Thread 16777973

Analytic continuation edition

Formerly >>16767261

sweaty, the proper term is meromorphic continuation
"analytic" is an outdated term from the XIX century

Does anyone know what Hartshorne means by [math] \mathscr{F}(d) [/math] in this proof? (See the 4th line of the proof.)

Sorry if this is a basic question, I haven't followed Hartshorne from the beginning so I don't know all of his notation.

anal lysis is aptly named, as it wrecks your pooper

>>16778271
>anal
>ysis
>isis
It terrorizes your pooper.

>>16778087
im still wondering where you even find the definitions for everything in that picture

>>16778087
[math] \mathscr{F} [/math] twisted by [math] d, [/math] i.e. [math] \mathscr{F}\otimes_{\mathscr{O}}\mathscr{O}(d) [/math]

>>16778087
>vomit

does the time evolution of the complex plane form a complex space?

>>16778604
Thanks very much anon.

>>16778651
Lol, out of curiosity which part made you vomit?

Hey em gee, let's collaborate on this problem, I think it's interesting and deceptively subtle.

Let [math] G=(V, E) [/math] be a finite simple graph, and let [math] \mathcal{O} [/math] be the set of acyclic orientations of its edges. Define the weight of [math] O\in\mathcal{O} [/math] as
[eqn] w(O)=\displaystyle\prod_{v\in V}\frac{1}{1+\text{deg}^+_O(v)}, [/eqn]
where [math] \text{deg}^+_O(v) [/math] is the outdegree of [math] v [/math] in [math] O [/math]. Let [math] S(G)=\displaystyle\sum_{O\in\mathcal{O}}w(O) [/math]. Conjecture: [math] S(G)\ge 1 [/math] for all [math] G [/math].

A couple observations to start:
- [math] S [/math] is clearly multiplicative over disjoint unions, so we can assume [math] G [/math] is connected.
- Equality holds when [math] G [/math] is a disjoint union of cliques.
- In particular, [math] S [/math] can't be generally monotone (in either direction) under the insertion or removal of edges. In fact, apart from adding a leaf which it's easy to see increases [math] S [/math], I don't see any other basic but non-trivial graph operations under which [math] S [/math] is monotone.
- An interesting approach is to try showing that [math] w(O)\ge\frac{L(O)}{|V|!} [/math], where [math] L(O) [/math] is the number of linear extensions of [math] O [/math]. This suggests a tempting probabilistic interpretation where you assign a random total ordering to [math] V [/math] and each term in [math] w(O) [/math] is the probability that that vertex is smaller than its out-neighbors, and it would suffice to show that these events are negatively correlated... but this is actually false in general. For example, in [math] P_3 [/math] with the middle vertex a sink, we have [math] w(O)=\frac{1}{4} [/math] but [math] \frac{L(O)}{3!}=\frac{1}{3} [/math]. So as nice as this approach is I'm not sure it's salvageable.

I heckin love linear algebra!

>>16779202
Let 0.999... be defined by an infinite sequence:
[eqn] r = \{ 0.9, 0.99, 0.999, \cdots \} \stackrel{\text{def}}{=} \left\{ \sum_{i=1}^{N} \frac{9}{10^i} \right\}_N [/eqn]
To say [math] 0.999... = 1 [/math], it means for any positive rational [math] \epsilon > 0 [/math] there exists a natural number [math] n \in \{1, 2, 3,\cdots\} [/math] such that [math] |r_n - 1| < \epsilon [/math].
If [math] 0.999... \neq 1 [/math], there would be some positive rational [math] \delta > 0 [/math] such that for all natural numbers [math] n [/math], [math] |r_n - 1| \nless \delta [/math], i.e.
[eqn] \begin{align} \exists 0 < \delta \in \mathbf{Q} \rightarrow \forall n \in \{1, 2, 3, \cdots\} &\rightarrow |r_n - 1| \geq \delta \\ &\rightarrow (1 - r_n) \geq \delta \end{align} [/eqn]
/mg/ please help me complete the proof so I can convince the popsci folks real numbers are real...

What kind of math should I take courses in if I'm interested in ecology? I presume stats.

>>16780839
Differential equations, statistics, Markov chains and stochastic processes. A pretty surprising amount of the math taught in modern curricula were developed to solve ecology and population modeling problems. It's sad that most ecology programs don't actually teach their students any of the math parts of their discipline when so many key contributions in statistics and differential equations were made by ecologists.

>>16779001
>which part made you vomit?
Everything in that vomit of symbols on the page.

>>16779397
>it means for any positive rational [math]\epsilon > 0[/math] there exists a natural number [math]n \in \{1,2,3, \cdots\}[/math] such that [math]|r_n - 1| < \epsilon [/math]
It's not enough to prove a single item is arbitrarily close to the proposed limit. You need to prove all items beyond a certain point are within that arbitrarily tight margin. If [math]r_200=1[/math] and the sequence converged on 1.01, then for any [math]\epsilon[/math] you could choose [math]n=200[/math] to meet the condition (but the sequence wouldn't converge on 1)

You need to introduce another variable:
[math]0.999\dots = 1[/math] iff for any positive rational [math]\epsilon > 0[/math] there exists a natural number [math]N \in \{1,2,3, \cdots\}[/math] such that for all [math] n \geq N [/math], we have [math] |r_n - 1| < \epsilon [/math]
Now we prove there's an N for any [math]\epsilon [/math]. Consider a natural number [math]d[/math] such that [math] 10^{-d} < \epsilon [/math]. We set [math] N = d [/math], and let [math] n \ge N = d[/math].
[math] |r_n - 1| = 10^{-n} \le 10^{-d} < \epsilon. \Box[/math]

I like to visualise this back and forth between epsilon and delta/N as a game or a flurry of endless one-upmanship

Dumping this stuff I made for a /pol/ post that got archived. It's the conditional probability question about that gold and silver ball.
I think conditional probabilities are too high for a suffrage test.
It's interesting how in the maths, it's the mixed case that gets halved (rather than the double gold case that gets doubled when you reason through it intuitively).

Did anyone study Boolean algebra? How much math do you need to know beforehand to get into it? Which book to read?

>>16781177
> How much math do you need to know beforehand to get into it?

Not much math at all. All you really need is some basic experience reading and writing proofs. A couple of weeks of self studying proof-based discrete math will do the trick.

> Which book to read?

That depends on your purpose. If you want to understand Boolean functions and more of the applications side of things, a digital logic textbook (E.g., Fundamentals of Logic Design by Roth and Kinney) would probably be a good place to start.

If you want to understand Boolean algebras from a formal pure mathematics perspective, Halmos' Introduction to Boolean Algebras is a fantastic choice.

>>16781187
>A couple of weeks of self studying proof-based discrete math will do the trick.
Then I ask the same question again, how much math do you need to know beforehand to get into that, and which book to read? Also "a couple of weeks of self studying" is very vague, as it can vary a lot how much that is depending on how much you read per day. It would be better if you said something more specific like a certain book.

>>16781167
Fun thread. I'm reading it now. I said 50%. Did I get it right? Also your pic is horribly grainy, but I don't know how to read or understand it anyway.

>I think conditional probabilities are too high for a suffrage test.
I don't understand this sentence.

What's the linguistic ambiguity?
>>>/pol/515244069

>>16781196
I don't think there is a linguistic ambiguity. Everybody knows to rule out the double silver box. The trip hazard is not an ambiguity but a subtlety. The subtle fact that the probability distribution between the boxes changes from uniform to 66% to 33%.
I think the average person (understandably) has a poor enough understanding of probability that they don't consider that a distribution might change shape (rather than just merely cut down on support) when some information is known.

>>16781200
>subtlety
empty word, cope for not actually understanding/being able to explain why it's 1/3 rather than 1/2

>>16781196
When you pull out a gold ball for the first time, you know you're working with either box #1 or box #2. The "trick" is in the fact that while you're equally likely to pull from each box, you're not equally likely to get a gold ball from each box. There are 3 cases where you pull out a gold ball, and only 1 of them involves you pulling it from box #2, so the chance that the one you've pulled is from box #2 - and thus the chance that the other is silver - is only 1/3

>>16781209
If you pulled a gold ball the first time, then you know that the box you're holding either contains nothing but gold, or nothing but silver.

>>16781211
Yes.
But there are two cases where it contains nothing but gold, and one case where it contains nothing but silver.

>>16781215
I agree with the guy who said
>Its not asking waht the probability of drawing gold twice in a row is.
The way you reason, and the others who think like you, doesn't take into account the new information you have after drawing a gold ball. I'm focusing on figuring out exactly what the linguistic ambiguity is, because I think he's right that there is one, and you fell for it, this is not so much a question of math as it is a question of language, and that's why I think it's interesting.

>>16781223
Perhaps the question could be worded to highlight the tripping hazard and hold your hand a bit more.
"You pick a box at random and pull out one ball after the other. What is the probability that the second ball is silver, given that the first ball was gold?" Not sure if this is clearer to a layperson.
Regardless even if there is room for improvement, I wouldn't say it's ambiguous because there is only one interpretation to a mathematically proficient person. It's unhelpfully worded or at the very worst, adversarily worded to misdirect your attention.

>>16781223
You might want to reread what I'm saying; I'm laying out exactly the reasoning that you're suggesting that I'm not

>>16781227
Well, I think your math knowledge is what's tripping you up. This is not a math question, it's a language question, and it's deliberately worded the way it is in order to trip up those who know about probability theory. You have to shift your way of thinking about the problem. I've seen this type of questions before, they have people thinking too deeply, and then when they finally get it they go "oh it's one of those dumb questions". Off the top of my head I'm thinking of picrel. People read it and think hard about it, but overlook the wording, "contract" vs "harmed by", which is what the whole question is about. This is that type of question. I could be wrong, but that's my take.

>>16781228
I don't understand what you mean.

Anyway, to both of you, one poster said this.
>One box has 1000 gold balls. The box next to it has 1 gold ball and 999 silver balls.
You just picked a gold ball. Chances are your hand is in the first box, right?
I think this is how people are thinking who think it's 1/3. But it's wrong because it's looking at it from the perspective of what is the probability that you have the first box, but that doesn't matter here. If you KNOW that you have either a box with 999 gold balls in it, or a box with 999 silver balls in it, then the probability of drawing a silver ball IN THAT SCENARIO is 1/2, regardless of what the probability was before you drew the first ball.

>>16781228
See what I said above, it's your math knowledge tripping you up, it's actually a "dumb" language question.

>>16781263
>You just picked a gold ball. Chances are your hand is in the first box, right?
>I think this is how people are thinking who think it's 1/3. But it's wrong because it's looking at it from the perspective of what is the probability that you have the first box, but that doesn't matter here. If you KNOW that you have either a box with 999 gold balls in it, or a box with 999 silver balls in it, then the probability of drawing a silver ball IN THAT SCENARIO is 1/2, regardless of what the probability was before you drew the first ball.
that's literally exactly how you should be thinking about it lmao
Yes, it absolutely matters that there's an overwhelming chance that you're looking at the first box. The question being asked - the odds that your next ball will be silver - are 100% if you're looking at the one-gold box and 0% if you're looking at the all-gold box. It's only 50/50 if you're equally likely to be looking at the two boxes - but you aren't, and you've already acknowledged why.

Try putting it into Bayes' theorem if you don't believe it. It's not 50/50.

>>16781263
>I've seen this type of questions before, they have
I guess I should have said "it has", since "type of questions" is singular. I proofread it but still missed that.

>>16781280
Well, don't know if you're the same person but what about this possible aspect of it?
>It's unhelpfully worded or at the very worst, adversarily worded to misdirect your attention.

>>16781227
>It's unhelpfully worded or at the very worst, adversarily worded to misdirect your attention.
>even if there is room for improvement, I wouldn't say it's ambiguous because there is only one interpretation to a mathematically proficient person
I think you're contradicting yourself here.

>>16781310
Here's a python simulation. If you disagree with the way runs have been counted up, please suggest how you would adjust the code.
I think this makes it a lot more clear that the primary thing at play is half of the GS picks getting thrown out.
>In 10000 runs, the first ball was was gold for 5115 runs and the second ball was subsequently silver for 1713 runs.
>Thus P(second silver | first gold) = 0.3349
>The outcomes are like so: {'GG': 3402, 'GS': 1713, 'SG': 1623, 'SS': 3262}
[code]
from random import choice, sample

run_count = 10000
boxes = [['G', 'G'], ['G', 'S'], ['S', 'S']]

first_ball_is_gold = 0
second_ball_is_subsequently_silver = 0
outcomes = {'GG':0, 'GS':0, 'SG':0, 'SS':0}
for _ in range(run_count):
chosen_box = choice(boxes)
shuffled_box = sample(chosen_box, 2)
first_ball = shuffled_box[0]
if first_ball == 'G':
first_ball_is_gold += 1
second_ball = shuffled_box[1]
if second_ball == 'S':
second_ball_is_subsequently_silver += 1

outcomes[''.join(shuffled_box)] += 1

print(f"""In {run_count} runs, the first ball was was gold for {first_ball_is_gold} runs and the second ball was subsequently silver for {second_ball_is_subsequently_silver} runs.
Thus P(second silver | first gold) = {second_ball_is_subsequently_silver/first_ball_is_gold:>.4}""")
print(f"The (first_ball, seocnd_ball) pairs ended up like so: {outcomes}")
[/code]

>>16781280
Forget the question in that thread for a moment. Let's say I were to say this to you:

>I'm holding two boxes. You can't tell them apart, but one of them contains 999 gold balls, and the other contains 999 silver balls. Pick one box randomly and draw a ball from it. What are the odds you will draw a silver ball?

What would you reply?

Reposting because I replied to the wrong post.
>>16781167
Here's a python simulation. If you disagree with the way runs have been counted up, please suggest how you would adjust the code.
I think this makes it a lot more clear that the primary thing at play is half of the GS picks getting thrown out.
>In 10000 runs, the first ball was was gold for 5115 runs and the second ball was subsequently silver for 1713 runs.
>Thus P(second silver | first gold) = 0.3349
>The outcomes are like so: {'GG': 3402, 'GS': 1713, 'SG': 1623, 'SS': 3262}
from random import choice, sample
run_count = 10000
boxes = [['G', 'G'], ['G', 'S'], ['S', 'S']]
first_ball_is_gold = 0
second_ball_is_subsequently_silver = 0
outcomes = {'GG':0, 'GS':0, 'SG':0, 'SS':0}
for _ in range(run_count):
. chosen_box = choice(boxes)
. shuffled_box = sample(chosen_box, 2)
. first_ball = shuffled_box[0]
. if first_ball == 'G':
. first_ball_is_gold += 1
. second_ball = shuffled_box[1]
. if second_ball == 'S':
. second_ball_is_subsequently_silver += 1
. outcomes[''.join(shuffled_box)] += 1
print(f"""In {run_count} runs, the first ball was was gold for {first_ball_is_gold} runs and the second ball was subsequently silver for {second_ball_is_subsequently_silver} runs.
Thus P(second silver | first gold) = {second_ball_is_subsequently_silver/first_ball_is_gold:>.4}""")
print(f"The (first_ball, seocnd_ball) pairs ended up like so: {outcomes}")

>>16781315
damn the indentation keeps getting messed up

>>16781313
In this setup it really is 50/50, because it is equally likely that the box I pick from is all-silver as it is all-gold.
This is not the case for the other question.

>>16781333
Then it's a matter of WHERE you calculate FROM, which is my point, that it's a matter of semantics.

>>16781339
And the original question very clearly outlines where you're calculating from. There is a very big difference between "what is the probability of B?" and "what is the probability of B given A?" and it's blatantly wrong to suggest that that distinction is just "semantics" - as we've seen, it changes the answer by a very significant amount

>>16781315
I just ran it with 40 golds in the all-gold box and 40 silvers in the all-silver box and I got the same results/outcomes. So it's not that the all-gold box is twice or 40 times more likely, but that the mixed box is half as likely.
If the mixed box has m golds and n silvers, then [math] \mathbb P(\text{Gold then Silver}) = \frac{m}{m+n}\cdot \frac{n}{m+n-1}[/math]
Here's the graph of it. I was surprised it's monotonic radially. I thought if you added more gold the mixed box becomes more likely (it does) but then in turn it's less likely to spit out a silver!
Very interesting

>>16781344
In the end it boils down to how you interpret words anyway. It's not an IQ test as the OP was implying, because what you put into words is not innate, it's learned.

>>16781353
eh verbal reasoning/vocab does form a decent chunk of IQ tests though
But I guess that's a result of not being able to test the g-factor directly and thus having to probe at it from multiple, synthetic angles.

>>16781367
I only took one IQ test and it was only visual patterns. That question is not an IQ test. People in that thread who are insulted are proving right whoever thinks whatever they answered shows their low IQ, not by the way they answered but by the fact they think it's a measure of intelligence, and by the fact they're insulted, regardless of what they answered.

>>16781381
Yeah I guess
>took an IQ test and it was only visual patterns
The WASI is a widely used IQ test. In addition to visual patterns, it has memory tasks and verbal tasks. WASI-V has the 'Similarities' and 'Vocabulary' tasks. It seems that the 'Information' and (supplementary) 'Comprehension' tasks in WASI-IV, got removed. I guess comprehension is harder to standardise than a vocab test or smth?

>>16780883
Since it made you vomit, do you have a preferred alternative book to Hartshorne on introductory algebraic geometry?

>>16780874
Thanks, I turned down grad school to go back and study applied math because it seemed I'd be useless for anything but bashing R scripts together if I continued on the normal ecology path.

>>16780874
>many contributions in DEs from ecologists
wow I wouldn't have expected that
nice

>>16781515
Görtz-Wedhorn

>>16781191
No reply, so to clarify:
>Then I ask the same question again, how much math do you need to know beforehand to get into that, and which book to read?
In this sentence "that" is referring to "A couple of weeks of self studying proof-based discrete math". In other words, how much math do you need to know beforehand to get into the proof-based discrete math in question? And when I said in the quoted sentence "which book to read" I'm asking about which book to read for this proof-based discrete math. Basically I studied math in high school, but it's many years ago, and I feel like I've forgotten almost all math. Can I read these below?
>a digital logic textbook (E.g., Fundamentals of Logic Design by Roth and Kinney)
>Halmos' Introduction to Boolean Algebras

Which book is recommended for proof-based discrete math? I have seen these two books, are they this kind of book and are they good?

>How to Prove it: A Structured Approach by Daniel Velleman
>Book of Proof by Richard Hammack

>>16781603
Predator-prey models were a huge source of 20th century results in both theoretical and numerical differential equations. While the original Lotka model was more relevant to organic chemistry, a lot of these types of models were developed and improved upon to solve ecological population modeling problems. In particular, a huge amount of the non-linear mutualism equations that get used in mathematical finance originally were developed for ecological population dynamics.

>>16781191
>>16781975
Sorry, I was at work and not on the thread.

I think you could start from essentially no formal math education on most of the recommended discrete math textbooks. Some comfort with high school level algebra will be helpful (really more to develop your patience with yourself as you work through parts you don't understand).

As far as what discrete math textbook, basically everyone recommends one of three. Susanna Epp's Discrete Mathematics with Applications, Rosen's Discrete Mathematics and It's Applications, or Grimaldi's Discrete and Combinatorial Mathematics.

None of these explicitly require any higher level math for the vast majority of their content. Grimaldi and Rosen both have sections on introductory Boolean algebras/Boolean functions at the end.

Having experience with calculus "will help" in the "I've spent some amount of time banging my head against things I've found difficult before" sense. You won't be calculating integrals or derivatives or any of that for discrete math books (for the most part).

>>16781975
> Which book is recommended for proof-based discrete math? I have seen these two books, are they this kind of book and are they good?

>How to Prove it: A Structured Approach by Daniel Velleman
>Book of Proof by Richard Hammack

Both of those books are fantastic. I also really like Jay Cummings Proofs book. It's not as highly regarded as the other two, but it's free on his website and covers all the same basic topics.

>>16781984
> but it's free on his website and covers all the same basic topics.

Was free on his website. Scratch that.

>>16781227
>Regardless even if there is room for improvement, I wouldn't say it's ambiguous because there is only one interpretation to a mathematically proficient person. It's unhelpfully worded or at the very worst, adversarily worded to misdirect your attention.
We didn't get into this further. As I said I think you're contradicting yourself. Anyway I think it's interesting this thing about "given that".
>>16781227
>given that the first ball was gold
>>16781344
>There is a very big difference between "what is the probability of B?" and "what is the probability of B given A?"
As I said earlier I insist that it is a matter of semantics, but whether it's "just" semantics... it's debatable what "just" even means in this case, and whether it is "just semantics" or not by whatever definition.

Anyway, I just think this matter of "given that" is interesting. The significance that you're putting into it is a little unfamiliar to me and seems to be a rather specialized language for math or probability. The way I thought about "given that" was as stating a premise. I sometimes interchangeably use the words/phrases "premise", "a given", "something given", "it is given that", etc, "given" there being either a noun or a verb but really pointing to the same thing. So I read it as "this is what we have", "these are the conditions", "these are the facts" etc. In other words just presenting the situation that we have to consider, without this having reverberations into the deductive structure etc. Think of it as Sherlock Holmes being presented with a boot with a scratch mark and a glass with an inch of water in it, etc, just these things in themselves, not any of the inferences.

>>16781984
>>How to Prove it: A Structured Approach by Daniel Velleman
>>Book of Proof by Richard Hammack
Are these books about discrete math?

>>16781992
> Are these books about discrete math?

No, they are proofs books. Discrete math involves a lot of proofs (and is one of the ways that a lot of people are first exposed to proof-based mathematics), but proofs books also include a lot of things you won't see in discrete math textbooks (and vice versa). There's overlap for sure, but they are different.

Proofs books are generally preparation for real analysis and abstract algebra.

Discrete mathematics is generally preparation for graph theory, theory of computation, boolean/discrete algebras or algorithms analysis.

They cover a lot of the same topics and ideas, but are facing different directions while doing so (if that makes sense).

>>16781999
So if I want to try reading Boolean algebra, do you still suggest preparing by reading one of these
>Susanna Epp's Discrete Mathematics with Applications
>Rosen's Discrete Mathematics and It's Applications
>Grimaldi's Discrete and Combinatorial Mathematics
rather than one of these
>How to Prove it: A Structured Approach by Daniel Velleman
>Book of Proof by Richard Hammack
>Jay Cummings Proofs book
?

.

>>16781991
'given that' indicates that conditional probability shenanigans is going on. However, the core of my point is that the problem's nature is more apparent if this disclosure (that the first ball is gold) was placed after the question than before.
(Let's just take the idea that "the first ball being gold is critical information" as a premise)
By placing the disclosure before, the reader might be more eager to group it with the other exposition and toss it away when abstracting the word problem down into mathematical form, whereas placing it afterwards (and separating it from the exposition), would cause the reader to deliberate over it more when deciding if it is relevant.
i.e. [math]\text{[exposition][question][critical info]}[/math] reveals the solution more than [math]\text{[exposition][critical info][question]}[/math]

We can take this line of thought further and note that appending "Use the definition of P(X|Y)" clarifies the problem even more.

>>16782026
Both discrete math and introductory proofing textbooks provide similar background in proofs in their problem solving strategies. Discrete math will provide more relevant background material for Boolean algebras specifically.

If you have other mathematics interests besides Boolean algebra, I'd say maybe try both. If you're only interested in the fastest possible path to Boolean algebra, skip the proofs book and just learn the proofs as you're learning discrete math.

>>16779397
[math] \displaystyle
\boxed{0 < p < 1} \\
p^n-1 = (p-1)(p^{n-1}+p^{n-2}+ \dots +p+1) \\
\dfrac{p^n-1}{p-1} = \sum \limits_{j=0}^{n-1}p^j \\
\displaystyle
\lim_{n \to \infty} \dfrac{p^n-1}{p-1} = \lim_{n \to \infty} \sum \limits_{j=0}^{n-1}p^j \\
\displaystyle
\dfrac{0-1}{p-1} = \sum \limits_{j=0}^{\infty}p^j \implies \dfrac{1}{1-p} = \sum \limits_{j=0}^{\infty}p^j
[/math]

>>16782661
I don't know about that stuff but my opinion is that this way that you're interpreting "given that" is a specialized mathematics usage of that term, and people get it wrong because they interpret "given that" as any other premise, and with the latter interpretation it doesn't have the same reverberations for the whole thing. Therefore it is a language problem first and foremost, and not a mathematics problem.

>>16783049
Have you read Euclid's Elements?

Fuck teachers fuck numbers they racist nikka
Summers not over light UUUUUUUUUUPPPPP

>The topology generated by the subbasis [math]S[/math] is defined to be the collection [math]\tau[/math] of all unions of finite intersections of elements of [math]S[/math]
How can I describe this symbolically?

so the whole basis of mathematics is less than 150 years old
this means I still have a chance to catch up

>>16777973 (OP)
what are groups ESSENTIALLY and where are they even used aside from QFT?

>>16783707
>what are groups
https://en.wikipedia.org/wiki/Group_(mathematics)

>where are they even used
https://en.wikipedia.org/wiki/Group_(mathematics)#Examples_and_applications

>>16783707
There's a 3b1b video that motivates and describes them. https://www.youtube.com/watch?v=mH0oCDa74tE
Dunno how they fit into qft, but the video linked motivates them as a tool to describe the symmetries of an object, and then extends them to cases where a symmetry isn't apparent.

Here's an alternative, more definitional viewpoint. Picrel are a handful of algebraic structures. An algebraic structure is a pair of a set of values and an operator (like how a graph is defined as (V,E) pair where V is a vertex set and E an edge set).
The magma is a barebones algebraic structure with scant properties. No axioms, no building blocks to make theorems. We can impose more constraints, such as associativity and the existence of an identity value and we end up with the definitions of more restrictive structures.
Groups have the inverse, identity, and associativity axioms. Turns out this set of axioms is enough to make quite a wide library of theorems.

An example of a group is the set of 3d rotations. I think some guy proved that this group matches the group of quaternion multiplication, so now animation software can use quaternions to compute the otherwise challenging 3d rotation composition. This leads to good rotation interpolation and avoids gimbal lock etc.
Cryptography uses groups. Concretely, RSA relies on the idea that, for a given number N, there is an m, such that for all values x (this is the message you send), [math]x^m \equiv x \text{mod} N[/math]. The existence of m is a result of group theory (or number theory i guess). *
There is a related structure, Rings, which are like groups except they have two operators not one. I think those are used more widely.
I don't think groups they get much use in practical settings generally though.
* Side note: Pick a large m first, derive an N from that (which will be even bigger), then tell a friend to send a partially exponentiated version of your message, and privately finish off the exponentiation to decode it.

>>16783707
If your background is physics then you should know that what physicists call a "group" is more specifically what mathematicians would call a "Lie group," and the term "group theory" can mean two quite different things depending on which one you identify as.

>>16783150
> Have you read Euclid's Elements?

Cover to cover? No. That's one of those books I want to eventually get around to, but I don't see much reason to study Euclid's Elements in particular.

If you're learning math for solving practical problems (rather than purely for the study of logic for its own sake), having coordinates is quite helpful.

>>16783784
I was thinking because you said one should study proofs as preparation for Boolean algebra, and Euclid's Elements is about proofs.

>>16783937
> I was thinking because you said one should study proofs as preparation for Boolean algebra, and Euclid's Elements is about proofs.

It might not hurt, but they are very different kinds of "proof."

Boolean algebra is about the combination and manipulation of Boolean (true or false) variables. These kinds of proofs are much more closely related to ideas of set membership (think: "does this object fit into the set of objects?") rather than proofs of relative relationships constructed from postulates.

These are both "proofs" but they use different kinds of logical relationships to construct their arguments.

>>16781167
g1 --> g2
g2 --> g1
g3 --> s

2/3

pack it up, AGI is finally here

>>16784547
I actually did ask it prove a very specific problem, which I couldn't find any mention of online, and surprisingly, it did manage to prove it.

https://chatgpt.com/share/68c5c12f-6d84-8006-9429-e7a5fe1731f3

>>16784567
I've played around with giving it a bunch of math problems and it's actually quite clever in general, bar the occasional bout of severe retardation like >>16784547
It's a bit tricky even to find problems with "quick" proofs that it can't solve, at least ones you'd expect a competent human mathematician to be able to knock out in an hour or so.

>>16784011
>"proof"
>"proofs"
Why the quotation marks?

>>16784740
I just think it's an ambiguous/non-specific term. A lot of different mathematics books use proof strategies in their instruction, but there's quite a few different strategies to proofing. Geometric proofs and analytic proofs are both proofs, but having practice with one doesn't necessarily make you better at the other etc. There are many "transition to advanced mathematics" books that focus almost entirely on teaching proof strategies, but they are very focused on the specific types of proofs a student will need for real/complex analysis, point-set topology, and algebra.

They are fairly distinct from the approaches to proof used in graph theory, non-analytic forms of geometry, and pre-WW2 approaches to combinatorics.

>>16783707
Groups motivate buildings, and buildings have galleries and apartments and so on.

Is there a sequence of sets A1, A2, A3, … etc. such that,
(1) An is not empty for any n
(2) A(n+1) is a subset of An for every n
(3) the intersection of all these sets is empty
?

>>16784853
[math] A_n=(0,\frac1n). [/math]

>>16784854
Ok I see. You can pick any point in some An and prove there is another Am which it is not in. Thanks

>>16777973 (OP)
I've been teaching myself group theory this past 1-2 months out of boredom. I thought about drawing diagrams to help my visual intuition on basic axioms, properties and theorems. Here's one I made.

>>16784954
Get mogged.

>>16784954
subgroups
>>16784967
Category theorists miss the point(object)

>>16785024
Any centralizer of 'a' is subgroup of G
The zentrum of G is subgroup of G

I've been thinking about Bertrand's box paradox. I'm the guy who talked about it being a semantic issue. I think I might have been wrong. And I think I kind of figured out what was wrong in my thinking. I was thinking that because I'm holding a gold ball in my hand, therefore I know that my next draw cannot be from the box which has two silver balls, but rather is going to be from either of the other two boxes.

My thinking is that the fault is in thinking in terms of having two boxes, and thinking you're going to be drawing from either the first or the second of these two. You don't have two boxes. You only have the one box you picked up. Therefore you can't think in terms of two boxes at all, you have to think only in terms of the balls that are in the two boxes which you know your box is one of. So you originally had 3 gold balls and 3 silver balls, and you were equally likely to draw any of these 6 balls. But what you know given the draw of a gold ball, and given that you have to draw the next ball from the same box, is that you drew your ball from a total of 4 balls, out of which 3 were gold and 1 silver, and that your next draw is going to be from a total of 3 balls out of which 2 are gold and 1 is silver.

>>16785024
All subgroups overlap and have subgroups.

>>16785044
When I say 'overlap' I mean more elements than just e. Also some groups don't have subgroups.

>>16785073
>Also some groups don't have subgroups.
False. Every group has the trivial group as a subgroup (and every group except for the trivial group has a second subgroup in itself).

>>16785078
>>16785024
How about this? I think I have expressed better what I wanted to tell

>>16784967
Is ι the morphism that takes an element to its inverse? That's the best I can tell from e: 1G

>>16784966
For more information, visit:
>>16784960

>>16785310
Yes.

I have two mathematical problems that I came up with and was unable to solve.

Problem 1)
If you roll a dice N times, what is the probability of getting a row of at least K consecutive sixes? Of course given that K < N.

Problem 2)
If you roll a dice N times, what is the average length of the maximum run of consecutive same digits? For example, if you rolled the dice a million times, and then looked up what the longest sequence of any consecutive digits was, how long would you expect that to be and what would it be on average for any number of dice rolls generally? Just to be clear here I'm talking about a consecutive sequence of any digit, not just sixes.

Zeta[2] = Integrate[1/(1 – a b), {a, 0, 1}, {b, 0, 1}]

Zeta[3] = Integrate[1/(1 – a b c), {a, 0, 1}, {b, 0, 1}, {c, 0, 1}]

Zeta[4] = Integrate[1/(1 – a b c d), {a, 0, 1}, {b, 0, 1}, {c, 0, 1}, {d, 0, 1}]

Zeta[5] = Integrate[1/(1 – a b c d e), {a, 0, 1}, {b, 0, 1}, {c, 0, 1}, {d, 0, 1}, {e, 0, 1}]

et cetera

>>16785573
>Zeta[3] = Integrate[1/(1 – a b c), {a, 0, 1}, {b, 0, 1}, {c, 0, 1}]

I'm having a hard time understanding peano's 5th axiom
I understand that without it, you could have junk elements in the set, but I don't see how the axiom explicitly excludes them, is it just based on the conclusion that it will always be the smallest subset of natural numbers possible, therefore the set without junk?

>>16785783
it's almost like a recursive definition which I think is what is making it hard for me to understand

>>16785783
>junk elements
>without junk

>>16785854
if you define N as 0,1,2... and a,b,c... then the subset of N without a,b,c... is equal to N as it satisfies the 5th axiom

>>16785783
It is helpful to visualize what happens to the possible models of your theory once you drop induction.

In picrel you can see the standard model of the naturals and two other models for PA minus induction. The designated zero of each model is underlined, the successor function is visualized by arrows. In all three models the designated zero is a non-successor (there is no incoming arrow to it) and the successor function is injective (there's at most one incoming arrow to each number).
These two axioms alone already force your models to become quite large in the sense that they must contain at least one copy of the naturals. Notice how you can't have a model that is just zero mapping to itself, or zero mapping to one and one mapping back to zero, or zero mapping to one and one mapping to itself or really any other attempt at a finite model like that.

Adopting induction in addition to those other two axioms makes sure that the process above stops without adding any "junk" (as you said) to the model. This leaves you with a unique (up to unique iso) model of the naturals, namely [math]\mathbb{N}[/math]!
(This isn't quite true due to the existence of nonstandard models for first-order theories of arithmetic. This answer is only concerning "full" models, i.e. written from the PoV of full second-order semantics. You can ignore this remark if you're not familiar with this stuff, or can read up one some of the details on Wikipedia's or nLab's page on second-order arithmetic).

part 1/2, this got longer than i expected

>>16786016
Dropping induction and you may end up with models like [math]\mathbb{N}+1[/math], which is essentially one copy of the naturals together with an isolated element that is its own successor. This model satisfies all axioms but induction, since the copy of [math]\mathbb{N}[/math] inside of [math]\mathbb{N}+1[/math] contains zero and is closed under successor, yet its a proper subset of our model, i.e. it isn't all of [math]\mathbb{N}+1[/math]. Notice how in this model the successor function actually has a fixed point, namely the isolated element. Whereas PA (with induction) proves that the successor function has no fixed point.

Similarly, the model [math]\mathbb{N}+\mathbb{N}[/math] also satisfies all axioms but induction. It is essentially two copies of the naturals (say a blue copy and a red copy). Since a model of arithmetic has only one designated zero one has to choose one of the two copies of zero to act as that designated zero of the model (the blue one in picrel's case). To see why induction fails, notice how the blue copy of the naturals contain the designated zero (the blue zero) and is closed under successor, yet it is not the entirety of [math]\mathbb{N}+\mathbb{N}[/math]. Regular PA proves that every natural is either a successor or equal to the designated zero. But notice how the red zero is neither a successor nor is it equal to the designated zero (the blue zero was chosen for that role in our model). This model is also a classical example to show that strong induction does not imply (weak) induction, despite a lot of elementary textbook accounts getting that wrong.

>>16785413
1. [math]1 - \frac{a_{N}}{6^{(N+K-2)}}[/math] where [math]a_{i}=5(a_{i-1}+a_{i-2})[/math] and [math]a_0=1, a_1=6[/math]
2. I spent half an hour on this and gave up.

>>16785413
1. [math]1 - \frac{a_{N}}{6^{(N+K-2)}} where a_{i}=5(a_{i-1}+a_{i-2}) and a_0=1, a_1=6[/math]

2. I spent half an hour on this and gave up.

edited for formatting

>>16786141
nvm, it's worse
I can't into LaTeX.

>>16786017
so basically because the induction axiom holds for the set of 0 and its successors, you can't construct a set N with junk elements because you can substitute 0 and its successors with N according to the induction axiom, and N cannot equal itself plus junk

How would you solve the following?

Licence plate is made from three of 26 letters and three numbers, like ABC 123. What is the minimum number of cars you need to have in a parking lot in order for there to be more than 50% probability for every letter and number to appear in the licence plates?

>>16784954
>>16785149
>>16785031
I am continuing with cosets. I believe this is the most important diagram so far. If I am in the mood tomorrow I will make a figure about bijections, homomorphisms and isomorphisms.

>>16786194
coupon collector problem (squared)

https://www.youtube.com/watch?v=JbhBdOfMEPs&list=PLybg94GvOJ9FoGQeUMFZ4SWZsr30jlUYK

Is this equivalent to induction:
If P(0) and for all n, P(n) <-> P(n + 1), then P(n) for all n.
Like induction but the implication in the induction premise goes both ways instead of just P(n) -> P(n + 1)

>>16786393
It's strictly stronger than induction.

>>16786274
When do you get to the Sylow theorems?

>>16786584
I haven't studied those but now that you mention them It should be worth making a diagram.

>>16786539
if anything it is weaker since the premise is stronger but the conclusion remains the same
ie that principle immediately follows from induction

i think it might be equivalent since a similar theorem holds for general well-founded relations but i can't see how at the moment

If you roll a dice ten times, what is the probability that you only roll at most three different dice numbers? For example, it could be something like 6,5,2,6,5,5,5,5,5,2. Or it could be just 4,4,4,4,4,4,4,4,4,4. But at most three different digits.

>>16784954
You say you made it in the past 1-2 months, yet it looks like it was made 20 years ago

>>16787317
I drew those in kolourpaint (ms paint for linux). I know it looks ugly or like a bad meme format but I want to get the essense first. The second reason I made them is to check if I have understood the concepts well enough.

Return 0

>>16787242
(6c3)*3^10/6^10
or 5/256 exactly

>>16788128
you're overcounting sequences with fewer than 3 values

>>16785037
It's more complicated than that.
The amount of balls *seems* to be the key, like in >>16784147
but think of the situation when the all-gold box has 1 million gold balls instead of two.
The answer doesn't change, it's still 2/3.

>>16788308
>the all-gold box has 1 million gold balls instead of two
What was the makeup of the other boxes? Anyway this question is about this particular problem, not any other problem, such as your example. Also I don't know what >>16784147 is saying.

This article says it's 2/3 for gold, which means 1/3 for silver, as I said.

https://en.wikipedia.org/wiki/Bertrand%27s_box_paradox

>>16788326
>What was the makeup of the other boxes?
Unchanged; the point is, the amount of balls in the all-gold box doesn't matter, so long as they're all gold. (Well, you need at least 2, but besides that). The reasoning presented in >>16784147 doesn't work because it suggests that it's specifically because 1/3 of the gold balls are in box 2, which, again, is not the case - the probability will be the same so long as box 2 has 1 silver and 1 gold.
The reasoning is essentially as follows:

>1/3 chance of drawing from each box
>100% chance of gold from box 1
>100% chance of silver from box 3
>50% chance of gold from box 2, 50% chance of silver from box 2

>therefore when drawing a ball, 1/3 chance of gold from box 1, 1/3 chance of silver from box 3, 1/6 chance of gold from box 2, 1/6 chance of silver from box 2

>drawing gold first rules out all options where you draw silver, leaving behind the unadjusted 1/3 chance you've drawn from box 1 and the 1/6 chance you've drawn the gold from box 2
>renormalising (setting total probability to 100%), 2/3 chance of the ball coming from box 1 and 1/3 chance coming from box 2

and "coming from box 2" is equivalent to the next ball being silver since it's the only option

>>16788278
How so? He said at most 3 values, not exactly 3.

>>16788341
I said it's 1/3 so what's your point?

>>16788278
>>16788377
Ah yeah, I see it now. Several ways of choosing 3 end up being redundant.

>>16788378
>I said it's 1/3 so what's your point?
>>16788341
>the point is, the amount of balls in the all-gold box doesn't matter, so long as they're all gold
have to admit, phasing out by literally the second word that isn't quoting you is something else

>>16788413
Speak English

>>16788413
>the point is, the amount of balls in the all-gold box doesn't matter, so long as they're all gold
Wrong. You have one box, not two. Your fault is thinking you have two.

I made a simulator for Bertrand's box paradox but with cards instead, which you run in your browser.
>open this link:

https://pastebin.com/ANBM3Vg4

>click "download"
>change ".txt" in the filename to ".html"
>click to open the html file
>it will open up in a web browser

>text might be jumbled but kings are black, queens red
>press button to shuffle
>press any stack once to reveal the top card, twice to reveal both cards, three times to hide both cards

>>16789003

I fixed the corrupted code, now it's showing king of spades and queen of hearts.

https://pastebin.com/SF0MDzgV

>>16789196

I tried two hosting websites so all you have to do is click to download and then click the file to run it. I was hoping for an html link but this is the best I could do. Let me know what you think about this game. I might try github later. Anyone know a good site?

https://limewire.com/d/1cWQ3#k8PD6k0Vpl

https://we.tl/t-IjHyXLc5JK

I made another game with actual boxes and balls.

https://limewire.com/d/B8mAD#b0he0jg4pg

I'm an EE who is close to finishing Zorich Vol 2 and just recently finished Axler's Linear Algebra, really enjoying myself so far, though Zorich kind of broke me a bit.

Should I jump into Aluffi Chapter 0 or Dummit and Foote for abstract algebra

>>16789503

Link that opens up the game directly in the browser:

https://litter.catbox.moe/jljwe6lnmcbwxtmx.html

>set of all real numbers is uncountable
>set of all finite strings from a countable alphabet is countable
>hence, set of all definitions is countable
Mathematicians are seemingly okay with this.

>>16790062
Mathematicians are lazy people. They want to achieve infinite results with the least finite effort

>>16778087
>>16778604
A downmarket question here from me because I'm still just a new undergrad grinding fucking calc, linear alg, and discrete math lmao. That seems to be Hartshorne's book on algebraic geometry if my Googling is anything to go by. OK, so what I want to know is, once you get to the level where you've read a few of these go-to books on algebraic geometry, is that enough familiarity to understand (not write) contemporary journal articles on algebraic geometry, or is there like this huge gulf between the books and the articles?

I'll give a lecture about multilinear algebra to a class of physics students (I'm a math student). What's a good lesson plan to follow?

Recently started combinatorics, is there a formal proof for Catalans Number satisfying Segner's recurrence relation without having to use the explicit expression for [math]C_n[/math]?? I managed to find it by drawing triangles, but I don't see how I could "translate" it into hard maths, and Wikipedia doesn't list such a proof.

>>16790527
>I managed to find it by drawing triangles
That is the formal proof

>>16790551
huh, good job me then lol, I can get behind that proof by doodling concept honestly.

How can i confirm some math textbook's ebook edition never existed?
>>>/wsr/1541342

>>16778651
>>16779001
>>16780883
I used to know a differential geometry grad student whose research involved fiber bundle gerbes, he said "gerbe" was French for "vomit."

>>16790062
What is your contention with it?

>>16777973 (OP)
i'm reading about lebesgue integration
the book develops the lebesgue integral using functions that have nondecreasing sequences of step functions that converge almost everywhere and whose sequence of step function integrals is bounded
evidently the difference of two such function is in the space of lebesgue integrable functions

i don't see how this is any different than riemann integration, which was defined in terms of the supremum/infinum of lower/upper sums.

i keep reading that riemann integration proceeds by partitioning the domain, while lebesgue integration proceeds by partitioning the range, which seems to make sense.
but i don't see how this is connected with the development of the lebesgue integral using step functions.

any insights?

>>16790696
It's the difference between adding up terms like
>(measure of fixed subinterval)*(some value attained on that subinterval)
and
>(value that is attained)*(measure of subset which attains that value)
To see the difference, try to integrate the characteristic function of irrational numbers on [0,1] in both ways.

>>16790704
>try to integrate the characteristic function of irrational numbers on [0,1] in both ways.
i'll think about this example more (in terms of rational numbers)
it's clear to me why it is not riemann integrable; the lower and upper sums converge to different values because the supremum and infinum of any open interval will be 1 and 0
i also see why the lebesgue integral exists and is equal to 0, particularly because if you enumerate the rational numbers as a sequence of step functions, then the sequence converges almost everywhere and the step function integral is 0 for all functions of the sequence (i.e. the rationals are a set of measure zero).
... but i still am missing something

i should add, the development of the lebesgue integral that i'm reading hasn't introduced measure theory yet, although it has introduced the notion of sets of measure zero and null sets (sets where there exists a sequence of nonnegative nondecreasing step functions that diverge for all elements of the set, but the sequence of step function integral converges)
i should just keep reading, because that gets covered in a section or two

what's weird is there isn't any mention of partitioning a range, or preimages, or anything like that in the development, and i haven't detected how those kinds of concepts related to each other w.r.t. lebesgue integration yet

>>16790713
Characteristic function is 1 at every irrational number, so integral is 1, not 0.
>>16790720
There is no reason to partition the range. The idea is that you approximate the function with a sequence of step functions; step functions are easy to integrate, and then the integral of your function will be the limit of the integrals in your sequence. Then you show that this limit does not depend on the choice of sequence, so the integral is well-defined.

>>16790696
>while lebesgue integration proceeds by partitioning the range
This is bullshit told by people who don't know about Lebsgue integral. This can hold true only in the case that the range is countable.

>>16790761
Actually, I shouldn't say only in the case, it can be true in other cases, but the point is this is true only if you select your simple functions in a a particular way, which is not necessary and may not exist for a particular function.

do characteristic functions exist in constructive math or can they not work with {0,1}

>>16790874
Yes they do exist, and in constructive math sets are defined in terms of their characteristic functions - for something to be a set, there must be a constructive procedure to determine whether something is or is not inside that set, which is the same as evaluating the characteristic function of the set.

>>16790696
>i keep reading that riemann integration proceeds by partitioning the domain, while lebesgue integration proceeds by partitioning the range, which seems to make sense.
>but i don't see how this is connected with the development of the lebesgue integral using step functions.
You're probably missing a bit of history. Lebesgue's original idea was to partition the range of some function [math]f[/math] and look at those [math]x[/math] with [math]y_i <= f(x) < y_{i+1}[/math] (just the preimage); call this [math]E_i[/math].
He then took as simple functions [math]\sum y_i 1_{E_i}[/math].
This is most certainly partitioning the range and I don't know what >>16790761 is on about.
Nowadays a lot of texts skip over this development and just define a sigma algebra and simple functions on it, merely requiring that they should be [math]\leq f[/math] pointwise.
See also, for example, https://old.maa.org/sites/default/files/images/upload_library/46/Barnett_TRIUMPHS_MiniPSPs/MiniPSP_Lebesgue_Integration_2023_01_01.pdf
>i don't see how this is any different than riemann integration
General measurable sets are a lot more flexible than intervals to use in a partition.

>>16790938
>General measurable sets are a lot more flexible than intervals to use in a partition.
Only in garbage non-constructive frameworks.

>>16790874
>>16790878
{0,1} doesn't classify subsets in constructive mathematics. P(1) does that job just fine though

>>16790713
On the reals, you can read into the McShane integral to see just how far you must broaden the Riemann idea to obtain the Lebesgue-equivalent power
https://en.wikipedia.org/wiki/McShane_integral

As for "why" Lebesgue is stronger (if more complicated), I think one key ingredient is to recognize that the countable union/intersection postulates in the setup (which aren't generally there in Riemann) translate to tools you can use with (inevitably) countable limit statements (Monotone/Dominated Convergence Theorem)


>>16791374
classifier detected


>>16790878
Those are nice sets - the sets where the membership predicate fulfills excluded middle - but constructive math broadly understood is not restricted to speaking of those sets.

E.g. if CH is the continuum hypothesis (or any other undecidable proposition of your choice), then the subclass/subset of the singleton {0} defined by
S := {x ∈ {0} | CH}
is generally still understood to be a set, in constructive math. Just not a nice one.
It's not a set where the proposition
(0 ∈ S) or not(0 ∈ S)
can constructively be proven. But we'd still call it a set.
Anyway, this is a bit semnatics of the word. I just don't want people to get away thinking there's suddenly unspeakable things, constructively. The domain of terms is indeed restrictived, especially if you also want predicative foundations, but broadly speaking you don't use too many objects, in those material foundations.

>>16791274
Likewise, at least if your (constructive) foundation is strictly weaker than a classical foundation (say ZF or ZFC), then if some object X fails to validate a property P(X) in ZFC, it by definitions also fails to validate it in your weaker theory.

That said, of course there's also frameworks deemed constructive that break with ZFC axiomatically (notably those of Brouwer and Markov II).

And the story is yet again different for Bishop and Type theories, which usually can be modeled with ZFC, but not as simply or directly.

Are (committed) constructivists the vegans of mathematics?

>>16791441
People who seem to know the stuff (at least somewhat) deeply like this guy >>16791382 are cool. People like >>16791274 are definitely akin to vegans.

>>16791441
The 'philosophically motivated' ones are at the very least a little bit annoying to deal with.
Then there's also those that act straight up condescending towards classical mathematics and can't seem to go one day without criticizing classical mathematicians for not working in their one trve foundation. Sadly enough this can range from the clueless zoomer discord tranny that read one blog post about types in programming languages and is now coninced that excluded middle is a hoax to actual working constructive mathematicians getting regular warns on mathoverflow for constantly stirring shit up and thinking that the entire mathematical community is against them and sponsored by big LEM (I won't name who I had in mind when writing this since I still respect him as a mathematician but the other constructivists here can probably guess who I'm talking about).

But: The vast majority of actual working constructive mathematicians are basically just like any other mathematician. Some may be a little too emotionally attached to their research subject and may sperg out here and there but not necessarily any more than some autist from a completely different field would.