𧔠/mg/ math general
Anonymous at Mon, 17 Feb 2025 22:29:29 UTC No. 16589624
Required Reading edition
Talk mathematics, formerly >>16562509
Anonymous at Mon, 17 Feb 2025 22:30:20 UTC No. 16589625
>>16589624
First for Verbitsky and Kaledin's Trivium
>>16568704
>>16569867
>>16573937
Anonymous at Mon, 17 Feb 2025 22:33:24 UTC No. 16589632
>>16589625
This is the other document. There's no order, you can approach both of them concurrently
Anonymous at Mon, 17 Feb 2025 22:47:37 UTC No. 16589647
>>16589409
>Why are compact spaces defined in such a peculiar way (space is compact iff every open cover of space has a finite subcover)?
Hey, I was taught that compact spaces are T2 (Hausdorff) + finite subcover condition. Was my life a lie?
Anonymous at Tue, 18 Feb 2025 01:38:24 UTC No. 16589755
>>16589625
>>16589632
where's the other one?
Anonymous at Tue, 18 Feb 2025 02:23:21 UTC No. 16589789
>>16589625
>>16589632
Algebra 9, question 18 is messed up but I'll write the question here as best as I can:
Let [math] v \in R [/math] be an Artinian ring over [math] k [/math] and
[math] P(t) [/math] be its minimal polynomial. Consider the subalgebra
[math] k[v] \subset R [/math] generated by [math] v [/math] and [math] k [/math]. Prove that [math] R_v [/math] is isomorphic to the ring
[math] k[t] / P [/math] of residues modulo [math] P(t) [/math].
Are there any other errors in either of them?
Anonymous at Tue, 18 Feb 2025 06:03:19 UTC No. 16589883
>>16589755
There's only two of them.
Anonymous at Tue, 18 Feb 2025 09:34:59 UTC No. 16589962
>>16589883
Why are the documents called trivium then?
Anonymous at Tue, 18 Feb 2025 09:40:17 UTC No. 16589964
>>16589962
>trivium: singular of trivia; anything of little importance
Anonymous at Tue, 18 Feb 2025 13:09:13 UTC No. 16590143
>>16589962
They are a "must-know" set of problems. V. Arnol'd also has a trivium but for applied math/physics students:
https://physics.montana.edu/avoront
Anonymous at Tue, 18 Feb 2025 13:12:12 UTC No. 16590146
>>16589647
No, that's a common definition. FSC by itself is sometimes called quasicompact to distinguish.
Anonymous at Tue, 18 Feb 2025 14:58:35 UTC No. 16590240
Can someone explain how term proofs work in lean 4? I get tactics proof but term proofs are so opaque to me.
Anonymous at Tue, 18 Feb 2025 23:30:49 UTC No. 16590876
>>16590240
This is too broad of a question to answer very meaningfully. Every proof in lean, even with tactics, is constructing a term of a specific type, it's just that tactics are some boilerplate corresponding to common proof patterns that generate terms for you. If you haven't read it already, this is the reference you need to start understanding type theory in Lean properly: https://lean-lang.org/theorem_provi
Anonymous at Wed, 19 Feb 2025 12:34:06 UTC No. 16591373
I envy people who love math.
Anonymous at Wed, 19 Feb 2025 15:10:29 UTC No. 16591461
>>16591373
I use to love math, but Iâm not sure if I do anymore. Itâs been beaten out of me by my career. Hard to find anyone in the workforce who cares about abstractions
Anonymous at Wed, 19 Feb 2025 15:25:07 UTC No. 16591471
>>16591461
Math is the ideal hobby T B H.
Anonymous at Wed, 19 Feb 2025 15:29:44 UTC No. 16591474
>>16591373
It's the only thing i genuinely like in the world. I wish i liked other things too.
Anonymous at Wed, 19 Feb 2025 22:09:54 UTC No. 16591888
>>16591759
i would have used different colors for the triangles
like red, green, blue and white
or magenta, yellow, cyan and black
Anonymous at Thu, 20 Feb 2025 01:11:23 UTC No. 16592026
Does Zorn's lemma imply excluded middle?
Anonymous at Thu, 20 Feb 2025 04:18:25 UTC No. 16592141
>>16592026
Ok, apparently it does not.
Anonymous at Thu, 20 Feb 2025 17:03:04 UTC No. 16592976
What's a good pathway to go from knowing only basics of probability to taking a graduate course on probability next semester? I know the analysis pre reqs but the actual probability i'm very sketchy on. I'd like to really learn it well if possible. Any text recommendations that include exercises would be greatly appreciated, thanks!
Anonymous at Thu, 20 Feb 2025 17:04:40 UTC No. 16592980
Do you think you have to be an intellectual masochist to be a good researcher? I'm starting to think so
Anonymous at Fri, 21 Feb 2025 07:54:36 UTC No. 16594532
>>16592976
I'd say measure theory is the main difference between intro probability and graduate level probability. I'm a big fan of "Probability: Theory and Examples" from Rick Durrett. Starts with strong foundations, works up to lots of useful topics. Lots of exercises
Anonymous at Fri, 21 Feb 2025 07:57:43 UTC No. 16594534
>>16592026
Yes, if we're assuming ZF set axioms.
Under assumption of ZF, the Zorn's lemma is equivalent to axiom of choice. We then have Diaconescu's theorem (aka Goodman-Myhill theorem) which shows that AoC -> excluded middle.
Anonymous at Fri, 21 Feb 2025 08:01:10 UTC No. 16594537
>>16589632
Why is this labeled "Geometry"? Reads more like a topology book to me.
Anonymous at Fri, 21 Feb 2025 08:06:30 UTC No. 16594543
>>16589624
Some stuff to add to the list: Aluffi's "Algebra: Chapter 0" for modern algebra with a category-theoretic flavor.
Linear Algebra Done Right for more advanced linear algebra stuff.
Hatcher for Algebraic Topology
Anonymous at Fri, 21 Feb 2025 09:03:04 UTC No. 16594573
>>16594534
Zorn's lemma implying choice requires excluded middle
Anonymous at Fri, 21 Feb 2025 11:52:29 UTC No. 16594690
Is there a weaker version of Zornâs Lemma that is equivalent to the Axiom of Dependent Choice? My intuition tells me that there has to be some restriction made, maybe every chain with an upper bound has to be countable?
Anonymous at Fri, 21 Feb 2025 16:41:49 UTC No. 16595034
Teach me non linear analysis sci
Chaotic introductions and advanced levels
Anonymous at Fri, 21 Feb 2025 20:50:37 UTC No. 16595431
>>16589789
Any other errors in the algebra and/or geometry pdf?
I only found one in algebra 9, but the others seem okay.
Anonymous at Sat, 22 Feb 2025 02:50:42 UTC No. 16595876
Why doesn't algebraic thing have property p?
>Here's the proof.
Do geometric things a,b,c, etc have property q?
>Here's the proof.
Why don't most PDEs have closed form solutions?
>They just don't okay? No we don't have any existence disproofs or impossibility theorems or any ways to measure the space of closed-form anythings, you've just got to accept that you can't do it on faith. If it was doable it would have already been done.
This is my prof, unironically. Where do I go to find the actual existence/nonexistence theorems about the big picture of PDEs?
Anonymous at Sat, 22 Feb 2025 03:30:46 UTC No. 16595895
>>16594537
>https://en.wikipedia.org/wiki/Orde
https://en.wikipedia.org/wiki/Order
>https://en.wikipedia.org/wiki/Metr
https://en.wikipedia.org/wiki/Categ
Anonymous at Sat, 22 Feb 2025 07:32:55 UTC No. 16596007
>>16595034
chaos is so 80s, gramps
just pretend everything is a harmonic oscillator and perturb away
Anonymous at Sat, 22 Feb 2025 07:34:23 UTC No. 16596008
>>16595876
You can prove existence without requiring closed-form solutions.
Anonymous at Sat, 22 Feb 2025 08:00:24 UTC No. 16596016
>>16595876
I'd like to point out that the world is even messier than your question suggests. As a matter of fact, there are proofs that most ODE's do not have closed form solutions in terms of elementary functions.
Consider that the solution [math] f [/math] to the 1st order ODE [math] \frac{df}{dx} - g(x) = 0 [/math] for some given function [math] g [/math] is equivalent to the computation of an antiderivative of [math] g [/math]. Well we all remember how hard it is to take integrals of arbitrary functions [math]\mathbb{R} \to \mathbb{R} [/math] or [math]\mathbb{C} \to \mathbb{C} [/math]. Many of us have taken multiple courses dedicated to the solution of that particular ODE. Moreover, Liouville's theorem in differential algebra essentially states that the only functions with elementary (closed form) antiderivatives are those we can write as the sum of rational functions and finitely many logs of rational functions.
Anonymous at Sat, 22 Feb 2025 12:56:11 UTC No. 16596289
>>16590876
Explain how the convert tactic works.
Anonymous at Sat, 22 Feb 2025 14:53:53 UTC No. 16596402
>>16596289
It works by taking your prompt and redirecting it into chatgpt until you have something less imbecilic to ask about.
Anonymous at Sat, 22 Feb 2025 22:21:41 UTC No. 16596819
>>16596402
I'm sorry anon.
I'm not smart like you
Anonymous at Sat, 22 Feb 2025 22:36:54 UTC No. 16596837
>>16589624
I just started reading the Mary Dolciani books and I'm surprised you all haven't shilled her books before, starts off with logic,set theory, and mixes proofs into a lot of the material.
Modern Algebra: Structure and Method Book 1
Modern Geometry: Structure and Method
Modern Algebra and Trigonometry: Structure and Method Book 2
Modern Introductory Analysis
All from the 60s-70s during the "New Math" movement, later revisions in the 90's drop the "Modern" in the title and aren't as good apparently. Throw on spivak and you have an Algebra 1 to Calculus curriculum.
Anonymous at Sun, 23 Feb 2025 17:06:06 UTC No. 16597422
>>16596837
Better to just use the 6 volumes from Hung-Hsi or the 4 grade8-12 Art of Problem Solving books.
Pedagogy has evolved substantially since then. I don't think they cover set theory or basic logic the way those seem to do.
Anonymous at Sun, 23 Feb 2025 17:13:56 UTC No. 16597429
>>16597422
>Hung-Hsi
redpill me on his work
Anonymous at Sun, 23 Feb 2025 18:17:04 UTC No. 16597484
>>16597429
He works on math education reform, fighting against a lot of the over emphasis on computation, âtricksâ and rote in modern American schooling, and advocates for a return to a âNew Mathâ-esque approach like we had in the 50-70~.
The 6 books he wrote in the volume are all of the math that must be taught in the K-12 common core, explained with proofs/logically explained with minimal/no tricks involved. Starting with place value and arithmetic up to calculus. The books are primarily intended for math teachers so they themselves actually understand what they are teaching their own students, but itâs accessible by anyone. Most recent one was released in 2020 I think. I used them to relearn everything I forgot in the 7 years I was out of high school to go back to college. He puts out papers every once and a while talking about the things he hates about modern math education. I got shilled him through a random YouTube comment I read when watching a how to relearn math video, then got the books off libgen. I plan on buying the whole set eventually.
Him and the AoPS Grade 5-12 are basically the gold standard for self teaching K-12 math IMO. Throw in an Intro to Logic/Proof book and youâre pretty much ready for any math concept past calculus and could just straight to Spivak. Not sure you could get a deeper understanding of K-12 math as efficiently any other way.
Anonymous at Sun, 23 Feb 2025 19:28:20 UTC No. 16597543
>>16597484
Thank you for the throrough reply, I think I'll go with the last three books of his.
Anonymous at Mon, 24 Feb 2025 00:50:15 UTC No. 16597824
Take the natural logarithm of (n+1)!. Take the natural logarithm of the result. Keep repeating natural logarithms iteratively until you get a negative real number and then stop. Color the interval containing that number red.
Will every negative interval eventually get colored red if you do the above process for all n?
Anonymous at Mon, 24 Feb 2025 02:53:03 UTC No. 16597916
>>16597824
No. Every penultimate iteration is a number between 1 and e, and the final iterations only get to the far reaches of the negative real line if the penultimate iterations are arbitrarily close to 1. Iterated logarithms of integers never approach an integer in the limit as iterations increase - case closed.
Anonymous at Mon, 24 Feb 2025 17:07:08 UTC No. 16598420
https://www.wolframalpha.com/input?
Anonymous at Tue, 25 Feb 2025 06:36:28 UTC No. 16599028
>>16598262
the magnitude of the bottom is about |z|^3 and the magnitude of the top is about |z|^2 as |z| grows large. how are apparently doing complex analysis and still missed that lol
Anonymous at Tue, 25 Feb 2025 12:19:21 UTC No. 16599201
Any good grad level textbooks on diff geo that cover Lie groups in great detail? The material on Lie groups and algebras is usually algebraic in nature and skips the topology and geometry bits.
Anonymous at Wed, 26 Feb 2025 16:00:21 UTC No. 16600202
>>16589624
I'm getting absolutely filtered by An Introduction to Optimization on Smooth Manifolds by Boumal ATM. The first 4 chapters weren't so bad, but I'm really stuck on the second order embedded geometry parts.
Do you think it's worth spending time going through a proper point-set topology/topological manifolds book? My only real topology exposure has been in the context of analysis and I think I would probably have an easier time understanding this if I was more well versed in general topological manifolds.
Not taking this for a class btw. Just looking into smooth manifolds methods for solving a research problem as an alternative to the classical convex analysis/NLP based approaches I'm used to.
Anonymous at Wed, 26 Feb 2025 16:33:01 UTC No. 16600222
Is there a way to define the integers up to (unique) isomorphism that's similar to how one can define the naturals as the unique model of (second order) peano arithmetic?
I'm aware that Z is the initial ring or that it's the free group on one generator (similar to how N is the free monoid on one generator) and that does define Z uniquely, but I'm looking for a definition that doesn't speak of addition altogether (like how N can be nailed down uniquely just by talking about zero and succession).
Anonymous at Wed, 26 Feb 2025 16:42:28 UTC No. 16600227
>>16600222
>N can be nailed down uniquely just by talking about zero and succession
throw predecession into the mix
Anonymous at Wed, 26 Feb 2025 18:31:20 UTC No. 16600296
I don't even like math that much but it's still somehow the most enjoyable thing in my life.
Anonymous at Fri, 28 Feb 2025 00:43:15 UTC No. 16601526
>>16591373
i love math. i make passionate love to her every night
Anonymous at Fri, 28 Feb 2025 13:53:26 UTC No. 16601909
>>16601526
prove it
Anonymous at Fri, 28 Feb 2025 18:43:53 UTC No. 16602109
Hello, can anyone link me that book list that people like to shill here since I've forgotten it's name and the exact contents? It starts out pretty normal, but it focuses on algebraic geometry quite early.
Anonymous at Sat, 1 Mar 2025 01:21:07 UTC No. 16602388
>>16602109
Take a look at the archive:
>>16534183
https://warosu.org/sci/thread/16534
Anonymous at Sat, 1 Mar 2025 04:00:06 UTC No. 16602464
>>16600222
It's very tedious but you can define spherical standing waves using just N and multiplication, then define Z as the unique inverse domain to certain types of fourier series on them. You need a lot of bookkeeping to make it work with infinite products instead of infinite sums but you get to Z without a single smidgen of addition.
Anonymous at Sat, 1 Mar 2025 13:48:15 UTC No. 16602693
(x â a)*(x â b)*(x â c) = x^3 â 2221*x + f
a, b, c, and f are integers
find the sum |a| + |b| + |c|
Anonymous at Sun, 2 Mar 2025 18:28:07 UTC No. 16604223
Greetings sci
I want to start studying formally Computational Topology, but I'm lost on which books, courses, notes, lectures, etc, are available online that may be helpful. I have background in abstract algebra, functional analysis, topology, and algebraic topology. I'm currently reading Jhon Harer and Wei Xiong, but I'm lost on the road I have to follow. Any recommendations may be invaluable.
Anonymous at Mon, 3 Mar 2025 05:48:40 UTC No. 16604710
>>16591373
I love math but math hates me.
Anonymous at Mon, 3 Mar 2025 05:55:19 UTC No. 16604718
>>16591759
why are the lines scrunchy
Anonymous at Mon, 3 Mar 2025 11:03:43 UTC No. 16604850
Given pic related, doesn't that imply that the continuum must also be lesser than [eqn]\aleph_{\omega_4}[/eqn] if [eqn]\aleph_{\omega}[/eqn] is a strong limit? Doesn't that contradict Easton's Theorem?
Anonymous at Mon, 3 Mar 2025 17:12:08 UTC No. 16605159
>>16604710
>but math hates me
true
Anonymous at Tue, 4 Mar 2025 07:17:33 UTC No. 16606715
>>16604850
If you make the [math]\mathfrak{c}[/math] bigger than [math]\aleph_\omega[/math], [math]\aleph_\omega[/math] stops being a strong limit. [math]\aleph_\omega[/math] being a strong limit means [math]\aleph_\omega = \beth_\omega[/math] and we always have [math]\beth_\omega > \beth_3> \beth_2> \beth_1 = \mathfrak{c}[/math].
Anonymous at Tue, 4 Mar 2025 15:28:29 UTC No. 16606981
>>16604326
Seems fine, though I don't know how Comm. Alg., Alg. Geom. and Lie Groups are applied; here economics are listed but I've never heard of such use.
Also you should add some books for courses. Here are some I've encountered with a more "applied approach":
Lawrence C. Washington - Elliptic Curves: Number Theory and Cryptography
Henri Cohen - Number Theory. 2 Volumes
Jean-Pierre Serre - Linear Representations of Finite Groups
Steven Vickers - Topology via Logic
For Stage I, almost all the topics are quite standard and pretty much any book works. Hope this is helpful
Anonymous at Tue, 4 Mar 2025 17:09:12 UTC No. 16607086
>>16606981
Vickers' text does not strike me as a particularly applied book, and I also don't think the average mathematician should need to read a book with such a specific focus (unless locales, sober spaces and all that stuff is relevant to whatever they're specializing in)
Anonymous at Wed, 5 Mar 2025 01:01:08 UTC No. 16607509
What math things do you want to be doing after you die?
Anonymous at Wed, 5 Mar 2025 01:59:14 UTC No. 16607567
(x â a)*(x â b)*(x â c) = x^3 â 2221*x + f
(x â a)*(x â b)*(x â c) = x^3 â 2143*x + f
(x â a)*(x â b)*(x â c) = x^3 â 2011*x + f
(x â a)*(x â b)*(x â c) = x^3 â 1933*x + f
(x â a)*(x â b)*(x â c) = x^3 â 1873*x + f
(x â a)*(x â b)*(x â c) = x^3 â 1831*x + f
(x â a)*(x â b)*(x â c) = x^3 â 1801*x + f
a, b, c, and f are integers
|a| + |b| + |c| = 98
Anonymous at Wed, 5 Mar 2025 03:00:38 UTC No. 16607608
>>16607567
a, b, c, a*b + a*c + b*c
Anonymous at Thu, 6 Mar 2025 00:05:27 UTC No. 16608242
I'm a 30 year old slow ass dude that has always hated math and only knows that 2 + 2 equals 4 but I want to learn, where do I start? where does a retard like me learn given that it takes me ages to learn things
Anonymous at Thu, 6 Mar 2025 00:25:06 UTC No. 16608258
>>16608242
>2+2=4
thatâs only true in certain algebraic structures, not all of them. The classic counterexample is the field of 3 elements.
Anonymous at Thu, 6 Mar 2025 00:27:24 UTC No. 16608264
>>16608258
I don't even know what the hell you just said, I just want to start learning math slow and easy on my own
Anonymous at Thu, 6 Mar 2025 00:52:20 UTC No. 16608276
>>16608264
Is 2+2=4 really all you can do or can you at least add basic fractions like [math]\frac{1}{4} + \frac{3}{8}[/math]?
If you can manage fractions self-study is viable, check out openstax. If you can't, your school system massively failed at their job to teach you and you need to attend some kind of adult remedial education program.
Anonymous at Thu, 6 Mar 2025 01:06:11 UTC No. 16608282
>>16608276
no, can't do fractions
Anonymous at Thu, 6 Mar 2025 01:41:36 UTC No. 16608305
>>16608264
>I don't even know what the hell you just said, I just want to start learning math slow and easy on my own
There's a guide for your. A little overkill. You can bootstrap the process by doing Khan Academy too (feel free to skip the vids, what you really need is the practice)
https://4chan-science.fandom.com/wi
Anonymous at Thu, 6 Mar 2025 02:50:19 UTC No. 16608364
If I'm unable to understand even basic stuff like Linear algebra, I should quit math and kill myself, right?
Anonymous at Thu, 6 Mar 2025 03:06:33 UTC No. 16608374
>>16608242
In-person community college classes
Anonymous at Thu, 6 Mar 2025 03:10:41 UTC No. 16608381
>>16608364
>I should quit math and kill myself, right?
No. Switch to an easier/gentler book and practice until you can achieve the same level as the original reference. That's all there is to it.
Anonymous at Thu, 6 Mar 2025 03:10:53 UTC No. 16608382
I was thinking about math too physically. Math is based on definitions and logic, not physics.
Anonymous at Thu, 6 Mar 2025 03:32:57 UTC No. 16608402
>>16608364
No, definitely not. You probably just need an easier book.
I'll give you an example. I'm a doctoral candidate whose research focuses heavily on optimization/convex analysis and applied functional analysis. A relevant research problem led me to checking out an Optimization on Riemannian Manifolds book and I figured it wouldn't be too hard to catch up given that the book states it doesn't assume the reader has differential geometry experience and provides all of the differential geometry background the student should need.
It kicked my ass, and I decided to eat my humble pie and start working through the suggested "undergrad" reference (O'Neill) until I stop being shit at diff geometry.
You probably just need an easier textbook and a bit more patience for yourself. Check out Lay's Linear Algebra book for one that's a bit easier and friendlier overall.
Anonymous at Thu, 6 Mar 2025 14:53:00 UTC No. 16608875
How do the serious mathematicians who have no interest in foundations avoid getting sucked in to studying it?
Anonymous at Thu, 6 Mar 2025 17:56:43 UTC No. 16609195
>>16589624
Retard here
Which book do I start with?
Anonymous at Thu, 6 Mar 2025 21:19:05 UTC No. 16609573
>>16609195
If retard: openstax
Anonymous at Thu, 6 Mar 2025 23:46:23 UTC No. 16610300
>>16609195
[math]\textit{Precalculus: Mathematics in a Nutshell}[/math] and [math]\textit{Calculus Made Easy}[/math] concurrently with Khan Academy (skip the vids, interative practice is the only thing that matters there). These are very thin books compared with the industry standard.
Anonymous at Fri, 7 Mar 2025 00:07:25 UTC No. 16610554
>>16607608
{-56, 21, 35, -2401}
{-55, 16, 39, -2401}
{-49, 00, 49, -2401}
{-56, 27, 29, -2353}
{-49, 01, 48, -2353}
{-53, 14, 39, -2263}
{-49, 03, 46, -2263}
{-53, 26, 27, -2107}
{-51, 13, 38, -2107}
{-49, 07, 42, -2107}
{-51, 16, 35, -2041}
{-49, 09, 40, -2041}
{-51, 23, 28, -1957}
{-49, 12, 37, -1957}
{-50, 19, 31, -1911}
{-49, 14, 35, -1911}
{-46, 05, 41, -1911}
{-50, 21, 29, -1891}
{-49, 15, 34, -1891}
{-49, 18, 31, -1843}
{-46, 07, 39, -1843}
{-49, 21, 28, -1813}
{-47, 11, 36, -1813}
{-44, 03, 41, -1813}
{-49, 22, 27, -1807}
{-43, 01, 42, -1807}
Anonymous at Fri, 7 Mar 2025 00:17:31 UTC No. 16610650
>>16610300
Thin = suitable for nontards only
Anonymous at Fri, 7 Mar 2025 04:10:38 UTC No. 16611211
>>16608402
This was very sweet to read. I still undeniably think I'm a dumbass but I'm a dumbass who finds math beautiful
Anonymous at Fri, 7 Mar 2025 04:12:52 UTC No. 16611213
>>16608402
Sorta related sorta not, why do we care about convex analysis? I know it's super useful and there's a ton of research in industry in places like finance but my adhd monkey brain is unable to see why convex functions should be cared about
Anonymous at Fri, 7 Mar 2025 04:21:03 UTC No. 16611218
>>16611213
Convex analysis is useful for two reasons:
1) Convex spaces are locally "stable." If you connect any two points in a convex space, your path between them will be in the space. So if your function is defined on a convex feasibility space, you don't need to be worried about the optimization algorithm wandering into a region where the function is undefined.
2) Convex functions defined on convex spaces have local minima. If your function is convex on your feasible space, you will have unique minima, which you can be certain you will get convergence towards. That is incredibly useful in basically anything that uses a mean-square-error loss function (whether it be finance, robotic path planning, localization of a distant whale or training a video game bot).
Anonymous at Fri, 7 Mar 2025 06:05:23 UTC No. 16611270
>>16611213
convex functions have global minima, are generally easy to optimize with numerical methods, for example gradient descent and eigendecomposition. there are tons of results guaranteeing optimality, convergence 'speed', etc. and so optimizing a convex function is generally a predictable/well-understood computational task. for example, polynomial regression and SVMs both have convex objective functions. there are probably thousands of applications.
all that said, deep neural networks have much more complex objective functions* yet gradient descent optimizes them effectively all the same.
* If I recall correctly, even a single 'neuron' w/ a soft threshold activation has a non-convex loss function w.r.t a typical dataset. It might even have bad local minima if the data aren't linearly separable, but don't quote me on that.
Anonymous at Fri, 7 Mar 2025 06:26:55 UTC No. 16611279
>>16611270
>convex functions have global minima
unique global minima, that is. i should also add, while gradient descent can optimize both convex and (some) non-convex functions, convexity permits the use of more effective optimization algorithms like newtons method, which uses first and second order derivatives. if the objective function is quadratic, then the first and second derivatives model the loss function entirely.
Anonymous at Fri, 7 Mar 2025 20:00:30 UTC No. 16612201
>>16611460
What is this
Anonymous at Fri, 7 Mar 2025 21:41:51 UTC No. 16612300
>>16612201
>What is this[?]
it's a parametric plot
https://www.wolframalpha.com/input?
Anonymous at Sat, 8 Mar 2025 01:13:30 UTC No. 16612462
Why can't we prove the consistency of ZFC using cut elimination?
Anonymous at Sat, 8 Mar 2025 02:26:30 UTC No. 16612525
Now consider the recurrence T(n) = T(2n/3) +1, which has a=1 and
b=3/2, which means that the watershed function is n^logba = n^log3/21 = n0 = 1. Case 2 applies since f(n) = 1 = Theta(n^logba*lg^0 n) = Theta(1). The solution to the recurrence is T(n) = Theta(lg n).
This is from the MIT algos book and I'm confused. How does this reduce to theta of lg n? It's part of the master method for recursion.
Anonymous at Sat, 8 Mar 2025 03:36:08 UTC No. 16612590
Retard here.
Is math mostly practice solving problems?
Anonymous at Sat, 8 Mar 2025 03:57:39 UTC No. 16612603
>>16612462
A lot of people really overcomplicate Godel's incompleteness theorems.
The basic result is that, no matter what axiomatic framework or language you use, you will always be able to write sentences like "this sentence is false." So long as you have a mathematical language which can express ideas whose very truth value is contingent upon the axioms of the language being true, you will never be able to verify them.
Truth can only be verified in relationship to consistency with axiomatic frameworks. You cannot verify whether an axiomatic framework itself is consistent.
Anonymous at Sat, 8 Mar 2025 04:56:31 UTC No. 16612637
>>16608364
Buy Linear Algebra by Axler and do it, it's easy as pie.
đïž Anonymous at Sat, 8 Mar 2025 13:41:48 UTC No. 16612908
Please help me understanding the following basic probability definition:
[ eqn ]Throughout, (S, P) will be some finite probability space. An event in this space
is a subset of S. If E â S is an event, the probability of E, denoted P(E), is
\Big[\sum_{s\inE} P(s)\Big] [ /eqn ]
S is the set of possible outcomes, P is the probability assignment to the (set of)outcomes.
The thing I don't understand here is probably just notation, since afaik the subscript below sigma is supposed to represent what you begin your iteration with, e.g. i=1. The second issue is if S is the set of possible outcomes, and E is now defined to be one of those possible outcomes, what the hell is s? It seems intuitive to be that if big S is the set then little s would be one of its members, but now we're told each member is E. And s has to be different from E if we're being told little s is an element of E. It's worth noting the textbook didn't define little s before although it had been using it, so I just assumed it was a member of the set S.
Anonymous at Sat, 8 Mar 2025 13:43:58 UTC No. 16612910
Please help me understand the following basic probability definition:
[eqn] Throughout, (S, P) will be some finite probability space. An event in this space
is a subset of S. If E â S is an event, the probability of E, denoted P(E), is
\Big[\sum_{s\inE} P(s)\Big] [/eqn]
S is the set of possible outcomes, P is the probability assignment to the (set of)outcomes.
The thing I don't understand here is probably just notation, since afaik the subscript below sigma is supposed to represent what you begin your iteration with, e.g. i=1. The second issue is if S is the set of possible outcomes, and E is now defined to be one of those possible outcomes, what the hell is s? It seems intuitive to be that if big S is the set then little s would be one of its members, but now we're told each member is E. And s has to be different from E if we're being told little s is an element of E. It's worth noting the textbook didn't define little s before although it had been using it, so I just assumed it was a member of the set S.
Anonymous at Sat, 8 Mar 2025 15:44:04 UTC No. 16613006
>>16612910
You're getting lost in the sauce here.
Let's consider a simple example, rolling a die. The set of elementary events from rolling a die is given by:
[math]
S = \{1,2,3,4,5,6\}
[/math]
Where we assume each of those elementary events have equal probability:
[math]
P(\{1\}) = P(\{2\}) = \cdots = P(\{6\}) = \frac{1}{6}
[/math]
Now, consider the event [math]E[/math] defined by "even numbers"
[math]
E=\{2,4,6\}
[/math]
The probability of this event is given by the sum of the individual (i.i.d.) elementary events:
[math]
P(E) = \sum_{s\in E} P(s) = P(2)+P(4)+P(6)
[/math]
That event [math]E[/math] can be made up of a union of many different independent elementary events. In that case you'd need to take the sum over all of them. Contained within E.
Anonymous at Sat, 8 Mar 2025 22:38:50 UTC No. 16613354
Shot in the dark here but is there a simplified book, series of books, or online learning sources that explains 'applicable' mathematics in layman's terms? When I say 'applicable' I mean something that can teach how to logically break down problems you would encounter in daily life. If you recommend a textbook I'm just going to throw it out of a window.
I know I'm going to get shit on by the purists and I'm not even sure it exists.
Anonymous at Sat, 8 Mar 2025 23:15:39 UTC No. 16613405
>>16613354
I think you'll have to give some specific problems if you want certain subfields of applied maths, but you might like "guesstimation" by Adam and Weinstein as a general useful book.
Anonymous at Sat, 8 Mar 2025 23:17:25 UTC No. 16613409
>>16613354
There's Oxford University Press' A Very Short Introduction series. You can start with A Very Short Introduction ("VSI", for short) to Mathematics by Timothy Gowers and A VSI to Applied Mathematics by Alain Goriely. Then you can explore the rest of their VSi titles on math and physics in any order. https://global.oup.com/academic/con
Anonymous at Sun, 9 Mar 2025 02:08:34 UTC No. 16613576
>>16606715
>>16604850
Wtf is this demon script
>>16591759
What is this image trying to convey?
>>16591888
>it's fun to stay at the Y-M-C-K!
Honestly, I'm so shit at maths, I couldn't tell you how to workout ratios, I couldnt rattle off my 6 or 8 times tables, and I have no idea what calculus, statistics or the different between pythagoras and triganometry.
>t.lead engineer of 11 years with foundation degree in mech-engineering funded by apprenticeship
So glad maths isn't a requirement for actual engineering work.
Anonymous at Sun, 9 Mar 2025 06:54:57 UTC No. 16613732
>>16613558
It's just a multinomial distribution problem since the exact values of the samples don't matter. All that matters is in which of the three intervals (0,0.25], (0.25,0.75) and [0.75,1) the values are.
Hell since the event of three or more values falling into (0,0.25] and the event of three or more values falling into (0.75,1] are mutually exclusive you can even reduce it into binomial distribution problems.
Anonymous at Sun, 9 Mar 2025 11:48:44 UTC No. 16613874
What would you call a measure of sine wave """energy""" that rises proportionally to both amplitude and frequency (but not squares)? So basically [math]k\sin(x)[/math] would have the same """energy""" as [math]\sin(kx)[/math].
Anonymous at Sun, 9 Mar 2025 12:03:07 UTC No. 16613877
>>16612525
Here (picrel) is the theorem I know (should be the master method or at least similar).
You have [math]\gamma=\log_b(a)=0[/math], and are in case (2), so you are [math]\Theta(n^0\log(n))[/math], no?
Or am I misunderstanding what you don't get?
Anonymous at Sun, 9 Mar 2025 13:27:36 UTC No. 16613921
[math]\mathbb{OpenStax} . \mathbb{org}[/math]
Anonymous at Sun, 9 Mar 2025 16:02:44 UTC No. 16614001
Is math based on definitions or physics?
Anonymous at Sun, 9 Mar 2025 16:50:32 UTC No. 16614026
Bros, I have the worst luck in existence.
Basically, I waited a year to be able to apply for a master's degree in mathematics at a foreign university, but during the application week the scholarship page was hacked, causing my application to be disregarded. I didn't know that. So I have been waiting for 4 months for an answer that will never come. The worst part is that I declined other options because my tutor, who has contact with the admissions department, had told me that I was very likely to get the scholarship.
And now I have lost a year, when I could have applied to another university and already finished my first semester.
Another funny thing is that one of the requirements for the scholarship was not to have worked, so I've basically been a neet for a year and justifying myself that I was going to get the scholarship. I feel like I let my parents down.
Anonymous at Sun, 9 Mar 2025 16:54:56 UTC No. 16614029
life becomes weird as I actually meet the people from my books and my papers.
I sort of feel like an asshole
Anonymous at Sun, 9 Mar 2025 16:56:22 UTC No. 16614030
>>16614001
holy shit I've never been more offended in my life
go suck cock and sell your anus you fucking prostitute slut whore bitch!
Anonymous at Sun, 9 Mar 2025 16:59:22 UTC No. 16614032
>>16613354
well to me the layman is an engineer, so yes there's like hundreds of shitty baby enginie who could books.
Anonymous at Sun, 9 Mar 2025 21:56:16 UTC No. 16614315
>>16613558
it's approximately 793/1000
https://www.wolframalpha.com/input?
Anonymous at Sun, 9 Mar 2025 22:15:47 UTC No. 16614330
>>16614315
What the hell is this?
Just do
[eqn]1 - 2 \sum_{k=3}^5 {5 \choose k} \left( \frac{1}{4} \right)^k \left( \frac{1}{4} \right)^{5-k} = \frac{203}{256} [/eqn]
Anonymous at Sun, 9 Mar 2025 22:29:26 UTC No. 16614347
Is there a way to intuit statistics? I'm just memorizing these formulas and I know that's not how you learn math but it feels like solving things like nth moments for a random distribution just can't be done easily analytically. I've never done applied math before and am starting to try for an applied phd after doing pure undergrad and just feel out of place
Anonymous at Mon, 10 Mar 2025 00:35:29 UTC No. 16614428
>>16614330
>What the hell is this?
"Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results."
>Just do [...]
Your equation is false!
The depicted equation is true.
But my 793/1000 and your 203/256 are approximately equal.
Anonymous at Mon, 10 Mar 2025 04:49:48 UTC No. 16614588
>>16613006
Thanks. However, how can it be that sâE if E is made up of instances of s? I could understand it if it was supposed to mean "sum the elements where s matches E" (e.g. s2, s4 and s6 in your example), but to my knowledge sâE being under sigma means that is supposed to be the first round of the iteration, the value with which the sum begins (and if there was something on top of sigma that would be the last value). I don't see how we go from there to saying s belongs to E.
Anonymous at Mon, 10 Mar 2025 06:18:52 UTC No. 16614614
>>16613558
Let L refer to [math](0,0.25][/math], C refer to [math](0.25,0.75)[/math], and R refer to [math][0.75,1)[/math], and l,c,r refer to the amount in the sample in each space. In a sample of 5, it can be ordered from least to greatest, where L is before C and C is before R. To have a median of C, it must be the third in the order. To have a median of C, in the case of no L's there are at least three C's, in the case of one L there are at least two C's, in the case of two L's there must are least one C. If there are three L's it's impossible to get a median of C. So from zero to two L's, the number of C's must be at least three minus the number of L's, and at most five minus the number of L's. So the summation is
[eqn]\sum_{l=0}^{2} \sum_{c=3-l}^{5-l}[/eqn]
The probability of a given l, c, and r is
[eqn]\frac{5!}{l!c!r!} \frac{1}{4^{(l+r)}} \frac{1}{2^c}[/eqn]
Set [math]r = 5 - c - l[/math] and combine it with the summation to get
[eqn]\sum_{l=0}^{2} \sum_{c=3-l}^{5-l} \frac{5!}{l!c!(5-c-l)!} \frac{1}{4^{(5-c)}} \frac{1}{2^c}=\frac{203}{256}[/eqn]
Anonymous at Mon, 10 Mar 2025 06:29:28 UTC No. 16614625
>>16613354
Mathematics and the Imagination
Anonymous at Mon, 10 Mar 2025 08:34:08 UTC No. 16614669
>>16614330
>Just do [...]
Your equation is a monkey wrench!
quote:
The largely US idiom "to throw a monkey wrench into (something)" means to sabotage something, equivalent to the British English "to throw a spanner in the works".
Anonymous at Mon, 10 Mar 2025 09:02:58 UTC No. 16614678
>>16614428
>>16614669
It's
[eqn]1 - 2 \sum_{k=3}^5 {5 \choose k} \left( \frac{1}{4} \right)^k \left( \frac{3}{4} \right)^{5-k} = \frac{203}{256}[/eqn]
The term [math]\sum_{k=3}^5 {5 \choose k} \left( \frac{1}{4} \right)^k \left( \frac{3}{4} \right)^{5-k} [/math] is the probability that the median lands in (0, 0.25] this happens if 3 or more samples land in the interval.
Same shit for [0.75,1).
Anonymous at Mon, 10 Mar 2025 09:43:08 UTC No. 16614686
I was up a couple of nights ago thinking about what would happen if the number line were made radial. Consider the red circle r=v and the blue line an upward vector at v. As you increase v the red line on the radial number line ascends and stays horizontal at all times while the blue line rotates around the center. (
gray radial lines on the right-hand side are not in correct scale, just an example)
As v approaches (I assume) the cardinality of natural numbers blue becomes horizontal at the equator (green line).
Since blue and red must always intersect at v they would also be overlapping when they reach the green line.
My question is what the fuck is on the other hemisphere if it cannot reasonably surpass the green line? Or am I just being retarded and this projection is inherently broken?
t. math noob
Anonymous at Mon, 10 Mar 2025 09:47:01 UTC No. 16614688
Anonymous at Mon, 10 Mar 2025 12:10:40 UTC No. 16614765
bi-nomial: >>16614678
tri-nomial: >>16614614
both bi-, and tri-, nomial: image
Anonymous at Mon, 10 Mar 2025 14:05:56 UTC No. 16614831
Consider a sample of size three from a uniform distribution over the interval (0, 1).
Compute the probability that the median is in the interval (p, 1 â p).
Where p = (sqrt(3)*sin(t) â cos(t) + 1)/2
and t = arctan(2*sqrt(10)/9)/3.
Anonymous at Mon, 10 Mar 2025 14:57:39 UTC No. 16614862
>>16614831
It works exactly like the other problem. The solution is
4 p^3 - 6 p^2 + 1
Anonymous at Mon, 10 Mar 2025 15:33:03 UTC No. 16614885
>>16614588
E is just some subset of the overall space of outcomes.
E doesn't need to correspond to only one elementary outcome, it can be a group of them together. In fact, in most cases E will be a collection of individual events s (as opposed to S which is the collection of all of the individual lower case s values in one set).
Anonymous at Mon, 10 Mar 2025 15:49:13 UTC No. 16614899
>>16614885
I get that E is a set of events "on top" of the set of outcomes, a separate thing, but then how is sâE, how does s belong to E? I can see that sum meaning "sum the elements where s belongs to E", which in your example would be s2, s4 and s6, but that is just me trying to find something that makes sense, like I said before I thought whatever was below the sigma was supposed to represent the first element in the sum sequence.
Anonymous at Mon, 10 Mar 2025 16:19:19 UTC No. 16614934
>>16614614
>>16614315
>>16613732
>>16614688
Thanks. Your answers were all slicker than mine. I ended up integrating the density function of the third order statistic and got 0.79248
Anonymous at Mon, 10 Mar 2025 17:26:05 UTC No. 16615012
>>16614862
>4 p^3 - 6 p^2 + 1
Correct, but you ignored the last sentence.
Anonymous at Mon, 10 Mar 2025 17:26:19 UTC No. 16615013
>>16614899
Okay, let's consider our set of possible elementary events to be [math]S[/math] given by:
[math]
S = \{s_1, s_2, s_3, \cdots, s_N\}
[/math]
(We'll keep the sample space finite for now, but it works the same way with infinite sets with some caveats).
Consider then what it means for [math]E[/math] to be a subset of [math]S[/math]:
[math]
E \subseteq S \Rightarrow E\in \left\{
\emptyset, S, \{s_1\}, \{s_2\}, \cdots, S-\{s_n\}
\right\}
[/math]
If E is a (non-empty) subset of S, it by definition must have its elements be some combination of the lower case s "elementary events" which define it. The point is that you can define the probability of E by summing/integrating over the elementary events that make up E.
It's really not all that complicated. You're overthinking it.
Anonymous at Mon, 10 Mar 2025 17:31:18 UTC No. 16615015
>>16614899
Oh, wait, I see where you're getting confused.
No, the Sigma sum notation does not necessarily mean that the starting index is the bottom. You can define a sum over some specific set of conditions simply by stating them at the bottom without it being explicitly indexed.
[math]
\sum_{s\in E} P(s) = \sum_{n=1}^N P(s_n)1_{s_n \in E}(s_n)
[/math]
Where the 1 function is the indicator:
[eqn]
1_{x \in E}(x) =\begin{cases}
1, & x \in E\\
0, & x\notin E
\end{cases}
[/eqn]
Anonymous at Mon, 10 Mar 2025 17:54:33 UTC No. 16615035
>>16614315
median = {0, 0, 1, 0, 0} . Sort[{RandomReal[{-2, 2}], RandomReal[{-2, 2}], RandomReal[{-2, 2}], RandomReal[{-2, 2}], RandomReal[{-2, 2}]}]
Anonymous at Mon, 10 Mar 2025 19:45:05 UTC No. 16615134
Anonymous at Mon, 10 Mar 2025 19:45:14 UTC No. 16615135
Suppose you take an abelian group [math]A_1[/math] and then consider the ring [math]R_1=\text{End}(A_1)[/math]. Let [math] A_2 [/math] be the additive group of [math]R_1 [/math]. Now consider [math]R_2=\text{End}(A_2) [/math] , let [math] A_3[/math] be the additive group of [math]R_2 [/math] and so on. What can happen here?
Anonymous at Mon, 10 Mar 2025 22:23:50 UTC No. 16615268
>>16614934
>I [...] got 0.79248
I don't believe, that you got something which is less than 0.7925.
I believe, that you're not being forthright.
Anonymous at Tue, 11 Mar 2025 10:23:26 UTC No. 16615598
Is there some sort of leetcode site for math? A collection of math problems of various fields, ideally highlighting one every day? I know about project euler but I was thinking more of a pen and paper approach.
Anonymous at Tue, 11 Mar 2025 10:40:51 UTC No. 16615601
>>16615589
>Cube
Isn't it just a regular octahedron?
Anonymous at Tue, 11 Mar 2025 13:42:48 UTC No. 16615724
Let I(a,b) be the integral of exp(-x^2) from a to b.
We got the "reduction"
I(-inf, inf) = I(0, inf) * 2
Is there some other box [u,v], possibly unbounded, and polynomial p such that
I(-inf, inf) = p(I(u,v))
?
(Generalization to a polynomial with multiple but finite number of inputs is okay)
(I came up with the question thinking of best use of a list of random data for Monte Carlo evaluation)
Anonymous at Tue, 11 Mar 2025 14:28:46 UTC No. 16615755
Without making any semantic assumptions about [math]\notin[/math], doesn't Russell's Paradox rely on a weak form of LEM to produce a contradiction? Specifically that [math]\forall x \forall y , (x \in y) \vee (x \notin y)[/math].
Anonymous at Tue, 11 Mar 2025 14:33:21 UTC No. 16615757
>>16615598
Brilliant.org is the closest i can think
Anonymous at Tue, 11 Mar 2025 14:39:24 UTC No. 16615759
>>16615755
I don't know what you intended to use your statement for, but the answer to your question is no.
You can prove
[math] \big(\forall x. xEs\leftrightarrow(xEy \land \neg xEx)\big) \to \neg(yEs\lor sEs\lor sEy) [/math]
in predicate logic, constructively, no matter the predicate E.
It follows, constructively after having introduced class notation for convenience,
[math] \{x\in y\mid x\notin x\}\notin y [/math]
Anonymous at Tue, 11 Mar 2025 14:48:30 UTC No. 16615765
(This does in fact prove more. It says that in a set theory allowing for separation with "not x in x", for ANY set y you can find a subset of y that's not in it. Conventionally, after adopting more axioms, that subset is y itself.)
Anonymous at Tue, 11 Mar 2025 15:30:11 UTC No. 16615791
>>16615135
possibly related https://arxiv.org/abs/math/9808014
Anonymous at Tue, 11 Mar 2025 15:38:42 UTC No. 16615802
>>16615759
I see, thanks for answering. The motivation behind my question is a little vague but it was brought on by learning that the Co-Russell Set exists in [math]\text{GPK}^+_\infty[/math] and its relation to itself is independent. I guess I'm curious as to what could salvage Unrestricted Comprehension, or as much of it as possible, but maybe Positive Set Theories are the best we can do.
Anonymous at Tue, 11 Mar 2025 15:40:39 UTC No. 16615805
>>16615759
recipe for that cake?
Anonymous at Tue, 11 Mar 2025 16:25:20 UTC No. 16615841
why bother with noncommutative rings when noncommutativity makes everything so much more complicated?
Anonymous at Tue, 11 Mar 2025 16:43:38 UTC No. 16615852
>>16615805
It's just a cake at Starbucks.
Don't even want to know how it comes about thb.
>>16615802
There's many ways around it, but all of the theories are rather odd I think.
Another I know of is just disallowing "x\in x" in comprehension by a condition on the types, where x\in x just doesn't fit in. That's a sort of stratification.
Yet another is paraconsistent logic, etc. I'm sure there's more.
When I say odd set theory, I mostly mean that the models of them are then not like a big graph, or that those are super complicated.
So e.g. I came to think of non-well founded set theroies as not even that odd, given I can few the \in relation in terms of a graph and infinite descending lines aren't all that wild.
>>16615841
For one, the theorems it proves will be true for matrix rings
>>16613874
I wouldn't call it energy, but I'm reminded of modular forms. So something with "morphic" in the name I guess. In any case, given the periodicity of sin, you'll have some consistency problems if you go out of the first interval with this. Take x=1 and k a positive zero of sin.
>>16608875
Learning what tools and theorems are accepted by the comminity, hasting to the next paper to write. Just my guess.
Anonymous at Tue, 11 Mar 2025 16:46:29 UTC No. 16615855
>>16615841
matrices dont commute, and linear algebra is useful irl, among other uses for matrices
Anonymous at Tue, 11 Mar 2025 17:07:40 UTC No. 16615871
I only care about math that can help me make money i.e. LinAlg, Calculus and Prob/Stats. The rest is just intellectual âartâ
Anonymous at Tue, 11 Mar 2025 17:34:18 UTC No. 16615905
>>16615871
There's a lot of math that can help you make money beyond that. There's a lot of useful applications of geometry (affine, convex and differential mainly) in probability/statistics, and theoretical statistics relies quite a lot on concepts from functional analysis for convergence and derivation of test statistics.
Anonymous at Tue, 11 Mar 2025 17:50:42 UTC No. 16615921
>>16615871
Based, for me it's signal processing.
Anonymous at Tue, 11 Mar 2025 18:29:58 UTC No. 16615951
>>16615601
>a regular octahedron
omg you're right
how embarrassing
that's my second (distinct) mistake on /sci/
Anonymous at Tue, 11 Mar 2025 19:05:25 UTC No. 16615990
>>16615852
How exactly does paraconsistent logic avoid Russel's paradox?
Anonymous at Tue, 11 Mar 2025 19:22:05 UTC No. 16616005
>>16615015
Alright so that dispels the conflict between my intuition and what I thought was the definition
Thanks!
Anonymous at Tue, 11 Mar 2025 19:25:10 UTC No. 16616009
>>16615990
I saw a paper once - I don't remember the detail but it's online somewhere.
But you know, I wouldn't bother with it. That paraconsistent set theory, I remember form the text, has theorems like
>The collection of all ordinals is not a set and, as a bonus just for you, it's a set also.
I don't have a good intuition for looney toons math like that. That said, I don't want to dismiss it, since usually nowever silly stuff sounds, if you get the right semantics then it's suddenly totally sensible.
Anonymous at Tue, 11 Mar 2025 20:36:04 UTC No. 16616097
how do you know it's time to give up on math?
Anonymous at Tue, 11 Mar 2025 21:42:17 UTC No. 16616193
Someone should write an Abstract Algebra book that just goes really into detail constructing the reals and the operations on them used in physics/science
Anonymous at Tue, 11 Mar 2025 21:51:47 UTC No. 16616206
>>16616097
when you become a centenarian. Math is "ages 9 months to 99 years" only.
Anonymous at Tue, 11 Mar 2025 21:56:58 UTC No. 16616217
>>16616193
>Abstract Algebra book that just goes really into detail constructing the reals
That's not in the scope abstract algebra courses nowadays. But there are lots of Number Systems books, especially from the New Math era. For example, this
https://archive.org/details/foundat
and this
https://archive.org/details/moderna
Anonymous at Tue, 11 Mar 2025 22:03:11 UTC No. 16616227
>>16616005
No problem. I hope your probability course goes well!
As a lowly EE, I love probability theory and when I'm working on probability it's one of the only real opportunities I have to flex my math minor/elective analysis courses in my professional life.
Anonymous at Tue, 11 Mar 2025 22:17:02 UTC No. 16616234
Do some of the groups you learn about in abstract algebra like matrices and congruence classes have any application outside of math?
Anonymous at Tue, 11 Mar 2025 23:08:18 UTC No. 16616270
>>16616234
Some of the basics of group theory get used in signal processing, especially signal processing on manifolds. Robotic path planners and localization use rotation groups like SO(3) and SE(2) to handle heading/curvature constraints.
Information Theory uses some amount of group theory for optimal discrete encoding of continuous sources. Group theoretic coding was a pretty hot topic of research even up until the mid-2000's. There's still a decent amount of information theory research to be done with Galois theory for information optimal encrypting codes and network info theory.
Matrix groups are used a lot in random matrix theory, which gets used in a lot of the more cutting edge of probabilistic machine learning.
Anonymous at Tue, 11 Mar 2025 23:36:58 UTC No. 16616294
>>16615921
>signal processing
It sounds like you work for Unit 8200, GCHQ, or NSA.
Anonymous at Tue, 11 Mar 2025 23:58:29 UTC No. 16616299
>>16616294
I can neither confirm nor deny
Anonymous at Wed, 12 Mar 2025 00:44:04 UTC No. 16616325
>>16614686
thinking more about my retarded projection I've realized two things so far:
1. I made the radial real lines equidistant in the example which is incorrect if the red circle were to be made a straight line, it would have to be exponentially increasing in distance as the imaginary component increases (not unlike looking at a sphere with lines placed equidistant on its skin, the lines would be squished closer to the edge)
2. the upper hemisphere has an equal amount of numbers as the complex hemisphere below it, and if I consider the lower complex hemisphere being the conjugate of R+- and I+, then it's maybe acceptable to consider the upper unknown hemisphere as being the conjugate of R+- and I-, which satisfies having an equal amount of numbers contained within. Though I then couldn't expand the projection into a sphere.
still looking for input on this however, feel free to point out that my thinking is incorrect/schizobabble if it calls for it
Anonymous at Wed, 12 Mar 2025 01:21:56 UTC No. 16616352
>>16616294
There's a lot of people who work in signal processing that don't work for those three specific agencies.
t. Works in a different spooky government agency
Anonymous at Wed, 12 Mar 2025 01:57:36 UTC No. 16616386
>>16615755
Unfortunately a lot of texts written by classically trained mathematicians make it seem like this would be the case, but it's not.
Most proofs proceed as usual until they get to [math]r\in r \leftrightarrow r\notin r[/math]. Some authors skip over the details and just claim that this is a contradiction, and indeed [math]P\leftrightarrow\neg P[/math] leads to a contradiction, but how? This is where a lot of texts unfortunately invoke LEM to get [math]P\lor\neg P[/math] and then show that each case leads to a contradiction.
But we can do better than that: If we assumed that [math]P[/math] holds, then [math]\neg P[/math] also holda by the forward direction of [math]P\leftrightarrow\neg P[/math]. We get a contradiction, hence [math]\neg P[/math] must hold (since we just assumed [math]P[/math] and ended up at a contradiction, we may discharge that assumption to conclude [math]\neg P[/math]). But by the right-to-left implication of [math]P\leftrightarrow\neg P[/math] we now get [math]P[/math], contradicting [math]\neg P[/math] and we're done.
Not classical logic reasoning needed!
Anonymous at Wed, 12 Mar 2025 14:01:47 UTC No. 16616852
shape of pond, color of curve
2-gon, blue
triangle, yellow
square, green
pentagon, orange
...
circle, purple
see the "N ducks in a square pond" thread (before it gets archived) for more information
Anonymous at Thu, 13 Mar 2025 01:41:23 UTC No. 16617399
hey guys I was on the collatz subreddit and someone mentioned this number:
14927931876218086035181556215237625
have any of you guys ever seen this number before? the OP said it was interesting
Anonymous at Thu, 13 Mar 2025 08:00:48 UTC No. 16617628
>>16617399
it has 8*35 + 6 = 286 digits
it is divisible by 10
it contains 666, 999, and 888
Anonymous at Fri, 14 Mar 2025 13:50:33 UTC No. 16618716
>>16618187
s = arc length
>>16618624
S = surface area
Anonymous at Fri, 14 Mar 2025 17:03:02 UTC No. 16618850
Why is [math]x^2-9/x+3[/math] indeterminate at [math]x=-3[/math], but can be simplified to [math]x-3[/math] and at this form it is defined at [math]x=-3[/math] ?
How can you use algebraic manipulation on a function and end up with with a new function with a point where the result is different than the starting function?
Anonymous at Fri, 14 Mar 2025 17:06:35 UTC No. 16618852
>>16618850
Because part of that 'algebraic manipulation' is the assumption (requirement) that at no point are you dividing by zero.
Anonymous at Fri, 14 Mar 2025 17:41:40 UTC No. 16618872
>>16618850
Didn't you forget about any parentheses?
Anonymous at Fri, 14 Mar 2025 18:09:12 UTC No. 16618886
>>16618852
Thank you kindly, I get it now
>>16618872
I don't think wrapping either or both numerator and denominator would change anything here
Anonymous at Fri, 14 Mar 2025 18:30:03 UTC No. 16618897
My combinatorics is starting to do weird things.
Is this related to spin?
Is there a graph theory interpretation for the 2x2?
What cool things can be done with the effortless square root reduction?
Anonymous at Sat, 15 Mar 2025 03:43:00 UTC No. 16619472
>>16618716
>S = surface area
Shouldn't it be (dxdy - dydx)^2 + ...?
Anonymous at Sat, 15 Mar 2025 03:57:45 UTC No. 16619477
>>16618886
9+9+3 isnt indeterminate
Anonymous at Sat, 15 Mar 2025 11:41:18 UTC No. 16619673
>>16619472
no, it shouldn't be 0^2 + 0^2 + 0^2
Anonymous at Sat, 15 Mar 2025 17:36:13 UTC No. 16619884
>>16619673
That's only if x and y depend one parameter. Surface area needs two parameters. Order matters
Anonymous at Sat, 15 Mar 2025 17:50:40 UTC No. 16619897
>>16589625
>>16589632
What's are good books to learn about these topics?
Anonymous at Sat, 15 Mar 2025 20:20:25 UTC No. 16620031
>>16619884
>Order matters[.]
Well then please post what (dS)^2 is equal to, according to you.
Anonymous at Sat, 15 Mar 2025 20:50:54 UTC No. 16620065
>>16620031
I already mentioned it >>16619472
Anonymous at Sat, 15 Mar 2025 21:05:13 UTC No. 16620075
>>16620065
>I already mentioned it
yeah, zero
Anonymous at Sat, 15 Mar 2025 21:56:22 UTC No. 16620098
>>16619897
Abstract Algebra (Hungerford, Dummit and Foote, Gorodentsev, etc) and Mathematical Analysis (Rudin, Amann and Escher, Zorich, etc)
Anonymous at Sat, 15 Mar 2025 22:02:36 UTC No. 16620099
Are there any mathematical systems that don't have left hand and right hand sides? "Sidedness" seems pretty fundamental to most kinds of logic, let alone math, but is there any way to have a system where every sentence has 1 side, or 3 sides, or n sides?
Anonymous at Sun, 16 Mar 2025 10:51:12 UTC No. 16620617
>>16618830
this guy's channel is pure kino lmao
Anonymous at Sun, 16 Mar 2025 14:28:34 UTC No. 16620740
>>16618850
Equality of functions means equality of formulas PLUS equality of domains and codomains. Your simplified function doesn't have -3 in the domain nor its image in the codomain
Anonymous at Mon, 17 Mar 2025 02:48:22 UTC No. 16621331
>>16618850
f = x - 3 (for x <> -3)
Anonymous at Mon, 17 Mar 2025 03:27:36 UTC No. 16621344
>>16618897
that's interesting. Obviously not every digraph is gonna give you a matrix that looks like a bunch of imaginary-matrix blocks. I wonder if there's a way to characterize the ones that do
Anonymous at Mon, 17 Mar 2025 03:30:59 UTC No. 16621347
Anonymous at Mon, 17 Mar 2025 14:53:47 UTC No. 16621645
Do all the members of a cover of a set have to be subsets of the "covered" set? Seems like that has to be the case, according to the definitions of cover I've found.
Therefore, the collection [math]\{(-n,n)\}_{n=1}^{\infty}[/ma
Anonymous at Mon, 17 Mar 2025 20:46:35 UTC No. 16621898
>>16621645
>according to the definitions of cover I've found.
define what you think what a cover is
Anonymous at Mon, 17 Mar 2025 20:57:49 UTC No. 16621909
>>16620115
Medication posthaste
Anonymous at Mon, 17 Mar 2025 21:19:33 UTC No. 16621923
>>16621898
A cover of a set [math]X[/math] is a collection of subsets of [math]X[/math] such that their union is exactly [math]X[/math]. I'd post a Wikipedia screenshot but this God-forgotten website does not allow uploading shit from private browsers.
It's obvious that written as that, my previous example doesn't qualify as a cover. But then again while reading analysis I'll stumble into something like this https://en.wikipedia.org/wiki/Outer
Anonymous at Mon, 17 Mar 2025 21:53:50 UTC No. 16621952
>>16621923
Some authors are probably using different definitions for it then. The family of subsets being subsets of the entire space rather than just the set being covered sounds like a better use for the word "covers" imo, but just know people use two diff definitions i guess. Either check if the author mentioned the def beforehand, or just be open minded
Anonymous at Tue, 18 Mar 2025 08:05:01 UTC No. 16622291
>>16618830
>he clickb8d a guy's death
toplel
Anonymous at Tue, 18 Mar 2025 09:08:16 UTC No. 16622308
When I take the product of a family of sets that are each equipped with relations and order the product component wise, many texts will say that the product is a well-founded relation whenever all the factors are well-founded. But isn't it sufficient if even just one factor is well-founded?
Anonymous at Wed, 19 Mar 2025 19:24:32 UTC No. 16623830
>>16589624
> math general
> literal ph*sics in OP
retard
Anonymous at Wed, 19 Mar 2025 19:25:40 UTC No. 16623831
>>16589625
> even on 4chan on other side of the globe people are worshipping this schizo
Anonymous at Wed, 19 Mar 2025 19:37:49 UTC No. 16623834
>>16589624
Which textbooks do you like to use for people who just wanna learn physics/math out of curiosity?
In my experience they are the flakiest, lowest discipline mfers who drop out as soon as it gets hard to any degree.
What are your go to calc/stats/abstract algebra/physics textbooks for this case?
Currently looking at Rudin and it's a bit too rigorous for somebody focused on physics
Anonymous at Wed, 19 Mar 2025 19:43:49 UTC No. 16623838
>>16594543
>Algebra: Chapter 0
>> Is actually a BSc in math full of algebra
What did he mean by this???
Anonymous at Wed, 19 Mar 2025 21:45:49 UTC No. 16623910
how can i learn to do long multiplication? i used to be able to do it in grade school fine but now i am re visiting the topic and im struggling to multiply 2 digit numbers together
Anonymous at Wed, 19 Mar 2025 22:13:33 UTC No. 16623938
>>16623910
retard lol
Anonymous at Wed, 19 Mar 2025 22:17:59 UTC No. 16623944
>>16623938
yeah i know but how can i learn to do it?
Anonymous at Wed, 19 Mar 2025 22:23:18 UTC No. 16623946
>>16623944
Literally just follow the algorithm? You can find.. FUCKING ANYWHERE. Ask gpt
Anonymous at Wed, 19 Mar 2025 22:29:01 UTC No. 16623955
>>16623946
But i dont understand it
Anonymous at Wed, 19 Mar 2025 22:49:17 UTC No. 16623973
>>16623955
You just have to follow it.
Anonymous at Wed, 19 Mar 2025 23:01:59 UTC No. 16623982
>>16623955
Long multiplication is mysterious and important.
Anonymous at Thu, 20 Mar 2025 00:07:57 UTC No. 16624017
>>16623618
More like
(dxdx+dydy+dzdz+dada)(dxdx+dydy+dzd
- (dxdx+dydy+dzdz+dada)(dxdx+dydy+dzd
+ ...
(actually out of order but im not gonna expand to figure out the order or use a program)
Anonymous at Thu, 20 Mar 2025 01:13:24 UTC No. 16624061
>>16624017
that's equal to 0 + ...
Anonymous at Thu, 20 Mar 2025 01:43:32 UTC No. 16624075
Who are some of the biggest genuine math schizos/cranks that you've encountered on the internet?
Anonymous at Thu, 20 Mar 2025 02:37:05 UTC No. 16624094
>>16624075
there are two classes of schizos I see way too often. Extreme finitists who spend years and years tying themselves in knots over the idea of "the biggest number" and people who insist the universe actually must be some discrete computer process because that's what makes sense to them.
So, I guess you can just say "computer scientists" and you'll capture both of them quite succinctly.
Anonymous at Thu, 20 Mar 2025 03:41:07 UTC No. 16624113
>>16623831
wdym? qrd?
Anonymous at Thu, 20 Mar 2025 03:49:58 UTC No. 16624116
>>16623955
>>16623944
>>16623910
Let "[math]ab[/math]" and "[math]cd[/math]" be a couple two-digit numbers, for example, [math]15[/math] and [math]67[/math].
[math]ab\times cd = (a\times 10+b)\times (c\times 10+d)=c\times 10(a\times 10+b)+d(a\times 10+b)=\cdots[/math]
[math]\cdots=c\times 10\times a\times 10+c\times 10\times b+d\times a\times 10+db=c\times a\times 100+(c\times b+d\times a)10+d\times b[/math].
Anonymous at Thu, 20 Mar 2025 03:52:27 UTC No. 16624117
>>16624075
Almost def he was trolling but some guy pushing for Terrence Howard's bs. Either Howard's a schizo or he was using people's ignorance to push a scam
Anonymous at Thu, 20 Mar 2025 12:15:52 UTC No. 16624221
>>16623834
>Which textbooks do you like to use for people who just wanna learn physics/math out of curiosity?
>In my experience they are the flakiest, lowest discipline mfers who drop out as soon as it gets hard to any degree.
>What are your go to calc/stats/abstract algebra/physics textbooks for this case?
Try:
>Precalculus_ Mathematics in a Nutshell - George F. Simmons
>Calculus Made Easy - Silvanus P. Thompson
>Calculus: An Intuitive and Physical Approach - Morris Kline
>Introduction to Concepts and Theories in Physical Science - Gerald Holton, Stephen G. Brush
>Foundations of Modern Physical Science - Gerald Holton, Duane H. D. Roller
>Statistics in plain English - Timothy C. Urdan
>A Book of Abstract Algebra - Charles C. Pinter
Anonymous at Thu, 20 Mar 2025 13:23:21 UTC No. 16624245
>>16602693
>>16607567
>>16607608
>>16610554
>picrel if anyone is still interesed
I've always been uneasy with this kind of brute force - like problems.
Anonymous at Thu, 20 Mar 2025 14:10:46 UTC No. 16624264
Why are we still using Euler angles and teaching them to the poor students? Theyâre absolute dogshit.
Anonymous at Thu, 20 Mar 2025 14:54:26 UTC No. 16624282
>>16623955
You just have to live it!
Anonymous at Thu, 20 Mar 2025 14:55:38 UTC No. 16624283
>>16624113
back when I frequented Russian 2ch he was huge on their /sci/ and regularly talked about
Anonymous at Thu, 20 Mar 2025 14:56:42 UTC No. 16624284
>>16624221
Thanks will check out. Currently gonna look at Stewart's
Anonymous at Thu, 20 Mar 2025 15:04:36 UTC No. 16624287
>>16624284
Stewart has standard exercises and examples but the exposition is goyslop
Anonymous at Thu, 20 Mar 2025 15:06:35 UTC No. 16624289
>>16624283
Did you read his trivium?
Anonymous at Thu, 20 Mar 2025 15:16:57 UTC No. 16624291
>>16624264
Quaternions are black magic and geometric algebra is a meme.
Anonymous at Thu, 20 Mar 2025 16:00:20 UTC No. 16624317
>>16623830
The Bible is there too. It is called a well-rounded education.
Anonymous at Thu, 20 Mar 2025 17:22:36 UTC No. 16624358
>>16624245
The question was answered first in this thread
>>16602716
Anonymous at Thu, 20 Mar 2025 17:55:16 UTC No. 16624372
>>16624289
nah never got to it. Only looked at the document itself in Russian. It was basically like a BSc in math worth of math but that you go through on your own before enrolling in a uni
Anonymous at Thu, 20 Mar 2025 17:58:06 UTC No. 16624375
>>16624317
lmao
Anonymous at Thu, 20 Mar 2025 17:59:01 UTC No. 16624376
Am I supposed to enjoy the grunt work required to understand math? I used to think I was normal and just a little lazy but after taking to enough researchers I think they not only bear the grunt work required to do research because they want the end result, but some of the great ones ENJOY the intellectual masochism of struggiling with a problem and get joy out of that frustration, similar to top athletes. Thatâs not me at all, Iâm very intellectually lazy. Is math not for me?
Anonymous at Thu, 20 Mar 2025 18:02:22 UTC No. 16624377
>>16589624
Why is exactly the axiom of Ch*ice so (((((problematic))))))? Isn't it fucking obvious?
Anonymous at Thu, 20 Mar 2025 18:03:23 UTC No. 16624379
Is graph with one vertex empty or complete?
Anonymous at Thu, 20 Mar 2025 18:05:21 UTC No. 16624382
>>16624376
It's learnable. But le grunt work is literally required to achieve anything in anything and all the greats learn to love it
Anonymous at Thu, 20 Mar 2025 19:04:12 UTC No. 16624422
>>16624379
>empty
yes, degenerate
>complete
yes, degenerate
Anonymous at Thu, 20 Mar 2025 19:12:16 UTC No. 16624428
(e^x)/(i * 2.718) = 9.81 * 2.718 * i
:^)
Anonymous at Thu, 20 Mar 2025 19:24:02 UTC No. 16624431
>>16624377
From Jerry Bona: "The Axiom of Choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's lemma?"
Anonymous at Thu, 20 Mar 2025 19:51:28 UTC No. 16624446
>>16624422
>yes, degenerate
Are you trying to offend me?
Anonymous at Thu, 20 Mar 2025 20:50:09 UTC No. 16624500
>>16624245
https://www.wolframalpha.com/input?
Anonymous at Thu, 20 Mar 2025 21:15:31 UTC No. 16624520
>>16589624
>>16624431
Le axiom of foundation is said to be what prevents sets from including themselves. BUT HOW THE FUCK IS THAT POSSIBLE TO BEGIN WITH? Isn't that a logical contradiction purely in terms of infinite never-ending definition? Also how does a set containing a set with which it's disjoint prevent it FROM ALSO CONTAINING ITSELF?
Wtf
Anonymous at Thu, 20 Mar 2025 21:39:23 UTC No. 16624530
>>16624446
yes degenerate :^)
ultimately whether you define complete as non-empty or not is pretty unimportant
trivial cases arent very interesting and if its not explicitly stated, it will likely be implied that you only want to talk about non-empty graphs [spoiler], degenerate[/spoiler]
your prof or whoever might only care about that for a 1 mark question at the start of a test, so make sure you check their course notes at some point
Anonymous at Thu, 20 Mar 2025 22:35:18 UTC No. 16624556
foundation/regularity essentially just state that [math]\in[/math] is a well-founded relation in the sense that every inhabited set has an [math]\in[/math]-minimal element. (*)
A well-founded relation does not admit infinitely descending chains and classically (and assuming dependent choice) the converse holds too. So indeed well-foundedness of [math]\in[/math] is equivalent to disallowing infinitely descending [math]\in[/math]-chains, i.e. ones of the form [math]x_n\ni x_{n+1}[/math]. Infinitely ascending chains, i.e. those for which [math]x_n\in x_{n+1}[/math] are fine and you can probably think of some simple ones.
Any well-founded relation is irreflexive, so if regularity holds and hence [math]\in[/math] is well-founded then no set contains itself. For suppose there was an [math]x[/math] such that [math]x\in x[/math]. [math]\{x\}[/math] is inhabited, so there is some [math]y\in\{x\}[/math] such that [math]\{x\}\cap y[/math] is empty. Since [math]y\in\{x\}[/math] we have that [math]x=y[/math]. And now we can quickly show that the intersection is actually inhabited: We already know that [math]y\in\{x\}[/math] but since [math]x\in x[/math] and [math]x=y[/math] we also have [math]y\in y[/math], hence [math]y\in\{x\}\cap y[/math], contradiction.
Anonymous at Thu, 20 Mar 2025 22:36:45 UTC No. 16624558
>>16624520
>>16624556
(*) This part isn't entirely true since a one-to-one translation of "[math]\in[/math] is well-founded" would turn into the statement that every inhabited *class* has an [math]\in[/math]-minimal element. This obviously implies regularity since every set is a class but it turns out the two are actually equivalent, so one gets away with just a single regularity axiom in ZF rather than having to turn it into an axiom schema. This is quite remarkable as for example constructive set theories, where one has to replaces regularity with [math]\in[/math]-induction, do not get away with this: [math]\in[/math]-induction for sets does not entail [math]\in[/math]-induction for classes. Hence these set theories are carry an entire axiom schema of [math]\in[/math]-induction in place of regularity.
Anonymous at Thu, 20 Mar 2025 23:18:10 UTC No. 16624597
>>16624556
> is a well-founded relation in the sense that every inhabited set has an â-minimal element. (*)
I wish I knew wtf â-minimal means
>Infinitely ascending chains, i.e. those for which xnâxn+1 are fine and you can probably think of some simple ones.
That would be empty set-constructed numbers like the axiom of infinity allows
The point is what's preventing you from having a set A ={A, {some shit that's not in A}} and then Aâ©{some shit that's not in A} is an empty set but A still contains A
Anonymous at Thu, 20 Mar 2025 23:22:44 UTC No. 16624601
>>16624556
>in the sense that every inhabited set has an â-minimal element
Does that mean that every set must contain either an element that's not a set or an empty set? And since technically every "element" must be a set, when we write {3,4,2} every number there is a set of sets of sets of sets etc. So does it mean that every set of numbers has a hidden empty set that we don't write so that this axiom is satisfied?
Anonymous at Thu, 20 Mar 2025 23:30:35 UTC No. 16624604
>>16624520
>Le axiom
Fuck off to whatever shithole you crawled from.
Anonymous at Thu, 20 Mar 2025 23:31:13 UTC No. 16624605
>>16624604
that would be 4chan
Anonymous at Thu, 20 Mar 2025 23:32:21 UTC No. 16624606
>Sequences of functions. Series of functions. Power series
>Fourier series of continuous functions. Pointwise and uniform convergence. Dirichlet kernel. Approximations to the identity. Fejér kernel. Fejér's theorem. Fourier series of functions in L^1.
>Normed vector spaces, Banach spaces. Hilbert spaces. Riesz lemma. Orthonormal bases. Separably Hilbert spaces. Isomorphisms and isometries. Operators in Hilbert spaces. Compact operators in Hilbert spaces. Fourier series in L^2
>Fourier transform in L^1. Properties. Inversion theorem in L^1. Fourier transform in L^2.
I'm familiar with most of the important topological concepts of analysis and I have a solid linear algebra background. How difficult would be a subject containing these topics?
Anonymous at Thu, 20 Mar 2025 23:35:23 UTC No. 16624610
>>16624597
> The point is what's preventing you from having a set A ={A, {some shit that's not in A}} and then Aâ©{some shit that's not in A} is an empty set but A still contains A
Nvm I get it now. Such A cannot exist because otherwise you COULD form a singleton that would get fucked by the axiom
Anonymous at Thu, 20 Mar 2025 23:52:09 UTC No. 16624617
>>16624556
>and assuming dependent choice
what's that
Anonymous at Thu, 20 Mar 2025 23:58:27 UTC No. 16624622
So basically ZFC is
>Axiom of Extensionality:
> Set = it's elements
> Axiom of Regularity (Foundation):
> There are no infinite turtles down there/Bottom exists
>Axiom of Pairing:
> ditto
>Axiom of Union:
> you can open containers and mix the content
>Axiom of Power Set:
> set of all subsets of a set exists
> Axiom of Infinity:
> you can stack turtles up forever
>Axiom Schema of Separation (Restricted Comprehension):
> you can filter by property
>Axiom Schema of Replacement:
> functions on sets exist
>Axiom of Choice (AC):
>you can pick at least 1 element from any number of sets and put them in a set
It seems like if anything the foundation axiom is the strongest and least intuitive. I assume it basically guarantees empty set but you could just guarantee it and have sets that contain themselves
Anonymous at Fri, 21 Mar 2025 01:24:01 UTC No. 16624667
>>16624245
oh now i get it
it wasn't a comparison of
the old a*b + a*c + b*c = â(9*231 + 142)
and
the new a*b + a*c + b*c = â(11*231 + 142)
it was a comparison of
the old |a| + |b| + |c| = 9*10 + 8
and
the new |a| + |b| + |c| = 11*10 + 8
this can mean only one thing:
that you want me to plug my fayv
Anonymous at Fri, 21 Mar 2025 04:21:09 UTC No. 16624740
are there any interesting topological results regarding finite sets {1, 2, 3, ..., n}? maybe some combinatorical hard facts, anything
Anonymous at Fri, 21 Mar 2025 09:06:10 UTC No. 16624815
>>16624667
>>16624500
>>16624358
* ab = c2 - 2221
I've made mistake, solution is (-49, 45, 4) respectively.
I apologise
Anonymous at Fri, 21 Mar 2025 09:11:02 UTC No. 16624817
>>16624740
>are there any interesting topological results regarding finite sets {1, 2, 3, ..., n}?
The only metric space on a finite set is discrete... (Wait, you wanted interesting.)
Anonymous at Fri, 21 Mar 2025 12:23:45 UTC No. 16624919
>>16624377
Because le funny paradoxes. Just use dependent choice if you donât like them.
Anonymous at Fri, 21 Mar 2025 16:25:22 UTC No. 16625117
>>16624919
What paradoxes?
Anonymous at Fri, 21 Mar 2025 17:53:10 UTC No. 16625189
>>16624376
>>16624382
Bumping this because it's important to me. I'm trying to not be lazy but my dumb adhd brain finds it hard. Not an excuse, just where I'm at.
Other note, if I wanna do applied math and do research in industry should I start throwing away elegance and mathematical beauty and just focus on results? There's some truly gorgeous stuff that comes from applications i'm interested in but it seems like numercial methods are in general more effective than the fancy models I wanna use. Is there any field that uses probability, convex optimization and/or stats but rewards working with proofs and stuff over numerical methods? Should I just grow up and embrace y data overlords?
Anonymous at Fri, 21 Mar 2025 17:57:19 UTC No. 16625196
>>16624245
In the image, the handwritten 1s aren't English.
They're German.
Anonymous at Fri, 21 Mar 2025 18:46:22 UTC No. 16625240
>>16624815
>ab = c2 - 2221
a + b + c = 0
a*b = c^2 â 2683
The foregoing system has 12 integer solutions.
6 of them feature â59.
6 of them feature +59.
proof:
https://www.wolframalpha.com/input?
a + b + c = 0
a*b = c^1 â 2221
The foregoing system has 48 integer solutions.
0 of them feature â59.
2 of them feature +59.
proof:
https://www.wolframalpha.com/input?
â59 is handwritten on that paper.
+59 isn't.
Thus your story doesn't hold up to scrutiny!
Anonymous at Fri, 21 Mar 2025 18:59:38 UTC No. 16625249
>>16624815
>[Hermann Wilhelm Goering]
"He committed suicide by ingesting cyanide the night before his scheduled execution."
Anonymous at Sat, 22 Mar 2025 12:48:15 UTC No. 16625763
>>16625117
Banach-Tarksi, well-ordering of the reals, unmeasurable sets, etc. Basically, choice makes uncountable objects behave in a way we consider âunphysicalâ or âunnaturalâ. Dependent choice is a weaker axiom, but it still allows you to develop the usual real and complex analysis.
Anonymous at Sat, 22 Mar 2025 13:50:21 UTC No. 16625790
>>16625763
First, it's Banach-Tarski, second, you mentioned unmeasurable sets twice, third, you got me curious: what we CANNOT develop in real analysis without an Axiom of Choice?
Anonymous at Sat, 22 Mar 2025 13:58:55 UTC No. 16625794
>>16624815
>[Hermann Wilhelm Goering]
was in charge of the Luft-Waffe
(literally "air weapon")
>ab = c2 - 2221
w + t + c = 0
w*t = c^n â 2221
If n = 2, then the foregoing system has 12 integer solutions.
If n = 1, then it has 48 integer solutions.
If n = 7, then how many integer solutions does it have?
>I've made mistake
what a bogus story
Anonymous at Sat, 22 Mar 2025 14:14:49 UTC No. 16625804
>>16625790
oh, no, a typo! I shall commit sudoku at the earliest convenience for such a grave mistake, you pedant. I didnât name unmeasurable sets twice either. Banach-Tarski is an implication of the former.
>what we CANNOT develop in real analysis without an Axiom of Choice?
Functional analysis. That every linear space admits a basis requires choice. Various weaker versions of choice restrict the cardinality of vector spaces where this theorem holds.
Anonymous at Sat, 22 Mar 2025 15:40:04 UTC No. 16625865
>>16625790
jesus, you sound like a pos
Anonymous at Sat, 22 Mar 2025 16:51:32 UTC No. 16625899
Should the idea of "space" be reworked as long as "topological space" be reworked and space is defined as the set of all transformations, when transformations of any kind don't apply to topology
Or is there something I'm missing
Anonymous at Sat, 22 Mar 2025 16:52:41 UTC No. 16625900
>>16625899
Accidentally something. "As long as topological space is a term"
Anonymous at Sat, 22 Mar 2025 17:52:20 UTC No. 16625975
>>16625804
But we know that all vector spaces have a basis so clearly axiom of choice is true.
Also regardless of le paradoxes HOW THE FUCK WOULD YOU NOT BE ABLE TO PICK ONE ELEMENT OUT OF EVERY SET? (Unless it's empty)
It MUST be true, simple as.
Anonymous at Sat, 22 Mar 2025 17:54:09 UTC No. 16625977
>>16625189
I literally fucking told you, just develop the skill/trait of working consistently
lowercasesage !!4DQphUM8gee at Sat, 22 Mar 2025 20:10:04 UTC No. 16626097
>>16625975
>But we know that all vector spaces have a basis
Yeah, especially [math]\mathbb{R}\text{ over } \mathbb{Q}[/math].
Anonymous at Sat, 22 Mar 2025 20:14:31 UTC No. 16626101
>>16626097
nobody ever builds a linear space on Q
Garrote at Sat, 22 Mar 2025 20:16:48 UTC No. 16626102
Redpill me on module theory.
Anonymous at Sat, 22 Mar 2025 23:37:55 UTC No. 16626229
I'm doing an undergrad second probability course. I really don't like it, I prefer my math more theoretical.
How bad would It be to use a measure theory texbook alongside my probability textbook just to spice things up?
Anonymous at Sun, 23 Mar 2025 03:15:43 UTC No. 16626340
>>16626229
Use Bremaud's new applied probability book as a self-study tool if you feel like it. It's got enough coverage of the basics of introductory probability to be relevant to your undergrad course.
Personally, I consider the measure theoretic stuff to be genuinely different in focus and emphasis than applied probability/statistics. For example, it's quite rare to see measure theoretic textbooks actually go into the process of actually solving for Radon-Nikodym density functions rather than just using the Rad-Nik theorem as an introduction to Martingales. Numerical approximation of Rad-Nik densities are very important in research level applied tasks where you know a density exists but you don't necessarily have a closed form for it.
Anonymous at Sun, 23 Mar 2025 04:26:59 UTC No. 16626370
>>16626330
PEMDAS
Multiplication and division are the same, addition and subtraction are the same. You're supposed to do M and D before A and S.
Anonymous at Sun, 23 Mar 2025 05:07:01 UTC No. 16626383
>>16626370
I forgot to do -53/45 Ă· 1.1 first. I accidentally did 2 2/3-53/45 first. I ignored how the division was supposed come first right after I solved what was in the parenthesis.
Ivan at Sun, 23 Mar 2025 07:15:47 UTC No. 16626438
>>16626333
use udemy
Anonymous at Sun, 23 Mar 2025 12:14:57 UTC No. 16626542
>>16626340
Thanks for the material anon. Just read the TOC, it'll be useful
Anonymous at Sun, 23 Mar 2025 18:26:42 UTC No. 16626792
>>16626383
Gj. Btw, in practice, it'll take too much of your time to write out 9 lines of math when you don't need to.
2 2/3 is better written as 2+2/3, .6 = 6/10, and 1.1=11/10, and you can put in all in one line of math instead of four separate lines. Turn everything into a fraction if there is nothing else faster to do. This way, when you combine them, you have better chances to cancel out terms in in numerator/denominator. Also, keeping numbers as a mixed fraction isn't amazing for avoiding mistakes esp with multiplication
Anonymous at Sun, 23 Mar 2025 19:35:02 UTC No. 16626840
>>16626792
>eeping numbers as a mixed fraction isn't amazing for avoiding mistakes
this, the first thing i do with this kind of calculation is just try to get everything into top heavy or regular fractions
đïž Anonymous at Sun, 23 Mar 2025 20:25:43 UTC No. 16626889
Starting with a space X, suppose S is a subset of X, and V is an open set in X. If I want to talk about the intersection of both, U = S && V, then U is also an open set RELATIVE to S, is that correct? I'm pretty sure it is, but if I'm wrong, please tell me.
Anonymous at Sun, 23 Mar 2025 22:06:49 UTC No. 16626963
>>16626542
Np. It's probably my favorite recent textbook in probability and it serves well as a "transition" text between the applied calculus based undergrad stuff and the more abstract measure theoretic stuff.
Anonymous at Mon, 24 Mar 2025 01:21:58 UTC No. 16627054
>>16621347
i drew pretty
Anonymous at Mon, 24 Mar 2025 11:01:26 UTC No. 16627211
Is there any way to create a separate version of addition that works the same for integers but does something different for non-integers?
Anonymous at Mon, 24 Mar 2025 11:04:47 UTC No. 16627214
>>16627211
Without just doing a piecewise rule for every pair of integers, I mean
Anonymous at Mon, 24 Mar 2025 12:04:38 UTC No. 16627234
>>16627211
You could throw in multiplication by periodic functions that take 1 at every integral value, or something along those lines
Anonymous at Mon, 24 Mar 2025 12:54:06 UTC No. 16627251
>>16627211
[math]#(a,b)= a+b(1+| sin (\pi a+b)|) [/math]
Anonymous at Mon, 24 Mar 2025 14:46:12 UTC No. 16627317
>>16627251
Hash signs aren't supported by mathjax, just use a blackboard bold character as the function symbol
Anonymous at Mon, 24 Mar 2025 14:53:14 UTC No. 16627322
>>16625975
>But we know that all vector spaces have a basis so clearly axiom of choice is true
Proof by uhhhh? What?
Anonymous at Mon, 24 Mar 2025 16:24:51 UTC No. 16627375
Moin. Im starting my next semester soon. I failed analysis 1 and 2. Basically real analysis concepts in both, discussed single and multi variable calculus, and some vector calculus/metric spaces/topology
What books and resources should I use to pass the same courses next semester. I was using a German one called Forster. I really struggle with the jumps in logic for following proofs for theorems/lemmas etc though
I tried rudin and got my ass kicked
Anonymous at Mon, 24 Mar 2025 17:14:28 UTC No. 16627419
>>16589624
Are these books worth to read?
Anonymous at Mon, 24 Mar 2025 17:38:53 UTC No. 16627440
>>16627322
it's fucking obvious ffs
Anonymous at Mon, 24 Mar 2025 17:43:43 UTC No. 16627443
>>16627440
And 2+2=4 is also obvious except that 2+2=11 in base 3. What did you forget ITT, enginigger?
Anonymous at Mon, 24 Mar 2025 18:20:41 UTC No. 16627479
>>16627443
'cept people assume numbers are in base 10 just like people assume log(a) is logairthm base 10
Anonymous at Mon, 24 Mar 2025 18:24:55 UTC No. 16627485
>>16627375
If you had issues with Rudin and read German then get "Differential- und Integralrechnung" by G. M. Fichtenholz.
Anonymous at Mon, 24 Mar 2025 18:26:31 UTC No. 16627486
>>16627479
>people assume log(a) is logairthm base 10
>logairthm
Suddenly I assume that you are retarded.
Anonymous at Mon, 24 Mar 2025 18:30:53 UTC No. 16627495
>>16627375
I also sucked at analysis until I managed to get it back together.
And Rudin, for all its fame and worth, is a trap...like someone
flushing the toilet while you're in the shower.
I can recommend some books for you to try:
Understanding Analysis--Stephen Abbott
Basic Analysis: Introduction to Real Analysis, Vols. 1 & 2--Jiri Lebl (2 pdfs related)
How to Read and Do Proofs--Daniel Solow (to help boost proof-writing, otherwise use...)
Analysis with an Introduction to Proof---Steven Lay
Anonymous at Mon, 24 Mar 2025 18:32:46 UTC No. 16627497
Anonymous at Mon, 24 Mar 2025 19:50:41 UTC No. 16627553
>>16627479
>base 10 log
certified enginigger moment
Anonymous at Mon, 24 Mar 2025 20:11:47 UTC No. 16627560
>>16627553
No idea what kind of shithole are you from, but in civilized places [math]\log[/math] defaults to decimal logarithm and natural logarithm is [math]\ln[/math].
Anonymous at Mon, 24 Mar 2025 20:12:42 UTC No. 16627561
>>16627485
Thanks anon I'll check this out
>>16627495
>>16627497
thanks for giving a lot of recommendations
Do you have any thoughts on Tao's book? Any point in referencing that too or would it be useless for the most part if I'm using Abbott?
Anonymous at Mon, 24 Mar 2025 20:21:30 UTC No. 16627567
>>16627560
No idea what shithole youâre from, but in civilized places people do calculus instead of counting squares like children and so log or ln or whatever the fuck always means the natural logarithm. Just like the argument of trig functions is in radians. Log base 10 is fucking useless aside from le plots.
Anonymous at Mon, 24 Mar 2025 20:21:55 UTC No. 16627568
>>16627553
>log = ln
which state are you in
Anonymous at Mon, 24 Mar 2025 20:23:57 UTC No. 16627570
>>16627568
https://en.cppreference.com/w/c/num
https://numpy.org/doc/2.1/reference
https://reference.wolfram.com/langu
Anonymous at Mon, 24 Mar 2025 20:33:28 UTC No. 16627582
>american
>american
>american
jesus christ dude, which state are you from
Anonymous at Mon, 24 Mar 2025 21:02:59 UTC No. 16627611
>>16627609
Next thread
Anonymous at Tue, 25 Mar 2025 00:19:13 UTC No. 16627804
>>16627561
>>16627495
Some people like Tao's exposition, some find it pedantic or dull.
I would say, go for what topics you need in a book by looking
through the table of contents. From what I heard, Tao doesn't
cover metric spaces, something that's needed in later analysis
courses. But if it reads fine for you, go for it, and cover any overlap
with Abbott.
Anonymous at Tue, 25 Mar 2025 03:04:04 UTC No. 16627940
>>16627804
Tao does metric spaces in Analysis II
Anonymous at Tue, 25 Mar 2025 03:07:42 UTC No. 16627943
>>16627940
>>16627804
That's good, then. Thanks for letting me know