>>16835273
>/mg/ - math general
>picrel
More like "/~mg/ - not-maths general", amirite.

>>16835273
does Mathchan still exist?

>>16835273
>Mathchan
it's still down btw.

Math is gay.

Q.E.D.

>>16835273
I'm a retard, when I was in university and school I could do Trigonometry, matrix and Inequation of second grade with no problem at all and now after 10 years of doing nothing I forgot everything, I took my old notebooks I wrote a lot, I took all mines from start to finish and re reading and re learning from what I wrote down.
Now I'm learning from first grade equations with fractions, because I noticed that after I took a job in Engineering I noticed that since I use software for calculation and many stuff I wanted to go back to the roots of when I needed to really know this stuff and calculate like that on the top of your head.
So I'm relearning and reloving math even if I feel like a retard but I noticing that is just like riding a bicycle but maybe slower, I noticed I even forgot binary and exadecimal calculus and I notice how everything makes so fucking much sense of life and logic and even programming that I started as an hobby last month, it's like rediscovering my own brain.
ESL sorry.
So how are you math anons?
Me I'm both sad but happy, sad because I turned off my brain after I took a job and happy because I'm rediscovering my brain.

>>16836318
keep at it anon. Doing math every day for over a year genuinely transforms your brain. I was also basically a retard at math before going to uni. Always make sure to challenge yourself, though. If a book is challenging to you, see that as a sign that you have a lot to learn from it, don't let that discourage you from reading it

>>16836378
>I was also basically a retard at math
How did you change that? Where did you start from? I’m solving linear algebraic equations, reading Elements by Euclid, books of propositional logic, Frege’s concept script, Logic gates , Thermodynamics , Quantum Mechanics and on set theory.

>>16836559
reading difficult books, analyzing proofs carefully until I've fully internalized the reasoning and could call them my own, and spending a lot of time on my own proofs (exercises, etc.).
When I first started out, these notes on Linear Algebra (technically also Algebra, maybe combine it with something like Shilov or Gorodentsev to get a full course) helped me a lot
https://www.ams.org/open-math-notes/omn-view-listing?listingId=110700
https://www.ams.org/open-math-notes/omn-view-listing?listingId=110701
If you want something on naive set theory that constructs the real numbers, I'd recommend the first chapter of Amann Escher's analysis series. Don't spend too much time on foundations
For physics, I always liked Landau Lifshitz. Volumes 1-3 and 5 are must reads imo, 4 is subsumed by later QFT books like Weinberg. Also check out these lectures https://www.youtube.com/playlist?list=PLPH7f_7ZlzxTi6kS4vCmv4ZKm9u8g5yic

what IQ do you have to stand a chance in doing a bachelors/Master/PhD degree in mathematics? im working on linear algebra at the moment and analysis I but its not my main subject. i just booked the course for fun but Lina is doing something to my mind i cant explain. the past 4 months i worked 5-6 times a week net 4-5 hours a day without a break. i jumpstarted from 0 to 100 workload. sometimes i get deep encoding fatigue but damn i love algebra.

>>16833126
>>16832694
use deepseek, you can compile code with shadertoy to visualize it (at least it tries to anyway)

does anyone know a good, quick resource to learn the basics about modules, tensor products, and fibred products? i'm going through the rising sea of algebraic geometry right now and the category section is whopping my ass, mostly because it's using these things that i have never worked with before

>>16836854
Roman's Advanced Linear Algebra has a ton of material on modules and a decent amount on tensors (if im remembering correctly, I'm sure about the modules but it's been a couple years since I've opened that book).

>>16836854
since other anon mentioned Rotman I'll point out that Introduction to Homological Algebra Chapter 2 has a great explanation of tensor products, as well as a QRD rundown on modules in general

>>16836854
>>16837205
Roman is pretty good. Lang's Algebra also touches on modules and the tensor product, it's worth checking out.

>>16835326
What happened to Mathchan?

>>16837205
>>16837212
>>16837427
thanks anons

>>16837680
I heard the guy running it wanted to do an entire redesign of the chan. Seems to have been procrastinating for months now though

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Any Geometric Langlandsfag here who can explain how you cope with this insane notation? shit's giving me an aneurysm

>>16836378
>Doing math every day for over a year genuinely transforms your brain.
Yeah I'm advancing a lot (i'm the OP you answered)
>reading difficult books, analyzing proofs carefully until I've fully internalized the reasoning and could call them my own, and spending a lot of time on my own proofs (exercises, etc.)
i'm basically doing this otehr than my old notebooks I took a book of general Mathematics, it's in italian (i'm italian) it roughly translate to
>Let's know Mathematics: From the theories of the ancients to the logic of the computers of today
It explain math IG from the past to today and not in a historian way but a mathematician way and it's really interesting and even i'm learning a lot, even the intro of the book explain:
>Many people study math by the rules but never from the theory this book is for people who are studying in school in a rule way and not in a mathematician way.
And I noticed that's true that many don't know the basics.

>>16838280
>"THICK"
Lol. At this level of math, idk why they don't just go the CS route and use variables to represent the names of other objects. It just doesn't seem like good practice.

I would appreciate help filling in the blanks.

Say you got a list of smooth [math] g_i: S \rightarrow \mathbb{R} [/math], where the functions are made into constraints so now you got a space to do calculus on, so [math] X = \{\vec{x} \in S |\ g_i(\vec{x}) = 0 \} [/math]. For all smooth curve [math] \gamma: I \subset \mathbb{R} \rightarrow X [/math] then, we have [math] \vec{\nabla} g_i\big|_{\gamma(t)} \cdot \tfrac{d\vec{\gamma}}{dt}\big|_t = 0 [/math].
Let [math] f: S \rightarrow \mathbb{R} [/math] and we wish to find the point or (or set of points) [math] \gamma_0 [/math] that max/minimizes f restricted to X, so [math] \max/\min f|_X [/math] means finding all points [math] \gamma_0 \ \ \text{s.t.}\ \ \forall \gamma \ \ \text{where} \ \ \gamma(0) = \gamma_0,\ \ \text{then} \ \ \vec{\nabla} f\big|_{\gamma_0} \cdot \tfrac{d\vec{\gamma}}{dt}\big|_0 = 0 [/math]

If there's only 1 constraint making a surface, the simple rationale given next is that both normal vectors [math] \vec{\nabla} g\big|_{\gamma(0)} \ \ \text{and}\ \ \vec{\nabla} f\big|_{\gamma(0)} [/math] are perpendicular to ALL possible tangent vectors [math] \tfrac{d\vec{\gamma}}{dt}\big|_0 \ \ \text{at} \ \ \gamma_0 [/math], which is a tangent space plane, implying that the normal vectors are parallel, or [math] \vec{\nabla} f\big|_{\gamma(0)} = \lambda \vec{\nabla} g\big|_{\gamma(0)} [/math].
If there's 2 constraints g_1 and g_2 whose intersection is a smooth curve, the simple rationale is that the tangent space is only a 1 dim line, and since the 3 normal vectors are perp to it, they're coplanar, or [math] \vec{\nabla} f\big|_{\gamma(0)} = \lambda_1 \vec{\nabla} g_1\big|_{\gamma(0)} + \lambda_2 \vec{\nabla} g_2\big|_{\gamma(0)} [/math].

But what's the general rationale? Why does [math] 0 = Df\big|_{\gamma(0)} \dot\gamma|_0 = Dg_i\big|_{\gamma(0)} \dot\gamma|_0 [/math] where [math] \dot\gamma|_t \in \bigcap_i T(g_i) [/math] imply that [math] Df|_{\gamma(0)} = \sum \lambda_i Dg_i\big|_{\gamma(0)} [/math]?

>>16835273
Hello everyone. I have a few questions for all of you. What kind of background do you have in mathematics? Do you do math on your own, as a recreational activity? Or are the posts here relevant to your employment?

>>16838280
The notation is well-known to be shit. But there aren't really any better options.

>>16838280
Filtered. Maybe webdev is more your speed.

>>16839202
see >>16840215

[math]\dfrac{x^n D^n}{n!}=\binom{xD}{n}[/math]

>>16840213
One thing would be to write C(-,-) instead of Hom_C(-,-) like some other cat theorists do. I just seriously dislike these concatenated sub-/superscripts

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How do I achieve math wizard aura like Grothendieck

>>16841010
>>16841010
Start reading EGA, SGA, FGA. Or go for the Stacks project in case you can't read French

is there something like a leetcode for math?

>>16841952
project euler?

>>16838280
Whew...
I'll need some sauce for that spaghetti?

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>>16841222
checkem

> also tfw you somehow got too relaxed

>>16841952
or vibe proving?

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>>16841492
teal = dark cyan
purple = dark magenta
olive = dark yellow

>>16843555
https://people.mpim-bonn.mpg.de/gaitsgde/GL/Arkh.pdf

>>16835326
damn...I hope it comes back

You can define a smooth manifold without defining a topological manifold first, and the topology will come from the smooth structure you define.
But how can I prove that a smooth manifold defined without its topology beforehand will be, in fact, a topological manifold?

>>16835273
I finally got off my ass and started relearning math. I've always been poor with math and get tripped up doing basic arithmetic. I've tried a couple times over the past decade--literally twice--but didn't stick with it for long due to various factors. Anyhow I'm starting with khanacademy again and working my way through the very beginning with its baby early math review and Kindergarten-through-grades
to IDK I was thinking of getting to something like Algebra THEN transitioning away from khan and following textbooks, as a curriculum.
If anybody has taught (themselves or others) math and has any ideas or suggestions on starting from absolute scratch, I'd love to hear them.

>>16845597
The best advice I can give is to be consistent, rigorous, and test yourself often (or just act like anything you learn is something you will be tested on). While I did not start basic as you, I did relatively successfully teach myself quite a bit of university-level math before going to uni

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>>16843850
The parameters were found by WolframAlpha.
The code was written by Grok.
The image was drawn by Cliprun.

What are some comprehensive books on 2-categories and bicategories

>>16846772
WolframAlpha found the parameters.
Grok wrote the code.
Cliprun drew the image.

I'm trying to show that Q has gaps using only open balls and failing and I reckon it's because I'm using an approach that works with bounds and trying to fit it to open sets instead. What should I be doing?

I've passed college Algebra and to apply to an Ivy League I need Calc 1 but the class has a precalc prerequisite.
Do you guys think it's possible to train in the winter break period and then enter it or should I bite the bullet and delay gaining my Associate's to take Calc 1 in the summer?

>>16845547
No you can't?

>>16849134
are you talking about training Calc 1 during winter break or training College Algebra? because precalc IS college algebra in all but name, maybe with some trigonometry. if you mean that then it's easy

Give me some interesting smooth manifolds

>>16849689
Like the manifolds themselves? Or books related to smooth manifolds?

>>16849790
The manifolds themselves

>0.999... (repeating) equals 1 (one)
surely people aren't retarded enough to believe this?
math is literally a meme

How much math can I learn realistically in 1 year, assuming 4 hours minimally a day?
Been thinking about getting back in the game:
>Logic
>Precalculus, Stewart
>Calculus, Stewart
>LinAlg, Poole
>Discrete math and its applications

>>16849994
you can learn all of what you've listed but discrete math is not a real field
Don't read a dedicated book on logic either, if anything, it should be on elementary set theory or foundations stuff. The book "Sets for Mathematics" by Lawvere is great for that.
For Precalc, read Lang instead.
For Calc, read Apostol instead.
For LinAlg, read Shilov instead

>>16847988

0 <= θ <= π/4

Tan[θ] = h/s

Cos[θ] = (2*r)/(s – h)

Tan[θ/2] = R/(s – R)

r*(Cot[(π/2 – θ)/2] + 2*(n – 1) + Cot[(π/2 + θ)/2]) = s*Sec[θ]

>>16850039
Considering the level of the books mentioned, I highly doubt that anon was referring to mathematical logic and foundations when mentioning logic. I assume that point was meant to refer to basic mathematical reasoning and proof writing skills.

But even if he was, that Lawvere text has to be a meme suggestion

>>16850039
Do you want rigor or applications?

>>16849994
With genuinely 4 hours a day, loads of it. Make sure you do the exercises

>>16850253
nta but Lawvere is a perfectly fine introductory text.
Also, nobody ever reads "proof-writing books". Just start doing real math and you'll eventually learn it, don't be difficult

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>>16850401
I'm aware most people don't actually read the books they recommend but genuinely recommending any of Lawvere's texts to somebody not specifically asking for a structural treatment of set theory/introduction to ETCS is done in incredibly bad faith.

This is especially true if that person is self-studying. The text is specific and non-standard enough that if they're confused about something in the text they can't just grab another similar text from their bookshelf because there simply are none. Their only resources would be other stuff written by Lawvere himself (which would be unlikely to help), actual treatments of elementary topoi (which is presumably above the reader's pay grade) or maybe Leinster's recent notes (which is fine but more of an alternative rather than a supplement as the two differ quite a bit in their approach and presentation).

This isn't a critique of the book itself but suggesting it to somebody who is not going to know what he's getting himself into just isn't right. Reserve that suggestion for somebody that is specifically asking about categorical treatments of set theory.
It's like telling somebody who naively asked for a book on constructive mathematics to read Taylor's Practical Foundations. It's not a completely misplaced suggestion but incredibly misleading as the book is very specified and non-standard to the point where any other resources will be by the same one author. That suggestion should similarly be reserved for somebody asking specifically about topics unique to the text (such as his general recursion theorem, constructive ordinals, well-founded coalgebras, etc).

any logi/sci/ans here? currently doing a minor in logic+computation and its comfy aight. Might just get into theoretical computer science

>>16850476
Theoretical computer science is awesome

>>16849994
You can definitely learn everything you listed and more. Instead of learning logic by itself, you should get one of the discrete math texts with mathematical logic as its first chapters. Discrete Mathematics and its Applications by Kenneth Rosen is good for self study.

>>16850476
>>16850544
what are some of the more prominent active research fields in the intersection of logic and theoretical computer science?

>>16851205
its all algorithms. Mostly catering to AI projects. Doing AI + logic is all the rage now instead of the OG math+comp sci stack (as every other logician already has done the latter)

I was a 28 year old boomer when I started my BS a few years ago. Not sure if you guys remember me. Just received a grad school acceptance letter.
Are we all gonna make it?

>>16850043

variables:
p = s/2 – R
T = Tan[θ]
u = r*Cos[θ]
v = r*Sin[θ]

equation of square:
|x – y| + |x + y| = s

equation of parallelogram:
|x – y| + |(1 – 2*T)*x + y| = (1 – T)*s

set of equations of n circles:
Table[(x – k*u)^2 + (y – k*v)^2 = r^2, {k, 1 – n, n – 1, 2}]

set of equations of 2 circles:
Table[(x + k*p)^2 + (y – k*p)^2 = R^2, {k, 1 – 2, 2 – 1, 2}]

Who cares about conic sections

Going to the desert soon. idk wtf I'm doing but well I guess I hope to up my skillz. in math league i just had to pump some blood to the math part of my head and it shit out the answer like jimmy neutron. i work in food and a lot of the problem solving like when people call off is the same. But for one I want more than a mind palace, I want a mind machine or something and a mind language. I guess to do anything you have to have a model of it in your head so I need to simply learn calculus and really it's already kinda like wut but you know it's as simple as imagining an abacus, or as out there as I guess creating an archetype/muse/vision, I guess this is what people do when they try to "find god" they just build one out in the desert through meditation and really I already have a muse like my thoughts often come to me with the title and then the description and I'm just reading didn't have to think it at all but maybe I can cultivate it or commune with my genius at least with some privacy

If [math]t:X\to X[/math] is a monad on X in a (strict) 2-category [math]\mathsf{C}[/math], that should turn [math]T:\mathsf{C}(X,Y)\to\mathsf{C}(X,Y)[/math] with [math]Tf=f\circ t[/math] into a monad on the [math]\hom[/math]-category [math]\mathsf{C}(X,Y)[/math], right? (unless I made a mistake somewhere)
Is this a common construction and does it have any name? In particular, is there anything interesting to say about T's algebras?

>>16852532
>Who cares about conic sections
Maybe yo momma do

>>16851669
gratz bro
as a fellow boomer I do hope the stigma of doing things "late" goes away especially considering that people live so long now
like its fine if im a virgin at 30 ill lose it some day

How did you guys prepare for grad school after being out of school for over a year? I don't think I forgot anything in terms of the structure of the argument or the name of the lemmas or theories, but I'm nervous that on a written exam I won't be fast enough. Looking at the syllabus it looks like most courses in grad school are take home exams with no limits on cooperation, is this normal?

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>>16852460
toodle-oo

I am ashamed of my scientific conduct.

>>16854345
>Looking at the syllabus it looks like most courses in grad school are take home exams with no limits on cooperation, is this normal?

Once you pass quals, yes.

>>16854377
How does this work in r3?

>>16855675
I feel pretty scared, now I'm getting anxious that I'm gonna show up day one, fail everything, shit my pants, and everyone I know will gather around me and tell me how they always knew I was a big headed fraud retard.

>>16855879
>How does this work in [R^]3?
I would have to think about it.
But I'm trying to forget this problem.

The square may become a cube.
The parallelogram may become a cylinder segment.
The cross section thereof may be triangular, hexagonal, or circular.
The 2 large circles may become 6 spheres.
Of 1, 2, or 3 distinct size/s.

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i'm repeating the first year of uni because i failed equivalent of calculus 1 and i've been doing barely good enough up until now and averaged like 55-60% on the past tests (i need 50% total to write final exam) and i had a minor fuck up so i'm gonna get 30-35% on a test now and should be around 50% +/- 3 percentage points but i can't really afford to not do at least average on the next 3 tests (1 test every 2 weeks) because they more important and i'm just so fucking scared and feel dread of failing again and i don't want to fucking work in construction but there's so much shit memorize it's all just computing limits, bounds, estimations, simple proofs (on numbers) and there's like 100 problems we go over before any test which is like 15-20 unique problems and the rest are somewhat similar and i just can't fucking remember all of that shit and i have mental paralysis on actually studying
i wish my course had more theoretical approach without all these fucking gay numbers and computational tricks i have to remember so i could just focus on understanding things but up until now we've basically done 0 actual calculus
i don't even know why i'm writing this i could use some help on how to study because i've always been the bright kid but i've burned out in high school and did absolutely nothing for 4 years and it's not like i've had to actually study before that either. i'll just mention that we have free education here so graduation rate is like 50% and professors will try to fail around 1/3 of students by the end of first semester

>>16856291
What helped me "cheat" was to do the course before actually starting. So if you knew you had calculus I this semester, you could have gone through the entire syllabus and practice exams and showed up to class ready to crush retards thinking they had to learn it all in one semester.

>>16856540
that's called studying

>>16856541
That's lame and boring, what I do is just time cheating. Another cool hack is to buy additional books rather than relying solely on the one listed on the syllabus, it makes you look at the content in a different point of view and the parts that overlap are emphasized.

>>16855974
The large circles lie on a line connecting opposite vertices

>>16856540
too late for that

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If you roll an N-sided dice, what is the minimum number of rolls needed so that you will throw two consecutive digits in two consecutive rolls at least once with greater than 50% probability?

>>16856822
That depends on how you label the sides of the die.

>>16856834
You label them 1, 2, 3, 4, ..., N

>>16856838
Do N and 1 count as consectutive? I will just assume they are.
If you do n rolls then the probability is
[eqn]1 - \left(\frac{N-2}{N} \right)^{n-1}[/eqn]
So you can just solve the inequality for n.
[eqn]1 - \left(\frac{N-2}{N} \right)^{n-1} \geq 0.5 \\
n \geq 1 + \frac{\log(0.5)}{\log\left( \frac{N-2}{N} \right)} [/eqn]

>>16856822
Let [math] M [/math] be the [math] N [/math]-by-[math] N [/math] matrix with [math] m_{ij}=0 [/math] if [math] |i-j|=1 [/math] and [math] m_{ij}=1 [/math] otherwise. Then the number of sequences of rolls of length [math] k [/math] with no consecutive numbers occurring consecutively is the sum of the entries of [math] M^{k-1} [/math].
There's no real hope of a closed form, but it's going to be dominated asymptotically by [math] ||v_{\text{max}}||_{L^1}^2\lambda_{\text{max}}^{k-1} [/math], where [math] \lambda_{\text{max}} [/math] is the largest eigenvalue of [math] M [/math] and [math] v_{\text{max}} [/math] is its [math] L^2 [/math]-normalized eigenvector. So if you take that as an approximation, you'd solve [math] ||v_{\text{max}}||_{L^1}^2\lambda_{\text{max}}^{k-1}>\frac{1}{2}N^k [/math], which gives [math] k>\displaystyle\frac{\log\left(\frac{\displaystyle2\displaystyle||v_{\text{max}}||_{L^1}^2}{\displaystyle\lambda_{\text{max}}}\right)}{\log\left(\frac{\displaystyle N}{\displaystyle\lambda_{\text{max}}}\right)} [/math].
If we use [math] \lambda_{\text{max}}=N-2+O\left(\frac{1}{N}\right) [/math] and [math] ||v_{\text{max}}||_{L^1}^2=N-O\left(\frac{1}{N}\right) [/math], then [math] k=\frac{\log 2}{2}N+O(1) [/math], which is essentially what you get when you assume [math] 1 [/math] and [math] N [/math] are considered consecutive. Probably a better spectral analysis would give a better approximation, but I don't actually have much expertise in that. For large fixed [math] N [/math] you can at least feasibly numerically approximate these values without too much computation.

>>16856970
Whoops, that first inequality should be flipped.

>>16855973
You probably just have imposter syndrome. However if you do show up and do terribly, my advice is to just call it quits. Do not try to push through a PhD as a mediocre student. It is just a waste of your time.

>>16855973
Unless you're going to one of the handful of tippy top schools that are a bit notorious for egotistical competitiveness, every single person around you will not only want you to succeed but do everything in their power to help you. Nobody wants to flunk you out, they actually actively want to not flunk you out because it'd be bad for them too. Exams aren't meant to be a weed-out like they can be in undergrad, the weed-out is the admissions process itself, if the department didn't think you could pass they wouldn't have accepted you in the first place. As long as you don't just isolate yourself at the first speedbump and passively let failures snowball, you know, as long as you're making an effort and keeping positive and getting help if you need it, you'll be completely and utterly fine.

>>16857013
Its Johns Hopkins.

>>16857021
I've not heard anything about JHU's culture but I'm sure it's fine, when I say "tippy top" I mean like Berkeley or MIT (with all due respect to JHU), just names that are bigger attractors to the sort of person who cares about going to the most impressive-sounding school.

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>>16854377

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>>16857029

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Posted this question on stack exchange. Why does surjectivity holds?

>>16857033
Just count the dimensions, bro. Two finite dimensional spaces with the same dimension over the same field are always isomorphic.

>>16835322
>>16837680
hello everypony, i made a copy of mathchan some time before it shut down. is anyone interested in it?

>>16857038
I wanna use the first isomorphism theorem

>>16836854
You should not even attempt at learning algebraic geometry without learning sufficient algebra first. As other suggest, go with Lang as it is the closest to actual alg geom; also Aluffi and Bourbaki work pretty well.

>>16838280
Reading the first pages, there is a lot of non-standard notation here. For categories I like \sf fonts, here the \frak{D} denotes a dualising object which is also weird, as it is typically written as \omega, maybe with a bullet to emphasize it is a complex of sheaves.
Anyways, while it looks convoluted two things should be pointed: the first is that Gaitsgory does Langlands in a different case than Langlands himself. His context is much more categorical and it shows, as it heavily relies on derived algebraic geometry, which is lots of fomalisms. The second thing is that most authors just have their sub- and supscripts implied, so that's why this paper may "stand-out" notation-wise, but not really.

>>16857038
The idea of this exercise is that the isomorphism is canonical and/or natural, and that is important for applications.

>>16856952
N and 1 would only count as consecutive on a two sided dice.

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Life became far more pleasant when I started just focusing on what I actually love and am good at (probability, PDE's) rather than trying to impress the goalpost moving algebraic geometers and pure math crowd. Simply put, nothing I do will ever be good enough to sit at the cool kids mean girls table, I think that's the point.... happy in my element with the stinky chinese kids and autistics that wear cargo shorts instead of tweed, and the engineers that validate me and tell me its okay to only have a masters.

I've forgotten high school level math. In college I took an applied calculus class and that was pretty much the peak of my math abilities (Not sure if college level statistics is harder than applied calculus). I'm wondering if you guys have any recommendations on books or lectures I can look into to ultimately help me understand real calculus.

>>16857620
spivak

>>16835273
>PhD candidate
>have to take semester off to go to rehab for alcohol addiction
>haven't thought about research in like 2 months
It's fuckin over boys

>>16855973
This >>16857013 anon is correct. From all my experiences, graduate programs will bend over backwards for you to be set up to do well. You will be okay. Your classmates will likely be willing to help too. No idea how JHU people are, but all the people I've met in other programs have a massive collaborative mindset and are always willing to help those who struggle if they ask--not saying that you will struggle, but it can be helpful to know that that's there for you. Remember, they admitted you for a reason.

>>16857054
drop links now dammit

Why should I write [math]{A^\mu}_\nu[/math] instead of [math]A^\mu_\nu[/math]

>>16857620
This is basic integration theory and you learn enough to do real analysis with measure after and generalized integrals like the Henstock-Kurzweil integral
http://classicalrealanalysis.info/documents/T-CalculusIntegral-AllChapters-Portrait.pdf

>>16857938
\nu is for vagina
\tau is for peepee

Lower subscript \tau must stick out. Assert space.
So in your case, second is correct.

thoughts on this book?

https://libgen.li/edition.php?id=136944873

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>>16840267

Why isn't there a simpler thing, maybe just from multigraphs or curvature, that does what chain complexes do?

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>The INTERNET Sorcerer

Brothers... not like this.... not like this........

>>16858552
Mathcels we are forgotten...

>>16858031
This book

https://annas-archive.org/search?q=boolean+algebra+goodstein

Please some advance or state of art books about Dynamic system and turbulences from a mathematic point of view.

>>16858552
Mathcel found asian girl.
White men can't be incels.

general system 1:
h ≤ r*H
h + H = s

general system 2:
h/r ≤ s/(1 + r) ≤ H
h + H = s

---

special system 1:
2*h ≤ 2/3*H
h + H = 230

special system 2:
h ≤ 115/2 ≤ H/3
h = 230 – H

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>>16858873

when did you realize maths is siren that will call for you and consume your whole life?
in other words- an addiction

>>16858798
Seems like a decent intro
It's weird that he calls exercises "Examples" though

>>16859880
When it started harming my marriage and my wife asked for a divorce because I spent too much time doing math.

>>16859880
When I realized math and meth aren't the same thing.

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Here is a maths problem I came up with.

You have a regular polygon with N sides. Draw a line connecting each vertex into the midpoint of the second side from that vertex, moving clockwise around the whole polygon. This will create a smaller version of the same polygon inside the original one.

In terms of N, what is the ratio between the area of the smaller polygon and the area of the original?

>>16860169
[eqn]\cos^2\left( \frac{\pi}{N} \right) [/eqn]

Why can't the perimeter of ellipses just be solved as the limit of inscribed polygons?

>>16860272
Because the polygons would be retarded

>>16860272
It can, there's just a continuum of ellipse shapes depending on the eccentricity compared to just one circle shape, so you don't have just one answer. You could in theory write down the [math] \pi [/math]-analog as, say, [math] \pi_e [/math] for each eccentricity [math] e=a/b [/math] (with [math] \pi_1=\pi [/math]), and then the perimeter is [math] 2\pi_e a [/math]. But the general [math] \pi_e [/math] don't show up anywhere nearly as often in broader math as plain [math] \pi [/math], so there's no real benefit to giving them a notation, and elliptic integrals are important on their own anyway, so no reason to hide them.

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>>16860185
Doesn't work. I know the answer is 1/5 for N=4 but your formula gives 1/2 for N=4. Even visually there is no way the grey square is half of the area.

>>16860313
Makes sense. High school GPA is complete bullshit. In 10th grade I was already getting 120WPM on type racer consistently, trained in high pressure typing situations by runescape PK'ing. I was given a C in my keyboarding class, which makes no sense given the class would gather around me to watch me crush any challenge.
There's also just most of the grade in other courses being busywork for girls rather than anything actually academic. Finally, teachers just hate smart kids, check out r/teachers, they have a seething hatred of kids that are going to be more successful than them or aren't social, and they gather like flies on shit to get validation from popular kids. Disgusting people, K-12 teachers.

>>16860425
>High school GPA is complete bullshit.
I don't disagree but selective colleges still heavily rely on it because it's an easy way of trimming the applicant pool and they have enough generally qualified applicants that they can afford to ignore the ones without 4.0s. There's 0.0% chance you get into Stanford nowadays with a 1.8 GPA unless your family is deep in someone's pockets or you've got rec letters from Nobel laureates or something.

>>16860439
You're correct. But GPA is so arbitrary that it's just a method for spiteful teachers to destroy kids or build up their favorites.
A big issue here too is overvaluation and social connections from undergraduate institutions. From a purely academic basis, there is little difference between an average state school and an Ivy League or public Ivy school. Some state schools like Florida have access to Fields medalists in their math departments...
I don't even know where I'm going with this, but I just think it's better for undergraduates to attend local universities with more family support, and try for ivies and T10 for grad school.
I'm a janitor and don't have my GED.

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>>16857031

>>16860169
Let [math] \omega=\exp(2\pi i/N) [/math]. Let [math] z [/math] be the intersection of the line between [math] \frac{1+\omega}{2} [/math] and [math] \omega^2 [/math] with the line between [math] \frac{\omega+\omega^2}{2} [/math] and [math] \omega^3 [/math]. Then the answer is [math] |z|^2 [/math]; pretty straightforward to compute.

how do i conceptualize morphisms between [math]Hom(\underline{\{p\}}, F)[/math] and [math]F[/math]? [math]\underline{\{p\}}[/math] here is the constant sheaf

it's kind of intuitively clear (though i may be wrong) that the sections in [math]\underline{\{p\}}[/math] are in correspondence with the open subsets of my topological space (since each [math]F(U)[/math] just contains the continuous map [math]U \rightarrow \{p\}[/math]), but the layers of elements/subsets and sections/F(U)/F is kind of making my head spin

can't really make out what exactly i'm even looking at, let alone how to map things nicely

>>16837205
>>16837212
>>16837427
thanks everyone

>>16857166
honestly, even though i've been sort of lax with my studies, it hasn't been too bad. that being said i am working in commutative algebra also, just to cover my bases. the plan's to stop after i finish part 1 of this book

>>16860439
>they can afford to ignore the ones without 4.0s
they're right desu
I was given a chance at a good school because I spoke with an admission guy there and made a good impression on him
it went fucking disastrously even on a social level, I couldn't interact with any of the richfags full of ambition trying to one up each other in every interaction
literally couldn't understand why the fuck everyone was so competitive and clinically insane but now I understand that's exactly the types of people elite schools search for
it's not even about GPA, it's more about your drive and ambition to pathologically work and be productive every single waking moment in order to impress others and achieve things

it's like those kids who do charity work as extracurriculars
everyone knows it's not honest and the only reason they do it is to bolster their application but that's all the university admission people care about
that they're willing to do the retarded charity work just to get into that university because that means they'll do whatever else is needed by the university

I cannot free myself from the demons telling me that I am ordinary if not stupid. The thoughts tell me I am unable to learn anything past basic arithmetic yet I desire to learn more and learn things at a faster pace. What books would you guys recommend for a mentally ill neet to take his life one equation at a time?

>>16860655
>I understand that's exactly the types of people elite schools search for
I wouldn't necessarily say that. I think any reasonable admissions standards these schools implement are going to be abused by people desperate to go to a top school and who will openly "play the game" to maximize their chances. I have plenty of complaints about the undergraduate admissions process at these schools but I don't think attracting type As with big egos is something you can really blame them for.

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>>16860169
f(n) = (4*Cos[π/n]^2 – 1)^2/(8*Cos[π/n]^2 + 1)

f(3) = 0
f(4) = 1/5
f(5) = (7 + √5)/22
f(6) = 4/7
f(7) = [...]
f(8) = (7 + 4*√2)/17

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>>16860715
(4*Cos[π/n]^2 – 1)^2/(8*Cos[π/n]^2 + 1) = (2*Cos[2*π/n] + 1)^2/(4*Cos[2*π/n] + 5)

>>16835273
any french anons could you tel me what your favorite undergrad textbooks are ? I've learned the language and i want to learn mathematics by mostly reading it in french.

>>16856676
In March and/or April, I got it wrong.

>>16856291
use periods and paragraphs

>>16860715
I also got 4/7 for N=6 by solving it manually so I trust this is correct.

>>16860715
Are N=3, N=4 and N=6 the only values where it's a rational number?

How do I prove a covering map is an open map?

>>16858552
You guys don't realize how crazy this is. Math Sorcerer has actually interviewed working mathematicians on his channel....His interview with Tom Garrity helped me get prepared for grad school.

>>16861537
I have literally interviewed Math Sorcerer. He answered a question I asked on a live stream he made. That counts right

>>16861545
You are an important person to me, that's awesome. Some Brit on X yesterday posted a reply he received from Tao on one of his papers. You're like, that famous to me.

>>16861548
In fact I'll try to find that video and time stamp my question

>>16861486
in my schizo consciousness stream?

Struggling to find the right Real Analysis textbook for a first exposure to the subject. I've just finished working through Velleman's How to Prove It and this will be my first foray into the world of real mathematics. Here's what I want:
>clear exposition
>not too difficult (no Rudin)
>has many examples
>has a full solutions manual

What I've tried and why I didn't like it:
>Binmore's book
Feels too barebones but it does have full solutions
>Abbott's book
I like his explanations but not enough examples and the solutions manual only has answers to odd questions
>Cummings' book
His explanations are good for my level (retard-tier level), but not enough examples and no solutions manual. If Cummings made a solutions manual, this is what I'd go for.

>>16858552
So that's why he started selling his books on ebay: to fund his Phillipines trip. Well, he looks happy.

>>16861565
>I've just finished working through Velleman's How to Prove It
Holy shit, are you me? I'm finishing the book as well. Is a calculus book (Spivak's) a mandatory read before I jump into real analysis?

>>16861569
Based Vellemanbrother. I'm not sure if a Calc book is mandatory. But I do know in certain European countries like France and Italy, they just jump straight into Analysis without doing the Calc 1,2,3 thing that North Americans do.

>>16861565
>>16861569
Have you already completed a calculus sequence? Generally you'll use something like Stewart, but you can supplement with Spivak and Rudin in the process. While some very advanced undergraduates can use Rudin and Spivak early on, if you have to ask here I wouldn't use them as primary texts there, wait until you take real analysis to really use them, for now they should just be references for you.

Now if you've already completed your undergraduate calculus sequence, nothing in Rudin should be too difficult.... You should have already taken a number theory course, maybe a discrete math course, and the basic copy paste introductory theorems and lemmas that apply should have also been reinforced in every single course up to this point. I hear quite often people complaining about real analysis in their junior or senior year, but frankly I found it pretty mid. Only thing I found actually difficult were modern algebra courses and Galois theory, but what is difficult, but something that we don't find interesting enough to devote plenty of time to?

>>16861577
This isn't true.... They just call their calculus sequence "Analysis"
The universities in Europe starting with real analysis are just the same as the top schools in the US where students can also start with real analysis and do all their graduate school prelims day 1 of grad school. So yes at Warwick a guy will start with real analysis, but that is not common for math programs.
I'm not sure why this keeps being repeated like a mantra, or frankly, why European math students have such a chip on their shoulder they need to continuously state such things.

>>16861586
Second this.
Most students don't do analysis until they have gone through a series of more computational courses in calculus.
No need to be a hero. Supplement with some number theory or combinatorics if you're craving more "proofy" feeling work.

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Use straight lines to connect all the vertices of a (regular) polygon which has an odd number of sides. A small version of the same polygon appears in the middle. What is the ratio between the area of the little polygon and the area of the whole polygon for the N:th polygon with an odd number of sides?

>>16861532
The AI assistant which I used wrote:

To determine for which integer values of n>2 the function f(n)=(2cos(2π/n)+1)^2/(4cos(2π/n)+5) is rational, we analyze the cosine values of specific angles and their impact on the function.

Key Angles and Cosine Values:
For n=3, cos(2π/3)=−1/2.
For n=4, cos(2π/4)=0.
For n=6, cos(2π/6)=1/2.

Evaluating f(n):
For n=3, f(3)=[...]=0.
This is rational.
For n=4, f(4)=[...]=1/5.
This is rational.
For n=6, f(6)=[...]=4/7.
This is rational.

Checking Other Values:
For other values of n such as n=5,7,8,10,12, the cosine values cos(2π/n) are irrational, leading to irrational results for f(n).

Conclusion:
According to Niven's theorem, the only rational values of cos(2π/n) for integer n>2 are 0,±1/2. These correspond to n=3,4,6.
Thus, the values of n for which f(n) is rational are 3, 4, and 6.

>>16861565
>has a full solutions manual
Seriously just fucking give up on math.

>>16861724
What he wants is Spivak, it comes with a solution manual. But it seems pointless when all of this stuff is trivial enough you can just google it or have an AI explain it to you.

>>16861725
Spivak is bloated garbage. Read that if you have no respect for your own time.

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What's the best resource to understand high dimensional spaces.

I have stats and CS in mind, but not only.

>>16861586
>>16861590
I'm >>16861565
I've worked through Thomas' Calculus book up to and including the Infinite Series chapter. The chapters after that are "Calc 3" stuff, which I haven't started on yet.
>>16861724
Nope. I'm not giving up. Two years ago I couldn't add or subtract fractions. In those 2 years I've done 8 chapters of Thomas' Calc book, all of Lay's Linear Algebra book, all of Velleman's proof book, all of Professor Leonard's ODE playlist, a bit of Group Theory and some of Silverman's Number Theory book. I will not give up. If you don't want to help me, don't reply.

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What's the morally right definition of the residual?

[math] r_1 := X\beta - y [/math]

[math] r_2 := y - X\beta [/math]

Suppose you're working with the motivic Hall algebra H associated to a quiver algebra A.
You have a nongeneric stability condition t on the category of A-modules. How do you characterise the class in H of the stack of all t-stable A-modules? What's an equation that involves it?

>>16861748
I mean there's a few of them.

I think "Bott periodicity" is probably the best introduction. It teaches you how spheres work in higher dimensions. Particle physics and the laws of the universe basically revolve around the 1 and 2-sphere.

A sphere is a fascinating object and I don't know if it's the best or worst introduction, but it's how the world works. A sphere is equidistant from every angle and every sphere is embeddable in the one that's one dimension above it. Fascinating stuff.

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>>16856291
i will fucking fail again because i just can't fucking study we didnt go over this shit i dont have fucking notes for this the dumb cunt just expects me to know how to solve this i cant even use llms to fucking explain this shit to me because it uses some black magic i fucking hate all of this what the fuck am i gonna do after i fail i guess i'll have to work construction or i'll starve to death

>>16861973
ive been so fucking depressed i couldnt even touch the books for 2 weeks i've been procrastinating the whole day today because knew what was gonna happen and it's happening all over again same shit

>>16861975
how the fuck am i supposed to know if i'm doing problems correctly if i don't even have the answers??????

i'm so fucking tired

>real analysis not offered during summer semester, have to take matrix theory canonical forms mumbo jumbo course instead

over status?

So i am a retarded and i don't know nothing about math, Where should I begin?

>>16862013
Lang's basic mathematics.

>>16861668
a/A = (s/S)^2

N = 3 —> a/A = 1
N = even —> a/A may = 0
N = 1/0 —> a/A = 0

>>16861748
>stats & CS
https://www.math.uci.edu/~rvershyn/papers/HDP-book/HDP-book.html

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Learn from uni or learn from tutor + ai and ace shit when i go back cause i know it already?

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You build this regular octagon shaped thingy out of one unit long rods so that two adjacent rods always make a 90 degree angle like in picrelated. Then you dip it in soap water and the soap creates this minimum surface.

How do you find the surface area of that minimum surface?

>>>/g/107417397
What is he on about?
My mind procures a bunch of things. It's like group theory mixed with game theory and the word "cyclotomic extension".

>>16862299
thx, looks quite nice

>>16858552
>next up
>the divorce and alimony sorcerer

>>16856291
study a subject that you actually like
not a subject that makes you feel miserable

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>>16862695
calculus 1 is mandatory
i've been studying for ~3 hours and i have about 7-8 left
if i'm not a retard it should be enough to do OK but i'm ANXIOUS

>>16862537
Jacobian integral

>>16862537
Minimal surfaces for arbitrary closed polygonal chains are very difficult and unsolved

>>16862537
Cut it into eight (or sixteen if you prefer) equal pieces symmetrically around the origin (like cutting a cake). Any individual piece can be solved using the helicoid rule.

>>16862705
yesterday i skipped lecture and seminar or whatever you call that in english, got home by 1pm, went to sleep, woke up around 8pm and have studied 9pm-7:30am then i had a test then i had 4 hours of calculus
holy fuck i am done, i think i did somewhat ok though but i'm too fucking demoralized and tired to care

Anybody use math in data analytics?

Let [math]X,Y[/math] be topological spaces and [math]A_1,A_2,\ldots[/math] be closed subsets of [math]X[/math] such that [math]X=\bigcup_{n=1}^{\infty}A_n[/math]. Let [math]f_n\colon A_n\to Y[/math] be continuous (where [math]A_n[/math] is provided with the subset topology) such that for all [math]m,n\in\mathbb N[/math] and [math]x\in A_n\cap A_m,\ f_m(x)=f_n(x)[/math]. Define [math]h\colon X\to Y[/math] by [math]h(x)=f_n(x),x\in A_n[/math]. Does [math]h[/math] have to be continuous?

>>16863338
No, consider [math]X = \omega \cup \{\infty\}[/math] where [math]\omega[/math] is discrete and [math]\infty[/math] is a limit point of [math]\omega[/math]; any continuous function from [math]X[/math] to [math]Y[/math] must send [math]\infty[/math] to the limit of the image of [math]\omega[/math]

>>16863129
can someone tell me i did a good job

>Statistical data analysis
>Matrix Theory
>Real Analysis
>Monte carlo methods
>Theory of machine learning
>Probability and Stochastic process I
>Probability and stochastic process II
>Stochastic differential equations
>Game Theory
>Stochastic optimization and control

uh bros, are we so back or so over?

>>16863382
you did a terrible job, cramming is stupid

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>>16863435
i have never in my life had to study for anything it's hard to do it
i'm also mildly autistic so i have the mental capacity to hyperfocus
i have 2 weeks before the next test and as always i will pretend that i'm gonna study and cram at the last moment as always anyway

>>16863433
Most masculine and testicular list I have ever seen.

>>16863439
admitting that what you're gonna do is stupid doesn't make it any less stupid

>>16863444
you think this is a choice?

>>16863447
Absolutely. You could study for 60 minutes starting right now -- you're about to choose whether you do or not.

>>16863382
You did very good, anon. You're very smart and your mother and I love you very much.

>>16863354
what

>>16863514
what part are you confused about?

>>16863523
>>16863514
desu I did overcomplicate it, any non-discrete countable T1 space will do it

>>16861802
The morally right way is:
[math]
r = h(X) - Y
[/math]
where [math] Y [/math] is the true random variable, [math] X [/math] is your observation, and [math] h : X \to \hat{Y} [/math] is your modeling function. The two forms you wrote are just rewriting of the linear-Gaussian special case.

>>16863523
how does the second half follow

>>16863562
because [math]\infty[/math] is the limit point of the sequence [math](0,1,2,\ldots)[/math] so if [math]f : X \to Y[/math] is continuous, [math]f(\infty)[/math] is a limit point of [math](f(0),f(1),f(2),\ldots)[/math] because that's how continuous functions be.

>>16863433
>data analysis
Automatically a shit list.

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can anybody explain the method of arranging the stack of columns based over the product

what method goes in their order?

>>16863653
Most feminine and vaginized post I've ever seen.

If an intro to linear algebra book has a calculus prerequisite what's the typical overlap with calc 1, 2, and 3? Can you get away with studying calculus at the same time or will it likely require stuff from calc 3 as soon as any linear algebra in a calculus context is introduced?

>>16861802
It doesn't matter, because you would be taking their norm.

>>16863543
No, the morally right way is [math] d(h(X), y) [/math], where [math] d [/math] is the metric in the space of regressand. The form you wrote is just Euclidean special case.

>>16863771
Linear algebra---even graduate level---mostly doesn't require calculus at all.

>>16863868
I know it's not a hard prereq but when reading the preface of many intro LA books they'll say something a long the lines of "calculus isn't a prerequisite for linear algebra but we've included many interesting and useful applications of linear algebra to calculus"

>>16863433
>monte carlo methods
>stochastic optimization and control
If they are applied, very nice.

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>>16835273
Wait, none of that is math?

Like, not even type theory? I thought proof assistants use type theory or some shit. I'm not a programmer, but someone once told me something like that.

>>16862320
I have a question related to what you're asking about. I want to learn as much as I can on my own, and I've already downloaded a lot of the recommended texts posted in the /sci/ sticky thread, and other threads. But I was wondering if there's any website out there with free tracks for learning mathematics that aren't as shitty as Khan Academy?

>>16864330
libgen to download books
LLMs sites to ask when you get stuck

Let [math]X[/math] be a topological space and [math]Z[/math] a dense subspace of [math]X[/math]. Let [math]Y[/math] be
a Hausdorff topological space. Show that if two continuous maps [math]f, g\colon X\to Y[/math] agree
on [math]Z[/math], they must agree on the whole of [math]X[/math]. Does this remain true if you drop the
assumption that [math]Y[/math] is Hausdorff?

>>16864367
What have you tried?
If the problem seems to difficult you can first try to prove a special case with specific spaces X,Y and Z and then try to generalize the proof for the general case.

>>16864368
I have tried nothing and I want you to solve the problem for me.

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Can someone please explain simply and plainly in qualitative terms the inuition behind Itō Calculus (and especially Itō's Lemma), how it works, and why it's needed?

t. non-mathfag retard

>>16864413
*intuition

>>16864345
Did you even read my post dude?

>>16864370
Why?

>>16864370
Suppose Z is a dense subspace of X. Let Y be a Hausdorff space and suppose f, g : X -> Y are continuous maps which agree on Z. Because Z is dense, it is obvious that if x is any element of X, it must be the case that f(x) = g(x). On the other hand, if Y is not Hausdorff, then it is not true that any two points in Y can be separated by disjoint open sets, which easily allows one to construct a pair of maps continuous maps f, g : X -> Y which agree on a dense subspace but not on all of X

>>16864367
Using both f and g you construct the diagonal function [math]h := (f, g) : X \to Y\times Y[/math] which is continuous (exercise, maybe?). The condition that they agree on [math]Z[/math], means that the image of [math]Z[/math] via [math]h[/math] lies in the diagonal, which is closed (because [math]Y[/math] is Hausdorff).
Now take preimage of the closed diagonal and is a closed set which contains [math]Z[/math], that is, is the whole [math]X[/math].

For a counterexample, take [math]Z[/math] to be any dense subset different from [math]X[/math] and [math]Y = \{ 0, 1 \}[/math] with the indiscrete topology. Any function towards [math]Y[/math] is continuous so just take one that has values 0 in [math]Z[/math] and 1 outside, vs the constant 0 function.

>>16860623
I don't quite understand your notation, but the constant sheaf [math]\underline{G}[/math] with values at the group (or set) [math]G[/math] has sections [math]\Gamma(U, G)[/math] the direct sum of [math]G[/math] for the connected components of [math]U[/math]. This is equivalent to say that it is the group of continuous functions [math]U \to G[/math] where [math]G[/math] is endowed with the discrete topology.

Suppose your base space [math]X[/math] is connected (if not, just take direct sum over the connected components as before).
I claim that the sheaf homomorphisms are in bijection with [math]Hom(G, F(X))[/math]. To prove it, notice that if [math]U[/math] is a connected open set, it will also have the same values, as each section of the constant sheaf is the restriction of the global section. If not, then it will be a direct sum of restrictions. Either way, you realize this is the only way to construct such homomorphisms, so it must be [math]\Hom(G, F(X))[/math].

Change "direct sum" for "disjoint union" (and interpret [math]Hom[/math] as the set of functions) if you are working with set-valued sheaves.

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Is it possible to prove this number is irrational?

>>16864592
https://www2.oberlin.edu/faculty/jcalcut/tanpap.pdf
see proposition 2

>>16864592
I guess the roots in the divisible would fold into something simple (look there are some patterns there), and then you'll apply arctan and get a multiple of Pi. Pi will eliminate and the answer would be rational.

>>16864413
I don't think I can but I'm willing to try something.
The rule to remember is "[math](dW)^2=dt[/math]" for the lemma.
People like the Ito integral because it's nonanticipating, but something like the Stratonovich integral has no correction term.

It seems to me that taking [math]X_t[/math] as the value to plug into a function [math]f[/math] when approximating over an interval [math][t,t+\delta t][/math] will underestimate when [math]f[/math] is convex.
BM will "wiggle" a lot over the interval [math][t,t+\delta t][/math] which should yield, on average, slightly higher values for eg [math]f(x)=x^2[/math].
The extra Ito term corrects for this by working out the bias of the Ito integral definition up to first order and just adding it.

>>16864778
I'd think the explanation would at least mention stuff like the diffusion equation.
desu I don't know why if someone asks for the intuition for the Ito calculus you'd list off technical aspects of it that distinguish it from other stochastic integral calculus variants.

>>16863865
I mean we can say "it doesn't matter", sure, but in the end we want to communicate and settle for convention, hopefully somewhat motivated, so that we don't have to search for the right minus signs in formulas everywhere when we switch between contexts or books.
E.g. if you write down how to compute the minimum of costs involgin ||r||^2, this distance will float around on the page and if you summarize it to "r" then you want to know if the formula is using -r or r.

That said I take >>16863543 into consideration but I also don't consider that post as having given an argument for Y last. It's a mere proclamation whether the estimator of Y can be generalized to some more abstract h or not.

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>>16864592
(18*√74 + 80)^(1/3) – (18*√74 – 80)^(1/3)
= 2*√26*Sinh[ArcSinh[10*(2/13)^(3/2)]/3]

Consider the functions [math]f,g:\mathbb R^2\to\mathbb R[/math] given by [math]f(x,y)=\begin{cases}\frac{x^2-y^2}{\sqrt{x^2+y^2}},\text{ if }(x,y)\neq(0,0)\\0\text{ if }(x,y)=(0,0)\end{cases}[/math] and [math]g(x,y)=\begin{cases}\frac{2xy}{\sqrt{x^2+y^2}}\text{ if }(x,y)\neq(0,0)\\0\text{ if }(x,y)=(0,0)\end{cases}[/math]. On [math]S^2[/math], consider the equivalence relation generated by [math](x,y,z)\sim(-x,-y,z)[/math] and let [math]X=S^2/\sim[/math] with the quotient topology. Using the map [math]F\colon\mathbb R^3\to\mathbb R^3[/math] where [math]F(x,y,z)=(f(x,y),g(x,y),z)[/math], show that [math]X[/math] is homeomorphic to [math]S^2[/math].

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If I have N samples of a random variable, how do I predict how much my estimate of the mean will change by adding M additional samples to my population?

>>16864791
I read the question as "why does Ito calculus have that extra term vs ordinary calculus" and probably it's good to mention that it's not necessary at all and is more a consequence of the particular way it is defined rather than the underlying stochastic stuff.

>>16838280
maths gore, goodness

>>16865027
What do you know about the distribution?
My naive thought would be to look at differences of standard errors/variances.

>>16864798
>you want to know if the formula is using -r or r.
No, you don't because it doesn't fucking matter. It's the same fucking formula.

>>16865027
(N*x + My)/(N+M) - x is the change of your est mean for the first N samples alone, x, and y is the est mean for the M additional samples alone.