[math]/\mathfrak{mg}/[/math]Lissajous editonTalk maths, previously >>16475762
First for Hugh A. Thurston's Differentiation and Integration https://archive.org/details/differentiationi0000thur
>inb4 Verbitsky fagHis Trivium is more useful than that late 20th century string theorist curriculum:>>16493957>>16493959
any grad school or beyond chads here?could you guys recommend the fastrack way to get up to speed on bachelor's level math? Kind of like the pareto-principle of math curriculum? I am a graduated EE fag peasant but want to get into grad school for statistics PhD. Heard they require a lot of background in pure math.
>>16515541Follow anon's suggestion. Feel free to skip (only) Lang:>>16509496
>>16515545>fudding lang
>>16515541you don't need most of the shit in a math bachelor. I assume you already did calculus and linear algebra. So if you can top that with some real analysis and measure theory you should have all you need.
>>16502194What is their twitter? I am curious.
>>16515589I think the basics of bachelor's level group theory end up being quite helpful in statistics later. As someone who did an EE bachelor's and is doing a statistics oriented PhD (technically still EE but the program I am going through requires real analysis, measure, smooth manifolds and convex analysis), the main thing I wish I had more background in from undergrad is abstract algebra. I don't need it enough to spend time taking a grad level course on it, but having some basics of group theory, rings fields and modules would be helpful as well.
Three funny names of international authors a found recently:>Dingding Donghttp://math.uchicago.edu/~may/REU2017/REUPapers/Dong.pdf>Semën S. Kutateladzehttps://ncatlab.org/nlab/show/Sem%C3%ABn+S.+Kutateladze>Arjum Nigamhttp://math.uchicago.edu/~may/REU2022/REUPapers/Nigam.pdf
>>16515813Are you doing a PhD in statistical signal processing by any chance?What core pure math curriculum courses would you recommend that is obviously not covered in EE undergrad curriculum?
>>16515545>>16515589Thanks for the suggestions anons.
>>16515541>Follow-up questionIs it advisable to learn and practice International Math Olympiad level math and problem solving skills for someone already graduated? Does it help to excel in grad school?
norman wildberger's youtube is working again
>>16516184thanks norm
>>16516184Why was it down?
>>16516458He got hackerd
>>16516083Yes statistical signal processing and non-linear optimal estimation. Ultimately, a good 90% of statistical signal processing is answering two questions "Can you find the eigenvalues/eigenfunctions?" and "Can you figure out how good/bad your estimates/decisions are?" Those are deceptively simple concepts in question, but can very quickly lead you into deep waters when your convergence requires bounding random complex eigenvalues. If you get into sensor array design, you basically need to act as a translator between graduate pure-math manifolds textbooks and your applied tasks in R^n.At an undergrad level? Take a proper real analysis course, an introductory functions of complex variables/analytic functions course, and a proper mathematics department probability theory course. If you have the ability to take one, I'd also recommend a course in measure theory, because convergence of statistical functionals will make a lot more sense of you understand the basics of measure. I'm assuming you've also done a linear algebra, ODE's, and some kind of signals and systems/Fourier analysis course. If you haven't, you should do that.
For some mesurable space [math](X,T,\mu) [/math] I'm considering the sigma algebra induced by the set [math]N [/math] of all elements of [math]T [/math] whose measure is 0. I've found that if the measure of [math]X [/math] is finite then the induced sigma algebra is the union of [math]N [/math] and the set of complements of the elements of [math]N [/math].Things get messy if the measure of [math]X [/math] is not finite however, does my union in the finite case still work as the induced sigma algebra in the infinite case?For [math]A_{1},A_{1}\in N [/math], [math] X\A_{1}\cup X\A_{2}=X\A_{1}\cup (X\A_{2}\X\A_{1})[/math]. The measure of the left hand side is the measure of X, the measure of X\A is also the measure of X but the last term could be anything. Any pointers?
If I want to visually represent a complete graph with n nodes in such a way that all edges are of the same euclidean length, can I only do so in n-1 dimensions? What's this called and where can I read about it?
Is there a name for this type of curve?It's simlar to an ellipse: You have two points and a given angle t, and you want all points that when you connect them to these two points, they make the angle t.For a right angle, the curve would just be a circle (with the two points just being antipodal points on it).
Nevermind >>16516955, I'm retarded, it's just a circle (sort of)https://en.wikipedia.org/wiki/Inscribed_angle
>>16516955It's still a circle just with the two points forming a chord.
>>16516466Isn't statistical signal processing being quickly phased out in lieu of machine learning? Basically spamming compute power in Bayesian statistics instead of wasting brain power in classical statistics?>At an undergrad level? Take a proper real analysis course, an introductory functions of complex variables/analytic functions course, and a proper mathematics department probability theory course. If you have the ability to take one, I'd also recommend a course in measure theory, because convergence of statistical functionals will make a lot more sense of you understand the basics of measure.I am not an undergrad anymore but going on to do my master's in signal processing. I was aiming to do courses similar to picrel (marked ones). Am I covered?I have done signals and systems, linear algebra, and ODEs yes.
>>16517215forgot to include picrel
>need to get a perfect score on my final exam to get an A in the course>would have to get a 0 to fail the course
>>16517251
>>16515848>Dingding Donghttps://youtube.com/watch?v=vzkp10zbgMA
im a undergrad first year and fucked up first quartile (Linear Algebra 1, Calculus, Set Theory)any tips man? In 4 weeks Ill do resits aswell as Analysis, linear algebra 2 and Programming. Probably wont pass LinAlg2.
>>16516088>Is it advisable to learn and practice International Math Olympiad“mostly no”, in the sense that the benefits are not that direct, and you are probably busy. instead, learn pure-mathematics-style problem-solving through material such as real analysis, group theory, and the other suggested core pure math coursesbut if you enjoy em, by all means do olympiad problems too
>>16517305I mean, I do enjoy them but it's more of a regret that I was too poor to know about these competitions when I was in HS. I would only like to seriously do them now if it would actually help with problem-solving skills in advanced math courses.
>>16516888You have [math]N = \{A \in T:\mu(A)=0\}[/math], and [math]S=\{A \in T:\mu(A)=0\text{ or }\mu(A^c)=0\}[/math].Since [math]N \subseteq S[/math], and [math]S[/math] is a sigma algebra, also [math]\sigma(N) \subseteq S[/math].The other inclusion is pretty easy as well.What are you worried about, precisely?Maybe you should try to show [math]S[/math] is indeed a sigma-algebra.
How many shots can you do before you start fucking problems up?
>>16517215> Isn't statistical signal processing being quickly phased out in lieu of machine learning? Basically spamming compute power in Bayesian statistics instead of wasting brain power in classical statistics?There are people who approach problems this way. In general I think it's a bit more nuanced than that because there are many aspects of classical signal processing that Machine Learning just doesn't easily accomplish. As an example, if you're doing multi-target tracking, you need some process for determination of track formation and track deletion. There have been some papers that have tried to use things like CNN's or some transformer sequence model, but they all suck and are super unreliable. What ends up actually moving the field forward are (generally adaptive) graphical models, with classical statistics based bounds for determination of the probability of rare events.Also, a lot of the people who are doing the best research in the mathematical theory behind Machine Learning are statistical signal processing types. It's not like these are separate disciplines in a meaningful sense. Signal detection is fundamentally a classification task. Estimation is fundamentally a regression task. Whether you use a linear filter or a Bayesian kernel approximation doesn't really change the fundamentals of the problem to be solved.The course list looks good. I think you'll be fine. If you need more, you'll do more.
>>16517298Just grind at practice problems. If you have to, try and get problems from other books (e.g., Schaum's outlines or any of the relevant springer problem books) and you'll be alright.
I keep fucking up problems because I forget to put negatives where they belong or stupid shit like saying 5 x 5 is 20 then the entire problem is wrongDo I need practice?
>>16517344That was the issue, wasn't able to show it was one, but funnily enough, the way you wrote the set helped me figure it out, cheers.
>>16517720Learn your times tables by heart.
>>16517720Visit neurologist. Seriously. I began making such stupid mistakes (like dropping signs, writing that cos(0)=0) and I was diagnosed with sclerosis multiplex.
>>165174850, as someone whose studying measure theory
>watch video about Crash Team Racing shortcuts>suddenly the homotopy definition pops up
>>16517994I mean, it is an optimal feasible path planning problem, which is a discrete geometry and geometric graph theory problem.
>>16517891Hell, I need a shot before I even open a measure theory book. Probably doesn't help with the comprehension of the material but it does help with the existential dread.
>>16517599>a lot of the people who are doing the best research in the mathematical theory behind Machine Learning are statistical signal processingI thought they were theoretical statistics people who did that.How different would a statistics PhD be from a PhD in (statistical) signal processing?
>>16518068It depends on the school you go to and the focus of the program. For my program you're required to do a "minor" (meaning 4 grad level courses before your defense) in either math, statistics, physics, computer science or another engineering discipline. Of the 12 courses you're required to do before defense, 4 of them could be statistics (which, at my school, means you're doing about half of the same lecture material as a stats PhD).Personally, I did mine in math focusing on analysis (class 1: baby rudin, class 2: measure, class 3: measure theoretic probability theory and stochastic processes, class 4: linear functional analysis and spectral theory). However, one of my lab-mates did his focusing on Bayesian statistics and actually has a stats professor on his advisory committee, so I'd say his degree is pretty similar.
How much of topology can I learn if I don't know about complex numbers?
>>16518068Oh, I forgot, there definitely are theoretical statistics researchers working on the frontiers of ML, but they aren't alone. Take a look at the major topics for IEEE's Fusion conference and you'll see dozens of engineering PhDs from different universities and national labs working on both the development of ML-integrated statistical signal processing systems and the statistical characterization/modeling of ML-systems in the context of signal processing tasks.If you look at the most cited recent papers in IEEE Transactions on Signal Processing, you'll find quite a lot of focus on trying to catch the theory up with the practice. That's not to discount the work by guys like Trevor Hastie. He just isn't doing this alone in a vacuum.
Is it ok to hand out Walmart job applications to dumb people in your uni
>>16518089why did you choose to go with math? wouldn't statistics make more sense for you?>>16518108So basically EEs with signal processing background can choose to go into theoretical ML research, like statisticians can? I knew SP folks did a lot of work in ML, but I reckoned they only specialized in the niche of wireless communications at that.
>>16518248> Why did you choose to go with math? Wouldn't statistics make more sense for you?I figured that if I had a strong analysis background and probability background, that would allow a much easier path towards self-teaching from theoretical stats books like Keener/Shao, than the other way around. My signal detection/estimation research also relies a lot on constrained convex optimization, and having taken analysis was a godsend for that. > So basically EEs with signal processing background can choose to go into theoretical ML research, like statisticians can?It depends on the researcher, the money they have access to, and their personal research interests. I'd say more SSP researchers in the ML space have a focus on "how do we apply ML to target tracking/signal detection/multi-user communications" etc., but a surprising number of those more "applied" tasks require you to get down in the mud with some fundamentals of probability theory/statistics. It's not an accident that the Covariance inequality for Bayesian estimation (which applies in either a classical or ML-based approx. inference case) was produced by an EE (van Trees). Variations on that same covariance inequality is the one used for performance bounding every single ML based regression system out there.
i'm trying out MathJaX for the first time sorry if this is not allowed[ math ] /sqrt{2*2} + /sqrt{2} [ /math ]
[ eqn ] /sqrt{2*2} + /sqrt{2} [ /eqn ]
>>16518373>>16518371Don't. Put. Any. Fucking. Spaces. Within. Tags.
>>16518380>Reddit-speak
>>16518381>autocapsphonefag
>>16518098What's there to know about complex numbers you can't brush up in a day? Do you mean complex analysis? If it's the latter, you only need elementary real analysis mainly because of maturity and concrete examples.
>>16518383Nice catch, he's a Redditor AND a phonefag.
>>16518431I mean the latter. I know the arithmetics of complex numbers, their polar forms, de Moivre's formula etc., but statements like "bounded entire function must be constant" are almost black magic to me (at least I understand every word.)
>>16518448>how much x do i need to know before learning yJUST LEARN y AND LEARN THE BITS OF x WHEN YOU NEEDComplex analysis gives some useful intuition for things like homotopy/homology but basically none of what most people learn in AT has anything to do with even complex numbers. Why? Because topological spaces are not contingent on some ground field, and even then when ppl speak of manifolds it is usually real manifolds. Stop procrastinating and just do the math you want to do.
What's a good scientific calculator for uni?Can't have graphing for cheater reasons
>>16518530>Can't have graphing for cheater reasonsngmi
I’m studying basic arithmetic and I’m already running into shit that has me stumped. The product of 65 x 26 is less than 64 x 27. That’s extremely counterintuitive to me. Why does the +1 increase from 26 to 27 have more weight than the decrease from 65 to 64?
>>16518575because the +1 for 26->27 is being multiplied by 64, while the +1 for 64->65 is only being multiplied by 26. it becomes a bit more apparent if you break it down slightly[math]65 \times 26 = (64+1) \times 26 = 64 \times 26 + 26[/math][math]64 \times 27 = 64 \times (26 + 1) = 64 \times 26 + 64[/math]
>>16518530I have a Casio fx-991EX that I got for like $15 during undergrad and it's been my main calculator since. I still use my inspire sometimes if I really need it, but that Casio is definitely worth the basically nothing that it costs.
>>16517298You should take a group theory course before linear algebra 2
>>16515464LaTeX is honestly my favourite software ever written. It allows me to do all my work from the comfort of my bed :) What editor do you guys use? I am still stuck on overleaf since I'm too lazy to configure nvim for it. Are any of the simpler all in one GUI programs worth using?>>16515541Get a book that goes over proving things. I get the impression that you're in the US where they make you sit exams during your PhD. I'd imagine the concepts on something like Real Analysis are not going to be too tough if you got through EE, however, actually going from a blank page to a proof is a daunting task. If you're looking to go into stats probably measure theory/probability stuff might be relevant. >>16517298Do not give up anon. I fucked up my entire first year and 1/4 of my second but am now on track to do quite well. Depending on where you are in the world and what university you're at, please make sure you know exactly how your university weights classes for you average. In my country first years (tend) to not actually contribute to your final average at all. >>16518530Check if your university recommends one. If you aren't taking a class with a calculator this semester, try check other classes you might be taking. Then, get the recommended one. Only one class in my entire degree used a calculator and honestly, I barely remember actually needing it in the exam. Honestly a bit strange that it was even required. Worse yet, my first one broke, I then lost my second and then found it after having just sat the exam and having bought a third. Whoops.
>>16519179>I'm too lazy to configure nvim for itnvim with snippets is worth it
>>16515464How conditioning on an "event" defined using measure theory?
>>16519179I use TeXStudio
>>16519329You have a probability space [math](\Omega, \mathscr{A},P)[/math]. Take any set [math]B \in \mathscr{A}[/math] with [math]P(B)>0[/math] then you can consider the probability space [math](B, \mathscr{A}', P')[/math] with[math]\mathscr{A}' = \{ B \cap A | A \in \mathscr{A} \}[/math][math]P'(A) = \frac{P(A \cap B)}{P(B)} [/math]
>>16518098There is no real obstruction to learning all of (algebraic) topology without knowing anything about complex numbers as far as I know, though you would miss out on a few (beautiful) examples. The double cover of S^1 has a very simple description in complex polar coordinates, for example
>>16519269Thanks I will get this set in one of these days>>16519360Thank you as well, I will use this for now.
>>16518321>It depends on the researcher, the money they have access to, and their personal research interests. I'd say more SSP researchers in the ML space have a focus on "how do we apply ML to target tracking/signal detection/multi-user communications" etc., but a surprising number of those more "applied" tasks require you to get down in the mud with some fundamentals of probability theory/statistics.That's good to hear. I was afraid I would pigeonhole myself into wireless communication if I go the SSP route.>>16519179>What editor do you guys use?I use VSCode. Effortless git backups.
I thought if I went to uni for math it wouldn't be just hammering roots and logarithms to shit out some limit for 80% of the time but that i'm gonna learn cool proofs and learn tons of definitions and try to apply them but nopeI'm less than 3 months in and I'm basically guaranteed to fail at this point. Also my fucking lecturer is such a senile retard i legitimately have no clue when he's rambling and when he's actually talking about shit i need to know he just doesnt make it clear.I'm just too tired to try to memorize 12 algorithms to solve textbook problems and then lose half of the points for not doing the problem explicitly the same way it was shown to me before.I don't even know what to do at this point I'm 1 year in the dump or maybe I should just give it up for good but I have no other besides hobbyist IT stuff. It's so humiliating but I just can't fucking work on this shit my i just can't handle 4 hours of solving boring problems that are identical to the shit i've been doing in high school
And it's not like I could've ever caught up because I'd have to start from the beginning
Let C be some closed solid regular cube in [math] \mathbb{R}^3 [/math].Is there a (2-)plane P in [math] \mathbb{R}^3 [/math] satisfying the following conditions?(1) P passes through at least 2 different vertices of C ; and(2) the intersection of P with any given edge e of C is a set of vertices of e . In other words, P does not touch the "interior" part of any edge of C .I'm pretty sure the answer is "no", but what's an efficient/clever way to show this?
>>16519539git backups are very helpful and I'm making the most of my univeristy's overleaf premium by making as many backups as possible>>16519567I understand this feeling anon, I nearly failed my first year. It was pretty boring and I only even went to campus maybe 3 or 4 times a month. Just stick through the boring shit and try keeping yourself stimulated with more interesting work. By the time the good stuff finally comes around you will find it easy since you've been self teaching. It isn't the best strategy for grade-maxxing but it's personally gratifying.
>>16519568You'll get your f***ing hands dirty like the rest of us.
>>16519581buddy i'm already on my way out i'd have to ace 2 tests and suck lecturer's cock to let me rewrite one of the old tests and maybe i could make it when the best i've done was to reach like 30% even if i tried i wouldnt get more than 60%linear algebra is palatable but calculus is 100% memorization and I'm pretty sure I've fucked my brain to a point where I could be diagnosed with "dyscalculia" i just cant focus on it i am too tired i cam barely manage to wake up in the morning i am already thinking of which minimum wage job i can get
>>16519592>calculus is 100% memorizationAgreed, honestly why even take calculus, Wolframalpha does all that for you anyway. It's basically just to exercise your algebraic and analytic thinking, but I've literally never needed to use it myself and I'm in my final year of grad school in math.
>>16519370>The double cover of S^1 has a very simple description in complex polar coordinatesMore than that, literally any finite connected covering of S^1
>>16519369What if event is of probability 0?
>>16519539> That's good to hear. I was afraid I would pigeonhole myself into wireless communication if I go the SSP route.It can happen, but it doesn't need to. It all just depends on your research interests and the funding available to you. My recent research is technically in a subset of sonar, but really it's about random matrix theory and convergence of non-linear programs with random variable Hessians.
Thoughts on OpenAI automating math research? Terry Tao did not expect progress to be made on FrontierMath for years, and now look at o3...
>>16519882I think, if `AI' will be able to generate code in some proof assisting language like lean, essentially formalizing its current sometimes-schizo ramblings, it's over.That's a big if, though, and they'd have to make the generating multiple orders of magnitude cheaper than it currently is for the impact to be significant.It's certainly interesting.
>>16519583if by getting my hands dirty you mean working construction for the foreseeable future than you're right >>16519680it's mandatorytruth be told i'm not doing too well with linear algebra either but failing it doesn't make me illegible to continue studying i like logic a lot which probably makes sense since i have some CS background
>>16519888The fact is, there's no breakthroughs needed anymore for that to happen. Just engineering + getting better / more gpus.I think a lot of mathematicians, especially the truly autistic ones, might kill themselves once they're made obsolete
>>16520063nta, but you won't get orders of magnitude cheaper with some basic engineering + better gpus. That would already require a pretty serious breakthrough. Moreover, I don't think the quality of the ai is there yet either. Not saying it's impossible, but it's also not in the state of "no breakthroughs needed".>I think a lot of mathematicians, especially the truly autistic ones, might kill themselves once they're made obsoletedo you have some weird investment in this? Are you the same tard that made the ai post in /scg/? Be honest, are you an undergrad/underaged/NEET?
what can i use the frobenius norm of a matrix for
TIL m comes *before* n in the alphabet
>>16520074o3 is just doing the same thing o1 does (generating tons of chain of thought solutions + having another model selecting the best one) and the leap between them is immense. I don't see why they can't just push the same for o4 etc, until the distilled and cheap models will be good too>do you have some weird investment in this? Are you the same tard that made the ai post in /scg/? Be honest, are you an undergrad/underaged/NEET?No idea of what scg is. Also I'm a postdoc but not in math. Math will fall first and other fields will follow
>>16520213Stop posting this meme.
>>16520078Yeah, math people prefer to use n before m, and only introduce m as a second variable, meanwhile a is preferred to b, making it seem like the reason is alphabetical order, while in fact it's preferred because n stands for "number".
>>16520213See:>>16515514String theory is dead. Update your curriculum.
>>16520063Surely formal verification is still a human issue though?
Math teachers should be curved graded against each other and the bottom 10% move down a grade level every semester
Why do they hit calc students with Delta epsilon rigor that isn't matched until literally 5 semesters later?
What the fuck is algebra
>>16520603>Why do they hit calc students with Delta epsilon rigor that isn't matched until literally 5 semesters later?Europe exist, they dont delay rigour there
>>16520607In (modern) algebra not only they use "x" and "y" with variable meaning but "+" (plus) and "*" (times) as well. For example, they study commutative and non-commutative operations and in the latter case they accept the entities being multiplied may not be real/integer numbers like usual.
>>16520615European universities confuse me. They clearly start out ahead of US universities in terms of math education, but seem to completely drop the ball. As someone who did his undergrad/master's in the US and then worked in Italy/Switzerland for a bit, their master's degrees in my field barely even covered what we did in my bachelor's program. Granted, the Bologna process is a 3+2 master's and the average US bachelor's length for a good stem program is 4.5-5 years now.
>>16520647Well don't for profit universities get most of their profit from undergrads?
>>16520646I told my shrink about non real entities multiplying and I got forced on meds
How come bijective homomorphisms between groups/rings/fields/modules are always isomorphisms,but e.g. bijective continuous maps between topological spaces aren't necessarily homeomorphisms?I understand how to prove these facts with symbols, but what's an "overarching" reason for this?
>>16520646For background, the non-commutative case can cover situations like e.g. matrix multiplication, which is of utility and interest in physics and engineering to say the very least>>16520941Sucks 2 suck
>>16520947>How come bijective homomorphisms between groups/rings/fields/modules are always isomorphisms,Perhaps it's because in algrebra isomorphism is *defined* as bijective homomorphism?>but e.g. bijective continuous maps between topological spaces aren't necessarily homeomorphisms?Because in topology there are continuous and bijective functions whose inverse is not continuous. Classical sample is an interval [0; 1) and a circle. You may easily turn an interval into a circle, but to turn a circle into interval you must tear it somewhere and thus introduce discontinuity.
>>16520976>Perhaps it's because in algrebra isomorphism is *defined* as bijective homomorphism?No, an isomorphism is a bijective homomorphism whose inverse is also a homomorphism. It's just that being a bijective homomorphism turns out to be sufficient.
>>16520976What I'm asking is, why are morphisms in the category of groups/rings/fields/modules whose underlying set-functions are bijective necessarily isomorphisms,but this is not the case for other concrete categories like the category of topological spaces?>Because in topology there are continuous and bijective functions whose inverse is not continuous. Like I already said, I know how to prove these at a basic level, so you repeating this is not helpful to me. But kudos to you for coming up with a proof yourself.Btw, another nice example is: on any given set X, the "identity" function is continuous from the discrete to codiscrete topology on X and is bijective, but its inverse isn't continuous.>>16520983Yes, this aligns with what I've been taught.
>>16520674> Well don't for profit universities get most of their profit from undergrads?This is kind of true but also kind of not. On the margin, the most profit is from international students who pay cash (largely because most of them are ineligible for subsidized loans/financial aid). This is a big reason why so many schools (even public ones) have been orienting themselves so much towards the luxury resort model. They want to attract rich Chinese and Indian people with money to spend (launder) by sending their kids to the US for a good education.My personal opinion is that US universities do a better job at providing a broad range of coverage, and if you go to a good university that coverage ends up fairly deep as well. This gets meme'd on by people who are pointing at the required humanities courses for accredited programs, but it also end up with a much broader coverage of whatever your major is. My bachelors was in EE, and by the time I'd finished it I'd done all of the basic calculus sequence, linear algebra, real analysis and an introductory analytic functions course, as well as probability theory, stochastic processes, linear dynamical systems and control theory. The Italian/Swiss colleagues I'd worked with had a depth of knowledge in a few particular niches that my education did not provide, but they entirely excluded really fundamental skills in favor of waiting until an elective Master's course to talk about them at all. One of the aerospace engineers I'd worked with did his entire master's degree at a fairly good Italian university and had never taken a formal probability course. You couldn't even take third year undergrad courses at my university without a probability prereq.
>>16520947I can't quickly find a reference, but iirc it is because groups/rings/fields/modules are all algebras (in the universal algebra sense) and in this general setting bijections are good enough. There are also some categories where some property + mono + epi = iso, but this is probably not the best line of thought to enter, since for example ring morphisms can be epi without being surjective.>>16520976isomorphism is *defined* as invertible morphism
>>16520947Because groups/rings/fields/modules are models of first order theories, and homomorphisms are simply maps between models that preserve truth claims in the first order theory. Any such bijection turns out to be an isomorphism of models.Meanwhile topology is a second order theory.
>>16520999>I can't quickly find a reference, but iirc it is because groups/rings/fields/modules are all algebras (in the universal algebra sense)The statement holds for all first order theories, not just algebras. For example graphs and partial orders also have the property that morphisms which are bijective are isomorphisms. As do all structures which are modelled by first order axioms.
>>16520947Another way to think about it is that with topology the structure is not just the underlying set X but also the topology t. So in a sense a "true" notion of a morphism would be a map (X,t) -> (Y, t') where t' is the topology of Y, considered as a pair of maps on the underlying spaces X->Y and the map from open sets to open sets t -> t' which respect each other. This would be the model theoretic way of looking at morphisms between models. A morphism is a bijection between models then if it's a bijection X-> Y together with a bijection t -> t'. In such a case, the map is a homeomorphism. However, this model theoretic notion of morphism is not very useful in topology, and instead another type of morphism is chosen, that of a continuous map, which is more complicated and doesn't respect the model theoretic structure in a nice way.
>>16521017>>16521022Awesome, this is exactly the kind of thing I was looking for! So, what distinguishes groups/rings/fields from topologies is whether they use 1st order or 2nd order logic. Thanks anon, I'll look into this further.
>>16521032>whether they use 1st order or 2nd order logicOr I guess to be more precise, whether they are *models* of these.>>16521027Thanks, this is another interesting perspective.
How quickly should I be going through a math textbook? When is it time to say you're done and move on to the next one?
>>16521070when u do the hw problems and feel you got a good understanding of it
>>16521032Nta, but no need to talk about 2nd order logic. It is enough to think that algebraic structures only involve algebraic operations in the structure signature, while other structures like topologies involve families of subsets in the signature. Yes, there's no way to talk about "subsets" in the pure first order logic theory taking operations as primitives, but we usually already have set theory when we want to study algebra and topology.
Any good but short calculus (single and vector) books you can recommend?I already have some proof experience, worked through most of Hammack and the first few chapters of Tao Analysis 1 and half of Axler.But I never had Calc classes.
>>16521530>Good but short single variable calculus>Proof/elementary analysis experienceConsider:>>16515492
>>16521530Get a classic "differential and integral calculus" book like Piskunov.
would learning math help a software engineer? what opportunities could it unlock?
>>16521832Software Engineer can mean a lot of things. Most programmers nowadays are essentially just library and framework plumbers.So it depends on what you actually do.
>>16518869>Other sciences seek to discover the laws that God has chosen; mathematics seeks to discover the laws which God has to obey.J. P. Serre>From a metaphysical standpoint, I would say that mathematics is exactly what God didn't have to create, because it was all there from the beginning and He couldn't but take it as it is. (Maybe it may be said that mathematics is just part of God's own nature, namely the part of it to which human reason has access just by its own feeble means ...) God had an infinite choice about how to build a universe, with its laws (spiritual, physical, biological ...), and maybe there are many or even infinitely many such, of which we only know (ever so little) one. But whatever way he thinks up his Universe, He's got to use the same mathematics, with 2 + 2 = 4, and not 3 or 5. It is not in His power to change this, any more than to change his own nature – and surely He never had any wish to do so!A. Grothendieck
I wish proof assistants were more user friendly. Solo learning math I never know for sure if my proofs are valid unless I ask and I don't want to ask for every exercise.
>>16521955play the natural number gamevery fun
>>16521866Allegedly there's a prohibitively long proof of the fact that 2+2=4. Where can I read (about) it?
ayo nigga we intergrating arrows
>>16522042I guess you are thinking about Alfred North Whitehead and Bertrand Russell''s [math]\textit{Principia Matematica}[/math]. The following is a contemporary equivalent:>Inspired by Whitehead and Russell's monumental Principia Mathematica, the Metamath Proof Explorer has over 26,000 completely worked out proofs in its main sections, starting from the very foundation that mathematics is built on and eventually arriving at familiar mathematical facts and beyond. Each proof is pieced together with razor-sharp precision using a simple substitution rule that practically anyone (with lots of patience) can follow, not just mathematicians. Every step can be drilled down deeper and deeper into the labyrinth until axioms of logic and set theory—the starting point for all of mathematics—will ultimately be found at the bottom. You could spend literally days exploring the astonishing tangle of logic leading, say, from the seemingly mundane theorem [math]2+2=4[/math] back to these axioms. Let [math]2,4\in\mathbb{C}[/math]https://us.metamath.org/mpeuni/mmset.html#trivia>The complete proof of [math]2+2=4[/math] involves [math]2,913[/math] subtheorems>These have a total of [math]26,323[/math] steps—this is how many steps you would have to examine if you wanted to verify the proof by hand in complete detail all the way back to the axioms of [math]\textrm{ZFC}[/math] set theory.
Fun generalization of mobius transformations. Can also use split quaternions to shorten things.[math]\pmatrix{a & b & c\\ \bar b & \bar a & \bar c\\ d & \bar d & r}\sim \dfrac{az+\bar az+c}{dz+\bar d \bar z +r},r\in \mathbb{R}.\\ \text{Introduce}\ j:j^2 = 1\ and\ \forall z \in \mathbb{C},\ jz=\bar zj.\ \text{This gives the split-quaternions (the usual quaternions have }j^2=-1).\\\text{The matrix representing the transformation can be reduced to a 2x2 matrix of split quaternions}:\\\pmatrix{a+bj & c (\frac{1+j}{2})\\ (1+j)d & r (\frac{1+j}{2})},\ r\in\mathbb{R},\ a,b,c,d\in\mathbb{C}. \text{Care must be taken when multiplying since the j does not commute.}[/math]
>>16521832The ability to graduate with a cs degree with a good gpa and stand out from mathlets
>>16522227should be az + bz* + c on top of the fraction
Is this problem solved:Let [math]\omega(m)[/math] denote the number of distinct prime factors of [math]m[/math]. Does there for every natural number [math]k[/math] exist infinitely many natural numbers [math]n[/math] such that [math]\omega(n)<\omega(n+1)<\dots<\omega(n+k)[/math]?
>>16515464get the fuck in here >>>/x/39491447
1 != 0 is an artificial restriction on what fields are, and there SHOULD be a field with 1 element, which is the zero ring.If your reasoning for not allowing it is "it causes too many exceptions in results" then that same reasoning can be used to say 2 isn't a prime.
>>16522492I want my fields to be integral domains.
>>16522512THE ZERO RING IS AN INTEGRAL DOMAIN VACUOUSLY YOU NEGRO
>>16522526but it doesn't have a field of fractionsinb4 it's its own field of fractions, you need to change the construction to fit this particular example
>>16521032>>16521098adding on to this, if you have relations in your language then you can have bijective homomorphisms that aren't isomorphismsFor example, there are bijective homomorphisms of partial orders that aren't isomorphisms
>>16522492>then that same reasoning can be used to say 2 isn't a prime.The reasoning is in fact why 1 isn't considered a prime.
Mathematicians are so stupid. There are only countably many formulae in FOL. So, just enumerate through them and you'll eventually prove everything. What's the big fuss?
>>16522594Why are plebs so consistently filtered by infinity?
>>16522594See:>>16522145If a simple fact like that takes so many steps in fully formal language, how much time would it take to enumerate all theorems up to that fact and verify it? (You can think of every step as new theorem which is the concatenation of all the preceding ones). Let's see your results
>>16522613>le it's too much heckin work Lmao. If you give a simple MBA grad the work of proving theorems, he would prove all the mathematics by the end of the year.
>>16522627How much time would it take her to yield and single out the shortest Godel sentence in your hypothetical system?
>>16515464I got one, what's the level of energy required to infer infinity?
>>16522643Like from a calculator battery for e.g. haha pfft
Question for my /mg/ nerds, but how do you guys feel about the new ai math news? Do you feel like it's overhyped? The one time I used chatgpt and told it to do a word count it couldn't even do it properly, and it would be different answers whenever I would ask it to redo it
>>16522729Yes and no. Currently available AIs are genuinely extremely impressive but they aren't actually PhD level and are very inconsistent which is a big problem.
>>16522593No it isn't.>>16522560It IS its own field of fractions, whether you like it or not.The field of fractions of an integral domain is the localization at the maximum set such that the map x -> x/1 is injective. Works for the zero FIELD.
>>16522743>The field of fractions of an integral domain is the localization at the maximum set such that the map x -> x/1 is injectiveWhere can i find this characterization in the literature?
>>16522754It's called "using your head", algebralet
>>16522801Touché, but i still want to read something similar
>>16522812Read Atiyah and Macdonald, chapter 3.
Atiyah is an Arabic name meaning "gift", because Michael Atiyah is a gift to the world.
>>16522743>No it isn't.Yes it is.
41^2 = 168142^2 = 176443^2 = 184944^2 = 1936and so oneach goes up a hundred and down a perfect square
>>16522899I'll go up your ass and down your mom
>>16522906There is no need for such vulgarity.
>>16522729Impressive, but still way overhyped. Personally I still don't see in it more than a glorified documentation search and/or autocomplete+. That can be genuinely useful of course, but is way different from the "ai will solve it" narrative.The thing is: all current llm fail miserably on very simple, original logical questions and calculations. They get some impressive things right too.Now if you work on anything that isn't monkey work or library plumbing then you will have to spend as much time verifying the AI's potential answer (if it isn't bogus right away) as you would have spent coming up with it in the first place.And I currently don't see anything changing in that regard except huge marketing buzzword babble.
Grant's making shitposts again
Has anyone read pic related, or the apparently similar book "Guesstimation" or something else that is similar? Are they any good? I find getting better at estimating arbitrary problems interesting. But the official descriptions sound cheesy.
>>16522899That's nice but check this469^2 = 219961470^2 = 220900471^2 = 221841...499^2 = 249001500^2 = 250000Each goes up a thousand and down a perfect square.
Sup, /mg/. What's your take on that book?>>>/g/103614046
>>16522978the wiki mentions a neat book called dead reckoning, check it out
>>16515464Can someone explain this post from /x/:>>>/x/39492043So, that was the foundational theory. Eventually, a mathematical proof was developed to prove it:Wave Function ExpansionPsi(x,t) = A exp(i k x - i ω t)= A exp(i k x) exp(-i ω t)= A [cos(kx) + i sin(kx)] [cos(ωt) - i sin(ωt)]Probability Density|Psi(x,t)|2 = Psi*(x,t) Psi(x,t)= A2 [cos2(kx) + sin2(kx)]= A2This shows the wave-like nature remains intact until a measurement occurs.Measurement ProcessFor an operator M acting on the state |Psi>, the possible outcomes are its eigenvalues mᵢ, and the probability amplitude for each outcome is ⟨ϕᵢ|Psi⟩, where |ϕᵢ⟩ are the eigenstates.Collapse MathematicsAfter measurement, the state collapses into |ϕⱼ⟩ with probability |⟨ϕⱼ|Psi⟩|2.Wave-Point TransformationPsi(x,t) can transform into a delta function delta(x - x0), normalized so that integrating its squared magnitude over all x gives 1.Uncertainty Principle ConnectionΔx Δp ≥ ℏ / 2This sets the fundamental limit on positional and momentum precision.Quantum EntanglementFor an entangled state like (1/√2)(|0⟩1|1⟩2 - |1⟩1|0⟩2), measuring one particle collapses the overall state instantaneously.Conservation Laws∫|Psi(x,t)|2 dx = 1 (total probability),E = ℏ ω (energy),p = ℏ k (momentum).Complete TransformationPsi(x,t) = A exp(i k x - i ω t)delta(x - x0) exp(-i ω t)delta(x - x0)Mathematical IdentityAs the width σ goes to zero, a normalized Gaussian becomes delta(x - x0), showing how wave packets can become point-like.This expanded proof demonstrates that wave functions hold complete information, and measurement collapses them. Position and momentum remain complementary. Probability is conserved. The transformation can switch from wave-like to point-like while preserving all conservation laws. In essence, both descriptions—waves and points—are two faces of the same coin, just like mass and energy.
>>16522729> ai now solves these mega hard problems real mathematicians have to spend hours on. The solutions totally can't be found online or in its training data!> no you may not see or evaluate any of the problems, or the ai output, just trust usConvenient
Can someone help me understand some of the points of this question? >Let [math]A[/math] be an arbitrary subset of the set [math]\{1,2, ...,2n\}[/math] of size [math]n+1[/math]>Prove that there exists two distinct elements [math]a[/math] and [math]b[/math] such that[math]a[/math] divides [math]b[/math] [math]A[/math] is a set of size [math]n+1[/math]; [math]A:=\{1,2, ..., n+1\}[/math]?[math]B:=\{1,2, ..., 2n\}[/math][math]A \subseteq B[/math], therefore [math]a,b \in A,B[/math]?Are the above assumptions correct?
>>16523345A is an arbitrary subset of B, so A is not necessarily {1, 2, ... , n+1}. A could be {n-1, n, n+1,..., 2n-1, 2n}, or any selection of n+1 elements of B.Think of it like this: the statement to prove is that at most half of the elements of B can be chosen such that none of the chosen subset A divide any of the elements in B which aren't in A, nor do any of the elements of A divide any of the other elements in A. Any subset of more than half of B (aka the subset A of size n+1) must contain an element that divides one of the elements of B that is not in A, or one of the other elements of A. (an element of A is still an element of B, as its a subset. Which is why dividing one of the other elements of the chosen subset is relevant.)
>>16523183I like it, it's a fun read.
>>16523345>>16523405Jumping into this conversation to say that it's not clear where "a" and "b" are from in the statement of the question.Most likely it's that both are in A, because otherwise the statement is either trivial or wrong.I'm guessing this is a question about the Pigeonhole Principle, and since A has n+1 elements, you need to define n "pigeonholes" and a rule to place the "pigeons" (elements of A) in them.
why are category theorists so annoying ;_;
>>16523604a coconut is just a nut
>>16522036My progress so far
>>16523692Became more frustrating than fun in the advanced multiplication world. At one point I had to use a previous theorem but it was not listed anywhere on the right panel which pissed me off.
>>16515464How do I get into mathematical proofs as a hobby? I have a book called Calculus by Michael Spivak, but it's a good bit over my head, not going to lie.
>>16523947>TextbooksJourney into Mathematics: An Introduction to Proofs - Joseph J. RotmanProof, Logic, and Conjecture: The Mathematician's Toolbox - Robert S. WolfAlice in Numberland: A Students’ Guide to the Enjoyment of Higher Mathematics - John Baylis, Rod HaggartyHow to Read and Do Proofs - Daniel Solow>Lecture NotesA Primer for Logic and Proof - Holly P. Hirst and Jeffry L. Hirsthttps://www.appstate.edu/~hirstjl/primer/hirst.pdfModicum Mathematicum: A Swath Through The Basic Language Of Abstract Math - Paolo Aluffihttps://math.hawaii.edu/~pavel/Aluffi_notes_321_Modicum.pdfProof, Sets, and Logic - M. Randall Holmeshttps://randall-holmes.github.io/proofsetslogic.pdfBasic Concepts of Mathematics - Elias Zakonhttps://www.trillia.com/zakon1.html
>>16523692>>16522036Are you doing the latest, dumbed down version? Because the superior version is this:https://www.ma.imperial.ac.uk/~buzzard/xena/natural_number_game/index2.html
>>16523984Looking at "Advanced Addition World" it's the exact same things I proved in the later version's advanced addition world. How is it dumbed down?
>>16524011Idk i was just quoting some anon from the lambdaplusjs /math/ board
Is the corestriction of a morphism onto its regular image always an epimorphism? I feel like this should be true but I'm completely blanking on a proof right now
I read some post by Joel David Hamkins somewhere arguing that the concept of a "standard" model of arithmetic is far more vague and poorly defined than most people seem to believe, and recently I've started to see things the same way. How do you know "when to stop" with induction? Isn't the concept of finiteness a little circular? Obviously unsoundness can make "non-standard" models undesirable, but hypernaturals and the like seem natural enough (pun not intended).
>>16525479first order logic problems
>>16525654The true theory of second-order arithmetic isn't definable in second order logic either, and the issue with higher order interpretation of first order theories is that consistency results become useless.
Is it too autistic too draw a black square after each exercise? I draw a black square for easy exercises and write QED for hard exercises.
>>16525788Exercises do not merit the tombstone symbol by merit of being an exercise. It is intended to be a punctuation mark at the end of a proof. The symbol plays a formal grammatical role, and is interchangeable with QED. Length and difficulty have nothing to do with it. You wouldn't drop your periods with proper grammar, would you?
Is there an uncountable subset [math]A[/math] of [math][0, 1][/math] such that [math]\forall a\in(0, 1], \{x\in A : a\le x\}[/math] is at most countable?That is, I want all uncountability to be concentrated near zero.I feel that maybe Cantor set can be modified to satisfy this. Like, instead of removing open intervals from the middles, remove them from the rightmost part: - remove [math](\frac23, 1)[/math]- then remove [math](\frac49, \frac59)[/math] and [math](\frac59, \frac23)[/math]- then [math](\frac8{27}, \frac9{27}), (\frac9{27}, \frac{10}{27}), (\frac{10}{27}, \frac{11}{27})[/math] and [math](\frac{11}{27}, \frac49)[/math]- etc.It should still be uncountable (proof that Cantor set is uncountable should work for this set as well? And it feels like there is an obvious bijection between those two sets). And if I choose arbitrary [math]a\in(0,1][/math], there will be only a finite number of points to the right of [math]a[/math], and therefore uncountable to the left.Is this correct?
>>16526076Pretty sure that the Archimedean property would guarantee this is impossible. Exists definitionally for the hyperreals though. Note of caution, I only thought about this for 45 seconds, so might be wrong. Try assuming A exists and then contradict Archimedean property for a fixed a.
>>16515464best algebra book if i want to get foolishly good at algebra, or just things you'd recommend reading after one semester of abstract algebra? i like algebra
Either I'm retarded or my professor is, I've got a function [math] f(x,y)=\frac{y^{2}x}{1+x^{6}} [/math] which can easily be shown to not be lebesgue integrable with respect to [math] \lambda_{\mathbb{R}^{2}} [/math]. The partial function for some fixed y, [math] f^{y}=f(\cdot,y)[/math] is certainly lebesgue integrable with respect to [math] \lambda_{\mathbb{R}} [/math], however my professor justifies this by saying that this partial function is continuous and [math] |f^{y}|\leq y^{2}\frac{1}{|x|^{5}} [/math] and states that the right hand side is lebesgue integrable in x over the reals. That really doesn't seem to be the case though, is the right hand side truly lebesgue integrable?
>>16525479Its not vague and poorly defined. Its the smallest inductive set, end of.
>>16523345Something something pigeonhole principle.
>>16526103If you want to be autistic then you can bound it like this[eqn]|f^y(x)| \leq \begin{cases} y^2 & |x| \leq 1 \\ \frac{y^2}{|x|^5} & |x| > 1 \end{cases} [/eqn]
>>16523692>>16523984I remember doing this a few years ago. Got filtered by one double induction question where I could not possibly know how to do it because the technique needed to prove it wasn't actually explained as possible. That was very frustrating and a major bad part of the natural number game.
>>16526155NTA, but that's a good one. Then you can just use DCT to show that it's Lebesgue integrable almost everywhere. I need to spend more time brushing up on measure. There's so many neat party tricks in analysis.
>>16526094I really liked Algebra: Chapter 0. It's actually the book that changed my mind about category theory being exclusively for CS-troons and schizos. Now I'm aware there's a third category of category theorists, galaxy brained algebraists who use category theory as a mechanism to solve algebra problems.
>>16526185Huh? How do you use category theory to solve algebra problems?
>>16526076No, since A can be written as countable union of at most countable sets.
>>16526115What do you mean by "smallest"? From the perspective of any non-standard model, it is working in [math]\omega[/math].
>>16526155Cheers, I think another way would be [math] \int |f^{y}|d\lambda=y^{2}\int \frac{|x|}{1+x^{6}}d\lambda(x)= y^{2}(\int_{[-1,1]} \frac{|x|}{1+x^{6}}d\lambda(x)+\int_{|x|>1} \frac{|x|}{1+x^{6}}d\lambda(x))\leq y^{2}(\int_{[-1,1]} |x|d\lambda(x)+\int_{|x|>1} \frac{1}{|x|^{5}}d\lambda(x))[/math]. The last integrals being obviously finite. What do you think?
>>16526094>foolishly good at algebra>just things you'd recommend reading after one semester of abstract algebraJacobson for both.
>There is no set [math]X[/math] such that [math]2^{X} = |\mathbb{N}| [/math].What if there was? Are there any abstract generalisations of power sets that enable us to do some kind of inverse power set operation for sets that don't normally have an inverse power set?
How long would it take for me to brush up on Precalc/Trig and Calculus 1 stuff? What's the best way of going about that?
>>16526308No. It's impossible for a power set to be a limit cardinal, by definition.
Does there exist a Lebesgue integrable function that does not almost equal a Riemann integrable function?
>>16526391Okay, is there a way to construct N so that it isn't a limit cardinal?
>>16526422Did you already browse >Counterexamples in Measure and Integration - Franziska Kühn and René L. Schilling
>>16526512No.
I have a BA in Math from a pretty chill school. I have been a public school teacher for the past few years but want to go back to grad school to escape the terrible pay and working conditions. My city has good programs in Financial Mathematics.The furthest I have gone is a few independent study courses in Graph Theory, Topology, and Category Theory. My background in Prob and Stat is severely lacking. I am now 27. Is it worth it, or am I just fucked? If the rest of my life is just staying up into the wee hours developing lessons to teach Algebra I and Geometry to kids who can't read or count, I'll be really mad.
Are there functions which are continuous in one direction and non continuous in another?
>>16526525Probability is just real analysis and linear algebra. You'll be alright. Just get a good "transition" probability textbook (or two, one that is all calculus/linear algebra but is a bit more advanced and one that is properly measure). If you self study them, you'll probably be fine.
>>16526534Thank you. Not to bother, but do you have any specific recommendations?
>>16526528"Continuity" in a classical sense doesn't really work that way. For something to be continuous, you have to be able to arbitrarily approach it in any direction.There are "smooth" maps in lower spaces that are continuous in that lower space, but that requires a bit of topology and doesn't always work. (Take, as an example, the function [math]f(x,y) = 1_{x^2+y^2=1}(x,y)[/math]This function is not continuous in [math]\mathbb{R}^2[/math] but is continuous in angle [math]\theta[/math] if you convert to polar. So, you can't really have a function that is properly continuous in some directions but not in others, but you can have a non-continuous function in one space that is continuous in a lower dimensionality projection.
>>16526534>Probability is just real analysis and linear algebra.If you are intention is to never go beyond baby's first measure theoretic probability course, then yes, that is an accurate description.
>>16526538Do you want something more applied or more analysis heavy? Casella and Berger's Statistical Inference covers a lot of probability and statistics from a primarily calculus/linear algebra perspective. If you want something more "pure math" there's no shortage of good textbooks. My favorites ATM are probably Le'Gall's Measure Theory, Probability and Stochastic Processes, as well as Bremaud's Probability Theory and Stochastic Processes.
>>16526547The kid is asking about entry level probability theory. Your autistic screeds about concentration inequalities for random eigenfunctions can wait.
>>16526550I have never actually taken a single applied course. My background, meager as it is, is entirely pure. Something tells me dipping my toes into applied would be good for me, but realistically, an analysis approach would be far smoother at this point.I actually have a copy of Bremaud already so I supposed I'll start there. Thank you.
>>16526559No problem, you've got this. Remember, there's always a deeper rabbit hole. Don't let yourself get chased into losing the forest for the trees. The basics will work for most use cases (read, statistics) and when you need something more sophisticated, you'll figure it out when you get there.
Anyone have a good rec for a calculus of variations resource/textbook? It can rely on some basic functional analysis/measure theory, but ideally it would be more applied than functional analytic in nature. Thanks in advance
>>16526500No. You can represent the power set of k as the set of all functions from k onto {0,1} - think of the 1s of a function as the elements present in some subset, and the 0s as their complement in k. The set of all finite binary sequences is countable, so for there to be a k such that P(k) = |N|, k would have to be larger than any finite set but smaller than N, which immediately produces countless obvious contradictions.
>>16526936Israel M. Gelfand's maybe, but havent read it
>>16527054I ended up ordering that one on Amazon for cheap. I keep running into problems in information theory that require calculus of variations and I only really understand it in a super heuristic way. I'm hoping by spending some time learning calculus of variations I'll be more comfortable with maximum entropy/minimum distortion distribution problems (which seems to require CoV by default).
>>16526195[math] \int |f^{y}|d\lambda=y^{2}\int \frac{|x|}{1+x^{6}}d\lambda(x)= y^{2}(\int_{[-1,1]} \frac{|x|}{1+x^{6}}d\lambda(x)+\int_{|x|>1} \frac{|x|}{1+x^{6}}d\lambda(x))\leq y^{2}(\int_{[-1,1]} |x|d\lambda(x)+\int_{|x|>1} \frac{1}{|x|^{5}}d\lambda(x)) [/math]
>>16526185thanks! i wish this book were cheaper, it has been recommended to me a few times. pdf will have to do for now>>16526188i guess you may have to read the book to find out, huh>>16526297thanks, i assume you mean Jacobson Basic Algebra I and II, right? any comments on what you like particularly about Jacobson? looks like used copies of this aren't too expensive>>16526339how much do you need to brush up on, and to what end? how thorough do you want to be? how much time do you have to dedicate to working on it?the best way of going about that would be to pick an appropriate textbook for precalc and for calc, and then to work through the exercises in relevant chaptersif you're reviewing you can probably get pretty sharp at this stuff in about a month
>>16527373>how much do you need to brush up on, and to what end? how thorough do you want to be? how much time do you have to dedicate to working on it?Enough to take a physics course. I'm about to enter my next semester for school in a week. I may only end up taking a couple of classes.>the best way of going about that would be to pick an appropriate textbook for precalc and for calc, and then to work through the exercises in relevant chaptersWould Khan Academy work as well?>if you're reviewing you can probably get pretty sharp at this stuff in about a monthDamn, I definitely should've started earlier then. lol
>>16527373>thanks, i assume you mean Jacobson Basic Algebra I and II, right? any comments on what you like particularly about Jacobson? looks like used copies of this aren't too expensiveIt treats a lot of advanced topics and is very comprehensive without being very slow like Aluffi. Aluffi has this patronising tone which I really dislike—treating the reader like a kid who needs jokes to be in their book. However, it's also a nice book if you don't mind the slow pace and its use of category theory from the beginning. Jacobson gets through the basic stuff quite quickly and doesn't bog you down with loads of superfluous exercises though Aluffi does mark the important exercises. This is not to say, it is terse. It has a lot of examples to motivate the definitions. In particular, it explains in detail the relationship between dihedral groups and regular polygons, which surprisingly some books like Herstein don't bother with at all. Clearly, one can't describe it as a terse book. Neither is it dry like Dummitt & Foote. Maybe this is pedantic, but it's the only basic book I have read that actually formalises the notion of polynomial expressions. I really like its treatment of polynomial rings as the ring generated by appending an additional element to a subring. I haven't seen it anywhere else. I can't say much beyond the first few chapters. I never got beyond PID since I don't care much for algebra but it's the only algebra book that made the subject interesting for me. That said, I have heard it becomes really difficult when you get to the more advanced topics especially the ones in the second volume.>looks like used copies of this aren't too expensiveMake sure you are getting the second edition. The first one doesn't talk about Chinese Remainder Theorem for some reason. Who knows what other basic stuff is missing.
>>16527373>i guess you may have to read the book to find out, huhCan you not give an example?
>>16527696>Would Khan Academy work as well?Nta, but at the basic level the best exercise sheet you can find is on Khan Academy, because you get instant feedback (they are interactive). And the videos are optional, it is better to look up things as you need them, from internet or your textbook of choice, instead of listening to the jeet.
>>16516921You can assume that one of the nodes is [math] v_0 = 0 [/math]. Let [math] v_1, v_2, \ldots, v_k \in \mathbb{R}^n [/math] be the other nodes. Then [math] v_i^{\top}v_i = 1 [/math] for all [math] 1 \leq i \leq k [/math]. On the other hand, the angle between different nodes must be [math] \frac{\pi}{3} [/math] which implies [math] v_i^{\top} v_j = \frac{1}{2} [/math] for [math] i \neq j [/math].We now need to show that the vectors [math] v_1,\ldots,v_k [/math] are linearly independent, as this implies that [math] n \leq k [/math] (i.e. there are at most [math] n+1 [/math] vectors). Consider the matrix[eqn] M = (v_1, v_2, \ldots, v_k) [/eqn]By considering the inner products see that[eqn] A = M^{\top} M = \begin{pmatrix}1 & 1/2 & \ldots & 1/2 \\1/2 & 1 & \ldots & 1/2 \\\vdots & \vdots & \ldots & \vdots \\1/2 & 1/2 & \ldots & 1 \end{pmatrix}. [/eqn]But you can show that [math] \det(A) = \frac{n+1}{2^n} \neq 0 [/math] which implies that [math] A [/math] has full rank [math] k [/math], so [math] M [/math] has full rank either.
>>16527834The determinant should of course be [math] \frac{k+1}{2^k} [/math], not [math] \frac{n+1}{2^n} [/math]. As this implies that the rank of [math] M^{top}M [/math] is [math] k [/math], the rank of [math] M [/math], having [math] k [/math] columns, must also be [math] k [/math].
>>16516921>>16527834It's called a simplex. It's pretty easy to prove that such a graph must be a simplex. Prove the following simultaneously by induction1) The n-simplex is the only complete equidistant graph with n+1 nodes2) The midpoint of the simplex is the only point that's equidistant to all the vertices in an n-simplex inside the affine plane spanned by the simplex, and its distance from some vertex is <1Base case n=1 easy.Inductive step:Take a n+2 complete equidistant graph. Remove a vertex v. By induction, what's left is a simplex. The vertex we removed cannot lie in the plane of the n-simplex. d(v,w) = sqrt(d^2 + a^2), where w is any vertex other than v, d is the orthogonal distance to the plane of the n-simplex, a is the distance from v' = v orthogonally projected onto the plane and w. Since all d(v,w)'s are equal, and d is constant, that means the orthogonal projection is the midpoint, so v is somewhere on a line orthogonal to the midpoint of the n-simplex. There are two unique points on this line satisfying the distances requirement, and both lie outside the plane of the n-simplex. Therefore we get a n+1 simplex.The midpoint of the n+1 simplex is also somewhere on that line so its distance to one of the vertices is <1.This has gotten a bit verbose but tl;dr you're forced to follow the inductive construction of the simplex as taking n-simplex, extruding its center to an orthogonal new dimension a bit to get a n+1 simplex.
>>16527735of course i can't! i haven't read the book yet!>>16527696just to brush up on what you'll need for physics, i don't think you need a full month. if you're able to lock in and really focus this week, you can make good progress.khan academy is alright for working on exercises--if you're just going in to take calculus based physics 1, i think it's fine to start with khan academy. but in the long run, the better way to study will be with textbooks.check out Paul's Math Notes and use khan academy for exercises like >>16527792 saida week is plenty of time, make sure you start yesterday though>>16527716but what if i am a kid who needs jokes to be in their book? jksounds like a good book, thank you for the rec
>>16528395>cant read the book yetIQ? Very important for math
>>16528397yh lol theres basically no point to math unless you have a tested iq of 135+
>>16528421>yh lol theres basically no point to math unless you have a tested iq of 135+I have a 118 IQ and I can do math just fine.
>>16528426No you can't if ur anything short of answering research questions in low dimensional topology stfu
>>16528426Idk, my IQ is 131 and my mathematical intuition is rather limited. The cutoff being slightly higher makes sense to me.
>>16528429Don't be ridiculous.>>16528434Cutoff for what?
>>16528439>ridiculousbeing below par for a doctoral student is not ""just fine""
>>16528446Not specializing in finite dimensional topology doesn't make one below par for a doctoral student.
>>16528397nice bait, comprehensionleti can't provide an example [because] i have yet to read the book, the fuck?my iq is at least 69 so i think i'm equipped just fine
>>16527716Personally, I kind of like when there's some jokes/levity in the book. It gives the text a bit more personality than being just being "Springer Graduate Texts in Mathematics: Introduction to Categories for People Who Already Know Categories." If anything, I'd say the more difficult the topic, the more necessary it is to sometimes include something a bit less serious and, dare I say, fun, so it isn't just a dry slog the whole time. Maybe that's just my dumb engineer brain talking, and I don't appreciate the "rigor" of three line elegant but incredibly opaque proofs with no instructional quality.
>>16528617one must develop the nebulous concept of "mathematical maturity" to really appreciate how hysterical the elegant but incredibly opaque three line proofs can be
>>16528619I guess if I didn't learn it in my applied math undergrad or the math electives I took in my PhD program, it might be beyond my capacity.
>>16528638i have yet to find a non-insane way (i.e. laughing my ass off when something is just completely over my head) to appreciate it in my pure math undergrad, so who's to say whose capacity it is really beyond
>>16528649My favorite is when they define a dozen pages worth of Lemmas so that the proof for their 2 line theorem can be 3 lines long instead of just having a longer theorem and longer proof. My spectral theory textbook does open with some joke like "This entire textbook is supporting lemmas for the spectral theorem" if I'm remembering like that.
>>16528847Remembering correctly.* Sorry, I need sleep.
Anybody else become a complete shell of themselves after learning so much maths?I can barely get two words out these days
>>16528869Grad school just did that to me in general.
Use your big quantitative brains to write an equation for x = distance from object where a magnifying glass flips an image
>>16529146You'd need to know how the lensing works.
>>16529166I don't know and you don't figure it out
>>16529180You'd need to know the lensing effects to start the problem (how is the lens curved and how does it change the image projection). Without having a formula for the way that light curves when passing through the lens, you'd have no way to measure projected distance.
>>16529209I wear glasses i know they are differentI'm asking what variables in the lens affect the output. No this isn't my homework.
>>16529298I'm not an optical physicist, so my answer is probably lacking some nuance. My understanding is there's basically three components that matter for lensing, thickness, curvature, and spectral properties (does it impact all colors/wavelengths uniformly or does it attenuate/distort some more than others?). Curvature will tell you both where the focal point is, and how steep the tangent lines are at any given point approaching the focal point. If you have a very sharp curvature, you can expect very large distortions at the edges (meaning you'll get a lot of non-linearity and stretching in the image). Beyond that, I'd recommend you look at an optical physics reference. It's been a while since I've opened it up, but I remembered liking the bit of optics coverage in Serway and Jewett's university physics book.
>>16528617I think it depends on the joke. There are some subtle deadpan jokes you can add it academic literature that may cause a nice chuckle. For instance, an actually funny joke is the statistical mechanics suicide one, or the "better known for other work" paper. But then you have shit like Jay Cummings which literally describes itself as a "meme filled book," which just makes cringe. Aluffi is not that bad, but it has an entire theorem environment dedicated to a joke, and then a paragraph having to explain the joke. That's not funny and just wastes my time. >the more necessary it is to sometimes include something a bit less serious and, dare I say, fun, so it isn't just a dry slog the whole time.A good book wouldn't need humour to be fun or not-dry. The exposition of the topics itself ought to be the primary aspect making the book fun.
>>16529146Everyone who takes an optics class learns this. Any introductory book will teach it. Optics is one of the original important uses for Electrodynamics. They start out with a certain reasonable approximation, talk about the index of refraction, then you go into basic 1 sided lens, then you do a 1 sided mirror, then you go into 2 sided lenses and 2 sided mirrors. The derivation is basic enough that I'd imagine you can just google the answer instead of looking in a textbook. On the Physics GRE there's like one or two questions on it, funny enough.
>>16529443*2 mirrors
>>16528847lmao that rules>>16529399yeah, there's an art to it, and deadpan humor plays very well with things that are presented more formallythe statistical mechanics suicide one is a hilarious examplemaybe math is ruining my sense of humor because lately the funniest thing in the world to me is just stating a tautology. normal people laugh at this too though
These typesetting mistakes bother me a lot.
I sucked at math in school and always had a suspicion that the people were good at it were better at mentally framing things. For instance when multiplying 20 x 40, you could either think of these numbers as 2 tens and 4 tens, or you could think of them as distinct entities. If you frame it in the first sense then you’d simply have to multiply 2 x 4 and add two zeros to get your answer. Frame it in the second sense and it becomes a seemingly bigger problem. Are there books that teach you how to fundamentally hone such problem solving skills?
>>16529946>20x4040 x 10 = 400, then double it.
>>16527037Is there no way to take a couple axioms away from ZFC so that P(finite set) isn't always guaranteed to be finite?What about the other option - is there a set of axioms where there is no smallest infinite set?
>>16530072You would have to take away most of the axioms in zfc or nbg to get either of those, idk more about logic than that.
>>16530072>Is there no way to take a couple axioms away from ZFC so that P(finite set) isn't always guaranteed to be finite?Technically depends on what you mean by finite, since ZF without C can have Dedekind-finite infinite sets (that is, infinite sets which aren't in bijection with any proper subset of themselves), and for any infinite set X, P(N) injects into P(P(X)). By the usual meaning of finite (in bijection with some Von Neumann natural), not in any sane set theory, no.>What about the other option - is there a set of axioms where there is no smallest infinite set?Sure, it's consistent with ZF (assuming the consistency of a measurable cardinal) that there is an infinitely decreasing chain of cardinals. Of course, those are incomparable with N so you can't meaningfully say whether they're bigger or smaller than it. As far as the well-orderable infinite sets go, no.
>>16529805Where is the mistake here?
>>16529805>>16530226Nevermind, this can indeed be confusing.
>try to learn lean from book>get question with stuff like "assume a - 5b = 4" and "b + 2 = 3" prove a = 9>Hmm I think I will conclude b = 1 and then substitute>Try to write that>can't do it>check book>there's no way to introduce that conclusion>totally confused>Check book again, must have missed out to do this>"So the way you do this problem is to write a - 5b + 5b, then 4 + 5b, then subtract 10 from 4 and then add 2 to b so you have -6 + 5(b+2) :). The rest is trivial">??????????????????????>delete lean
>>16530248Which book, which chapter, which exercise?
>>16530253Book is "The mechanics of Proof"
>>16530248The mathlib theorem you are looking for to cancel out subtraction like that is eq_sub_of_add_eq: https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Basic.html#eq_sub_of_add_eq. You have to get good at mathlib-fu to learn Lean.
>>16530248>>16530349Want to add also, I'm not sure what this retarded book you're using is, but if you're already familiar with how to make proofs on paper this "Mechanics of Proof" book doesn't look great. I recommend "Mathematics in Lean" by Avigad and Massot if you already have the 'mathematics' part down and are just trying to learn Lean: https://leanprover-community.github.io/mathematics_in_lean/mathematics_in_lean.pdf
>>16530349>>16530350The book is called "Mechanics of proof" by Heather MacBeath I jumped into it because I finished the Natural Numbers Game without too much trouble and thought a proper textbook would be the next step. My mathematical skills are fairly below average but I didn't think The NNG was difficult.
>>16530365Well I recommend "Mathematics in Lean" to you as a serious sequel to the NNG. The problem of subtraction cancellation that you are having even is an exercise in that book, so I'm sure you'll find out how to solve your problems from that one.
Bros what is it like to think about math? Is it fun?
>>16530376NTA but going through that one myself. It's very nice.
>none at present
>>16530726Why are my tax dollars going to fund homological algebras if they have no practical applications?
What the fuck are subtraction and division?
>>16530749Mathematicians are still trying to figure that out.
>>16530749>What the fuck are subtractionAddition of additive inverse.>and division?Multiplication by multiplicative inverse.
>>16530770Fuck you
>>16530376I am that anon and I am working through it as well and I find it much easier than Mechanics of Proof, which is odd because Mechanics of Proof is advertised as the easier of the two.
I used to fantasize about being given an abel prize or a fields medal but recently I've found out they call you in advance to ask you if you're going to accept it to avoid another Perelman's situation (a mathematician who has a field's medal has revealed this on numberphile). That means it's possible that there have been mathematicians who were about to be awarded a fields medal or an abel prize but who rejected it but nobody knows about it except themselves and the prize committee. Which is an interesting thought.
>>16530758How long will it take?
>>16530785Fuck you, he gave the correct answers.
>>16531092Depends. They're still debating if Thrembo exists. Could be years.
>type "simp">proof clears>No clue why>move on to the next question
>>16531079No, because anyone who seeks so much attention that's they reject such prizes would publicly reject it.
>>16531206That's fine, the goal of mathematics is not personal understanding but knowing that it's been rigorously checked to hold.
>>16531209It's amazing how many problems I solve without knowing exactly what I'm doing. It's totally opposite from how I think of math normally. Almost like I'm feeling my way through the problem instead of thinking.
>plug hw into wolfram>never learn anything>fail exam
>>16531216Not wolfram, chudcel. it's a PROOF ASSISTANT. A little out of your league I bet huh? heh...well we can't all be geniuses.
>>16531206It works because someone else already proved the theorem in the standard library or in mathlib and the simplifier found and applied that proof. That's proof by "its in the textbook."
>>16530734you have a problem with constructing a giant pair of tits you can walk into??what are you, a fag?!?!
>Tits "buildings"
>>16530726source?
>>16531646
>>16531513Is there a way to see exactly which lemmas `simp` is applying?
>>16531658'simp?' with a question mark
>>16531659Thanks!
The question of "what is subtraction" has interesting answers but I'll only say one here. It's obvious that any two integers can be subtracted from each other but it isn't obvious how two group homomorphisms can be subtracted from each other.There are many different operations out there that reduce to ordinary subtraction in the case of the integers but are inequivalent for other "types" of input. I don't think it's possible to get a good bigger answer as to which one is "true subtraction" unless we have a way to map the space that contains all of these different operations.
>>16531750Also, if category theory could map spaces like that it would have already been done by now. Type theory can probably do it, though.
>>16531750Now do division.
>>16531750Even understanding adding 1 to a number is beyond current mathematics. Division is like light years away.
Need cleo gf
>>16532231you post here? what the fuck?
This is literally me omg
Is a person here able to suggest a quality university catered to mathematical foundation, an alternate request would be for a particularly fine public course's syllabus.
Bros, How can I get a scholarship for a master's degree in mathematics?My university has an agreement with the University of Paris Saclay, but since my country is garbage, they only let me apply for the master's degree in applied mathematics, but I would like to do pure mathematics.I have been searching on different websites for scholarships but the only thing I have been able to find have been scholarships that only include tuition fees.What I'm looking for are scholarships like the Sohpie Germain foundation, (https://www.fondation-hadamard.fr/en/our-programs/transversal-programs/graduate-program/apply-for-a-sophie-germain-scholarship/) , which give you money while you study so you can live.
I hate lean's syntax so much.Wondering if Metamath might be better to go into 2bh
>>16534183
