Analytic continuation editionFormerly >>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sweaty, the proper term is meromorphic continuation"analytic" is an outdated term from the XIX century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>isisIt terrorizes your pooper.
>>16778087im 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?
>>16778604Thanks very much anon.>>16778651Lol, 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!
>>16779202Let 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.
>>16780839Differential 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.
>>16781167Fun 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
>>16781196I 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>subtletyempty word, cope for not actually understanding/being able to explain why it's 1/3 rather than 1/2
>>16781196When 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
>>16781209If 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.
>>16781211Yes.But there are two cases where it contains nothing but gold, and one case where it contains nothing but silver.
>>16781215I 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.
>>16781223Perhaps 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.
>>16781223You might want to reread what I'm saying; I'm laying out exactly the reasoning that you're suggesting that I'm not
>>16781227Well, 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.>>16781228I 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.>>16781228See 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 lmaoYes, 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 haveI guess I should have said "it has", since "type of questions" is singular. I proofread it but still missed that.
>>16781280Well, 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 personI think you're contradicting yourself here.
>>16781280Forget 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.>>16781167Here'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, samplerun_count = 10000boxes = [['G', 'G'], ['G', 'S'], ['S', 'S']]first_ball_is_gold = 0second_ball_is_subsequently_silver = 0outcomes = {'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)] += 1print(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}")
>>16781315damn the indentation keeps getting messed up
>>16781313In 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.
>>16781333Then it's a matter of WHERE you calculate FROM, which is my point, that it's a matter of semantics.
>>16781339And 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
>>16781315I 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
>>16781344In 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.
>>16781353eh verbal reasoning/vocab does form a decent chunk of IQ tests thoughBut 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.
>>16781367I 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.
>>16781381Yeah I guess>took an IQ test and it was only visual patternsThe 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?
>>16780883Since it made you vomit, do you have a preferred alternative book to Hartshorne on introductory algebraic geometry?
>>16780874Thanks, 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 ecologistswow I wouldn't have expected thatnice
>>16781515Görtz-Wedhorn
>>16781191No 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 AlgebrasWhich 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
>>16781603Predator-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>>16781975Sorry, 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 HammackBoth 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 HammackAre 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).
>>16781999So 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 Mathematicsrather 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.
>>16782026Both 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]
>>16782661I 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.
>>16783049Have you read Euclid's Elements?
Fuck teachers fuck numbers they racist nikkaSummers 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?
