🗑️ Anonymous at Sat, 20 Jul 2024 14:42:41 UTC No. 16289618

First for information geometry: https://mathweb.ucsd.edu/~mleok/courses/math273b/

Anonymous at Sat, 20 Jul 2024 14:54:37 UTC No. 16289635

>>16289553
Let
[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

Anonymous at Sat, 20 Jul 2024 14:59:54 UTC No. 16289641

>>16289578
>>I know/remember faulhaber's formula / how to expand sums of powers of integers
You 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 >>16289635
In both cases, it involves looking at the antiderivative of [math]x^p\lnp[/math].

Anonymous at Sat, 20 Jul 2024 15:01:46 UTC No. 16289644

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

Anonymous at Sat, 20 Jul 2024 15:25:45 UTC No. 16289681

knuth's concrete math vs rosen's or cl liu's discrete math, which is better ?

Anonymous at Sat, 20 Jul 2024 21:48:27 UTC No. 16290140

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?

Anonymous at Sat, 20 Jul 2024 22:31:33 UTC No. 16290177

>>16290140
I haven't done a PhD but my instinct tells me that you have to run. By staying you are implicitly endorsing his leadership.

Anonymous at Sun, 21 Jul 2024 05:19:25 UTC No. 16290442

>>16289681
dont read books on fake meme subjects like discrete math

Anonymous at Sun, 21 Jul 2024 08:49:37 UTC No. 16290642

>>16289614
you didn't even fill the subject field retard

Anonymous at Sun, 21 Jul 2024 12:10:22 UTC No. 16290768

>>16290140
how 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

Anonymous at Sun, 21 Jul 2024 14:24:40 UTC No. 16290863

>>16290140
Sounds 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.

Anonymous at Sun, 21 Jul 2024 22:23:55 UTC No. 16291526

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?

Anonymous at Mon, 22 Jul 2024 00:14:29 UTC No. 16291637

/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.

Anonymous at Mon, 22 Jul 2024 00:17:53 UTC No. 16291642

>>16289681
Concrete math -> the art of computer programming
and 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 problems
and start doing real mathematics

🗑️ Anonymous at Mon, 22 Jul 2024 00:20:38 UTC No. 16291647

>>16290140
>>16290863
when he inevitably get fired you can get his position lmao.

Anonymous at Mon, 22 Jul 2024 06:39:39 UTC No. 16291843

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]?

Anonymous at Mon, 22 Jul 2024 10:09:06 UTC No. 16291954

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]?

Anonymous at Mon, 22 Jul 2024 10:25:55 UTC No. 16291968

>>16289614

I smell some serious fire power around here.

Anonymous at Mon, 22 Jul 2024 10:35:55 UTC No. 16291973

>>16291843
I mean the only way it to output 3 things like that and make sense if its a 3-tuple so yeah (2) definitely

Anonymous at Mon, 22 Jul 2024 10:36:59 UTC No. 16291974

>>16291954
The former is fine

Anonymous at Mon, 22 Jul 2024 10:54:17 UTC No. 16291980

>>16291974
Thanks.

Anonymous at Mon, 22 Jul 2024 13:11:32 UTC No. 16292078

>>16291843
>>16291973
Nobody will care if you use (1) instead.

Anonymous at Mon, 22 Jul 2024 16:22:23 UTC No. 16292278

Anonymous at Tue, 23 Jul 2024 12:15:20 UTC No. 16293347

>>16291954
I strongly prefer the former, but I also often use [math] x \notin \{ a_1,...,a_n \} [/math]

Anonymous at Tue, 23 Jul 2024 12:25:28 UTC No. 16293351

>>16293347
Ahh that's actually a good way as well. Thanks.

Anonymous at Tue, 23 Jul 2024 13:31:24 UTC No. 16293401

just caught up to the latest chapter of csm. Life outside mathematics is meaningless now

Anonymous at Tue, 23 Jul 2024 16:34:58 UTC No. 16293563

>>16289614
I 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?

Anonymous at Tue, 23 Jul 2024 16:55:24 UTC No. 16293585

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.

Anonymous at Tue, 23 Jul 2024 17:33:11 UTC No. 16293648

>>16293563
The "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.).

Anonymous at Tue, 23 Jul 2024 18:28:27 UTC No. 16293713

>>16293563
in 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

Anonymous at Tue, 23 Jul 2024 20:25:52 UTC No. 16293881

>>16293563
Why 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.

Anonymous at Wed, 24 Jul 2024 01:36:48 UTC No. 16294207

damn i hate summer vacation. it's as if all mathematics discussion stops

Anonymous at Wed, 24 Jul 2024 05:46:53 UTC No. 16294377

>>16289681
bump for this

Anonymous at Wed, 24 Jul 2024 06:22:35 UTC No. 16294396

>>16293401
I WANT TO HAVE TONS AND TONS OF SEX!!!!!!!!

Anonymous at Wed, 24 Jul 2024 11:22:30 UTC No. 16294551

>>16294377
That depends what your goals and objectives are for studying discrete math?

Anonymous at Wed, 24 Jul 2024 15:28:09 UTC No. 16294803

>>16293401
What is that?

Anonymous at Wed, 24 Jul 2024 15:36:38 UTC No. 16294820

>>16294803
its just a wholesome slice of life manga

Anonymous at Wed, 24 Jul 2024 15:41:33 UTC No. 16294824

>>16293401
I love fami

Anonymous at Wed, 24 Jul 2024 17:57:23 UTC No. 16294962

>>16293881
yeah 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

Anonymous at Wed, 24 Jul 2024 21:35:46 UTC No. 16295287

>>16291526
>differential geometry (either classical or modern) with nice-looking pictures and drawings?
Spivak

Anonymous at Thu, 25 Jul 2024 05:52:29 UTC No. 16295680

>>16291526
Kobayashi-Nomizu

Anonymous at Thu, 25 Jul 2024 08:21:24 UTC No. 16295795

>>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.

Anonymous at Thu, 25 Jul 2024 12:39:29 UTC No. 16295951

>>16295795
PhD? Cryptography
Less than PhD? Machine Learning

The 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.

Anonymous at Thu, 25 Jul 2024 13:00:13 UTC No. 16295982

do the exercises they said

Anonymous at Thu, 25 Jul 2024 18:33:50 UTC No. 16296288

>>16295951
>PhD?
yup, either in cryptography or machine learning

Anonymous at Thu, 25 Jul 2024 18:35:36 UTC No. 16296289

>>16296288
and something that can help me be get into the industry after completing my PhD if need be

Anonymous at Thu, 25 Jul 2024 18:35:37 UTC No. 16296290

>>16296288
Final Ok.

Anonymous at Thu, 25 Jul 2024 18:39:09 UTC No. 16296296

>>16296290
>Final Ok.
huh?

Anonymous at Thu, 25 Jul 2024 21:31:14 UTC No. 16296503

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

Anonymous at Thu, 25 Jul 2024 21:42:56 UTC No. 16296515

>>16296503
What a silly question! You're such a silly guy you.

Anonymous at Thu, 25 Jul 2024 21:52:58 UTC No. 16296524

>>16296503
It is in fact correct

Anonymous at Fri, 26 Jul 2024 01:24:51 UTC No. 16296766

>>16289614
hello /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 picrel
does 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

Anonymous at Fri, 26 Jul 2024 01:32:06 UTC No. 16296778

>>16296766
there are also more complex shapes such as this

Anonymous at Fri, 26 Jul 2024 01:36:06 UTC No. 16296786

>>16296766
just with trial and error on desmos I came up with this
still not sure I can do this >>16296778 though

Anonymous at Fri, 26 Jul 2024 11:14:58 UTC No. 16297162

>>16296288
>>16296289
Seems like you read one word of >>16295951

Anonymous at Fri, 26 Jul 2024 11:35:26 UTC No. 16297170

>>16297162
Still don't get it m8.
Crypto or ML?
knuth's concrete math or something else

Anonymous at Fri, 26 Jul 2024 22:30:07 UTC No. 16298006

>>16297170
Just read one. You could've read a lot of pages in the week since the first post.
I like knuth's book.

Anonymous at Sat, 27 Jul 2024 02:02:36 UTC No. 16298267

>>16296766
That shape in that chart you have looks similar to these shapes:

Oval of Cassini: https://en.wikipedia.org/wiki/Cassini_oval
or...
Hippopedes of Proclus: https://en.wikipedia.org/wiki/Hippopede

The difference between the two despite similar shapes is one
is made with 2 foci, the other none. For the graph >>16296786 ,
the equation fits the hippopede.

Anonymous at Sat, 27 Jul 2024 02:58:14 UTC No. 16298333

>>16298267
I came up with this polynomial ultimately
but Im running into a problem where if I plug in theta values into the equation the radius value comes out all wrong
even on desmos, look at what happens when I enter pi/3 rad for example
shouldnt the answer be on the line?
am I just misremembering how polar coords work?
someone please tell me whats going on here

Anonymous at Sat, 27 Jul 2024 02:59:15 UTC No. 16298334

>>16298333
>>16298267
FUCK I forgot the picrel

Anonymous at Sat, 27 Jul 2024 03:30:50 UTC No. 16298358

>>16298334
>>16298267
You have to define the function as r(theta), not r.
So, when you use the function at pi/3 you'll have the value

Anonymous at Sat, 27 Jul 2024 04:07:05 UTC No. 16298386

>>16298333
>>16298358
I 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. Increase
the angle and it proceeds counterclockwise but from
the opposite side of the first quadrant (negative r).

The value at pi/3 is -6.8951

Anonymous at Sat, 27 Jul 2024 04:41:43 UTC No. 16298441

>>16298358
that doesnt matter and has nothing to do with it

>>16298386
you 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 now
let me know if theres a way to simplify the function

Anonymous at Sat, 27 Jul 2024 05:19:46 UTC No. 16298488

>>16298441
You could expand the sines using the angle sum
formulas and get them down to single angle sines

Anonymous at Sat, 27 Jul 2024 05:43:34 UTC No. 16298516

>>16298488
forgot about those, thanks

Anonymous at Sat, 27 Jul 2024 06:14:04 UTC No. 16298561

>>16289614
1/n diverges, classical proof: illustrated with bqn.

Anonymous at Sat, 27 Jul 2024 06:34:24 UTC No. 16298574

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?

Anonymous at Sat, 27 Jul 2024 10:52:07 UTC No. 16298746

>>16298561
that is a sequence at most, which converges to zero.

Anonymous at Sat, 27 Jul 2024 11:06:58 UTC No. 16298756

>>16298746
My capital sigma got deleted by 4chans filter

Anonymous at Sat, 27 Jul 2024 11:33:35 UTC No. 16298787

>>16290140
Big 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.

Anonymous at Sat, 27 Jul 2024 15:03:49 UTC No. 16298994

I got an idea for this question from another thread that had this pic.

If you have a venn diagram which is made of five identical shapes by overlapping them, and you wanted to minimize the difference between the surface areas if the biggest and smallest sub-region, what would that venn diagram look like and which shape should you use to construct it?

Anonymous at Sun, 28 Jul 2024 01:05:04 UTC No. 16299768

>>16298756
fair

Anonymous at Sun, 28 Jul 2024 12:48:48 UTC No. 16300307

>>16289614
I'm doing discrete mathematics and I'm only just learning about implications and this doesn't make sense it's making my brain hurt

say p is "it has wheels" and q is "it is an animal"

according to the truth table "It has wheels therefore it is an animal" returns true but that does not make sense. Also "It does not have wheels therefore it is an animal" also returns true but that isn't right either

Anonymous at Sun, 28 Jul 2024 13:13:31 UTC No. 16300325

>>16300307
logic is above worldly common sense.

Anonymous at Sun, 28 Jul 2024 13:13:51 UTC No. 16300327

>>16300307
Implication is classically (aka materially) defined to be truth functional:
>implies(true, true) = true
>implies(true, false) = false
>implies(false, false) = true
>implies(false, true) = true
don't think about it, just do it

🗑️ Anonymous at Sun, 28 Jul 2024 15:14:30 UTC No. 16300449

>>16300307
>according to the truth table "It has wheels therefore it is an animal"
It is not. Car has a wheels but it is not an animal. Car is a counterexample to p implies q, making p implies q false. What is true is that: cow does not have wheels but it is an animal. In this case p is false, so p implies q is trivially true in the case of cow in the sense cow does not contradict that p implies q. So, p implies q is false in general but true specifically in the cases where p is false (cow).

Anonymous at Sun, 28 Jul 2024 15:16:06 UTC No. 16300451

>>16300307
>according to the truth table "It has wheels therefore it is an animal"
It is not. Car has a wheels but it is not an animal. Car is a counterexample to p implies q, making p implies q false. What is true is that: cow does not have wheels but it is an animal. In this case p is false, so p implies q is trivially true in the case of cow in the sense cow does not contradict that p implies q. So, p implies q is false in general but true restricted to the cases where p is false (cow). In a universe, where nothing has wheels, p implies q would be true since nothing contradicts it.

Anonymous at Sun, 28 Jul 2024 17:42:04 UTC No. 16300586

>>16289614
How do you cope with the fact that there are people around you who are just straight up geniuses? People who you'll never be able to compete against just because they're so damn smart? These feelings of inadequacy are really difficult to deal with especially in math because you'll encounter so many people like them.

Anonymous at Sun, 28 Jul 2024 18:33:27 UTC No. 16300654

>>16300586
I don't believe such people exist.

Anonymous at Sun, 28 Jul 2024 19:00:30 UTC No. 16300681

>>16300586
i don't do math to compete

Anonymous at Sun, 28 Jul 2024 19:07:37 UTC No. 16300697

/sci/ plays /x/ shortly after the current /toy/-/sp/ game finishes up >>16300672

Anonymous at Mon, 29 Jul 2024 02:28:01 UTC No. 16301298

>>16300654
Well that's clearly not true, unless you're Terence Tao himself.

>>16300681
I guess yeah that should be the outlook, I shouldn't really compare with others, and just do math for its own enjoyment. Thanks anon, you've made my day a little better.

Anonymous at Mon, 29 Jul 2024 14:04:45 UTC No. 16301827

>>16301298
I don't believe anyone is more naturally talented than I am. I think I could be as good as they are if I invested enough time.

Anonymous at Mon, 29 Jul 2024 14:41:58 UTC No. 16301863

Does anyone know what should my latex recipe be if I want biber with latexmk.
I looked it up online and I found
latexmk -> biber -> latexmk*2
But that doesn't seem right. The point of latexmk is for it to automatically decide how many times to run. So, running it twice at the end seems to go against it.

Anonymous at Mon, 29 Jul 2024 16:08:49 UTC No. 16301946

Reading through Tao's Analysis II, he defines bounded sets as follows: in a metric space (X,d) let Y be a subset of X. Then Y is bounded iff for every x in X there exists a ball B(x,r) in X of some finite radius r which contains Y.
All good, it's not the same as previous definitions I've seen but it's perfectly understandable.
He then states that the metric space (X,d) is bounded if X is bounded. This last part is confusing me relative to the definition since then X is bounded iff for every x in X there exists a ball B(x,r) in X of some finite radius r which contains X?
Am I missing something here? That doesn't make much sense to me. Is this set X supposed to be relative to some larger set?

Anonymous at Mon, 29 Jul 2024 16:17:31 UTC No. 16301960

>>16301946
No.
In the metric space [math]([0,1], d_{\text{euclidean}})[/math], what does the ball of radius [math]\frac12[/math] centered at [math]\frac1{10}[/math] look like?
What about the ball of radius [math]2[/math] centered at [math]\frac1{10}[/math]?

Anonymous at Mon, 29 Jul 2024 16:25:27 UTC No. 16301971

>>16301960
Ahh right you are, so the ball of radius 2 centered at [math] \frac{1}{10} [/math] is just the whole space [math] [0,1] [/math]. So X is bounded simply because the open balls are X themselves (when the radius of the open ball is finite).
Cheers

Anonymous at Tue, 30 Jul 2024 19:01:39 UTC No. 16303540

>>16295982
challenge accepted

Anonymous at Tue, 30 Jul 2024 20:02:22 UTC No. 16303615

Suppose there are [math]n[/math] identical looking items (usually coins), with one of them being counterfeit, and having a different mass (weight) from the rest. There's a balance scale where you can compare which side is heavier. If the counterfeit is known to be either heavier or lighter than the rest, then you need at most [math]\lceil \log_3{n} \rceil[/math] weighings to identify the counterfeit. This works for any positive integer value of [math]n[/math].

But what if we don't know if the counterfeit is lighter or heavier than the rest? What is the minimum number of weighings required to identify the counterfeit? And if you also need to identify whether it's heavier or lighter than the rest, would it change the formula? I only know that with 12 items, you need 3 weighings, and also that it's easy to work out that it's impossible to tell with only 2 items, and with 3 items, you need 2 weighings. But that doesn't tell me the general formula.

Anonymous at Wed, 31 Jul 2024 06:50:43 UTC No. 16304319

Somehow I ended up doing quantum algebra. I wanted to be a geometer. Is it over?

Anonymous at Wed, 31 Jul 2024 09:35:27 UTC No. 16304447

>>16303615
(sketchy)
Each weighing has three possible outcomes (tilt to left, right, or stay level), so the information gained is [math]I_w = \log3[/math].
You need to identify one (say, heavier) item from a set of [math]n[/math], which requires [math]\log n[/math] information.
In [math]w[/math] weighings, you thus need [math]w I_w \geq \log n[/math], so that [math]w=\lceil \log_3 n \rceil[/math].

This works in your second case too: you need to identify one item from a set of [math]n[/math], requiring [math]\log n[/math], and you need to identify whether or not it is lighter or heavier, requiring [math]\log 2[/math].
Together you then need [math]\log n+\log2[/math] information and the same inequailty gives [math]w=\lceil \log_32n\rceil[/math], which agrees with your values for [math]n\geq3[/math].

Anonymous at Wed, 31 Jul 2024 09:39:38 UTC No. 16304450

>>16304447
It doesn't work. It's very easy to see that with 13 items you need at least 4 weighings despite the fact that
2 * 13 < 3^3

Anonymous at Wed, 31 Jul 2024 09:56:40 UTC No. 16304460

>>16303615
On the internet you can find algorithms that work for up to
[eqn]3 + 3^2 + \ldots + 3^{w-1} = \frac{3^{w} - 3}{2} [/eqn]
items assuming you're allowed [math]w[/math] weighings.

Anonymous at Wed, 31 Jul 2024 10:09:20 UTC No. 16304465

>>16304450
Hmm, you're right.
I tried some other stuff but I can't something fitting with this idea, right now.

Anonymous at Wed, 31 Jul 2024 10:48:36 UTC No. 16304490

>>16303615
I guess you have the bounds [math]\lceil\log_32n\rceil\leq w\leq\lceil\log_3(2n+3)\rceil[/math] where the first comes from >>16304447 and the second from >>16304460.

Anonymous at Wed, 31 Jul 2024 17:18:07 UTC No. 16304771

>>16289614
Is it true that all lebesgue integrable functions on a closed interval almost equal a riemann integrable function? The Dirichlet function satisfies this for instance.

/mg/ mail boy at Wed, 31 Jul 2024 17:32:09 UTC No. 16304786

Author of the legendary EYE document here. Attached is a highly condensed short note on the notation only for certain maps called "accumulators" of generalized free monoids known as "message spaces"

The EYE document effectively ended online Gödel-flavored "math patent medicine" along with my sustained attack on mathematical logic, Tarski's semantics, and general Russian / German / Eastern attacks on English language mathematical logic discussion, taking down Thomas Jech, Herbert Enderton, David J. Barker-Plummer, John Etchemendy, and Jon Barwise

EYE document link: https://imgur.com/a/eye-Gzp2Ile

American history link: https://en.wikipedia.org/wiki/Patent_medicine

Anonymous at Wed, 31 Jul 2024 17:45:26 UTC No. 16304798

>>16304771
No.
If you take a fat cantor set (https://en.wikipedia.org/wiki/Smith%E2%80%93Volterra%E2%80%93Cantor_set) [math]C\subseteq[0,1][/math], then [math]f=1_C[/math] is Lebesgue integrable.
A function is Riemann integrable iff it is a.e. continuous, but any function [math]g[/math] which is a.e. equal to [math]f[/math] must be discontinuous at almost all the points of [math]C[/math] (because it is nowhere dense), and we have the measure [math]\lambda(C)>0[/math].

Anonymous at Thu, 1 Aug 2024 13:22:14 UTC No. 16305807

25, only undergrad math degree, no papers, no research experience, working as a codemonkey. Is it over?

Anonymous at Thu, 1 Aug 2024 13:26:30 UTC No. 16305812

>>16305807
>employed
be honest, you only came here to brag

Anonymous at Thu, 1 Aug 2024 13:43:28 UTC No. 16305818

>>16304786
Wikipedia, while being both state sponsored and propaganda, is not history anymore than a tabloid is history.

Anonymous at Thu, 1 Aug 2024 13:56:27 UTC No. 16305831

>>16305812
No. I am unironically considering quitting my job and going for a masters/PhD in math. But I feel like I'm too old and will be laughed at by everyone there, as well as waste money.

Anonymous at Thu, 1 Aug 2024 16:30:05 UTC No. 16305960

>>16305831
you're not too old. But gradschool is a massive opportunity cost. I don't think you'll like going from gainful employment to poverty. Why exactly do you want to go back to school?

Anonymous at Thu, 1 Aug 2024 16:39:19 UTC No. 16305972

>>16304798
Nice.

Anonymous at Fri, 2 Aug 2024 08:14:22 UTC No. 16306732

>>16305960
Because I want to spend more time doing math around people who do math, instead of writing code.

Anonymous at Fri, 2 Aug 2024 08:43:14 UTC No. 16306753

Mathematicians today: Therefore..., hence...., thus...., in particular....,

Mathematicians of yore
"The pulsations of the air, once set in motion by the human voice, cease not to exist with the sounds to which they gave rise. Strong and audible as they may be in the immediate neighbourhood of the speaker, and at the immediate moment of utterance, their quickly attenuated force soon becomes inaudible to human ears. The motions they have impressed on the particles of one portion of our atmosphere, are communicated to constantly increasing numbers, but the total quantity of motion measured in the same direction receives no addition. Each atom loses as much as it gives, and regains again from other atoms a portion of those motions which they in turn give up. The waves of air thus raised, perambulate the earth and ocean's surface, and in less than twenty hours every atom of its atmosphere takes up the altered movement due to that infinitesimal portion of the primitive motion which has been conveyed to it through countless channels, and which must continue to influence its path throughout its future existence."

Anonymous at Fri, 2 Aug 2024 12:00:36 UTC No. 16306926

>it's another "you have to have a turnitin similarity of lower than 25% for submission"
I can't believe this shit is still being practiced.

Anonymous at Fri, 2 Aug 2024 13:41:17 UTC No. 16307036

>>16305960
>I don't think you'll like going from gainful employment to poverty
One of the major upsides of going back to grad school after some time in the workforce is that you (hopefully) have a bit of savings. It's a way more enjoyable experience when you can boost your income up enough to afford a basic adult quality of life

Anonymous at Fri, 2 Aug 2024 13:48:41 UTC No. 16307042

>>16306753
Everything the Mathematicians of yore were saying is now quantified in ZFC. Therefore hence thus and in particular are the only logical quantifiers necessary to connect statements in ZFC.

Anonymous at Fri, 2 Aug 2024 15:04:55 UTC No. 16307112

>>16306732
There are some things you have to accept
1- in terms of career prospects you would be better served not going back to school
2- the probability of you getting a stable job in academia is very low
3- as you start to finish your degree you should already be making preparations to pivot back into industry (unless you're entirely sure you will make it as an academic)
Despite all of these, I'd say go for it. I love doing math, I wouldn't trade it for any other job.

Anonymous at Fri, 2 Aug 2024 15:20:56 UTC No. 16307135

I'm having a mental retardation moment, how do I prove that if there exists a primitive root mod m then it is also a primitive root mod every divisor of m?

Anonymous at Fri, 2 Aug 2024 17:01:46 UTC No. 16307291

>>16307135
A primitive root mod m is also a primitive root mod any divisor of m.

Anonymous at Fri, 2 Aug 2024 17:56:15 UTC No. 16307342

>>16306753
True, we've been able to get past the ambiguity of natural language and your quote is a great example

Anonymous at Fri, 2 Aug 2024 18:00:36 UTC No. 16307352

>>16307291
T-thanks

Anonymous at Fri, 2 Aug 2024 19:04:26 UTC No. 16307442

>>16307042
>>16307342
The Bourbaki fanatic failing to understand the distinction between exposition and proof dilutes mathematics into soulless symbols, thus dooming them to possessing solely a symbolic assimilation of mathematics, never to receive the light beyond their self-imposed turbidity of formalism.
When asked to paint a forest, the mathematician starts not with a loose sketch of the trees but instead, draughts the leaves, taking great care into ensuring every leaf is in the right place, every stem returning to a branch, and in doing so, they confuse themselves with this complex entanglement not reaching even the trunk.

Anonymous at Sat, 3 Aug 2024 03:03:34 UTC No. 16308017

Paging algebra-bros.....
I have a question on common notation. Let [math]R[/math] be a ring. Are there any common uses of the notation [math]I(R)[/math] and [math]U(R)[/math]? I want to use these two for my thesis to denote the set of idempotents and units in [math]R[/math], but I would prefer if they weren't already common notations for something else.

Anonymous at Sat, 3 Aug 2024 03:11:55 UTC No. 16308024

>>16308017
https://en.wikipedia.org/wiki/Unit_(ring_theory)
>The set of units of R forms a group R× under multiplication, called the group of units or unit group of R.[b] Other notations for the unit group are R∗, U(R), and E(R) (from the German term Einheit).

Anonymous at Sat, 3 Aug 2024 03:37:40 UTC No. 16308049

>>16308024
Whoops, I should've checked wikipedia first before asking here. Thanks anyway.

Anonymous at Sat, 3 Aug 2024 14:11:16 UTC No. 16308471

Let [math] (X,d_{l^{1}})[/math] be the metric space with [math] X= \{(a_{n})_{n=0}^{\infty}:\sum_{n=0}^{\infty}|a_{n}|<\infty\}[/math] and [math] d_{l^{1}}((a_{n})_{n=0}^{\infty},(b_{n})_{n=0}^{\infty})=\sum_{n=0}^{\infty}|a_{n}-b_{n}| [/math]. Consider the set of sequences [math] E=\{e^{(n)}:n\in \mathbb{N}\} [/math] in [math] X[/math] of the form [math] e^{(n)}=(e^{(n)}_{j})_{j=0}^{\infty}[/math] for [math]n\in\mathbb{N} [/math] such that [math] e^{(n)}_{j}=1 [/math] if [math]n=j [/math] and [math] e^{(n)}_{j}=0 [/math] if [math]n\neq j [/math]. For a given sequence [math] (a_{n})_{n=0}^{\infty}[/math] of sequences in [math] E [/math] we have that [math] d_{l^{1}}(a_{i},a_{j})=2 [/math] for any [math]i\neq j [/math]. So this sequence is not cauchy and hence cannot converge. Then the statement " for any convergent sequence in [math] E [/math] the limit of the sequence also lies in [math] E [/math] " is vacuously true.
Does this seem right?

Anonymous at Sat, 3 Aug 2024 14:29:59 UTC No. 16308487

>>16308471
There are convergent sequences in E. Every sequence of sequence that eventually becomes constant converges.

Anonymous at Sat, 3 Aug 2024 15:24:24 UTC No. 16308547

>>16308487
Yeah I missed that completely, thanks. I'm trying to show that E is closed in X, seemed too good to be true.

Anonymous at Sun, 4 Aug 2024 10:51:30 UTC No. 16309627

Any libgen alternatives? It has become very slow.

Anonymous at Mon, 5 Aug 2024 00:04:59 UTC No. 16310370

>>16304771
Well, any function almost equaling a Riemann integrable function is itself Riemann integrable, due to the other replier's notes.

Anonymous at Mon, 5 Aug 2024 05:50:56 UTC No. 16310615

>>16310370
There is literally a counterexample in the reply itself.

Anonymous at Mon, 5 Aug 2024 07:19:43 UTC No. 16310665

>>16310615
Yes, I'm giving a little more intuition.

Anonymous at Mon, 5 Aug 2024 10:09:40 UTC No. 16310752

>>16310665
Your intuition is a false statement?

Anonymous at Mon, 5 Aug 2024 10:31:45 UTC No. 16310770

>>16310752
The statement in question is whether every Lebesgue integrable [math]f[/math] is equal a.e. to some Riemann-integrable [math]g[/math].
The counterexample proposed uses [math]1_C[/math], which is Lebesgue integrable and not Riemann integrable.
My claim is that if [math]g[/math] is Riemann integrable and [math]f=g[/math] a.e., then [math]f[/math] is Riemann integrable.
[math]1_C[/math] is not Riemann integrable, so I don't see how it serves as a counterexample to my statement.
I can't rule out that I'm profoundly retarded, but I'm not sure where I'm going wrong.

Anonymous at Mon, 5 Aug 2024 10:37:22 UTC No. 16310772

>>16310770
The constant zero function is Riemann integrable.
[math]1_{\mathbb{Q}} = 0 [/math] almost everywhere.
[math]1_{\mathbb{Q}} [/math] is not Riemann integrable.

Anonymous at Mon, 5 Aug 2024 10:56:23 UTC No. 16310792

>>16310772
Alright, I'm just retarded then. I think you need something about the inequality set not being dense.

Anonymous at Mon, 5 Aug 2024 11:00:26 UTC No. 16310795

>>16310792
Mouf. Now.


GIANTS

Anonymous at Mon, 5 Aug 2024 11:40:22 UTC No. 16310823

>>16309627
why would someone make a libgen alternative instead of just contributing to libgen? it's not like it's a competitive for-profit space

Anonymous at Tue, 6 Aug 2024 03:46:49 UTC No. 16311749

summer so slow for math

Anonymous at Tue, 6 Aug 2024 04:23:14 UTC No. 16311781

>>16310795
my daughter saw your dog pic and compelled me to draw this

Anonymous at Tue, 6 Aug 2024 11:35:50 UTC No. 16312086

Why do math people not like talking about the semantics or meanings of math? Is that useless?

Anonymous at Tue, 6 Aug 2024 15:08:30 UTC No. 16312283

What has your experience been with Abebooks?

Anonymous at Tue, 6 Aug 2024 15:24:56 UTC No. 16312296

>>16312283
every cheap book comes from india and they get lost for months, if you're lucky some package will reach your house smelling like dog shit

Anonymous at Tue, 6 Aug 2024 15:26:01 UTC No. 16312298

>>16312296
Okay and if I don't order it from India? Retard?

Anonymous at Tue, 6 Aug 2024 15:55:17 UTC No. 16312316

>>16312298
they ship from india, retard

Anonymous at Tue, 6 Aug 2024 15:57:23 UTC No. 16312317

>>16312316
Why would a seller from US ship a used hardcover from India you dumb nigger? Indians don't even have hardcovers.

Anonymous at Tue, 6 Aug 2024 16:17:08 UTC No. 16312335

>>16312317
you're so dense

🗑️ Anonymous at Tue, 6 Aug 2024 17:31:09 UTC No. 16312385

>>16312283
I think I used them for a analysis book from Royden
and Fitzpatrick. The delivery was rather slow and
as a result they did not charge me for it.

Then, my book came out of the blue
(I really wasn't sure it will come) and was happy for that
--it's a legit Pearson edition as well.
Then, they had emailed me after wondering if the book came, and if so, give the card info over for charging
(about $87.00). I raised my head in the air in thought.
My fingers slid around the phone for a bit. And as soon
as I thought to look for the email again, it disappeared!

Really didn't know what happened if anyone in /mg/asked of me. At least I got a good book for my course.

Anonymous at Tue, 6 Aug 2024 17:36:59 UTC No. 16312391

>>16312283
I used them for a analysis book from Royden
and Fitzpatrick. The delivery was rather slow and
as a result they did not charge me for it.

Then, my book came out of the blue (even I didn't know)
and was happy for that--it's a legit Pearson edition as well.
Then, they had emailed me after wondering if the book came,
and if so, give the card info over for charging (about $87.00).
I raised my head in the air in thought.
My fingers slid around my phone for a bit. And as soon
as I thought to look for the email again, it disappeared!

Really didn't know what happened if anyone in /mg/ asked of me.
At least I got a good book for my course.

Anonymous at Wed, 7 Aug 2024 09:22:57 UTC No. 16313211

Anonymous at Wed, 7 Aug 2024 11:23:08 UTC No. 16313282

Folland is the best book for real analysis.

Anonymous at Wed, 7 Aug 2024 21:59:46 UTC No. 16314161

>>16313282
yes, and it's complemented well by Durrett's probability book

Anonymous at Thu, 8 Aug 2024 01:17:19 UTC No. 16314431

What's an example of a connected non-singleton topological space X such that every continuous real-valued function on X is constant?

Or does such an example not exist? Does anyone know a reference that discusses this?

If such an example does exist, can we also find such an X which is furthermore path-connected, not just connected?

Anonymous at Thu, 8 Aug 2024 10:40:56 UTC No. 16314860

>>16313282
waste of time

Anonymous at Thu, 8 Aug 2024 11:10:07 UTC No. 16314885

>>16289614
is this whole D-module microanalysis stuff actually worth studying? Keep on hearing about how great it is for pde theory but all the books on it seem way too abstract

Anonymous at Thu, 8 Aug 2024 14:28:28 UTC No. 16315103

>>16314860
While true, it is the best book to waste your time.

Anonymous at Thu, 8 Aug 2024 17:07:19 UTC No. 16315364

according to classical logic, math is a separation of statements about the world into true and false, and yet real math is a stunning resolution of a mystery that leaves us amazed with the brilliance of mathematical insight
so how is it then that mathematical statements have real meaning, emotional meaning that elevates us beyond the sense realm sending us flying into outer space
how can this sorting of statements into true and false touch infinite magical beauty

Anonymous at Thu, 8 Aug 2024 17:11:59 UTC No. 16315371

I know learning by reading books is not optimal. But what the fuck is the alternative when I'm not immersed in a mathematical community?

Anonymous at Thu, 8 Aug 2024 17:12:14 UTC No. 16315373

>>16315364
Classical logic is a religion. Mathematics is centered around proof, and as such it is only natural that its logic should be centered around proof. Thus intuitionistic logic is the proper foundation for mathematics.

Anonymous at Thu, 8 Aug 2024 17:18:04 UTC No. 16315379

>>16315364
so, to continue, Tarski, Gödel, Wittgenstein, and Arendt referee the line that separates mystical hypnosis from actual fact, each part of a layer or strata of social adjustment to the ambient background mystic arts as practiced, the taking of people away into fantastic possible worlds
math itself has this strata, with modal logic and more lyric and mythic, less literal meaning or precise meaning used as a milestone of mathematical or marker delineating a vast landscape of both fantasy and fact that is, ultimately, subject to the whims of school administrators, chairs of math departments, tenure policy, and pecuniary interests
indeed, this separation of statements into true and false does create mystic beauty, and it is the curriculum designer and textbook author who must find out how far and deep into the arcane art of calculation the student must go to earn an 'A'

Anonymous at Thu, 8 Aug 2024 17:27:46 UTC No. 16315398

>>16315371
you can upload a picture of your fake homework that you assigned to yourself and let us laugh at you for skipping steps and not understanding things while we deduct fake internet points from your permanent internet fake education record

Anonymous at Thu, 8 Aug 2024 17:40:43 UTC No. 16315419

>>16315373
'foundations of math' is just German and pan-European ZFC hysteria stemming from the misuse of logic as an instrument of blunt power at the height of colonial mania
the blackboard is no foundation, and the chalk only 'builds' in the sense of Dewey's theory of learning
indeed, math is the very essence of world domination
your children will become ZFC-bots, the cruel logic of history dictates this much
foundations of math? more like chains of love

Anonymous at Thu, 8 Aug 2024 17:43:14 UTC No. 16315420

>>16315371
>I know
I don't

Anonymous at Thu, 8 Aug 2024 17:57:01 UTC No. 16315430

>>16314860
you're talking to people who have enough time to assign themselves fake homework but not enough time to grade each other's fake homework
like...just take pictures of the fake homework you assign yourself and upload the pictures to imgur and let us laugh at you
like what are you afraid of...that strangers on the internet will laugh at your fake homework and deduct fake internet points from your permanent internet fake education record
oh my god...you might not graduate summa cum fake laude from fake internet university
your fake internet diploma is at stake
you might not get your fake job

Anonymous at Thu, 8 Aug 2024 18:13:37 UTC No. 16315452

>>16315430
>>16315398

Anonymous at Thu, 8 Aug 2024 18:19:53 UTC No. 16315456

>>16313282
>>16315103
Just read Rudin you hipster.

Anonymous at Thu, 8 Aug 2024 18:22:49 UTC No. 16315459

>>16315452
extend your neck in front of your console so that when I swing my fake internet scimitar, reaching through the internet tubes into my console and out yours, the blade will slice your neck and your head will fall into your lap, infidel

Anonymous at Thu, 8 Aug 2024 23:09:28 UTC No. 16315779

Thoughts on vector spaces over the rational numbers?

Anonymous at Thu, 8 Aug 2024 23:15:12 UTC No. 16315791

>>16315779
No one cares about the rationals anymore.

Anonymous at Fri, 9 Aug 2024 00:09:57 UTC No. 16315893

>>16315456
Read Folland. It will change your life.

Anonymous at Fri, 9 Aug 2024 05:13:11 UTC No. 16316302

I have abandoned research in my PhD to focus on making USA weightlifting open nationals.

Anonymous at Fri, 9 Aug 2024 05:17:00 UTC No. 16316306

>>16308017
What are you writing your thesis on? Idempotents in group algebras are key to my research currently (mainly in K[Sn])

Anonymous at Fri, 9 Aug 2024 05:50:09 UTC No. 16316321

>>16315456
>>16315103
The reason analysis is hard is because the ideas are presented in a way that is isolated and general. The same will go for algebra and topology. The books on those subjects should be read, but not be the center of your study. They are there for reference when needed.
I studied analysis and ignored everything else and it failed me. I then changed my approach, got a book on differential geometry (sternberg) and began to read that, referencing books on analysis, algebra, geometry and topology when it was necessary. Math is a slow process and the time you put in will significantly outweigh the knowledge you gain.
Also a little side note, just read complex analysis. Both (real and complex) deal with the same ideas, just complex analysis is more tangible due to its geometric nature. I recommend Ahlfors book.

Anonymous at Fri, 9 Aug 2024 07:34:58 UTC No. 16316454

If I have a graph with iid edges, how can I show that the probability of there being an infinite cluster is 0 or 1? In particular, how do I show it is a tail event.

Anonymous at Fri, 9 Aug 2024 13:57:53 UTC No. 16316840

>>16316321
>Both (real and complex) deal with the same ideas
No they don't. Also every Complex analysis book assumes Real analysis. In fact, real analysis isn't even restricted to real numbers. Real analysis also talks about complex functions but not the same topics that are discussed in Complex analysis. Real analysis talks about sequences, series, measurability, continuity, differentiation, etc., while Complex analysis talks about Holomorphy.

Anonymous at Fri, 9 Aug 2024 15:31:15 UTC No. 16317038

>>16289614
Aside from quant research jobs

What are the best ways to make money with applied mathematics?

What sort of business could one make with little capital?

Anonymous at Fri, 9 Aug 2024 15:37:25 UTC No. 16317046

>>16316840
Ahlfors doesn't. Complex analysis is also much more useful (and not just confined to holomorphy lmao). There's a reason shit like GMT is a historic subject nowadays that went nowhere

Anonymous at Fri, 9 Aug 2024 20:23:10 UTC No. 16317547

>>16316321
>>16317046
There is a common core to real and complex analysis, sure. But no, basic properties complex numbers, complex functions, complex sequences and series etc. are not a part of real analysis per se, it's more like "fundamental analysis" but it's just convenient to treat it in part of a "real analysis" course because that's what people usually study first. Complex analysis is very important (and I heartily agree with your recommendation of Ahlfors), but to ignore real analysis results in a narrow perspective. Analytic functions are very rigid and thus many interesting phenomena of real analysis don't happen in the complex domain. For example, test functions can't exist because of the identity theorem and so there isn't a theory of distributions in complex analysis. And geometric measure theory is not dead, to say that would be just as ignorant if I said value distribution theory or quasiconformal mapping theory is dead.