ITT: Discussion of mathPrevious thread: >>16835273
What are some other people like:>Cumrun Vafahttps://www.physics.harvard.edu/people/facpages/vafa>Semën S. Kutateladzehttps://ncatlab.org/nlab/show/Sem%C3%ABn+S.+Kutateladze>Dingding Donghttps://math.uchicago.edu/~may/REU2017/REUPapers/Dong.pdf>Arjum Nigamhttps://math.uchicago.edu/~may/REU2022/REUPapers/Nigam.pdf
>>16876118I hate the euler identity
[math] \displaystylef(x) = e^{-ix}(\cos x + i \sin x)\\ f^{\prime}(x) = e^{-i x}(i \cos x - \sin x) - i e^{-i x}(\cos x + i \sin x)\\ f^{\prime}(x) = e^{-i x}(i \cos x - \sin x) - e^{-i x}(i \cos x + i^2 \sin x) \equiv 0\\ f^{\prime}(x) = 0 \;\;\; \forall \; x \in \mathbb{R}\Rightarrow f(x) \text{ is a constant}\\ f(0) = e^{0}(\cos 0 + i \sin 0) = 1 \cdot(1+0) = 1 \Rightarrow f(x) = 1 \;\;\; \forall \; x \in \mathbb{R}\\ \\1 = e^{-ix}(\cos x + i \sin x) \Rightarrow e^{ix}=\cos x + i \sin x \;\;\; \forall \; x \in \mathbb{R}[/math]
Has anyone here self studied for the Calc I CLEP exam? I need to do it within 2 weeks so I can take the exam in time to get into calc 2. I understand derivatives and integrals and can do basic ones. I'm working through the Klein Intuitive Approach book but I'm only like 60 pages in. Any advice?
Gromov bros, how do we respond ?
I love this board. At 31 years old, I am newly married and have a kid on the way. Tons of stability in my life that I never had, and unfortunately my love of all things esoteric has always, and keeps leading me to, math. I think the door is closing on me to pour myself into mathematics as a pursuit or even a hobby, and if you're one to believe in a framework like the Myers Briggs, I'm a somewhat stereotypical INFP -- oriented towards language/literature, the arts, social sciences. Those simply aren't cutting it anymore. I want to explore, and math (or my perception of what it is, but I've only ever taken up to Calc II) always seemed like something to scratch that itch.
>>16876474>oriented towards language/literatureMathematical Linguistics, Art of War being Linguistic Vector Analysis, if they here (vector), then be there (polaric anti-vector). More words add dimensionallity.>the artsProportion, cadence, perspective, ratio.>social sciencesThermoEconomics (as an example), from there you can deduce Psychology from MetaPhysical (broad, loose, Physics, at the human scale and timeframe) properties.
>>16876160You dont like the +1....because youre negative.;^)
>>16876474>I want to explore>oriented towards language/literature>mathTry:>A Mathematician's Apology - G. H. Hardy>Calculus: A Liberal Art (a.k.a. Historical Approach) - William McGowen Priestley>Alice in Numberland: A Students' Guide to the Enjoyment of Mathematics - John Baylis & Rod Haggarty>Proofs and Refutations: The Logic of Mathematical Discovery - Imre Lakatos>The Enjoyment of Mathematics - Hans Rademacher & Otto Toeplitz>The Pleasures of Counting - Thomas William Körner>Mathematics and the Imagination - Edward Kasner & James Newman>Jorge luis Borges preface and review of Kasner & Newman>Jorge Luis Borges works>Logicomix: An Epic Search for Truth by Apostolos Doxiadis & Christos Papadimitriou>Mathematics Made Difficult: A Handbook for the Perplexed - Carl E. Linderholm>Surreal Numbers: A Mathematical Novelette - Donald Knuth>Récoltes et semailles - Alexander Grothendieck
It's over bros. An ex-Google DeepMind engineer solved Naive-Strokes and Hedge conjectures with LLM.
I'm about to finish Stewart. What are the best strategies to understand math on a deeper level? I'd like to pursue a PhD in Physics later.Also I find pure math so fucking boring, I want to like it but I only find it interesting in applied form.>>16876626Proof?
>>16877467>What are the best strategies to understand math on a deeper level? I'd like to pursue a PhD in Physics later.There's a whole genre of books to read after Stewart if you want to pursue physics. I will name some and rhem so you can extrapolate:>George Arfken, Mathematical Methods for Physicists>Sadri Hassani, Mathematical Methods: For Students of Physics and Related Fields>James Nearing, Mathematical Tools for Physics>Michael Reed and Barry Simon, Methods of Modern Mathematical Physics>Paul Bamberg and Shlomo Sternberg, A Course of Mathematics for Students of Physics,Read this too:https://www.goodtheorist.science/BUT you will need to learn about statistics, measurements, laboratories, data analysis, computer simulations. This is equally if not more important than the methods and the theory.>Also I find pure math so fucking boring, I want to like it but I only find it interesting in applied form.Why are you so sure? Do you hate synthetic geometry or proof-based linear algebra? (The latter being a misnomer for abstract vector space theory).
>>16877519Yes when I find proofs or exercises about reaching a given result such as a double partial derivative but no use, I feel my eyes getting glassy with boredom. But when its related to say the change in a gas or an electromagnetic field, I get excited. I've been reading Stewart as an exercise in order to truly master it, but I feel a craving to move onto Griffith's EM or Circuit Analysis already, though I don't wanna move from Calculus until I'm ready to tackle higher Differentials and math modeling.
>>16877521>double partial derivativeI think mathematicians find those boring too. As i said, you wont know until you have studied abstract vector space theory, which is a crucial prerequisite for quantum mechanics. Try reading Georgiy Shilov's Linear Algebra and skim through AI Kostrikin and Yu. I. Manin's Linear Algebra and Geometry.Also, anons here like dropping this reading listhttps://sheafification.com/the-fast-track/And thishttps://theportal.wiki/wiki/Read
>>16876160
what’s your favorite johnson solid
>>16877519>>16877542What is the difference between Applied Maths and Physics?
If you select n independent random numbers uniformly distributed between 0 and 1, the expected value of the second-largest among them is (n − 1)/(n + 1).
>>16877644Don't know about the difference, but neither do math that's for sure.
Guys I think I discovered something amazing!
Am I retarded or is probability theory really hard?I still can't wrap my head around what the fuck a martingale is. The lectures are awful, it's all cryptic set and measure theory>The borel set equipped with a sigma algebra inducing a probability measure on a filtration of the complete probability
>>16878243It's very unintuitive. I don't think it's easy for anyone, at least anyone more or less average.
what is the best book for differential equations?
>learn le advanced mathematics for finance to become a millionaire from trading >"Ummm akshually the market is efficient so you can't beat the market long term, but here's how you can calculate the ito integral of a wiener process and price premiums" My grades are shit because the courses are hard and it's a noname uni so zero chance for quant. Don't fall for the finance meme
>>16878729Should have majored in physics or pure math like a real man. Only pussies go for anything related to economics, also the point of a finances major is that you already are a millionaire otherwise its pointless.
>>16878645Evans
>>16878729I'm not into the quant space but based on what I have heard/read there are a lot of quant companies nowadays that aren't doing rigorous work, their models are probably ML-based shit. Only the top firms might still want to be trying more advanced things but I would bet that even on those firms the groups doing such work aren't that big.
ok please don't hate on me but is there an easy way to remember formulas for the area of circles spheres and cones?I think it's pi r for circle, 4 pi r 2 for sphere and pi r2/2 for cones or something?
>>16878933You can just calculate the integrals.
>>16878938I don't know what that means, but please explain it.
>>16878938Yes, because that's so much easier than just memorizing a few compact formulas.>>16878939It's beyond your math level, but the tl;dr is that there's a few convenient methods out there for calculating the area enclosed by a function. It's not gonna be helpful to you at this point in your education.>>16878933To answer your original question:Cone is the easiest. It is 1/3 of a cylinder with the same radius and height.A cylinder is just the area of the circle at the base times its height (which I would hope is obvious) so take 1/3 of that to get a cone.The area of said circle is pi*r^2. The lame joke I was told in grade school was "pie aren't square! Pie are round!" It's dumb as fuck but it might help.The volume of a sphere is kinda hard to get an intuition on (at least without the aforementioned integrals). You're probably just gonna have to remember 4/3 times pi*r^3.
>>16878945>pie aren't square! Pie are round!" It's dumb as fuck but it might helpI like that.ok socircle is pi r 2 I just have to remembercylender was obvious it's just h pi r^2cones are 1/3 of a cylender so 1/3 h pi r^2where does the 4/3 come from exactly? I get the pi r 3 is because it's a 3d shape but the 4/3 doesn't make sense to me
>>16878948>where does the 4/3 come from exactlyThat's where the integrals thing comes in, which is calculus. Long before the invention of calculus, Archimedes discovered this relationship empirically by manually slicing up a sphere into thin circles. He declared that his proudest moment.For now, just remember 4/3.
>>16878951ok, thanks I'll just keep this in mind
>>16878948>4/3 come from exactly?This might be helpful or it might just be more confusing:Slice a sphere in half. The resulting dome is twice the volume of a cone of the same dimensions (therefore, 2/3 of a cylinder). So we get 2/3h*pi*r^2. Since h = r, we substitute and simplify to 2/3*pi*r^3Because we were just looking at one half of the sphere until now, double the result:2*(2/3*pi*r^3)Therefore: 4/3*pi*r^3So if you can remember the progression as:>cone = 1/3>semisphere = 2/3>cylinder = 3/3Then you might have a good intuition going forward.
A semi circle is inside of a quarter of a circle. The yellow region has the same area as the pink region. What is the angle alpha?
>be 40yo looking at science clips on YT>hmm interdasting but how do they know?>decide want to get back into math so I can learn physics and understand it fundamentally>habing a bit of fun relearning prealgebra and algebra concepts i havent thought about in 30 years>decide want to go full bore and learn advanced mathsWould learning math to a high level and even taking it up as a hobby help me get smarter? Like would I be able to better solve real world problems at my job (I work as IT admin)?
im retarded how do i prove that - for a given positive integer (say, z) it can't have a prime factor (say, p) which exceeds both values (say - m,n) at once of any of its factor pairs?in other words, prove m*n is not divisible by a prime p if p is greater than both m and n. however, avoid using proof by contradiction (unintuitive), or the concept of the fundamental theorem of arithmetic // unique factorization (cheating and unintuitive), or reference to euclid's lemma (unintuitive), or the concept of gcd in general (i.e., incl. bezout's identity). ideally just explain it in terms i can understand, that is, a number is always conceived of as a product representing buckets and balls (e.g., 3 x 2 = 3 buckets with 2 balls in them = 2 buckets with 3 balls in them). thanks
>>16879505First I will assume it is obvious that any given number will only have one prime factorization. Next, if you multiply any two numbers, it is the same as multiplying their prime factorizations together (see: commutative property). Example: 6*8 expands to 2*3 * 2*2*2 which has the same product: 48Since p > m or n, then we know it is not present in the factorization of either individually. Therefore, p is not a factor of their product either.
>>16879498Maybe you would be able to call bullshit more easily if you don't already, at least. There's a lot of myths about transferable skills.
>>16879505>>16879527Sorry, didn't read the rest of your post.You're saying "don't use x, y, or z" kind of proof because they're "unintuitive" but even going the "m buckets of n balls" route would end up being a proof by contradiction. Eg. Add k empty buckets such that m+k = some prime p and p > n then attempt to redistribute the balls. If the number of balls you had to remove from m buckets were evenly divisible by k buckets then m+k wouldn't be prime unless it was a factor of n, but if that were the case then p wouldn't be greater than n. But that's a proof by contradiction.
>>16877644Applied Maths :3 < Pi < 3.2Physics ;Pi = 3
>>16879583I don't remember a single time when Pi was approximated as 3 during my entire physics bachelor's degree.
>>16879527>>16879537retard here - ok i like except the part where u start with unique prime factorization is obvious. perhaps but there's gotta be an even more glaringly intuitive way of considering it, like maybe modular arithmetic.i was thinking of it like this (i phrase it so retardedly to make it more intuitive to me as I read it)Given m,n and some number p greater than both m and n, p cannot be a divisor of m*n under the following 3 circumstances -1) m,n are prime (and p=/=m*n), 2) p is prime, or 3) m,n,p are all prime. Recall that multiplication of any m,n will only ever result in some collection which is capable of being broken into some # of equal sets of whole parts (it is "composite"). In general, for m*n the # of whole parts in the equal sets can be either: 1) 1, 2) m, 3) n, 4) m*n, 5) some composite factor of m*n, or 6) some prime factor of m*n. Since p is greater than m and n, and prime, this rules out possibilities #1-5 by definition. However, the only way for a factor of m*n to exceed both m and n is if it is composite (?? - this is where i was thinking of balls & buckets) because otherwise that would mean it is possible to break m*n up into a certain number of equal parts of that prime factor... where the process of multiplication results in a product which can only be broken evenly by the prime factors of m and n (else those equal parts can be broken further)... but prime p, which is larger than m and n, can only ever be broken unevenly (e.g., p = 7 = 2+2+3) into prime factors of m and n, if at all ...
>>16879584Your iq seems to be 2 digits like your Pi approximation.In reality :Applied Maths :3.1415 < Pi < 3.1416Physics :Pi = 3.1416Maths :Pi = [math]\frac{4}{1}-\frac{4}{3}+\frac{4}{5}-\frac{4}{7}+\frac{4}{9}... = 3+\frac{4}{2*3*4}-\frac{4}{4*5*6}+\frac{4}{6*7*8}-\frac{4}{8*9*10}+\frac{4}{10*11*12}...[/math] = 1(base Pi)
>>16879527>>16879537>>16879587o shit - retard here again, i found Gauss's proof of this at the start of Disquistiones. the fact he resorted to his own inductive proofs using prime factorization instead of just invoking euclid's algorithm (even avoids euclid's algorithm when discussion gcd) suggests he also wanted something more direct
>>16879604oop forgot pic
>>16879605also theorem 16
>>16879605>The product of two positive numbers, each of which is smaller than a given prime number, cannot be divided by this prime number.That seems trivial.a, b and Prime are in N+*a < Prime, then a/Prime is always lesser than 1, therefor no integer divisor.b < Prime, then b/Prime is always lesser than 1, therefor no integer divisor.a*b < Prime*Prime, then a*b/Prime*Prime is always lesser than 1, therefor no integer divisor neigher.
>>16879590>Your iq seems to be 2 digits like your Pi approximation.You said 3 before, then changed it in this post, and posted an overhead joke image, I'm assuming you're projecting
>>16879614>You said 3 beforeThat's an hyperbole, an exaggeration as a rhetorical device or figure of speech.It is often used to make a joke.Are you an AI ?
>>16879613Oopsy, should've add a couple of lines at the end :a, b and Prime are in N+*1) a < Prime, then a/Prime is always lesser than 1, therefor no integer divisor.2) b < Prime, then b/Prime is always lesser than 1, therefor no integer divisor.a*b cant have Prime*Prime as a divisor because of 1) and 2)Prime*Prime has Prime as a divisor obviously : Prime*Prime/Prime = Primea*b < Prime*Primea*b/Prime < Prime ..., then (a*b/Prime)/Prime is always lesser than 1, therefor no integer divisor.
>>16879616>a/PrimeI'm pretty sure Gauss doesn't want to use rational numbers here to prove a basic fact about integers. Much less arguements that depend on the order relation of those rationals.
>>16879620So it's a typo in pic in >>16879605?
Ok a manifold is paralelizable its tangent bundle is isomorphic as a vector bundle to the trivial bundle right? Now a basic question should be then to give an example of a non trivial vector bundle that is nonetheless diffeomorphic (but not linear on fibers) to the trivial bundle, right? I can't find anything about this online besides some articles showing cases where it is enough for the total spaces to be diffeomorphic. Fuck differential topologists and their handwavy bullshit.
>>16877519don't forget V. I. Arnold Mathematical Methods in Classical Mechanics.
>tfw brainlet and had shitty teachers so now I'm so afraid of maths I've become unable to learn and should kms
>>16879441One solution at 0, and another one I don't know how to squeeze a number out of
>>16879709[math]\sin \alpha \cos \alpha = 0.5 \ sin ( 2 \ alpha ) [/math]
By Hopf-Rinow, if a riemannian manifold is geodesically complete it is also a complete metric space. Considering that the metric (in the metric space sense) coincides with the metric (in the riemannian sense) in a normal neighborhood, can I say that a geodesically complete riemannian manifold is locally a Banach space?
>>16879775Looks trivial.Yes, if their number of dimensions, their number of unit vectors, are the same locally. Or converge to have he same unit vectors.
Does one ever reach a point where doing proofs becomes like doing regular calculations, in that it's a straightforward and you know what you need to do to find the answer. Or will it always be like feeling clueless and lost in the dark and trying random shit from similar problems and hoping you haven't forgot some assumption?
I've made a thread on a math data project at >>16879862. If you got some idea or question about Wikipedia statistics, you can ask there I can look it up. Doesn't have to be something small.>>16879914Well there's mechanical tasks, like taking derivatives, where you can follow a path. With theorems, there's a broad spectrum and given there's unsolved conjectures, it can't be as mechanic. With textbook questions of which you know upfront that there's gonna be a solution to which you can get within a sensible time, it's often just a matter of being willing to writing down all the definitions and putting things together. Some things are tricky and come with exercises, some things are naturally less intuitive to people than others. Trying ideas will probably always be part of the game - to a degree it's a feature. better to love the bomb than taking it to you being in the dark.
>Look up how to find the area of a triangle>"It's just base times height divided by 2, bro">Realize I don't know what the height of my triangles are>Look up how to find height of triangle>"It's just 2 times the area divided by base, homie"ARRRRRGHI know this is a basic math problem. But come on.
I am struggling to understand the difference between in probability convergence and almost sure convergence.a.s. means that X_n\to X for all sets of measure greater than 0.i.p. says that the the probability (a measure) of X_n and X being different is zero. More formally P(|X_n-X|≥\varepsilon)=0.Isn't this the exact same thing as the a.s. ? I.p. convergence means the intervals where X_n and X are different have measure zero, same definition as the a.s.
>>16879920Thanks that's reassuring.So I guess, besides understanding basic logic stuff, the only way to get better at proofs is to grind exercises
>>16879622retard here - are you referring to the fact that he says "number"? he refers to integers as "numbers". plus his modular arithmetic wouldn't make sense otherwise
>>16879965it really is a crime that they don't teach Heron's formula more aggressively
>>16879965Pick a side as a base and draw a perpendicular line from that side to the opposite corner. That's your height. Depending on what information you do have about your triangle, you'll do some form of pythag fuckery or using trig functions.
>>16880027All I know, are the lengths of the 3 sides. Nothing else.Intuitively, I can draw the perpendicular line. Or with a tool, I can draw it that way. But supposing I only have numbers to work with, how does one get the height?If you're curious as to why I even bring this up? I'm doing some stuff in Blender, and need to know the area between 3 points. I can actually get the height using vector math to project 1 of the points onto the vector of the other two. And then measure the distance between the projected position and the original position. But I wanted to know how to do it otherwise.
>>16880045>>16880004I should also mention, I looked up Heron's formula, and it doesn't make sense to me. I followed along with a video, and I got weird results and don't know where I went wrong.
>>16880046Can you post an example and your calculations?
>>16880046Step by step for heron's formula:>label each side a, b, and c>(a+b+c) ÷ 2, label this new quantity "s">(s-a)*(s-b)*(s-c)*sThe square root of that product is your area.
>>16880059>>16880056Thanks guys, but I got it now. I repeated the steps and it worked. I don't know where I fucked up the first time.
>>16879654>Now a basic question should be then to give an example of a non trivial vector bundle that is nonetheless diffeomorphic (but not linear on fibers) to the trivial bundle, right?The tangent bundle on [math] S^1 [/math] is the easiest example, and the next easiest is the tangent bundle on [math] S^1\times S^1 [/math]. For [math] S^1 [/math] you can prove this by writing down a two-set atlas on [math] S^1 [/math], writing the tangent bundle in local coordinates on each chart (where it's necessarily locally trivial), and showing that the transition functions on the overlap are the identity. If you want something you can visualize as "looking non-trivial but is trivial" you can add two half-turns to [math] TS^1 [/math].Geometric intuition is unfortunately something that's hard to approach pedagogically since everyone responds to something a little different, and a lot of authors don't even try. It takes time and effort to consolidate everything you read into your own mental picture, and if it makes you feel any better, the process will at least give you some chest hair.
>>16879966Both of your definitions are wrong. The former means the set on which the sequence does not converge is of probability zero.This is equivalent to saying that for every epsilon > 0, the event that X_n is epsilon away from X infinitely often is of probability zero.The latter means that for every epsilon > 0, the probability that X_n is epsilon away goes to 0. The latter is much weaker because there is only one n in the probability, while the former is considering the intersection of many events involving n. It is entirely possible for X_n to be within epsilon to have very high probability for every n, but low probability for all X_n to be within epsilon.
What is usually meant when an author speaks about a partial function from a A to B? I get that intuitively it's a function that's not necessarily defined for the entirety of A but how are they usually encoded in different foundations (considering most don't take partial functions as a primitive)?
>>16880153How is a function (just a function) encoded in the different foundations?
>>16880181The usual definition is that a function is a triple [math](A,B,\Gamma)[/math].[math]A[/math] and [math]B[/math] are arbitary sets and [math]\Gamma \subset A \times B[/math] such that for every [math]x \in A[/math] there is exactly one [math]y \in B[/math] such that [math](x,y) \in \Gamma[/math].Some authors however just say that the function is the graph [math]\Gamma[/math] and make the domain and codomain an implicit part of the definition instead.
>>16880109>the event that X_n is epsilon away from X infinitely often is of probability zero>the probability that X_n is epsilon away goes to 0>The latter is much weaker because there is only one n in the probability, while the former is considering the intersection of many events involving n. So a.s. is making a statement about the entire sequence while i.p. is only making a statement about the final resulting X_n?My problem is that the way I learned this is that a.s. is too strong and only used for theoretical stuff, i.p. is for saying that an rv goes to constant and i.d. is for saying that an rv converges to an rv. But the true def of all three is about convergence of sequences of rvs X_n to an rv X.I know a.s. \implies i.p. \implies i.d.. I am struggling to see why the implications don't work backwards
Are there uncountable sets that can be well-ordered with Dependent Choice?>inb4 ask muh AIIt’s hallucinating.
>>16880227Both are about the entire sequence, but the probabilities involved take different number of terms in its argument. A.S involves the entire sequence in the argument of the probability, but I.P. involves exactly one term in the argument of the probability at one time and then takes the limit. A.S [eqn] P \{ \omega \in \Omega \mid \lim X_n (\omega) \neq X (\omega) \} = 0 [/eqn]or equivalently, for all [math] \epsilon > 0 [/math], [eqn] P \{ \omega \in \Omega \mid |X_n - X| (\omega) \geq \epsilon \text{ i.o.} \} = 0 [/eqn]In both the cases, the probabilities involved take intersections of events involving several [math] n [/math].It's the most natural extension of the calculus definition to random sequences. A sequence converges almost surely if and only if for almost all omega when the sequence is realized at omega it converges in the calculus sense.I.P For all [math] \epsilon > 0 [/math], [eqn] \lim P \{ \omega \in \Omega \mid |X_n - X| (\omega) \geq \ epsilon \} = 0 [/eqn]The probability is only about exactly one [math] n [/math] at a time. The different [math] n [/math]s come into the picture when taking the limit. The distinction comes in the fact in that the latter is only checking one X_n at a time and is ignoring how the X_n behave together. If you consider an independent sequence of 0s and 1s where the nth term is 1 with probability 1/n, then the sequence obviously converges in probability to 0, but it does not converge almost surely because using the fact that the harmonic series diverges you can show that almost surely the sequence will hit 1 infinitely often, which prevents it from converging to 1. So, you have an example of a sequence that converges in probability, but will for almost all omega, not be converging in the calculus sense.
>>16880272tl;dr yeshttps://math.stackexchange.com/questions/1833322/a-well-order-on-a-uncountable-set
>>16880300Converges in distribution is even weaker since it completely ignores the probability spaces and only measures how close in distribution each X_n is to X. It completely ignores how far apart the actual metric distance between X_n and X is in the probability space. A.S. and I.D are most useful from my experience, with the latter being especially useful in statistics: it allows us to approximate an unknown distribution. I personally have never seen a use of I.P. and I find it to be the most unnatural definition. Strangely enough, in every probability space there is a metric whose metric convergence agrees with convergence in probability while no such metric exists for a.s.
>>16880302I’m asking more in line of well-orderable by Dependent Choice, but not Countable Choice. I’m trying to understand Dependent Choice in terms of transfinite induction type arguments. I already know that it’s related to Baire spaces, but I want to get a “pure” set-theoretic intuition for DC.
>>16878111>neither do math that's for sure.Exactly because that would be like doing linguistics. A pointless exercise for pussies. Real men apply to nature in order to force it to reveal its secrets, that's the Physicist way.
>>16880192Well, if you want a partial function you can change this definition easily. For different foundations, see:https://web.archive.org/web/20250901103543/https://ncatlab.org/nlab/show/partial+function
>>16880348What secrets has string theory revealed?
Starting grad school on the 20th. Scared. >>16878729Who cares about money, that's the exact math you need to build better weapons and optimize killing.
>>16880435Weapons that will go to Israel.
>>16880447? I don't care. Making better weapons is awesome. Not everyone is morally charged or is attached to an interest group.
>>16880454When you stand before God, you cannot say that I was told by others to do thus, or that virtue was not convenient at the time; it will not suffice.
You roll four dice, three of which are normal dice and the fourth one has S sides. The sum of the digits you rolled is 14. What is the probability that S=6?I think this problem is weird because of the fact that S could be any integer out of infinite possibilities. So it seems like the probability for S being six should be zero. But then on the other hand, it feels pretty reasonable that S would be six beause of the fact that 14 would be the expected sum if S was six. And the bigger the number, the less likely it would be for S to be that number.How do you deal with this problem? Is there a solution?
>>16880718What is the distribution of S?
>>16880747It could be any number with equal probability.
>>16880765There is no uniform probability on the entire natural numbers.
>>16880952>entire natural numbersBro, any number means 4,6,8,12 or 20 since obviously there are no platonic solids with a differnet amount of sides. Have you ever seen a 3 sided die?
>>16880971There is no uniform probability on any infinite subset of natural numbers.
>>16870812
>>16880952What if you restrict S to be an integer between 1 and 1000, how would you solve the problem then?
>>16880765>It could be any number with equal probability.Obviously not. That's whole point of the problem.>>16880976>There is no uniform probability on any infinite subset of natural numbers.This is irrelevant.
>>16880718>S could be any integer out of infinite possibilitiesWe know the lower bound is 1.We can calculate an upper bound by looking at the case where all other dice roll a 1.We know we rolled a 14. If 3 dice rolled a 1, then the remaining die is 11.Not that it really matters all that much, what you would do is construct a sample space of all values for the three known dice where their sum is less than 14, call the number of possibilities n. Then count the number where this sum is 6 less than 14, call that m.Your probability is m/n.
>>16881283This is irrelevant.
>>16881259Let N be the maximum number of sides for the mystery die.For a fixed k, you can calculateP(D is n-sided | D = k) = P(D = k | D is n-sided) * P(n-sided) / P(D = k) = [1 / n if k <= n, 0 otherwise] * 1/N / P(D = k) You can find P(D = k) as sum_n (D = k and D is n-sided) = sum_n 1/n * 1/N = H_n / N, where H_N is the nth harmonic convergentSeems hard to calculate this probability exactly, but for large N you can approximate H_N as ln(N).Then P(D is n-sided | D = k) is approximately(1/n * 1/N) / (ln(N) * 1/N) = 1 / (n ln(N)).For your original problem, we're taking n = 6 and summing up over k ranging from 1 to 11 (and multiplying by the probabilities that 3 normal-sided die sum up to k); as N gets larger, these probabilities all go to 0, so in the limit, the probability that S = 6 is 0.
>>16881293Incorrect.
>>16880423Oh nothing much except HOW FUCKING EVERYTHING WORKS! Really, math alone is just an exercise in masturbation like philosophy. Physics is the real man's way
>>16881004Those filthy jiggers deleted my pretty images!
Since [math]T\in \mathrm{Hom}(V,W)[/math] is actually a tensor [math]T\in V^*\otimes W[/math], shouldn't we write [math]{T_\mu}^\nu[/math] instead of [math]{T^\mu}_\nu[/math]?
>>16881780Leaving space for “phantom” indices only makes sense if you’re working on Riemannian manifolds where the metric allows one to “move” indices via the musical isomorphisms. For generic vector spaces there is no such canonical isomorphism between a vector space and its dual, so you can just write T_/mu^/nu unambiguously.
>>16882072Suppose we are working on a riemannian manifold. Which notation makes more sense?
>>16882095Then there is a canonical isomorphism between the tangent and cotangent bundles, so it doesn’t really matter. The only context I can think of where it would is Hamiltonian mechanics, which is done exclusively on the cotangent bundle. But 95% of physicists don’t care and abuse notation anyways.
>>16882095>>16882126contI guess my point is that it’s all just a notational convention and a really inconsequential one at that. People who take differential geometry seriously avoid tensor notation because it requires descending to charts, which need to be glued together, and it all becomes a mess that is entirely avoided in the coordinate-free language. On the other hand, people who don’t care (ie physicists) don’t give two shits about rigor anyways and raise indices willy-nilly.
>>16882126>Then there is a canonical isomorphism between the tangent and cotangent bundlesHow?
>>16880300>>16880309Thanks for the answer.from my experience so far, i.p. is used alot in statistics to show consistency and for asymptotics in general
>>16876118Is the probability book by the Kolmogorov (pbuh) any good?It would be cool to learn probability from the words of the big K himself
>>16882462No. It has no exercises, and it's the first complete work in rigorous probability. You are basically going to be learning from a long paper. Probability and its exposition has advanced a long since His time.
>>16882435Most things that can be shown to converge in probability can also be shown to be converge almost surely albeit with greater difficulty, and the latter is significantly more operational. Even if you can show it converge in probability despite not almost surely, it's not particularly useful in deriving secondary results outside of what can be shown also from convergence in distribution. There is one result I can think of: convergence in probability of a uniformly integrable sequence implies L1 convergence, but even then, every time I have seen that result used, the sequence also converged almost surely. And when it comes to question of practical use, none of the convergence are useful really. In practice, it's concentration inequalities that's useful since they give exact operational bounds.
>>16881259As you increase the number of allowable sides, the probability will go to 0. Otherwise, you can use probability generating functions and write a simple program to calculate the coefficient of the 14th degree term for different values of S. The 14th degree term for S = 14 and above are going to be the same, so you only need to calculate up to S = 14 because of which the time complexity of the algorithm would be constant. Either that, or you approximate.
>>16882463>Probability and its exposition has advanced a long since His time.Like what? I just want to learn rigorous probability at an entry level, the advanced stuff can come later
>>16882578Listen man, if you think learning from an ancient foundational paper on probability theory is going to be more effective than reading a book that's the culmination of years of combined experience of experts teaching it to their students, I really don't think you're intelligent enough to study probability, or at least make any productive use of it. In the future, try not to be this contrarian trying to find some special golden book that will unlock the vast mysteries of the universe as opposed to just going for the usual suggestions like Durrett or Kallenberg or whatever.
>>16882578It's not about "advanced" vs "entry-level." It's about the fact that teaching a subject is, itself, a skill to be honed. And later works often come scarred with lessons from the past.
>>16882593Something tells me you think learning is restating a collection of facts at the end of a semester.
Can you help me figure out an equation for purple area?There's three functions f1, f2 and f3.f1 and f3 are linear, f2 is constant, set to 1 for simplicity's sake.t1, t2 and t3 are abcissa of points where f1=f2, f2=f3 or f3=0. I should maybe add t4 which would be the abcissa where f1=f3.The difficulty is that f3 can move. In picrel I show two cases, either t1<t2<t3; or t2<t3<t1.Naively I want to calculate the integral of min(f1, f2, f3) if such a thing exists, sadly I didn't find anything regarding primitives of min() with three functions, only two.https://www.youtube.com/watch?v=vB14KK8_OdEI understand the logic with two functions, it's pretty easy to find the intersection point and get an equation of the integral, but f3 moving left means two cases. Is there no way to get a general equation? An equation that would account for all positions of f3?
>>16883364I'm not gonna do your homework for you, but you can start by taking the minimum of the two sloped functions. Once you have a general equation for that, take the minimum of your new function and the constant. As with may things in life, big problems get easier when you break them down into smaller and more manageable parts.
What the fuck is up with a design on a soccer ball.Why does a 2-sphere allow for perfect tilings of pentagons surrounded by modulating hexagons in all directions.What the fuck is going on.Other people seem to consider this insignificant and have just accepted it.
>>16883455I realized that my variable names are bad, I call both negative slope functions f3 but they aren't the same function at all : they haven't the same y-intercept. So that y-intercept is a variable that is part of the equation.Drawing the problem is helping a lot. Not homework btw, just fun.
>>16883488What I'm saying is you calculate the integral of the triangle as if it's case 2 globally. Ignore the constant. Once you have sufficiently generalized that equation, then you take the minimum of that and your constant. If it helps: min(min(f1,f3),f2). Work inside to outside.
>>16876160i hate that it's pronounced Oiler
>>16883487There's nothing really special about that particular orientation. One way to think about it is to imagine the hexagons were all originally triangles whose vertices meet at the center of the pentagons. The soccer ball is just what happens when you cut those tips off.You can construct arbitrarily many similar patterns by adjusting the angle at which those triangles meet.See: https://en.wikipedia.org/wiki/Geodesic_polyhedron
>>16883509How much of a blackpill is tilings on spherical geometry in general
>>16883518I'm not sure if I'd call it a blackpill. But any convex polyhedron technically works as a spherical tiling. Start with any of the regulars and procede to cutting off vertices. Or you can work the opposite direction and model a sphere as arbitrarily many triangles then use those triangles as "pixels" for whatever shapes you want your tiling to consist of.The fringes of the field are just those simple concepts taken to logical extremes.
>>16883556Seems too specific to be called a coincidenceI will be researching this intently
>>16882468Are there any graduate books that properly cover sequences of random variables? I am doing mathematical statistics and this is the one thing that never sat right despite it being pretty much all of large sample theory.
gemini dynamic view is pretty good
>>16883364Integrate dy from 0 to min(intersection, f3). Now you're only playing with two functions since your constant just becomes the endpoint of your integral.
>>16883700Any grad level book on probability?
>>16835273Well no fucking shit the notation is going to be a visual mess, jackass. Math papers are notoriously substance over style and mathematicians don't give a fuck about making their work look pretty. Aside from that, Langlands theory is one of the most if not the most interdisciplinary subfields of mathematics.
>>16883924Meant to reply to >>16838280, which was someone whining about Geometric Langlands.
>math fags be like: if you have a bucket with balls and you can choose any ball from this bucket then you can you can spawn a ball out of thin air
ok, i kinda fucked up and started a master's in applied mathematics without having even taken real analysis or any course above it.I somehow scraped by and only failed one course first semester but my gpa is beyond fucked. Is it feasible to self learn everything along the way or should I drop out and comeback later
>>16884069Can you not just schedule the courses you skipped for the next semester without dropping out or changing your major? Set up an appointment with your counselor or something.
FINITISTS ROCKED!INFINITISTS SHOCKED!
>>16884158>definitions are finiteSome reals are infinitely long.
>>16882152via the metric. Read my posts, ADHD kun.
>>16884161>infinitely long definitionAbsolute STATE!
>>16884158> how come reals are uncountableAre you sure they are?https://arxiv.org/pdf/2404.01256
>>16884203>ain't no transcendentals in my neighborhoodYeah, finite ghettos be like that.
>>16884069imagine failing a course in the age of AI teacher
>>16884273How do you write down an "infinitely long definition" retard?
>>16884275I got fucked over by AI hallucinations too many times to trust it
>>16876439Nice, I haven't seen this one before.
Is this graph planar?This graph was described to me by ChatGPT, which said it should be planar, but I have no idea how to make it so. Is ChatGPT just hallucinating?>>16884706My thoughts exactly
>>16884812There's a linear time algorithm that tests planarity.
>>16884812Anyone using ChatGPT for anything remotely computational really needs to learn the magic words: "Use Python."
>>16884569Start with the digits of pi. Let me know when your done.
>>16884838Perfect. Thanks for your help anon
>>16884569[math]e^(r) = 1+ \sum_{\ell=1}^{\infty} \frac{r^{\ell}}{\ell !}[/math]Pick any r you want in the rationals. That's a real number that can be arbitrarily well approximated by rational numbers, but is nonetheless not a rational number.
>>16884967Bruh. Fucking typos. [math]e^{r} = 1+ \sum_{\ell = 1}^{\infty} \frac{r^{\ell}}{\ell !}[/math]
>>16884967That's a finitely long definition by virtue of the fact that you could write.
This might be a very specific soft question but have there been some notable mathematicians that suffered from persistent health issues throughout their lives (think physical disabilities, chronic illness or possibly a serious substance addiction or something like that)?
>>16885705Euler nearly died of a fever before going blind in his right eye from an infection. About 30 years later he got a cataract in his left eye and on the day he died he wwas almost completely blind.
>>16885705>>16885710Lev Pontryagin made major lasting contributions to geometry despite being blind since 14 (I do wonder how developed his "mind's eye" was).John Nash's schizophrenia is also a very famous case.
>If [math]Y[/math] is finite dimensional, then [math]T:X\to Y[/math] is continuous iff [math]\ker{T}[/math] is closedWhere do I even begin?
Aaaaand there goes the last bit of respect I had for him.
>>16885886By writing down all definitions. All involving images as sets, set operations, inverses, stuff like this.Then choosing either direction, forward backward.As usual, if you've written down all definitions of a problem at hand, you've solved the problem or have a better question at least.It might also help to look at examples. What are two different T you can think of and how do they behave?So what's kerT by definition? What's continuous by definition?That's were you begin.
>>16885926
What kind of a sick fuck takes a transpose of their own child?
>>16883700Any measure theory book that covers fourier transform and sequences or look up a good resource on the central limit theorem that uses them.Another way is skim a book on alternative probability foundations then they explain the original definition and present a new one which for me helps such as this book which you can freely find https://www.probabilityandfinance.com/2019_book/index.html or their original book "It's only a game" which replaces probability in stats with game theory instead so there's no more stochastic nonsense everything is a simple betting strategy instead of pvalues and shit
>>16885886Your first step should be clearly writing down whatever the fuck it is you’re doing. What are X and Y? TVS? Any extra structure assumptions? Banach spaces?
>>16886059>What kind of a sick fuck takes a transpose of their own child?If your child is orthogonal you can take the transpose to invert your child. even if they are not, if they have full column rank, you can take the inverse of the product of the child and transpose of the child multiplied by the transpose of the child to get a close estimate to the inverse of the child.Worst case scenario you decompose your child with SVD then apply the former method on the diagonal part of the child and the transpose the entire product.
>>16877644Physics is applied math but applied math is not physics
>>16885886If it's finite (closed) then ker t is a normed space meaning every vector has a length ergo it's a measurable space and continuity can exist here
>>16885886I'd like to do this exercise myself, but I'm pretty sure more assumptions are being used beyond what you've written. For example, take X=Y = (ground field but with the indiscrete topology), and T the identity map. This is continuous, but kerT={0} is not closed.
>>16876500>Oh u want to learn math?>Read all these faggot books
>>16886480>you want to learn [subject]?>read books written on [subject]This is how it works for literally anything.
>>16886477Note: here I'm using the definition of topological vector space that only requires continuity of vector addition and scalar multiplication.
>>16885705Descartes invented analytical geometry while laying in bed everyday while ill for like a year. It's only an appendix in his book discourse on the method because it's supposed to only be an example of the work he was able to do while following his 'method for the sciences' yet that part became the most famous.
>>16886496Don't lay down when thinking frens. It builds very bad habits.
>>16886497Please don't call me out like that. I'm doing that right now.
>>16886096>>16886477X,Y are normed spaces and T is linear.
>>16886497While lying in bed and watching a fly on his tiled ceiling he conceived the idea for analytic geometry by envisioning a coordinate system to describe the fly's location.
>>16886480Ramanujan never read a book in his life
>>16886725Who told you that, retard?
>>16886725False, he obsessively read this through and through:https://en.wikipedia.org/wiki/Synopsis_of_Pure_Mathematics
color cube netresolution: 7How many smallest squares are there?That's a rhetorical question.
Curious how transverse waves occupy 3D space but have no volume
Im trying hard to proooompt chat jibbity to the best of my ability for proofs. So far, ive been telling it to always use search before answering. Ive also uploaded yhe books im studying to its context. Any other advice? Feel like this is a great time for us nobodies with no connections to great mathematicians to get valuable feedback while learning.
>>16886480You dont have to read all of them. Try some of them. They are free on the internet, so if you don't like your first pick, try the next one, but take a look at the bibliography on the one you are skipping. If you like to read Wikipedia, go to the references and further reading section. It's simple.
>>16887076I agree, but sometimes it fucks up basic things, see >>16884812 and >>16884838 . This makes me uncertain as to how correct it really is.
