing a dry, but well-organized modern textbook one chapter at a time.
>>
Why does the epsilon-delta definition use "... < epsilon" instead of "... <= epsilon"? They are obviously equivalent, but sometimes it could be easier to find a N for which it holds only with equality
>>
>>10196149
They're not equivalent. It's the difference between an open interval and a closed one.
>>
>>10196155
But they are. If you can find a N such "... <= epsilon" an arbitrary epsilon > 0, then you can also find a N' such that it holds for epsilon/2, so you have "... <= epsilon /2 < epsilon"
>>
>>10196165
then why you have to choose that as epsilon in the first place
>>
>>10196099
Assume it doesn't for a subset of measure K of measure k. Then the Lebesgue integral across all integer multiples of K diverges.
The integral can still diverge if it just converges.
>>
What is the intuition behind topological data analysis from statistical perspective? I mean, besides that it's just fancy, the fact that we found some n-dimensional hole in our "data cloud", what does it mean qualitatively? Does topological complexity provide some deep knowledge that coudn't be found with traditional data analysis tools?
>>
>>10196090
nice someone revised it
>>
>>10194191
I know, but it would feel kind of weird citing some guy looking for a solution to his homework problem
>>
>>10194158
Pretty sure literally any advanced calculus book has that. It was the Jacobian something something inversibility condition or whatever.
>>
>>10196646
just checked rudin and it doesn't seem to contain it
note that i'm not looking for the inverse function theorem, since I need global invertibility
>>
>>10196665
>Rudin
I said a Calculus book, not an Analysis book.
>>
>man do I love it when the book does this
Anyhow: Assume that f(z)=w has no solution. By Casorati-Weierstrass, we can find a sequence $a_n$ such that $f(a_n)$ converges to w. Assume it converges in the boundary of the disk. Then we reduce the disk's radius and apply Casorati-Weierstrass again. Assume it converges in the one point in the center. That's the $w_0$. Laurent series expansion guarantees the uniqueness of $w_0$. Second part:
Assume we've enumerated all $a_n$ that satisfy $f(a_n)=w$. Set $g(z)=f(z) \Pi_{i=1}^{n}(z-a_i)$. Casorati-Weierstrass guarantees at least one more value.
Did I miss anything particularly important?
Please respond, I genuinely don't have people irl to ask this to.
>>
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>>10193714
>i don't think it's a long shot to assume that this turmoil killed his passion.
>>
>>10193556
THIS.
>>
Mathematics without some exterior goal is demotivating and pointless. It's like mining the ground without knowing what you are mining for.
>>
is this book any good? https://infinitedescent.xyz/
>>
>>10197303
A mathematician always knows what they're looking for. Worrying about applications and other trivialities should be left to the sciencelets who can only think in the context of the real world.
>>
>>10185459
Good, not too complicated book on linear algebra?
Just out of interest.
>>
>>10196259
statistical methods would say all of these are the same
https://en.wikipedia.org/wiki/Anscombe%27s_quartet
it's easy to see that they aren't but that's only since you can look at all the data at once since it's only two axes
you can't plot out data points with 100 things being measured per data point, so how do you notice patterns like the above?
the answer is you teach a computer to not abstract structural bits and report them back to you
asking what can be inferred from that is the same as asking what can be inferred from the quartet, which is all sorts of things
>>
riemann BTFO
>>
>>10197506
Lay's Linear Algebra and Its Applications is what we used for my first course in linear algebra. You can also check out Strang's book of the same name along with his MIT lectures.

>>10185459
Also requesting a good intro book to Abstract Algebra.
https://www.amazon.com/Abstract-Algebra-Course-Dan-Saracino/dp/1577665368/ref=sr_1_fkmr0_3?s=books&ie=UTF8&qid=1544234435&sr=1-3-fkmr0&keywords=abstract+algebra+saccarino
This is the required text for my class next semester.
>>
>>10198004
Durbin is the gold standard.
>>
>>10197190
Can't he just teach?
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>>10198121
>wanting this mess of an individual as your teacher
???
>>
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>>10198159
He's kind of cute tho
>>
>presented in undergrad math seminar today
>chose a cool problem from analysis
>two classmates presented on cool statistics problems
>rest of classmates presented on bullshit topics
>methods of proof
>the formulas of Gauss
>the abacus
What a waste.
>>
>>10198193
normies have no imagination.
>>
>math: column vector
>physics: ket
what the fuck!?
>>
>>10198274
Damn I think your mom dropped you on your head a few too many times when you were a baby anon. Sorry
>>
>>10197303
>It's like mining the ground without knowing what you are mining for.
Isn't this fun in its own right? You don't always know what you'll get even if you do set yourself a goal.

>>10197361
That book seems to skim over all the topics and never really goes into detail. You might be happy with it if you only want a casual reading, but if you really want to learn the topics shown in the book you're better off reading separate books for every subject (i.e. the first chapters of a book on logic, maybe a book about proofs, one on elementary number theory, another on elementary analysis etc.)

>>10197990
source?
>>
>>10198314
Bressoud - A Radical Approach to Lebesgue's Theory of Integration
>>
How do I revive Euclidian Geometry?
>>
>>10198274
>Column vector
>Restricting yourself to finite dimension

lmao
>>
>>10197190
well that's good to know i suppose, but i don't think it precludes that having a part in it. he seems to be fabricating his issue at least a little bit. from the little i know of the berkeley grad program, it really does not seem to me like anyone is "assigned" any problems to work on with no input. maybe he was just a pushover.
>>
>>10198314
>Isn't this fun in its own right? You don't always know what you'll get even if you do set yourself a goal.
My problem is that I cannot value mathematics for its own sake. I can understand why people do it: same reason they value music and art and Chess. But for me, motivation is only possible if I know the work is directly applicable to a physics.
>>
Choose any 3 digit number. The only constraint is that the first digit within it has to be bigger than the third. (Eg, 321). Reverse the order of its digits, and subtract this from the original number. (Eg, 321 – 123 = 198). Reverse the digits of this new number, and add the result of that, to the new number. (Eg, 198 + 891 = 1089). You’ll notice that the answer has been written on the whiteboard behind you for the duration of the interview. Why is it always 1089?
>>
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>>10197303
>Mathematics without some exterior goal is demotivating and pointless. It's like mining the ground without knowing what you are mining for.
>>10198314
>Isn't this fun in its own right? You don't always know what you'll get even if you do set yourself a goal.
Mathematics is like Minecraft...
>>
>>10198789
201 - 102 = 22
22 + 22 = 121
In base 3
>>
>>10198789
1089 = 9.11^2
It is easy to check that the first operation yields a multiple of 9 and that reversing the digits preserves that condition, hence the second step also yields a multiple of 9
Secondly, the first step yields a multiple of 11 (same argument), and that given a multiple of 11, say with decimal expression abc (a+c=b), then abc+cba is a multiple of 121 that is at most 1089:
Write abc = a(11-1)^2 + b(11-1) + c.
Then abc = a*11^2 + (b-2a) 11
Similarly, cba = c*11^2 + (b-2c)11 and thus abc+cba = (a+c)11^2
Moreover, since a+c = b <= 9, we have abc+cba <= 1089.
Since the result of the process is divisible by 1089, it has to be equal to 1089.

Now would I have come up with that in 5 minutes during an interview, I'm not sure..
>>
Since I know there is some other frenchfags in here, I'd reccomend the Jacques Gabay librairy. It's next to Polytechnique and it's just an old guy making high quality reprints of classical texts in Physics and Math.
>>
>>10197982
Okay, so maybe the contrary? Is it true that any traditional statistical metric we can derive from topological properties of the dataset?
>>
>>10200183
I'm not very informed, but I'd assume so, you can see a mode, skew, deviation, or an average for data sets that can be visualized. it follows that computer could 'see' these sorts of things for higher dimension data-sets through TDA
>>
What's the intuition behind the Carathéodory criterion for measurable sets?
>>
>>10199046
Secondé
>>
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>>10200257
The idea with Caratheodory's criterion is that we should be able to cut up any set, no matter how strange and weird (S = the "test" which can be as funny and weird as we want) into two parts such that the measure of the two parts add up to the total (outer). As you can see in my diagram, with the measurable set; there is nothing funny going on so even though S is weird and ugly, we can divide the set up into two pieces (area where the black and red overlap and otherwise). But if A was also weird and ugly, S and A can interact in their weirdness so that we "lose" some information or measure when they interact, i.e. when A is sufficiently weird, we pick a likewise ugly and weird S such that

$\mu(S) > \mu(A \cap S) + \mu(A^C \cap S)$

i.e. we lose measure because of the how S and A interact. Remember to fail Caratheodory's criterion we only need to find ONE (1) set ugly and weird enough such that when you interact with A you lose this information. If no such set exists, then the set is not ugly at all and so it is measurable.
>>
Does anyone have a good book to read on measure theory? Someone in the last thread said it was better than the typical darboux and riemann definitions you learn in analysis. How does it relate to typical integration and why are people so willing to throw darboux and riemann integrals in favor it?
>>
>>10200440
not the anon who asked but thanks, this really cleared it up, I never fully grasped it in functional analysis and just memorized it
>>
>>10200440
This sounds good in theory, but how would you apply it in practice? Could you give an example?
>>
can somebody explain the intuition behind the Jacobian determinant giveing the ratio of a non linearly transformed area to the original area pre-transformation?
>>
>>10201918
>intuition
nice meme
>>
>>10201918
Simple answer: It doesn't. The Jacobian determinant at some point $p$ describes the "infinitesimal" change of volume near that point, and infinitesimal changes are linear.
>>
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What if I don't?
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>>10202097
My guess would be that then there exist rings where you simply cannot tell if they have no prime ideals. Kind of similar to infinite-dimensional vector spaces for which without the axiom of choice you cannot say if they have a basis or not.
>>
How can I show $\displaystyle f_{n}(x) = \begin{cases} \frac{1}{x} & , \frac{1}{n+1}\leq x < 1 \\ 0 & , \text{ otherwise } \end{cases}$ is not Cauchy in $L^{1}[0,1]$?
>>
>>10202122
Direct calculation, check $\| f_{n+1}-f_{n}\|$.
>>
>>10202134
I'm an idiot, check $\|f_m-f_n\|$ for $n$ fixed and $m$ getting bigger.
>>
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>>10202141
This is pretty brainlet tier, but the actual $f_{m}-f_{n}$ is what is confusing me here. Would it just be $\frac{1}{x}$?
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>>10202148
Check how $\|\cdot\|$ was defined.
>>
>>10202150
I know the norm $||f||_{1} = \int_{0}^{1}|f|$ it's just the actual subtraction of $f_{m}$ and $f_{n}$ that's confusing me.
>>
need a good book on abstract algebra for absolute idiot brainlet faggots, but is also somewhat comprehensive
i know the wiki exists but i want Your recommendation
>>
>>10202173
Durbin is the gold standard.
>>
>>10202158
Just play dumb and write it down. You will see what kind of integral you need to evaluate.
>>
>>10202189
What do you mean 'play dumb' . I just want to know what $f_{m}-f_{n}$ is equal to.
>>
>>10202195
Write it down. If you don't see it immediately make a sketch of, lets say, $f_3$ and $f_1$.
>>
>>10185459
Man I'm dumb.
>>
>>10202207
$f_{3}-f_{1} = \frac{1}{x}, \frac{1}{2}\leq x < 1$, $0$ otherwise?
>>
>>10202173
Dummit & Foote is very comprehensive, and starts from the basics.
>>
>>10202231
m8
Drink a coffee, relax, and draw it again. If it doesn't work after that I'll save you.
>>
>>10202248
If $m>n$ , $\displaystyle f_{m}-f_{n} = \begin{cases} \frac{1}{x} &, \frac{1}{m+1}\leq x < \frac{1}{n+1} \\ 0 &, \text{ otherwise } \end{cases}$ ?
>>
>>10202277
Nice! Now integrate it and you're basically done.
>>
>>10202288
irish coffee does the trick. god help my liver.
>>
>>10202329
don't drink and derive
>>
>>10185459
Anyone care to explain greens theorem for me?
>>
>>10202556
too late
>>
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>>10202559
When you integrate a normal function, you calculate the values of the antiderivative on the edges to get the integral along the middle. Same general idea. Except now the edges are a curve, and the middle is an area. It's generalized later with Stokes.
>>
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>>10202628
arigato senpai uWu
>>
>>10201918
what more do you need than your picture ? that's literally all
>>
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Intro to Lebesgue Integral final tomorrow.
>>
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>>10202879
>intro to Lebesgue
Good joke my fellow math PhD.
>>
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>>10202895
grug no joke
>>
>>10202904
>the math department can't even arse itself to lie and call it introduction to measure theory
Good luck, fella.
>>
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>>10202911
dr. grög no teach grug measure.
>>
>>10193589
>Generally when you're self-studying you want to shoot for the shortest book available
Goddamn, I never thought of that. Any recommendations for books that are short but good? Not just financial math, but any subject of math.
>>
>>10195971
when did you first get this passion about math?
>>
>>10196090
I'm behind schedule, do I an hero?
>>
>>10202938
>I'm behind schedule, do I an hero?
How far behind?
>>
>>10198193
what were the cool problems?
>>
Is statistics math?

Can someone explain to me whether a pearson's chi-square would hold any relevance if used on interval level data? The issue I've run into is that my chi-square test found that the data was a bad fit, but the t-test I ran on it found that it was statistically significant data. I imagine this means I disregard the t-test result and accept the null hypothesis but should I exclude the chi-square from my dissertation?
t. stem but non "em" major
>>
>>10203070
>Is statistics math?
No.
>>
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>>10203073
can you help me anyway
>>
>>10203080
Yeah you just plug the numbers into the normal distribution and it finds the answer for you.
>>
>>10203070
You better not be an economist.
>>
>>10203104
xd

>>10203107
I might actually send this to my old stats professor. This is capstone review of stats for an MS. I passed stats with a 79.51 rounded to a B. I'd much rather write a paper than try to understand the mess that is the required statistics program for that and now this course
>>
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>>10203234
who is this?
>>
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>>10203268
>who is this?
>>
>>10203268
>>
>>10188210
Anyone who's on the level of Khan Academy probably shouldn't be posting here... Undergraduate STEM major in progress should be the bare minimum to post, and at that point you should be self-studying textbooks.
>>
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>>10203234
>>10203268
>>10203274
>>10203367
Did this popup bring you here?
>>
pray for me
>>
>>10196183
equivalent as in one implies the other obviously
>>
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who /final exam/ today
>>
>>10204159
Saturday for me
should be easy
have to write an essay from scratch by tomorrow though. rip
>>
>>10204159
for me, it's /final examS/
>>
What do I need to learn before starting real anal?
>>
>>10204661
remember to use plenty of lube
>>
>>10204661
Depends. Learning Calculus before is usually the bare minimum.
Calculus -> "Real Analysis" -> Set theory, Topology, Measure theory, some Functional Analysis -> Real Analysis -> Transcendent elder knowledge -> Realest Analysis.
>>
did no homework, none of the projects, didn't even take the final, coming out with an A
thank god i can rely on the idiots in the class
>>
>>10204677
>usually the bare minimum
>usually
what kind of uni would allow taking real analysis without calc
>>
>>10204687
Italian unis usually skip Calculus and just throw Real Analysis at you, I've heard. Maybe some italianon can confirm.
>>
How does one find the work required to pump out water from a non-inverted conical tank?
h=0.8m, r=1m
>>
what are the subjects one usually go through in sophomore class in mathematics regarding physics and engineering studies?
>>
>>10204835
>conical tank
These textbook problems fucking kill me.
>>10204850
What?
>>
>>10204835
I got this solution, but the answer is not true.
>>
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>tfw absentmindedly left computation running on the department cluster for a week
oh no oh no oh no oh no
>>
>>10193384
How so? t. Eurofag
>>
>>10203811
How did you do anon?
>>
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>>10185459
>study 12 hours a day for the past 6 months
>do almost all the problems from the book and past quals
>take qual
>walk out feeling 100% confident on maybe 3 out of 10 problems

Anyone else here currently getting wrecked by quals?
>>
>>10205728
at least you got into grad school
>>
>>10205615
Good enough, thanks.
Fucked up an elementary question that was probably really easy because I never developed any talent for those group theory problems where the solution is just arbitrarily shuffling around variables until the answer pops out.
Should still end up between 80 and 90% though, which is fine.
>>
>>10205763
There exist problems where that isn't the solution?
>>
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>>10205894
You mean the zero ring?
>>
>>10205894
all the theory around $\mathbb{F}_1$ is a lovely example of what happens when you let algebraists sniff their own farts for too long
>>
>>10201412
>why are people so willing to throw darboux and riemann integrals in favor it?
Because measure theory is the "natural" way to define the size of sets, and is the basis for probability. Lebesgue integral (as in integrals using measure theory, not necessarily integrals using the Lebesgue measure) is the natural way to define an integral given a measure, and since you can have measures on very abstract sets you can define integrals on spaces where Riemann and Darboux constructions wouldn't even make sense. There's also the fact that Lebesgue integrals have nice properties (work well with limits under weak hypothesis among other things).
>>
>>10195653
If A = (Q, <, c_n) n in N, and T = Th(A). I want to show there are only 3 countable models of T. Another one being B = (Q, <, 1/n) n in N. How do I show there are exactly 3 models (not more)?
>>
>>10206015
Disagree, a one element field-like structure crops up in lots of interesting contexts. It would not surprise me at all if fields were not the ideal setting for a lot of studies and we see the theory recharacterized in terms of a slightly more general structure in the next few decades.
>>
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Is he a brainlet or a future mathematician?
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>>10206044
Actually, even if a Jordan measure is enough for a lot of sets, just going to 2 dimensional integrals shows why a measure theoretic approach is better when you want to integrate over weird domains.
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>>10200198
Ok.
Continuing on the topic, is the way we look at homologies of 2-dimensional complex on metric graph of data set is universal in some sense? Why, for example, we don't obtain a n-dimensional complex instead gluing n-simplex on any (n+1)-clique of the graph and compute corresponding homologies?
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>>10190004
In the ring ZZ the ideal (2) is the lattice generated by the vector 2 in ZZ.

In general lattices are a good intuitive picture for ideals
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>>10201412
Lebesgue integration, in spite of what undergrads would have you think, isn't strictly better than Riemann or Riemann-Darboux integration. You can't Lebesgue measure thinks that don't absolutely converge like (sin x)/x from 0 to infinity.
But it's nice because of what it does let you measure, and also as an introduction to manipulating other measures and integration along them, besides how it cleans up a lot of theorems in say, probability, nicely.
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What does it mean for a line in the real plane $g_2: A_1x+B_1y+C_1=0$ to be isotropic? I couldn't find anything on google, and to me the condition implies that $A_1^2=B_1^2=0$ since that's the only real solution, and this relationship is modelled on that the length of a vector is invariant under linear transformations of a plane, so that it probably means that the line is not a line but just the origin?

But if it were just the origin, then why would you call the line isotropic?
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>>10206587
>real solution
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Will this https://openstax.org/details/books/calculus-volume-3 be enough of a prerequisite to learning real analysis? My university requires courses similar to this but also requires multivar calc, Fourier series, point set topology, and uniform convergence. However, I'd like to be able to teach myself RA well before I actually start my uni's RA course
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>>10206698
>his uni requires point set topology for Real Analysis
Absolutely based.
Also, no. No way. Learning topology is your main concern.
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>>10188415
(you)
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>>10206698
>Will this https://openstax.org/details/books/calculus-volume-3 be enough of a prerequisite to learning real analysis?
Why don't you try it and find out?
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>>10185490
You shouldn't wail on Khan Academy. It wasn't made for >year 2 college math.
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I have two questions on using the Feynman method to study math. I'm switch from this method from my previous read and chug because it seems I'm not truly understanding the material

1. Is there an algorithmic way I can subdivide a section of my book into a checklist of subjects that I need to give in order to ensure competence or do I have to actually exert effort on this?

2. When using the Feynman method to explain a proof, I find that as the material becomes more and more advanced, more I'm forced to rely on using terminology that is not layman level in order to keep things things succinct. Is there a way around this or am I going about it the wrong way?
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>>10206587
it means that it passes through a circular point at infinity
https://en.wikipedia.org/wiki/Point_at_infinity
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>>10208207
>Is there an algorithmic way I can subdivide a section of my book into a checklist of subjects that I need to give in order to ensure competence or do I have to actually exert effort on this?
No. Learning to determine on your own what is core material and what is fluff is an extremely important skill to develop, especially for self-studying and especially for math.

>I'm forced to rely on using terminology that is not layman level in order to keep things things succinct. Is there a way around this
If you are going to follow this method you should be much more concerned about writing up explanations for definitions and concepts than you should be for proofs. If you already did an explanation of, say, what a manifold in this style, I think it's fair to use that word in a proof as if your imaginary reader knows what it is.
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Lads my prof is a Jew who is teaching us straight up graduate algebra/algebraic number theory in an undergrad number theory class. I've been able to find most of the proofs for the homework problems in graduate number theory books but I haven't found this one or anything else online. Wikipedia and other places state that the ring of integers of a number field have factorization into irreducibles, but they don't ever prove it. There was an earlier question that had us prove factorization into irreducibles for the algebraic integers of an imaginary quadratic field, but that case was trivial when you use the norm function. I don't feel comfortable mentioning norms in a general ring of integers.

My only lead is that a ring of integers of a number field is a commutative Noetherian ring, and that in general those types of rings have factorization into irreducibles, but we've never gotten close to the topic of Noetherian rings or even rings in an abstract sense. When I look up the topic it all looks like Cantonese to me. Help lads.
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>>10208503
It's hard to give you advice because I don't know how much algebra you know or how much algebra Shekelberg expects you to know.

All factorization proofs go basically the same way: if a isn't irreducible, then you break a into a=p1p2 and apply the same recursion to p1 and p2.
The only nontrivial part of this argument is proving that you eventually hit an irreducible element this way, that is, you can't have an infinite chain such that p2 | p1, p3 | p2, p4 | p3... and never hit the end. Basically, you're proving it's Noetherian.

How easy this is depends on how much you know about rings of integers. If you're okay saying they're finitely generated rings, it's pretty easy. If you can't say that much, then I don't know how you would approach this without a sledgehammer from algebra.
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>>10185459

> had to do a research paper and presentation for my freshmen writing class
> did it on the hilbert-brouwer controversy
> talked about intuitionism and applications to software engineering and cited a talk by doron zeilberger about limits of mathematical knowledge
> it was ok but I was one of the only presentations that wasn't about food or sports
> I looked like a fucking sperg

bros this is why you never leave the math library
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>>10208596
fucking fag kys. You seem like a fedora intellectual. to bad your classmates didn't spit on you for boasting about your knowledge of math.
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>>10208503
>Lads my prof is a Jew who is teaching us straight up graduate algebra/algebraic number theory in an undergrad number theory class.
Any examples?
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>>10208577
I took undergrad algebra and it helped a lot in the class, but he hardly talked a lot about ideals until the end of the semester and in general the ring theory covered wasn't super deep. The ring theory that he's covering would have came right after where my algebra class left off with prime ideals and how ideals can generalize the multiples of integers and what not.

I have a feeling he wants us to use some properties of prime ideals, and the Fundamental Theorem of Arithmetic for ideals since that was what was covered right before he assigned this homework. I'll play around with it more when I come out of the shower.
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>>10208637
>The ring theory that Shekelberg's covering
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>>10185459
I have a Discrete Structures exam on Friday. What are some good things to read (i missed a few too many lectures.) that could help me understand this stuff a bit better.
Literally any resources you guys got are good. I did find one textbook on the wiki, but I felt there was no harm in asking.
My textbook is kind of hard to digest
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>>10208674
>Discrete Structures
this is maths general
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>>10208503
Easy. Rings of integers are noetherian and hence in particular factorisation domains. Although you say you don't know that. I assume you've seen norms.

Let a be an element that is not factorisable into irreducibles, and choose it with minimal integer norm. In particular, a is not irreducible so you can write it as a product a=bc with b and c not a unit, and by assumption also not factorisable into irreducibles. But then b has a smaller norm than a and is not factorisable into irreducibles, contradicting minimality of the norm of a
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>>10206565

You can actually use the gauge integral for that.
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>>10209535
sin(x)/x from -infinity to infinity is Riemann integrable, that's why I used the example.
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>>10206706
>Absolutely based.
This. Doing Analysis with a point-set topology is the right way to do Analysis. It's the modern way to study the material and it's way easier and clearer than the traditional way.
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How do I do this?
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>>10209940
Think of the conjugates senpai
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>>10209940
Now this, this is underage posting.
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>>10209940
first of all tell your teacher to use latex
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>>10206493
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>>10196149
You said it yourself, they're equivalent so it really doesn't matter

After the introductory analysis course no one cares what version you choose
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>>10208501
>If you are going to follow this method you should be much more concerned about writing up explanations for definitions and concepts than you should be for proofs.
Thank you for the heads up on this. I was under the assumption that proofs would be more important then concepts or definitions, but I think I understand how concepts and definitions are more core to a subject then it's proofs.
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>>10209940
>Post on 18 or over Website
>Can't factor
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Gotta give a comprehensive lecture on T3 spaces as an assignment, introducing them, basic examples etc, but I have a hard time really understanding them. I understand that their structure is more "strict" than a Hausdorf space, but why exactly? What difference does it make that we want a closed set to be inside a subset that doesn't contain a point? Also, are there neat results/equivalence relations for T3 spaces like the "all singletons of a space X are closed iff the space is T1"?
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>>10211028
>Kolmogrov's meme classification
>useful results
T3 has it's pros, to be honest. For example, an open set either contains another open set or is both open and closed (verify by yourself).
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>>10209940
Hey anon, when you see two squares and one is substracted from the other like this: $a^2-b^2$, this equals $(a-b)(a+b)$. You can see that it's true if you expand it: $(a-b)(a+b)=a(a+b)-b(a+b)=a^2+ab-ba-b^2=a^2-b^2$.

You can use this in your simplifications. I'll do an example with you (the first one at the top left). Since $x^2-16=x^2-4^2$ is a difference of squares, we can do the same thing as we did in the pararaph above, replacing $a$ with $x$ and replacing $b$ with $4$. We get: $x^2-4^2=(x-4)(x+4)$. Then we take a step back and consider the whole expression: $\frac{(x+4)^2}{x^2-16}=\frac{(x+4)(x+4)}{(x-4)(x+4)}=\frac{x+4}{x-4}$. Can you see why the last equality is true?
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>>10211028
Hey man check this out, https://en.wikipedia.org/wiki/Regular_space

I know, Wikipedia, but the article is unironically pretty good.
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>>10211096
I've already read that and thought the same thing, how come the wikipedia nerds didn't ruin it? The style is not wikipedia-ish. A math professor wrote this.
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>>10204850
At the most you will need calc/multivariate calc/vector calc, LA, ODE/PDE
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>>10211951
I noticed that the term 'trivial' a lot comes up in these subjects. Can we use something else, because this is a blatant microaggression.
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>>10212078
>wahhh why is the book calling stuff I find hard trivial
Get good.
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>>10212097
Retarded animeposter

>>10212078
You're either memeing or dumb too
"It's easy to show that", "It's trivial that", "One can easily show that..." means either 3 things :
>I run out of space/time and I don't want to typeset it in Latex ;
It's just some algebraic manipulations that are easy to do and don't actually show something interesting, who cares
>It's not that trivial, but well within what the reader is (supposedly) able to do ;
Just view it as a disguised exercise. Usually it's here for pedagogical reasons because it helps you to see some connections
>It's some cumbersome calculation that would distract the reader's attention from my main point
Usually it's like that in physics textbook, where the details of the math steps is not really important compared to the final result (and its physical meaning).
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>>10212078
it means trivial within the context, not necessarily trivial out of context
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>>10208674
johnsonbaugh
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>>10211046
Thanks
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>>10212097
>>10212305
>>10212309
You need to stop trivializing the experiences of those who identify as a minority
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>>10212078
We use the word "trivial" when we forgot the proof to something but don't want to admit it.

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