/mg/ Hurewicz editionTalk maths, formerly >>16271237
>>16289553Let[math]f(x) = x^{p+1} \ln x / (p+1) - x^{p+1}/(p+1)^2[/math][math]f'(x) = x^p \ln x[/math][math]\ln (1^{1^p} \ldots n^{n^p}) = \sum_{k=1}^n f'(k)[/math]By MVT and the fact that [math]f'[/math] is monotone,[math]f'(k) \leq f(k+1) - f(k) \leq f'(k+1)[/math]Therefore summing[math]f(n) - f(1) \leq \sum_{k=1}^n f'(k) \leq f(n+1) - f(1)[/math]Now using the fact that[math]f(n+1) - f(n) \to 0[/math]and using the form of f(x) quickly gives you the result
>>16289578>>I know/remember faulhaber's formula / how to expand sums of powers of integersYou need way less than faulbaher for that expression. All you need is an integral of [math]x^n[/math]. Otherwise, nice solution. This is essentially what I did here >>16289635In both cases, it involves looking at the antiderivative of [math]x^p\lnp[/math].
For anons who didn't know: Witold Hurewicz (of Hurewicz theorem fame) died by falling down a pyramid while at a math conference.https://en.wikipedia.org/wiki/Witold_Hurewicz#:~:text=He%20died%20after,to%20his%20death.%22
knuth's concrete math vs rosen's or cl liu's discrete math, which is better ?
Bros I am really conflicted with my PhD advisor. I mean he is the typical egotistical asshole who has just taken over my life, but at the same time he is actually promoting me with every single one of his peers. I mean he is pulling strings to get me to enter the department and I know it is not bullshit because he already managed to get in another of his students. But I mean he is just getting crazier and crazier and I really don't know how to work with him. And not only his students, other academics are also shocked at how dysfunctional he is becoming. Its weird because he is focusing on working on really dumb shit and he has become quite unproductive since he uses us for bullshit shit. Idk I know a lot of advisors are really difficult to deal with. but in this case he kinda compensates for it (consciously or not) and I don't really think I have better chances with a more normal advisor. But also I think he will be cancelled any time know because of his behavior. What do?
>>16290140I haven't done a PhD but my instinct tells me that you have to run. By staying you are implicitly endorsing his leadership.
>>16289681dont read books on fake meme subjects like discrete math
>>16289614you didn't even fill the subject field retard
>>16290140how far are you in? If you're almost done, just shut up and finish. If you're far off and expect shit to seriously hit the fan before you finish, you better plan for it
>>16290140Sounds like a great situation for you. You can build up your cv now and then when he self-destructs you will receive a lot of pity and people will help you find a new position.
What are some good books on differential geometry (either classical or modern) with nice-looking pictures and drawings? Also, is there any book exhibiting a gallery of parametrized curves and surfaces?
/mg/ doesn't even come with any of the benefits of brevity or focus. It's like a giant joke competition of who can describe every trivial thing in the most abstract and abstruse way except it got out of hand and the participants forgot it was supposed to be a joke.
>>16289681Concrete math -> the art of computer programmingand you can finally escape the hell of retards larping like it's the 19th century spending years learning shit like topology just so their wetware with is 2 byte ram can solve nontrivial problemsand start doing real mathematics
Question on language/notation: if an equation [math]\gamma[/math] has variables [math]x,y,z[/math], what kind of grammar am I using when discussing the solution? Which is more correct:[math]\texttt{(1)}[/math] - The solution [math]x,y,z[/math] to the equation [math]\gamma[/math]~ or ~[math]\texttt{(2)}[/math] - The solution [math](x,y,z)[/math] to the equation [math]\gamma[/math]?Like, do I treat the solution as an ordered triple? Or is it fine if I just list it down like in [math]\texttt{(1)}[/math]?
If I want to state that [math]x[/math] cannot be equal to any [math]a_1,a_2,a_3[/math], can I do like this:[math]x\not=a_1,a_2,a_3[/math]?Or is it better to write:[math]x\not=a_1,x\not=a_2,x\not=a_3[/math]?
>>16289614I smell some serious fire power around here.
>>16291843I mean the only way it to output 3 things like that and make sense if its a 3-tuple so yeah (2) definitely
>>16291954The former is fine
>>16291974Thanks.
>>16291843>>16291973Nobody will care if you use (1) instead.
>>16291954I strongly prefer the former, but I also often use [math] x \notin \{ a_1,...,a_n \} [/math]
>>16293347Ahh that's actually a good way as well. Thanks.
just caught up to the latest chapter of csm. Life outside mathematics is meaningless now
>>16289614I am currently in dilemma in how to introduce complex numbers to students. Initially I had decided to introduce them as the natural extension of arithmetic of $\mathbb R$ to $\mathbb R^2$ and formalisation of 2d geometry. Essentially visualising complex numbers as composition of scaling and rotation. Obviously, this is a lot more intuitive and motivated as opposed to the usual way of just defining them out of nowhere.However, recently I have been feeling I should introduce it the "usual" way as the algebraic closure of $\mathbb R$. The main problem is that the former approach is not in line with how complex numbers was discovered and gives the false belief that everything in math has some deep visual motivation. The truth is complex numbers were defined because they were convenient, and we often define things in math as according to convenience. This is something students have a hard time grasping (like defining $0!$ as $1$, $1$ as composite, defining $0^0$ differently etc.). Even if one thinks complex root to be nonsensical, complex numbers are still used to find real root since they often cancel each other out. Another problem is that former approach requires quite a bit of analysis with sequence of functions to be rigorous about it. I think too much handwaving and saying that "this will be proved later in an analysis course" is just going to make things less satisfying.So how do you think they should be introduced?
Gonna have to start restudying calc 1 and 2 in prep for my calc 3 class in the fall. I have about 20 days. Please pray for me.
>>16293563The "closure" argument pushes 19th century math (real numbers) into 16th century math (algebraic roots). At that point, you might as well show how purely algebraic ring extensions work (really, just comparing Q[sqrt 2] and Q[sqrt -1]), which gives you enough to do arithmetic. After that, you can point out that we can approximate sqrt 2 with (e.g.) the Babylonian method, but not sqrt -1.Without analysis, the "scaling and rotation" story gets hard to grasp (why is angle measured from the positive real axis, need to understand trig, the complex exponential looks scary, etc.).
>>16293563in terms of what makes sense didactically I am strongly of the opinion that no ad hoc definitions should ever be made. Giving a definition before telling the student its purpose and some fundamental examples is confusing and frustrating, precisely because mathematical structures are inherently arbitrary and hence meaningless without context or knowing how to think of them. (e.g. the definition of spectral sequence is hard to follow in isolation) I highly encourage you to do the experiment in Sfard's "on the dual nature of mathematical conceptions" to understand what I mean. Following her article, I suggest studying the historical development of complex numbers first as intermediate steps to solving equations, then as variable quantities and functions with Riemann. Since they're clever, the conceptualization of complex numbers, i.e. as an abstract set, should be a process that any mathematically minded student can come up with anyway, so they should benefit from you introducing the abstract definition only in the second lecture after they've had time to think about complex quantities
>>16293563Why do you *have* to introduce complex numbers? I'll tell you something funny, I did a whole phd in math (analysis specialization) and no one in the department used complex numbers willingly. on the occassion when you *had* to work with a complex number (like in a very few spectral decompositions), we usually only cared about its real part. usually the intuition we were taught is that if you're spending time on complex numbers then you're probably not phrasing the problem in an intelligent way. I understand how you *can* use complex numbers in some places where you mean R^2, but if you mean R^2 then there's almost no worthwhile convenience afforded to you by using C.
damn i hate summer vacation. it's as if all mathematics discussion stops
>>16289681bump for this
>>16293401I WANT TO HAVE TONS AND TONS OF SEX!!!!!!!!
>>16294377That depends what your goals and objectives are for studying discrete math?
>>16293401What is that?
>>16294803its just a wholesome slice of life manga
>>16293401I love fami
>>16293881yeah you specialized in a field where they're not used a whole lot, nobody cares. I used quaternions and symplectic vector spaces a lot which people in less geometric areas will hardly ever encounter
>>16291526>differential geometry (either classical or modern) with nice-looking pictures and drawings?Spivak
>>16291526Kobayashi-Nomizu
>>16294551>That depends what your goals and objectives are for studying discrete math?um.. I'm currently unsure if I should pursue cryptography or machine learning.
>>16295795PhD? Cryptography Less than PhD? Machine LearningThe reason is that you want a job that allows for your creativity. There's more creativity in a low level machine learning job that a low level cryptography job, but there's more creativity in a high level cryptography job than a high level machine learning job.
do the exercises they said
>>16295951>PhD?yup, either in cryptography or machine learning
>>16296288and something that can help me be get into the industry after completing my PhD if need be
>>16296288Final Ok.
>>16296290>Final Ok.huh?
How do I take the euclidean norm [math]||(\textbf{x}, \textbf{y})||[/math] of a vector [math](\textbf{x}, \textbf{y}) \in \mathbb{R}^n \times \mathbb{R}^n[/math] My quick guess would be to take [math]||(\textbf{x}, \textbf{y})|| = \sqrt{||\textbf{x}||^2 + ||\mathbf{y}||^2}[/math] but I don't know if this is correct
>>16296503What a silly question! You're such a silly guy you.
>>16296503It is in fact correct
>>16289614hello /mg/Im making a sailing game and I would like to implement a function that models the speed of a sailboat at each point of sail such as the diagram in picreldoes anyone know what this shape is called and how I would create a polynomial function to approximate it?if you dont know exactly how to do it, could you point me toward resources such as textbooks for understanding how to create such approximations? thanks
>>16296766there are also more complex shapes such as this
>>16296766just with trial and error on desmos I came up with thisstill not sure I can do this >>16296778 though
>>16296288>>16296289Seems like you read one word of >>16295951
>>16297162Still don't get it m8.Crypto or ML?knuth's concrete math or something else
>>16297170Just read one. You could've read a lot of pages in the week since the first post.I like knuth's book.
>>16296766That shape in that chart you have looks similar to these shapes:Oval of Cassini: https://en.wikipedia.org/wiki/Cassini_ovalor...Hippopedes of Proclus: https://en.wikipedia.org/wiki/HippopedeThe difference between the two despite similar shapes is oneis made with 2 foci, the other none. For the graph >>16296786 ,the equation fits the hippopede.
>>16298267I came up with this polynomial ultimatelybut Im running into a problem where if I plug in theta values into the equation the radius value comes out all wrongeven on desmos, look at what happens when I enter pi/3 rad for exampleshouldnt the answer be on the line?am I just misremembering how polar coords work?someone please tell me whats going on here
>>16298333>>16298267FUCK I forgot the picrel
>>16298334>>16298267You have to define the function as r(theta), not r.So, when you use the function at pi/3 you'll have the value
>>16298333>>16298358I double checked the values, and it seems consistent.Graphing is in radian measure. Starting at zero radians,r is negative meaning it is in the negative x-axis. Increasethe angle and it proceeds counterclockwise but fromthe opposite side of the first quadrant (negative r).The value at pi/3 is -6.8951
>>16298358that doesnt matter and has nothing to do with it>>16298386you explained that in the most convoluted way possible but I see now that the intercept is in the opposite quadrant (so pi/3 intercepts at 4pi/3 and vice versa)I modified the function to the one shown in picrel and it seems to work as intended nowlet me know if theres a way to simplify the function
>>16298441You could expand the sines using the angle sumformulas and get them down to single angle sines
>>16298488forgot about those, thanks
>>162896141/n diverges, classical proof: illustrated with bqn.
Does anyanon know if Hn ∼ 0.5log(n^2+n+0.5)+γ−(1/12(n^2+n+0.5))is always a over approximation? If not when does it flip and does it keep fliping? Also is it the best approximation for Hn?
>>16298561that is a sequence at most, which converges to zero.
>>16298746My capital sigma got deleted by 4chans filter
>>16290140Big deal lol. Not only is he promoting to with his peers, but those same peers are also well-aware of his spiralling behaviour. If he 'blows up', you have a network he helped you build, and they'll be understanding of your situation. Stick it out.
