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[
"Mathematician Carina Curto Thinks Like a Physicist to Solve Neuroscience Problems | Quanta Magazine"
] | [
"math"
] | [
"8sarej"
] | [
21
] | [
""
] | [
true
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false
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0.83
] | null | I'm sure I'll be lambasted for disagreeing, this being a maths forum and all, but I'm not aware of any instances in which the level of rigor demanded by mathematicians was necessary for successful modeling in physics. Our objective is not to appeal to the sensibilities of mathematicians, but to model reality. If the model works, why should physicists care about rigor? Understand, I have the utmost respect for mathematicians and their work, but I take issue with your arrogant dismissal of the massively successful methods of physics. | SM fails spectacularly at matter v antimatter, at total mass in the universe (dark matter), at expansion (dark energy), and quite possibly even at describing neutrinos (recent confirmation of sterile and all). Sure, it's great. But we know it's wrong. you're really splitting hairs here. you're saying something is wrong if it isn't 100% absolutely correct, but by this standard all of physics is wrong. it's not the heawood conjecture, you don't get to demand that everyone else disregards it because it isn't correct in this one instance. Also what does it mean to fail "spectacularly"? You are making this more dramatic than it actually is. Why don't you list everything that it doesn't "fail spectacularly" at modeling? Probably because that would take you way too long. | And here I was thinking that it was their complete and utter disregard for rigor and for the fact that many of their beautiful models are mathematical gibberish that was holding them back. | It's when physics people pretend that such a thing exists and looks like the 2D version that things get screwy. Yes, but the amazing thing is that it does work anyway! Well enough at least to make some impressive predictions. I agree that it is important to recognize the issue and to eventually solve it. But physics is not math and I do not think it has to be. I don't think that the current confusion about the right "next step" is caused by the lack of mathematical rigor | the massively successful methods of physics But that's my whole point, they aren't massively successful. In fact, if I understand correctly, the issue is that you folks actually have a serious mass problem (couldn't resist that one). I'm not dismissing the nonrigorous aspects of physics as useless; I'm dismissing the clinging to nonrigorous mathematical models as hindering all of us. I am firmly convinced that the mathematical issues with formalizing QFT are intricately connected with the physical issues of QFT. We know that "Standard Model + Gravity Somehow" doesn't quite work. We know that both physically and mathematically. I am quite certain that the resolution to the problems on both sides is the same. Please continue your nonrigorous explorations of reality, we need that. But please don't get lost in faulty mathematics just because it looks pretty; that helps no one. I mean, I am a mathematician who, for purely physical concerns, is more or less entirely convinced that the very axioms on which the mathematics of the past century has been built are simply "wrong" when it comes to modeling reality. When I say don't cling to faulty math, I mean it. Edit: to put it more bluntly, I'm not arrogantly dismissing non-mathematically-rigorous physics, I'm arrogantly dismissing non-physically-relevant mathematics. Edit 2: I hope you don't get lambasted for that comment, I think it was exactly on point and I'm glad to hear a physicist perspective even if I do think you misunderstood (equally, I didn't do a very good job of explaining) what I was trying to say. |
[
"Could Positive/Negative numbers be differentiated from Addition/Subtraction?"
] | [
"math"
] | [
"8sa6ak"
] | [
0
] | [
""
] | [
true
] | [
false
] | [
0.3
] | I am interested with idea that positive and negative numbers be written differently than when using addition and subtraction. For example if there were Uppercase and Lowercase numbers to differentiate between positive and negative. I understand that there is a lot of overlap. But would having separate notations for each add a level of clarity? Also, are there any other cultures were they don’t use the same symbols (+ & -) for both positive/negative and addition/subtraction? Edit: spelling | The way we define addition today is that a-b is really a+(-b). The point here is that the same symbols are used because they describe exactly the same phenomenon. It's not a coincidence. | Why would you gain clarity by giving two names to the same phenomenon? And how does the real world come into this? | 1/2 + 1/3 = 5/6 and 1/2 x 1/3 = 1/6 Don't be a dick, especially when you're the one who is wrong. Multiplication as repeated addition stops working the moment you leave the naturals/get past fourth grade. | Those should not be thought of as different. | Addition is not the same as multiplication. Compare: 2 + 5 = 7, 2 × 5 = 10, but 7 and 10 are not the same number. |
[
"Measure theory and probability"
] | [
"math"
] | [
"8s9qjd"
] | [
11
] | [
""
] | [
true
] | [
false
] | [
1
] | I am currently writing a small assignment on non-measurable sets, Vitali sets to be exact. It's for the course introduction to probability theory, and instead of it being an essay on measure only, I want to incorporate probability theory. Does anyone have any sources which explain the need for the sample space \Omega to have measure, or even the event space \mathcal{F}? Thanks in advance | Nonmeasurable sets aren't nonmeasurable for every measure. On any space you can define all sorts of measures which make every set measurable, for instance just pick an element of your space and declare that any set containing it has measure 1 and the rest have measure 0. The nonmeasurability issue comes up when we specifically want to consider the space [0,1] and we want to say that Prob([a,b]) = b-a, i.e. that the measure of an interval is its length. Once you make that assumption, things like the Vitali set can be shown to not have a well-defined value for their measure. Edit: once you know there are sets which aren't measurable and are trying to do probability theory, you have basically concluded that there are sets of real numbers which cannot be thought as events (since they have no well-defined probability). This is why probability theory is much more about the study of a sigma-algebra equipped with a measure than it is about analysis. | Are you asking how to show the Vitali set is not measurable? Wikipedia proves it in about a paragraph. Are you asking why being nonmeasurable means the probability isn't well-defined? A probability is a measure so having a well-defined probability == being measurable. | Maybe it will help if I just do the Vitali argument at the level of probability. Let V be a set in [0,1] such that for every r in [0,1] there exists exactly one v in V s.t. r = v + q for some rational q, aka a Vitali set. For each q in Q, write V_q = { v + q mod 1 : v in V }. By construction, [0,1] = Union[q in Q] V_q and the V_q are disjoint. Suppose Prob(V) = c is well-defined. Then Prob(V_q) = c also since the uniform probability measure on [0,1] (aka Lebesgue measure) is translation invariant. So 1 = Prob([0,1]) = Prob(Union[q in Q] V_q) = Sum[q in Q] Prob(V_q) = Sum[q in Q] c. But this is impossible since an infinite sum of the value c is either infinite (for c ≠ 0) or zero (for c = 0). So such a c cannot exist and we conclude that V has no well-defined probability, i.e. is nonmeasurable. | No, there's no contradiction at all. If F has no measure, that's fine. There is nothing contradictory about the existence of non-measurable sets. They are in some sense pathological, but they're not false. But if you want to do probability theory, that situation is useless to you, because "probability" just means "a measure where 𝜇(X)=1". The issue with the Vitali set in particular is that it shows that not all subsets of R have a well-defined "length". There is no value you can assign to the Vitali set which will obey measure properties and remain consistent with 𝜇([a,b])=b-a. You could prove by contradiction - assume the Vitali set has measure m, and deduce something false. Moreover, given an event space F, there's always way to assign a measure to it. For example, the Vitali set has a perfectly well-defined measure if you take 𝜇 to be the counting measure. It's only when you want measure to represent length that the Vitali set is a problem. | Sure, but that measure won't be the uniform measure on [0,1]. When people actually use probability for anything, they want it to match up with intuition. If someone speaks of choosing a random number between 0 and 1, they want the answer to be that half the time it will be in [0,1/2] and half the time it won't. As soon as we make that assumption, we end up needing to use the Lebesgue measure to define probability. Your measure would somehow say that when I pick a random number between [0,1] that it is always rational. This is clearly not what anyone intends. |
[
"Is there a fast way to determine if a root exists in a set of intervals for a multidimensional polynomial?"
] | [
"math"
] | [
"8s9f2f"
] | [
5
] | [
""
] | [
true
] | [
false
] | [
0.74
] | Say that we have an n dimensional polynomial of degree m. Are there any methods to check whether a root exists when x_k∈[a_k,b_k] other than directly attempting Newton's method to solve for one? Again I am only trying to prove the existence of at least one root in the set of interval | The word for this is non-negative :) | It requires convexity to be sure. Suppose you have two minimums and we are aware that there is a positive point. one lying above zero while the other is below. Newton finds only one of them, so your conclusion depends on which is found. You may be interested in the intermediate value theorem though. | Sub in x = a and x = b. If one result is above the x axis (positive) and the other is below it (negative) then you know there is a root somewhere in between. If they are both positive or both negative, then pick some other numbers in the interval until you find a pair of numbers that are different signs. | I see. Newton's method isn't ideal for finding a root in this case, since the gradient of f is 0 wherever there's a root. You could try solving for when the gradient is 0. This will give you the extreme values, and then you can check if one of them is 0. | In this case all the roots will be at points where the gradient is 0. (As the gradient will be going from negative to positive as it passes the point) So find the derivative of your polynomila and do what I said in the previous comment :) EDIT: PLEASE NOTE not all points of 0 gradient will be roots, but all of your roots will have 0 gradient. |
[
"Definition of a set"
] | [
"math"
] | [
"8s8c1a"
] | [
0
] | [
""
] | [
true
] | [
false
] | [
0.5
] | null | But then the elements would be related to each other as they all share a set! | not sure why so many downvotes. this is absolutely the right answer. Want sets which only contain related elements? You can have that. Eg set theory with urelements or structural set theory. Want sets with no restrictions on whether elements of sets share any properties? You can have that too, eg material set theory. Neither of these examples meet your needs? Come up with something else. It's mathematics, you can do whatever you want! | Yes. | OP's plan foiled again by you scheming mathematicians. | It's mathematics, you can do whatever you want! :) |
[
"Can someone explain this to me?"
] | [
"math"
] | [
"8s8zoi"
] | [
0
] | [
"Image Post"
] | [
true
] | [
false
] | [
0.42
] | null | This sounds like a homework question, for which there already is a subreddit for. (I think the sidebar gives directions, am on mobile so I can't see). I'm unsure of exactly what you're asking, but have you tried inserting y into your root equation? | yep | Yes, it is. When any function intersects another they must be equal at that point. Say we call f(x) + g(x), h(x). So when h(x) intersects g(x) then h(x) = g(x). But h(x) = g(x) + f(x), so we're saying that at the intersection g(x) + f(x) = g(x). Obviously that can only be the case if f(x) = 0. The same logic will apply to f(x), f(x) will only intersect h(x) when g(x) = 0. The logic makes sense but I wouldn't expect you to have to figure it out on a GCSE paper without it being taught to you | If you're looking for approximate solutions, you should re-plot the y-curve as it has a root in 0. Then it should be of more help. | For further clarification I'm given the two equations listed and have to find approximate solutions to x - 4x + 2 = 0 To do so I have to find a line that will help me find the roots of x - 4x + 2 = 0. How do I find that line? |
[
"Found this in my old notes and wondered what kind of curve it is. How would one prove/disprove whether it is space-filling or not?"
] | [
"math"
] | [
"8s8zzd"
] | [
511
] | [
"Image Post"
] | [
true
] | [
false
] | [
0.95
] | null | 3Blue1Brown created a YouTube Video on exactly this subject. You'll see an explanation and application of space filling curves in his video. As a programmer myself, I enjoyed the application of what appeared to be a pure math topic to solve a practical problem in a way to help the disabled. I hope this helps out! | Already aware of the video, but thanks! :) It's a great introduction to the concept of space-filling curves. He explains halfway through that a curve has to be to be a space-filling curve (coincidentally, /u/Oscar_Cunningham , this is why a Z-order curve is not a space-filling ). However, I don't know enough about the maths involved to determine if this is or isn't. My hunch is that it is continuous, we ease our definition of curves to allow for a single diagonal step. But I'm not sure if that is "allowed" or not, mathematically. Even if it isn't, I guess this is at least very close to space-filling: taking the limit essentially results in lots of H-curves, connected by a diagonal seam. The H-curve is known to be space-filling, so that part we don't have to worry about. In that case, only points connected diagonally would not be space-filling. | Context: I'm a programmer. Last year I was investigating space-filling curves for data-visualisation purposes, and came up with the bottom... curve? Is it even a curve, mathematically speaking? The idea came from this paper[0]. Through that I found another one[1] describing the "H-curve" (which indirectly lead to a submission on /r/programmerhumor , but I digress). For my own needs I preferred a curve that would have the start and end point as far removed from each other as possible - like with the Z-curve, but with great locality otherwise, like the H-curve. So this was the result. I never got around to implementing this, but I'm still curious about the mathematics of space-filling curves. I just don't have any clue where to start. [0] Martin Wattenberg. IBM Research http://hint.fm/papers/158-wattenberg-final3.pdf [1] Rolf Niedermeier, Klaus Reinhardt, and Peter Sanders. International Symposium on Fundamentals of Computation Theory. Springer, Berlin, Heidelberg, 1997. http://users.informatik.uni-halle.de/~ahyjb/meshind.pdf | The diagonal move that cuts across the entire width of the curve is non-continuous in the limit. A space filling curve maps the unit interval [0;1] to the unit square [0;1] . As you use more and more iterations, the sub-interval that maps to the cross-cutting segment gets smaller and smaller, but the length of the cross-cutting segment remains the same size. In the limit, an infinitely small sub-interval maps to a non-infintely-small segment of the curve, which is the same as a discontinuity. | He said he was using it for data visualization. I know that the Hilbert Curve is used in this visaulization of the IP address space: https://www.xkcd.com/195/ If you start in the top left corner at 0 and then draw a line between consecutive numbers you will draw a hilbert curve (or maybe technically it is only called a hilbert curve in the limit, where it is filling all space, but then a sort of proto-hilbert curve at least) I guess the general pattern is that if you have some locations (IP addresses in the example) which are one dimensional (IP addresses are just numbers) and want to draw a two dimensional map of them, you can lay the numbers out according a curve. If you choose a curve with good locality, you will get the property that close numbers will generally be close to each other, which is a good property for a map. |
[
"Fourier Transform question"
] | [
"math"
] | [
"8s8urr"
] | [
5
] | [
"Image Post"
] | [
true
] | [
false
] | [
0.64
] | null | fourier transform answer | Your question is how to compute Fourier decomposition by just looking at a graph?? | If you're just given the graph, then you can take a sample of data points from the graph and do a fast fourier transform | Can you give that in terms of a summation of answers with different periods? | It doesn't make sense. If you are given the resultant, and told to find the constituent harmonics, your question is how to find the resultant? You said it was given... |
[
"Statements about convergence of products"
] | [
"math"
] | [
"8s7m5i"
] | [
1
] | [
""
] | [
true
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false
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0.67
] | Today in our complex analysis class we had the Weierstrass factorization theorem and as a application sin(Pi*z). So my question is, if there is any formal theory or criterias for the convergence of infinite products and the quality of the approximation if you stop adding terms, analouge to the one of series. I've looked around for a bit already but couldnt find anything. | Well you can always take a log of it to make it a sum (and the log is continuous) so in a sense sums and products are similar as far as limits go. | For real numbers, an infinite product of positive terms is related to an infinite sum by exponentials and logs. The product converges if and only if the sum converges . I'm not sure what happens for complex-valued products. | Wikipedia is your friend : The key idea is that infinite products (with positive factors) behave just like the infinite series of the natural logarithms of those factors, and if the series converges, then the infinite product converges to e to the power of that sum. | Non-Mobile link: https://en.wikipedia.org/wiki/Infinite_product#Convergence_criteria /r/HelperBot_ /u/swim1929 | Thanks for the reply. I started searching from the factorization theorem and products in general. It didn´t come to my mind to just search for the products. |
[
"I'm trying to learn basic maths but I always get stuck 20 pages in."
] | [
"math"
] | [
"8s7ln1"
] | [
2
] | [
"Removed - try /r/learnmath"
] | [
true
] | [
false
] | [
0.75
] | null | Concrete Mathematics is not an easy book. It would be slow and tough going even if you were working with a group or tutor or class. Be prepared for progress when self-studying math books to feel very slow. Some problems might take you hours or days (or longer) to solve. American high school books are intentionally written in a fluffy style full of easy exercises. | What are you reading? | Just now I started Ficktenholz calculus 1 because I got stuck in Concrete Mathematics (even though I really liked the way it was written). I thought I understood most of these things, but I couldn't remember them or felt like I wouldn't be able to explain it to someone 20 minutes later | Reading a maths book is very different from any other book. These books are very dense with information, so it's normal to have to spent hours (or even days) digesting a particular passage. It can be frustrating, but struggling with the material is really the only way to learn things. Everyone has their own suggestions of how to read a text, so I suggest you try various options and see what works. Here's a relevant discussion on Math.SE . I would emphasise that it's often encouraged to read the book in a non-linear fashion; you can jump ahead and skim the big results, go back and reread parts you're unsure about, or just skip uninteresting parts altogether (and maybe revisit it later). Edit: It seems like this was removed while I was trying my response; I personally think /r/math is a suitable place for this question, as the ability to effectively learn from a book is an important skill that most people learn during their time in university (both at undergrad and grad level). | Can confirm the last statement, I've found that although it will take time, dedicating that time to solve problems and really understand, is worth it. In highschool, my physics teacher would give us extra challenge problems and other proofs to look at which really helped understand the math and get a firmer grasp on problem solving |
[
"Real Integral through complex numbers"
] | [
"math"
] | [
"8s7lcb"
] | [
14
] | [
""
] | [
true
] | [
false
] | [
0.89
] | I woke up one day wanting to solve a real integral through complex numbers. So i chose 1/(x +1) from 0 to 1 because the denominator can be factored as (x-i)(x+i) and that can sometimes yield nice and easy natural logarithms. I solved further and used eulers identity and bla bla i came to infinite solutions with 1 of them being the actual solution. How can I narrow it down to the legitimate solution? Can someone explain why there are infinitely many? Thanks, sorry if I get something wrong but Im new to complex numbers (not to analysis though) | Integrals in the complex numbers of functions with poles (like how that function blows up at i and -i) are path-dependent. The answers correspond to a path through the poles and paths wrapping around one of them clockwise or counterclockwise a number of times. Wrapping around both of them doesn't change anything, so you can always reduce to the case of wrapping around just one. The actual details are taught in complex analysis. | that actually sounds reaaally intresting, thanks | Complex numbers can be written as a pair of coordinates on a plane which has a real number axis and an imaginary number axis. Normally, complex numbers are written like a+bi and can be given coordinates of (a,b). a is the real number coordinate and b is the imaginary number coordinate. This is just like normal rectangular Cartesian coordinates. Or, you could convert from Cartesian coordinates to Polar coordinates, and now every complex number can be written as (r, theta) where r is the magnitude and theta is the angle of the vector pointing to the number in the complex plane. In this case, where (a,b) is written as a+bi, (r, theta) is written as r * e . You may recognize that from seeing the Euler's identity 0 = 1 - e . However, this means that if I wanted to point to a number in the plane, i could give you the Polar coordinates (r, theta) OR (r, (2 pi to my angle, I'm just making a full rotation back to where I started. So I could also add 4 pi or so on and so on, giving me infinite ways of writing one complex number. It seems like as long as the angles are off by 2*pi, it doesn't matter. However, this is not the whole case, because if I were to have the complex number z = e and take the natural log of that, ln(z) = ln(e ) = i pi it DOES matter! So we need to discern the angles of these complex numbers. In summary, you will find that many applications of complex numbers result in countably infinite answers, all complex numbers whose angles are off by some constant. These are all valid solutions, but are also separate numbers. When doing even more complicated complex integrals (like the path-dependent ones that u/cards_dot_dll mentioned) the angle of the complex numbers becomes super important as you rotate around in the plane. | alright, we need to discern the angles but how does one do so. thanks for the detailed answer by the way | It just knowing whether the complex number has angle theta or (2* pi)+theta. They might both be answers, but different numbers. In the plane, they seem to be the same number, but are different. |
[
"Phinary Arithmetic and Native Number Representation"
] | [
"math"
] | [
"8s78pl"
] | [
3
] | [
""
] | [
true
] | [
false
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0.72
] | I recently stumbled on while wondering about irrational bases and thought it was interesting. I understand Standard Form, and addition and subtraction in "phinary", but when I started thinking about multiplication, I realized that phinary has some really strange properties. For example: 2 > 10 2 = 10 – 1 3 > 100 3 = 1000 – 10.01 At the same time, while you represent any rational number in phinary, the number of digits needed to represent integers grows fast enough that it's impractical. Which got me wondering if a sensible non-standard representation of phinary is possible, eg. Would "native" phinary users need to represent numbers in Standard Form or might they settle on some number of symbols greater than 2 to represent numbers? I'm not sure how to even approach this question. Help? | The best representation of these numbers in my opinion is just + , where and are integers. Writing them out using only 0 or 1 for coefficients of arbitrary powers of is a pain in the butt. If you need a field, you can use ( + ) / . while you can represent any rational number in phinary Maybe you mean represent any integer? If you want arbitrary rational numbers you either need infinitely many digits or some way of encoding the denominator. Anyway: ( + ) + ( + ) = ( + ) + ( + ) ( + )( + ) = ( + ) + ( + + ) Extending this to division where you represent numbers as ( + ) / isn’t too hard either. [( + ) / ] = / ( + ) = ( + – ) / [( + )( + – )] = ( + – ) / ( + – ) i.e. the three new coefficients are ( + ), (– ), and ( + – ). Eliminate any common factors from those three if you feel like it. | The best representation of these numbers in my opinion is just + , where and are integers. Writing them out using only 0 or 1 for coefficients of arbitrary powers of is a pain in the butt. If you need a field, you can use ( + ) / . I agree but to write + you have to use symbols that aren't part of phinary but our own decimal system. I'm trying to imagine if there's a way to have a genuine phinary number system that uses more than just 0 and 1. | aren't part of phinary but our own decimal system Well I used binary (stored in a computer) for each coefficient, but sure. There are a bunch of papers about doing arithmetic in this golden ratio base if you want. I guess the question is: what are you trying to accomplish? For me personally, I was trying to understand the symmetry system of the icosahedron/dodecahedron. | I run a large Dungeons and Dragons game (40+ active players, 10 DMs), and was exploring the possibility of having a lost civilization that used this base as their number system. But if people natively used phinary, would they have to use standard form to represent values or could they have 3 or more symbols in their representation system? In Phinary: 2=10.01 20=100–0.1 30=1000.01 So I started wondering if integer representations could be compacted with more than 0 and 1. | You'll want to have a lot of dodecahedra, penrose tilings, etc. all around any civilization using the golden ratio field instead of rational numbers and square grids. |
[
"The Unreasonable Effectiveness of Quasirandom Sequences"
] | [
"math"
] | [
"8s75uu"
] | [
19
] | [
""
] | [
true
] | [
false
] | [
0.93
] | null | The title is kind of lame, but the subject is interesting, a generalization of all the golden-ratio-has-optimally-low-discrepancy-in-1D posts that pop up here all the time, e.g. in the form of numberphile videos about spiral phyllotaxis.] There are a lot of related applied math problems with applications throughout science and engineering. | The Uninspired Ineffectiveness of Parodying the Title to Wigner's Seminal Paper | I thought the title was kind of nice personally | What proof (outside of small scale experiments) do we have that this sequence has good properties wrt discrepancy? The useful thing about low discrepancy sequences in high dimension is that they have bounds on discrepancy. I don't see what's so (provably) special about the irrational numbers after the golden ratio that the author has chosen. See chapter 2, section 3 of https://www.cs.princeton.edu/~chazelle/pubs/book.pdf to get a good intro on what this method is basically doing. | You might ask the author for some proofs. From the bottom of this link it looks like further material is forthcoming. Based on the construction method, the improved low discrepancy of such sequences in arbitrary dimension vs. other 'open' sequences (i.e. having good properties for any number of points, as points are incrementally added) seems at least plausible. |
[
"Matrix question"
] | [
"math"
] | [
"8s5beg"
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2
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""
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true
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false
] | [
0.76
] | Is there a way to take two matrices, A and B, of any order and apply a function f to them so that: if f(A,B)=f(C,D), then A=C and B=D? I am in search for that for a computational science project. | Sure, depending on what sort of domain/codomain you’re considering. If the codomain is pairs of matrices, you can map (A, B) to the ordered pair (A, B). Not interesting, but it works. Or.. if A and B have the same dimensions and have real entries, then you can map (A, B) to A+i*B, where i = -1. The codomain is matrices of the same dimensions with entries in the complex numbers. Still not great. If your codomain is the same as the space where A and B live, then I think you’re out of luck. If something is going to work for any dimension, then it’ll work for dimension 1. So I suggest playing around with 1x1 matrices, aka scalars. | If your codomain is the same as the space where A and B live, then I think you’re out of luck. You can do dumb things again. Let A, B be positive integer matrices and let p(i) be the i-th prime. Then set f(A, B)[i, j] = p(2 * A[i, j]) * p(2 * B[i, j] + 1). In general, if you have matrices over some set F, all you need is some bijection between F and F x F, and then apply it element-wise. This is easy when F is infinite. For matrices over a finite set, this is not possible thanks to a pigeonhole principle argument. [Edit] Correction to the formula. Fixed the problem where it would be f(A, B) = f(B, A). | You do not say what the range of f should be or over which ring you take your matrices and the answer depends on both of these. So I assume that you consider real entried matrices and ask for a real valued function. For any natural number d, R and [0,1] have the same cardinality. It follows that there exists a bijection f(d) between them. Being a bijection f(d) is certainly injective. Now the space dxd matrices over R can be identified with R and the space of pairs of dxd matrices with R The injection f(2d will then have the required property. So the answer is yes. One could also consider replacing [0,1] with the sequence space {0,1}x{0,1}x... of infinite binary sequences. | Your construction reminds me of the popcorn function. It has a similar kind of "dumb" feeling to me. | Damn i read this question and thought we were talking “The Matrix” and was thinking the possibilities 2 matrices- wow!! Endless possibilities- two Neo’s |
[
"Clustering of Probability Distribution"
] | [
"math"
] | [
"8s5l17"
] | [
1
] | [
""
] | [
true
] | [
false
] | [
1
] | Is there any measure of how "clustered" a probability distribution is? Suppose we're looking at a pdf over the vertices of a graph - one might say the distribution is clustered if the probability that two randomly selected vertices are adjacent (or the same) is high. This would be pretty low for a uniform distribution and would be maximized if all of the probability mass was at a single point. Could this sort of idea be extended to the unit interval, or some region in R^n, using convolutions or something? | You can use ideas from covering spaces. Say a measure mu on a compact metric space X is (n,eps)-clustered when there exist sets A_1,...,A_n all of diameter less than eps such that mu(Union A_j) = 1 (or possibly 1-eps). This will give that a measure which is a finite combination of point masses is (n,0)-clustered where n is the number of point masses. On the other hand, the uniform measure is (n,eps)-clustered only when n eps > diam(X). Not sure how to extract a value of out that, it depends on what you're trying to do. But something like looking at inf { n eps : the measure is (n,eps)-clustered } should capture what I think you're aiming for. | Not exactly, at least standard deviation isn't what I want. Say we're looking at probability distributions over the unit interval. A point mass would have SD 0, good so far. But then two point masses at 0 and 1 would have a greater SD than the uniform distribution on the interval, when I want the uniform distribution to maximize dispersion. | Not exactly, at least standard deviation isn't what I want. Say we're looking at probability distributions over the unit interval. A point mass would have SD 0, good so far. But then two point masses at 0 and 1 would have a greater SD than the uniform distribution on the interval, when I want the uniform distribution to maximize dispersion. | I don't think that would evaluate to 0 for point masses. Those correspond to a mixture of delta functions for f, which will convert the expression into -sum_i pi log pi, where pi is the probability of event i. Also, that doesn't seem to capture the spatial relationship between points in the space (although I might just be sleep deprived). | I don't think that would evaluate to 0 for point masses. Those correspond to a mixture of delta functions for f, which will convert the expression into -sum_i pi log pi, where pi is the probability of event i. Also, that doesn't seem to capture the spatial relationship between points in the space (although I might just be sleep deprived). |
[
"What is the significance of so(3) being isomorphic to su(2)?"
] | [
"math"
] | [
"8s50kc"
] | [
9
] | [
""
] | [
true
] | [
false
] | [
0.78
] | I have been reading "Quantum Theory for Mathematicians" by Brian Hall. We encode a quantum state into some projective Hilbert space. Spin operators can be identified with elements of su(2). so(3) can be identified with su(2) in such a way that does not arise from a map between SO(3) and SU(2). I don't understand the necessity of the identification between so(3) and su(2). Even we didn't know about so(3), we could still pull the spin operators out of su(2), no? Why should I care if the representation arose from SO(3)? Is there more reason than showing that spin is not a classical observable? EDIT: There seems to be some confusion, either by me or the commenters. Motivated by Noether's theorem, I have infinitesimal symmetries giving rise to conserved quantities. SO(3) gives symmetries in a classical sense, so I look at it's representation as elements of SU(2). But it is really the infinitesimal symmetries I am after, so I want to see what so(3) looks like in su(2). There exists an so(3) - su(2) identification (usual so(3) generators with spin operators) that does not arise from a map from SO(3) to SU(2). I am asking what is the significance of this identification not coming from a map from SO(3) to SU(2)? | It's a low dimension accident, don't pay it any mind. spin(n+1) = su(n) doesn't hold outside of n=2. They have different dimensions. | A priori, SO(3) is what you should care about. It acts as rotational symmetries of your system and so it should act on the Hilbert space, but you only get a projective representation. SU(2) is the universal cover of SO(3), and it turns out that you can lift a projective representation of SO(3) to a genuine representation of SU(2). The relation between the lie algebras so(3) and su(2) just comes from differentiating the double cover SU(2)->SO(3). The lie algebras can only see what happens in near the identity and the covering map is locally an isomorphism. There are a few ways to actually go about constructing the map SU(2)->SO(3). The easiest to actually "see" imo is to take the action of SU(2) on C and descend to an action on projective space P Then you view P as the riemann sphere and check via some fun calculation that unitary matrices act as rotations (basically because antipodal points on the riemann sphere correspond to orthogonal lines in C and unitaries preserve pairs of these). | Spin(3) or even Pin(3) is perhaps what we should most care about. There are two pin groups ; which one are you thinking of? | Arguably SO(3) is too restrictive. There seems to be something really physically important about the double cover; the two seemingly same orientations turn out to be physically distinguished. https://en.wikipedia.org/wiki/Plate_trick etc. Spin(3) or even Pin(3) is perhaps what we should most care about. | 𝔰𝔬(3) can be identified with 𝔰𝔲(2) in such a way that does not arise from a map between SO(3) and SU(2) This is a little funky to me; there is a map SU(2) -> SO(3) which induces the Lie algebra isomorphism between 𝔰𝔲(2) and 𝔰𝔬(3). Namely, we can realize SU(2) as the unit quaternions; then, SU(2) acts on the 3-dimensional (real) vector space of imaginary quaternions by . = . This has determinant 1 and preserves the inner product for which { , , } is an orthonormal basis, hence acts on this 3-dimensional vector space as an element of SO(3). and - act by the same transformation, and this map is actually a double cover, which implies that their Lie algebras are isomorphic. I'm sure you could construct an isomorphism without knowing this, but to me, this is the reason "why" they're isomorphic. |
[
"Unsolved problems in mathematics"
] | [
"math"
] | [
"8s4k8c"
] | [
7
] | [
""
] | [
true
] | [
false
] | [
0.77
] | There is also a well-known open question if 2 is the largest power of two that does not contain every digit in its decimal expansion. As much as I know, there are no additional terms up to 2 | I'm not sure what area of math this question really should be considered as lying in. Recreational. | I had not heard of this open problem before. From a computer algebra package I see that 2 does not have the digit 2 anywhere in its decimal representation (all other digits in 0, 1, ..., 9 appear). Curious. Questions about how something looks in base 10 are not really mainstream issues that people think about in number theory. I'm not sure what area of math this question really should be considered as lying in. | I had not heard of this open problem before. From a computer algebra package I see that 2 does not have the digit 2 anywhere in its decimal representation (all other digits in 0, 1, ..., 9 appear). Curious. Questions about how something looks in base 10 are not really mainstream issues that people think about in number theory. I'm not sure what area of math this question really should be considered as lying in. | Unimportant. | "prime numbers are cool and Collatz is cool so here's a random question that invokes both" |
[
"Can you be good at math but bad or average at (timed) math exams?"
] | [
"math"
] | [
"8s4mdr"
] | [
6
] | [
""
] | [
true
] | [
false
] | [
0.71
] | Recently did a math exam and didn't properly do four things. On the train back home, I manage to do them independently - all I needed was more time. But timed math exams always seem like a rush especially since I don't memorize much and have to do things from first principles. I've done a lot harder questions before. Math exams seem like they are just regurgitation of a course and advantages those with a good memory. Especially in this particular exam when questions were just copied exactly from questions done in the course. Its not like I'm even close to failing, I still estimate 70%+ score, but given more time I could easily do every question and get 90%+ accounting for 'careless mistakes'. So, in general, can you be good at math but bad or average at (timed) math exams? Are there people working in math fields (especially academia) who got above average results but not top results? | I'm about to get my PhD in math and I almost had to repeat the 5th grade because I was terrible at those timed arithmetic tests. The ones where you had to do 100 simple addition/multiplication/whatever problems in like 2 minutes. I still do arithmetic on my fingers. My students make fun of me for it all the time. :-) | Addendum: try to avoid the mindset of "I am good at this, and bad at this." The problem with this sort of language is that it conflates immutable personality traits with learnable skills. The title question is pretty similar to asking "Can you be good at the trumpet, but bad at the french horn?" Well, yes, if you've worked at the trumpet but not the french horn. That doesn't mean that you're inherently a trumpet person who's not meant to play the french horn. It just means you should work harder at the french horn if you'd like to learn it. (cf. ideas of growth mindset) | Being good at math and getting good grades are 2 somewhat different things. Yes you can be good at math and get poor grades because the class is also testing other things, including your ability to do what your told and what you need to do. If you are good at math I feel you should be able to ace the test in no time, especially if it’s regurgitated material. Maybe you can “figure out” the answer but honestly a test isn’t there for this. You are supposed to have already figured everything out and become familiar enough that you can do it relatively quickly. Basically it boils down to both having natural math ability as well as working hard. If you do both you really should be breezing past these tests. | Brusque advice from someone who has been through this: But timed math exams always seem like a rush especially since I don't memorize much and have to do things from first principles. Memorize the things that are important to memorize. There is nothing shameful about memorization. Just be sure to understand what you memorize. Math exams seem like they are just regurgitation of a course and advantages those with a good memory. "Good memory" is largely a myth. Unless you have some kind of cognitive abnormality, you are capable of memorizing definitions and theorems for an exam-- you just need to do it. Write them out on paper, derive them many times, work more problems, and you will remember. People who are "good at math" are people who put in a hell of a lot of effort. | Maybe, but assuming this is the problem sounds like a way to set yourself up for failure. |
[
"[Number Theory] So I was looking at patterns for twin primes and..."
] | [
"math"
] | [
"8s4ebn"
] | [
1
] | [
""
] | [
true
] | [
false
] | [
0.57
] | Well I found that the twin primes (up to 1700 at least) that had 9 as the last digit in the first number and 1 as the last in the second “surrounded” multiples of 30. Not all multiples of however it seemed, but I was working with limited data. The pattern for the differences between the sequential multiples of thirty in relation to these primes seemed random, but with an overwhelming difference being 30. Is this just a coincidence or not? | (9,11) mod 30 and (19,21) mod 30 both are guaranteed to have a number that is divisible by 3, so (-1,1) mod 30 is the only pair that can possibly contain two primes ending in 1 and 9. edit: never mind, misunderstood your post. | Think of it as a remainder when dividing by 30 in this case. This is probably the most important operation when learning basic number theory so you really aught to learn this. | All prime pairs are of the form 6n-1 and 6n+1 for some n. If 6n-1 = 10k+9, then 6n = 10k+10. Thus 3n = 5(k+1). This means n must be divisible by 5. So, in this case, you must have primes of the form 30n -1 and 30n +1. (If you look at 6n+1 = 10k+1, you will get the same conclusion.) Edit: It should be 3n = 5(k+1) but originally I had 3n = 5(k+2). (Hey. It's late. Whadda ya want.) | Also, what I meant is that like you have a multiple of 30, like 570, and that if you add and subtract 1, both will be primes. I am not sure if that is one, I forget, but I hope you see what I mean. | Yes. I just saw this and it is known as Wilson’s theorem. But this will not work for all twin primes correct? 3&5 is just one pair, but I am sure there are others. Also, you meant 6n = 5(k+2) right? |
[
"Is there a hierarchy of mathematical subjects or fields of study?"
] | [
"math"
] | [
"8s41x5"
] | [
0
] | [
""
] | [
true
] | [
false
] | [
0.5
] | [deleted] | I'm curious if there's a "hierarchy" to the branches of math Pedagogically, some subjects must be learned before other subjects, or at least it's better to do so. For example, you should learn basic algebra before calculus. You should learn commutative algebra before scheme theory. In some sense that may be taken as a hierarchy. Is each branch a subset of another? Not really. Some branches are subsets in an obvious way, eg complex curves is a subset of complex manifolds. You might say group theory is a subset of universal algebra, but it's not especially useful or helpful to do so. Each area has its own intuitions and techniques, and they need not be shared by any other. Can all modern mathematical concepts be derived from any specific area of study? I mean perhaps you know that standard procedure in constructing mathematical theories is to construct them as sets using the axioms of set theory. In that sense most mathematics can formally be thought to follow from axioms of set theory. But that's only a formality that doesn't include all the intuitions and pictures and techniques. The fact that 3-manifolds admit purely set theoretic descriptions is completely useless for thinking about Ricci flows. Also the axioms of set theory are necessarily incomplete, and may need to be augmented to capture all mathematics. | Yup totally. That checks out. My graduate program divided topics between algebra and algebra. | Yup totally. That checks out. My graduate program divided topics between algebra and algebra. | but some algebras are more about "anlysis" than others | You're close to the distinction between "applied" and "pure" mathematics; also, analysis is one of the main branches of pure mathematics (it started off with the attempt to make calculus rigorous) and the historical undergirding of much of applied mathematics . |
[
"Can anyone explain this 10/11 proportion at the end?"
] | [
"math"
] | [
"8s3l32"
] | [
0
] | [
"Removed - ask in Simple Questions thread"
] | [
true
] | [
false
] | [
0.2
] | null | You're not going to like this, but as the 10/11 does indeed look really wierd and interesting, without knowing anything about the problem you're working on, and without seeing the equations written down or typeset properly... ...the simplest explanation is that you probably made a mistake. | That depends what velocity it was doing at the start of the 10 seconds. | You have that the opposite way around, a=d s/dt and is a const so a dot dot = 0 The convention I was taught over 20 years ago was: t time elapsed s displacement u velocity at start v velocity at end a acceleration Apparently they're even called the SUVAT equations. https://en.wikipedia.org/wiki/Equations_of_motion#Constant_translational_acceleration_in_a_straight_line http://www.mash.dept.shef.ac.uk/Resources/em1_9constantaccelerationequations.pdf | Well I have tried putting in different numbers and comparing it to integrated formula and the results are the same. Problem was how much distance car travels if it accelerates 5m/s for 10 seconds. | Interesting, I forgot to take that into a count. I didn't take that into a count as always started from zero velocity. |
[
"[Contest Results] Average Number Guessing"
] | [
"math"
] | [
"8s3o7m"
] | [
22
] | [
""
] | [
true
] | [
false
] | [
0.84
] | So I made a poll where the goal was to correctly guess what half of the average guess would be. A lot of people (21 to be exact) guessed as if everyone were perfectly logical, putting 1 as their answer. But unfortunately, not everyone is perfectly logical, and a lot of people intentionally tried to skew the results by putting in higher numbers. Several people put my name in as a username, and those answers were not counted. Additionally, all guesses which were not integers were not counted. In the end, there were about 75 valid guesses, and the average guess was 19.8875. Half of that is 9.94375, which, for the purposes of this poll, will round to 10 as our final answer. So who guessed 10? Four people! , , , and , you are the winners of this poll. Congrats! | Why do you say that? Lets say there's two people, who guess 1 and TREE(3). Half the average rounds to TREE(3)/4. The person who guessed 1 would be off by TREE(3)/4 - 1, which is better than 3 * TREE(3) / 4. | Uhhhh can I downvote myself? You are obviously correct. I don't know what I was thinking. [Edit] Yes I can. | That's neat, if you do a similar thing, I'll try and not miss it. | Isn’t it a name the second biggest number game? | Number had to be between 1 and 100 |
[
"Two basic linear algebra questions"
] | [
"math"
] | [
"8s3hwu"
] | [
0
] | [
""
] | [
true
] | [
false
] | [
0.5
] | Can somebody provide an accessible definition and motivation for what an inner and outer product is? Many spaces admit these. Usually definitions are provided without motivation. Are they simply useful for applications or do they have pure math motivations too? Determinants: in a similar vein, other than for solving systems of equations, the representation as a matrix and related uses of trace seems mysterious. | Once you have defined the notion of a vector, which is like a line, how do you define the notions of length, angle, volume and expansion of volume, all concepts related to lines, in terms of vectors? First length: Visualize a vector A = (x,y) in the plane. Clearly ||A|| = \sqrt{x + y is the length of A. Next angle: Visualize two vectors A = (x,y) and B = (x',y') in the plane. Since we are doing linear algebra, how would you give a linear description of the angle between these two vectors? The angle between the two vectors is, from s = r 0, the length of the arc of a unit circle passing through these two vectors, s = 0, a curved line. The cosine of this angle, however, is the length of a line segment. Thus, if a is the angle from the x-axis to A and b is the angle from the x-axis to B then cos(a - b) is a linear representation of the angle between these two vectors, however from cos(a - b) = cos(a) cos(b) + sin(a) sin(b) = (x/||A||)(x'/||B||) + (y/||A||)(y'/||B||) = (x x' + yy')/||A|| ||B|| = (1/||A||) A * (1/||B||) B we see that A * B = (x,y) * (x',y') = xx' + yy', the inner product of A and B will provide a linear representation of the angle between A and B. Next volume (area since we're in the plane): Determinants as explained here https://www.khanacademy.org/math/linear-algebra/matrix-transformations/determinant-depth/v/linear-algebra-determinant-and-area-of-a-parallelogram and visualized here https://davidroodman.com/wp-content/uploads/2016/02/determinant-as-area-1.png link determinants to area, and you can abstract properties that area must satisfy to give a general definition of a determinant to give volumes in n-dimensional spaces http://www.cs.uleth.ca/~holzmann/notes/det.pdf Trace is the derivative (linear approximation) of the determinant https://math.stackexchange.com/questions/2437848/why-i-should-believe-that-the-derivative-of-the-determinant-is-the-trace note the parallelepiped comment there and it's relation to the expansion of volume... | Thank you | I don't think there's any operation whose standard name is "outer product". There is the vector product and exterior product and tensor product. The inner product is the projection of one vector onto another. The exterior product is the biplane spanned by two vectors. The determinant is the volume of the box spanned by several vectors. | It's best to use the same definitions as everyone else on your course. The inner product is the simplest natural abstraction of the dot product on vectors, but others are available too. Determinants are crucial when you're solving square linear systems of equations, to know if there is a unique solution or not. | The outer product usually refers to u v^t (that's a rank 1 matrix), compared to the inner product u^t v. |
[
"Machine Learning for Parameters of Differential Equations"
] | [
"math"
] | [
"8s37m1"
] | [
2
] | [
""
] | [
true
] | [
false
] | [
0.67
] | Hello everyone. So as we all probably know, its generally pretty simple to machine learn parameters to fit data in a function. However I recently came up with a way to use machine learning with to find the parameters that fit a set of data points for a given differential equation with unknown parameters. This process uses Runge Kutta 4th Order. I was wondering if anyone had seen anything like this before, and if not if you think it'd be worthwhile to write up a blog post/article about it? | I have seen a few, that sounds like system identification. Google around for machine learning system identification. I've seen "extreme learning" applied, for example. | Yeah, it's called data assimilation. Most basic topic is probably the Kalman filter but you can also learn about particle filters if you're comfortable with monte carlo. | You may be interested in Nathan Kutz 's work. For example, here is one of Kutz's lectures involving neural networks for this course . | "Inverse problems" is probably the right keyword here. | You might want to check out this paper: https://rss.onlinelibrary.wiley.com/doi/full/10.1111/j.1467-9868.2007.00610.x Section 1.3.1 gives a quick overview of parameter fitting through numerical integration. Check out the references mentioned there for more information. If there's a professor at your school who researches things like this, you should talk to them if you think you have something interesting. That's the best way to get guidance about publishing and to make sure what you're doing is worthwhile. |
[
"How to best learn (up to) analysis in limited time?"
] | [
"math"
] | [
"8s2tof"
] | [
0
] | [
""
] | [
true
] | [
false
] | [
0.5
] | Cross-posted from , tell me if this doesn't belong. Assume for the sake of argument that I will take a graduate sequence in real analysis with little background in proof based maths (except discrete math for CS) (also did probability, multivariable calc). Probably a bad idea. But if I were to do this, what would be the best way/resources/path to self-study in the next two months that would bring me closest to the level of understanding that I would need to be well-equipped for this course? Books I'm currently considering/starting to use are Velleman's How To Prove It and Abbott's Understanding Analysis. (I started using Rudin and felt somewhat comfortable with it, but stopped because I read that it omits many basic ideas) Specific questions: 1. Is it better to build up proof competency (eg. with How To Prove It) before attempting analysis, or attempt analysis and build proof competency along the way? 2. What resource would you recommend the most in studying analysis? 3. Rudin. y/n for self study? | That is not just probably a bad idea. It is a truly misguided idea. Don't do it. If you are at the stage where you have to read a book to learn how to prove things (like Velleman's) then you have absolutely no idea what those later courses will really be about and there is no way you could get anything out of a graduate real analysis course anytime soon. Learn undergraduate analysis well ( in a rush over the summer) before even thinking about a graduate course in that topic. | You should take the undergraduate analysis course before the graduate analysis course. It will provide more basic experience with concepts like compactness or connectedness, which the graduate course is likely going to assume students have worked with in an undergraduate course already. Unless, perhaps, the "graduate" course at your school is really an undergraduate-level course in disguise somehow. | oh god don't do it. just take the undergrad version | The thing I would be concerned about, is your instructors not cutting you ANY slack taking a graduate math course. I took a 400/500 math course and watched the instructor absolutely belittle the graduates in that class at times and show no mercy grading wise. | Yeah the professor was somewhat disparaging when I emailed him about the course... I see how that could be a problem. |
[
"Math puzzles that I solve at work when I'm bored"
] | [
"math"
] | [
"8s2n22"
] | [
2
] | [
""
] | [
true
] | [
false
] | [
1
] | Hello all, I'm new here, but was encouraged to post some math puzzles I enjoy.Where I work I'm stuck standing around doing mindless repetitive work, and I can't be caught writing anything down. Years ago I began printing off randomly generated math puzzles, because I can stick them on the shelf I face and solve them while I work. I started with base 10 versions of this puzzle, but eventually I could do them too quickly in my head and I've moved up to base 16 ones. I had 2 or 3 people who attempted the base 10 puzzles, but I doubt they really enjoyed them. Now that I've moved to base 16 I can't find anyone to share these puzzles with, even though I'm sure there are people out there that may love them as much as I do! So here I am, new to Reddit, looking to share my common past time and see if they are enjoyed by anyone else. This is a standard long division problem, where each of the digits was replaced with a letter. The starting numbers, the letters of the alphabet used, and the value of each were randomly generated with a little computer program. For me this adds in an extra element of fun, because it isn't known 100% that a solution can be found with the starting information. I've only ever given up on a few, believing they may be impossible, but it's nice doing a puzzle that wasn't constructed with a solution already in mind. I finished this puzzle a few days ago, and quite enjoyed it. If you're anything like me, you'll spend the first several minutes just totally stuck and convinced that it's impossible. I tend to start out most of the puzzles totally stuck, but somehow they get solved. Good luck! | Hello, can you share an easier problem of that type? It seems interesting. | Well this was one of those moments where I feel like a fool for over complicating things. Yesterday when I noticed that contradiction with Z, my brain immediately switched out of math mode and started to view it as a programming error with whatever you had written to generate the puzzle. I'm especially surprised I didn't get jarred to my senses when I noticed the 2 K's dropping down, right at the end as I was typing my message. I see now that you just skipped one of the subtractions altogether to save on space. I've been so used to seeing each step written out (since that's how I wrote mine 8-10 years ago), that it didn't occur to me. So my apologies threenplusone, I made a very silly math mistake. Now that I woke up and looked at your puzzle correctly for the first time, I think you've done a fine job and it seems quite straight forward. The steps the puzzle takes to skip the unnecessary zero subtraction are so obvious to me now that I'm still kicking myself for not seeing it before. | I'd be happy to. I've put the easier one in the puzzles sub reddit. https://www.reddit.com/r/puzzles/comments/8s3qb8/a_math_puzzle_i_love_to_do_at_work/ By easier, I mean it is in "regular" base-10 numbers. This does make it much easier to solve and less intimidating to start. | Hello, the link for the base-10 numbers puzzle is removed. | https://www.reddit.com/r/puzzles/comments/8s3qb8/a_math_puzzle_i_love_to_do_at_work/ Strange, it seems to work for me. Here's the link again anyway, just in case something is screwy with the first one. |
[
"Free Probability Distribution Calculator App for Students"
] | [
"math"
] | [
"8s1y23"
] | [
23
] | [
""
] | [
true
] | [
false
] | [
0.77
] | Hey guys, I created an Android application that has probability distribution calculators and flash cards for several common distributions. The app is useful for students enrolled in probability courses, or for those looking for a quick way to calculate certain metrics from probability distributions Hope this helps a few people out. If you find it useful, it would be great to hear some feedback! | No use for it at the moment, but with any luck I'll be in front of my first classroom of my own next fall, so I will definitely save this for later! | Awesome! Best of luck in the future, I hope it proves to be useful for you :) | I’m in a probability course right now and just started covering distributions, this is extremely helpful. | Excellent! Glad to hear it's helping you out :) | Will it become available on iOS? |
[
"Friend is taking an open book placement exam, we don’t know how to create an equation, [help]"
] | [
"math"
] | [
"8s1dtq"
] | [
0
] | [
"Removed - this isn't a subreddit for cheating"
] | [
true
] | [
false
] | [
0.11
] | null | If you don't know how to do this getting put into a class which assumes you can will be setting you up for failure. It would be a disservice to you to answer this. | It’s just confusing the shit out of us for some reasons. We’re very past the question already, but I understand if you feel it would be a disservice. Also, this is legitimately for my friend so he can enroll, I’ve already been to college lol | I’m not cheating, the question is answered and already passed. I just don’t know how to do it since we just guessed our way through. I’d like to know the actual equation. That’s why I came here...to r/math ....for help...on math... | Okay, good to know. Nevertheless, this is a subreddit for mathematical links and discussion and not a math help subreddit. Try the simple questions thread or /r/learnmath as per the sidebar then. Based on the title alone it sounded like you were trying to cheat. | Thanks |
[
"To people with higher levels of math under their belt, do you sometimes struggle with very simple math?"
] | [
"math"
] | [
"8s2oaa"
] | [
522
] | [
""
] | [
true
] | [
false
] | [
0.95
] | This dawned on me a day or so ago. I was working on a project with lots of nth dimensional equations and such and having no problems whatsoever, and then thought about how at work earlier that day I couldn't subtract 1.7 from 2.2 in my head without second guessing myself. Since then I've been paying a little more attention to how easily I can do simple operators in my head. I've found multiplication is my quickest, while subtraction is my slowest. I took up through linear algebra in school and experimented with higher levels since then, although rarely need to. Just curious what everyone elses take on this is. | Graduate-level math. Works at a cafe. Checks out. | That's not too hard: Just do a polynomial division and then use the evaluation homomorphism. | I wrapped up a graduate-level class on algebraic geometry last semester. I frequently make mistakes when closing down my register at the cafe I work for. | I was somehow able to obtain a 1st class masters degree from a reputable University without being able to do long division. | you could have stumped me with a uniform convergence problem while I was doing analysis |
[
"My take on the infamous frog riddle..."
] | [
"math"
] | [
"8s1dso"
] | [
0
] | [
""
] | [
true
] | [
false
] | [
0.45
] | So I might be wrong, but here's my take on the Ted-ed riddle. If you have two frogs, then here are the odds: M/M = 25% M/F = 25% F/M = 25% F/F = 25% We know that one of them has made a croaking noise, so we can get rid of F/F. That means that there's a 2/3 chance we get a female, hurray, I'm great at math. Except for one small thing; we can determine based on the question's setup that we have Left!Frog and Right!Frog. In M/F left is male, and in F/M right is male. Now, if one of the frogs croak , then we can determine that one of F/M and M/F is removed from probability space immediately. . Conditional probability is all well and good (and much better demonstrated by Monty Hall), but either Left!Frog croaked OR Right!Frog croaked in the setup of the riddle, even if the riddle withholds from us the identity of the croaking frog. This changes probability space. Assuming that Left!Frog croaked, we have M/M and M/F as possible outcomes. 50% chance of finding a female if you lick both. Assuming that Right!Frog croaked, we have M/M and F/M as possible outcomes. 50% chance of finding a female if you lick both, again. But M/F and F/M cannot coexist in probability space here. They're mutually exclusive both in reality (naturally!) and in probability space, since the setup makes it clear that one of them has croaked. These aren't schrodinger's frogs; one of them has certainly croaked in the setup and one of them certainly hasn't. That information lets us that conditional probability cannot apply. Thoughts? p.s. Imagine that you were watching the riddle play out from another viewpoint. You see Left or Right croak (which does not matter), and you see the lost man turn and face the two frogs. Would you claim that he has a 2/3 chance of surviving with the duo? Us being behind a veil of ignorance doesn't actually change the probability, at least in problem. | Us being behind a veil of ignorance doesn't actually change the probability Of course it does. If we had perfect knowledge of the situation then we'd simply know the gender of the frogs. You are mistaking probability-as-modeling-imperfect-information for probability-as-actual-description-of-reality. | You have no way of distinguishing left and right. For all you know between when the croak happened and when you turned around they switched places. Just draw out the whole tree of possibilities if you distrust the calculation, you'll see it. | Actually, it occurs to me that there is a more direct resolution, which doesn't require throwing out your framing of the problem: you're considering left/right for the mixed-gender pairs, but not for the same-gender pairs. This leads to overcounting. Say we run this experiment 1000 times. In those trials, there are 250 each of M/M, M/F, F/M, and F/F, yes? The F/F trials have no croak, so they are not relevant to us, leaving us with the other 750. Out of those, half the time it's a left croak: either M/M or M/F. . That's 125 M/M and 250 M/F. The other half of the time, it's a right croak: 125 M/M and 250 F/M. In either case, you have 250/375=2/3 where there's a female in the pair. | We don't know which of M/F or F/M we can exclude, Right, exactly. That's why we give them each a probability of 25%, instead of giving one 50% and the other zero. You are deeply confused here. This is like saying that a coin flip is either 100% heads or 0% heads, we just don't know which. The whole point of probabilities is being able to write down that uncertainty as a number. | I wrote a quick java program that simulates this riddle a million times. Generates 2 random numbers and, if at least 1 is them is a male (1=male, 0=female) then it checks of the pair has a female (0). I got 500405 successes and 249983 failures, the other ~250000 not being counted because there was not at least one male (a croak would’ve been impossible.) There were, as expected, twice as many successes (pairs with a female) than failures (pairs without). I don’t know exactly where your analysis errs, but the riddle works in practice. |
[
"Is carrying your password in caesar-chiffre safe?"
] | [
"math"
] | [
"8s14fd"
] | [
0
] | [
""
] | [
true
] | [
false
] | [
0.38
] | Edit: Actually i meant Vigenère cipher, which is a letter by letter caesar shift, see here So before i start using the idea that just came up to me, i wanted to make sure i am not doing something dumb. First of all the problem. I know people that cannot memorize their passwords. Until know i tried to teach them the method, where you remember a sentence and turn the starting letters into a password. However there are so many things where you need password, this method does not solve the memorization problem. So i thought the people should carry their passwords with them in written form. Of course it needs to be a chiffre version of the password. For example you can use caesar chiffre for that. You would only need to remember 1 key sequence for chiffre and dechiffre. The password could be fully random, since they don't need to remember it anymore. Of course some might say, but encrypting/decrypting is as complicated as memorizing, but i have tested with some people. And since they use the same method over and over again, they can easily remember the process. Is there a pitfall? Of course if you find out 1 password + chiffre you can reconstruct the key. However from multiple chiffres for example you could not make any conclusions about the key, right? | Brute force? Takes like only some secs | I'd like to take this opportunity to derail the conversation for a PSA: Use a password manager. KeePass and LastPass are good options. That way, you memorize one hard password, and the computer makes as many strong, passwords as you like. Super interesting password cracking video: https://www.youtube.com/watch?v=7U-RbOKanYs Follow-up just about what to do ideally: https://www.youtube.com/watch?v=7U-RbOKanYs | First of you had to lose your written password. Then the guy finding it would have to identify that it even is a password. After he has done that he still doesn't even know what it is for. So I'd call that already "safe" enough for most purposes. Caesar-chiffre doesn't make it harder at all, the "safety" actually comes from not knowing that you have used it on the password. Once he has the paper he has infinit time to try and solve it so no, caesar-chiffre won't do any good | I have the database file on my dropbox, and dropbox on my phone. If I need to log into a school computer or something, I open it on my phone and type it in by hand. It's a little annoying because it's 20 characters' worth of numbers and upper/lowercase letters, but it's not too bad, and it gets around typing your master password into a potentially-keylogged computer. And if putting the database on the cloud makes you uneasy, then you can additionally lock it with a key file, so you need to have that file and know the password to decrypt it. You transfer the keyfile between computers/phone once via offline methods, transfer the database via cloud (because it gets updated every now and then), and you're good. | If i'd find a wallet with a credit card and a paper with 4 digits on it, i'd know what they stand for. (Could also be mobile phone, or anything else with pw/pin etc.) So i would not rely on someone not knowing what the pw is for. And it should be safe in case of a robbery. |
[
"The Most Efficient Way To Find The Area Of A Triangle"
] | [
"math"
] | [
"8s10ia"
] | [
9
] | [
""
] | [
true
] | [
false
] | [
0.74
] | null | Wow! Why didn't my school show us this. It's so much simpler now. Thank you for opening my eyes to why math is so beautiful. :) | I'm sober and it took me until he started using polar coordinates | I am drunk right now so it took me 2 or 3 minutes to realize that I was being trolled. | ...watch the video | I haven't watched the video but I found the proof without words presented in "A Mathematician's Lament " to be quite insightful. Granted in only works for a particular case but its simplicity is quite powerful however simple the result may be. |
[
"If math is like a language, what does changing the exponents on either/both variables either separately or together communicate? How does that fit into graphing circles/ellipses?"
] | [
"math"
] | [
"8s122m"
] | [
0
] | [
""
] | [
true
] | [
false
] | [
0.5
] | I’m learning Pre-Calculus/calculus, but my teachers never explained this very well, or how it works, they just fed us equations to spit back out onto test sheets, but it seems like there are both exponent changes and fractions involved in graphing something like a circle. How would you change a basic x into graphing circles, ellipses, or what I think is called sinusoidal graphs. It seems like they are all connected somehow, but I can only find advice on each separate subject, but not how they might fit together. I figured this is the best place to ask, and as to why, I learn best when I understand how different variables effect the system as a whole, and again, these different parts seem linked together, but it’s too scattered in learning resources to be able to quite put it all together. | no that wasnt my point | You may want to ask for help at /r/learnmath . You could start by reading https://en.wikipedia.org/wiki/Conic_section | "math is like a language" is only part true. math is an extremely formalized way of thinking about the general concept of "systems" and the language, the equations and the symbols and all that, lets us talk about it in a formal but still casual manner. it's like how learning chinese doesn't teach you kung fu. it just makes learning kung fu from a chinese person a lot easier. you can do "math" without learning any equations, but you'll find yourself inventing new notation/equations regardless because that's just the kind of language that suits the subject. | Honestly, you shouldn’t try to take an analogy so literally. | Everything about conics is deeply related to the pythagorean theorem. |
[
"Some stuff I was scribbling while bored at work, based on tools I learned from watching Mr. You Math’s videos on the Zeta Function. Is this sound? It seems reasonable, and Wolfram Alpha appears to agree at every step."
] | [
"math"
] | [
"8s02xz"
] | [
27
] | [
"Image Post"
] | [
true
] | [
false
] | [
0.85
] | null | Honestly I didn't check everything you wrote. Seems to me you could simplify the calculation by changing the order of summation: /sum {k>1} 1/p Use the geometric series to solve the inner sum Edit: Sorry don't know how to typeset properly on Reddit | I actually did do that, summing over p gives 1/(p(p-1)), because the series starts at k=2. I changed from summing over the primes to all positive integers by incorporating the prime counting function, specifically pi(n)-pi(n-1), which is 1 when n is prime, and 0 when it isn’t. | I actually tried that! The reason I use that difference for the integration is because that’s just the Fundamental Theorem of Calculus at work: treat 1/n(n-1) - 1/n(n+1) as F(b) - F(a), and the find F(x). What you suggested leads down another path surely, but I couldn’t think of anything past what you got. | Well apart that in the first steps you divide by zero so you have to take out the first terms before calculating the sum and that before writing a sum up to infinity you have to prove that it is summable, I have nothing to complain about. | I did not quite get the part from [1/n(n-1) - 1/n(n+1)] to integrates, I did not check if it was right but it seems complicated. I would first factorise by 1/n then put on the same fraction that would give 2/n(n*n-1) Then I would use integration with that. EDIT : Nevermind it is also a good and probably better solution. |
[
"What Are You Working On?"
] | [
"math"
] | [
"8s0pdo"
] | [
19
] | [
""
] | [
true
] | [
false
] | [
0.85
] | This recurring thread will be for general discussion on whatever math-related topics you have been or will be working on over the week/weekend. This can be anything from math-related arts and crafts, what you've been learning in class, books/papers you're reading, to preparing for a conference. All types and levels of mathematics are welcomed! | My math homework | Damn. Shout me out when you win the Fields medal. | Hmm... I won't discredit you or discourage you from publishing. That being said I would advise that you look into what has already been done to ensure that your work isn't 'redundant' for a lack of a better term. If, however, you have managed to rederive important results then that is an impressive feat on its own. | Figuring out how to best explain functorial TQFT and the ideas behind invertible field theory to physics grad students. They know a lot of math, so that will be fine; in fact, they've heard these definitions many times, so I will focus on explaining why I find these definitions to be useful. | Trying to finish up proving a lemma I need in a paper that I thought I had a complete proof for but when writing up discovered that it needs more work. The Lemma is that there are no primes x,a,b,c with x=a=b = 1 (mod 5), and x +x+1 =(a +a+1) (b +b+1)(c +c+1) with also a +a+1, b +b+1 and c +c+1 prime. Also, helping office staff clean up organize paperwork because some people who finished their positions left their desks disorganized and we need to go through and figure out which things (like old exams) need to be saved and what can be tossed. |
[
"Question about integrating the differential of inverse trig."
] | [
"math"
] | [
"8rzxov"
] | [
0
] | [
""
] | [
true
] | [
false
] | [
0.25
] | [deleted] | The derivative of arcsin(x/(-2)) = arcsin(-x/2) is -1/sqrt(4 - x ), not 1/sqrt(4-x ). | No, that gives you a function whose derivative is not what you started with. That's ultimately what's going on here: The derivative of arcsin(x/a) (for a > 0) is 1/sqrt(a - x ), so we know the integral of the latter is the former. And if a < 0, then the derivative is -1/sqrt(a - x ). | Not quite. The minus sign in the sin example "goes outside" because sin is an odd function. The minus sign here "goes outside" because we're differentiating to go from arcsin to the sqrt thing, so by the chain rule we must get that minus sign in front. | Also could you not take the negative a value and just get a negative arcsin, meaning there’s two integrals? | So it just negates the whole function, like how sin(-x) = -sin(x) ? |
[
"Non-Euclidean geometry"
] | [
"math"
] | [
"8rzpy6"
] | [
5
] | [
""
] | [
true
] | [
false
] | [
0.86
] | Does a globe have parallel lines that cross each other at the poles? | The name for geometry with no parallel lines is elliptic. If any two lines intersect exactly once, it's projective geometry. Exactly twice, it's spherical. | Parallel lines do not exist in spherical geometry. Two separate straight lines on a sphere would intersect each other twice, the intersection of two lines at the poles is one example. | the general term is elliptic, not spherical. See comment, below One way to distinguish euclidean, spherical, and hyperbolic geometry is by the existence of parallel lines. You can definite a bunch of axioms that work for all three. Then, you come to a decision point. Given a line L, and a point A not on the line, how many lines through A do not intersect L? You axiomatically define the answer. 1 means euclidean, 0 means spherical, and more than 1 means hyperbolic. | Oh okay | Might be a stupid question, but can we define a geometry where any two lines intersect exactly n times for all n? |
[
"Math stack exchange question about topology/geometry meme"
] | [
"math"
] | [
"8rzx2a"
] | [
129
] | [
""
] | [
true
] | [
false
] | [
0.91
] | null | In short, topology is beautiful, geometry is ugly. The topologist bird is enjoying itself thinking about some beautiful topological concepts, when along comes an annoying geometer crow squawking ugly formulas. This illustrates the sentiment that many topologists feel when are forced to think about geometry concepts. In long, topology is the study of properties of spaces which are invariant under continuous deformations likely stretching and bending. Concepts like connectivity, how a knot is tangled, and something called the "fundamental group" which describe how closed loops in the space can be deformed into one another. In particular, topology ignores details like length and curvature, which leads to a great deal of simplicity. On the other hand, geometers study spaces while caring about these details. Usually, geometers study manifolds, which are spaces which locally look like Euclidean n-dimensional space. Since n-dimensional space is described by matrices, and maps between them are described by multivariable calculus, things can get complicated pretty fast. | Holy shit that's a great meme. | Somebody explain it for us lowly undergrads. | Nothing much, it’s just a bunch of formulas generally used by topologists and geometers. | Nothing much, it’s just a bunch of formulas generally used by topologists and geometers. |
[
"What are some fun maths exercises you like to do on your down time?"
] | [
"math"
] | [
"8rz145"
] | [
8
] | [
""
] | [
true
] | [
false
] | [
0.91
] | I just finished a course on finite representation theory, and finding character tables are quite fun! | The second one : it follows that the derivative is 0 everywhere, so the function is constant. | I like to try to compete proofs in my head sometimes. Here a few fun ones I've done: Show that 3 divides m -m. Find all functions F:R to R that satisfy the inequality F(x_1)-F(x_2) <= (x_1-x_2) Show algebraically the expression n!/k!(n-k)! is an integer for all integers n,k where 0<=k=<n. | How do you know the function is differentiable, let alone continuous? | If it wasn't, then the inequality couldn't be satisfied at the discontinuous points. This follows from the formal definition of continuity and limits by extension. | (F(x_1)-F(x_2) )/ (x_1-x_2)<= (x_1-x_2), taking limits ax x_1 approaches x_2 gives the derivative on the left and 0 on the right. |
[
"Learning Math from Elementary to Advanced"
] | [
"math"
] | [
"8ryqqg"
] | [
6
] | [
""
] | [
true
] | [
false
] | [
0.8
] | [deleted] | All you need to be able to pass a typical college calculus course is solid algebra and know how to evaluate trig functions. The vast majority of people that struggle with calculus are usually struggling with the algebra | Hey. Maybe this video will help you choose math field to make progress (it's map of mathematics): https://www.youtube.com/watch?v=OmJ-4B-mS-Y I think some universities share their courses for free, so research that. Here you have MIT: https://ocw.mit.edu Next search through Great Courses Library. Maybe you will find something useful here. https://www.thegreatcourses.com Also, I heard about www.brilliant.org - site/app where you learn through solving actual problems and puzzles. Good luck! =] | Khan Academy is amazing. | /r/learnmath has a stickied post with a lot of resources: https://www.reddit.com/r/learnmath/comments/8p922p/list_of_websites_ebooks_downloads_etc_for_mobile/ Scroll down to the e-books. The subjects are approximately in the right order. | Here is one search engine and here is another for online courses. Doubtful they're all free, but hopefully they let you filter or easily spot it. Maybe after you feel more comfortable look at a course like this , which is more about the thinking of math (not vouching specifically for that course--it was one of the first that popped up on one of the search engines I linked). I think you should continue to do Khan or similar, at least until you are through some basic algebra, geometry, and trigonometry courses (back when I was in high school we had to take/test out of algebra I, geometry, algebra II, trigonometry, linear algebra, and analytical geometry before we could take AP calculus), but I would encourage you to get into reading and practicing proofs as soon as possible. In undergrad, that was with our modern algebra course. This was after I had taken calc 1-3, ordinary differential equations, linear algebra, numerical methods, and real analysis but I really wish I had taken it earlier (bypassing the modern algebra pre-req to to take real analysis over the summer as my first proof course was pretty rough). Once I actually worked with proofs, my thinking and understanding of math changed a lot. It has tied (and continues to tie) together concepts in my field that I think most others miss out on, but it's only with repeated effort and challenging myself to learn more. Which means pushing myself out of my comfort zone and learning to be okay with feeling frustrated while something takes awhile to sink in. It sounds like you're dedicated, so keep it up! 5 years ago I was restarting college after failing out without a solid idea of what I wanted to do. Earning a math degree is one of my proudest accomplishments because of how much I feel I've opened up for myself in the world. Good luck in your learning, I hope you grow to love and appreciate math like many others. |
[
"A quick guide to find square root of y."
] | [
"math"
] | [
"8rxxfw"
] | [
0
] | [
"Image Post"
] | [
true
] | [
false
] | [
0.48
] | [deleted] | Or just a link, https://en.wikipedia.org/wiki/Methods_of_computing_square_roots#Babylonian_method | Why make this an image when it would be infinitely more convenient as text? | I apologize to all. I'll try my best to follow the unspoken rules to better convey my message in the times to come. | No need to apologize. Don’t take our comments personally. Your post ran into a wall of curmudgeonly regulars before you got through to the wider mass of upvote-anything-that-comes-packaged-as-an-image casual reddit browsers. This technique is one of the oldest tricks in the book (actually, from before the invention of books), and all of the regular readers here have studied this and related ideas in great depth. Your post is pitched at a ~early high school level, which is not inherently bad in itself, but a lot of regular readers here are looking for something meatier or more advanced. If you submitted to a forum about quick mental math tips, you would probably get a better response. Or if you want to talk about this subject (root finding using Newton’s method), there are a variety of more interesting ways of tackling it, e.g. by discussing the history, talking about how this method compares to alternatives, etc. etc. Here’s a fun related story, https://www.ee.ryerson.ca/~elf/abacus/feynman.html | Infinitely? |
[
"What happens when you put 7 triangles to a vertex [OC]"
] | [
"math"
] | [
"8rwomr"
] | [
41
] | [
"Image Post"
] | [
true
] | [
false
] | [
0.88
] | null | I'd say this looks like a total clumpy mess, but I don't want to be hyperbolic. | Most hyperbolic plane constructions are great fun to play with in real life, but their greatness appears impossible to share on photos -- is it the case here too? | I see you one paper hyperbolic plane, and raise you one hyperbolic blanket. http://geometrygames.org/HyperbolicBlanket/index.html | Yes, it just looks like a blob, rather than a hyperbolic plane. | You could make an infinite 3D lattice with 9 triangles at a vertex, like this: Start by dividing 3D space into cubes, with cube centers at the integer points. Take away all cubes whose center has an odd x-coordinate and an odd y-coordinate; similarly, remove all cubes whose center has an odd x-coordinate and an odd z-coordinate; and remove all cubes whose center has an odd y-coordinate and an odd z-coordinate. You're left with an infinite Swiss cheese, with 6 squares meeting at each vertex. So the total angle at each vertex is 540 degrees. The Swiss cheese has a 2 x 2 x 2 unit cell, which repeats infinitely (or you could identify opposite sides and make it finite). In each unit cell, there's a cube with center (2n, 2m, 2l); a cube with center (2n+1, 2m, 2l); a cube with center (2n, 2m+1, 2l); and a cube with center (2n, 2m, 2l+1). The other cubes have been taken away, so the Swiss cheese is half filled with cubes. The cube at (2n, 2m, 2l) has no exposed faces. The cubes at (2n+1, 2m, 2l), (2n, 2m+1, 2l) and (2n, 2m, 2l+1) each have 4 exposed faces. You can triangulate this lattice by drawing a line diagonally across each exposed square face. This can be done so that at each vertex there are 9 triangles, in a pattern of 90, 45, 45, 90, 45, 45, 90, 45, 45 degree angles meeting at the vertex. Each triangle has angles of 90, 45 and 45 degrees. The cubes with 4 exposed square faces would have diagonal lines on all 4 exposed faces. If you consider one non-exposed face to be bottom and the other non-exposed face to be the top, all the diagonal lines on the 4 exposed faces would be going counterclockwise from bottom to top. So what happens if you turn each triangle in this lattice into an equilateral triangle? Would it still fit nicely into 3D space? I hope so ... The columns of cubes would twist. Something with 7 triangles at a vertex might not fit neatly into 3D space like this, though. |
[
"In category theory, is the union of commutative diagrams commutative?"
] | [
"math"
] | [
"8rwpna"
] | [
0
] | [
""
] | [
true
] | [
false
] | [
0.25
] | If not for all, then for what conditions? | I suspect that if you manage to formalize your idea of what a "union of diagrams" is, you'll most likely find a trivial answer to your question. | That doesn't mean anything. | What is a union of diagrams? | What is the union of two commutative diagrams? How are you attaching them together? Formulating this in the most obvious way gives you a negative answer but if you are more specific about how exactly you are allowed to glue things together then maybe the answer would be yes. | Therefore, I am the Pope. |
[
"Should we reconsider how the order of operations is taught in school? (video)"
] | [
"math"
] | [
"8rw7g1"
] | [
0
] | [
""
] | [
true
] | [
false
] | [
0.22
] | null | Simply put: No because notation is a convention and is not defined by any kind of logic therefore you would be able to use any kind of notation as long as anyone who would be reading it would somehow know what conventions you are following and by that understand what expression you intended | I am not since it follows a special rule which isn't used generally | Using commas is also convention, yet you don't follow it. (/s) | I agree but what is outlined in the video, is still following the convention of Pemdas with that additional rule about fractions. | Does it really? If we stick to the rule that expressions only containing operations of the same priority should be evaluated from the left, then 1/2 x = (1/2) x = x/2. If we want it to evaluate to 1/(2x), we need to add a special rule along the lines of "monomials not written with a multiplication sign is evaluated before M and D". Moreover I'd say this really is a rule that we generally use, even though it doesn't show up in elementary school arithmetic. When it does show up, students have already developed enough mathematical maturity and intuition to understand these expressions from context without being explicitly taught the rules, which might explain why it's never made explicit. |
[
"If you look at an infinite amount of digits of pi could it repeat?"
] | [
"math"
] | [
"8rvy8x"
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0
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"Removed - ask in Simple Questions thread"
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true
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false
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0.3
] | null | It's suspected but not proven. A normal number is a number where you can find any finite sequence you want. Being irrational is not equivalent. For example 0.101001000100001... is irrational but 2 never shows up. | it is possible to find sequences of any sort in pi. That's believed to be true, but it's an open problem (and does not follow from pi being irrational). https://en.wikipedia.org/wiki/Normal_number Edit: Then could it be possible to find a sequence infinitely down the digits of pi where it repeats itself an infinite amount of times. You can't go infinitely far down the digits of pi. Every digit of pi corresponds to a unique place. There's a 10th digit of pi, a 1000th digit of pi, a 10000000000000000000000000000000000000000000000th digit of pi, and so on. But there's no such thing as the infinityth digit of pi, or going infinitely further down the sequence or anything like that. | It's possible there could be something point in pi where it looks like it's repeating up to a really large number (something kind of like that happens in the first 10 digits of e). But that wouldn't actually make it a repeating decimal. | The mathematical term you are looking for is "arbitrarily", not infinitely. what /u/jm691 is describing is that you can take an arbitrarily "small" digit of pi (1000000000000000th digit). And that number has no upper bound - you can go as far as you want. But you can't take "infinite"... | What you are describing here doesn't exist. Every integer smaller than infinity can be written, given enough paper. They are all just finite numbers. |
[
"Any computer assisted proof software work like debugger and syntax checker?"
] | [
"math"
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"8rvqid"
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3
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""
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true
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false
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0.64
] | I've been having trouble proving as exercises because there's no one to validate each single statement for me, and I've arrived at proofs that are bogus because some little error in the proof. Does there exist a computer-assisted proof software that can validate the 'syntax' and 'debug' proofs? | No, there aren't proof compliers that will do your abstract algebra homework for you. | At your level, you should go line by line of the proof and find any errors. The proofs are simple enough that, with time, and if you understand the material, this should be doable. If you are making errors, that means you are misunderstanding what you are writing. | Based on your comment, I would guess at least undergrad and at most 2nd year grad. You shouldn’t be submitting or accepting a proof without being 100% certain you are correct. | Coq is pretty powerful in terms of allowing higher-order tactics. Mizar is closest to standard mathematical formulations. It would help a lot if you could specify an example of a theorem you would like to verify. Please be aware that discrete mathematics and computer verified proofs form a good marriage. Continuous mathematics and computer verified proofs are still in a tinder-stage, so to speak. | 'abstract algebra' how about for linear algebra, statistical and probabilistic methods? |
[
"I am bad at geometry."
] | [
"math"
] | [
"8rvpxm"
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2
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""
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true
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false
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0.62
] | [deleted] | Can you give an example? usually have no idea how to approach the problems This comes from lack of experience. would like a little direction so I can learn the ins and outs You could try meeting with a tutor face to face. Or you could take a course. Or you could try self-studying from the type of problem book that has difficult problems and provides hints and full solution write-ups to compare your work to. To start you could work through Kisilev’s , or possibly Hadamard’s . Or that book mentioned elsewhere in the thread. After that you might enjoy e.g. Polya’s book , or Yaglom’s books about . | You don't have to be good at everything.... make sure to explore lots of subjects. | I'm the same way, and really you just need to practice more. I would recommend this book - https://artofproblemsolving.com/store/item/intro-geometry And follow up with something like this - https://www.amazon.com/Challenging-Problems-Geometry-Dover-Mathematics/dp/0486691543/ You can download the Euclidea app if you want to practice constructions. | We used the textbook "Elementary Geometry from an Advanced Standpoint" by Edwin Moise in my Geometry course. We covered Absolute Geometry (Euclidean w/o parallel postulate), then transitioned into full Euclidean. There is more in the book, we only covered about half the book, but it's pretty simple for an introduction and presents stuff without a lot of necessary prior knowledge. | Well, the idea is to be good enough to maintain my GPA |
[
"Why is a function labelled concave when the set of points under it is convex?"
] | [
"math"
] | [
"8rvewe"
] | [
8
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""
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true
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false
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0.85
] | I know it's an arbitrary naming thing and that you can say that the set of points above a convex function is convex but this has always been unintuitive and gnawed at me | At the end of the day you have to decide whether you care about the points above the function or below the function. It was decided to go with the points above. The decision is fully arbitrary. You can speak of concave functions where the set of points below them is a convex set. Any statement about convex functions has an analogue for concave functions. | Part of it is a question of whether you're more interested in finding a minimum or a maximum. If the points above are convex, then minimization is nice; if the points below are convex, then maximization is nice. In physics applications, the Principle of Least Action means that we're often interested in minimizing. Problems from physics were a major motivation for a lot mathematical methods and even today a lot of textbooks treat minimization as the default and trust to the reader to adjust things for maximization problems. If economists are more interested in maximizing, I can see why this could get frustrating. | Aww fuck I think I got one of the questions in my calculus exam wrong cause of this. I assumed it referred to the points below as well, goddamnit. | And to further add to the confusion of economics students, a convex preference relation is represented by a (quasi) concave utility. Of course this is the same thing, since convexity of a preference relation refers to the set convexity of upper contour sets. Notice that if f is a concave function then {x | f(x) > a} is a convex set. This speaks as to the reason maximization is so nice with concave functions. | Okay. From my learning of math for economics it would seem the points below are more interesting because they have an economic interpretation as "feasible" whereas points above in all of our functions aren't |
[
"Software developer moving to machine learning. Need some math advise, thanks!"
] | [
"math"
] | [
"8rv55l"
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5
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""
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true
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false
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0.59
] | I have done math a long while back in college. I have recently decided to start moving into machine learning space and going few relevant courses/programs. Given that Math/Stats is an absolute essential to understanding how it all hangs together, more specifically - linear algebra, Probability Statistics , vectors. I want to ask all the people who love math here. How do I start loving math again? I was an okish math student but nothing more than that and while I can do just enough math to cruise through the courses, I want to be good at it. When I go through the concepts I feel like I get it and then once all the symbols come into play, I get confused again. | I don't know how exactly to put it into words. Loving math doesn't mean loving the little symbols and how you move them around and such. Loving math is about appreciating and wanting to understand the patterns we see in the world around us, the concepts we invent in our minds. The symbols, the names, all of that is just words to describe the patterns. if you want to love math you need to learn to find the beauty in the patterns. If you want to understand math you need to understand that the symbols are just shorthand to describe things that take many words to describe. Sorry if this is worded weirdly english isn't my first language. | I don't have a whole lot of time for math these days. However I still enjoy every 'Numberphile' (channel) video on youtube. They talk about a lot of really interesting problems and topics. | Thanks for the reply, What resources do you recommend to read to for the " find the beauty in the patterns " bit. I generally like the here is a problem, use this tool/concept/formula and it solves it approach..helps me apply it and then understand it. I guess a lot of material online is in reverse, and hence very dry. I would love to read some real world problems -> applications of these concepts-> problem solved kind of a thing..That would help me think of other similar problems it can be applied to. | 3Blue1Brown is an excellent recommendation, thank you! | 3Blue1Brown is an excellent recommendation, thank you! |
[
"Can you recommend non-visual but confounding mathematical theorems or paradoxes like Russel Paradox i can tell to my blind friend"
] | [
"math"
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"8rvd0c"
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25
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""
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true
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false
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0.85
] | null | I assume things like Banach-Tarski are what you're aiming to avoid. The prisoner paradox is an equally confusing consequence of the axiom of choice: https://cornellmath.wordpress.com/2007/09/13/the-axiom-of-choice-is-wrong/ | The Paris-Harrington Theorem . Something which is "true" about integers but cannot be proved using Peano's axioms. Quoting from the page above: "For any positive integers n, k, m one can find N with the following property: if we color each of the n-element subsets of S = {1, 2, 3,..., N} with one of k colors, then we can find a subset Y of S with at least m elements, such that all n-element subsets of Y have the same color, the number of elements of Y is at least the smallest element of Y." (Emphasis is mine.) A related example is Goodstein's Theorem which says that a certain naturally created sequence starting with any positive integer terminates. Again, this can be proved to be "true" but not if one restricts oneself to Peano's axioms. Both these results can lead one to wonder about whether we can solve the Collatz Problem . Moreover, it suggests that one need not go "as far as" the Axiom of Choice to demonstrate problems between mathematical intuition vs logic. "Large" integers are "good enough"! | Suggesting we might want to question AC is far from outside the mainstream. It's not like I linked a constructivist manifesto My understanding is that undergrad is about critical thinking so some absurd dogmatic insistence we not talk about the axioms is just stupid. | Tl;dr: nonmeasurable sets are and not just because of Banach-Tarski. | Tl;dr: nonmeasurable sets are and not just because of Banach-Tarski. |
[
"OC Math Problem - Easyish - Based Off Of Finding The Area Of The Portion Of A Hinge On My Door"
] | [
"math"
] | [
"8rv0op"
] | [
0
] | [
"Removed - try /r/learnmath"
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true
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false
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0.14
] | null | Yep, mathematics is definitely all about finding the area of shapes. Also, without any specifics, all we can do is estimate. So, the area is about 10. | Fairly sure this can't be calculated exactly without knowing the shape of the curve on the sides. | What they’re saying is since the lines are hand drawn and not the same on both ends, we can’t just assume anything, especially in a math-specific forum. | You have not given enough info. You’ve given us a crude drawing. We can’t give an exact answer without knowing the exact construction of the shape. | I can prove it can't by creating two different shapes that fit the diagram. Suppose the ends are semicircles with radius 1. Then the two ends together form a circle with area π, and the remaining part of the shape is a 2x4 rectangle, so the area is 8 + π. Suppose instead that the ends are formed from sections of the curve y = 2 - 1, which can be easily verified to pass through (0, 0) and (1, 1), thus fitting between the two points as needed. Then the area is 4(2 + 0∫1 2 - 1 dx) = 4 + 4/ln 2, which is not equal to 8 + π. |
[
"A nice analysis problem I made"
] | [
"math"
] | [
"8ruops"
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21
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""
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true
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false
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0.82
] | Let {a_n} be a sequence of positive real numbers such that sum a_n converges. Find a necessary and sufficient condition on {a_n} such that if r is any positive real less than sum a_n, then there exists a subsequence {b_n} of {a_n} such that sum b_n converges to r. | Hmm, good try! But there's an obvious counterexample. For example the series 10000 + 1 + 1/2 + 1/4 + ... has no subsequence that converges to 9000. | The terms are all positive so there is no conditional convergence going on. | Solution (I hope) First, observe that the limit as n->infty of a_n is 0. Now, I claim a subsequence always exists if r <= sum: If r = sum, you're done. If r < sum, consider sum - r > 0. Find the first element of a_n smaller or equal to this, and remove it. Now, our sum is closer to r. Call the sum of this new sequence sum'. Then, look at sum' - r and delete the first a_n smaller or equal to this (but after the previous element), etc. Note that, if we ever get that the new sum = r, than we just stop removing and we're done. As the difference will always be non-negative and as we can find arbitrarily small numbers, this procedure can always be carried out. Now, to show that this procedure produces a sequence which converges to r, denote this sequence by {b_n} and say i_1, i_2, ... are the indices of removed a's (so if a_5 and a_10 are deleted in the above algorithm, i_1 = 5 and i_2 = 10). Note that the partial sum S_n of b_n will be the sum of a_n - sum of removed terms. Call this T_n - R_n (T_n = sum of first n terms of a, R_n = sum of all a's removed up to first n terms--so if a_5 is the first element removed, T_6 = a_1 + a_2 + a_3 + a_4 + a_5 + a_6 and R_n = a_5). Now, focus on R_n. If removing finitely many removed terms makes us good, stop. Otherwise, we need infinitely many removed terms. As all terms are positive, R_n <= R_(n+1). Also note that R_n is bounded above by the sum of a's, and hence we have a bounded monotonic sequence and thus R_n converges. And it must converge to sum a's - r, since if it will eventually be larger than any number smaller than that. | WLOG, we can take a_n+1<= a_n for all n and we'll start indexing at n = 1. Then a subsequence {b_n} such that sum b_n = r exists for any r in (0, sum a_n) if and only if a_n <= sum(j>n) a_j. Suppose that a_n <= sum(j>n) a_j and choose r in (0, sum a_n). Define {c_n} by c_0 = 0 and c_i = a_i, if a_i + sum(j=0 to i-1) c_j <= r, c_i = 0, otherwise. Then take {b_n} to be the nonzero entries of {c_n}. Clearly sum b_n = sum c_n and sum c_n <= r. Suppose that sum c_n < r. Then there exists k such that c_k = 0 and c_n = a_n for n > k. But sum(j>k) a_j >= a_k, so it's a contradiction to include all the terms after a_k, but not a_k itself. So sum b_n = r. On the other hand suppose there exists m such that a_m > sum(j>m) a_j. Let r be between sum(j>m) a_j and a_m. That r cannot be the sum of a subsequence, because a_n > r for n <= m, and sum(j>m) a_j < r. | All praise the magic conch |
[
"What’s the answer?"
] | [
"math"
] | [
"8rtvu7"
] | [
0
] | [
"Image Post"
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true
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false
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0.17
] | [deleted] | I’d say 6 Edit: but I see how he got 9 | This is not the correct subreddit for this kind of content. | X3=9 | Y=(x-1)(x) Y=6 | The answers are false, false, false, false, and 3. Don't use the equals sign if you mean function application. |
[
"Is it continuous?"
] | [
"math"
] | [
"8rt76c"
] | [
0
] | [
"Removed - ask in Simple Questions thread"
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true
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false
] | [
0.33
] | null | No, in general you need the limit to hold uniformly for it to be jointly continuous. | I think as a counter-example, it's actually possible to construct a function which satisfies what you say, but lim(t->0) f(u * t, v *t) is not = 0 | How about f(x,y) = slope of line through (x,y) times radius. f(x,y) = y/x √(x + y ). Set f(0,y) = 0. Clearly not continuous, despite approaching zero along every line through the origin. Or think about it in polar coordinates. You're asking whether a function f(r,𝜃) is continuous if it is only continuous with respect to r, since you are only taking radial limits. Clearly no-go. | For another example, consider f(x,y) = x y/(x + y ). It approaches 0 along any line through the origin, but if you approach along a parabola, it goes to 1/2. This is on wikipedia Can we at least say "if f(p(t)) is continuous for all paths p, then f is continuous"? Yes, I think that should be true, at least for R (multivariable functions of two variables), since it is locally path connected. | since it is locally path connected. According to this m.se answer , you need the space to be locally path connected first countable. That seems entirely reasonable. In any event, yes, it holds for the plane. |
[
"Does 0 divide 0?"
] | [
"math"
] | [
"8rt3pf"
] | [
3
] | [
"Removed - ask in Simple Questions thread"
] | [
true
] | [
false
] | [
0.8
] | null | m divides n if there is an integer k such that n = km. Some authors may require k ≠ 0. But either way, there is a k such that 0 = k ∙ 0. Eg k = 1. So yes, zero divides zero. | 0 is not invertible; you may not divide by 0. But yes, there are infinitely many solutions to 0 = 0 ∙ x. That's true. | x divides y whenever there exists an integer z such that zx = y. For x = y = 0, this is true. | But can't I do something like this then: 0 = 0*x 0/0 = x So 0/0 has infinit results? Is 0/0 = 0? Or is it 1 because a number divided by itself is 1? Or is x arbitrary? | The relation m|n is the number n/m. Just because the fraction n/0 is undefined does not mean a relation 0|n is undefined. To have 0|n means n = 0c for some integer c, and that holds exactly when n= 0. |
[
"How delayed will your flight be if you are seated in 10A. Use S for seats and M for minutes delayed."
] | [
"math"
] | [
"8rsyrb"
] | [
101
] | [
"Removed - see sidebar"
] | [
true
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false
] | [
0.81
] | null | The intended answer is 40 minutes, to check for students who just plug in numbers without thinking. | Here’s the story (Sorry I could only find the story on The Independent) | Similarly, if it take 8 programmers to complete a project in 60 days, how long will it take 4 programmers? 30 | T=40 | T = 40 x (P-1 < P < P+1) |
[
"🤔 Isn’t a translation just a rotation with the center of rotation at infinity 🤔"
] | [
"math"
] | [
"8rvfrv"
] | [
1321
] | [
""
] | [
true
] | [
false
] | [
0.93
] | null | Yes! Including a point at infinity allows you to represent affine transformations as linear ones without lifting to higher dimensional space. This can be useful for eg. computer graphics. See this paper for a detailed comparison of different computational models of geometry, including ones that use this trick: https://www.staff.science.uu.nl/~kreve101/asci/GAraytracer.pdf | uh lemme get uhhhhhh rotation | isn't a rotation just an infinite series of translations with transpositions :thinking_face: | What am I doing with my life reading this. | what kinda cartesian plane got in it |
[
"An incorrect definition of fairness."
] | [
"math"
] | [
"8rrnm0"
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0
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""
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false
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0.47
] | [deleted] | I didn't read your paper very closely but is the gist of what you're saying that the existing definition of symmetric game (that the payoff matrix is symmetric across the diagonal) isn't a good notion for "fairness" of a multiplayer game when opponents are anonymous and instead "fair" should mean equal payouts under optimal play? That seems reasonable but I am not seeing anywhere that anyone is using the word "fair" other than you. It does seem worth considering the distinction between what you term standard symmetric and fully symmetric but I'd suggest that describing what you're doing as showing that the existing definition of "fairness" (a word which doesn't even appear in the page you claimed has the incorrect definition) is not a good strategy for getting people to pay attention. | I'm saying that I don't see anyone using the word fair in game theory, or at least not trying to use it in any technical sense, and it does not appear on the page you linked. So it's a bit strange for your post to say you're trying to correct an incorrect definition for a word that does not appear to be defined anywhere. My knowledge of game theory is fairly cursory but your paper did seem to have interesting things in it, I have no idea how much of it is new vs existing of course. It just struck me as weird for your post to make it sound like there was some agreed upon definition of fair that needed fixing when you gave no evidence of anyone defining that word at all. | I'm saying that I don't see anyone using the word fair in game theory, or at least not trying to use it in any technical sense, and it does not appear on the page you linked. So it's a bit strange for your post to say you're trying to correct an incorrect definition for a word that does not appear to be defined anywhere. My knowledge of game theory is fairly cursory but your paper did seem to have interesting things in it, I have no idea how much of it is new vs existing of course. It just struck me as weird for your post to make it sound like there was some agreed upon definition of fair that needed fixing when you gave no evidence of anyone defining that word at all. | I do still consider symmetric to at least partially be about ascribing some notion of fairness to games It is. But I feel like this is similar to when probabilists discuss "impossible" (you can see my long winded post about this from a few weeks ago if you look for it). Generally speaking, in probability, we define the word impossible and most probabilists will simply say that's outside the purview of probability theory. I sort of feel the same holds for "fairness" in game theory. Sure, we have well-defined notions such as symmetric just as we have things like almost never in probability but the mathematical model doesn't actually capture the intuition of "fair". there is an 'agreed upon' definition of fairness/symmetry (by agreed upon, here I mean published with over 1000 citations which is also being used as the multiplayer definition of symmetry on wikipedia) that needed fixing I didn't realize the paper used the word "fair". The wiki article, notably, does not, leading me to believe that whoever wrote it is aware of the issues. You may be correct that the existing definition is simply wrong then and that you have a better one, just as the naive notion of impossible in probability (outside the "underlying set") is simply wrong and the only viable definition of impossible is measure zero. But I'd suggest that the real conclusion ought to be that "fair" and "impossible" ought not be used at all in our respective fields. | I do still consider symmetric to at least partially be about ascribing some notion of fairness to games It is. But I feel like this is similar to when probabilists discuss "impossible" (you can see my long winded post about this from a few weeks ago if you look for it). Generally speaking, in probability, we define the word impossible and most probabilists will simply say that's outside the purview of probability theory. I sort of feel the same holds for "fairness" in game theory. Sure, we have well-defined notions such as symmetric just as we have things like almost never in probability but the mathematical model doesn't actually capture the intuition of "fair". there is an 'agreed upon' definition of fairness/symmetry (by agreed upon, here I mean published with over 1000 citations which is also being used as the multiplayer definition of symmetry on wikipedia) that needed fixing I didn't realize the paper used the word "fair". The wiki article, notably, does not, leading me to believe that whoever wrote it is aware of the issues. You may be correct that the existing definition is simply wrong then and that you have a better one, just as the naive notion of impossible in probability (outside the "underlying set") is simply wrong and the only viable definition of impossible is measure zero. But I'd suggest that the real conclusion ought to be that "fair" and "impossible" ought not be used at all in our respective fields. |
[
"Is there an easy-to-understand explanation for the Modulo Operation?"
] | [
"math"
] | [
"8rquss"
] | [
2
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""
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true
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false
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0.58
] | Everywhere I look, I can’t find an explanation that’s easy to understand. Anyone know where I can find one? | The modulo operation gives you the remainder after division. For example, (10 Modulo 3) = 1 3 x 3 = 9 which is 1 away from 10. Another example, (21 Modulo 10) = 1 10 x 2 = 20 which is 1 away from 21. Does that help? | Thank you so much. It helped me a lot! | Think of a clock. 11 o'clock + 1 hour = 12 o'clock. 12 o'clock + 1 hour = 1 o'clock. In modular arithmetic we usually write 0 for 12 o'clock, but either way is fine. | would this help you with negatives? like -21 mod 10? | For negatives it helps to use the division algorithm: n=qm+r, with 0<=r<m. So in this case with n=-21 and m=10, q=3 and r=9 so -21 is 9 mod 10. You can also see this by repeatedly adding ten until you end up in [0,9] . |
[
"Understood and Aced my first proof based class!"
] | [
"math"
] | [
"8rquc2"
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0
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""
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true
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false
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0.25
] | [deleted] | Relevant . | Thanks! Can't wait for until I take harder classes! | Hahahahaha. I'll delete this sorry~ | Sorry that was not my intention, your post is fine in my opinion, I just found the timing of their comment and your post quite hilarious. Deleting your post means less people will see the comical timing :( | ah. well too late now haha. I'm just glad I did well in my proofs class this quarter~ |
[
"Is it possible for a line to exist without any possible parallel lines (excluding itself)?"
] | [
"math"
] | [
"8rr8dq"
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6
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""
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0.69
] | Title. Sorry if this is the wrong place or anything for this, I'm not great at math but I was wondering if such a line could exist (inwhich no lines that follow the definition of a parallel line exist for an equation of a line). If not, is there any mathematical proof that such a line does not exist? | In the standard Euclidean geometry, each line with a distinct point off the line has exactly one line through the point parallel to the first line. The other two geometries create disjoint circumstances that cover all cases, those non-Euclidean geometries are Hyperbolic and Elliptic whose parallel postulates allow for at least 2 parallel lines rather than exactly one for Hyperbolic geometry, and as for Elliptic geometry, there exists no parallel lines through a distinct point off a line. Thus, if you are looking for a line to exist without any possible distinct parallels, you must look to Elliptic geometry, of which all lines fit the bill. You can read more on Elliptic Geometry here . It is worthwhile to note that Elliptic geometry isn't nice like Euclidean or Hyperbolic geometry because it cannot follow the rules of Neutral geometry as it requires parallel lines to exist. Though a good model to get intuition is looking at the surface of a sphere, where lines are defined as great circles. In this model it is impossible to find two great circles that do not intersect, and thus there do not exist parallel lines. However if you don't restrict yourself to great circles you can have non-intersecting lines which becomes Hyperbolic geometry. | I believe elliptic geometry stipulates only that any two lines intersect, but not how many times. Both spherical (two intersections) and projective (one intersection) are elliptic. You don't have to identify antipodal points if you don't want to. | You have to identify antipodes in spherical geometry in order to make every two distinct lines intersect at a point, instead of two points. : or not. | The Fano Plane is an example of a finite geometry with no parallel lines. It has only seven points and seven lines. It is the smallest example of a projective plane. All projective planes have no parallel lines. | Oh ok I didnt know |
[
"Minimum cards that you need to see to tell if a game of solitaire is doable or not"
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"8rr8iy"
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237
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false
] | [
0.95
] | null | Maybe I’m missing something, but if all you had left was four columns with the kings stacked on queens, but the other three columns are free, doesn’t the game become winnable? Why don’t we just bury the Aces instead? | Maybe I’m missing something, but if all you had left was four columns with the kings stacked on queens, but the other three columns are free, doesn’t the game become winnable? Why don’t we just bury the Aces instead? | That’s a really difficult question because there are so many ways for a game to be unwinnable, I’m not a big solitaire player so it would help if you could list all those different ways so that I could account for all of those in an algorithm | And in the best scenario, you can tell the game is unwinnable right at the start. For exemple, if the game start with four queens and three 9 revealed | Revealing 51 cards is essentially the same as revealing all 52 cards, since you can determine what the last card is by elimination. If you know what every card is, you have full information, and you can always tell whether a game is winnable or not. (If nothing else, by an exhaustive search of all possible moves.) On the other hand, consider these two positions: K Q J 10 9 8 (7) (6) 5 K Q J 10 9 8 (6) (7) 5 Cards in parentheses are not revealed, all cards are of the same suit, there are no cards left in the deck. Both positions have 50 revealed cards, and in both positions the player sees the same board. Yet the first position is winnable, but the second is not. There is no way to tell which is the case without revealing another card. Thus revealing 51 cards is always sufficient to determine whether a game is winnable; but even after revealing 50 cards it might still be ambiguous. |
[
"How is the derivative of the area of a circle the same as the circumference, and similarly the derivative of the volume equal to the surface area of a sphere?"
] | [
"math"
] | [
"8rqk3l"
] | [
31
] | [
""
] | [
true
] | [
false
] | [
0.8
] | null | You can obtain the area of a disk by integrating thin circular strips of circumference 2πr, see this link for a visual explanation. The same applies to the volume of a sphere, which can be obtained by integrating thin spherical shells of surface area 4πr . Since the infinitesimal change in the area of a disk with respect to the radius is equal to the circumference (from the integration above), the derivative of the area is the circumference. | This is how it's described in by Morris Kline. | This is how it's described in by Morris Kline. | More generally this comes down to Stoke's theorem. If you think of a 3d ball with radius r what is its boundary? The 2d sphere with radius r. What Stoke's theorem yields is that taking a derivative is dual to viewing the boundary of a body. The way I have described it, is a large oversimpilfication but sufficient for now. If this subject interests you, try learning about differential geometry. It deals with these subjects. | I'll have to take a look at that. I think Calculus, and Linear Algebra for that matter, is too intuitive and beautiful for universities to teach it computation heavy and with a brute force approach. Thanks for sharing! |
[
"Negative base algorithms"
] | [
"math"
] | [
"8rq7e0"
] | [
66
] | [
""
] | [
true
] | [
false
] | [
0.91
] | Looking at : Negative numerical bases were first considered by in his work , published in 1885. Grünwald gave algorithms for performing addition, subtraction, multiplication, division, root extraction, divisibility tests, and radix conversion. Negative bases were later independently rediscovered by A. J. Kempner in 1936 and Zdzisław Pawlak and A. Wakulicz in 1959. I scrounged around some Google book sites and found Mr. Grünwald's paper. Either I found the wrong paper, looked in the wrong section, or mistranslated (with Google), because I couldn't find the actual algorithms, just examples. I guess he assumed the reader could figure out the precise methods themselves, but I have to admit I may not be that smart. I'm assuming integer operands from now on. Addition, subtraction, and multiplication seem easy; just reverse the sign on your carries/borrows. It's the other ones I can't seem to wrap my head on. I guess for division (quotient & remainder style), you try to reduce the trial dividend to the smallest absolute value, but I don't have the whole picture. And I have no idea on roots, divisibility, and radix conversion. (Well, divisibility and radix-conversion can be done once division is solved, but Mr. Grünwald may mean some sort of faster method.) Anyone have any ideas on these algorithms? If anyone can translate Italian mathematics, there's a . | I had written a paper on imaginary number bases, but sadly never got it published (it’s on arxiv though). It gives an overview of number bases and conversion methods, and then expands those to work with any purely imaginary number as a base. In that, I give an example of doing division in a negative base, quoted as follows: The rules for doing a division in such a base -B (where B is positive and real) are as follows: And here are two examples in base -10 It’s been a while since I’ve done these, and I recall division being harder to grasp, even when the steps are written out like this. As a side note, I am working on a python module for number base conversion but it's not yet quite ready. | I spent a bunch of time working on the base 2 + exp(πi/3). It’s pretty fun. There are 7 possible digits at each place: 0 or exp( /3). Any Eisenstein integer is expressible in this base, and if we take fractions we can get arbitrarily close to any point in the complex plane, but no (real, non-integral) rational number has a terminating expansion. Division is a pain in the ass; I was just doing it by hand, and it was a bit tricky for me to figure out how to turn my manual heuristics into a reliable algorithm. Then I got sidetracked by other projects. | For my own sake, I'd need to see a proof. It's not obvious at all. | https://i.imgur.com/KJr9X6d.png Source Why? Creator ignoreme deletthis | but no (real) rational number has a terminating expansion. Aside from 0. :P |
[
"Should I get a PhD in Math?"
] | [
"math"
] | [
"8rovqn"
] | [
42
] | [
""
] | [
true
] | [
false
] | [
0.76
] | So it goes without saying I love math: it's one of the best subjects out there and it's fascinating how it's used in so many different fields. I remember back in undergrad being deeply invested in courses such as Linear Algebra, Abstract Algebra and Probability and Statistics just to name a few. I even got the opportunity to do research on Tropical Geometry during my senior year. A year has passed since then, and I'm currently finishing up my Master's in Education to become a high school math teacher. However, I've been having thoughts about pursuing my math knowledge even further by either going for my M.S. or possibly PhD in the subject. I love learning about new topics and wish to teach it to others or possibly use it as an application source with topics from Linear or Stats. I'm wondering if a PhD is really worth it if my main purpose is to use it for academia. I was wondering if most professors do work on the side from teaching, such as research, as their primary source of income and see how it all balances out. | A few years ago, I'd recommend getting a Ph.D. without hesitation. However, now I recommend hesitating. Jobs in academia are very difficult to come by and you need to be somehow different from the other Ph.D. holders to get one. I'm a professor and I'm one of about 20% of all my fellow students from my B.Sc./M.Sc./Ph.D. that ended up in academia. It's a good life, but also not so great. The travel is amazing, and I love writing about my work, and teaching is fun too. But there are also huge demands on my time that amount to annoyance and stress. Dealing with unprepared/uninterested students, and lots of them, can drain your interest from teaching. University bureaucracies are stifling; essentially, you, as a faculty member, will be a machine for answering emails. And this is the one career out any I've encountered where it is very difficult to strike work/life balance. Some do it well; most do not. What I'd say is, liking and being good at math (which is learned!) is the bare minimum requirements for pursuing a Ph.D. If you want a job in academia, you need extreme multitasking skills, you must love writing and have potential to do it, and do it well, during the little cracks that open in your schedule, you must be personally likable - unless you have astounding talent in mathematics - since academia is primarily about interacting with people, and you need to be ambitious, which manifests as actively pursuing and securing funding (my weakness). You also need self-determined focus - I see many people pursue their Ph.D. advisor's research program without formulating and answering their own questions and are suddenly adrift after graduation. This shouldn't be surprising. A Ph.D. is about becoming an independent thinker. Not everyone gets this. I went to university with people who were astounding in math; I was good, but not astounding. Everyone of that top tier, except two, totally failed at getting into academia. For numerous reasons, all related to the above. It takes a special mix of personal, professional, and intellectual qualities to make it. If you find you develop this mix, by all means pursue the Ph.D.! Of course, you can also go into industry with a Ph.D. My colleagues in industry make twice my salary and have time to go hiking on the weekends. If you want to teach higher-level math, teaching at a community college, which seldom requires a Ph.D., is a option. The pay is often on par with many professorships, your scope of work is interesting, and you have far less demands on your time. | Teaching is usually a required duty of their job, even if it's not their main focus. It's not just an optional extra thing they can do. | I was wondering if most professors do work on the side from teaching, such as research, as their primary source of income and see how it all balances out. I maybe reading this incorrectly but a professor's primary source of income is research. They teach whichever courses they have to. | This is HIGHLY dependent on the type of school. Liberal arts colleges, for example, put much less emphasis on research productivity and quality. Many are happy if you simply apply for grants, whether or not you have any chance of actually getting it. At departments that place more emphasis on research, applying for grants is expected and failing to get one can hurt your chances of getting tenure. | the people you're talking to are talking about research positions many college faculty across the country are paid just to teach at a reasonable salary |
[
"Universal integer triples"
] | [
"math"
] | [
"8rohcm"
] | [
4
] | [
""
] | [
true
] | [
false
] | [
0.7
] | Hello reddit/math! I'll start with a disclaimer: My main discipline is software, whatever math I picked up in school is beyond rusty. And my interest in math is very flimsy: I like general problem solving, pattern matching and number games; but fall asleep standing as soon as the discussion strays too far from practical application. That being said. My social ID is slightly unusual in that it only contains three unique digits: 4, 5 and 7. Year of birth is 77, which means that it took me roughly 41 years to start playing with the numbers. Doing that, I quickly realized that I can represent 1-9 using only those three numbers and addition/subtraction. 1=5-4, 2=7-5, 3=7-4, 4=4, 5=5, 6=7-(5-4), 7=7, 8=7+(5-4), 9=5+4 This property seemed unusual enough to warrant further exploration. My intuition and random testing says that it's unlikely to find another triple that can do the same thing. So I thought I'd throw out the question to hopefully get some more context, I'm sure someone somewhere spent a lifetime exploring these kinds of patterns. Over, out | This is actually a decently common property. It turns out that there are 19 such triples: (1, 2, 6) (1, 2, 7) (1, 3, 5) (1, 3, 6) (1, 3, 7) (1, 3, 8) (1, 3, 9) (1, 5, 8) (1, 6, 9) (2, 3, 6) (2, 3, 7) (2, 3, 9) (2, 5, 6) (2, 6, 7) (2, 6, 9) (3, 4, 5) (3, 4, 9) (3, 5, 9) (4, 5, 7). Some of these extend even beyond 1-9. For example, the pair (1,3,9) lets you represent any natural number up to 13 using only addition and subtraction: 1 = 1, 2 = 3-1, 3 = 3, 4 = 3+1, 5 = 9-3-1, 6 = 9-3, 7 = 9-3+1, 8 = 9-1, 9 = 9, 10 = 9+1, 11 = 9+3-1, 12 = 9+3, 13 = 9+3+1 This is the best you can do, since there are only 3 = 27 possible totals you can construct using addition and subtraction, and one of those will be 0 while half the rest will be negative. However, if you expand past triples into larger tuples, you can reach larger consecutive series. For example, (1,3,9,27) is the unique quadruple that lets you represent any natural number up through 40 using only addition and subtraction, while (1,3,9,27,81) is the unique quintuple allowing for representation of any natural number up through 121. In general, the first powers of 3 forms the optimal -tuple, allowing you to represent any natural number up through (3 -1)/2 using only addition and subtraction without repeats; in fact, this is the principle behind the balanced ternary system. | Balanced ternary is a non-standard positional numeral system (a balanced form), used in some early computers and useful in the solution of balance puzzles. It is a ternary (base 3) number system in which the digits have the values –1, 0, and 1, in contrast to the standard (unbalanced) ternary system, in which digits have values 0, 1 and 2. Balanced ternary can represent all integers without using a separate minus sign; the value of the leading non-zero digit of a number has the sign of the number itself. Different sources use different glyphs used to represent the three digits in balanced ternary. | See, I knew someone noticed :) Thanks, that helps a lot by providing some mental hooks to hang the fact on. My social ID triple goes on to 16; skipping 10, 13, 14 and 15. Whenever something looks like a sequence my brain switches into pattern matching mode, but I can't make any sense of this. | Whenever something looks like a sequence my brain switches into pattern matching mode That’s what brains do, pretty much :p | Agreed :) But, have you written any code yourself? There are different levels of pattern matching, writing code definitely changed something for me. I don't know anyone who's not a programmer or math addict who'd even notice something like this. |
[
"Exotic metrics on \\mathbb{R}^2"
] | [
"math"
] | [
"8ro3d5"
] | [
2
] | [
""
] | [
true
] | [
false
] | [
0.75
] | There are the euclidean metric and the manhattan metric. What other metrics are there for \mathbb{R}^2? | It’s exotic in the sense that it induces a weird topology on the plane: each point is isolated and has no nearby (<1) neighbors. | The discrete metric between any two points A and B is 1 unless A=B, in which case it is 0. | I’m not sure I know what I’m talking about but here’s a potentially exotic idea: Take the real line and wind it up in Hilbert curves, 1 unit square per unit length or something, to fill the entire plane. Then take the distance between two points in the plane as the distance between two inputs from the real line. Can anyone who knows more about Hilbert curves tell if this is possible? I’m worried about running into issues that there might be zero or multiple points on the line corresponding to a given point in the plane. Edit: yeah those overlap points at binary fractions are a problem, and there can’t be a continuous bijection between [0,1] and [0,1] . There can however be a non-continuous bijection: interleave the decimal digits of the two coordinates in the plane (starting with the small coordinate), then define a metric as the difference of these combined numbers for two points in the plane. This is now pretty exotic everywhere. | And now that I'm thinking about it, space filling curves aren't rectifiable, so the proposed metric would give either 0 or ∞ as the distance between any two points. Unless I'm missing something, of course. We are talking about the distance of the points in ℝ, not the arc-length of the graph in ℝ². I think one actually could make a metric with that by setting: [; \text{dist}(x,y) = \text{max-dist}(f^{-1}(x), f^{-1}(y)) ;] Where we define [; \text{max-dist}(A,B) = \text{sup}\{|a-b| : a\in A, b\in B\} ;] | Jungle river metric and the french railway metric are pretty exotic I guess. |
[
"I think if you extend Elliptical Curve cryptography up to three dimensions, using an Elliptical Bulb, you could create a trap door function with multiple public keys corresponding to a single private key."
] | [
"math"
] | [
"8ro71l"
] | [
0
] | [
""
] | [
true
] | [
false
] | [
0.29
] | null | This is nonsense. Elliptic curve cryptography uses elliptic curves over , not the real numbers. There is no axis to rotate a curve around, and even revolving an elliptic curve over R around an axis does give you an abelian variety. | You realize that an ellipse is an elliptic curve, right? | Okay. OP, right now you're doing the equivalent of going 'hey, stars are made out of gas, so they're farts' because there are two different uses of the word involved. "Field", as used in "finite field", has absolutely nothing to do with the word as used in "vector field". The former describes an object on which particular algebraic operations are defined, the latter describes a function from a manifold to its tangent bundle. | What is a 3-dimensional field? (Yes, you've already said nonsense.) | A field isn’t a literal field, it’s just a term for a 3-tuple (A, +, *) whose operations and elements satisfy certain properties. If you don’t have any background in algebraic geometry I’d recommend you stop trying to come up with novel ideas. |
[
"Algebraic structures using 3 sets?"
] | [
"math"
] | [
"8rnobs"
] | [
6
] | [
""
] | [
true
] | [
false
] | [
1
] | I found an informal taxonomy of algebraic structures online (I cant remember where) and it had them separated by number of underlying sets and then by number of operations. E.g. 1 set 1 operation Magmas, groups, semi groups, etc 2 operations Rings, fields, etc 2 sets Vector spaces, modules, etc It got me thinking about structures on a larger number of sets, as well as combinations of these structures (afaik vector spaces are a field (scalars) and a group (vectors)?). Is there some term for this kind of classification/is there somewhere I can read more? | There are things out there called bimodules which have 3 objects involved. An (R,S)-bimodule is an abelian group A which has an action of R on the left (so it's a left R-module) and an action of S on the right (so it's a right S-module) and such that the actions of R and S commute with each other in the obvious sense: r(as) = (ra)s. Another example could be representations. If G is a group and k is a field, then a "representation of G over k" is a group action of G on a k-vector space V that's compatible with the vector space structure. Equivalently, it is a homomorphism from G into GL(V) = Aut(V). So in some sense, a representation knows about G, V, and k. We can generalize this last example slightly, although this might be a bit of a stretch, but let's say that k is a field, and R is a k-algebra, and M is an R-module. In some sense, R is a k-algebra, so it "remembers" k, and since M knows R, M also "remembers" k, so you could think of (k, R, M) as some sort of triple. (This is the same thing as the last example when we let R = k[G] be the group ring of G over k and M = V.) You might also be interested in universal algebra . | The set theoretic underpinnings of complex structures are usually not that interesting, and no one is going to classify structures by the number of sets in them. I'll mention two things which may be of interest though. A variety in universal algebra is an abstract algebraic structure with an arbitrary number of operations, called its signature. In category theory, there is a criterion to tel whether a structure has extra properties, extra structure, or extra sets, by whether its underlying set functor is faithful, fully faithful, or neither. see stuff structure properties at nlab . | A group action by itself is just a group G acting on a set X. If X has n elements, then you can think of that as just a homomorphism of G into the symmetric group S . For representations, absolutely! Looking at homomorphisms from groups into, say, GL (C) (where C is the complex numbers) can tell you quite a bit about the group. Here are a couple weird properties. Call a representation V if no subspace of V is preserved by the action of G, except for the trivial subspace and V itself. Then, a finite group G is abelian if and only if all of its irreducible representations (over the field C) are 1-dimensional. Additionally, the number of irreducible representations of G is exactly the number of conjugacy classes in G. This is related to character theory . Also, if I tell you all the representations of a finite group, then you can recover the group itself from that data. Sometimes we might have a group that's very hard to understand, so we instead try to understand it's representation theory, and then use that to recover information about the group itself. | Thanks for the detailed response! I think universal algebra is along the lines of what in looking for. On the group action, does that mean that groups can be studied as matrices, and does that reveal any interesting properties about groups? Edit: on that last point wikipedia led me to representation theory, so answer solved I suppose | I think universal algebra is along the lines of what in looking for. What you're looking for is "many-sorted universal algebra" (If you look up just "universal algebra", then it is most likely gonna be just single-sorted). |
[
"Mathematics symbol dictionary?"
] | [
"math"
] | [
"8rnp4n"
] | [
20
] | [
""
] | [
true
] | [
false
] | [
0.83
] | One thing that really bothers me is how sometimes certain symbols that seems to mean something are just variables like x, y and z, but you can't know that unless you're told so. So I was wondering if there was a sort of dictionary that would allow you to look up their meaning and that's divided into different categories like variables, operators, etc. | Lots of symbols are context-dependent; there's not going to be a perfect solution to this question. A decent first approximation would be to use Detexify to find out what some symbol is called (at least in LaTeX) and then googling that. | Well, there's Wikipedia's list of mathematical symbols . | What I wrote above was just a joke. You had said "things like SO(2) that you might not think of as anything special" and, well, SO(2) is indeed literally "special". | The groups SO(n) are indeed special. That's even part of their name. | I have seen someone, in the context of complexity theory, simplify O(n² + mn) into On² + Omn. |
[
"Did I just proved Fermat's Last Theorem? Am I doing something wrong missing something here?"
] | [
"math"
] | [
"8rmavq"
] | [
0
] | [
"Removed - see sidebar"
] | [
true
] | [
false
] | [
0.14
] | null | NullandRandom, I have found the error in one of your calculations, however, this comment box is too small for me to explain it. | Your proof doesn't work for any c, only for the specific one you chose. It is indeed true that if c = x+y then x + y =/= c but in the statement of the theorem c could be any number at all. You have to be able to choose x, y and c independently. | Yes, got that. Really missed on choosing a,b,c independently. Thanks | Let n = 2, xy < 0 You don't have a positive value on line 5 | and are supposed to be positive integers so show can be negative? |
[
"The History of Non-Euclidian Geometry - A Most Terrible Possibility - Extra History - #4"
] | [
"math"
] | [
"8rls69"
] | [
432
] | [
""
] | [
true
] | [
false
] | [
0.96
] | [deleted] | FTR the sum of the measures of the angles in a triangle is than 180° in hyperbolic geometry; what Daniel probably meant (and what the illustration was similar to) is spherical geometry, where the triangle angle sum actually is greater than 180°. Also, Daniel's example of two points on the sphere through which multiple lines pass is less general than he implies; on the sphere, two distinct non-antipodal points determine a line, and this fact is related to the concept of the real projective plane, which can equivalently refer to a space where each line through the origin in Euclidean 3-space is treated like a point, or to the unit sphere in which antipodal pairs of points are identified with each other. | Hey, that shape. I recognize it! | A visual sense of the geometry is a great pedagogical tool. This is an intro-level YouTube channel. I think its an understandable stance given the target audience. | I've probably been a subscriber for 6+ years. It feels weird when he does commentary with a non-shifted voice, that's too deep in comparison. | Spherical geometry is not a relevant counterexample. It does not satisfy all of Euclid's first four postulates , so the fact that it also doesn't satisfy the fifth doesn't prove the fifth independent. In any case, the reason you're asking this question is that your entire worldview of what mathematics has already been deeply shaped by the results you're mentioning and those that came from the 19th-20th century. The model theoretic idea of axiomatic reasoning is relatively recent, especially in its full scope. It wouldn't occur to most mathematicians for most of that time period to try to show that an obviously true fact didn't follow from four other obviously true facts by reinterpreting the definitions of all the words in those facts in some weird new way so that the first four facts were still true but the fifth wasn't. Also, that isn't even what geometric proof . It was about straightedge and compass constructions. The goal was to that parallel line without having to use a new postulate to invoke it. How would you even prove that could come up with such a construction just because you haven't? (Use algebra and proof by contradiction, just like in number theory, I hear you cry!) But that's also a thought that has the benefit of 400 years of Cartesian analytic geometry, which for the first millenia of this problem was not a tool available. In fact, even thinking of "geometries", plural, is an after-the-fact framing. Spheres were shapes that lived space; shapes themselves were not spaces with some type of intrinsic geometry. (How could space itself be curved? Curved into ?) Space was obviously an idealization of the table in front of you or the air around you. It was a basic, obvious concept that didn't need definition, like point or line, much less a concept that could be defined to mean something different. A lot of questions seem obvious in retrospect, but that's because their answers fundamentally changed the way we see the world. |
[
"Any update on ABC error?"
] | [
"math"
] | [
"8rljdz"
] | [
107
] | [
""
] | [
true
] | [
false
] | [
0.92
] | I remembering seeing something maybe 5-6 weeks ago about a serious error that had been discovered in the IUT papers and that an exposition would be shortly available. Has anyone seen an update to this? | Mochizuki and his followers have yet to address the problem. I would not recommend holding your breath. | In general, it is quite dangerous. | There is a paper detailing the error in progress right now, I expect it'll show up sometime this summer. | It's the mathematician's "general" but the physicist's "quite" | I guess you are referring to https://totallydisconnected.wordpress.com/2018/05/09/the-latest-hot-abc-news/ |
[
"Can anyone explain what the pictures on these coasters are?"
] | [
"math"
] | [
"8rl279"
] | [
5
] | [
"Removed - see sidebar"
] | [
true
] | [
false
] | [
0.86
] | null | In mathematics, an Apollonian gasket or Apollonian net is a fractal generated starting from a triple of circles, each tangent to the other two, and successively filling in more circles, each tangent to another three. It is named after Greek mathematician Apollonius of Perga. | They are Apollonian gaskets! | Steve Butler (the source of the coasters) has a page on his site here that explains them | The lower left one is an Apollonian gasket by the way. | What about the ones starting with 4 and 5 circles? Still Apollonian gaskets? |
[
"Predicting Prime Numbers"
] | [
"math"
] | [
"8rl1ls"
] | [
0
] | [
""
] | [
true
] | [
false
] | [
0.17
] | null | The guy is a crank. He seems oddly obsessed with doing what amounts to a sieve in base 24 and then somehow fixates on 2 and 3 being special (of course they are: they are the factors of 24). There is to this. | It has no implications to anything. It's just some crank who's rediscovered some basic facts about modular arithmetic and is acting like it's some huge breakthrough. | Lol. Wrong sub. | Can you explain? I was hoping to get some help understanding what implications this could have on number theory and cryptography. | Thanks :) |
[
"If neural networks are so complicated that it’s impossible to know how they got from A to B, could they be used for encryption?"
] | [
"math"
] | [
"8rkqsh"
] | [
3
] | [
""
] | [
true
] | [
false
] | [
0.57
] | [deleted] | The point of encryption is that the intended recipient decrypt the message, but everyone else cannot. If you just want a message that nobody can retrieve, that's easy: burn it. | I'm no expert on cryptography, but I'm a CS student and pretty informed about neural networks. The answer is no, these are two very different things. What is meant by "neural networks are so complicated that it's impossible to know" is that essentially the how a neural network makes its "decisions" is incomprehensible to humans from the outside, from a semantic point of view. Neural networks are also trained to optimize a certain function, in a certain sense to maximize the probability of getting the right answer (e.g. guessing what is on a picture) - they are thus probabilistic. Cryptographic functions are not probabilistic, but instead deterministic. This is why they exclusively make use of integers and not fractions / floating point numbers. Neural networks are closely related to approximating a manifold, implying that a small (tiny) perturbation in the input will only have a small perturbation in the output. Cryptographic functions are designed to do the exact opposite, to avoid any approximation being possible. From a security standpoint it is impossible to use neural networks for encryption. However, if you want to hash in a non-security related way, for example storing items in a database and having fast lookups, you can actually use the properties of neural networks to find hashes that carry a certain representation, therefore finding better hash structures than deterministic ones. You can read more on this topic here | I would want to be able to read it myself, at least. Then burning wouldn’t work. I’m realizing my question was perhaps stupid. Thanks for answering. | Don't feel bad about asking as long as you are open to answers. | Google has tried something like that. If I understand it correctly they used three network A B and C and trained them such that A and B must be able communicate messages without C understanding them and C should try to understand the messages. Training multiple networks by letting them ‘fight’ is a common technique in AI and is called adversarial networks. You can find the paper here is you want to look into it yourself https://arxiv.org/abs/1610.06918 Edit: I might add that this work is just a proof of concept and I doubt is will ever be safer than current state of the art encryption. I highly recommend looking into modern encryption techniques. There is some very interesting number theory involved. |
[
"A One Parameter Equation That Can Exactly Fit Any Scatter Plot"
] | [
"math"
] | [
"8rjw5m"
] | [
15
] | [
""
] | [
true
] | [
false
] | [
0.8
] | null | You can obviously pack as much data into one "parameter" as you want, if the parameter has infinite resolution. Forget scatter plots, you can put any mathematical statement into one parameter if you want, https://en.wikipedia.org/wiki/Gödel_numbering | I think there's at least a little value in showing you can do it with a fairly innocuous looking formula like f(x) = sin(exp(ax+b)), which is basically the formula in the paper, because a lot of people probably have an intuition that you can't do that kind of 'obvious coding' in a simple analytic expression like that. | With four parameters I can fit an elephant, and with five I can make him wiggle his trunk. John Von Neumann Hold my beer. Steven Piantadosi | The title of the blog post is slightly misleading. It can't actually fit exactly, only arbitrarily approximately well. Also the function in the paper sort of has two parameters. | Aside from the wonderment at the result, the paper also tells us that Occam’s Razor is wrong. Today on "mathematicians forgetting that heuristics live in a non-arbitrary world". |
[
"Can someone explain the intuitive idea behind cos (180+x) b eing negative sin x ?"
] | [
"math"
] | [
"8rjly3"
] | [
0
] | [
"Removed - try /r/learnmath"
] | [
true
] | [
false
] | [
0.5
] | null | cos(180 + x) = cos(180)cos(x) - sin(180)sin(x) = -cos(x) (not -sin(x)) If you look at the way cos(x) is defined on the unit circle, you'll see that adding 180 degrees to the angle causes the value on the x axis to negate e.g. 0.5 -> -0.5 Since the value of cos(x) is the x axis value, adding 180 degrees to x gives you the same value but negated i.e. cos(180 + x) = -cos(x) | Then you're rotating the circle by a multiple of 90 degrees. It's still easy to what happens from the picture. | You can also get from even simpler considerations in a right triangle (although trying to generalize the argument to general angles in the unit circle will just lead to the argument already given to you). No manipulation using other complicated identities (like angle subtraction) is needed... just understanding the meaning of these trigonemetric words, how they depend on an initial choice of angle to refer to, and how they transform when you change your angle of reference. In a right triangle, there's no "pre-fixed" determination of which leg is "opposite" and which is "adjacent". It depends on which of the two non-right angles you chose. If you chose the other angle, the role of opposite and adjacent would flip. Hence whether you got a certain value from a cosine or a sine depends on you initial choice of angle to call "your angle". This would lead to: The sin of your angle is the cosine of the other angle because opp/hyp now becomes adj/hyp. The cosine of your angle is the sine of the other angle because adj/hyp now becomes opp/hyp. (And similar results for tan <-> cot and sec <-> csc) But, because angles in a triangle add up to 180, you have your angle + other angle + right angle = 180 Rearranged: other angle = 90 - your angle Plugging this into the results above, you get sin(your angle) = cos (other angle) = cos(90-your angle) And so on to generate six other identities. | Thanks | It isn't unless x is 90°. Do you mean why cos (180 + x) =-cos (x) ? |
[
"Countable and uncountable infinity doesn't make any sense to me."
] | [
"math"
] | [
"8rknl0"
] | [
9
] | [
""
] | [
true
] | [
false
] | [
0.63
] | null | The resulting number won't be natural because it would have infinitely many digits. | Don't precede your post with spaces like you did, it makes it not readable. In any case, if you did that, you would produce an object with infinitely many digits. That object will not be a natural number because all natural numbers only have finitely many digits. Since the object is not a natural number, there is no problem with it not appearing in the list of natural numbers. | Every natural number is finite in size. that is, they have a certain number of nonzero digits. Every real number is also finite, and thus can only have a finite amount of digits on the left of the decimal point, but can have as many as you like to the right. 0.12121212121212... (repeating forever) is a legitimate real number. I can say things about it, like it's bigger than 0.1, it's smaller than 0.2, it's bigger than 0.121211, it's smaller than 0.121212125. There is a place for it on the number line. 12121212121212... (repeating forever) is not a natural number. Such a thing would have to be bigger than a million, bigger than a billion, bigger than a googol (which only has 100 digits). Bigger than a googolplex (which has a googol digits). There is no number that's bigger than every other number. This is basically infinity (which isn't a number). If you want to specify a natural number, you need have something finite in magnitude. The real numbers have the property that you can convert infinite sequences of digits into finite numbers, because each digit contributes a smaller and smaller amount of "size" to the number. The natural numbers don't, because each digit contributes more and more size and thus any infinite sequence contributes infinite size and doesn't converge to an actual number. | Thanks for explaining why my question doesn't work! It's always nice to know that I can get my questions answered. | Your proof does, however, show that the set of all leftward-infinite strings of digits is uncountable, which is at least something somebody might be interested in in its own right. |
[
"This small youtuber called Flammable Maths has light hearted yet articulate maths related videos. Thought maybe some of you would enjoy this video as well."
] | [
"math"
] | [
"8rjgk5"
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161
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""
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true
] | [
false
] | [
0.96
] | null | Papa Flammy is love ❤️ | We call him Papa Flammy | I could be doing something wrong but there seems to be a much faster way of doing this. ∆ABC is equilateral with side 1 and area √3/4 (along with ABD, but only need one for now). Angle B is 60 so wedge ABC has area π/6. This gives x = 2(π/6 - √3/4) = π/3 - √3/2 y = ∆ABC - x/2 + 2(x/2) = π/6 By imagining another circle at the intersection of B and C (also with radius 1) you can use the same argument to show z = π/6 | Haha, used to be called Fapable Maths. Not sure why he chose that name, but changed it now. | What a |
[
"Make life easier"
] | [
"math"
] | [
"8rihnv"
] | [
211
] | [
"Image Post"
] | [
true
] | [
false
] | [
0.82
] | null | /r/casualmath is a better place for this content. | (1+x) ) | When you integrate and you use 1 step to only rewrite the inverse square root into -1/2 power because otherwise it seems impossible | Sadly that sub is dead as a doorknob. | niceme.me |
[
"Additive number theory?"
] | [
"math"
] | [
"8rin1s"
] | [
14
] | [
""
] | [
true
] | [
false
] | [
0.84
] | Can someone explain me what you do in additive number theory? As you would explain it to an early PhD student in an algebraic field? To be precise: what are the goals/conjectures (which people actually work on, so probably not Colatz) in this field? What's new? what tools do you use? Non-commutative algebra, Category theory? No clue. is it an active field of research? what are the intersections to other fields? | Additive number theory is about the properties of numbers that are the sum of other numbers. A useless description, no doubt. Let me elaborate with some examples. The classical problem of additive number theory for which we have positive solutions is Waring's problem. In short it asks: in how many ways can an integer i be written as the sum of n k-powers. With n = 2 and k = 2 the problem is to ask which integers are the sum of two squares. Formally speaking this description isn't exactly Waring's problem, but the crux of the issue is: What is known about: n = a_1^r + a_2^r + ... + a_k^r ? The Goldbach conjecture asks if every even number greater than two can be written as the sum of two primes. I recommend skimming the book of Nathanson on this topic for a more thorough exposition. To answer your question of what tools we use, and intersections to other fields, the answer (as always in number theory) is that we use a hodgepodge of methods from all over mathematics. | I just want to add that recently some people (including me) have been doing additive number theory using techniques from formal languages and automata theory. For example, Additive Number Theory via Approximation by Regular Languages Waring's Theorem for Binary Powers When is an automatic set an additive basis? Lagrange's Theorem for Binary Squares Sums of Palindromes: an Approach via Automata | Then post your solution already, or shut up about it. At this point, it feels like you're just trolling. You've claimed to have 80% of a solution, and that the other 20% seems "promising". That is quite different from having a proof. Most likely it means that you just pushed the hard part of the problem into that 20% you haven't checked yet, and so you have no progress at all. | Then post your solution already, or shut up about it. At this point, it feels like you're just trolling. You've claimed to have 80% of a solution, and that the other 20% seems "promising". That is quite different from having a proof. Most likely it means that you just pushed the hard part of the problem into that 20% you haven't checked yet, and so you have no progress at all. | No one cares. |
[
"Lambda function of integrating π(x) made by me"
] | [
"math"
] | [
"8ribie"
] | [
6
] | [
"Image Post"
] | [
true
] | [
false
] | [
0.67
] | null | Interesting that you use 7 as an integration variable. | That's an upside-down t | That's an aesthetic t | Yeah, seems like it😂 | "Therefore" |
[
"Math.SE discussion about recreationally trying \"to generate, rather than solve, cool integrals\""
] | [
"math"
] | [
"8rhk0f"
] | [
66
] | [
""
] | [
true
] | [
false
] | [
0.96
] | null | Pah, this isn't the cool stuff, the cool stuff is generating closed form solutions of recurrence relations. That way you can search for nice relationships between the differential equations governing the analouges and the reccurence relations themselves <3 | Man I'm taking Diff Eq over the summer and when we were doing power series solutions and had to solve a recurrence relation to get the nth terms I was interested in learning a bit more about em and it was really cool to see the characteristic equation pop up and lead to very similar looking results. Really makes me look forward to learning more about why and how that happens. | Presumably the readers of /r/math , some of whom might be more knowledgeable about this topic. | Presumably the readers of /r/math , some of whom might be more knowledgeable about this topic. | Anyone knows where could I find a solution with steps to one of the problems in the post: https://cdn.discordapp.com/attachments/372771010224717826/457596215773626368/unknown.png Would be very appreciated, I can't find it despite my best efforts |
[
"Modern Algebra"
] | [
"math"
] | [
"8rhiv5"
] | [
3
] | [
""
] | [
true
] | [
false
] | [
0.64
] | There is an interesting problem on ring and I think it can potentially be a popular puzzle for everyone like sudoku . The problem is giving a complete addition table for the ring, and some multiplication, fill in the remaining slot for multiplication table. Have anyone make a game out of this before? | It goes deep , especially if you don't require the rings to be unital. | The 3p+50 refers to the nonunital rings and those are not isomorphic to matrices. | Non-Mobile link: https://en.wikipedia.org/wiki/Finite_ring#Enumeration /r/HelperBot_ /u/swim1929 | This is very cool: The occurrence of non-commutativity in finite rings was described in (Eldrige 1968) in two theorems: If the order m of a finite ring with 1 has a cube-free factorization, then it is commutative. And if a non-commutative finite ring with 1 has the order of a prime cubed, then the ring is isomorphic to the upper triangular 2 × 2 matrix ring over the Galois field of the prime. The study of rings of order the cube of a prime was further developed in (Raghavendran 1969) and (Gilmer & Mott 1973). Next Flor and Wessenbauer (1975) made improvements on the cube-of-a-prime case. Definitive work on the isomorphism classes came with (Antipkin & Elizarov 1982) proving that for p > 2, the number of classes is 3p + 50. I don't quite understand what's going on though, because it sounds like there are 3p+50 iso. classes of rings all of which are iso to a ring of 2x2 upper triangular matrices over a certain field. | Makes sense, thanks. |
[
"Any good equivalent of Theoretical Minimum Book Series in Physics but for Math?"
] | [
"math"
] | [
"8rgue6"
] | [
8
] | [
""
] | [
true
] | [
false
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0.91
] | I liked the Ian Stewart's Concepts of Modern Mathematics, but I was wondering if there were more books like that, or books that were wider in scope and covering much more mathematical fields, or perhaps books with a little more math, but without exercises. I am also looking for Mathematics books that just explain what some of the symbols you encounter in university mathematics and you don't know what they are and explain when and how to use these symbols, so they don't confuse you when you encounter them in a university course. | God Created the Integers - Stephen Hawking What is Mathematics? - Courant, Robbins Road to Reality - Roger Penrose Mathematical Methods for Physics and Engineering - Riley, Hobson, Bence | Stewart/ Tall Courant/Robbins http://web.evanchen.cc/napkin.html https://www.amazon.com/All-Mathematics-You-Missed-Graduate/dp/0521797071 Felix Klein wrote one but I can't find it https://www.amazon.com/Course-Higher-Mathematics-Adiwes-International/dp/1483169278 | If you put two spaces at the end of your lines, you will get line breaks: God Created the Integers - Stephen Hawking What is Mathematics? - Courant, Robbins Road to Reality - Roger Penrose Mathematical Methods for Physics and Engineering - Riley, Hobson, Bence | Seconding the recommendation of Penrose; it’s a solid overview of a loooot of math and physics. | Did not know that. Thanks. |
[
"What interesting properties does this palindrome have?"
] | [
"math"
] | [
"8rgws0"
] | [
5
] | [
""
] | [
true
] | [
false
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0.69
] | So I was thinking about the idea of a "very palindrome" palindrome, and I came up with the below number and was wondering what kind of properties the community might be able to find. It has several palindromes embedded within it, but that was kinda the point to make a palindromey palindrome. 121232123432123454321234565432123456765432123456787654321234567898765432123456787654321234567654321234565432123454321234321232121 | Up to the 9 (in fact to the following 1), it matches infinite sequence http://oeis.org/A004738 , which what happens if there's no palindromic reflection. Take every even digit and divide by two and we have the sequence 1, 11, 121, 1221, 12321, 123321, etc. all concatenated, which is the sequence of odd and even length palindromes generated from the infinite sequence 1,2,3,4... etc. truncated at various points. Take every digit, subtract 1 from each and then divide by 2 and we have the sequence 1, 0, 11, 0, 121, 0, 1221, 0, 12321, 0, 123321, 0, etc, which is the previous sequence with zeros thrown in between the elements. Taking every (1+2 )th digit extracts a palindrome (of odd digits) from the original (for 1 ≤ n ≤ 7). Ninja edit: It's divisible by 131, which happens to also be a palindrome. Dividing this out leaves a very large composite factor that I currently have something trying to break down further, but it's not a simple one by the looks of it. 5 days later edit that no-one will see: It took a few days to factor this behemoth, but its factorisation is: 131 × 354061877691228084759376874743974094949 × 2613769273236542487564820447363116841746647727097291332614798842825752352158598066892759 | It's not divisible by 3. (The sum of digits is 473 ≡ 2 (mod 3).) | It’s divisible by 3. That’s all I got. | If you just used a large number calculator, you really didn't do any math, so I can only be agnostic about the former disjunct. The latter disjunct seems likely. | That’s fair. |
[
"I want to write a random number by hand. How big does it have to be to ensure that no human or computer has created it be for?"
] | [
"math"
] | [
"8rgmqh"
] | [
0
] | [
""
] | [
true
] | [
false
] | [
0.5
] | null | Right, exactly! Even a careful enumeration like the above only manages to hit four of them. | Globally Unique Identifiers are based on the idea that just randomly generating a large enough number is almost guaranteed to be unique. Those use 128 binary digits, which comes out to about 38 decimal digits. | You'd be surprised how many numbers less than about 30 haven't been considered before. | There are about 10 atoms in the observable universe. I think it's safe to say that most 80 digit numbers have never been written down before by anyone. You could probably get away with something much smaller even. | Find one of those sites that list the first 10,000 digits of pi or something like that. Copy them all down, but at some point in the middle change one digit. I can almost guarantee that nobody has ever written down that exact number before. If you write down the first 10,000 digits of pi, it is almost certain that you are doing this to specifically write down pi. And in that case, you will not make just one error somewhere in the middle. |
[
"I started a collaborative Wiki to share motivation behind theorems and definitions"
] | [
"math"
] | [
"8rgzbc"
] | [
709
] | [
""
] | [
true
] | [
false
] | [
0.98
] | null | I await the day when it becomes standard for the subheadings of every textbook to be . | Inspired by mozartsixnine's post here: https://www.reddit.com/r/math/comments/8raj87/idea_for_authors_make_a_math_textbook_consisting/ A lot of people were commenting that they would be keen to collaborate on something like this, so I have set it up! Please help by adding pages! p.s. I knew nothing about setting up a wiki a couple of hours ago so apologies if it's not very good! If you're keen to be an admin, let me know (although not sure what admins can do that normal members can't) If you want to edit this, click the membership tab at the top and click the button to become a member. Then explore via the categories tab at the top. It's pretty empty so far, and the topics aren't necessarily arranged in the most logical way – I'm only a mere undergrad ;) | Not sure how this will work out, but it sounds interesting! I always thought that all things, especially math, have a secret "magic", a sense of wonder, that sometimes gets lost. Maybe this will help it surface... | Thanks! If it doesn't gain so much publicity this time around I'll try to work on some entries and post again when it's more developed :) | Thanks! If it doesn't gain so much publicity this time around I'll try to work on some entries and post again when it's more developed :) |
[
"Definitive Texts of Necessary Areas for a Modern Mathematician"
] | [
"math"
] | [
"8rg5ri"
] | [
1
] | [
"Removed - see book thread (sidebar)"
] | [
true
] | [
false
] | [
0.57
] | null | I'm sorry I should have clarified, the point of my question is to find interesting areas of modern Mathematics to learn for the sake of learning. So yeah, if you work in geometry or mathematical physics, number theory is probably useless. But regardless of the fact of whether or not it is useful depending on your area, I believe that it is still an essential part to shaping a mathematician's knowledgeability of the modern math. Edit: Also I'd be interested to hear what you think would be definitive texts in mathematical physics or geometry, I would love to learn those more in depth, especially mathematical physics. | Isn’t the zeta function used somewhere in theo physics? I think number theory finds place in string theory too. If you care about locally symmetric spaces, then there is “some” number theory there. But you can of course completely avoid number theory in general. Don’t mathematical phycisists care about Riemann surfaces too? | Imo number theory is something most mathematicians do not spend any time on. Functional analysis (Yosida), Lie theory (Chevalley, Jacobson, Knapp), and geometry/topology (Lee, Poor) are pretty crucial for a lot of people. Knowing category theory (Freyd) and algebraic geometry are likely nice to know. Parenthetical notes are authors I like. | Alright, those sound nice, thanks. | Why would number theory help me as someone interested in geometry and mathematical physics? Honest question. |
[
"193,939 is a Circular Prime: regardless of what way you arrange the digits, the resulting number is prime."
] | [
"math"
] | [
"8rhl34"
] | [
827
] | [
"Removed - see sidebar"
] | [
true
] | [
false
] | [
0.92
] | null | The title is misleading. You can't arrange the digits however you want. For example, 999313 = 7*142759 is not prime. What you can do with this number is cyclically permute the digits, and those will all be prime. So if the number is abcdef, then bcdefa and cdefab and ... and fabcde are all prime. | Fancy! (Fun fact: 2 is also a circular prime) | Seems to fail in base 2. Base 10 is not special. Let me repeat: base 10 is not special. | So is 3 | Whenever I realise that something is specific to base 10 is the moment I completely lose interest. |
[
"Tensor Algebra for Linear Algebra Students"
] | [
"math"
] | [
"8rg242"
] | [
14
] | [
""
] | [
true
] | [
false
] | [
0.89
] | [deleted] | If you want to introduce them for someone with only a little mathematical maturity, just introduce the idea of a multilinear map. Show that the determinant and dot product are examples of this, and ask what the natural domain is for a multilinear map. Once they see this, it should be fairly easy to motivate the defining equations that you quotient by to get the tensor product. EDIT: Since you also cover dual spaces, it would also probably be a good idea to introduce the explicit isomorphism of [;(V \otimes W)^*;] with [;V^* \otimes W^*;] , since that is the connection to the more widely known notion of tensors. | I think tensor products of modules might be a little much but having them work with tensors on R and maybe differential forms on R in not too much detail as examples of alternating tensors would be doable. | I think tensor products of modules might be a little much but having them work with tensors on R and maybe differential forms on R in not too much detail as examples of alternating tensors would be doable. | The best informal, discursive treatment of differential forms and tensors I've seen is still in Misner/Thorne/Wheeler, . No, you don't want to ask your students to read it, but you might look at it for exposition ideas. | Non-Mobile link: https://en.wikipedia.org/wiki/Einstein_notation /r/HelperBot_ /u/swim1929 |
[
"Can we ban cranks?"
] | [
"math"
] | [
"8rg0wk"
] | [
103
] | [
""
] | [
true
] | [
false
] | [
0.84
] | Fairly often the following interaction happens on this sub. User: I've proven [insert famous open problem here] Reddit: post to arxiv or something so we can find out what's wrong with it. User: It's not wrong and I won't post it or someone will steal it. Ad nauseam. A couple times I've actually seen proposed proofs which are wrong, but fairly often it's this shit. I'd like to propose a ban on people who do this. If anyone has a proof for a theorem that they're not willing to post under their own name, it's pretty safe to assume it's bullshit. If this describes you, post something publicly with your name attached, your idea thus won't be stolen and people can check your proof. There really isn't a place on a math subreddit for people who cannot admit they are wrong. For more relevant information on mathematical crankery: | I don't think being a crank should be bannable ipso facto. Once it's clear that they're not willing to engage in critical discussion or learn, then sure whatever... | Cranks are literally the reason I bother browsing this sub. So much better than "I took prooofs and now I <3 math guys!!!", "look, fractals!", or "high school math education is bad!" | I would be okay with a rule saying "if you are proposing a solution to some well-known open problem, you need to link to your solution in the post" or something like that. | Anyone who thinks badmath is an elite math forum clearly doesn't visit it. | This reads like an exceptionally poor freshman interpretation of poststructuralism. The problem you're encountering is that you're attempting to frame imprecision in colloquial language as imprecision in logic. "It is after three" does not necessarily imply "it is not much after three". So yes, when you take something that does not follow, it's contrapositive does not follow either. |
[
"Reflections on Autism, Ethnicity, and Equity | AMS inclusion/exclusion blog"
] | [
"math"
] | [
"8rexyx"
] | [
42
] | [
""
] | [
true
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false
] | [
0.79
] | null | It's a very collaborative endeavour, so interpersonal skills are important. This is often true even at the undergraduate level, and certainly for the graduate level. While the requirement is that solutions are written independently, discussing problems and concepts with colleagues is an important part of the learning process and the strength of your colleagues is one of the things that makes the difference between a lower and higher ranked program. But this isn't the first time you've learned this about mathematics as I've definitely seen you make the same comment about math research in the past. | are you really calling an autistic person, one who's struggled severely throughout their entire academic career due to their disability, a hypocrite about privilege? because he's hispanic? | He actually does. "I was surprised at the conference by how many fellows were in fields directly related to their own race, ethnicity, or gender, and how few in STEM fields. I assume that’s why I was selected, despite my less-than-stellar academic record." In addition to him acknowledging that his ethnicity may have played a role in his selection for the fellowship, he also recounts how his ethnicity has negatively affected him in other ways, most notably through details of the discrimination that he has faced. "The kids at my school would say things like, 'We forget you’re Hispanic because you’re smart.' " "Interactions with police officers devolved into frightening ordeals; once, for instance, during a routine traffic stop, my (white) future wife was taken aside and questioned as to whether I was kidnapping her or transporting knives, guns, or drugs. That sort of thing occurred on campus as well, but in less obvious ways..." | O.O this one is an eye opener, so it seems one of the most important aspects of Mathematical research is interpersonal skills. | Yeah, there are treatment options for people on the autism spectrum disorders. I'm sure there are behavioural therapies and the like which could help you improve social skills or other skills that may be underdeveloped. This isn't just if you have autism, you could look into stuff like this regardless. Think of it this way, if you had adhd, even a mild case, you'd likely want treatment options just so you could get work done. With your social skills it may be less obviousl how it affects your career and day to day life, but it does and it's something you may want to improve on to give yourself the best chance in your career and in life. |
[
"Topology without tears page 24 example 1.1.5 http://www.topologywithouttears.net/topbook.pdf is the infinite union of sets not equivalent to the set N? I don’t understand why it says it doesn’t belong to T"
] | [
"math"
] | [
"8rdjzz"
] | [
1
] | [
"Removed - ask in Simple Questions thread"
] | [
true
] | [
false
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1
] | null | can you put a link not in the title | Link for others. The infinite union given in the example is equal to {2,3,4,...}. In particular, it does not include 1. So no, it is not all of N. | Damn you are totally right how am I gonna be a mathematician if I don’t see something so obvious XDDD | I've watched tenured professors get single-digit arithmetic wrong. You will be fine. | Do you mind taking a look at our sidebar? You're consistently posting questions that belong either in the simple questions thread or the career and education thread. |
[
"Is pure mathematics not interdisciplinary enough?"
] | [
"math"
] | [
"8re8rn"
] | [
0
] | [
""
] | [
true
] | [
false
] | [
0.21
] | Pure mathematics is a closed subject. There are many powerful and beautiful ideas, techniques, and results which we can use from one field to solve problems in another field, but we stop there and don't look for anything outside of the hedge-walled garden where we reside. We have seen many examples of illuminating connections between fields throughout the history of mathematics - today, we have the unifying Laglands program which connects analysis, number theory, and geometry - and this makes pure mathematics an intradisciplinary subject. But when offered ideas from subjects outside mathematics, like physics, as a possible approach to solve pure mathematics problems, we either become skeptical, dismissive, or even ashamed or embarrassed to present these ideas. Perhaps, this is understandable, given how detached pure mathematics can be from reality. How could ideas about concrete objects help solve problems about abstract ones? But I think this is a prejudice we need to overcome, because one story after another, we have examples of ideas adopted from subjects outside of pure mathematics which have assisted in solving important problems. We have examples of pure mathematicians turning to quantum physics for insight on the Riemann Hypothesis: We have a statistician who was ignored by mathematicians for years when in fact his statistical proof of the Gaussian Correlation Inequality was correct: Even the field of knot theory was initially a failed attempt at explaining what matter consists of, and it later flourished as a mathematical subject. Pure mathematics needs to be more interdisciplinary, because there are so many potential ideas we could use from other subjects which could help us solve our problems - they provide a fresh and different perspective. I am not saying is that using the Standard Model of Particle Physics will help prove theorems. What I am talking about are ideas. Maybe if there is even overlap between subjects, we could also use the same tools and techniques from other domains. There is just so much possibility that it shouldn't be wasted on pretentiousness, prejudice, and closed-mindedness. Write your thoughts about this. | 5 months ago , you barely passed Real Analysis. Why are you suddenly making blanket proclamations about how the entire field works? I am pretty skeptical about most of the assertions in your post. For example, Royen's result went unnoticed . Experts weren't ignoring him; nobody knew he had done it in the first place. Besides, Royen is a professor emeritus in statistics, hardly an outsider when it comes to proving a conjecture in statistics. | I think you'd have to discount large swaths of mathematics (analysis of PDE, dynamical systems, geometry/topology off the top of my head) in order to come to the conclusion that mathematics doesn't work with professionals in other disciplines or borrow ideas from them. You're assuming the compartmentalization of the undergrad curriculum carries over to all research level mathematics, which is kind of ridiculous. | Your suggestion that ideas from physics are dismissed by mathematicians sounds like you came out of a time machine from before the 1990s. Algebraic geometers have long since learned to take suggestions from physics to answer their enumerative geometry questions (just google mirror symmetry and algebraic geometry). At the same time, your link to the Bristol page does not reflect what you suggest: the random matrix theory links between physics and zeta-functions have not led to real progress on the Riemann hypothesis at all. Zero. Zilch. Nada. In a sense all that stuff is mostly relevant for a post-RH world, since it's concerned with the vertical spacing of the zeros on the critical line while RH is concerned with the horizontal spacing (real parts of the nontrivial zeros). My main thought is that you should learn a lot more about the state of modern math before making some of your grandiose claims. Edit: I see you couldn't resist posting exactly the same thing on MO at https://mathoverflow.net/questions/302892/is-pure-mathematics-not-interdisciplinary-enough . That was a mistake. | "But when offered ideas from subjects outside mathematics, like physics, as a possible approach to solve pure mathematics problems, we either become skeptical, dismissive, or even ashamed or embarrassed to present these ideas. Perhaps, this is understandable, given how detached pure mathematics can be from reality. " This is honestly absolutely false. I study algebraic geometry in the context of mirror symmetry/other areas of mathematical physics. A lot of the geometric ideas I think about tend to have their origin in physics. No one is embarassed about this fact. In general, I've never seen a negative reaction to employing an outside technique or idea to solve a particular problem. The example of Royen that you give is really a stretch. The problem isn't that he was a statistician. He was no longer employed at a university (so in general no longer publishing), he published a paper in a really questionable journal of which he was an editor, and he posted it to arxiv written in Microsoft Word. These are all huge red flags, especially if the paper is a purported solution of a significant open problem (as there are many many cranks on the internet who post this exact kind of thing). All this leads to the paper as having a lot of signal for crankery, if you read the article one of the people who would've seen his proof didn't because he found 2 other false proofs of the same theorem on arxiv. No one is obligated to look at the proof of every rando on the internet who claims to have proven the Riemann Hypothesis or Collatz Conjecture or whatever. Royen did the right thing and communicated to some people he knew about it, and that's how it came to be accepted. The point is people did eventually find the proof and accept it, I don't think that the fact that it took so long is a failing of mathematics, or has anything to do with any kind of prejudice against statistics, it's that the paper was submitted sufficiently outside normal channels. These kind of things do happen every once in a while, Aubrey de Gray recently published a paper on the chromatic number of the plane. Again people weren't sure what was going on at first because he's mostly known for writing books about anti-aging, but people did read his proof and found it was correct. It's very true that ideas from other fields have contributed a lot to mathematics, and I'm sure they'll continue to do so. However I don't really see mathematicians being resistant to this. | I agree, it was rather blunt but OP is hardly asking for advice though. It's someone with no knowledge pretending to know what they're talking about to seem smart (reverse imposter syndrome where even if most of us did know about something we would still doubt ourselves :/) followed by Write your thoughts about this... I mean who ends with that |
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"How is this even possible , please someone explain"
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] | I think i found something that proves euler's formula wrong We know e = cosx + isinx And e =cosx - isinx Adding both we get (e + e ) /2 = cosx Also A.M >= G.M So (e + e ) /2 >= √(e ) I.e (e + e ) /2 >= 1 So cosx >= 1 But cosx varies from -1 to 1 so cosx = 1 ?.. | AM GM is for non-negative numbers, not complex numbers. This only applies when x = 0, in which case it gives a correct result. | for example, try AM-GM on –1 and –1... | AM-GM applies to non-negative numbers, and e is real and non-negative when x is pure imaginary, so the OP inequality cos x = (e + e ) /2 >= 1 applies when x is pure imaginary. | Thanks i learned alot by this | Thanks i learned alot by this |
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"Extra Multivariable Calculus Subtopics"
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] | I’ve recently just finished teaching myself Multivariable Calculus and I️ wanted to know if anyone had suggestions as to extra subtopics that wouldn’t normally be taught in Multivariable Calculus. The reason I️ ask this is because I️ really find Leibniz’s rule super interesting and not too complex and I️ know it’s not generally taught. Are there any other topics like this? | One extra topic in multivariable calc that is not usually taught is the bordered Hessian, with is Lagrange multiplier version of a second derivative test. I have used it as an extra credit problem on exams. | differential forms | Manifolds are fun. | Agreed, they so interesting and useful to understand. There is even a book which handles multivariable calculus at the same time as differential forms. I find that book to have by far the most easily understood definition of the exterior derivative, even if it does take some effort to prove the algebraic properties. | I'll take a look, thanks! |
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] | This recurring thread will be for general discussion on whatever math-related topics you have been or will be working on over the week/weekend. This can be anything from math-related arts and crafts, what you've been learning in class, books/papers you're reading, to preparing for a conference. All types and levels of mathematics are welcomed! | I’m reading Lee’s introduction to Smooth Manifolds over the summer. It’s very interesting! Geometry has always held a special place in my heart, even back with the basic Euclidean geometry in junior high and high school. I only have gotten through the first few chapters, but I am enjoying the subject a lot. I am also reviewing Munkres and Pintor to get ready for my topology and algebra classes next semester. I have already read both, at least to a certain extent, (perhaps not as closely as I should have, and I’ve only read the non algebraic stuff in Munkres). I’m also going to look over a book on complex analysis to get ready for that class too. Not sure which one yet. | Fighting imposter syndrome while preparing a talk that could get me a great PhD position starting this fall. Other than that, marking some first year linear algebra homework and trying to stay awake while doing it. | I'm fresh out of school because I'm a high school grad, so I'm going through Velleman's How To Prove It to prepare myself for my math classes next year. The university I'm going to only lets people into the CS program if they do very well on the intro to CS and math for CS courses, and for the latter I think I'd do much better if I already know about proof techniques. I also have the opportunity to take Analysis 1 instead of Calc 1, but I've been told it's much harder so I'm getting ready. I am excited about analysis 1 though! The course uses Spivak's Calculus, which I've been told is awesome. | Lee is a common summer thing. Some subreddit's from previous years . If you find any of the resources/discussions there helpful even though they are no longer active. | Thanks! I’ll be sure to look through those. |
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"Are there any pathological languages that don't obey Tarski's undefinability theorem?"
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] | Godel's incompleteness theorem is known to have a couple of "loopholes". For example, you can have a formal system or that are consistient and complete. Likewise, are there any examples of a language that does not satisfy , due to not meeting the conditions in its premise? | I don´t know for sure. Tarskis undefinability theorem states that there´s no predicate T, such that for every formula phi, phi is equivalent to T(Gödel number of phi). For that statement to be even meaningful you need to be able to talk about Gödel numbers of formulas. And I would say that talk of Gödel numbers is only meaningful if the theory can formalize its provability predicate, because otherwise the Gödel numbering is just a compltely arbitrary assignment of numbers to formulas without any properties. This severely restricts the kind of theories we have to consider. I recommend looking at self-verifying theories or paraconsistent theories. Self-verifying theories can formalize their own provability predicate and prove their own consistency, but are barely weak enough to escape Gödel, so maybe they also can formalize a truth predicate. Paraconsistent theories are theories that allow for contradictions, but which have an underlying logic where a contradiction doesn´t immediately make everything trivial. If you have a complete paraconsistent theory that contains Peano Arithmetic, then the provability predicate probably is a kind of truth predicate. | Yes, one example is Hintikka's independence-friendly (IF) logic, which is bi-interpretable with the existential fragment of second-order logic. ESO can easily be seen to contain its own Skolem sentences (truth definitions), contradicting undefinability. | Is your apostrophe key broken? You are using accents instead. | I think it's cool | Independence Friendly Logic |
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