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sweaty, the proper term is meromorphic continuation"analytic" is an outdated term from the XIX century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?
so the whole basis of mathematics is less than 150 years oldthis means I still have a chance to catch up
>>16783707>what are groupshttps://en.wikipedia.org/wiki/Group_(mathematics)>where are they even usedhttps://en.wikipedia.org/wiki/Group_(mathematics)#Examples_and_applications
>>16783707There's a 3b1b video that motivates and describes them. https://www.youtube.com/watch?v=mH0oCDa74tEDunno 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.
>>16783707If 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.
>>16783784I 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.
>>16781167g1 --> g2g2 --> g1g3 --> s2/3
pack it up, AGI is finally here
>>16784547I 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
>>16784567I'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 >>16784547It'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?
>>16784740I 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.
>>16783707Groups 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]
>>16784854Ok I see. You can pick any point in some An and prove there is another Am which it is not in. Thanks
>>16777973I'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.
>>16784954Get mogged.
>>16784954subgroups>>16784967Category theorists miss the point(object)
>>16785024Any centralizer of 'a' is subgroup of GThe 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.
>>16785024All subgroups overlap and have subgroups.
>>16785044When 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>>16785024How about this? I think I have expressed better what I wanted to tell
>>16784967Is ι the morphism that takes an element to its inverse? That's the best I can tell from e: 1G
>>16784966For more information, visit:>>16784960
>>16785310Yes.
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 axiomI 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?
>>16785783it's almost like a recursive definition which I think is what is making it hard for me to understand
>>16785783>junk elements>without junk
>>16785854if 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
>>16785783It 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
>>16786016Dropping 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.
>>167854131. [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
>>16786141nvm, it's worseI can't into LaTeX.
>>16786017so 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>>16785031I 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.
>>16786194coupon 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)
>>16786393It's strictly stronger than induction.
>>16786274When do you get to the Sylow theorems?
>>16786584I haven't studied those but now that you mention them It should be worth making a diagram.
>>16786539if anything it is weaker since the premise is stronger but the conclusion remains the sameie that principle immediately follows from inductioni 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.
>>16784954You say you made it in the past 1-2 months, yet it looks like it was made 20 years ago
>>16787317I 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)*3^10/6^10or 5/256 exactly
>>16788128you're overcounting sequences with fewer than 3 values
>>16785037It's more complicated than that.The amount of balls *seems* to be the key, like in >>16784147but 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 twoWhat 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 2and "coming from box 2" is equivalent to the next ball being silver since it's the only option
>>16788278How so? He said at most 3 values, not exactly 3.
>>16788341I said it's 1/3 so what's your point?
>>16788278>>16788377Ah 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 goldhave to admit, phasing out by literally the second word that isn't quoting you is something else
>>16788413Speak English
>>16788413>the point is, the amount of balls in the all-gold box doesn't matter, so long as they're all goldWrong. 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
>>16789044
>>16789003I fixed the corrupted code, now it's showing king of spades and queen of hearts.https://pastebin.com/SF0MDzgV
>>16789196I 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#k8PD6k0Vplhttps://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
>>16789503Link 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 countableMathematicians are seemingly okay with this.
>>16790062Mathematicians are lazy people. They want to achieve infinite results with the least finite effort
>>16778087>>16778604A 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 trianglesThat is the formal proof
>>16790551huh, 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>>16780883I used to know a differential geometry grad student whose research involved fiber bundle gerbes, he said "gerbe" was French for "vomit."
>>16790062What is your contention with it?
>>16777973i'm reading about lebesgue integrationthe 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 boundedevidently the difference of two such function is in the space of lebesgue integrable functionsi 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?
>>16790696It'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 0i 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 twowhat'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
>>16790713Characteristic function is 1 at every irrational number, so integral is 1, not 0.>>16790720There 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 rangeThis 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.
>>16790761Actually, 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}
>>16790874Yes 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 integrationGeneral 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
>>16790713On 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_integralAs 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)>>16791374classifier detected>>16790878Those 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 byS := {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.>>16791274Likewise, 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?
>>16791441People who seem to know the stuff (at least somewhat) deeply like this guy >>16791382 are cool. People like >>16791274 are definitely akin to vegans.
>>16791441The '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.
>>16791472I'm certainly not committed, I just like the stronger (compute-translatable) proofs it gives you, if you go through the non-LEM pain. Incidentally, the pic I posted is from a cafe where there's semi-militant vegan baristas. I find it extremely annoying to be asked if I want "cow milk" instead of just "milk". Well anyway, different story. Live and let live.If anything, measure theory is one of the reasons why constructive math is particularly tedious and (unless you're super dedicated or just really only interested in developing pure math) difficult to spend all your time on, be "committed". I think it's "better", just like going to that niche remote restaurant out of the city is "better". Just because it's better, doesn't mean you can go there all the time, practically speaking. (What goes for measure theory also goes for topology, but beyond analysis, this isn't imho as hands on relevant as measure theory is.)People have seen my handles, I think I've interacted with most internet-active constructive people in one way or the other, except maybe some type theorists who never touch pure math academia much. Real coders. So as for >>16791498, I don't even know. The more prominent people should be Bauer, Blass, and the nLab asian (forgot name), but I don't know that _they_ would get warnings. There's Shulman, who got into his own trouble, but I don't take him to be a philosophical constructivist, not in the old school sense. I know there's more fringe scholars like Weaver, as far as fight-willing people are concerned, but I also don't know if he's a constructivist per se. Even if he always seems to end up defending predicativism.
I've just learned about what you guys get up to (math that stems from uncountable infinity). I can't believe that tax payers fund this claptrap. Engineers don't need any of this. Define a number like googleplex and get on with things.
>>16791580Most taxpayers contribute a net negative to the world. You should be grateful that some amount of their money goes to something pointless.
i cant save the fag or his baby
>>16790689Are you fucking serous?
If your "math" cannot be computed by a MMIX machine, it's not real math. (Turing machines don't count)
>>16791671Yes. Acting incredulous isn't going to convince anyone of anything. Are you simply unable to explain the issue?
>>16791680You people make me sick. I know normies are too dumb to care about anything but the fact they are such apathetic people in math circles makes me lose all faith in everything.
>>16791826I'm not the guy you reply to (rather, the guy on a weekend photo spree), but let me point out, to anyone, that the "uncountably many undefinable reals" is more intricate than people generally assume. The argument >>16790062, also called "tea table argument" I think, doesn't actually go through as presented, at least not formally. The issue is that, related to Tarksi undefinability, the predicate "the string s defines a real number" is not definable. A variant of the arugment can be formalized, but it inherently involves two levels (i.e. it can be given in form of a metalevel analysis only.) It's annoying headbanding shit and (but this may just be my limitation), I wouldn't believe anything that seems intuitive to me, when it comes to this. Not without reading into the (annoying) literature.
>>16791842I spied on your photo! That gay couple sitting behind the tree is near:Gelati AlbertiPrater Str. 401020 ViennaAustria
>>16791826If you can't answer the question, you should just say that.
Is there any algebraic structure that models pointer arithmetic?
>>16792839The group of integers acting on set of pointers come pretty close. The only thing missing is the axiom of metric between pointers.
Probability is a homomorphismFrom monoid of sets under disjoint unionTo monoid of closedinterval(0,1) under addition.
>>16792884Well [0,1] is not closed under addition but probability reduces sets down to numbers while preserving order. What is the word?[latex]O \subseteq disjointunion(A,B) \subseteq Ω\implies0 \leq add(x,y) \leq 1[\spoiler]
If you roll a dice a hundred times, what is the probability that you never throw the same digit seven times in a row?
>>16792925Calculate the result of binomial distribution (100,7)And substract it from 1
>>16792925Since every event is independent, It doesn't matter if you throw it 100 times or 7 times.1 - (1/6)^7I misread your question and thought 7 successes in general. You wanted in sequence
A long time ago somebody posted here an image with Kurisu Makise and "You should be able to solve this". It was a probability theory problem with die.Does anyone have the img?
Let [math] a \in \mathbb{Z}, a>1 [/math]. Does there always exist [math] b \in \mathbb{R} , b>1[/math] such that [eqn] \lim_{n\to\infty} \frac{1}{b^n} \binom{an}{n} [/eqn]exists and is finite?
>>16794290Just pick [math]b > a \cdot e [/math] and it will converge to zero by the squeeze theorem.
>>16777973> tfw you reject all the othe women for the Queen> ftw you become her beta orbiter anyway because She is at the end like the other women> tfw
>>16790316hide everything behind meaningless formalismthat will teach them
>>16794209spent like 1.5 hrs looking for this shit from 2023 but found it in the end, lol.simply like the problem in the image
>>16794315Could you explain how the squeeze theorem argument would work? Right now the only bound I can figure out for [math] \binom{an}{n} [/math] is [math] \binom{an}{n} \leq a^n n^n [/math], but I'm not sure how to use this.
>>16794515Obviously [math] \binom{an}{n}<2^{an} [/math]. But you can get the optimal [math] b [/math] using Stirling's approximation.
>>16794712Could you explain how you know the bound [math] \binom{an}{n} < 2^{an} [/math]?
>>16794443well, any M - just throw M x N_1, parition the set of outcomes in equal M classes, the probabiblities are equalis this a trap?P.S. unless all dices are 1-sided
1 14 27 9 22 5 18 31 13Find the pattern
>>16795049Add 13 to each term then wrap after 31 or 30, is it some length of month thing?
>>16795150Yea nice
>>16794902Nvm I figured it out: [math] \binom{an}{n} [/math] is the number of size-n subsets of a size-an set S, which is less than the total number of subsets of S, which is [math] 2^{an} [/math]
>>16794712>you can get the optimal b using Stirling's approximation.Could you give a hint about how this works? When I try using Stirling's approximation on [math] \binom{an}{n} [/math] for large n, I get [math] n^a / a! [/math], but this doesn't make sense, since [math] \binom{an}{n} [/math] shouldn't have just polynomial growth in n
What's a good motivation to give before teaching tensors? Maybe the riemannian metric?
>>16795281Sorry ignore this, apparently I'd lost my ability to do algebra when I posted it
I have to study everything about Analysis up to differentiation until monday, the first two chapters and half of the third chapter of Do Carmo's Differential Geometry of Curves and Surfaces until tuesday, and everything about Rings until wednesday. Is it fucking over?
Do you hate engineers?Looks like ChatGPT wrote this but is bet ChatGPT isn’t as low IQ as the ME who actually wrote it.Yes I’m mad because my hot as fuck ME gf left me. I will never do as good ever again.
>>16795592How did you find yourself in this position? Qualifying exams or something? A bunch of different classes that all have exams at the same time?
>>16795619>A bunch of different classes that all have exams at the same time?Yeah.
>>16794908>parition the set of outcomes in equal M classeswhat do you think the cardinality of that set of outcomes is
>>16795622Well, I have faith in you. Remember, if you really couldn't do it, you wouldn't have gotten this far to begin with.
>>16795430Talk about the differential when doing a variable change, and the partial derivative when doing a variable change, and notice how they transform differently. Talk about the moment of inertia tensor, or some other physics tensor (I think feynman did something on crystals?)
>>16795603I don't get the issue. I guess it would've been better to say the non-zero diagonal entries?
>>16795689>Talk about the differential when doing a variable change, and the partial derivative when doing a variable change, and notice how they transform differently.What does it have to do with tensors?
>>16795430simulation of dick-in-ass dynamics
>>16795825What exactly are you hoping to convey about tensors if you aren't using Riemannian metrics and you aren't modeling covariance/contravariance
>>16795694Dumbass
>>16796107Well you have given zero evidence, so it looks like you're the dumbass.
>>16796191I’ll leave it as a trivial exercise to the reader to explain why Wikipedia is obviously incorrect
how do i not fail uni after 1st semester againi failed linear algebra and calc 1 because i didnt study at all, i never had to study in high school how the fuck was i supposed to figure out how to put real work in by myself?
>>16796598> how do i not fail uni after 1st semester againStep 1: Stop blaming others for your own failures. That's not easy, and was a thing that took me a while to really understand, but I think you need to understand that too.Step 2: Actually study. I never really had to study until I got to grad school. Needless to say, I felt like I got hit by a truck the first time that I took a course I really couldn't do without tons of studying. Try to make a habit of doing practice problems from the book. You don't need to grind yourself to dust, but a good rule of thumb is three hours of self directed study (practice problems without chegg) for every hour of in class time.Step 3: If you find you aren't getting it, be proactive. Attend office hours. Get tutoring if you need to. Seek out youtube videos from other instructors if you don't understand the way your prof. is teaching the subject. Find other (free if possible) textbooks that are on the topic, and see if those textbooks cover the material in a way which is easier for you to understand. Step 4: Try to read ahead if possible. You'll get a lot more out of lecture time if you've already skimmed the material ahead of time. That way you can use the in class time to clear up misunderstandings.
>>16796688i wan to fucking kill myself
>>16796688i mean thanks
>>16796692>>16796694Don't kill yourself. You've got this. Start with easy stuff and try to just get in the habit of a couple problems every day or two. You'll get there. Schaum's outlines are also great.As much as people shit on "homework threads," if you're genuinely looking for help and have tried, people are usually helpful here.
>>16795629I am retarded
>>16796698it's hard to study when the stupid fucking cunt of a lecturer doesnt even give you anything to study from
what a scam, algebra is bullshit, I want a refund
Does topology make any sense regarding things that aren't smooth surfacesThey call it "rubber sheet geometry" because anything that can be continuously deformed is equivalent. Continuous deformation doesn't apply to sharp create, though, it doesn't work like that. It doesn't get continuous deformed and then just smooth out any sharp bends.There's are "discontinuities" like 1/x that produce a spike. I'm pretty sure a "discontunity" isn't compatible with "continuous deformation" though.
>>16797900*creases
>>16797900>Continuous deformation doesn't apply to sharp create, though, it doesn't work like that. It doesn't get continuous deformed and then just smooth out any sharp bends.Sure it does; why wouldn't it? Creases are only a concern if looking for diffeomorphisms, not homeomorphisms in general
>>16797907Cuz it ain't smooth.A function works the same everywhere unless it just isn't defined at a certain point.How do you deform something that isn't smooth into something that's not smooth?
>>16797910You're thinking of an analytic notion of "smooth", not a topological one. The topological notion is more properly outlined by homotopy equivalence.For example, y=abs(x) isn't smooth at the origin in the analytic sense, but it still continuously deforms into a straight line topologically because we can define y=z*abs(x) where z is some parameter ranging from 1 (y=abs(x)) to 0 (y=0)
>>16797912>not a topological oneYeah but why.Where does the topological notion of "continuous deformation" come from.>abs(x)I'm not sure that abs is derived from anything. It seems like a weird piecewise function with one specific condition. Maybe by squaring something and always producing something that's positive.I'm going to end this conversation here because it's "one of those". I tend to do this. I have extremely abstract ideas and I'm 100% sure that I'm onto something but like 20% sure that it makes any literal sense. We're gonna get bogged down in the weeds trying to define Euclid's elements before I can explain what I'm talking about.
>>16797915>Where does the topological notion of "continuous deformation" come from.Again, I suggest you look into the concept of homotopy, because that's what it actually means in this context, and you're going to find a lot more than I can fit into a single 4chan post
Can you please suggest a good Pre-calculus book? I like to have physical books.Thank you.
>>16797900>Does topology make any sense regarding things that aren't smooth surfacesYes, look up e.g. "topological manifolds", or look up "topological spaces" in general>Continuous deformationThis is called "homotopy" in the study of topology, you should look it up>There's are "discontinuities" like 1/x that produce a spike. I'm pretty sure a "discontunity" isn't compatible with "continuous deformation" thoughI believe algebraic geometry calls such discontinuities "singularities". You can try looking up singularities in algebraic geometry, but algebraic geometry can get rather complicated
>>16794255s = side length of pentagonr = radius of circle inscribed in pentagonR = radius of circle circumscribed in 5-gonIf s = 8*Sin[π/5] = √[8*(5 – √5)], then s/2 = 4*Sin[π/5] = √[2*(5 – √5)], r = 4*Cos[π/5] = 1 + √5, and R = 4.Thus it's better to set R = 4, than to set s = 1.
>>16797912In the 3rd row and 3rd column, you should have written a ^ (1 ×^(–1) b) instead of a ^ (1 ^^(–1) b).Only your first two columns are entirely correct.
>>16798076>circumscribed in 5-goncircumscribed about 5-gon
>>16797815In the 3rd row and 3rd column,you should have written a ^ (1 ×^(–1) b)instead of a ^ (1 ^^(–1) b).Only your first two columns are entirely correct.
guys, please help me. how do you _actually_ read a book? i mean literally, what do you do?i have no idea how to start and i feel stuck in this time-wasting loop.take for example a very terse book where each phrase is essential and the rest is definitions, theorems and proof, how do you proceed?do i just read it? but the i forget everything the moment i turn the page overdo i write everything down? it takes so much time to do it and it's going to impact my desire for learning.
s = 8*Sin[π/5]r = 4*Cos[π/5]R = 4The foregoing assignments are superior.WolframAlpha command:Flatten[{Table[Sin[(2 k – 1) (π/5)] x – Cos[(2 k – 1) (π/5)] y + 4 Cos[π/5] = 0, {k, 5}], x^2 + y^2 = (4 Cos[π/5])^2, x^2 + y^2 = 4^2}]The following assignments are inferior.>>16794255>s = 1>r = Cot[π/5]/2>R = Csc[π/5]/2
>>16798132that's my whole point, the pattern is inconsistent
>>16798181Read it and try to rewrite it in your own words. This also means coming with new examples, proofs, alternate definitions, etc. Basically you don't need it passively, you have to play with it. >it takes so much time to do it and it's going to impact my desire for learning.You meant the actual learning is...discouraging you from learning? Have you tried to not be a faggot?
>>16798222>the pattern is inconsistentIt's only a make-believe pattern.Not a real one.You shouldn't have written that 3rd column.Hopefully someone else will chime in.
>>16798338algebra is make-believe>uuuh rings stop after 2 hyperoperators, because.... it just is, OKAY?
>>16798271>You meant the actual learning is...discouraging you from learning? Have you tried to not be a faggot?yes, reading the proof that x+0=x and all the other ones is, in fact, boring.
Let [math]C_p(M)[/math] be the set of curves on a manifold [math]M[/math] going through [math]p\in M[/math] with [math]\gamma(0) = p[/math] for all curves. For [math]\gamma_1,\gamma_2\in C_p(M)[/math] and a chart [math](U,\phi)[/math], we say that [math]\gamma_1 \sim \gamma_2 \iff (\phi\circ\gamma_1)'(0) = (\phi\circ\gamma_2)'(0)[/math]. Then we can define the tangent space as [math]T_pM := C_p(M)/\sim[/math].How is this definition equivalent to the usual space of derivations one? And how can I define a vector space structure in this definition?
>>16798367See Jänich Vector analysis chapter 2.
>>16798181Not gonna give a real answer, but one helpful tip is to use Chatgpt or others to help break things down for you. Note, however, that this shouldn't stop you from actually thinking - you need to verify that what it's saying is actually correct. This means going through their logic and seeing if it matches up with the text given to it, or finding another source, etc.But yeah, use LLM's as long as you know how to use them as a tool properly.
>>16798367https://en.wikipedia.org/wiki/Tangent_space#Equivalence_of_the_definitionshttps://en.wikipedia.org/wiki/Tangent_space#Definition_via_tangent_curvesNot super detailed or work shown, but it's a start
>>16798181Writing has a central role in learning. It is the medium of research. And learning/studying is independent research by itself. When we write down stuff we off-load cognitive work to the environment. What cognitive work? Holding our ideas, keeping thoughts constant and concentrated.As for my experience, when I was an ignorant person, I started to write down stuff I learned and didn't read those notes again. It improved my knowledge. However, after some time it stopped helping me with learning. And now I use note taking systems for studying and learning (Zettelkasten/slip-box).It must be noted that if we talk about mathematical books, then they cannot be read quickly. And some hard parts can be skipped and returned later.So to read a book and learn from it:1. Read some pages and make some fleeting notes (thoughts, words you liked etc.).2. At some point you stop reading (for example after reading 5 pages) and look at your fleeting notes. You improve this notes into cohesive permanent notes that you will look upon later. 3. After making permanent notes you put them inside some storage. For example, if you made digital notes (text files), then you put them in a folder.4. Afterwards you continue reading. Maybe stop reading for today, even if you read 5 pages, to coninue in the future.5. You repeat the process, but look upon your previous permanent notes to remember stuff and develop ideas.You can have different takes on this system: linking notes to each other, brainstorming all your notes after some point etc. Anyway, reading will be slow, yes, but it will have a real effect. You can also read only parts that you need, not the whole book.This is a general topic about knowledge management, and there are different KM systems that you can implement. This is if you read nonfiction and academic literature, of course. If you are reading nonfiction only for yourself then what's the problem of just reading it?
How can I prove the universal property for free vector spaces?>For any set [math]S[/math] and any v.s. [math]W[/math], every map [math]A:S\to W[/math] has a unique extension to a linear map [math]\bar A:\mathcal{F}(S)\to W[/math]
>>16798743>https://math.stackexchange.com/questions/2929907/universal-property-of-free-vector-spaceDecompose it with a basis of Kroenecker delta functions
How do you generalize the bug problem for any a*b*c box where a,b,c are width, height and depth of the box?By "bug problem" I mean the problem where a bug starts walking from one vertex of the box and it finds the shortest possible route to a point that you chose on its surface. The problem is to ask what is the maximum distance that it would be possible to force the bug to walk.
Let [math]X,Y[/math] be sets and [math]\sim[/math] be an equivalence relation on [math]X[/math]. Let [math]\pi:X\to X/\sim[/math] be the canonical projection and [math]f[/math] be a [math]\sim[/math]-invariant function. How can I prove that the map defined by [math]g([x]) = y \iff f(x) = y[/math] is unique? I can already prove that it is well defined
>>16798825If both [math]g,h:X/{\sim}\to Y[/math] satisfy [math]g\circ\pi=f[/math] and [math]h\circ\pi=f[/math] respectively, the [math]g\circ\pi=h\circ\pi[/math] and hence [math]g=h[/math] since [math]\pi[/math] is surjective (and hence epi).
>>16798861>(and hence epi).I don't think I get this part
>>16798863Surjections may be cancelled on the right, as in if [math]f:X\to Y[/math] is surjective then for any [math]g,h:Y\to Z[/math], whenever [math]g\circ f=h\circ f[/math] then [math]g=h[/math].So see why, take an arbitrary [math]y\in Y[/math]. Since [math]f[/math] is surjective we have some [math]x\in X[/math] for which [math]f(x)=y[/math]. Now since [math]g\circ f=h\circ f[/math] we have in particular that [math](g\circ f)(x)=(h\circ f)(x)[/math], but that's just [math]g(f(x))=h(f(x))[/math] and since [math]f(x)=y[/math] we finally get [math]g(y)=h(y)[/math]. Hence [math]g=h[/math].
>>16798868I get it now. Thanks!
>>16798772If the final point P is on the same face as the vertex V, the path is trivial. If it isn't on the same face, then there's always a way to orient the box so the vertex is on the bottom face, and P is on the top face T. The shortest path is always moving upward.There are 4 sides S in between the two faces, with S1 and S4 being adjacent to V. You open up the box and flatten it into a 2D cardboard layout - of which there exist multiple - and draw a straight line from the vertex to the point of interest - that is the shortest path for that particular layout. From this layout logic, there's no way for the line to go all four sides like V -> S1 -> S2 -> S3 -> S4 -> T -> P since V -> S4 ->T -> P is always faster, meaning that layout (and the one with S1 instead of S4) is useless to us. Notice too that if I try to enter T through S2, it's always faster to go from V -> S1 -> S2 than V -> S4 -> S3 -> S2, so this layout (and the one with S3 instead of S2) is useless.This means there are only 4 useful layouts to flatten the box, which means you should be able to partition T into 4 connected sections (the boundaries of the sections may be opened or closed, and only shared/non unique if T is a square) or 4 colors so that if P is in one of those sections, you know which way to layout the box and draw the straight line to the shortest path. Clearly the edges of T next to S1 and S4 are their colors. If I were to guess, the partition is just 2 diagonal lines.Thinking about the edges of T, seems one question to ask is when does W^2 + (H+c)^2 > (W+c)^2 + H^2 or H > W. So this means that the edges of T at S2 (and S3) are their colors only if the Height of the box is greater than the Width (or depth), which is does show in the OP picture; otherwise, the colors are S1 (and S4).So my final guess is that if it's a tall box, it's gonna be 2 diagonal lines, and if it's a short box, it's gonna be 1 diagonal line
whats the word for setting different values to some standard so you can compare them? is it normalization or are there other words for it?
>>16798877Basically if you always force the bug to walk as long of a distance as possible, then what is that distance for any box with any dimensions?I have calculated that for a box with dimensions 1x1x3, the answer is (1/3)*5*sqrt(5).Here is this if this helps:https://www.desmos.com/calculator/0a29cdad67
should i get a separate desk just for studying?t. >>16796598
>>16799183holding some of them constant?
>>16798772Lagrangian? It's pretty general.
>>16798877>>16799308Like you suggest but don't quite explicitly arrive at, you're maximizing the minimum of four quadratics corresponding to the four possible "straight line" routes. Should be fairly straightforward if a bit messy; I had ChatGPT do it for me and it gave that if [math] a\le b\le c [/math], then the maximum distance is [math] \sqrt{\frac{(a^2+2ac+2c^2)(b^2+c^2)}{2c^2}} [/math], which seems to align with the simple values. The maximizing point will always lie on the [math] a\times b [/math] face opposite the starting vertex, on the line through the antipodal vertex to the starting vertex and making an angle of [math] 45^\circ [/math] with its two adjacent edges.The exact partition of this face into which path gives the minimum distance is essentially a Voronoi diagram. Specifically, it's the Voronoi diagram on the points [math] (-c, 0), (0, -c), (a+c, -a), (-b, b+c) [/math] restricted to the rectangle [math] 0\le x\le a, 0\le y\le b [/math]. There's no reason to expect this to have any particular symmetry, certainly not just being one or two diagonals, the diagonals don't even play into it at all unless [math] a=b [/math].
>>16799417no because in this case all the values are adjusted
Does anyone have any good information on the J(x) function (which is related to the Reimann-Zeta function) I am trying to get a solid grasp on the J(x) function but there is basically no information on it online unless I am like googling the wrong thing and it has a better name.
>>16799796Which one? Riemann defined two different functions that were labelled J(x) in his original paper, and afaik neither have a specific name.
>>16799675Omg, I got mogged.
>>16799675The depicted expression is equivalent to ChatGPT's expression.
>>16800337If c is XL, then:c*Sqrt[(1 + (1 + a/c)^2)*(1 + (b/c)^2)/2] ≈ c*Sqrt[(1 + (1 + 0)^2)*(1 + (0)^2)/2] = c*Sqrt[(1 + (1)^2)*(1)/2] = c*Sqrt[2/2] = c
>>16795629>>16794443dude lmao any dice
Let V_1, V_2 be vector spaces. If [math]\mathcal{F}(V_1\times V_2) = \{f:V_1\times V_2 \to \mathbb{R}\;|\;f(v_1,v_2) \neq 0\;\text{for finitely many}\;x,y\}[/math], then why the fuck does a tuple [math](v_1,v_2)\in \mathcal{F}(V_1\times V_2)[/math] if this is a function space? Refer to pic related
>>16799308You're trying to maximize the walking distance? The bug can basically walk as far as it wants to. If you start at point V on side A and are trying to get to a point P on B, with C ~ A and D ~ B, then just stack A,B,C, and D repeatedly on top of each other as far as you want. This means that the slope from V to P just keeps getting bigger and bigger.
>>16800558Your definitions are not the same? Did you even read it?
>>16800664Everywhere I read about the definition of a free vector space, it is constructed using functions. I'm really lost here
>>16800558Let A be a set.The free vector space F(A) can be defined in either of the following equivalent ways:1. The elements of F(A) are formal linear combinations of elements of A. Vector addition and scalar multiplication in F(A) are obvious.2. The elements of F(A) are functions [math] f : A \rightarrow \mathbb{R} [/math] such that f is 0 on all but finitely many elements of A. Vector addition in F(A) is "pointwise", and scalar multiplication is by multiplication by a constant.Try to show definitions 1 and 2 are equivalent by showing they're naturally isomorphic.
>>16800720>Vector addition and scalar multiplication in F(A) are obvious.How so?
>>16800720To add to this, for background, the free vector space F(A) on a set A should be viewed as a vector space having A as a basis
>>16800722Do you know what a "formal linear combination" means? How do you add two of these together?
>>16800724I honestly don't know. How would I define sum and scalar multiplication on an arbitrary set?
>>16800725The set of formal linear combinations isn't an arbitrary set, it comes with a natural notion of addition and scalar multiplication.
>>16800726>it comes with a natural notion of addition and scalar multiplication.This is what I'm having trouble with.Should I just see the elements of [math]A[/math] as basis vectors and treat sum and scalar multiplications as I would in any other vector space over the same field?
>>16800725I think anon forgot to add in 1. that formal linear combinations are over [math]\mathbb{R}[/math] (that is they use reals as scalars). In [math]F(A)[/math] linear combinations induce vector addition and scalar multiplication, because addition is jut liner combination of two elements, and scalar multiplication is a linear combination with only 1 non-zero element. Thus it is obvious.
>>16800731>Should I just see the elements of A as basis vectorsYes you should see the elements of A as basis vectors in F(A).A "formal linear combination" of elements of A means an expression of the form [math] t_1 a_1 + t_2 a_2 + \cdots + t_k a_k [/math] where [math] t_1,t_2,\ldots,t_k\in\mathbb{R}[/math] and [math] a_1,a_2,\ldots,a_k\in A [/math].We can add two formal linear combinations of elements of A in the obvious way:[math] (t_1a_1 + t_2 a_2 + \cdots + t_k a_k) + (t_1 'a_1' + t_2' a_2' + \cdots + t_{k'}' a_{k'}')= t_1 a_1 + t_2 a_2 + \cdots + t_k a_k + t_1' a_1' + t_2' a_2' + \cdots + t_{k'}' a_{k'}' [/math].Similarly, scalar multiplication on a formal linear combination is defined as[math] s \cdot (t_1 a_1 + t_2 a_2 + \cdots + t_k a_k)= (st_1)a_1 + (st_2)a_2 + \cdots + (st_k)a_k [/math].
>>16800670See >>16798771. Just decompose it into a basis and as >>16800720 said, they're naturally isomorphic. Guy, whenever you see shit like "only finitely many non-zero", you should think about decomposing into a basis.
>>16800740>>16800734>>16800764Now it makes a bit more sense. Thanks!
>>16794255(0, 0) = center(pentagon)(h, k) = center(large circle)h = 0k is depictedr = radius(large circle)r – k ≈ 1/11
>>16799966[math] J(x) = \frac{1}{2} \left[ \sum_{p^n < x} \frac{1}{n}+ \sum_{p^n \leq } \frac{1}{n} \right] [/math]I want to get a better intuitive grasp on this and look up more about it, but you are right, the function doesn't have a name.
>>16800879That's the Riemann prime counting function though I think I've only ever seen it written as [math]\Pi_0 (x)[/math].https://mathworld.wolfram.com/RiemannPrimeCountingFunction.html
I seriously hope you guys don’t trust WikipediaWith the help of ChatGpt I’ve been able to spot over 200 errors in math and physics articles as well as criticize the terse nonsense they’re written in opposed to a friendly approachable format.Also literally anyone can edit Wikipedia.Yeah I still trust my 100 year old textbook thanks
I just discovered this way to represent functions as polynomialsAll you need is an infinitely smooth function in an intervalAnd then when you use the FTC on every derivative and sub in you get a nested integral which with linearity pulls out a polynomial with coefficients of derivatives evaluated at a point divided by the factorials a fact I invented from nested integrals.You end up with this infinitely stacked integral where you can just change the order of integration and take its limit, which you can move into the integrand, convergence is dependent on the derivatives.Where is my Nobel prize?
>>16800965>https://mathworld.wolfram.com/RiemannPrimeCountingFunction.htmlThank you, although it seems to be more specifically the "power prime counting function." Regardless you helped point me in the right direction.
>>16800994>With the help of ChatGptembarrassing post
I hope this is okay (not homework), I've been stuck trying to simplify this term for a couple of hours for a paper: [eqn] \int dr \langle r | \phi^\dagger(r) \phi(r') |r \rangle[/eqn]This feels like the right step but I don't know if I can go further, inserting [math]\sum | r \rangle \langle r | [/math] gives[eqn] \int dr \langle r | \phi^\dagger(r) \phi(r') |r' \rangle \delta(r-r')[/eqn]I know about [math] \int \phi(r) | r \rangle dr = | \phi > [/math] but the integral vanished because of the delta or not?
>>16778087Isn't a sheaf kind of a function? Thus[math]\mathcal{F}(d)[/math] the image of [math]d[/math] by the sheaf?
>>16801627I don't think your 1st line correctly simplifies to the 2nd line
>>16801630You sound like you have zero knowledge on this matter, and idk why you bothered to try to answer the question
>>16800994>t. future ceo of america
>>16801627In your integral wrt r, the phi(r') is a constant
How can I prove the universal property for quotients? i.e., given sets [math]X,Y[/math] and an equivalence relation [math]\sim[/math] on [math]X[/math], how can I prove that for each [math]f:X\to Y[/math] there exists a unique [math]\varphi: X/\sim \to Y[/math] s.t. [math]f = \varphi \circ \pi[/math] where [math]\pi[/math] is the canonical projection?
>>16802170First get the statement you want to prove correct because what you wrote is not right.
>>16802205How so? That is the universal property as far as I'm concerned
>>16802208Forgot to add, but [math]f[/math] is [math]\sim[/math]-invariant
>>16802170>>16802209Should be easy then. Given f, how do you construct [math] \phi [/math]?
>>16802208Consider any equivalence relation except equality and [math]f[/math] injective. Then the projection is not injective so [math]\varphi \circ \pi[/math] can't be injective either.
>>16802212[math]\varphi([x]_\sim) = f(x)[/math]? But that's just restating the universal property
>>16802170The uniqueness part is directly implied by (and indeed equivalent to) [math]\pi[/math] being surjective. See >>16798861 for the proof.The existence part doesn't actually hold for all arbitrary functions as you've stated it, we require [math]f[/math] to satisfy that [math]f(x)=f(x')[/math] whenever [math]x\sim x'[/math]. For if we could lift [math]f[/math] to a [math]\varphi[/math] making the relevant triangle commute then whenever [math]x\sim x'[/math] we have [math]\pi(x)=\pi(x')[/math], hence [math]\varphi(\pi(x))=\varphi(\pi(x'))[/math], i.e. [math](\varphi\circ\pi)(x)=(\varphi\circ\pi)(x')[/math] and so [math]f(x)=f(x')[/math].As for the existence proof, consider the relation [math]\Phi[/math] between [math]X/{\sim}[/math] and [math]Y[/math] such that for all [math]q\in X/{\sim}[/math] and [math]y\in Y[/math], [math]\Phi(q,y)[/math] iff [math]\exists x\in X\,(\pi(x)=q\land f(x)=y)[/math] (*). This relation is functional in the sense that for every [math]q\in X/{\sim}[/math] there is a unique [math]y\in Y[/math] s.t. [math]\Phi(q,y)[/math] holds (prove this!). Hence there is a function [math]\varphi:X/{\sim}\to Y[/math] whose graph is [math]\Phi[/math], i.e. [math]\varphi(q)=y[/math] iff [math]\Phi(q,y)[/math] for all [math]q\in X/{\sim}[/math] and [math]y\in Y[/math]. Verify that this function makes the relevant triangle commute and you're done.(*) [math]\forall x\in X\,(\pi(x)=q\to f(x)=y)[/math] also works if you fancy that more.
>>16802234Makes sense. Thanks!
Does anyone know where I can find Riemann's "Partielle Differentialgleichungen" book in English? Literally the only translation I am finding online is to spanish...Math ain't the universal language if I don't know what the hell λ is lol
>>16798194Are you trying to kill us all?
>>16781200>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%.The basic reason the gold ball box question is 50/50 is far more simple than you think...DID YOU *CHOOSE* THE FIRST BOX?Or was that choice made for you?So the next CHOICE... YOUR CHOICE.Is between 2 boxes. Therein, a 50/50 outcome.====There are 3 boxes. Each box contains 2 balls. One box contains 2 gold balls, another box contains 2 silver balls, and the final box contains one gold ball and one silver ball.You pick box at random. You put your hand in and take a ball from that box at random. It's a gold ball. What is the probability that the next ball you take from the same box will also be gold?(Note: You can't see into any of the boxes)====>>You pick box at random. You put your hand in and take a ball from that box at random. It's a gold ball. The line declares that you "had a choice". That you "selected" 1 of the 2 boxes with the gold balls inside. However, a PROPER 2/3 probability outcome is "you picked a silver ball". Since you didn't have that choice outcome option, you exist in a predetermined situation where your ONLY LITERAL choice is a 50/50 selection between two boxes.A true 2/3 choice is:There are 3 boxes. Each box contains 2 balls. One box contains 2 gold balls, another box contains 2 silver balls, and the final box contains one gold ball and one silver ball.You pick box at random. You put your hand in and take a ball from that box at random. IT'S EITHER A GOLD OR SILVER BALL. What is the probability that the next ball you take from the same box will also be gold?(Note: You can't see into any of the boxes)====By preselection of "it's a gold ball" that "you picked" the probability drops to 50/50.Imagine a line of coins, you can only flip one at a time.
Another way to look at the GOLD/SILVER BALL BOX QUESTION is this way...There are 2 coins. Each coin has a silver or painted black side.I flip one coin and conceal that result under a wooden block. If you spin the visible coin in the air and let it land on a random face, what are the odds that both coins match?For a proper 2/3 probability I need to give you 2 coins and conceal one. Note that all coins could be:SSSSSBSBBBBBThe GOLD BALL question frames it as.First fake choice:GG or GS2ND CHOICEIf GG then you pick between GS or SS.If GS then you pick between GG or SS.Which is 50/50.A proper X/3 choice is...First choice:GG or GS or SS2ND CHOICEIf GG then you pick between GS or SS.If GS then you pick between GG or SS.If SS then you pick between GG or GS.It's a shitty worded puzzle otherwise.Like I said, THE FIRST CHOICE was made for you. So the probabilities are false. The real choice is picking two gold balls out of 3 silver & 3 gold, or flipping 3 coins randomly. The odds of a 2/3 outcome are higher than a 3/3 outcome (all heads or all tails). By concealing the outcome of the first coin flip, you don't influence the natural probability of 3 coin flips. You just tweak the visual outcome to the coin flipper.
>>16798367https://webspace.science.uu.nl/~crain101/manifolds-2024/DG-2024-Dictaat.pdfSee chapters 2.3, 3.1 and 3.2.2.
NTA but how does one restate the universal property in >>16802170 in terms of they're defined in https://en.wikipedia.org/wiki/Universal_property?
New /mg/ thread: >>16803023
>>16782288Great recommendation
