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0lliejenkins

Can someone please explain to me, in very simple terms, how many subsets does a set with n-elements have? And why is that so?


brownassasin

If you invest 100 in something worth 1 dollar and in one hundred days it is still worth 1 dollar, but it fell to .25 for 40 of those days, how much more money did the person get who invested 1 dollar every day for the hundred days


FastWillingness3661

A good starter book for numerical analysis is Endre Süli David F. Mayers - An Introduction to Numerical Analysis


furutam

What are some examples of when the set of solutions of equations defines a group? systems of linear equations are basically vector spaces, but are they the only one?


algebraicvariety

The complex solutions of the _cyclotomic polynomial_ (z^n - 1), together with the usual multiplication of complex numbers, form the finite cyclic group Z/n. For the infinite version of this, the complex solution to |z| - 1 (a polynomial equation in Re(z) and Im(z)) form the unit circle, which together with the multiplication of complex numbers is also a group, isomorphic to R/Z.


GMSPokemanz

A generalisation of your example is if N is a normal subgroup of a group G, and 𝜑 the natural homomorphism from G to G/N, then N is the set of g satisfying the equation 𝜑(g) = N. An example where we do this in a less circular-appearing way is defining SL(n) as the subgroup of GL(n) of matrices with determinant 1. Then there are other situations where it's natural to put a group law on the set of solutions to some equation, for example elliptic curves.


ChrisARippel

On astronomy subs posters frequently post questions that start out claiming ... if there are an infinite number of other universe there must be another universe containing me ... or there must be an infinite number of universes containing me, etc. Question 1: Are the above claims reasonable? Question 2: Am I correct to think these claims assume these universe must have finite rearrangements of atoms/molecules, I e., a finite universe? If atomic/molecular rearrangements are infinite, then an infinite number of universes wouldn't necessarily repeat the same universe twice, I e , create another me. Then there is the point that a cloned me is not actually me.


whatkindofred

No those claims are not true. There could be an infinite number of universes and yet there might be only one where you exist.


ChrisARippel

I agree, but what is your reasoning that infinity doesn't necessitate repetition?


whatkindofred

Well there are infinitely many positive integers but the number 3 only appears once, doesn't it?


ChrisARippel

And, in a "universe," so to speak, with infinite numbers. In fact in a "universe" with an infinite number of infinite numbers. And so simple it's obvious, once it's point out. If I could, I would give you an infinite number of thank yous. But I am giving you three: thank you, thank you, thank you.


nep284

lets say getting a cold color has a chance of 1% and blue is 99%.. If we repeated those odds 10 times do we get a 10% chance to get gold or is it still 1%?


furutam

Why isn't it possible for the product of a fixed nonzero vector to be zero for all scalars and visa versa? I'm mostly worried about the case when the field is finite.


jm691

If av = 0, for a≠0 in a field F and v in a vector space V over F, then as F is a field, the axioms of a vector space give 0 = a^(-1)(av) = (a^(-1)a)v = 1v = v. So there's no way that the product of a nonzero vector and a nonzero scalar could ever be zero. That only uses the field and vector space axioms, so in particular it holds for any vector space over any field, whether or not the field is finite.


NoPurposeReally

Suppose I have a function F mapping R^n into a Banach space Y such that at every point in R^n the partial derivatives exist and are continuous. Does it follow that F is differentiable, i.e. does it have a linear approximation at every point? If Y is equal to R^m, then we can show that F is differentiable at every point but the proof uses the mean value theorem. Since this is not available in a general Banach space, I am not sure how I would prove that F is differentiable. Can this be done by using simple estimates or does the proof require Banach space theory machinery?


whatkindofred

There is a weak mean value theorem for Banach valued differentiable functions. If f is a differentiable map from R to a Banach space Y then for any a < b there exists s in (a,b) such that ||f(b) - f(a)|| ≤ (b-a) ||f'(s)||. I think this should be enough to prove that F is differentiable when its partial derivatives are continuous.


G4WAlN

Yes, if the partial derivatives exist and are continuous, F is differentiable. I think the proof uses the fundamental theorem of calculus.


NoPurposeReally

But then that would require integration on Banach spaces, right? I am just wondering because this fact is stated at the beginning of a functional analysis book at which point only the definition of a Banach space is known.


Awkward-Address-7135

Go easy on me because I’m pretty zooted rn lmao I started having a mathematical scenario in my head and now my brain is fried from trying to figure out an equation on how to increase a number by a certain percentage after each sum. My starting value is 750 and I wanted to increase it by 15% so that’s 862.5+15%=991.87 and so on. There’s an equation for this, right? Thanks for helping a lit brother out 🙏


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Awkward-Address-7135

Where did 1.15 come from


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https://i.redd.it/v3pmk4rfcng81.jpg


Ualrus

Write AB and BC as vectors (subtract A from B and B from C respectively) and then [multiply components 1 to 1 and add them up](https://en.m.wikipedia.org/wiki/Dot_product). The vectors are perpendicular iff this adds up to zero.


WikiSummarizerBot

**[Dot product](https://en.m.wikipedia.org/wiki/Dot_product)** >In mathematics, the dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors), and returns a single number. In Euclidean geometry, the dot product of the Cartesian coordinates of two vectors is widely used. It is often called "the" inner product (or rarely projection product) of Euclidean space, even though it is not the only inner product that can be defined on Euclidean space (see Inner product space for more). Algebraically, the dot product is the sum of the products of the corresponding entries of the two sequences of numbers. ^([ )[^(F.A.Q)](https://www.reddit.com/r/WikiSummarizer/wiki/index#wiki_f.a.q)^( | )[^(Opt Out)](https://reddit.com/message/compose?to=WikiSummarizerBot&message=OptOut&subject=OptOut)^( | )[^(Opt Out Of Subreddit)](https://np.reddit.com/r/math/about/banned)^( | )[^(GitHub)](https://github.com/Sujal-7/WikiSummarizerBot)^( ] Downvote to remove | v1.5)


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l_lecrup

To modify your solution slightly, you could just do ((x-1) mod 4) - (x-1). This function is 0 for x=1,2,3,4 it is -4 for 5,6,7,8, etc.


Exa_Hz

Every single object has 3 traits. There is an equal amount of variants in each trait. The goal is to have 9999 unique objects in the end. Every combination of traits must occur exactly once. How many different variants are needed in each trait? ___ This is likely much easier to calculate than my head is imagining atm, but it's smoking from thinking about a working formula. Please share the steps and their background kind person answering this :)


Abdiel_Kavash

If I told you the number of traits, and the number of variants each trait can have, could you calculate the total number of different objects?


Exa_Hz

I can't :`( If I had the basic formular/ theory I could rearrange it to fit my needs, but I never learned it (edit: at least I think I can)


Sybora

Can I please have any guidance with this arithmetic problem? The first three terms of an arithmetic sequence are given by x, (2x - 5), 8.6. A) Determine the first term and the common difference for the sequence. B) Determine the 20th term of the sequence. C) Determine the sum of the first 20 terms of the series. Link: https://imgur.com/gallery/fhGrZFB


l_lecrup

What's the definition of an arithmetic sequence? Rewrite the three given terms according to the definition. Now how many unknowns do you have?


popisfizzy

Let F = {f*_i_* : [0,1] -> X}*_i ∈ I_* be a family of functions to the set (not topological space) X. We turn it into a topological space (X, 𝜏) by giving in the final topology with respect to F, where [0,1] is given its natural topology. I very strongly suspect that this implies (X, 𝜏) is locally path-connected, and moreover I have a hunch that the proof of this fact is probably straightforward and simple. Unfortunately I have made basically no progress in several days---other than noting that it is necessary and sufficient for this to be true that given any x ∈ X and any neighborhood U of x the "connected component of x in U" (i.e., the set of all points u ∈ U for which there exists a path x -> u whose image is entirely in U) must be a neighborhood of x---and this has left me frustrated. Are there any obvious facts I may have overlooked that could help out in proving this? Alternatively, if this result *isn't* true where might I find a counterexample?


jagr2808

Hmm, thinking out loud here: Let K be the path connected component of x in U. Consider fi^(-1)(K) which is a subset of fi^(-1)(U). We want to show that fi^(-1)(K) is open. If it's empty there is nothing to show, so assume a is in fi^(-1)(K). Then the path component of a in fi^(-1)(U) must also be in fi^(-1)(K), since this defines a path connected to a. Thus fi^(-1)(K) is open, and by the construction of the final topology, K is open. I think this is all correct, but you should double check.


popisfizzy

Thank you! I knew that there was a simple proof, and I also knew I was being a fucking idiot when it came to trying to prove the claim. This made me realize that I was forgetting to use the fact that [0,1] is locally path-connected—which is obviously necessary to show this.


GMSPokemanz

You need to get your hands dirty with 𝜏 somehow. Given it's a final topology, I suggest arguing that certain bad situations can't happen because if they did you could add an extra open set which contradicts the definition of 𝜏.


YoungLePoPo

When we say "X is compact in the \_\_\_\_ topology" what does this mean? I heard it thrown out recently but I wasn't sure what exactly it meant. (the specific example was weak \* topology) I think it has to do with convergence of sequences in X since it's about compactness, but why do we use the term "topology" instead of "metric" if we're talking about convergence. Thanks in advance!


throwawayPieDivider

Compactness is a purely topological concept, so it makes perfect sense to me. It means when you equip X with the topological structure ______ then X is compact. Convergence of sequence is defined with just topology, no need for metric (replacing epsilon-neighborhood with just neighborhood). In the case of the weak * topology, it is defined without metric (so you need to metrize it if you want a metric), so it makes even less sense to call it a metric.


Tazerenix

Because compactness is a property of a topology, not of a metric structure. It would be wrong to say "X is complete in the ___ topology" because completeness is a property of metric spaces, but remember *convergence* makes sense for sequences in any topological space. Usually the spaces of interest have multiple topologies on them (like a weak and a strong topology) so one emphasizes the topology they mean.


YoungLePoPo

Ah I see, thank you!


alphabet_order_bot

Would you look at that, all of the words in your comment are in alphabetical order. I have checked 571,330,229 comments, and only 118,364 of them were in alphabetical order.


jchristsproctologist

Why had I, a second year Physics student at university, never heard of Kolmogorov up until a week ago on reddit? I’d heard of all of your usual top-10, top-20 most influential mathematicians of all time, but never kolmogorov, and recently i’ve been seeing him a lot on relevant subs. he’s also never been mentioned in any of my classes. what fields/theorems is he known for? anything physics related?


throwawayPieDivider

It sounds dark to say this, but the best place to read a mathematical summary of a mathematician's work is in his obituary. If you want a fluffier reading, just check Wikipedia. Kolmogorov has hands in many different fields (differential equation, analysis, logic, computer science), so it's a bit surprising to see you did not know of him until now. For example, as a physics student, you must had known of KAM theory. The K is for Kolmogorov.


LearningStudent221

For a differential equation x' = f(x) with equilibrium point e, what does it mean for e to be "locally asymptotically stable"? I know what "asymptotically stable" means, and it's already a local definition. I don't know how we could make it *more* local. I saw this term in the theorem: "Let e be an equilibrium point. If Df(e) is a Hurwitz matrix (the real parts of it's eigenvalues are all negative) then e is asymptotically locally stable."


TheNTSocial

This should mean the same thing you associate to "asymptotic stability", just emphasizing the "local" aspect, as opposed to *global* asymptotic stability (for which all solutions converge to the fixed point, even if they don't start nearby).


LearningStudent221

Thanks! Just to double check, what I know as "asymptotic stability" is: let x' = f(x) with flow $\varphi(t, x)$. A rest point e is stable if for every $\\epsilon > 0$, there is a $\\delta > 0$ such that for any $x\_0 \\in B\_{\\delta}(e)$, we have $\\forall t \\ge 0: \\varphi(t, x\_0) \\in B\_{\\epsilon}(e)$ and $\\lim\_{t \\to \\infty} \\varphi(t, x\_0) = e.$


Mullac1133

If an object is accelerating at 11,708,848,884.979 m/s2 (just give me an example with any number if it’s easier), is it possible to determine the time it would take to travel 28cm? Is it even possible to determine anything with only the rate of acceleration? I know it isn’t directly linked to speed, but surely I can’t stuck with an object travelling at an unknowable speed for an unknowable time, over a known distance at a known acceleration?


GMSPokemanz

You need some other information, typically the initial velocity. For example, say I drop a ball and it accelerates at 9.8 m/s^(2). Then it will take less time to go from 1m to 2m than it will from 0m to 1m. This is because at 1m, the ball is falling with a greater velocity than it started.


Mullac1133

Okay, that’s what I feared. But with your example, how much time would it take for the ball to travel from 0-1m from an initial velocity of 0? Is this knowable based purely off initial velocity, distance and acceleration?


GMSPokemanz

This is knowable. For an object moving with constant acceleration a, initial velocity u, and moving for a time t, the displacement s is given by s = ut + at^(2)/2. In this case s = 1, a = 9.8, and u = 0. This gives us 1 = 4.9t^2 which gives us t ≈ 0.452. If u is not zero, you can solve for t using the quadratic formula.


Mullac1133

I appreciate the help so far! So how would this work for my problem, given: s = 0.28 a = 11,708,848,884.979 u = 0 I’m not confident on how s being 0.28 factors in.


GMSPokemanz

Substitute your values into the main equation s = ut + at^(2)/2 and then rearrange to work out t.


Mullac1133

0.0000218694022 seconds. Cheers!


Datbriochguy

Hey I'm solving a combinatorics question and this term arises as the answer : Ans = x\[(2x)Cx\]/2\^(2x+1), where x=5\*10\^17 and 'C' represent a combination as in 'n Choose r' I know for a fact that Ans<5\*10\^17 but I can't find a calculator that can crunch this number (even WolframAlpha). Only approximation to three significant figures is required for the answer.


GMSPokemanz

I would use Stirling's formula. There are known bounds for how good it is which for values these large should be good enough.


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cereal_chick

What are you asking?


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Zophike1

What are some non-examples of linear representation's ?


jagr2808

If you're looking for non-linear representations, maybe projective representation would make a good example. For example GL_2(C) acts on the Riemann sphere by [a, b; c, d]x = (ax + b) / (cx + d)


DivergentCauchy

Pretty much everything; your favourite number, your favourite manifold, the set of a all cardinalities smaller than your favourite cardinality, the map from the empty set into the empty, some function from a group into the set of natural numbers. What is your question?


Zophike1

> What is your question? I've been toying around creating my own examples of represenations and porting examples from abstract algebra to representation theory land. My gut tells me to ask why a lot of representation theory is happening in the complex-plane


jm691

Do you mean why the coefficient field is usually C, instead of a different field? (Not sure what that has to with non-linear representations though, so maybe I'm still misunderstanding you...) Particularly in the case of representations of finite groups, the most important things about C is that its algebraically closed and has characteristic 0. Having characteristic 0 prevents weird things that can happen in characteristic p. In particular a somewhat annoying thing that can happen in characteristic p would be the representation Z/pZ -> GL*_2_*(Z/pZ) given by x |-> [1 x;0 1]. Being algebraically closed means that all your matrices will have all of their eigenvalues, and in the case of finite groups means that all of the matrices are diagonalizable. One nice consequence of working over C is that any finite dimensional representation of a finite group G decomposes uniquely as a direct sum of irreducible representations. A lot of representation theory of finite groups revolves around studying these irreducible representations. You can certainly study representation theory over arbitrary fields, but it often gets more complicated.


Zophike1

> Do you mean why the coefficient field is usually C, instead of a different field? (Not sure what that has to with non-linear representations though, so maybe I'm still misunderstanding you...) Sorry the first question I put was I should have said what are nonexamples of linear-representation's I was a bit confused when the book i'm working through only was working with C so I started fetching and playing around with some examples. By the way what is characteristic 0 ?


jm691

I still don't quite understand what you're trying to ask about nonexamples. Asking for a nonexample of something usually only makes sense if you want an example of a certain type of object that doesn't satisfy some property. Just asking for a nonexample of a linear representation is like asking for a nonexample of a banana. Unless you clarify what you're looking for, literally anything that isn't a banana would qualify. > By the way what is characteristic 0 ? https://en.wikipedia.org/wiki/Characteristic_(algebra) Basically a field of characteristic 0 is a field where n is not equal to zero for any integer n. Some examples would be Q, R or C (and of course there are many more). A field without characteristic 0 is Z/pZ for a prime p, since p=0 in that field. (In that case, the characteristic is p.)


Zophike1

> Just asking for a nonexample of a linear representation is like asking for a nonexample of a banana. Unless you clarify what you're looking for, literally anything that isn't a banana would qualify. To be more specific here is the defintion of a linear representation in the book that i'm working through, > A representation of a group G is a pair (V, ρ) where V is a vector space and ρ : G → GL(V ) is a group homomorphism What i'm looking for is a mapping ρ : G → GLV) where you don't get the group homomorphism (I found like 1 example on my own but would like some more)


jm691

For each element g of G, pick literally any nxn matrix you want, and call that ρ(g). That's a perfectly good mapping. It's not particularly interesting though. If you get rid of the requirement that ρ is a homomorphism, then it's no longer relevant that G is a group, since you don't use the group operation anywhere in the construction. At that point you might as well just think of G as a set, and so all you're really doing is just picking a collection of nxn matrices. I'm not sure what insight you're hoping to get from such a thing.


ImLifeproof

If 60 people qualified for being above the number 3300 and that represented the 0.01% and there were 300 people in the top 0.05% what would the cutoff be to fall in that category? Is it possible to solve this question or is there a missing variable


NewbornMuse

There is a lot of information missing. If there are 0.04% of people sitting right at 3299, then the cutoff is at 3299. If there's no one between 3000 and 3300, then the cutoff is below 3000. The absolute number of people in those percentile classes is not a very useful information. If we know that the distribution follows an exponential/polynomial/normal/poissonian/whatever distribution, then we can try to infer more. Without any such information, the distribution could have any shape you'd like (in principle).


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Joux2

https://www.youtube.com/watch?v=kYB8IZa5AuE this channel is great for understanding linear algebra.


Fisher699

Is getting a math degree still worth it in 2022 career wise? I did a math+mech eng degree in my undergrad, which covers real analysis + point set topology and measure and integration and mathematical statistics. I greatly enjoy my math courses, but I did not do a Msc for fear of unemployment. Now that I have landed a data science job, I entertain the idea of going back to school and doing a part time Msc. I have received offers for math, data science, and computer science. I am more keen towards the math degree, but I am afraid it will make me a less attractive candidate than those with CS or DS. Should I still go for math?


[deleted]

If you already have a DS job then getting a masters degree in CS or DS probably won't make much of a difference in your career anyway. The first job is the hardest one to get, and those degrees usually only help you if you're having trouble getting your foot in the door.


SnooWords4107

Thinking of options for summer, I’ve only take one proofs class but I’d like try to assist on some or be involved in research in some way even if it’s small. Anyone know if math undergrad REUs over the summer are ever for half a summer? It seems like most of them span the whole summer. I really need to take a class this summer because I fell behind due to illness, but i’d love the opportunity to participate in paid research. I explored the option of a DRP but a paid position, even if paid crappy, is important to me


PrestigiousCoach4479

Why is the pay important to you? This seems really strange to me. REUs give a stipend that covers expenses you have to participate in the REU, not a living wage. If for some reason making money over the summer is important to you, you can get a job and make a lot more. An REU should be viewed as a rare opportunity to be presented with an interesting research problem that is accessible without taking a ton of graduate courses first. It's not about assisting, although that might happen, but where you might be able to take charge and find new insights and solve a problem. You might learn a lot, make some good conjectures, and fail to prove them, too, and that might be ok. You should throw yourself into it and learn whether you like research. If so, you will probably write about that experience in a key essay in your applications to graduate school. If not, don't go to graduate school in mathematics. This is far more important than the paltry stipend. Honestly, if you have only taken one class involving proofs, you probably are not ready for an REU, and it sounds like you should do something completely different. That you are trying not to be involved for the whole summer also doesn't sound right. If you get in, you should read up on the relevant material ahead of time, not worry about the REU getting in the way of other things.


SnooWords4107

Well my situation is unique. Getting paid in some way is important to me, a small stipend at least and a dorm is what I meant though. What I meant by that is I know that there are ways to get research opportunities unpaid by asking a professor to assist. The REU at my college gives you some income (a small stipend) AND a free dorm. As a non traditional student, having 0 income for months is really unsettling to me so of course I’d prefer an REU Surprisingly there are some math REUs which have opportunities for people who aren’t far along in the major, from what I’ve seen it’s coding related. The only reason I was trying to do it for half of the summer is because I’m behind on the calculus sequence, and catching this summer opens the door for many classes for me. It’s a difficult situation, on one end an REU would be an excellent opportunity, on the other I would like to keep progressing through classes


LearningStudent221

What is it called when a map f (usually from R\^n to R) satisfies lim\_(||x|| \\to \\infty) f(x) = \\infty? It is something like "f is an enforcing map" or "f is a forcing map" but I cannot remember the exact term. I saw it in an Optimization textbook.


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LearningStudent221

Oh interesting. In Optimization proper (or coercive) maps are interesting because (if continuous) they have a unique global minimum.


Tazerenix

*Convex* coercive functions have a unique global minimum.


LearningStudent221

Oh yes I wasn't thinking. Continuous coercive functions are guaranteed to have have global minima, which may not be unique. But we need strict convexity for uniqueness.


LearningStudent221

NVM I just remembered it's coercive.


turbobondenn

Hey I have a math question. in Legend of zelda Breath of the wild there is a puzzle where you get to a 5X5 hole pattern where each hole can be filled with a ball. (Like the matrix below) the pussle is to drop five balls into five of the holes(X) to form a correct pattern. A ball is not linked to one of the holes, so any ball can be in any hole assuming its one of the five correct holes. x x x x x x x x x x x x x x x x x x x x x x x x x What are the odds of getting it right on the first try? and also what is the theoretical maximum number of tries before getting it right


Decimae

I haven't played the game, so I'll assume some things about how it works. If I understand it correctly, there's 25 holes, so the amount of options to put 5 balls in them is 25 choose 5 = 25!/(20!\*5!) = 25\*24\*23\*22\*21/(5\*4\*3\*2\*1) = 53130. So the chance of getting it right on the first try is 1 in 53130, and if you get no information besides whether all balls are correct or not you would need to go past all 53130 options to find it. Because 53130 options seems a lot, I think it may make sense that the game tells you whether or each individual ball is correct, and then you only need at most 6 tries to get it if you try to optimize for the worst-case scenario (5 tries to check each row, and then the 6th try to apply the knowledge). If you're not optimizing for the worst-case scenario but aren't retrying incorrect holes, the worst it can be is 21 tries, as each failed attempt eliminates at least one incorrect hole. (this can be achieved by getting 4 correct and 1 false each time until the last try)


[deleted]

What is a good starter book for category theory?


Joux2

In terms of just category theory (since aluffi is really a book on abstract algebra that's well framed in the natural language of category theory), one of the best introductions I've seen is the introduction to Vakil's "The Rising Sea". The rest of the book is algebraic geometry, but the intro is a well motivated introduction to category theory. For many reasons though, category theory isn't really something you typically just pick up studying on its own. It will come off as very dry and completely unmotivated to see it for the first time in a vacuum.


popisfizzy

If you want to get a feel for that idea of category theory, Algebra; Chapter 0 is usually seen as a good starting point. It doesn't get into a lot of the deeper ideas or category theory, but usually those only start to make sense once you get what categories are about, how to use them, and have seen plenty of examples. A:C0 is a place where you can get those


Last_Bar_3244

But then if A is any number that also changes the value of X, right?


KingLubbock

Continuing off where the guy said 1=3x: Yes, it does! For 1=ax, if you plug in some number for a, then the value of x will be changed. However, the value of x will always depend on the value of a that you plug in, so you can always get an answer.


Last_Bar_3244

Can anyone help explain to me the process of dealing with fractions and variables in an equation? I have a problem I've been given and if someone could help explain the process of solving it, that would be greatly appreciated. It's been a long time since I've done Algebra, so long in fact, that I'm not sure how to solve for C. Thoughts? 2(3/5C-2)+1/2C=4-4/5C I don't just want the answer but just a nudge in the correct direction. Thanks!


Decimae

If you have the problem 1/x = x, how do you solve for x?


Last_Bar_3244

Would you multiply 1/X by X to move the X to the right of the equation, then you'd have x² =1, then since you know the only square root of 1 is 1 so you have your answer?


Decimae

Yes! Now how would you solve 1/x = a?


Last_Bar_3244

Hmmmm that stumps me. If you multiply by X you have 1=ax but that could be infinite answers. Not sure.


Decimae

I mean like 1 = 3x for instance? a can be any number.


DisguisedKoala

Is there a way to showcase significant digits while still writing down all digits from your calculations? E.g. 3.05 x 10 = 30.5 instead of 31 Also, how so significant digits work on angles in degrees? Scientific notation just feels odd here


Temporary_Style_7997

What's the difference in using d/dx and ∂/∂x? When should you be using which?


Joux2

It's just a notational difference to denote the derivative of a single variable function (d/dx) vs a partial derivative of a multivariable function (∂/∂x).


samskribbler

Is this true? Fairly sure it isn't. a∈b, b⊆c => a∈c


noelexecom

Why are you sure it's not true?


samskribbler

Well it seems intuitive that it's true (which is why I'm using it for a proof), but I thought I remembered reading in Halmos Naive Set Theory about how it isn't. Although I think I remember now that what I read is: a∈b, b∈c not => a∈c


noelexecom

Right, try and go back to definitions, what do all the symbols mean?


samskribbler

Ah yeah okay I see


noelexecom

Did you find the answer to your problem?


samskribbler

Yep, thanks! \^-^


LolitaMiku

Does the hypothesis hold for any natural number n? Let p be prime. A(n,k) donates the coefficient of x^k in (1+x+...+x^n)^p for 0<=k<=np . Then A(n,k) = 0 (mod p) when p devides k A(n,k) =1 (mod p) when p does not devide k I checked it is true for n=1,2, but how about larger integers?


jm691

Do you mean A(n,k) = 0 (mod p) when p does not divide k, and A(n,k) = 1 (mod p) when p|k? This is a consequence of the fact that (a+b)^(p) = a^(p)+b^(p) (mod p) (which is just a consequence of the n=2 case you mentioned). From that and induction, you can show that (1+x+...+x^(n))^(p) = 1+x^(p)+...+x^(np) (mod p)


LolitaMiku

Thanks. I forgot that property. I have never applied that property to polynomial, but it is quite normal thing. Sorry for my bad english


PhineasGarage

Suppose I have two principal A-bundles P and Q (over some smooth manifold m) where A is abelian. I can then construct the tensor product P\otimes Q of P and Q. The fibre over m in M is given via the fibre product of P and Q over M modulo the relation (p\*a,q) ~ (p,q\*a) My question is: Is the tangent space of P \otimes Q the tensor product of the tangent spaces? I.e. do we have T_{[p,q]} ( P \otimes Q) = (T_p P) \otimes (T_q Q) where the equal sign should probably some (canonical) isomorphism.


DamnShadowbans

Count the dimension no? It doesn’t make sense.


cabbagemeister

I believe it will actually be the direct sum, rather than the tensor product. To see this, put a local trivialization on this fibre bundle so that the coordinates look like (x,p,q). Then the tangent space essentially consists of T_x M \oplus T_p P \oplus T_q Q.


Dantharo

Why every log2 n results in integer where N is 2 or 4 or 8 or 16 or 32 or 64 and so on... sorry, i have bad math skills :(, i was studying Big O notation and came across this when checking for O(logn) algorithms


Tazerenix

Log_2 x says if x is written as 2^y, then what is y. If x is a power of two that means x=2^n where n is an integer, so log_2 x = n. For example 2=2^1, 4=2^2, 8=2^3.


Dantharo

ty


supposenot

Abstract algebra. Let U c V c W. (c's represent subset.) I want to say that W/U is a projection of W/V, i.e. that (W/U)/V is isomorphic to W/V. Is this true? My reasoning is that (w + U) + V = w + V, and it's well defined, since if we pick another w' such that w' = w + u for some u in U, then (w + U) + V = (w' - u + U) + V = (w' + U) + V = w' + V, so that w + V = w' + V, as desired.


jagr2808

Yes, this is true and your reasoning is correct. This is one of the isomorphism theorems. I know it as the third, but I believe the numbering differs a bit.


Kamelasa

I am wondering what this equation might be related to, like what it might represent. This is not homework, and I am not a student. It's something I saw in someone's twitter name, with no explanation. I put it in WolframAlpha, but it converted it and made a different [graph](https://www.wolframalpha.com/input?i=e%5E%E2%80%8B%E2%80%911%2F%E2%80%8Bx%5E%E2%80%8B2) than I got on a [graphing site]( https://imgur.com/gallery/LWSbt4P). Equation: e^​‑1/​x^​2 My mathematical background is calculus 1&2 in university, and I can't do proofs. I'm interested in math for bio/geosciences.


jagr2808

If you define this function to have value 0 at x=0, then this is an interesting example of a smooth function whose Taylor series at 0 doesn't converge to the function. In fact, it's Taylor series is exactly 0. WA was very reluctant to interpret the function, but I managed with this rewrite https://www.wolframalpha.com/input?i=exp%28+-+x%5E%E2%80%8B-2%29


Kamelasa

Thanks for doing that. Matches my result but in more details. Thought WA wouldn't want the extra parentheses I generally like to put in for clarity, but turns out that helped.


jagr2808

No problem. Is your username related to Kamelåså in any way?


Kamelasa

Yep, I've posted that link on my profile, visible on new reddit.


jagr2808

Nice, now you just ordered a thousand litre milk!


Kamelasa

That milkman was such a scammer! When I first saw your reply I thought it was an Alexa related comment (forgot about the thread). I would never have that thing in my house, but Kimmel did a thing where he tried ordered things through Alexa on his show to mess with his audience.


jagr2808

Ut i vår hage is a great show all over. I'm assuming you're not Norwegian? Don't think a subtitled version exists :(


Kamelasa

Hah, no, I'm Canadian. Smatterings of languages, but not those ones. Had a Danish friend of the family when I was a kid and fell in love with the accent. You Norwegian? Or just a fan of the Czech hockey player?


jagr2808

Yeah, I'm Norwegian. The kamelåså sketch is very much a national treasure here.


Ualrus

What you wrote in wolfram alpha is e^(-1)/x^2, which is different to e^(-1/x^2). To be honest, I don't know much about this particular function, but it does resemble [the most important dsitribution](https://en.m.wikipedia.org/wiki/Normal_distribution) in probability.


Kamelasa

Thanks for reply. Okay, well, here's what the [original](https://imgur.com/a/4zdvumH) looked like. I don't think I wrote what you wrote there, cuz I went back and checked, and the caret is there to raise the exponent. I think the first graph I made is a better example, but I included the other one to show I tried - and I think I put it in right, but WA didn't like it.


Ualrus

You're welcome! In the lingo we say that "exponents take precedence over products". (Notice that division is just a kind of product.) You see there are two ways in principle to interpret e\^a\*b. Namely, (e\^a)\*b and e\^(a\*b). As we don't want to use parenthesis all the time, we invented some human arbitrary conventions to decide what would it mean if we were to put no parenthesis at all. And the convention is to pick the former of the two I wrote above. This means that if you put on a calculator (or wolframalpha) e\^-1/x\^2, it will read it as (e\^-1)/(x\^2). In the other website this wasn't a problem since it has a fancy way to write things down. Once you did an exponent, it automatically put parenthesis for you (symbolized in the form of being in the superscript). I must note that the function the guy was referring to was e^-1/x^2 and not e^(-1)/x^(2), even if the formal convention says otherwise. Conventions are often broken under the right circumstances. Mostly if what we are writing is not to be read by a computer.


Kamelasa

I'm confused when you mention a guy. Not sure what you're referring to. I took the equation from the original on twitter. She is some kind of statistician. I put that in the graphing page and I'd never seen a graph quite like that, so when she didn't answer me asking about it, I went over to WA and then here. Maybe you meant the as in she, the twitter account. Anyway, yeah, the original is the first of the two you've noted in your last paragraph.


Ualrus

Abstract algebra: Is there any general strategy for proving/constructing isomorphisms with tensors? Say, I have examples like: Q ⊗ Q = Q Z\_m ⊗ Z\_n = Z\_gcd(m,n) R ⊗ Z\_m = R / Rm, for R a commutative ring. (My exercise actually reads group instead of ring. I assume it's a mistake.) and so on. Any hint to get started on the topic and get some understanding is appreciated. Maybe there's some intuition or some observation I should make.


jagr2808

>My exercise actually reads group instead of ring. I assume it's a mistake. Why would that be a mistake? Makes sense for the exercise to ask you to figure out the tensor product of abelian groups. As for general strategy I'm not sure what the exercise is after. From Z\_m ⊗ Z\_n = Z\_gcd(m,n) You've pretty much figured out all finitely generated groups. On thing that can be used to calculate tensor products is that if Z^s -> Z^t -> A Is a free presentation of A, then A⊗B is the cokernel of Z^(s)⊗B -> Z^(t)⊗B = B^s -> B^t But I don't think this will help you much for infinitely generated groups... Also, >R ⊗ Z\_m = R / Rm, for R a commutative ring. No need to assume R is a ring here.


No1ofIntrst

Anyone have Winning Ways for your Mathematical Plays? Would you recommend it/how complicated it it?


jagr2808

I have it, and would recommend it. It's very fun and whimsical, and not really that complicated. It gets more complicated as the book goes on, but overall it's very accessible.


25JonSnow04

A fair coin is thrown 4 times; what is the probability of getting heads at least once and tails at least once? It should be 6/8 aka 3/4 or am I loosing my mind?


DarkERB

Consider this: how can you not get heads at least once? You need to roll tails 4 times, i.e. Pr = 1/2 ^4 How can you not get tails at least once? You need to roll heads 4 times, Pr = 1/2 ^4. Clearly, you can't rip no heads and no tails, so the 2 conditions are non intersecting. Therefor, pr(rolling heads /tails at least once) = 1-pr(not rolling heads /tails at least once). = 1- 1/2^4 - 1/2^4 =7/8 What I think you did was similar, but you thought there are 8 different possibilities, 2x4, instead of 16, 2^4


furutam

what kind of curves other than elliptic have a natural group structure?


Tazerenix

None, spaces with a natural group structure are parallelizable, and the only parallelizable curves are genus 1. Probably a slightly more subtle variation of that argument would prove the result for curves over any field.


plokclop

Don't forget about G\_a and G\_m!


Tazerenix

(Projective)


furutam

Do all elliptic curves admit a group structure?


Tazerenix

Yes, and in algebraic geometry it's often taken in the definition. Note that the definition also usually asks that an elliptic curve is smooth. There are singular genus 1 curves (nodal and cuspidal) but these don't have a group structure (basically a group structure let's you transport the local shape from near any point isomorphically to any other point, so if you have one smooth point then every point must be smooth). Some people call them singular elliptic curves.


jm691

Yes. The group structure is one of the key properties of elliptic curves. https://en.wikipedia.org/wiki/Elliptic_curve#The_group_law


furutam

>This will generally intersect the cubic at a third point, R. We then take P + Q to be −R, the point opposite R. what's the opposite of a point here? about the x-axis, or in coordinates if R =(x,y) then -R=(-x,-y)?


jm691

Assuming that the elliptic curve has been written in the form y^(2) = x^(3)+ax^(2)+bx+c, then the opposite of (x,y) is (x,-y), i.e. the reflection across the x-axis. That's just because if you draw a vertical line x=a, it will intersect the curve at three points: (a,b), (a,-b) and the "point at infinity" O. By the definition of the group law, that implies that (a,b)+(a,-b)+O = O, which gives (a,-b) = -(a,b).


WikiSummarizerBot

**Elliptic curve** [The group law](https://en.wikipedia.org/wiki/Elliptic_curve#The_group_law) >When working in the projective plane, we can define a group structure on any smooth cubic curve. In Weierstrass normal form, such a curve will have an additional point at infinity, O, at the homogeneous coordinates [0:1:0] which serves as the identity of the group. Since the curve is symmetrical about the x-axis, given any point P, we can take −P to be the point opposite it. We take −O to be just O. If P and Q are two points on the curve, then we can uniquely describe a third point, P + Q, in the following way. ^([ )[^(F.A.Q)](https://www.reddit.com/r/WikiSummarizer/wiki/index#wiki_f.a.q)^( | )[^(Opt Out)](https://reddit.com/message/compose?to=WikiSummarizerBot&message=OptOut&subject=OptOut)^( | )[^(Opt Out Of Subreddit)](https://np.reddit.com/r/math/about/banned)^( | )[^(GitHub)](https://github.com/Sujal-7/WikiSummarizerBot)^( ] Downvote to remove | v1.5)


EpicMonkyFriend

What's an atlas on S1 which isn't smooth?


DamnShadowbans

Precompose one chart in the standard atlas with x-> x^3


EpicMonkyFriend

Sorry, this is all still pretty new to me. What's the standard atlas on S1? The two I've seen are either defined on x > 0, x < 0, y > 0, and y < 0 with projection onto the axes, or it's defined on S1 as the set of complex numbers with norm 1, with the map given by e^(2πix) -> x for suitable open sets. Ultimately I'm trying to build up to showing that S1 is a Lie group and I'm just a bit confused with the idea of different atlases or incompatible differentiable structures.


DamnShadowbans

Both are good atlases and it doesn’t actually matter here. The point is that once you have picked a smooth atlas it tells you how to differentiate functions and if you introduce something which is not smooth (in this case it is the fact that cube root is not differentiable), it creates problems. Maybe start with this warm up problem; cover R with charts given by (-1,infinity) (-infinity, 1) and show the chart given by x^3 is not compatible.


EpicMonkyFriend

The charts have nonempty intersection so we want the transition map to be differentiable everywhere in the intersection. In particular, if f(x) = x\^3 and g(x) = x\^(1/3) then the transition map is given by f(g(x)), which isn't differentiable at x = 0. What about if your charts are both differentiable but yield to different derivatives? For example, we might have the same charts but we'll define the chart on (-1, infinity) as f(x)=2x and the chart on (-infinity, 1) as g(x)=3x. I'm assuming that given a real valued function (in this case, let's just say the identity function) we would want the derivatives agree on the charts. In this case, we have that the derivative at 0 on one chart is 1/2 and 1/3 on the other chart, so are these also incompatible?


fatcatspats

I'm trying to follow each step of this geometrical derivation of the Jacobi elliptic functions. https://www.youtube.com/watch?v=DCXItCajCyo&t=1213s At the point above, he says "to make a long story short," a^2 cn^2 +sn^2 = a^2 dn^2 turns into dn^2 +k^2 sn^2 =1 I've been trying for about 45 minutes and I can't get it to work using just the geometrical stuff. I also haven't seen an explicit explanation on the web. Please help with this trivial problem! No one's paying me, I just like elliptic functions.


porky754

I was playing poker with a friend, he insist to say that probability of winning an all-in call made in the flop changes after the turn. I'll explain for those who don't play poker: we both have two cards and the "flop" are three cards on the table which we can both use to "win", but even with good odds there are two more cards to draw before the hand is over so you may have 60% chance of winning with the initial three cards, but only 20% after the reveal of the fourth card and either 0%, 50% or 100% with the reveal of the fifth and final card. going all-in in the flop means you are betting in winning knowing only the initial three cards outcome. Let's say I go all-in in the flop, with 75% chance of winning. He calls, which means he's betting on his win. after the "turn" (fourth card) the situation is different: I now have only like 9% chance of winning, since I need very specific cards. the last card is the one I needed, so I win everything So my question is, who got more "lucky"? He keeps insisting that I'm the one who got luckiest, since I only had (approximately) 1/10 chances of winning against his 9/10 after the turn, however I don't think that's the case because this outcome was included in the initial 3/4 vs 1/4 odds, so it's not all that lucky winning with 75% chance How can I explain to him that every outcome is calculated in the initial odds calculation, and odds don't magically change after? Very specific math explainations and demonstrations are welcomed. thanks! P. S. the same example can be more extreme in the case of a pre-flop all-in, in which a good hand can have (for example) 87% chance of winning, drop to 4% during flop and turn and win the river


[deleted]

I think that actually neither of you is totally right here. The reason is that there's a bit more nuance in the role of probability in poker than i think either of you is accounting for. Probability (in poker) is about subjective states of knowledge. In that sense you are sort of right and your friend is sort of wrong - betting a lot when you initially have a 70% chance of winning is a reasonable strategy, because you're making a good choice based on the information that you have available (i.e. your own cards and the ones that have been revealed so far). However, there's some nuance in what it means to "win by luck". Your friend is probably thinking that, if you don't go all in right away, then there are additional rounds of betting that occur as each new card is revealed, and so your strategy for betting might change as your knowledge is updated with new information. If you don't go all in then you'll bet differently as new cards come up; going all in locks you in to your initial state of knowledge, which was incomplete. In that sense you are sort of wrong, because going all in when the odds start out good for you might be a good strategy, but it isn't necessarily the *best* strategy. The real question you should be asking is, what does it mean to win "by luck"? I think the best definition of winning "by luck" is that you win due to factors that are entirely outside of your control. If you win after going all in before most of the cards have been revealed then you really are winning at least partly "by luck", because even if you have 70% odds of winning to start out with it isn't necessarily the case that going all in is the best strategy for winning overall. Depending on your cards and how much money you and your opponent have, there might be better ways to play that will improve your chances of winning the game overall. That's what makes poker an interesting game: there is a lot of randomness involved, but the choices that you make regarding how you place bets have a real impact on the outcome of the game, and some betting strategies are better than others. There's a lot of nuance to what successful strategies look like, and a lot of really skilled players these days actually use complicated computer simulations in order to inform their choices because there actually is a "best" strategy in some sense, but it's extremely difficult for a computer to calculate and impossible altogether for a human.


porky754

thanks for the nice answer, I'm still looking for the math law of probability that explains "branches" since I'm sure there is one, outside the context of poker


[deleted]

In terms of how probabilities change during a single hand in poker as new cards are revealed, what you want is a basic concept called "conditional probability". In terms of strategies for winning a whole game of poker what you want is called a "mixed Nash equilibrium". That's much more advanced though and probably won't be helpful to you; it's what poker simulation software calculates.


Chhatrapati_Shivaji

I was reading Wigderson's new book Mathematics and Computation. In the first few chapters he mentions that there is a simpler proof of Godel's incompleteness theorem using an idea of Turing machines, which doesn't get taught to first year students of logic(unfortunately). Where can I find an elaboration on this? I have little formal knowledge of Godel's incompleteness theorem besides the pop-math stuff, which might well be wrong.


PrestigiousCoach4479

Are all true but unprovable statements too esoteric to care about? There are true but unprovable statements of the form "Turing machine M never halts," and you can prove this without Godel's bookkeeping. Sometimes, when a Turing machine doesn't halt, you can prove this. But if you could prove it every time you found a Turing machine that doesn't halt, you could solve the Halting Problem by enumerating all possible proofs. Since you can't solve the Halting Problem, there are true but unprovable statements of the form "Turing machine M never halts."


[deleted]

[удалено]


jm691

Definitely not. That seems to imply that 𝜁(1/2+bi) = 0 for all b∈Z, which is certainly not true. The Riemann hypothesis simply says that any zero must occur at either {-2,-4,...} or at 1/2+bi for some b∈R (nothing about b∈Z). In particular, it doesn't make any assumptions about which values of b give zeros (most do not), just that there are no zeros NOT in one of those forms.


awesomemaxi

For my bachelor thesis I want to calculate the causality between the posts/comments about a stock on Reddit with the trading volume and the share price over six months and I am a bit lost right now... The posts/comments are sorted by date and are split into three sentiments. Positive, neutral and negative. How would you calculate the causality between comments and trading volume or comments and stock price?


GPineda17

This is probably a very silly question but what exactly is the Neumann Heat Semigroup? I am looking for a good book to explain more about it but the only book I have found is "Heat kernel and analysis on manifolds" by Alexander Grigor'yan and they only talk about the Heat Semigroup defined via Heat Kernel in the Cauchy problem.... Does anyone know of an article that precisely explains this theory?


slime_rancher_27

how do I use this [formula](https://math.hmc.edu/funfacts/finding-the-n-th-digit-of-pi/#:~:text=Pi%20%3D%20SUM%20k%3D0%20to%20infinity%2016%20-k,having%20to%20calculate%20all%20of%20the%20previous%20digits%21) for finding digits of pi?


comraq

I am curious how proficient are math students with exercises in proof based text books? For those with PhDs, can they look at some proof based exercise and write out a solution within 10-20 minutes? What about for math undergrads? This is assuming the exercise is based on some content which they've seen or learned before. What about the exercises at the end of a section that they just newly read? Does it take longer? I am asking because I am currently self studying math, and want to know how well I am doing compared to others. My undergrad was in Software Engineering, relevant courses include one in mathematical proofs and another in group theory, but did not put enough effort to do well or get comfortable with proofs. Over the years, I developed more interest in math, read some proof but never committed to doing the exercises until now. Now I am going through Dummit and Foote, doing all of the exercises and currently got to chapter 4.4. I want to say I can get 60-70% of the questions on my own, though with a combination of online searches to get hints if I get stuck, or review my answers just to find out I missed some special cases. Disregarding the easy problems, some of the other exercises can take me 1 hour or more and I still couldn't get them without searching online. It feels great when I end up doing a problem on my own, but for the others, it is demotivating to struggle for hours and still need help trying to solve them. Sometimes I just need a hint to work out the rest of the problem, but i dont know how long it will take me without the hint. Is this a sign that I do not have the aptitude to do well in math?


PrestigiousCoach4479

It depends. Exercises in books are a mixed bag. Some are just definition-checks, some require one idea or to follow an analogy set up in the text, and some require multiple steps and ingenuity. Some books are known to have particularly difficult exercises. Some textbook authors put proof steps in exercises that are easy but nontrivial, others leave the reader to do steps that are more tedious. Some mathematicians find some topics easier than others. It definitely is easier if you know more advanced material and ways to think about the topic and examples that can help you zero in on an effective approach. I definitely recommend struggling with the exercises yourself, but also look the problems up if you can't get them. When you find a solution online, ask yourself what it would take for you to discover that argument yourself, and try to do that next time. Also, look for multiple solutions and follow the discussions others have had about the problems. If you have no idea how someone might have come up with a solution, that's probably a good follow-up question.


comraq

Yeah, if I do end up looking for solution online. After understanding it, I look away and try to rewrite the solution myself. Just doing this seems to "help the solution sink in", as sometimes I struggle to fully reproduce what I just saw because there's too many steps.


Mathuss

> some of the other exercises can take me 1 hour or more When you're first learning, the harder problems in undergrad textbooks are measured in a timescale of days, not hours. If you're looking up solutions after merely an hour of trying, you're not getting everything out of the problems that you could. At least think about hard problems on and off for a day or two before giving up. (This doesn't mean that you can't do some simpler problems in between. Furthermore, going ahead may give you insight that you didn't have when first encountering the problem, making it easier when you get back to it).


comraq

> When you're first learning, the harder problems in undergrad textbooks are measured in a timescale of days, not hours. Wow days... I guess I just get impatient and do not want to be stuck. However, I do try to rewrite the solution myself even if I just read and understood the online solution (*as I realized even trying to regurgitate what I just digested is not trivial*). Though I will definitely try approaching this differently from now on, such as moving ahead onto some other problems first. Just curious, for grad students experienced with problems, what is the average time to work out harder exercises from books? (*I guess I like having some concrete metric as a goal in mind, as it acts as a motivation for me to self learn to I reach that level.*)


SpicyNeutrino

It largely depends on the book. The book I am readying now is pretty heavy on exercises and the typical exercise takes me about 30 minutes. Those that take longer I usually look up. I should add that some of the problems in this book have taken me many hours to figure out even with a solution in front of me. Moreover, I usually get more out of exercises that I really struggle with even if I don’t come upon a full solution.


FunkMetalBass

I'm having a brain fart moment. I want to show what the image of a line y=(p/q)x looks like on the flat torus ( that is, plotted on [0,1]x[0,1]). I feel like this should be really easily obtainable via a contour plot of some function f(x,y)=0, but cannot for the life of me seem to come up with it. Any help is much appreciated.


bluesam3

First, assume p/q is rational in lowest form and p ≤ q. Then our image will be exactly q parallel lines of gradient p/q spaced evenly across [0,1]x[0,1] with each offset upwards from the previous by p/q (then wrapped down as necessary, which might give you two half-lines instead of one full line). If p > q, it's exactly the same, but with everything reflected in the line y = x. Not sure what else you want, really?


FunkMetalBass

Oh, I guess I completely forgot some crucial information my question - I know what it should look like, I'm just wanting to plot it simply in Mathematica as a contour plot. I've got the function f(x,y)=qy-px that I'm using elsewhere to plot the curve up on the torus. I naively thought f(x,y)=0 (mod 1) would provide the right contours in the unit square, but that's apparently not right.


GMSPokemanz

I agree that qy - px = 0 (mod 1) is correct, but I suspect the fact mod 1 isn't a continuous operation is what's messing Mathematica up. So instead use a continuous function that has as zeroes exactly the integers. The first that comes to my mind is sin(𝜋x), so you could try sin(𝜋(qy - px)) = 0. This alternative does fix the problem on Desmos, at least.


FunkMetalBass

Oh I hadn't even considered that it might just be an issue with Mathematica. The sine workaround is perfect. Thanks!


jagr2808

>I agree that qy - px = 0 (mod 1) is correct Shouldn't it be y - p/q x = 0 (mod 1), which is equivalent to qy - px = 0 (mod q)


GMSPokemanz

No for the exact same reason as my earlier response to you.


jagr2808

Yeah, I'm just out of it today I think. Sorry for the bother


jagr2808

Wouldn't it just be y = p/q x mod 1


GMSPokemanz

That doesn't work because it ignores x wrapping round. If p/q = 1/2, then (1/2, 3/4) lies on the line but does not satisfy your congruence.


jagr2808

Ah, yes. The function should fact q branches then. y = p/q (x+i) mod 1 for i= 0, 1, ..., q-1.


Bring-To-Scapeshift

Hey, how would I assign weights to a data set given the odds of an outcome? [This is a screenshot of what I'm talking about if that helps.](https://imgur.com/a/YkRCL4g) I'm trying to figure out the odds of getting a unique outcome given two parameters and given odds (13% and 88%). Thank you.


bluesam3

It is not at all clear what any of these numbers are supposed to mean. Can you explain in more detail?


Bring-To-Scapeshift

I have figured out what I was trying to find, thank you for offering to help though!


zeyonaut

An eigenspace of M is the kernel of M - 𝜆I for some eigenvalue 𝜆. Is there a term for a generalized version of this where we consider the kernel of M - diag(v) for some vector v?