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[
"Speed and percentage"
] | [
"math"
] | [
"zgpudp"
] | [
0
] | [
"Removed - see sidebar"
] | [
true
] | [
false
] | [
0.14
] | null | If I’ve got you right, you’re just arbitrarily calling 50mm/s 100% and scaling to that? So 100mm/s would be 200%, and so on. If that’s what you mean, then 65 mm/s = 100% * 65 / 50 = 130%, and 105% = 50 mm/s * 105 / 100 = 52.5mm/s | I guess it’s the printer that’s being arbitrary 😁But the specific answer you want is 65mm/s is 130% | I wouldn’t call it arbitrary! But for the sake of this, I guess yes? I have a 3D printer moving at 50mms, which to the printer is 100% of its speed. But, I’d like to adjust the speed to equate to 65mms, as what it’s printing, the flow rate, and it’s retraction settings can tolerate having a moving speed of 65mms (tested - I’m sure it can go faster, but 65 is a comfortable setting). The issue is I can only adjust the printer based off of percentages. So, it’s at 100% speed at the moment. Which I know is 50mms. But I’d like to move it to 65mms, and I’m nor sure that percentage in relation. What my dumb brain is telling me is that if 50mms = 100%, then 65mms equals 106.5% | Thank you! I meant to add that, I don’t deny your original response of course, as you answered my question originally. I guess I’d just very much love to know how you got to that result? My guess is 50mm/s x 2 = 100, so 65mm/s x 2 = 130? | So if I’m understanding correcting. Let’s say 40mms = 100%, and I wanted the result of 65mms, then it’s be 162.5%? - ultimately, I’m finding what 100 divided by 40 is, which would be 2.5, then multiplying my desired mm/s (65mms in this case) by that to reach said percentage - 162.5%? Thank you so much |
[
"Tips to get better in constructing objects"
] | [
"math"
] | [
"zgg0ob"
] | [
7
] | [
""
] | [
true
] | [
false
] | [
1
] | null | Often in these scenarios, if the solution doesn't come to you immediately you should work through some simple examples or reason about some properties that the solution should have. Once you've gotten a sense of the general shape a solution might take, that helps give direction to your guesses. | What do you mean? Writing down sqrt(2) is not constructing anything. It is just notation which is possibly not defined depending on what sqrt and what 2 means for you. If you however have the assumption that numbers are lenghts / areas etc. of geometric objects and that a number a is the square root of a number b if a*a = b, or even geometrically if b is the area of an square with side length a, then you can construct sqrt(2) by giving a right triangle with short sides 1 and 1 and prove by pythagoras that the hypothenuse must have lenght sqrt(2). Thus you have constructed sqrt(2) and at the same time proven sqrt(2) to exist. What is valid as a construction and what is valid as an existence proof is dependend on the current mathematical paradigm. Right now the paradigm is that everything which can be "constructed" in set theory using the ZFC axioms / which exists in a universe of ZFC also exists mathematically | Try stupid examples and standard examples first. By stupid examples I mean things like the empty set; standard examples vary by topic, but often there is a classic, almost canonical example of whatever object you are looking at. These examples are very easy to come up with. If they don't work, you should think about why they don't work, and how you might modify them to get them to work. This approach often won't get you all the way to a solution, but it is usually a good starting point. | Example? | Constructing an object is not the same thing as proving its existence 🙃 (e.g.: sqrt(2)) |
[
"A multiplication algorithm that's more efficient than lattice multiplication"
] | [
"math"
] | [
"zh3tui"
] | [
1
] | [
"Removed - incorrect information/too vague/known open question"
] | [
true
] | [
false
] | [
1
] | null | When implementing both of these algorithms in code But we're not talking about code. You claim that your algorithm is "mentally-efficient", so it stands to reason that your algorithm should need to perform better with a human. You claim that there are "no hidden steps", but in reality, there are. To a human, 8 + 9X9 + 9X9 requires four additions; two on the units digits to get 0 with a carry of 1, and two more on the tens digit to get 17. In that sense, the amount of effort required to add 8+81+81 is more than the amount of effort needed to add 1+1+1. The end result is that the total number of additions you have is 0 + 4 + 7 + 4 + 2 = 17 additions, exactly the same as lattice multiplication. | When implementing both of these algorithms in code But we're not talking about code. You claim that your algorithm is "mentally-efficient", so it stands to reason that your algorithm should need to perform better with a human. You claim that there are "no hidden steps", but in reality, there are. To a human, 8 + 9X9 + 9X9 requires four additions; two on the units digits to get 0 with a carry of 1, and two more on the tens digit to get 17. In that sense, the amount of effort required to add 8+81+81 is more than the amount of effort needed to add 1+1+1. The end result is that the total number of additions you have is 0 + 4 + 7 + 4 + 2 = 17 additions, exactly the same as lattice multiplication. | Burden of proof is on you. You do it and explicitly show us the additions. | Burden of proof is on you. You do it and explicitly show us the additions. | Burden of proof is on you. You do it and explicitly show us the additions. |
[
"Is there a generalization for mathematical connections; such as the Levi-Civita connection?"
] | [
"math"
] | [
"zh9nqd"
] | [
5
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""
] | [
true
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false
] | [
0.67
] | null | Could you clarify a bit what you are asking? The Levi-Civita connection is not a of connections; it is an of connections. If you want to see more examples of connections, a big topic in geometric analysis is to find out when an awesome connection exists (Yang-Mills, Hermitian-Einstein, etc.) If you want to see general formalism of connections, besides the standard definition there's also a beautiful interpretation of them via parallel transport due to Grothendieck [EGA4], which is what led to the invention of crystalline cohomology. | I think this is referring to viewing a connection on a vector bundle V → X as descent data to the first-order nilthickening of the diagonal inside X x X. e.g. if X = Spec k[t] is the affine line, then X x X = Spec k[x, y], the diagonal is Spec k[x, y] / (x - y), the first-order nilthickening of the diagonal is Spec k[x, y] / (x-y) . | [EGA4] gang | The most general kind I've heard about is a connection on a (fiber or vector) bundle. The Levi-Civita connection is a special case of this - a linear connection on the tangent bundle. See https://en.wikipedia.org/wiki/Connection_(vector_bundle) ) or https://ncatlab.org/nlab/show/connection+on+a+bundle | I’m not familiar with EGA but the relation between connections and parallel transport is very well known and goes back at least to Ehresmann |
[
"Relationship with professors?"
] | [
"math"
] | [
"zh9ner"
] | [
69
] | [
""
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true
] | [
false
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0.92
] | So i’m looking to do a phd and one of the main aspects of the application is the recommendation letters. But my problem is, I dont have any kind of relationship with any of my professors, I dont know them and most of them dont know me. Although i tried to talk to a few of them, i cant see how i can progress the relationship any further. So my question would be is that do people usually have personal relationships with the professors? And how do they go about doing so?. Thanks. | One of the easiest ways to develop a relationship with a professor of a current class is to make a lot of use of their office hours. If you're struggling in a class, go in to get help. If you're doing well in a class, come in to ask them about related questions that came up while you were learning about the material (if you don't have any, spend some time coming up with questions and trying to answer them yourself, then come to the prof with the resulting questions; or you can always ask about recommendations about where to learn more about topics which interest you). This helps you learn a lot more and it develops a relationship over time. Another possibility is to approach a professor and ask them about what they do. But don't just literally do that; research what they do ahead of time and spend some time looking those topics up on Wikipedia or something similar, so that you can actually try to have a meaningful conversation and learn something. But of course, any real relationship takes time to develop. If you're trying to develop relationships for recommendations needed for applications due soon, this is hard to do. You may not have much of a choice other than to approach professors whose classes you did your best in and ask them. If they say yes, they will probably conduct a little interview to get to know you a little better and then do what they can. Ultimately many people don't know their professors very well, so it's not unusual to be in this position. | Thankfully I still have some time to work on that, and will definitely take your advice and use office hours. Another thing is, where i’m studying (UK), no REU’s are available, and generally research opportunities are scarce. No such thing exists at my uni for instance. So how would you suggest i go about getting research experience in hopes to publish something to boost my application? | Office Hours. Go talk to them. I actually had a professor once that made everyone book an office hour day, where you just sat and talked about your final project, specifically because he wanted people to get comfortable doing more than just showing up to class and doing homework. Even if you have an A, the best way is to talk to them outside of class. I had a math teacher once that I'd talk to in line at the cafe almost every day. He was great. Look for events they're going to or talks the university is putting on. Even if you don't understand the subject. I once went to a math lecture from an outside speaker and I didn't even understand the Lecture name lol | You should edit your main post to say you are in the UK because in my experience the dynamic is much different there compared to the US where it is easier to develop relationships with professors as an undergraduate. | > Although i tried to talk to a few of them, i cant see how i can progress the relationship any further. Just want to make a brief point: if you talk to them and it's not vibing, drop it. Find another professor to talk to. Also, the relationships don't have to be particularly deep. If you're doing well in their class, just going to their office hours even as little as once or twice can be enough to get a letter of recommendation. |
[
"What is your favorite number?"
] | [
"math"
] | [
"zgqknd"
] | [
50
] | [
""
] | [
true
] | [
false
] | [
0.78
] | null | I'm a sucker for Champernowne's constant, 0.123456789101112131415..., formed by writing down 0. and then every natural number in order. It's irrational, transcendental, and normal - every possible finite string of digits of length L occurs with equal proportion 10 Y'know how people say that "the entire works of Shakespeare is encoded in the digits of π"? Well we don't actually know that (it's widely believed to be true but nobody has proven it). But this can definitely be said about Champernowne's constant. Its the first such number proven to have these properties. As for why this perhaps isn't as well known as π, e, the golden ratio, etc.? I suppose it's because this number was constructed to be normal, whereas π, e and the golden ratio are "universal" constants that arise from naturally occurring phenomena. Maybe I'm biased because normal numbers are part of my research interests at the moment. | 8 because I randomly decided upon that as a child and I’m sticking to it. | 24=3*2 for intrinsic properties https://youtu.be/vzjbRhYjELo ; because a lot of mathematical objects related to it have special properties. 379 for coincidences because it is a prime number and it has the property that 3*37*379=42069 which depends on cultural constants, the fact that we use base 10 etc. | At least it’s not 1729… such a boring number | To me it is 7. Nothing to do with math here. The number just simply comes up so frequently in my life that I start to like it. |
[
"Fun ways to realize symmetry groups as permutation groups"
] | [
"math"
] | [
"zh2g8h"
] | [
15
] | [
""
] | [
true
] | [
false
] | [
0.89
] | Hi great people of . Today I was wondering if anyone had any fun examples of how one can realize a symmetry group as a permutation group. For example the symmetry group of a cube can be seen to be S4 by how it permutes the diagonals of the cube. Another fun way to visualize this same phenomenon is that if you take an octahedron and color opposing faces the same color, then every permutation of the four colors appears around one of the verticies. Another example is to take a dodecahedron and split the verticies into 5 tetrahedrons. Then the symmetry group can be seen to be A5, by how it permutes the tetrahedrons. Analogously, if you color the faces of an icosahedron appropriately, then every even permutation of five colors appears around one of the vertecies. Can anyone else think of any other fun examples? | when turning two adjacent sides of a rubik's cube, the group of permutations of the corners is one of the transitive S5 subgroups of S6. specifically, it's the group <(1234), (3456)> which is isomorphic to S5. | You can always represent a (finitely generated) group as a geometric object by looking at its cayley graph (with respect to a generating set). Then left-multiplication defines an action of the group onto itself which is bijective (i.e. a permutation of its elements) and can visually be interpreted as a graph automorphism or graph isometry of it's cayley graph. https://en.m.wikipedia.org/wiki/Cayley_graph | Indeed you can, but I don't think this is very "fun", mostly because it tends to generate "too large" examples. For example A4 naturally permutes the four vertecies of a tetrahedron. I don't want to think about this as a subgroup of S12, and if I want the action on the Cayley graph to rigid transformations, then I don't think the graph fits into 3-space (does it even fit into 4-space?). | Putting the Rubik's cube aside, here are a few extra thoughts. This "weird" copy of S5 in S6 induces an action of S6 on its cosets. There are 6 cosets, so it's an action of S6 on six things. But it's the usual action of S6 on six things! So this gives another answer to your request for fun examples: there is a fun [nontrivial] way to realize S6 as a permutation group of six things! (In other words, S6 has a nontrivial outer automorphism.) A "weird" copy of S5 in S6 can be realized by producing any transitive faithful action of S5 on six things. One quick way is to have S5 act by conjugation on its order-5 subgroups, of which there are six. Do these observations piece together into a neat realization of the "fun" action of S6 on 6 things? This isomorphism doesn't correspond to any map X -> Y. Here I was hoping to put these pieces together into some nice visual description of the relationship between X and Y, ideally one that reflects the symmetry between X and Y in the end result, but after banging my head against it I couldn't come up with something nice myself. Instead I searched around and got this reference, which gives a pretty satisfying answer for the "weird" action of S6 on 6 things, but doesn't make it super clear how to get there from the above: https://cameroncounts.wordpress.com/2010/05/11/the-symmetric-group-3/ . | The obvious undergrad-level answer is Galois groups of polynomials/field extensions. An interesting higher-level answer is the monodromy of enumerative problems in algebraic geometry (which is really just an example of monodromy in general which could also answer this question). It is a classical result that a smooth cubic surface in P (so a smooth surface defined by a degree 3 equation in 3-dimensional complex projective space) has exactly 27 lines on it, and more generally a general hypersurface in P of degree 2n-3 will have a finite computable number of lines on it. I'll call this number L(n). As you perturb your hypersurface (i.e. just change the coefficients of the defining polynomial equation) these finitely many lines come along for the ride. If you start off from a hypersurface X, perturb it around, and then return to your original hypersurface X, your collection of lines will get mapped back to itself, but not necessarily via the identity - it's possible that by jumbling around your collection of lines by perturbing the hypersurface that when you return back to X the lines will have been nontrivially permuted. One might then call the group of all permutations that you can get by perturbing X along different paths and returning back to X the monodromy group associated to the enumerative problem of lines on hypersurfaces - this monodromy group will be a subgroup of the symmetric group on L(n) elements. For cubic surfaces in P it turns out you don't get the full symmetric group on 27 elements, but rather a certain orthogonal subgroup. For n >= 4, you do get the full symmetric group on L(n) elements for degree 2n-3 hypersurfaces in P - this is part of Harris' 1979 seminal paper on the subject. For instance, in the next case of lines on a quintic threefold in P you'll get the full symmetric group on L(4) = 2875 elements. One can apply this idea more generally to many enumerative question in algebraic geometry - questions of the form "how many subobjects of a certain type of a given space X exist" if that number is finite, and ask what the corresponding monodromy group is. |
[
"I want to become a mathematician, what are some serious steps I should take to do so?"
] | [
"math"
] | [
"uzb1ql"
] | [
2
] | [
""
] | [
true
] | [
false
] | [
1
] | null | "becoming a mathematician" is a hard target to help. Are there any particular fields of maths your interested in? Or anything in particular driving you to want to become a mathematician? There's too many different directions on can go in their study of mathematics to give a one size fits all answer, and understanding why you're approaching mathematics will give us a better idea how to direct you. | Find a copy of "Calculus" by Spivak. I believe there is a free version of the third edition somewhere. Although it's referred to as "Calculus", the book is a perfect introduction. Try and work on some exercises on your own, and post any questions on the Math Stack exchange. | I posted this is in response to a similar question a few days ago. To really build up to high level maths, I'd suggest taking some courses in Analysis and Algebra, then getting into category theory. I'm much more an algebraist at than an analyst so my suggestions are heavily skewed that way. Presuming you have a high school education and have encountered calculus and linear algebra before I have a few books to suggest. For introduction to abstract algebra, the study of groups, rings and fields, I suggest Gallians "Contemporary Abstract Algebra". It's a little light on the formality in some places, but it's a remarkably straightforward read and great for beginners. I used the seventh edition, but I've flipped through every edition from 5 to 9 and they all cover basically the same material. I've also heard amazing things about "Visual Group Theory" by Nathan Carter, but I've only looked at a few pages. Follow up with Dummit and Foote's "Abstract Algebra" to fill in the holes specifically on group extensions, and group actions. D&F reads like a brick though, and is basically a very bland and dry read that's not particularly great for anything but a reference book. Having flipped through a half dozen different algebra textbooks at this point I like Gallian the most as an introductory textbook, and having helped a few dozen students through intro algebra I've yet to have someone not like Gallians book more than whatever textbook their course used. Most people then suggest branching into Aluffi "Chapter 0" to then get into category theory. I'm not a fan of that, it retreads a lot of algebraic grounds under category theory and didn't feel particularly beginner friendly to me. It kind of felt like it assumed you were already comfortable with Algebra (as in senior undergrad or grad student who's been working with algebra for a few years, not a second year undergrad who just passed intro algebra). I'm, personally, a huge fan of "Categories for the working mathematician" but it felt like it too was not targeted at beginners. Great resource though. Emily Riehl's "Category Theory in Context" ( https://math.jhu.edu/~eriehl/context.pdf ) is fantastic, at least the parts I've read, but it's definitely not beginner friendly. It immediately dives into pretty powerful algebra, and assumes you already know what's going on. If you can hack your way through any of those from start to finish you should definitely be comfortable with category theory. This, https://arxiv.org/pdf/1612.09375.pdf , seems like a pretty gentle introduction to category theory. I've only looked at the first chapter, but it doesn't seem to expect you have spent years playing with Algebra nor does it seem to assume you've taken a course on formal logic and/or theoretical foundations of modern maths. It seems to brush over a bunch of the high level algebra, as a note for people who've run into it, but it doesn't seem to expect you to necessarily follow along. | well I'm most interested in pure mathematics, I like applied math as well but I'm most drawn to pure math. | Oh awesome! Thank you. |
[
"Conjectures and Currently existing Axioms"
] | [
"math"
] | [
"uz8lrw"
] | [
0
] | [
"Removed - ask in Quick Questions thread"
] | [
true
] | [
false
] | [
0.5
] | null | All over math do you think there is/are any conjecture(s) which cannot be proved or disproved with the help of currently existing set of axioms in mathematics? Yes, certainly. There have been many examples of this in the past (most famously the Continuum Hypothesis), and there will be many more in the future. Also how close you think we are near perfection(sorry if it sounds too abstract) in the field of math? Infinitely far away, since there is no such thing as a 'perfect' axiomatic system. The notion is not even coherent. | The number of true but unproven statements is infinite. There is no hope of ever discovering everything. | You should look into Godel's incompleteness theorems. | Unfortunately, your submission has been removed for the following reason(s): /r/learnmath books free online resources Here If you have any questions, please feel free to message the mods . Thank you! | Woah the second answer makes me feel like I shouldn't have asked it. But why do you think we are nowhere near perfection? Even though I agree that there might be a lot of things left to even state let alone prove, aren't the number of proved statements in math more than the number of unproved ones as of now? |
[
"Why is *all* of the ML research in the CS department?"
] | [
"math"
] | [
"zgpcjx"
] | [
310
] | [
""
] | [
true
] | [
false
] | [
0.92
] | I'm looking into doing a PhD and I was hoping to do a PhD in Applied/Computational Math with a focus in Machine Learning. But it seems that all of the ML research is done in the Computer Science department at the universities that do that sort of thing. You'd think that there would be professors in the math department specializing in optimization and/or statistics that would be studying ML, but they seem few and far between. Any ideas as to why that is? | To be fair, it’s not ALL in the CS department. There’s a good amount of it in electrical engineering departments, and occasionally mechanical engineering. But I suspect it’s mostly a historical thing tbh, it originated in computer science | There is research in ML at (applied) math departements. It usually focusses on investigating when and how it works, i.e., actually proving stuff instead of the usual "we throw ML at everything and hope for the best". E.g. Prof Dahmen, a renowned numerical analyst started working on ML recently - he became famous for doing research on FEM. At the numerics/optimization departement at our uni are also 1-2 people working on this. The main problem is really that in contrast to the widey-spread opinion that ML is the cure for everything, it is really rather a "niche" application. ML/DL are black-box optimization/fitting algorithms which can (!) work well in too complicated systems - say modelling of human brain or learning to translate natural languages. For applications where "classical" algorithms exist, they will often outperform ML/DL and offer guarantees, thus are preferable. | Most machine learning research is quite empirical and you don't really need to be very mathematically rigorous because you are not proving theorems. Being a wizard at pure math is not going to help you that much in doing machine learning research compared to having a strong knowledge of machine learning methods and techniques. The pioneers of the field are mostly computer scientists and the field originated in computer science. | It may be a history-driven thing but I think another essential aspect is that industry/government funding is easier to acquire for CS, so the more high-profile staff/attractive positions will be there instead of in (applied) math. | I did my ML research in the music department, but there are far more ML researchers in the CS than any other departments in general. |
[
"Just finished taking linear algebra"
] | [
"math"
] | [
"zh1y0g"
] | [
446
] | [
""
] | [
true
] | [
false
] | [
0.94
] | I took this course as an elective… my god, what a fun class. I heard about linear algebra’s reputation as one of the most difficult courses you could take / the equivalent of suffering, but I didn’t really see it that way? It was really fun to study, probably the most fun class I’ve had in all of college, save for maybe my French classes. Playing with matrices is like doing small puzzles and seeing how everything we learned stacks up really is amazing… I easily enjoyed linear algebra way more than upper level calculus classes because it just seems to make more logical sense. I could go on for hours about eigenvectors, bases, transposes and determinants but shut down when you ask me about L’Hopital’s rule. Wish I could take it again tbh | when taught correctly (which is rare), linear algebra should be one of the easiest university level math classes that there is. | Wish I could take it again tbh If you took an intro level computational linear algebra class, try taking an upper division theoretical linear algebra class. Then try numerical linear algebra. | Try taking an abstract algebra course! | when taught correctly You might say when seriously buy the book it's great | Came here to say this. OP is an algebraist. OP must avoid taking Differential Equations. OP eats corn on the cob side-to-side like a typewriter, not in circles like a phonograph. |
[
"is aleph null to the power of aleph null the biggest infinity ?"
] | [
"math"
] | [
"uz2crh"
] | [
0
] | [
"Removed - ask in Quick Questions thread"
] | [
true
] | [
false
] | [
0.25
] | null | No, because 2 to the power of is even bigger! | you probably mean "2 to the power of that", k^2 = k for any infinite cardinal right? | Aleph null to the aleph null is just 2 to the aleph null, the cardinally of the real numbers. | With ordinal arithmetic you can just keep counting after infinity until you get what is essentially infinite infinities. Then you can keep counting infinite infities +1 etc. | Ah, yes, that is what I meant. Thanks for spotting that. |
[
"Hello, I have a question about maths."
] | [
"math"
] | [
"uyy99c"
] | [
0
] | [
"Removed - ask in Quick Questions thread"
] | [
true
] | [
false
] | [
0.33
] | null | I don't think you can do it on a scientific calculator, at least not natively. I suspect you'd have to download an app, though I don't have experience with any app that has this functionality. After some brief research, I've come across this webpage which seems to offer an app with this functionality. However, if you're looking for a free resource to simplify algebraic fractions, why not try the Wolfram Alpha website ? I believe they have a lot of features for symbolic mathematics, including simplifying fractions. Though, you'll have to register and pay to see their full working. | Wolfram alphas good. Mathematica is even better if OP has access | I see. Then I guess you should try installing the app I linked to. It requires a TI-83 or later though. | I see. Then I guess you should try installing the app I linked to. It requires a TI-83 or later though. | I have a TI-84 but I am not able to do it. Nor do I know if there are any calculators out there that are able to. Correct me if I’m wrong. What do you mean specifically by algebraic fractions? In terms of simplifying or fractions that deal with certain mathematical operations? |
[
"Is there any algorithm that can increase a starting number towards infinity by applying double-or-nothing on well chosen portions of your number?"
] | [
"math"
] | [
"uyw932"
] | [
1
] | [
"Removed - ask in Quick Questions thread"
] | [
true
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false
] | [
0.6
] | null | My intuition says no, unless maybe you can change the portion depending on the size of the number (not bet the same each time), but even then I'm not sure. You could try finding the expected value of the log of the number after N iterations. Say you fix the portion at 10% (but I think this works with any fixed portion). After 1 itération, there is a 50% chance you multiply by 1.1, and 50% chance of 0.9. The expected value of the log of what you multiply by is thus -0.01 (approx.), but importantly is less than 0. The expected value of the log of what you multiply by doing N times then, is N*(-0.01). Importantly, multiplying every time means the logs will add. No matter how big your starting number, the law of large numbers (yes this is real) says the process will tend toward the expected value, i.e. the log goes to - infinity, so the number tends to 0 (but this part is probably crucial and you'd have to work out exactly how fast the LOLN says it will converge to the EV; I can't remember off the top of my head). Nevertheless, for a fixed portion I think it's impossible to make it grow long-term over a long period of time, and in fact goes to 0 as you said. If you wanted it to grow you'd have to probably switch "double or nothing" for a bet that makes the log EV positive -- I think 47% chance of nothing, 53% chance of doubling would work. Or changing "doubling" to a higher factor (i think 2.12 or greater works). This is where I think it MIGHT be possible to make the log EV positive if you change the portion each time in some way but even then I'm not sure (complicated betting stratégies like this usually lead to losing all the capital but here since it's a percent who knows) | You might be looking for something like this: https://en.wikipedia.org/wiki/Kelly_criterion | Because X > (X - 10 %) + 10% = (X + 10%) - 10% = X *1.1 * 0.9 = X * 0.99 It should be easy to see that the sequence W-L-W-L... or L-W-L-W... leads to a smaller and smaller number. Or in general: The amount after L losses and W wins is: X * 1.1 * 0.9 I think it evens out if loss has a chance of 49.5% and win one of 50.5%. | This is the correct point (the fact that the bets are only apparently symmetrical due to the fact that 1.1 * 0.9 = 0.99 != 1). Changing that 10% is not going to help: (1+x)(1-x) < 1 for (0<x<=1) 49.5% and 50.5% I think are not enough given that 0.9 1.1 = 0.9959862 You can find more suitable values by solving (9/10) (11/10) = 1. This gives x = log(100/99)/log(121/81) = 0.025041862 = 2.5041862 %, so to equilibry things the chances of loss should be about 47.4958138% and those for win about 52.5041862%, but in case you use computer simulation takes values a little more extreme, like 47.495% and 52.505%, to avoid potential problems due to finite representation of floating point numbers. | Also, I see an apparent paradox here: obviously no casino would ever give its players a 50.1% chance to win a double or nothing, but from these simulations it shows that players would eventually lose all their money if they kept betting 10% of their money, so what is the answer to this apparent paradox where it seems both the player and the casino cannot win? You are seeing the player lose all of their money in outcomes. The player's amount of wealth is growing with each bet. It's just that that expected value is concentrated in unlikely outcomes of very high value. |
[
"Comparing infinites, are the more complex numbers than real numbers"
] | [
"math"
] | [
"uyplq0"
] | [
1
] | [
"Removed - ask in Quick Questions thread"
] | [
true
] | [
false
] | [
0.6
] | null | There is a one-to-one correspondence ℝ²→ℂ given by (x,y)↦x+iy. The stack exchange post explains the bijection between ℝand ℝ². | They are the same size Actually, R for all natural numbers n(not including 0) is the same cardinality as R | Is there a proof? | https://math.stackexchange.com/questions/243590/bijection-from-mathbb-r-to-mathbb-rn | Help me a little here, where does this show that you can 1 to 1 map all complex numbers to real numbers? |
[
"Does mathematics demonstrate the highest form of human intelligence?"
] | [
"math"
] | [
"uypbd8"
] | [
0
] | [
"Removed - incorrect information/too vague/known open question"
] | [
true
] | [
false
] | [
0.21
] | null | Wtf no | What are the different forms of human intelligence? | If you want the highest form of intelligence, find someone who can consistently fold a fitted sheet nicely. That shit is way harder than algebraic geometry can be. | Even the statement that math is the most important thing for the advancement of society historically is very, very questionable. | If you are trying to say that math is the most important thing for the advancement of society historically, then the answer is yes. However that's probably not what you are asking. I would say that the peak of any skilled craft is the peak of human intelligence. I am rather good at Computer Science and ok at math, but someone who plays the trumpet at a world class level has achieved something just as remarkable, if not more, than I have. Any field that is not solved, is just as hard as math, importance is irrelevant. |
[
"Scared about attending the Ross Math Program this summer"
] | [
"math"
] | [
"uzb50k"
] | [
15
] | [
""
] | [
true
] | [
false
] | [
0.9
] | Hey - I was accepted to the Ross Mathematics Program for the summer about a month ago. However, as the date draws closer I'm becoming increasingly apprehensive about it: I found some past problem sets and I quite literally didn't understand how to begin the questions. I'm afraid that I'll be less intelligent and capable than the other attendees, and it'll become obvious. PLUS IM NOT GOOD AT CHESS WHICH I HEARD IS LIKE A BIG THING THAT THEY DO THERE IN THEIR FREE TIME..... If anyone has attended in the past, could you tell me about your experience with the program? | I can't speak to what things are like there, but forget chess or music whatever other stuff they say people do there. It's irrelevant. You don't know what the questions on the problem sets mean because you don't know that math yet. You'll learn about it when you get there. You surely are not expected to already know what they'll be teaching you and what you'll be discussing with other students. So relax. If you think you're supposed to understand how to solve a math problem on math you've never studied before, then you haven't been studying particularly challenging math yet. Things will change this summer: it should be really interesting for you, a first glimpse of what math beyond school courses is like. Googling for 10 seconds brought me to the page https://rossprogram.org/students/course-topics/ , so if you genuinely want to do some preparation ahead of time then just look up material related to the first two topics. Watch out: each of those first two topics contains within it a of material, so don't expect to master it in one hour or even one week. Modular arithmetic is a deal, so if you only have time to get used to one thing before showing up there, let it be that. | The program is designed so that anyone with any background can approach the problem sets. My experience there was that other participants were very friendly and competitiveness was not really an issue. Also from personal experience chess and other games were barely played. I'm not sure where you heard this from but doing non-pset related activities like playing games is actively discouraged. | Thanks for this response | Ah alright, that's a relief. Do you mind if I dm you with a couple questions? | sure |
[
"Are there ways to create new trig systems?"
] | [
"math"
] | [
"uz2kwy"
] | [
7
] | [
""
] | [
true
] | [
false
] | [
0.69
] | I'm finishing up Calc BC and one of the topics we recently revisited was trig substitution to solve integrals. This method uses the identites involving the squares of trig functions to rewrite a quadratic under a square root as a trig function. This got me wondering if one could create a "new" trig system involving cubes of the functions to evaluate cubic functions under a cube root. If we have the identities abs(sinc (x) + cosc (x)) = 1, and the derivative of sinc(x) is cosx(x), and the derivative of cosc(x) is -sinc(x), then these equations should be solvable to create "cubic" trig functions. These indentities could then be used to solve for the antiderivative of cbrt(1 - x ) through a "cubic" trig sub of x = sinc(x). Is this an already existing concept or something that even works? Could it be applied to further powers? Creating a qasin and qacos for 4th powers, qisin and qicos for 5th powers etc. I'm curious as to what applications could come from these functions. | The trig functions sin(t) and cos(t) = sin'(t) parametrize the curve x + y = 1. There is a nonnegative integer associated to curves defined by a polynomial equation, called its genus, and the reason you can successfully integrate certain expressions involving square roots is because of the trick of completing the square and the fact that the curve x + y = 1 of degree 2 has "genus 0". This means that curve can be parametrized by rational functions -- look up the Weierstrass or tan(t/2) substitution -- and what's happening with trig substitution is that you're sort of integrating rational functions in disguise. In higher degree this whole chain of ideas breaks down. First of all, there is no such general trick as "completing the cube" (cubics have too many parts) and for n > 2 the curve x + y = 1 has positive genus, which means it can be parametrized by rational functions. To use your terminology, there is no "easy" antiderivative of (1-x ) . That is a theorem, not a reflection of some lack of creativity on mathematician's part. There are analytic functions describing the points on genus 1 curves, and these functions are called functions (the name "elliptic" is used for historical reasons, and these functions don't parametrize an ellipse). They are a sophisticated generalization of trigonometric functions. But it is not anything that anyone taking calculus is going to find user friendly at all. | These functions don't exist: your condition on the derivatives means you want sinc and cosc to satisfy y'' = -y. All solutions to this differential equation can be written y = A*sin(x) + B*cos(x), so your sinc and cosc are just normal sin and cos. | Does easy mean writable in terms of elementary functions It means the integral is not an elementary function. you're only ever going to get an approximation to apart from at certain known values If all you care about with the sine function is making estimates then I don't think I can convince you that exact formulas for it have any value. But outside of numerically computing functions, matter. You can't differentiate an approximation, for instance. The sine function is a solution to the differential equation y'' + y = 0, which is not just some approximation. From this differential equation, together with the initial conditions y(0) = 0 and y'(0) = 1 (those are not just approximations), you can completely characterize the sine function and derive all of its properties. Or you can go the power series route; take your pick. | Does easy mean writable in terms of elementary functions? Like there is surely a power series representation of that integral gotten by integrating the binomial expansion. What I've wondered about is that surely sin you're only ever going to get an approximation to apart from at certain known values. So why is sin considered better than say a power series approximation? | The concept of an elementary function is mainly of historical and pedagogical interest: they're basically the kinds of functions you can describe without using calculus: powers, roots, trig functions, exponentials, logarithms, and things you can build from those with arithmetic operations and composition. I should have included algebraic functions in the building blocks too, and they are more sophisticated than just powers and roots, such as a function f(x) that satisfies f(x) - (x +1)f(x) + 7x = 0. (Root functions are solutions of equations like f(x) - x = 0.) Are elementary functions an arbitrary choice? Well, all concepts in math could be called arbitrary. We give names t things we are interested in. There is one interesting aspect of the notion of an elementary function: it gives a precise sense in which certain functions (like the error function in probability theory) have "tough" antiderivatives. That is, it is hopeless to search for an antiderivative of some functions, like exp(-x ) or (sin x)/x, among the pre-calculus functions because all pre-calculus functions are elementary and those tough antiderivatives are provably not elementary. I don't think there is any reason in modern math that the concept of an elementary function is still important. For similar reasons, whether or not some given polynomial is solvable by radicals is no longer important even though it is a good historical and pedagogical example of the power of Galois theory to answer a problem that had been unsolved for around 300 years (since the discovery of the cubic and quartic formulas) |
[
"Any good or equal alternatives to Microsoft Mathematics?"
] | [
"math"
] | [
"uyvbct"
] | [
0
] | [
""
] | [
true
] | [
false
] | [
0.36
] | Hello I have recently changed my laptop from Windows 10 to macOS Moneterey. While I like macOS, I really need this very sophisticated Math calculator. I don't understand why this program is not very popular even if it conveniently almost answers any mathematical problems plus its graphing capabilities. I find it impossible to use Wine or Crossover to run Microsoft Mathematics on the newest macOS Monterey using an M1 Macbook. And I find it very difficult to search for an alternative either an app compatible on Mac or online via web. It's so frustrating that calculators as convenient as Microsoft Mathematics is so difficult to find. Even the web version Microsoft Math Solver, it's not as good/sophisticated as the standalone app Microsoft Mathematics. Do you know any alternative to this wonderful Math calculator? Where it is almost similar to the interface and features of Microsoft Mathematics? Apps like this that I can run on Mac or use online via web? Cheers! | I don't understand why this program is not very popular even if it conveniently almost answers any mathematical problems You must have a very specific idea of what mathematical problems can be, because there are tons of mathematical problems that can't be answered by a calculator. For all that you wrote in your post, you never gave even a single example of the scope of problems you need to answer that can't be run on other calculators. | Come on. You obviously know what calculators do. Do you really need to chastise this person for not spelling it out? In any case, Wolfram’s Alpha is state of the art. | Wolfram Alpha? | https://en.wikipedia.org/wiki/List_of_computer_algebra_systems | Wolfram alpha, or geogebra, or mathematica if you want something extra strong |
[
"Soon: A five page proof of the 4 Color Theorem (Jackson & Richmond)"
] | [
"math"
] | [
"uz7iu3"
] | [
625
] | [
""
] | [
true
] | [
false
] | [
0.98
] | null | Let the font size ε go to 0... | I won’t be impressed until they generalize to an +1 page proof of the color theorem. | Math speedruns. | Ok, this is pretty big. Never have I ever been excited by a tweet than now. While doing my Phd, I've always heard from some of my older peers about the controversies surrounding the classic Appel and Haken computer based proof (there were more controversies other than the use of computers for the proof, please don't ask me for details). Also, some more recent problems, like the Hill conjecture in crossing numbers, are also stuck in the computational hell, that is, even using computers it still took years to advance. Hopefully this proof will shed some light that may help with these problems. | Meanwhile let the size of one piece of paper go to infinity. |
[
"Any recommendations on books about Clifford Algebra?"
] | [
"math"
] | [
"uz05f8"
] | [
19
] | [
""
] | [
true
] | [
false
] | [
0.95
] | I'm completely new to the subject, but it seems a bit less than intuitive. There any books that explain it from a basic level? | What is your background and what is the goal? If you want to learn about geometric algebra (a certain real Clifford algebra), then I recommend Alan Macdonald's books for a simple introduction that tie in linear algebra and some multivariable calc. They are and . If you just want a short simple introduction to Clifford algebras then by Vaz and da Rocha is good. | Can you explain what you mean by a basic level? | The only reason I’ve seen people care about Clifford algebras is to talk about spin representations. So anything that introduces the spin group should talk about Clifford algebras. | I enjoyed Clifford Algebras and Lie Theory by Meinrenken | If he's teaching MAT367 (differential geometry of manifolds), you should definitely take it; he did a wonderful job, making it one of my 3 favorite classes in my 4 years! |
[
"Any good math podcasts?"
] | [
"math"
] | [
"uysxgk"
] | [
36
] | [
""
] | [
true
] | [
false
] | [
0.86
] | Title says it. There are two kinds of podcasts I am looking for: I couldn't find a podcast similar to any of the above. Thanks | Podcasts aren’t exactly suited to geometric discussions generally. However, My Favourite Theorem is a great podcast and there have been some episodes which have personally been very insightful to me (my favourite being episodes 38 and 44). Edit: there are also episodes that discuss geometric ideas like Varignon’s Theorem and projective geometry. | As Gandhi said, start the podcast you want to see in the world | I like the 3b1b podcast by Grant Sanderson. The episode with Steven Strogatz was particularly good. | You should probably watch lectures if you want something technical. Kind of hard to convey mathematical idea without images of formulae | Thanks. Will look into it. |
[
"What are some curiosities or interesting things that every person interested in mathematics should know?"
] | [
"math"
] | [
"uz3fgn"
] | [
188
] | [
""
] | [
true
] | [
false
] | [
0.97
] | Our math teacher told us about the story of how Fermat wrote on the margin of a book page that he had found the proof for an important theorem, but that the page was too small to write it on. As a result, I began to think about how many stories and curiosities like that there would be out there, so if you want to share some interesting story or fact I'd really appreciate it. | The story of 1729--the Ramanujan-Hardy number--is a good one. More real numbers than natural numbers is a good one. | My favourite part of that story is the bit that isn’t told very often. You’ve got that 1729 = 12 + 1 and that 1729 = 10 + 9 - does anything jump out at you? No? What if I rearrange it to be 10 + 9 = 12 - anything there? Decades later, someone discovered that equation written down in one of Ramanujan’s notebooks along with scribbles about modular forms and elliptic curves. . 1729 is the byproduct of a Fermat near-miss. This part of the story of 1729 is too often missed out, but it’s my favourite bit. | Marden's Theorem! "Suppose the zeroes z1, z2, and z3 of a third-degree polynomial p(z) are non-collinear. There is a unique ellipse inscribed in the triangle (whose vertices are z1, z2, z3) and tangent to the sides at their midpoints. The foci of that ellipse are the zeroes of the derivative p'(z)." Edit: The zeroes of p(z) are in the complex plane, which is how they can fail to be colinear. | Dirac's delta function basically started as an abuse of notation by physicists but then mathematicians figured out what the hell it was | I'm surprised no one has mentioned the life of Évariste Galois. While just around 19 years old, he invented a whole new branch of mathematics (group theory) and discovered a connection to which polynomial equations are solvable in radicals. This was a long-standing open problem for centuries. His personal life was also interesting, he was involved in political acts related to the French Revolution, and he died in a duel aged 20 years old. |
[
"What is your favorite function?"
] | [
"math"
] | [
"uzee6c"
] | [
49
] | [
""
] | [
true
] | [
false
] | [
0.9
] | Often people ask people about their favorite number but I am interested in hearing what people consider their favorite function to be. It can be a continuous function, a discreet function or even a higher order function. Personally my favorite at the moment is the successor function. | ∅:∅→∅ | The function that maps rationals to 1 and irrationals to 0 | It doesn't look like anything to me. | Sheldon Axler disliked this message. | f(x) = x Identity function is the best function. Change my mind. |
[
"How good are you at coding?"
] | [
"math"
] | [
"uz38wl"
] | [
42
] | [
""
] | [
true
] | [
false
] | [
0.85
] | Obviously, if you are more in the applied side of things, this may be a bit more relevant. Still, many people seem to think that picking up coding is easy with a math background because it's "just logic." Personally, I have found it very difficult to get competent, however. Anyone with a degree in CS would beat me any day. I can get the job done, but my code is probably incredible inefficient. Even just shifting to the paradigm of OOP vs. procedural was a huge leap. | Logic is only part of coding. The other parts are the syntax and accepting the language designer’s quirks and esoteric nonsense. Then again, syntax and notation are a big impediment in math as well, so we should be used to it. | It's just a matter of practice. Also, there are many different kinds of being good at coding. There's being good at making a small program correct; there's being able to structure a medium-sized program so that it's clean and bug-free in the important ways; there's being able to structure a large project so that many people can work on it at the same time having different skill levels; there's being able to make a tool that can be reused by hundreds of people with different goals. | CS majors also specifically study algorithmic design from a strictly mathematical viewpoint, as well as efficiency purposes. In fact, most CS majors couldn’t care less about making web apps, they don’t learn such things in their core curriculum. Coding is seen as a means to an end to understand computer science as a whole. However, a math major will be more experienced in these areas, finding themselves right at home with the rigorous proofs in discrete math in application to theoretical CS. Really it’s more of a superset-subset relationship, at least in theoretical CS. | Probably because his mental model of what happened when a program executed didn’t include a stack, heap and a program counter. He might have intuitively thought of functions like processes with allocated shared address space, where running them in parallel leads to race conditions and crashes (as they’re trying to access the same memory addresses to do different things at different points in execution). It’s the fact that the current state of everything gets stuck on the stack and popped back later that makes recursion work. But it’s usually not described that way when you’re referencing the algorithm, and there’s no real reason to assume that’s what’s happening. I find a wall you periodically hit in CS is incorrectly inferring the underlying mechanics of something based on your observations of the parts you interact with, combined with a common sense guess at how you’d implement that behavior. Sometimes the reality is more sophisticated than your guess, and sometimes it’s far stupider than your guess. But to make the switch over from math you have to get good at catching yourself when making assumptions about how things work. | I think one issue for me personally is the assumption that syntax can be learned very easily, or that autodidacts will always be just as good as people who learn in class. One example is that someone I worked with genuinely didn't understand recursion in a program I wrote. He couldn't understand why a function calling itself didn't break the program, merely because he had never seen recursion before. |
[
"Recommendations for books on ZFC"
] | [
"math"
] | [
"wv888o"
] | [
5
] | [
"Removed - ask in Quick Questions thread"
] | [
true
] | [
false
] | [
0.73
] | null | It may not serve as very "introductory", but the book "From Frege to Godel" has the translations of the source material of foundational mathematicians from 1879 to 1931, and covers the motivation and development of ZFC to NBG, and so on. For something simpler, or for someone only starting out, maybe "Naive Set Theory" by Paul Halmos. | It may just be my ignorance, but I thought ZFC was precisely a rigorous alternative to Naive Set Theory. Am I wrong? | Halmos's book's name is a misnomer. He talks about axiomatic set theory in a Naive way. | Thanks! | Naive Set Theory by Paul R. Halmos |
[
"Repeating fractionals and Bases"
] | [
"math"
] | [
"wv2mvg"
] | [
1
] | [
"Removed - ask in Quick Questions thread"
] | [
true
] | [
false
] | [
1
] | null | What you say is essentially correct, except you don't need the denominator to have the prime factors as the base, but only that its prime factors are those of the base (some prime factors may be missing). For instance, in base 10 we have 1/4 = .25 and the prime factors of 4 are among those of 10, not the same as those of 10. Probably that correction is what you meant to say anyway. Here is a restatement of this result. . A positive fraction m/n in reduced form has a finite expansion in base b if and only if every prime factor of n is a prime factor of b. As for who discovered this, in base 10 it was observed by Wallis in the 1600s. Patterns in decimals are fairly straightforward to guess if you generate enough examples, but actually proving the patterns in any generality is not easy if you don't already know the connection between decimals and modular arithmetic. This is what Gauss had discovered, and in 1801 he used modular arithmetic to give a complete analysis of decimal expansions of fractions. He only treated base 10 since writing numbers in other bases at that time would have seemed pretty bizarre. Even today it would be pretty weird. While binary expansions come up in connection to computers, there is no practical reason to care about writing numbers in a general base. But if someone had actually asked Gauss about bases other than 10, he would have found it straightforward to extend all of his work to other bases (and not just because "he's Gauss"). As for references, the history of this is discussed in https://halshs.archives-ouvertes.fr/halshs-00663295/document , and if you want to find a textbook treatment then I advise you to find books on number theory that have a section on decimal expansions. | It's not too basic to have someone known as the discoverer: there is an historical record of earlier work and proofs can be found in Gauss' . See the link in my answer. | Probably that correction is what you meant to say anyway. Yip yip. And very thank you ^^ | Got removed. TvT. I fairly new to the world of Math so i was expecting that to happen | Got removed. TvT. I fairly new to the world of Math so i was expecting that to happen |
[
"How to make math less mainstream?"
] | [
"math"
] | [
"wv5hqh"
] | [
18
] | [
""
] | [
true
] | [
false
] | [
0.75
] | null | I think the Pythagoreans had it right, math should be a cult. Nobody except the most devout will want to do math if it comes with all kinds of weird rules about semen and beans. And if the cult gets too popular anyway? Just do a schism, maybe even have a holy war. | Force math departments to start teaching hard skills that are relevant outside of academia under threat of being merged with the philosophy department. Academics: expect every student to become an academic Also academics when there aren't enough jobs in academia: surprised Pikachu face | This might work - I wouldn't mind being in a cult if it means I get tenure. | I think instead if talking about how useful and practical certain mathematical concepts are for solving other mathematical or even real world engineering problems, we should focus on some abstract sense of beauty and constantly emphasize how etherally beautiful these things are. | It's a reaction to the recent post https://www.reddit.com/r/math/comments/wtt0l1/how_to_make_mathematics_more_mainstream/ |
[
"What are the levels of math?"
] | [
"math"
] | [
"wuz8z4"
] | [
0
] | [
"Removed - ask in Quick Questions thread"
] | [
true
] | [
false
] | [
0.33
] | null | Generally, as a couple final courses that can be taken before your first Analysis class are Discrete Mathematics and Differential Equations. Your first Analysis course is probably Real Analysis 1. This separates the people who can do proofs from the people who can operate algorithmically. Once you get past this point your limit is really how far your mind can stretch into abstraction from what you know. | branches of mathematics stack up Branches don't really "stack," do they? They branch. It'd be better to look at it that way. One can be the leading expert in one field of mathematics and have the understanding of another on par with an undergraduate student. I'd agree that the "trunk" of these branches is about the same for everyone, and most people really only diverge from one another after linear algebra and differential equations (and even then, you won't find many people with advanced math skills who haven't taken classes like real analysis or abstract algebra). But once you start getting into higher-level math, things start to specialize quickly, and it wouldn't make much sense to compare two fields with an expectation that one is more advanced than another (i.e., they don't "stack" past a certain point). | Not a full picture but a short overview for the topics I have already covered in video form: https://thebrightsideofmathematics.com/startpage/ | In American university, Real analysis and Abstract algebra is the gate to real math. You would be required to take both of them for a math major. But really, there are no limitations. There are so many branches of math available in university. The absolute basic of math you need before doing high level math are, arguably: Linear algebra: it's used everywhere you go, probably the most useful area of math. Set theory: the language of set theory appear everywhere, but you don't need anything deep. Topology: the language of point set get used everywhere, but not very deep. It comes after analysis. Group theory: you will learn in abstract algebra. Also appear very often in math. But other than these, the branches are quite disjoint from each other, to the point that you can be an expert in one area and don't even know the basic of another. Even seemingly basic topic like calculus get used a lot less than you think. | Probably immediately after multivariate calculus is differential geometry. However to get a full understanding of differential geometry one needs to know basic topology and the linear algebra you're learning. But that's just one of many directions you can take. Going the number theory / algebra route will be something else. |
[
"On naming all the fractions - a question of aesthetics."
] | [
"math"
] | [
"wvnl48"
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2
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""
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true
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false
] | [
0.52
] | Hello, math people. I have an old but still fresh math degree with a bunch of grad school classes but I mainly write programs these days. As a sort of art project, I am writing a program to immutably and uniquely name... pretty well everything. You get the idea. (The final names will be different, there is one big change to come.) Now I'm going to name all fractions. The mechanics are a little fiddling but almost elementary, Cantor did it first. So it isn't a math-math question. It's either a philosophical or aesthetic question. :-) The question is rather what exactly should I define a "fraction" to be, and this is more aesthetic than anything else. Is it a fraction in least terms, or do I distinguish between and ? Is the negative a property of the whole fraction or of the numerator and denominator independently? Do I distinguish between and ? Do I allow the representation of the indeterminate or undefined forms and where ? Do I allow ? EDIT: If I select "Yes, no, no, no", I get , but I'm leaning away from that being what I want. I'm pretty well decided on "no" for the last one, reason being "too annoying", but I'm still vacillating on the others. Your ideas are solicited! EDIT: Thanks, I appreciate the people who actually tried to think through what I wrote, though I was a lot less fond of the ones who treated me like a confused undergrad. | If by fractions, you mean rational numbers, then finding a unique representative for each rational number might be a good idea. Such a unique representation is, for instance, given by a/b with b positive and |a| and b coprime (in particular, 0 is represented by 0/1). This ensures that the denominator is nonzero, the fraction is in reduced form and the sign is carried only by the numerator. | If you want all fractions to be distinct and 1/0 is allowed, then what you really want is simply a tuple of pairs of integers, perhaps with rules that only allow the first of the pair to be negative. | I am representing the human idea of "fractions". The human idea of a fraction is still a fraction, unless you are using "human idea of..." as a synonym for "someone who doesn't understand what they're talking about". Why should a human idea of a fraction not be a fraction? If you want to talk about ordered pairs of numbers, so (20,100) and (1,5) don't mean the same thing, then do that and don't call them fractions. | No, I am representing the human idea of "fractions". This can mean different things for different people. There is no "one" human idea of "fractions". Like you mentioned, this is more a philosophical question. The "correct" answer (if there is one) will depend on your purpose for giving names to these entities. In my opinion, without knowing your end goal, the number 20/100 is the same as 1/5 is the same as -1/-5 and should therefore all be given the same name (or names! I understand you don't want two different things with the same name, but are you ok with giving two different names to the same thing?). The strings "20/100", "1/5", and "-1/-5" however are all different and should be given different names. So it's up to you to decide what you mean by "THE human idea of fraction". Do you mean the number, the string used to refer to that number, or something in between (like the numerator-denominator pair used to refer to that number). | your regular person will be dissatisfied if they input 100 / 16 and get back 24 / 4 and say, "They might be equal but they aren't the same." Well, 100/16 does not equal 24/4 (it'd be 25/4), but you simply made a typo. Anyway, someone who responds the way you indicated to the input and output is exactly making a distinction between fractions and ordered pairs. Maybe you are uncomfortable with the term "ordered pair", but that describes what the regular person was thinking about while "fraction" does not. I don't want to go down the rabbit hole of 1/0, so I won't comment on that. |
[
"Process for taking notes from a textbook"
] | [
"math"
] | [
"wvalvm"
] | [
6
] | [
""
] | [
true
] | [
false
] | [
0.72
] | Okay, so I am curious how other people take notes from books, if they do it at all. Attain a general working knowledge of a subject, to the point where I can answer any "standard" type problem on that topic. Textbooks contain of information, and only a fraction of it is information that I want to be able to quickly access later. I haven't found a good workflow for quickly capturing essential information in a textbook such that it can be easily retrieved later. I've sort of distilled this into two problems: do you capture the information? information do you capture? Regarding #1, plain ol' paper works decently well, but it's not good for retrieving things you wrote weeks or months ago. On the other hand, it's a lot faster to write things down, especially if there's a lot of notation involved (e.g. matrices). Something like Obsidian is great b/c it's searchable, but it probably takes longer to typeset it all, and you can't have lots of useful note pages "in front of you" the same way you can with physical paper, especially if you want the textbook PDF open simultaneously on your laptop. As for #2, I've often found it easy to write far too when taking notes into some software like Obsidian. It's hard to know what to write down: every last theorem and proposition in the book? Probably not. Important definitions and theorems? How do you know what could be important later? How about proofs? Do you save past work of exercises you did, or just toss them? Happy to hear what everyone's workflow is for this process (or, if you don't take notes at all from books, what the reasoning is behind that). | "Problem: Textbooks contain a lot of information, and only a fraction of it is information that I want to be able to quickly access later" I disagree. In other subjects maybe, but in math, depending on your end goals, usually the problem is that people aren't remembering *enough* content, not that they're being exposed to too much that they don't need. | Once you get used to typing in Latex, it is generally faster than writing math, with the main exception of pictures and/or diagrams (though I’ve heard of people that are pros with these as well). I usually annotate the textbook and then make a Latex doc summarizing important definitions and results. Usually, after reading a few sections, you start to get a good idea which theorems are important. More immediately, authors will emphasize important theorems, and often these theorems will have names. | How does one type matrices and lots of intricate fractions quickly in LaTeX? Use a better text editor. I use Emacs and do something along the lines of this . He's pressing the tab key to navigate between fields. So he can enter a matrix element, press tab, then enter the next element. The author links to a similar configuration in Vim made by Gilles Castel. It takes a while to learn how to use those text editors and get them the way you want, but in my opinion it's totally worth it in the end. Someone has probably done similar things for VS Code, so that could be worth looking into as well. | For #1 stick with paper. It's true that there is no way to do a regex search on your notebook, but actually you don't want to. You shouldn't be mindlessly copying things down, you should be rewriting them in your own words, annotating them, drawing pictures coming up with examples. If you want something later, you will have to do a bit of flicking through the notes, narrowing in on the desired concept by various processes of elimination (e.g. "I'm looking for X, this page is a special case of X, therefore X must be on an earlier page"). This is not a *waste of time* it is an important part of learning and understanding. As for #2 write down the definitions of course. Try to come up with an example or two, especially the smallest example, the smallest "non trivial" example (whatever that might mean in context), an example of something that satisfies all but one condition of the definition etc. focus on statements of results and key techniques and constructions, every step of an argument is perhaps unnecessary, but deciding that is part of the much discussed (but never defined) "mathematical maturity". Mainly you should spend your time solving problems, and also solving *around* the problems, i.e. trying to see how far you can push the solution, what's the natural next question, can you solve it? what happens if you drop this condition or add some, extend to more dimensions, to hypergraphs, to more general algebraic objects, or what have you. | I think I might see what you're saying, but the vast majority of textbooks do contain lots of information that will not be relevant to a significant portion of its readers. And not information that needs to be stored and accessed later, like proofs of random lemmas and stuff like that. It's fine to read and understand them, but that certainly wasn't what my question was about. |
[
"Anyone else find visualizations of the Mandelbrot Set oddly unnerving?"
] | [
"math"
] | [
"wvesqg"
] | [
24
] | [
""
] | [
true
] | [
false
] | [
0.79
] | This is coming from someone who would watch too many space documentaries as a kid and then have nightmares about the eerie sounds naturally emitted by Jupiter. Something about visualizations of the Mandelbrot set have always just seemed vaguely off-putting to me. The shape of it when zoomed out almost seems alive, like some sort of insect, and the pitch blackness of its inner body compared to the rest doesn't sit right with me. Especially in images with lots of color and contrast the thing feels like a cosmic horror entity you'd meet in a psychedelic fever dream that would try to swallow you whole, forcing you to fall endlessly through its infinitely recursive tendrils. Obviously I don't have a literal fear of this piece of math, and in fact I still think it's really cool and interesting to look at, but the otherworldliness of its appearance triggers some sort of reaction for me. Was wondering if anyone else had a similar reaction. This is just a weird thought I've had for a while that I wanted to share. | I think I know what you mean; for me, it's mostly caused by the detail. Zooming in on these fractals is like an acid trip. Even the smallest length of these visuals contains a universe of complexity; it's almost hypnotic. I think that's what can feel so unnerving: we can remember the basic outline of the Mandelbrot or Julia or other sets, but zooming in even slightly reveals that these images are too complicated for the human brain to get a complete picture. The sense of scale given by zooming in so far into an image can also be disorienting, and definitely feels like being lost in space.(I actually had nightmares about infinitely zooming into functions when I was studying limit proofs for the first time!) I definitely feel like I'm brushing up against something Lovecraftian or "unknowable" when I see those images. However, I think it's actually very reassuring to think that despite how peculiar they look when we visualize them, these sets have been deeply studied and they're much more "knowable" than they look. I think it's actually really cool that math allows us to effectively study something that looks so fantastical; it also serves as a reminder to not get so caught up in visual aids. | If you're looking for an explanation, you might have better luck in a psychology-related subreddit. My guess is, it kinda looks like a bug, bugs are icky. But I have never even touched a psychology textbook. | I have a fear of depths (think about the depths of the ocean). Zooming into the Mandelbrot set is like falling into infinite depths. | Yes, it’s like it’s so tiny and so immensely big at the same time, and I feel so alone and so connected at the same time, there’s weird emotions there. It’s stronger after seeing fractals during DMT trips, some regions/gradients now remind me of terrifyingly weird moments that I’m STILL not sure how to make sense of. It’s good though, I still love looking at Mandelbrot zooms, but sometimes it definitely kinda gives me the willies. | Intimate? Or infinite? (I have a fear of depths too.) |
[
"What Are You Working On? August 22, 2022"
] | [
"math"
] | [
"wuxfzr"
] | [
7
] | [
""
] | [
true
] | [
false
] | [
0.72
] | This recurring thread will be for general discussion on whatever math-related topics you have been or will be working on this week. This can be anything, including: All types and levels of mathematics are welcomed! If you are asking for advice on choosing classes or career prospects, please go to the most recent . | I'm on the penultimate week of my summer research project, in additive combinatorics. I've been thinking about the cap set problem, which asks the following question: If A is a subset of a finite field of characteristic 3 so that A contains no non trivial solutions to x+y+z=0 (ie other than x=y=z), then how large can A be? The reason this problem is interesting is that in characteristic 3, this is equivalent to asking when a set has no 3 terms in arithmetic progression. Finite fields of characteristic 3 are the simplest setting where we can ask this question of when sets have no non trivial arithmetic progression. Excitingly, I have managed to prove a new lower bound on this problem, which is better than the current world record! I have a few ideas for things to investigate for my final couple of weeks, and then my result will hopefully be distributed. I have never done any research before, so I am not sure how the process works - I'll follow the advice of my supervisor on this! At the moment, I have a couple of things I'm still investigating. Possibly the most exciting of these is trying to apply the probabilistic method, to prove that certain structures exist. I have an outline attempt at doing this, which has a couple of holes, but they may be fixable - I've got a meeting with my supervisor tomorrow, so I'll discuss this then! If I can get this probabilistic method to work, I'll have an even better result than I already do, which is very exciting. At the same time, I am trying to think about the problem with a clean slate, to see what other approaches may be worthwhile. In particular, my work is a generalisation/stronger version of a previous result. I've been thinking about whether there are other ways to extend the general ideas of this paper - this has been quite tricky, and hasn't yet given me any useful results, but it has given me new results which are worse than my current one! I've also been doing a fair amount of analysis and asymptotic estimation recently, which has been surprisingly good fun. In addition, I've had to learn a little finite geometry in order to understand the structures of certain objects which I can then use in my work. Trying to essentially translate work in finite geometry into additive combinatorics has been a fun challenge, and there's still some more things to think about here. | I got to dig into it a little bit when I took an undergraduate Number Theory course last year. My professor recommended this book , by H. M. Edwards, which is a great introduction for a motivated undergraduate. It delves into the method that Riemann himself used in investigating his prime number problem. Worth a read if you'd like to explore the zeta function and the Riemann Hypothesis in more depth and from a historical perspective. On a personal note - I do really enjoy reading the traditional monthly post here where somebody has claimed to solve it. The drama is always entertaining. | Just started my first year of grad school, working on a masters. Taking Analysis 1, Algebra 1, and probability theory, so lots of measure theory but some algebra in there to keep me happy. I’m also doing some independent reading/research on differential topology which has been neat. | Writing a new paper about the finitistic dimension conjecture. Yes, my flare says commutative algebra, but this topic, and this paper is actually noncommutative algebra. So I am stepping a bit outside my usual expertise, simply because I found a cool proof of a new result about this problem. | You should post the scariest looking diagram that you've seen recently! |
[
"Mathematical interest in the Ising model"
] | [
"math"
] | [
"wv0k3k"
] | [
107
] | [
""
] | [
true
] | [
false
] | [
0.96
] | Given that their have been two very recent fields medals awarded for work in relation to the Ising model, I was wondering if anyone could shed some light on its relavence to pure math. I'm a physicist, so I'm already well aware of why it is of interest from a physics point of view. My previous understanding of the world of mathematical physics was that it was mainly concerened with topics like formalising QFT, or things like mirror symmetry arising in string theory. Specifically: 1) My understanding is that prizes such as the fields medal are only awarded for work with wider importance in math, does the Ising model/statistical physics have many connections to other areas of pure math? 2) What are the big open problems in this area (or have the recent proofs killed the field so to speak)? | You're right that a lot of mathematical physics deals with quantum-y stuff. But a tremendous amount is also about "rigorous statistical mechanics," broadly construed. There is a lot of overlap here with probability theory and stochastic processes, and these papers are often published in probability journals, which is perhaps why you haven't come across them. Spin glasses, random matrices, the KPZ equation, and lattice models have been popular for decades. (Mathematical interest in KPZ might be slightly more recent? I don't know the history well.) This overview of Hugo Domnil-Copin's work might answer some of your questions: https://arxiv.org/abs/2207.03874 . In particular, the first section describes how the author (a mathematician) thinks about "mathematical physics." Regarding connections to other parts of mathematics, there has been a lot of interest in applying spin glass models and methods to problems in statistics and computer science. The Montanari–Mezard book "Information, Physics, and Computation" deals with this in an accessible way (though there is a need for a newer reference, I think). | Studying the critical Ising model in 2D connects it to complex analysis, conformal field theory and a new area called random conformal geometry. The central object here is the Schramm Loewner Evolution, a notable recent instance of mathematicians discovering something physicists didn't even suspect. It's a stochastic process that describes the interface lines between + and - spin clusters in the critical scaling limit of the Ising model. It is part of a one parameter family that also includes scaling limits of many other interesting 2d critical models like percolation. It also turns out to have deep connections to complex analysis. Smirnov and Werner got their Fields medals for work laying the foundations of that area. In general dimensions, the expected phase diagram for Ising that you see in physics books, including exponential decay of truncated correlation functions at any temperature and continuity at the critical point, turns out to be surprisingly hard. Duminil-Copin got his Medal in part for completing the picture. You mention rigorous QFT: this is related to critical Ising because one of the ways to try to construct a QFT is to construct a corresponding Euclidean field theory (and then somehow Wick-rotate it to get a Lorentz invariant theory). For phi fields one discretization looks like a critical Ising model. Aizenman and later Aizenman and Duminil studied this and showed that all possible limits of this type of discretization are trivial, so if phi exists ("non asymptomatically" in physics language), it can't be constructed by obvious discretizations. Critical Ising in 3D or high dimensions is very much open: for example, proving the existence of critical exponents, etc. is unknown. There is something on the more theoretical side of physics called the conformal bootstrap that seems to be interesting in 3D but it has not been mathematically understood yet. | Kardar-Parisi-Zhang. These slides have nice pictures and give a decent overview, or at least a starting point to Google things: https://www.math.columbia.edu/~corwin/IHPTalk1.pdf . Martin Hairer won the Fields medal in 2014 for his contributions to understanding the KPZ equation. | Probability is one of the larger areas of research in pure math and as far as I can tell the two biggest sub areas that people conduct research on are related to physics and finance. | As some other folks have mentioned, Hugo Domil Copin gives a very readable answer to you question, intended for a general mathematics audience, here: https://arxiv.org/abs/2208.00864 |
[
"Interesting problem I was thinking about and couldn't figure out."
] | [
"math"
] | [
"vgark1"
] | [
2
] | [
"Removed - ask in Quick Questions thread"
] | [
true
] | [
false
] | [
1
] | null | This is called a biased one-dimensional random walk (i think), and you will make it to 0 with probability 1! At least I'm pretty sure. I remember reading about it and working it out once, but I don't remember exactly how it worked. Edit: just realized I was thinking about the probability being weighted instead of the distance being weighted, which might make a difference | The average score for flipping a coin is +499.5, so your score is essentially guaranteed to always diverge toward infinity. Theoretically it's possible to win 1000 times in a row, but every time you lose the odds of you getting down to zero is divided by 2 . | Yes, but you should win eventually right? You always have a chance of winning and you can flip infinite times. | All you need is one flip out of 1000 to set you back significantly. This creates a "drift" factor towards infinity. You are very very unlikely to ever get to 0 | Just because you get infinite attempts does not mean every sequence happens. While it is technically possible I don't think you would win in practice even given literally forever. |
[
"\"Easy\" Putnam Questions"
] | [
"math"
] | [
"wvcywj"
] | [
124
] | [
""
] | [
true
] | [
false
] | [
0.93
] | First off, apologies if this doesn't belong here. I'm an engineer by trade but I got a side-gig coaching Math Olympiad for high school students. Out of curiousity, I was looking up old Putnam questions in my boredom at work, and I found a few where the question writers must have been asleep on the job. Or written the question at the last minute. For example, this one from 1988: Show that every composite integer is expressible as xy+xz+yz+1, where x,y, and z are positive integers. Turns out that z has to be 1, and in that case you can factor xy+x+y+1 as (x+1)(y + 1). (not my own) pretty much came to the same conclusion I did, so the solution almost seems trivial. Then I found this one: from 2021. A grasshopper starts at the origin in the coordinate plane and makes a sequence of hops. Each hop has length 5, and after each hop the grasshopper is at a point whose coordinates are both integers; thus, there are 12 possible locations for the grasshopper after the first hop. What is the smallest number of hops needed for the grasshopper to reach the point (2021, 2021)? The way I see it is that if you take a 3 up, 4 right, and then immediately take one of 4,3 you will be at (7,7). If you do that 288 times, you end up at 2016,2016. From there you it's all you need to do is hop (5,0) and (0,5). (again, not my own) that did it in an even simpler way: Divide 2042/7 and round up. Hence, you basically needed to know about 3-4-5 right triangles It's said that with the Putnam, scores are awarded on HOW you get the solution rather than the solution itself. So, is there something missing from these solutions? Also, has anyone ever seen anything easier on a Putnam or any competition similar to that? | I don't have any answers for the 1988 one, but for the 2021 one you have come up with a candidate for the minimum, but you haven't shown yet why this is the minimum number of steps. Perhaps this too is easy, but as it stands I think your solution would be incomplete. | Out of curiousity, I was looking up old Putnam questions in my boredom at work, and I found a few where the question writers must have been asleep on the job. Or written the question at the last minute. For example, this one from 1988: This is an awfully harsh judgement of someone who probably had reasons for what they did, that you just don't know. In this case, I suspect that over the past 35 years, students just tend to learn a lot more of what we think of as olympiad mathematics in high school. That problem would have been a lot harder if you had never seen a diophantine equation before, let alone had whole classes devoted to them as some current students do. You can see olympiads getting more difficult, for similar reasons. Sometimes famous difficult IMO problems inspire the widespread teaching of a particular technique (Vieta jumping and Muirhead's inequality come to mind), and people now are surprised that those problems were considered difficult. Your second question seems easier, though I think the argument for minimality isn't necessarily immediate to come up with. | I reasoned since the shortest path is along the diagonal, that taking steps of (3,4) and (4,3) and (5,0) and (0,5) keeps you on the diagonal This is a good assumption and the right one, but you haven't proved it. The video on the other hand, by introducing the formalism of the Manhattan distance, proves that the lower bound is actually 578. Then the example provided shows that the lower bound can be actually achieved, which completes the proof. | There was one like "prove that, given any 5 points on a sphere, there is a closed hemisphere which contains 4 of the points" This is the only Putnam problem which I was able to solve in less than a minute, it just seems like such a trivial application of pigeonhole. | Am I missing something? Take any 2 points and draw a great circle through them. There are 3 left, so at least 2 of them must be on one side of the circle. Take this side and you have such a hemisphere. |
[
"Was anyone else 'close' to proving Fermat's Last Theorem?"
] | [
"math"
] | [
"wv2gnr"
] | [
435
] | [
""
] | [
true
] | [
false
] | [
0.96
] | Would it have gone unproved for much longer (10+ years) without Wiles or were others 'close' as well? (I have absolutely no idea what I'm talking about with respect to maths. I'm just curious how different his approach was to others.) | I had the pleasure of taking linear algebra from Wiles in the fall of1992. It was an accelerated class for freshmen who'd done the AP thing and were likely to study math. On the last day of class, he drew several unsolved problems on the chalk board to inspire us to pursue math careers. When he got to FLT, his words became a bit spacier and he mentioned that it was a very interesting problem. I recall thinking that he must be secretly working on it and therefore nuts. Saw him once again on campus in the year between proofs. He looked stressed out. | In the mid-1980s, and not any earlier, there were two different plausible approaches to attacking FLT: (1) as a consequence of modularity of elliptic curves over Q and (2) as a consequence of the abc conjecture. The eventual proof by Wiles and Taylor followed the first method, establishing enough of modularity of elliptic curves over Q to get FLT. Serge Lang, in the preface to his 1991 book "Number Theory III: Diophantine Geometry", wrote about FLT Although it is not proved, it is not an isolated problem any more. It fits in two main approaches to certain diophantine questions. [...] Some people might even see a race between the two approaches: which one will prove Fermat first? Watch the documentary on FLT by the BBC ( https://vimeo.com/58907900 ) and you'll see everyone who was interviewed other than Wiles said it was inconceivable to all of them in the 1980s and early 1990s that FLT was anywhere close to being proved. Wiles says that he didn't know if it was possible at the time to prove FLT, but that at least he was sure he wasn't in serious competition with others because nobody had any idea where to begin. | There’s the wonderful anecdote about Ramanujan, the taxicab story. Hardy goes to see him in hospital, says he came in taxi number 1729 and that the number is boring, Ramanujan says that 1729 is the smallest number to be expressible as the sum of two cubes in two different ways, yada yada. Years after he died, some of Ramanujan’s notebooks surfaced. He had scribbled down on one page that 10 + 9 = 12 + 1. See what that is? It’s a Fermat near-miss. Then they flicked through the pages and found notes and sketches on elliptic curves and goodness knows what else. These ideas for attacking Fermat weren’t even realised to be related in Ramanujan’s day… and here he was, closer than anyone else, going down the path that Wiles would eventually go down himself decades later. It’s a right shame that Ramanujan was taken from us too early. | Haha, this is excellent! I recall thinking that he must be secretly working on it and therefore nuts. and Saw him once again on campus in the year between proofs. He looked stressed out. | the year between proofs By this do you mean, after the gap in his initial proof was discovered but before it was filled? |
[
"Resources for Toposes in Algebraic Geometry that aren't in French"
] | [
"math"
] | [
"wuuwty"
] | [
97
] | [
""
] | [
true
] | [
false
] | [
0.94
] | I was put in a position of having been exposed to concepts of Topos theory through books like Topoi by Goldblatt, Elementary Categories, Elementary Toposes by McLarty, and Sets for Mathematicians by Lawvere long before I started to learn algebraic geometry so it's odd to me that a lot of books on Algebraic Geometry actively shy away from the concept. In the first section of Vakil's The Rising Sea, he says 'This I promise: if I use the word “topoi”, you can shoot me.' In the preface for Milne's Étale Cohomology, he says "Only enough foundational material is included to treat the étale site and similar sites, such as the flat and Zariski sites. In particular, the word topos does not occur." So I ask, where can I find a good reference for toposes in their original context that isn't SGA4? | Firstly, I would like to say that I do agree that’s very strange to see people in algebraic geometry shying away from topoi. Even when you don’t need it, thinking in terms of topoi clarifies some stuff. Now, perhaps the book by Mac Lane and Moerdijk is closer to what you want? There’s no étale cohomology there, but quite a lot of geometry. About SGA4, I heard many people saying that it’s impossible to read, huge, etc… But I really had a great time reading it. Obviously you’re not going to read it like a roman. But it’s very well written, and has a lot of incredible explanations! I would not recommend avoiding it. | Would you be willing to translate 1623 pages of SGAIV for me 😉 | Would you be willing to translate 1623 pages of SGAIV for me 😉 | This is a stretch by far, maybe if you want to read the original context of of SGA and EGA, i.e if youre interested in motivic stuff. But I promise you that you do not need to learn French if you want to dive into t-structures, stability conditions, and derived torelli theorems | A number of reaons. Modern Algebraic geometry got its start through the likes of Serre, Grothendieck, Deligne, etc and the treatises and seminars run by Grothendieck effected a phenomenal reshaping of the entire field. Grothendieck, however was quite the crotchety fellow and actively sought to hamper any republication of his work and translation of his work to other languages. |
[
"Why is ratio of circumference and diameter constant?"
] | [
"math"
] | [
"vfwgae"
] | [
0
] | [
"Removed - ask in Quick Questions thread"
] | [
true
] | [
false
] | [
0.43
] | null | The reason that ratio is constant is that all circles in the plane are similar: they are scaled versions of each other. for the same reason, since all squares are similar figures the ratio of the perimeter to diagonal of a square is the same value for all squares. (You can't do this for all rectangles since rectangles are not all similar to each other, but it would work when comparing all rectangles having a common length to width ratio, since that makes them similar.) I don't think there is a reason for the particular value of pi. Conceptually you can see the value is the same for all circles, but to compute or estimate it you need to work it out and see what you get. The Veritasium video "The Discovery That Transformed Pi" explains starting at 1:33 why pi is between 3 and 4 using an inscribed hexagon and circumscribed square. If you look at circles not in the plane, but on a sphere (or other curved surfaces), and define the diameter of that circle by working entirely on the sphere (so it is the length of the shortest arc connecting two opposite points on the circle, which is not the usual straight line path in space), then the ratio of the circumference to diameter of a circle is not the same for all circles on that sphere. This is an effect of the curvature of a sphere. A plane is flat and that makes all circles in the plane similar in the geometry of the plane. So you could say the constancy of pi (that it is the same for circles of all sizes) is due to working in a flat plane. | I think you may have misunderstood the last comment (or I may have misunderstood yours). The point was not that different spheres are similar or not in 3D space, but rather that not all circles are similar . Consider a circle of radius 1m around the North Pole. It will be very close to a circle drawn on a flat plane and hence the ratio of circumference to diameter will be very close to pi. Now consider a circle around the equator. The shortest arc (or a shortest arc, they are not unique anymore) connecting opposite points could travel through the North Pole. Using this circle, you'd get a circumstance-to-diameter value of 2. So the circumference-to-diameter ratio of circles drawn on a sphere is variable. It depends on which circle you draw. The circumference-to-diameter ratio of circles drawn on a flat plane is constant: pi. So the constant-ness of the circumference-to-diameter ratio is actually indicative of the flatness of the space we're drawing circles on. | Why is it contstant? Well if you scale a circle to be twice as big, then the diameter is about twice as long, but so is the circumference, so the ratio betweem them doesn't change at all. Why is it 3.14...? Well, the best I can say is that when you measure it, or when you calculate it with math, that's just what it turns out to be. | I agree up until your last comment. All spheres are similar, just like all circles are similar. The key point is how lengths/areas/volumes change under scaling. In particular, if you have two lengths defined in a shape, their ratio doesn't change under scaling. Similarly, the ratio between two areas doesn't change. The square of a length is an area, and thus the ratio between an area and the square of a length doesn't change under scaling. So, in 2d, the circumference (length) to radius (length) ratio is a constant for circles. In 3D the surface area (area) to the square of the radius (length squared, so area) is a constant. This isn't particularly related to curvature, rather the scaling of the quantity of interest. The same argument could be applied to a cube (its surface area to side length squared ratio doesn't change), and this is a shape without curvature on its faces. | It is a consequence of Euclidean geometry in non Euclidean geometry, it can have other values. |
[
"Is any pure/theoretical knowledge required for applied math studies and industry work?"
] | [
"math"
] | [
"vfzv6s"
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2
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true
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false
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] | null | From what I know, industry jobs tend to require very little in terms of advanced mathematics of any kind. For many of them, it's mostly being good with Excel, possibly programming, and creative problem solving skills. Some abstract mathematics might help train the latter, but it obviously isn't required. | I would develop a theoretical background due to how it would help with the cs/programming side of things. I found theory very helpful when making programs | A lot of consultancy companies that hire graduate level applied mathematicians are looking for people who specialize in statistics and operations research. So the math used is mostly probability theory, optimization, algorithms, and numerical analysis (note that this is my viewpoint as a probability theorist, it is of course different if you're into PDE or cryptology etc.) Edit: "pure"/theoretical courses that would help in this area would be measure theory and functional analysis (but many of my colleagues only take applied OR/statistics courses) | Most industry jobs will favor calculus/analysis so you will be fine. | Unfortunately, your submission has been removed for the following reason(s): Career and Education Questions /r/mathematics /r/matheducation If you have any questions, please feel free to message the mods . Thank you! |
[
"Nees help on book choice"
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"math"
] | [
"vg3ccz"
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2
] | [
"Removed - ask in Quick Questions thread"
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true
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0.76
] | null | by Stewart | I got rudin and stewart both. Foubd them cheap on ebay. Thank you for the advice. I will read stewart first. | I will look into it. Thank you | My brother recommended the same book. Thank you | You are welcome |
[
"How do I become a mathematic genius at the age of 18?"
] | [
"math"
] | [
"vfogrl"
] | [
0
] | [
"Removed - ask in Quick Questions thread"
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true
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false
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0.5
] | null | You shouldn’t set out to become a math genius, your goal should be to learn as much as you can, that is if your passionate enough. Like others have said enrol into a maths major if you’re serious about it, and find out what math really is about, and proceed from there. | If you have to ask.. | Find a good mentor | Find a problem that no one else can solve, like the Clay Institute problems (ok, it doesn't have to be so difficult) and focus on learning the math tools that will help to solve that problem. If you succeed, you will be recognized as a math genius. | Life advice: don't focus on who you are, focus on what you do. Who cares if you're a math genius or not? There are plenty of prodigies who burned out and whose greatest accomplishment is an article in a local news outlet. Loving math (or whatever) is way more important than a label. You can do amazing things with math, whether or not you are called a genius. So study, find what you love, and focus on it. That's more than enough. |
[
"Which branch of math will I enjoy the most If I just like numbers and statistics?"
] | [
"math"
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"vfg1xw"
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] | null | By your question, I am going to assume that you have already studied a course in basic descriptive and inferential statistics (Standard deviation, t-tests, etc.) If not, that is the place you should be. There is a nice free textbook on OpenStax. "Real" statistics starts with a calculus background. If you have not already studied the calculus of one and several variables, I strongly suggest that next. Again, free textbook on OpenStax. Now you are ready for the theory of probability and statistics. After that, possibilities abound based on your specific interests and curiosities. | I agree with the other comments. If you like statistics... you would like statistics. | A statistics course sounds like it would be a good starting point, you will likely get more information on whether you want to go further or change topics. By a course I am referring to any options you currently are aware of, or you might consider Coursera or similar. | Statistics. | If you like numbers i dont know if math is the right fit for you. Math is more about reasoning than numbers. |
[
"Best math book which consists of the best and most important work of the greatest mathematicians till date? (With explanations)"
] | [
"math"
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"vfri1w"
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0.77
] | null | Closest one I can think of is the God created the integers by Stephen Hawking..he adds commentary about his favourite theorems from their work and explains why those are his favourite and their importance. | I can think of two books which are pretty close to what you are looking for: | A great book that I recommend that is in line with what you ask is Journey through Genius by William Dunham. Goes through the history of mathematics by selecting some of the most famous and/or important proofs and mathematicians and places them in historic context as well as explains the proofs in an accessible manner. | It's a while since I read it but Simon Singh Fermat's last theorem is a good read and IIRC it covers some important work on the way to telling the story of FLT. Not a book, but the BBC had an excellent series on the history of mathematics called 'The story of maths' with Marcus du Sautoy. | I will say that it's not an easy read. Hawking does not edit the original texts, so the notation and language in some of the papers can feel very archaic |
[
"Need references on where to start with algebra papers"
] | [
"math"
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"vg4cq4"
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false
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0.67
] | I have covered topics like commutative rings, modules , little bit of tensors. In field theory I covered till galois extension. In terms of topology the I covered till homotopy homology. I was wondering if people can suggest some good papers I can begin with which would help me get direction. I have never read any papers in my life so any help or guidance is appreciated. | Why do you want to start reading papers? I think it would be a lot better for you to read a textbook covering more advanced topics--maybe start learning some commutative algebra or cohomology--and then read a paper as you naturally get interested in something more advanced. Papers are generally a lot harder to read then textbooks, and you don't seem to have much need for a paper at this moment. | What are you interested in? Although at this stage in your life probably textbooks are still more appropriate than papers! | Math papers cover very specific topics that need a good knowledge of the general matter the topic of the paper is about. In some cases extensive knowledge of other papers. At your level I think you are better off with advanced textbooks on the matters you mention if those are the fields you are interested in. The fields you mention are so vast that is difficult to get an idea about where you are headed. If Algebra is what you are interested in I suggest Algebraic Geometry, Number Theory and Noncommutative Algebra or even Category Theory for instance. | Interest is what I am trying to find, as I don’t have enough knowledge, but I feel algebra is I can do that's why I asked for it. I understand textbooks in the way to go but I want to start reading papers but don't know how to start. | Depends on what you want to learn more about — for example, if error-correcting codes are interesting to you, and you like algebra, then Stichtenoth’s book “Algebraic Function Fields and Codes” could be a good next step :) It’s all up to what your desires and interests are! |
[
"Are non Noetherian rings useful?"
] | [
"math"
] | [
"vfyz0d"
] | [
71
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""
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false
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0.94
] | I'm learning AG by going through Ravi Vakil's notes on the subject. He tries to keep the hypotheses of theorems general and include non-noetherian situations. I've always wondered- is the study of non-noetherian rings useful as itself anywhere in mathematics? Or we might just as well restrict ourselflves to the Noetherian situation. | By far the most important ring in class field theory is the ring of adeles of Q (there are also adelic rings attached to other 'global fields'), which is non-Noetherian. This and the rings the other commenter mentioned are very important, but in algebraic geometry, they matter not so much for scheme theory. Noetherian assumptions are OK in algebraic geometry because a lot of the time algebraic geometry is reasoning about varieties, which are Noetherian schemes since (by Hilbert's basis theorem) polynomial rings over a field in finitely many variables are Noetherian. Where non-Noetherian schemes really arise is in arithmetic geometry. Perfectoid spaces, which you may have heard a big buzz about, are not usually Noetherian! Additionally, even if you are considering only varieties and curves, oftentimes moduli spaces of natural objects are non-Noetherian, even if you were working in a Noetherian context to begin with. | Yes. The easiest example is a ring of polynomials in infinitely many variables. The chain (x_0) < (x_0, x_1) < (x_0, x_1, x_2) < ... does not terminate. The ring of continuous real functions is also non-Noetherian, as is the ring of algebraic integers. It should also be noted that Noetherian rings can have non-Noetherian subrings (if I remember correctly). EDIT: I guess I didn't quite address utility in any way. Hopefully it is clear that those three rings are useful at times. I simply wanted to point out that there are useful non-Noetherian rings and that restricting yourself to Noetherian rings does exclude some naturally occurring rings. | Yes, and i'll give you an example of geometric nature. There are geometric situations where you want to prove a certain statement about all schemes, not just noetherian ones. A typical situation is something like "a certain functor from Sch/S to Set is representable". I don't know how much AG you have seen, but trust me that some of the most important theorems in AG are of this form. Since you need to work with all S-schemes to prove statements like the one above, you need to be able to juggle with non-noetherian objects The other answers give you other reasons, arithmetic geometry being another big one topic where non noetherianness comes into play | More concrete examples: Puiseux series Generalizing the first example a bit, you can consider classes of algebraic extensions. Like let's say we adjoin every square root of an integer. Generalizing the second example, if you have a valuation ring which is Noetherian then it has to either be a field or a DVR (discrete [e.g. integer-valued] valuation ring). For example, in tropical geometry, one wants to have a valuation ring where the range of the valuation is a dense subset of the reals (like the rational numbers). That way you can take a set of solutions to some algebraic equations, take the valuation coordinate-wise for every solution, then take the closure (in the standard topology) and you get some nice sets (piecewise linear complexes) which you can study. This wouldn't work the same if your valuation only gave you integers. | Other people have mentioned natural examples from number theory. There are also tons of examples in geometric representation theory. We study various moduli spaces related to algebraic groups - a basic example is an algebraic loop group, which parameterizes maps from a punctured formal disk into an algebraic group. They are analogous to p-adic groups in arithmetic, and relate to the geometric Langlands program, QFT, etc. These kinds of spaces are infinite-dimensional, and accordingly rings of (algebraic) functions on them are non-Noetherian. TL;DR: lots of moduli spaces that occur in nature are infinite-dimensional, so algebraic functions on them form non-Noetherian rings. |
[
"Do math courses get less detailed/rigorous as the courses become advanced?"
] | [
"math"
] | [
"vg31q0"
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305
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""
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false
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0.92
] | I've been really disappointed with the way my courses were taught (calculus 3, group theory) this semester: A lot, if not most of the material was just written on the board and accepted as true, as the professors were too lazy to prove them (by their own words). One of the main reasons I love math is because of the rigour and precision, and I feel like when it's taught the way it is, that is a bit missing the point. We even briefly had a substitute world renowned professor for a week, and he taught the same way ("the proof is the same as you've seen in previous courses"). It didn't feel like this in my previous courses, and I was wondering if this is because of the teachers or it's just natural because as the courses get advanced they expect more self work from the students? Unfortunately I don't have the time to prove to myself all of the things I see in class... Sorry for the rant, I just feel like this can really affect my goal for masters in math. for clarification: Here (not US) you have to decide your major before you start your education. Thus all of my courses were proof-based. Starting from linear algebra I and calculus I, the courses I took are only meant for math majors (and some of the first ones for CS majors as well).Thus, I am required for formal proofs in my homework in both courses. What bothered me is that the professors skip a lot of the details in their proofs - some of which I don't think are trivial and I don't understand why they are true. This is especially hard when introduced to a new subject, where I felt I don't yet have the tools to formulate a proof myself. | For calculus 3, this makes sense--in the US, at least, calculus 3 is usually taught not for math majors, but for general STEM majors, and little is proven. For group theory this surprises me--what in an introductory group theory class did you need to take on faith?!! What possibly could have been too hard for your professor to prove?!? But there's a second issue you describe. "The proof is the same as you've seen in previous courses" is a way to save time. Nobody can do any interesting math by going back to first principles for every single argument; as much as people clown on professors for saying "x is trivial," oftentimes they say that to mean "you should be able to figure this out on your own at this point in your education!" This is not an issue of rigor, it is an issue of the professors assuming you have the competence to fill in details. Maybe you can think they're unreasonable in what they want you to fill in, but that's a different issue. | They get more rigorous. | Without knowing the exact course and materials, my thought is maybe not too hard to prove, but too time-consuming. | At some point in your math studies, you have to advance from the rigorous stage to the post rigorous stage. Terrance Tao has an excellent blog post about this: https://terrytao.wordpress.com/career-advice/theres-more-to-mathematics-than-rigour-and-proofs/ | Yeah in my experience group theory proofs are either elegant “book” proofs (in the sense of erdos), or are really long and annoying. There seems to be no real in between |
[
"Unexpected appearance of Reuleaux Triangles"
] | [
"math"
] | [
"vfj253"
] | [
30
] | [
""
] | [
true
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false
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0.96
] | The other day I was trying to drill a couple holes into a plastic bin so that I could hang it. I didn't have a drill on me at the time so instead I stuck my pocket knife into the plastic and starting rotating it, shaving the plastic to make the hole larger and larger until it was as large as my blade. Doing this I was expecting the hole formed to be a circle, but instead what I found was that this method consistently produced holes in a shape that I have since learned is called a Reuleaux Triangle. If I understand the Wikipedia article about this shape it seems they are in some way related to the circle. But I'm still a bit confused as to why this action produces this shape instead of a circle. Any thoughts? | The knife edge wasn’t perfectly centered about the axis of rotation - that slight offset combined with the circular rotation produces the effect. Believe it or not, you can use a similar trick to drill a nearly square hole, using a bit that has a Reuleaux triangle profile with an ellipsoidal offset rotation. See: https://www.mikesenese.com/DOIT/2011/10/drilling-square-holes-with-a-reuleaux-triangle/ | I think basically the tip gets stuck, causing the back to slide in a circular arc, then they switch and the back is stuck with the tip scraping away an arc and so on. As to why it is a triangle and not some other polygon with arcs on the sides, i have no idea. | That’s nothing… I can make perfect pentagonal holes by using my Reuleaux square drill bit with an ellipsoidal offset. | not good enough joke. reuleaux square cannot exist | Cool post, and astute observation! I side with u/MathThatChecksOut 's explanation, back and sharp edge alternating circular movements from corner to corner. As for why a triangle, and not some other polygon, my hunch is that: it's the minimum number of sides to effectively get a hole (2 doesn't work! 😀) more than three sides would require fine control to avoid skipping a corner, so three is the only stable process |
[
"What's your favourite group?"
] | [
"math"
] | [
"vg0fwp"
] | [
55
] | [
""
] | [
true
] | [
false
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0.93
] | Or "class" of groups (using class loosely here), e.g. S_n. My favourite is probably Z_p (prime p) with multiplication. Using a simple rule, you can make multiplication of integers behave in really unintuitive ways. Though two integers normally don't multiply to 1 (except 1 and -1), in Z_p there are plenty of examples where this happens. Also, Z_p is used to prove some important theorems in number theory, like Euler's theorem. When I took a Rings and Fields class too, we did a little bit on RSA Encryption and how modular arithmetic comes into play. Blew my mind. Bonus points: (x+y) = x + y in Z_p! (I know the field structure is required here as well as just the multiplicative group, but it's still a fun fact) | The finite simple group of order 1004913. It doesn't exist, but it's really annoying to show that | Linkin park | I like GL(n,R) cause i love linear algebra and you can both analyze this group (and it's subgroups) using algebra and using topology | SO(3) is my favorite. It has so many weird properties. It pretends to be a 3-sphere, but it isn’t. It’s a subset of projective space. And yet it’s so useful in engineering. really the first physical non-Euclidean manifold we’re all introduced too, yet it isn’t an n-sphere, it’s non-commutative, non-constant curvature, yet it has so many nice properties like compactness, connectedness, it’s a Lie group, and clearly it has so many physical interpretations. Not to mention it’s completely represented by the space of orthogonal matrices with unit determinant. | Absolute Galois group of Q. (related: My first thought was “Z_p doesn’t form a group under multiplication”. Then I remembered I’m a number theorist. Don’t forget to exclude 0, btw.) |
[
"Primes: Something maybe interesting. Can somebody approve or debunk this?"
] | [
"math"
] | [
"vg0joo"
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68
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""
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true
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false
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0.94
] | EDIT: I have read replies from everybody. To make it shorter: What I wrote is "partial" golbach conjecture. That means that if goldbach´s conjecture is false, my statement can be correct. A bit on lighter note. I guess I will be cheering for goldbach to be wrong. Just kidding. I would also like to thank every single person that contribute comment to this post. You people are very knowledgeable and you people know a lot. ________________ Hi math people, recently I was a bit bored. I was experimenting with primes a bit. This is what I got. I do not know if this is new, but in a case it is, I just want to share it here. So: For better explanation what I am trying to say: a.)Let us say: 34 We see that if we 34+3=37, and if we 34-3=31, I Both, 37 and 31 are prime numbers. b.) 402044 +63=402107, and 402044-63=401981 Our same is 63. And our primes are 402107 and 401981. I do not know this, this sentence is just a guessing, but maybe this distance number can always also be prime number. I am not mathematician. Sorry if I did not use some correct wording. I hope it is understandable. Thanks for possible reply. ________________________________ EDIT2:"I do not know this, this sentence is just a guessing, but maybe this distance number can always also be prime number." This sentence is not correct. It does not work for at least number 28 as some redditor pointed out. | This is a special case of the Goldbach Conjecture, a famous unsolved problem. If an even number 2n is halfway between the two primes 2n-a and 2n+a, then that's equivalent to the number 4n being the sum of those two primes. So your observation is the same as the Goldbach Conjecture for multiples of four, rather than for all even numbers. | Do you see why 2n being halfway between two primes is the same as 4n being the sum of those two primes? | Thanks for this. I thought that goldbach conjecture is only this :It states that every even natural number greater than two is the sum of two prime numbers. I did not know that it has other variants. | If 2n is halfway between two primes, then we can call those two primes 2n-a and 2n+a. Then they add up to 4n. If 4n is the sum of two primes, then we can call one of those two primes 2n-a and then the other is 4n - (2n-a) = 2n+a, so 2n is halfway between those two primes. | Correct. If it happens that Goldbach's conjecture is true for all multiples of four but not for some even numbers which are not multiples of four, then your statement would be true even though Goldbach's conjecture wasn't. I don't know enough number theory to speculate as to whether that is likely. |
[
"What are obscure textbooks that you have liked or think has good quality?"
] | [
"math"
] | [
"vfoqpj"
] | [
70
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""
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true
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false
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0.96
] | There are tons of textbook but sadly only few seem to get spotlight. For me those would be Viro's Elementary Topology Shirali's Multivariable Analysis Bhattacharya's Basic Abstract Algebra J Yeh's Real analysis(Ive only done first chapter) Feel free to suggest some books with short reviews preferably. Thanks. | Duistermaat and Kolk's multidimensional real analysis. Perhaps the theory portion isn't the greatest, but that selection of exercises is. Half the book consists of just problem statements from all kinds of different domains, like physics, the Riemann zeta function, complex analysis, differential geometry. Even browsing through the different problems is amazing. Bloch's the real numbers and real analysis. This is an introductory real analysis text, but what makes it stand out is its careful focus on foundations. It starts by giving the Peano axioms, and rigorously constructing Z, Q and R in a way no other book does (although Tao kind of gives the idea, but imo doesn't work it out much). The book then continues by rigorously going over the properties of the real numbers, including a proof that decimal expansions of the real numbers exist and are "unique". Then it continues to usual analysis topics, but even there there is a focus towards the foundations, for example he proves that the intermediate value theorem is equivalent to completeness of the real numbers, and there are many other neat bits like this. Carothers, real analysis. I just really love how this book does real analysis. He assumes you are familiar with real analysis on R and a tiny bit of linear algebra (vector spaces and linear maps, nothing more). Then he goes on doing metric spaces, function space and measure theory on R. There are a ton of problems which really help a lot in internalizing the theory. Freitag & Busam's complex analysis. Here are two books on complex analysis that I really like. Great collection of problems (most of which are solved in the back or with hints!), and nice theoretical development. Anderson & Feil, abstract algebra. Not too different from a standard AA text, but does rings first and only then groups. I always felt this to be better since for example quotient rings are more intuitive to grasp than quotient groups. Brannan's geometry: Does geometry from the point of view of Klein: covers affine and projective geometry (and motivates it), and hyperbolic, elliptic and inversive geometry and relates them to eachother using the language of group theory. The problem selection isn't the greatest though. Any book by Keith Kendig, especially his book on plane algebraic curve. He throws rigor out of the window so it's not meant as a standard textbook, but he does offer a damn lot of intuition and nice pictures! Cox, O'shea and Little's "ideals, varieties and algorithms" is in my opinion the best introduction to algebraic geometry. There is a heavy focus towards algorithms (for example Grobner bases) though. But he really gives intuition to why we should care about algebraic geometry and why some questions arise naturally. Silverman's "Friendly introduction to number theory". Like all of Silverman's books this is a gem. Not sure if it's obscure, but I really like Simon's comprehensive course in analysis, and Harthorne's "Euclid and beyond" too. | This is neither a textbook nor is it obscure. | The way Lebesgue theory is taught in Jones’ is fantastic! The first chapter actually deals with a full intuitive construction of the lebesgue measure starting with intervals in R then building through the with polygons, open sets, compact sets, then finally arbitrary measurable sets and the properties of the Lebesgue measure are slowly developed along th way. Then the second chapter demonstrates the invariance of the Lebesgue measure. Finally after all this build up you get to start with Lebesgue integration. I really love this exposure to the subject and the exercises are phenomenal and so enlightening! | Werner Ballmans' is one of the best references I've come across for basic complex geometry. | I am really fond of Azriel Levy's Basic set theory. If you are at a certain level of mathematical maturity I feel like its the perfect book to have a feel around the basics of set theory. For my tastes there is a little bit of a gap between "set theory for undegraduates" (Jechs introduction to set theory, Goldrei's book) and "real set theory" (Kunen, Jech), where the first one doesnt care too much that all of the proofs should (theoretically) have an equivalent deduction in ZFC and the second one is comfortable and careful enough with the material to just believe that everything has an underlying ZFC statement. As a consequence I never felt that I have solid ground under me when reading either, and Levy's book fits neatly into that gap between pre-rigor and post-rigor. |
[
"Primitive Sets"
] | [
"math"
] | [
"vfj8ap"
] | [
11
] | [
""
] | [
true
] | [
false
] | [
0.88
] | Recently I saw Numberphile's video about a conjecture from Erdos being solved about primitive sets, and I found the topic quite interesting. I'm a third year math undergraduate, and I was hoping to know if anyone has some material to read about the subject. Thanks in advance!! | Quanta Magazine covered the story about a week ago - their articles are usually pretty good entry points to deeper topics. Here’s the link: https://www.quantamagazine.org/graduate-students-side-project-proves-prime-number-conjecture-20220606/ [Side note: Quanta won their first Pulitzer Prize for explanatory journalism this year - the prize committee singled out Natalie Wolchover’s coverage of physics, but they could just have easily cited their maths coverage. The award is well earned.] | Yess, I love Quanta Magazine, but I’m lookin some material to study a little the topic, it’d be good if it had some theory, theorems with proofs, and I’d like perhaps some exercises, but I couldn’t find much study material on google | Ohh, nice, I didn’t think about checking the references, thank you!! | Google is a bit dominated by Lichtman’s work on the topic, where you could springboard off his references. This paper has a decent listing: https://math.dartmouth.edu/~carlp/primitive6.pdf | read all Jared Duker Lichtman work about primitive sets , he is fucking awesome! |
[
"What is the difference between a Binary Relation and a Binary Operation?"
] | [
"math"
] | [
"by82oz"
] | [
3
] | [
"Removed - post in the Simple Questions thread"
] | [
true
] | [
false
] | [
1
] | null | I've seen relations defined as functions into the set {1,0} before. This kind of definition isn't super common, but I don't think it is particularly unusual either. | A binary relation of two sets A and B is a subset of the Cartesian product of A and B. A binary operation is a mapping from AxB to C, and it's generally used to represent a type of mathematical operation (such as a multiplication or addition) in abstract algebra. | Couldn’t both be thought of as functions but the latter specifically returns a Boolean? | Couldn’t both be thought of as functions but the latter specifically returns a Boolean? | I don't think that's the reason (see dot product), I think it's because we don't usually consider True and False to be objects in our universe. |
[
"Please refresh my rusty mind and don't make fun of me: \"17 divided by 21\" can be written as \"17 ÷ 21,\" and \"17/21,\" right? Or is it \"21/17?\""
] | [
"math"
] | [
"by842j"
] | [
0
] | [
"Removed - post in the Simple Questions thread"
] | [
true
] | [
false
] | [
0.31
] | null | Hot damn. Yes, one of my attempts had a 0.8 in it or something like that. Move the decimal and there it is! Thank you. | 17/21 is 81%! | No problem! :) | Whenever you are trying to determine a percent, remember you have to multiple by 100. So if you divide 17 by 21 ( which is written as 17/21) you will see 0.809 on the calculator. You have to multiply that by 100 to get 80.9%. When you got “1.2%” for typing in 21/17 ( which is 21 divided by 17) that is actually 120%. | Thank you. Now that you've all refreshed my memory I'm going to memorize the following formula from now on because for some reason I use this a lot in daily life: Question: a is what percent of b? (Let x = "What percent") Answer: x = 100 (a / b) |
[
"Different Perspectives of Numbers: layman vs mathman"
] | [
"math"
] | [
"vfonz1"
] | [
234
] | [
""
] | [
true
] | [
false
] | [
0.86
] | No matter how detached one is from math, numbers are broad mathematical objects that always show up in every aspect of life. From counting how many bags of milk you need to buy at the grocery store (sorry canadian) to heuristically calculating the how much you will have to tip your waiter. Despite how frequently we informally use numbers in day to day life, in mathematics, numbers have very well defined definitions and understandings. The average person probably uses numbers exclusively informally. They really have no definition, they're just used to measure quantity of "things" ranging from money to kids to distances. They most commonly use \mathbb{Z} and \mathbb{Q}. The whole real line isn't necessary because irrationals have measure zero in a normal person life. Probably can be finitist because who really needs more than a morbillion numbers? The set theorist is probably the first person to come to mind when thinking about numbers. They view numbers as just sets of sets of sets ... of sets of the empty set. These guys are just logicians with number-looking-trenchcoats. They are probably the only people to deal with every single possible set of numbers because they're looking for counter-examples to the continuum hypothesis. They probably have such an abstract view of numbers that their algebraic properties are beneath them. Combinatorialists deal with finite sets and structures so they mostly work with \mathbb{Z} and maybe a little bit of \mathbb{Q} . Numbers are just [n], the set of n elements. These mathematicians probably have the closest understanding of numbers to the average person: basically they count things. Its not that simple though, because they have generatingfunctions, which imo is Mark my words though, once highschools start teaching instead of calculus, people will appreciate how cool combinatorics is. (I can just imagine the board meeting pitch; "Yes, I don't think our children need calculus, but trust me bro they need this math called 'generatingfunctionology.' yes I know it sounds stupid, but that's the name of the book"). Topologists strangely only have to know \mathbb{N}. It's either an n-dimensional space or n holes in a manifold or n-th homology group. You would think this is ignorant, but no! Topologists have seem to cleverly avoid using any larger set of numbers by solving all their problems using nice algebraic/homological methods which have nice natural numbers. General Topologists/ counterexample-topologists probably deal with more numbers. An analyst (anyone in functional, harmonic anal and etc) has a very concrete view of numbers as well. They are simply the scalars/objects of a field in their vector spaces. They measure things like measures and they metricize things like metrics. They do use algebraic extensions like the complex numbers for their nice algebraic properties, so for the most part they are just working with \mathbb{R} and \mathbb{C} . They're probably not dealing with a lot of interesting sets of numbers, but they're probably working with infinite dimensional spaces of numbers. An algebraist doesn't really think about numbers as describing quantities or things. They have the very categorical paradigm that numbers are only as important as the relationship to others. Hence they study numbers with binary operations. Numbers are only as important as their algebraic properties. Highkey doing crazy things in AG tho. Hurr durr, I like primes... queen of maffs. numbaphile. what a numbah? its just n + 1 of a anotha numberg. (I actually have no idea what number theorists do I just thought it would be weird to talk about numbers without including them. I'm sorry for being mean to you guys, but I seriously don't understand why you do what you do). Are there any other branches of math that have cool perspectives on numbers in general. | Saving this post just to see the dumpster fire in the comments over the number theory remark | I'm surprised you say topologists don't need more than the natural numbers. \mathbb{R} and [0,1] are two of the most importabt topological spaces. How do you find an n-dimensional manifold without reference to \mathbb{R} | Some things you‘re saying are just utterly wrong. | When I think in terms of topology, I don't think of \R as a set of numbers, but as a line / 1-manifold. Its elements are "points", not "numbers" and you can't do arithmetic with them because we're looking at \R only up to homeomorphism. | Differential geometry: 0 means I've just applied Stokes' theorem. 1 I don't care about. 2 is useful. 2pi means I'm making something an integer. 3 and 4 are cool, but 4 is too hard. 1/n! means I've messed up a summation convention or computed a volume. No other numbers exist. |
[
"My brother is actually dumb in math"
] | [
"math"
] | [
"by6d0h"
] | [
0
] | [
"Removed - post in the Simple Questions thread"
] | [
true
] | [
false
] | [
0.13
] | null | Lol what | How old are you (and how old is he), and did you ask him to go through his reasoning? Or are you just here to pwn your brother on the internet? | Well, show him the order of operations and use algebra. Note that 3x3 can be written as a or b, and then the equation simplifies to a-b Or write it in a calculator. | Sorry for forgetting to mention. The reason i posted this is because I am 17, he is 15 | And why does he think 3x - 3x is not 0? |
[
"How does the fundamental theorem of algebra justify this?"
] | [
"math"
] | [
"by66h2"
] | [
3
] | [
"Removed - post in the Simple Questions thread"
] | [
true
] | [
false
] | [
0.81
] | null | This does not constitute a polynomial. From the point of view of algebra, it can only be regarded as a formal series. Think about it this way, algebra can not have "infinite sums". Polynomials are algebraic objects which can be represented as sums, but those sums must be of finite support, which means that only a finite number of the coefficients of the polynomial can be different from zero. So the power series expansion of the exponential is not really a polynomial. | How do u express exponential as a polynomial function? | Yeah I didn't know polynomials had to be finite. Thanks! | Well I'm sorry if I annoyed you. | Well I'm sorry if I annoyed you. |
[
"What if I think I have a proof for a famous problem, can produce evidence, but havent formulated a rigours proof yet?"
] | [
"math"
] | [
"by3gw0"
] | [
0
] | [
"Removed - incorrect information"
] | [
true
] | [
false
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0.22
] | null | This kind of thread shows up once a week. The OP refuses to post their actual work, which is guaranteed to be invalid because they don’t actually study mathematics. You could at least tell us what open question you’re working on. | Be honest with yourself that the chances are overwhelmingly high of your proof being incorrect. This site regularly gets people with minimal background posting about having potentially solved a famous math problem (Goldbach's conjecture, the 3x+1 problem, etc) and being worried about the idea being stolen. And when the person is convinced to explain the argument it has ALWAYS turned out the proof is wrong for an elementary reason. If, as you write, you want your proof to be proven wrong then just post it here (by a link if needed) and you will get your wish. The other day someone posted here asking their proof of Goldbach is wrong rather than saying they had really proved it (see https://www.reddit.com/r/math/comments/bxl7y4/where_did_i_go_wrong_in_my_proof_of_the_goldbach/ ) and that attitude is a much better one. | Having some numerical evidence can still be very far away from having an actual proof. Share your work here if you want specific feedback. | I've produced alot of evidence using computers As has everybody else who seriously tackled this problem. The evidence I have corresponds with exactly what a rigorous proof would imply, and there doesn't appear to be any reason that the proof is false In other words, you don't have a rigorous proof and your evidence hasn't produced a counterexample. Just like everyone else who studied this problem. For something like the Riemann hypothesis, for example, computer searches have been used to compute trillions of zeroes thus far, and every one of those agrees with the conjecture. Yet, this is not a proof. | . I cant share my proof here because that risks exposing my method, which I really dont want to do. It is extremely unlikely you have a method which no one has thought about for a famous unsolved problem. If you do post here, and your idea is original, no one is going to steal from you. There will be a permanent Reddit thread being a useful witness that the central idea was yours. |
[
"What does ln^2(x) mean"
] | [
"math"
] | [
"by2akj"
] | [
0
] | [
"Removed - post in the Simple Questions thread"
] | [
true
] | [
false
] | [
0.29
] | null | It's context dependent. It probably means (ln(x)) Can you share the context with us? | It's almost certainly (ln x) | what does the side bar mean? | From my experience in computer science, people typically say "log log n" for log(log(n)) and "log squared n" for (log n) . However, if you're writing something and have the opportunity, it's probably best to either use more explicit notation or explicitly state which you mean. | Unfortunately, your submission has been removed for the following reason(s): /r/learnmath books free online resources Here If you have any questions, please feel free to message the mods . Thank you! |
[
"Is an infinite straight line impossible?"
] | [
"math"
] | [
"bxzxzm"
] | [
0
] | [
"Removed - post in the Simple Questions thread"
] | [
true
] | [
false
] | [
0.5
] | null | Probably the person has an incorrect understanding of what "theoretical" means. | Mathematics allows for infinite straight lines. | Okay, thanks. Any idea where the idea that its impossible came from? | The surface of Planet Earth is a sphere and so you cannot draw an infinite line or plane on the surface of Planet Earth? | Yeah, I said that to this person, but they kept stating it was a purely theoretical idea, not based on actual events. I'm just going to assume that they misunderstood something, as they could not give me any explanation anyway. Thanks though |
[
"isomorphic"
] | [
"math"
] | [
"bxwxna"
] | [
0
] | [
"Removed - post in the Simple Questions thread"
] | [
true
] | [
false
] | [
0.4
] | null | If V is finite dimensional, then it's true. If the dimension is infinite, it's false. Edit: Because, if the dimension of V is infinite, f can be injective and not be surjective (so not an isomorphism) . | Thank you very much | Do you give me an example? | Sure, Take V as the vector space of the polinomials (in R) of one variable. And define the library function f(p(x))=x*p(x). It is really linear and injective, but it's not surjective because it doesn't reach the constant polynomials. | Unfortunately, your submission has been removed for the following reason(s): /r/learnmath books free online resources Here If you have any questions, please feel free to message the mods . Thank you! |
[
"Is or was anyone here \"hiddenly adept\" at math in highschool?"
] | [
"math"
] | [
"bxxei9"
] | [
0
] | [
""
] | [
true
] | [
false
] | [
0.5
] | null | If you are studying advanced topics for your age, it's understandable that you wouldn't find very many like-minded peers local to you. I would recommend looking into online communities such as the Art of Problem Solving forums or trying to find math circles in your area. Find out what other math offerings the local community college has, and if you've already exhausted them, maybe look into state universities close to you. I would also encourage attending events such as state and national math contests . If you are interested in research or competitions, there are very many amazing math summer camps available for high school students to find communities. | I seriously don’t get the point here. You’re just quoting topics you’ve read about - many many people in the world know these topics inside and out, they’re second year undergrad topics here in the U.K. It’s cool that you’re interested in higher level math and wanting to do something with it. Get yourself into a good university and go from there. Don’t make self congratulatory posts on reddit - it doesn’t come off well. | Many people are good at math and unfortunately no one really cares until you come up with something new and novel or use it to make money. It's certainly natural to want to be recognized for your talents and I feel that's what you are venting about here. I learned that, since schools and society put so much importance on math and use it almost exclusively to define intelligence, people get jealous when you understand math better than them. Extremely jealous. You may want recognition for your abilities, but from your peers who struggle with math, all you'll get is jealousy. For this reason, math and science are extremely solitary pursuits. There's a reason for the shy, loner nerd stereotype. It's a difficult life, but if you are really passionate about math, it's worth it. Cause what feeling is better than that feeling of mathematical understanding? You are naturally going to want to share that feeling with everyone, but you can't just talk about high level math with anyone out of the blue. People will disengage with you immediately. You just need to be patient. Keep studying for yourself and not to prove to anyone that you are smart. If you do, you can then go to college, and there you will meet others more like you who share your passion. | The word you're looking for is "secretly". Currently I am studying Galois theory from Milne's course notes and diff geo from Lee, and I've written over a hundred pages of (semi-decent) mathematical exposition about my thoughts learning along the way. Can you solve the problems? That's the only metric by which one is to measure success in a science. I feel like I am misunderstood since I am considered someone who gets top grades in their math classes, but no one would know/care/believe that I am capable of doing higher math. You're not misunderstood if you're going out of your way not to reveal what you've learned in self-study. You know how being secretly Superman does fuckall for Clark Kent? I'm just venting. Why? Why is your nose out of joint about the people around you not divining that you are studying some advanced topics on your own? What do you believe you deserve? If you believe you are exceptional at a topic but you do nothing to explore that and expand on it, what good does it do you? Here's a hard life lesson that you would do well to learn: if you wait for other people to tell you that you're special, you'll live and die ordinary. Unless you are a prodigy - and you're not, because you would definitely know it if you were - you need to seek out opportunities to expand yourself. Getting an A in college calculus is not going to inspire a professor to invite you to be his youngest research assistant ever or whatever fantasy you might be holding on to. YOU need to take control of YOUR life. Or just be ordinary. The world needs ditch diggers too. | I feel like I am misunderstood since I am considered someone who gets top grades in their math classes, but no one would know/care/believe that I am capable of doing higher math. I wouldn't say I am to anyone, as it would just come off as needless bragging. You will be misunderstood if you allow people to misunderstand you. I too felt that I was more proficient at math than many people in my graduating class (hehe, though seemingly not nearly to the degree as yourself, having looked through your expository notes--I myself simply liked computing lots of definite integrals!) and did kind of feel similarly; but, at the same time, I did go out of my way to assert to both teachers and classmates alike that I loved and enjoyed math. This wasn't done in a "bragging" kind of way... it happened fairly naturally. I'd come to my AP Calc teacher after class, say, and show her a random definite integral I was working on. Or, I'd use a method on a test that we weren't taught and still get the problem right. And then, I can't even count the number of times I'd be sitting in silence working on a problem unrelated to class and my classmate next to me would express interest, and so I would give an honest attempt to explain it to them. If you act in this way, there will probably be some people who think you are being pompous or whatnot... we humans are complex creatures, and I cannot summarize all the possible reactions reasonably in a single sentence. If you are genuine, those people are not worth your attention (easier said than done?). However, people who care about you will notice your passion, and encourage and support you. At least, that's what happened in my situation. You also need to recognize, as I often did not, that you probably know many specifics that your high school teachers either do not know at all/have heard of vaguely, or learned once but are no longer up to speed on the details. For example, I doubt my subpar algebra II (to clarify, high school, NOT abstract algebra, so fair enough...) teacher knows what an ideal is. I'm not trying to generalize per se... like, there are many high school math teachers who still actively do mathematics in their spare time and further their education. But not all of them. Anyway, this applies to everyone, not just math teachers. When you go to college, besides the fact that you will most likely excel, you will also find yourself in a different boat. There be more people like you, and at that point it is up to you to talk, ponder, and collaborate with them to further your passion and knowledge of mathematics. EDIT: Also, I'm not suggesting that you explain, say, Galois theory to some random person; they will quickly realize that they probably don't care, and aside from that it is very time consuming to explain "higher math" topics to people who are not versed in the frameworks that math often uses. But actually, even so, if you are enthusiastic in your explanation it goes a long way in kindling the listener's interest, especially if they are again someone who cares about you. And that's not even mentioning that it is essentially a valuable exercise to try to explain abstract things in a more concrete way. P.S. if you ever want to talk about any of this stuff more, please P.M. me. Would be happy to chat. |
[
"Ok,I’m bout to sound dumb as shit"
] | [
"math"
] | [
"bxvuk8"
] | [
0
] | [
"Removed - post in the Simple Questions thread"
] | [
true
] | [
false
] | [
0.5
] | null | No one will judge you for being inquisitive in 8 grade, but try to avoid using language such as "retarded" when asking academic questions. You are correct. Perhaps discuss this with your math teacher and ask why they believed it was 8. They could have misread the notation. | but try to avoid using language such as "retarded" when asking academic questions. Perhaps it's time that us academics break through this retarded formality. | It’s -8. -2(-2) = -2*4 | A negative number times any square will always be negative (because the square itself will always be positive). [This assumes no imaginary numbers] Your problem is a negative number (-2) times a square (namely the square of (-2)) so the result will be negative. | Unfortunately, your submission has been removed for the following reason(s): /r/learnmath books free online resources Here If you have any questions, please feel free to message the mods . Thank you! |
[
"Terrance Tao Caught 'Mind Streaming' Nuclear Prime Numbers to Chinese Government"
] | [
"math"
] | [
"bxse9e"
] | [
0
] | [
""
] | [
true
] | [
false
] | [
0.1
] | null | If you are going to make up insane fake news stories about Terence Tao, you could at least try to spell his name correctly. | If the US military is using 499, 503, 509 for cryptography then Terence Tao is the least of their worries. | Link won't work | Is this satire? I really can't tell, world's gone mad. | I usually don't do this, but:🤡 |
[
"The best way to fill a square with diamonds"
] | [
"math"
] | [
"bxq4nx"
] | [
2
] | [
"Removed - incorrect information"
] | [
true
] | [
false
] | [
0.75
] | null | Isn't there only one way? You just tile it. Then you can move your tiling around but that won't have any effect on the end result. | Squares, but ones whose sides make 45 degree angles with the sides of the big square. | Rhombus? | Yes, but the angles are all 90° i know this is a square but i mean squares rotated 45° | Essentially you just put one in the corner, add neighbors along the top edge, add more neighbors under them to make a second row, etc, until you've filled all the space you can. There's really only that one way. You don't end up filling the whole square, though. The question that's more interesting is, how many ways are there to do it with some set of distinct sizes of diamonds? |
[
"Question that has me curious"
] | [
"math"
] | [
"bxpgqr"
] | [
1
] | [
"Removed - post in the Simple Questions thread"
] | [
true
] | [
false
] | [
0.99
] | null | By squaring a+b=c, a + b + 2ab = c But by the first statement, we are left with 2ab=0. So there are no solutions with both a,b>0 | a + b = c = (a + b) a + b = a + 2ab + b 0 = 2ab No. | Random question. I may just be stupid but for your first statement, what if I had a 3,4,5 triangle? (3+4) gives 49 but c would be 25. | You are correct: that would be a contradiction. However, my statement was formed by the substitution c = a + b that the question initially asked. I was assuming initially that such an a, b, c exist and showed that they do not. The assumption was not necessarily true. | I understand now. Thanks |
[
"Trying to figure out an equation"
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"math"
] | [
"bxmt7a"
] | [
0
] | [
"Removed - see sidebar"
] | [
true
] | [
false
] | [
0.33
] | null | Yeah, probably. Brand new to Reddit really, figuring it out | Turning to the sidebar: If you're asking for help learning/understanding something mathematical, post in the Simple Questions thread or /r/learnmath . It's probable that this post will be downvoted quite badly and you deserve at least an explanation. | That's fair, it'd get tedious to always read the sidebar of a new subreddit in the early days for sure, when every other page you might wanna post on is a new subreddit to you, but it's definitely good practice, especially if you want the precious precious Karma. | Makes sense for sure. Good to know. | Unfortunately, your submission has been removed for the following reason(s): /r/theydidthemath If you have any questions, please feel free to message the mods . Thank you! |
[
"Math degrees aren't only for teaching math."
] | [
"math"
] | [
"by2b6m"
] | [
0
] | [
""
] | [
true
] | [
false
] | [
0.5
] | [deleted] | I thought math degrees were for people who wanted to know what dying at 50 of chronic stress is like | the people here know that you can do more with a math degree than teach. No i dont | You mean I have 25 more years of this shit? | Most people don't know anything about what studying in math is like or what people do with it (I doubt math is the only field that suffers from these kind of problems but we definitely get a unique set of reactions). It's something you have to deal with. If you're talking to a random person who you won't see again and really want to avoid playing the standard game of bingo just say you major in something else. | So... this isn’t a place to talk about your experiences concerning math? I am intrigued because I also just switched to a B.S. in Mathematics with a Data Science option from CS because I was falling asleep in every single CS class and can relate to maths being more enjoyable then another profession thought to be much more lucrative and receiving skepticism for it. |
[
"Problem on ellipses and elliptical integrals"
] | [
"math"
] | [
"bxwrak"
] | [
3
] | [
""
] | [
true
] | [
false
] | [
0.8
] | [deleted] | My intuitions tells me there is a way of doing this using only a single integral, instead of one (or two) per bin, but I cant put my finger on it | Can you draw a picture? | Yes, sorry i'm still searching how to do this on reddit | Done. | For an infinite number of bins, I'd have thought the arclength of the ellipse in each area of the bin would be proportional to the dot product of the unit tangent vector of the ellipse with the vertical vector in the plane... |
[
"More optimal solutions to the Numberphile Cat and Mouse problem"
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"math"
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"bxywux"
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232
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""
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0.97
] | [deleted] | Waits an arbitrarily short amount of time That time would be epsilon, so we could assume it's zero. The very instant the mouse start moving, the cat would change direction. EDIT: Since this is the top comment right now, I'd like to clarify: OP is right in that this strategy works. It was just sloppily worded. It would be better, instead of the mouse waiting for the cat to move, have the mouse immediately dash away from the cat. As soon as the cat starts moving, apply OP's strategy. Even a much simpler strategy of moving away then immediately left/right, would allow for a K of about 4.58. | Its a good approach, but the elegance of the solution presented in the video comes from the fact that is is successful independently of how the cat behaves. The strategy you presents is cool but it assumes that the cat chooses one particular strategy, namely minimizing the angle. To find a real solution to the problem you would have to assume the cats strategy as given and then show that you can find a strategy for the mouse that leads to success for every possible strategy the cat could choose | When (if) the cat decides to switch directions, the mouse has travelled a nonzero radial distance from where it started. From this point, the mouse can respond by switching directions and has gained an advantage. | You came up with the same answer as this guy did | If it's 0.0001 units of time, then the cat can move 0.00005 units left, then 0.00005 units right. Whatever time the mouse is waiting for, the cat can move back and forth in half that time. |
[
"If every odd number can be seen as a sum of three primes, then why can't we just \"subtract\" a prime to prove the Goldbach conjecture? Is it because the properties of even and odd numbers are too different?"
] | [
"math"
] | [
"bxmxqi"
] | [
0
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""
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true
] | [
false
] | [
0.5
] | [deleted] | then why can't we just "subtract" a prime to prove the Goldbach conjecture? How do you know that you'll get even number this way? | This is what I came to say, as well. | Counterexample: Let "oddish numbers" be the set of all positive odd numbers except 5 and 7. Every positive odd number > 1 can be written as the sum of three oddish numbers, but 8 cannot be written as the sum of two oddish numbers. | What are the Vileplume numbers then? | There are no Vileplume numbers, but the Vileplume Delta is an operator that takes in a vector as an input, and releases lots of pollen as a result. |
[
"Is a 1x1 matrix tridiagonal?"
] | [
"math"
] | [
"bxtqqy"
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4
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""
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true
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false
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0.59
] | null | Yes. | The theorems about tridiagonal matrices are all of the form "_ can be computed quickly". So 1×1 matrices would definitely satisfy them. | Yes. A tridiagonal matrix can have nonzero elements only on the main diagonal, the first diagonal below this, and the first diagonal above the main diagonal. In [ a b ] [ c d ] the main diagonal is (a,d), the first diagonal above this is just position b, and the diagonal below the main one is position c. These positions can have any values for a tridiagonal matrix, and for 2×2 those are all positions, so any 2×2 matrix is tridiagonal. | and so are 0x0 and 2x2 matrices. | That's not how math works... Regardless of theorems, a 1x1 matrix is tridiagonal by definition. |
[
"\"Who's more likely to claim [mathematical] expertise? Are you familiar with proper numbers? How about declarative fractions? If you claimed that you knew these concepts, you're probably fooling yourself\""
] | [
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"by2ve2"
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false
] | [
0.73
] | null | Makijg sure that the nonexistent terms were actually nonexistent must have been an interesting job. | The thing is, actual mathematicians are going to assume that whoever made up the test is really trying to gauge mathematical knowledge but uses idiosyncratic terminology. To help the test-maker, instead of giving a passive aggressive 'no' about something like declarative fractions, he/she is going to try to un-twist whatever terminological error the social scientest test-writer has made deciding e.g. maybe it refers to rational functions over the reals. There's no possibilty that such a phrase refers to an area of knowledge he/she hasn't learned about yet. In that sense, the test-writer screwed up. The terms had to be things that could have sounded like legitimate areas of math/science even to an expert. The issue is similar to how if you ask doctors, "Have you heard of shaky-hand syndrome" they'll think, this must be a colloquialism for parkinsons syndrome and answer 'yes.' If you ask a doctor, "Have you heard of old-timer's disease," he'll assume you mean `Alzheimer's disease.' Not to be pretentious, but to try to avoid allowing the test-writer's own lack of expertise to ruin their own test. | That's what I was thinking. You know someone out there has a Master's thesis where on page 14 they've got An integer n is said to be if... I'm sure that in spirit it's reasonable to find something that should be considered nonexistent to a layman, but it's still a bit strange. | Well, if I was asked on the street about "proper numbers" a chance to clarify what they mean, I would think they probably meant rational numbers or something. Lol. | Aside from the flaw of the fake term "proper numbers" potentially sounding to some people like "proper fractions," a legit concept for fractions, the full list of math concepts that the authors used (see page 8 at http://ftp.iza.org/dp12282.pdf ) includes "congruent figure," which they treated as a real math term but it is strictly speaking nonsense in the same way as "equal number" or "isomorphic group" since the congruence relation is between two objects at a time, not just one; you can't say a particular triangle is a congruent figure by itself. (See the answer by Asaf Karagila at https://mathoverflow.net/questions/53122/mathematical-urban-legends/53288 for an amusing instance of such incorrect usage for isomorphisms.) This whole research project on BS seems like BS itself. |
[
"Discussion: what's the best way to represent a well-connected network of pre-requisites? (Specific example of RPG quests)"
] | [
"math"
] | [
"bxz205"
] | [
0
] | [
""
] | [
true
] | [
false
] | [
0.5
] | [deleted] | It sounds like you want to topologically sort the graph. If you sort the graph from left to right with left being starting quests and right being late game quests, this will make sure that nodes have no dependencies except nodes that are further to the left. Algorithms on how to implement this are everywhere on the web, I’m not sure how the outcome will look, but if you are bent on using a graph it is probably the best option. | I may be wrong, but this reminds me of directed acyclic graphs (DAGs). I would guess that visualising these is well-studied, but maybe not under the constraints you have in mind. For example, Git (a type of version control system for computer source code) stores commits (a single unit of code change) in DAGs. The notion of in your case would be analogous to in Git. For an example of how this DAG is visualised, see something like Sourcetree . I'm not sure if that sort of layout fits your criteria though. | Is there a better way to organize this information that's more accessible than the image of a giant directed graph or a table that doesn't show the connectivity of the quests? Since we can assume there is no cycle of quests that have each quest depend on the previous quest in the cycle, we can sort the quests by "longest distance from root to this quest" in increasing order and display these quests in "tiers" with the same value. I'm thinking something like the tech tree from Civilization IV . (UI-wise, of course, OP would probably want to highlight all the quests that can be done given the player's current progression, and clicking on an uncompleted quest should tell the player what quests they can currently complete that are a prerequisite of said quest.) | Is there a better way to organize this information that's more accessible than the image of a giant directed graph or a table that doesn't show the connectivity of the quests? Since we can assume there is no cycle of quests that have each quest depend on the previous quest in the cycle, we can sort the quests by "longest distance from root to this quest" in increasing order and display these quests in "tiers" with the same value. I'm thinking something like the tech tree from Civilization IV . (UI-wise, of course, OP would probably want to highlight all the quests that can be done given the player's current progression, and clicking on an uncompleted quest should tell the player what quests they can currently complete that are a prerequisite of said quest.) | I'm not hard set on the idea of a graph, but it seems the most obvious to me. Tables clearly don't work since they're only good at sequential sorting instead of relationship sorting like a 2d graph could do. |
[
"Looking for advice from an expert in distribution theory"
] | [
"math"
] | [
"bxv9gk"
] | [
0
] | [
""
] | [
true
] | [
false
] | [
0.4
] | I have a particularly involved question about distribution theory and i'm curious if I could be pointed towards a professor willing to answer some inquires, which will help me understand a certain concept from distribution theory (essence of my question). I'm looking to establish email correspondence, and maintain a short discussion there, but I can't seem to find any with the time to help. Any ideas? I've already gotten in contact with my local math department at my college, but i'm not getting much help over there. I'm an engineering student, so it is more difficult to get help when i'm not enrolled with any math classes, and the engineering department doesn't seem to care about the mechanics of the logic, but rather the logics application. I'm quite frustrated and feel like I have no options. Thank you! | Post the question here | What do you mean by distribution theory? If it's in the context of PDEs or Fourier analysis, I may be able to help. However, I'm not doing this over email. | Go to math overflow | Or math.stackexchange if it is not a research-level question. | Just ask the question. |
[
"Fun problem I thought of: given a unit circle, divide it into N equal parts by intersecting it with N-1 other circles, such that the total area of all of the circles used is minimized."
] | [
"math"
] | [
"bxvltu"
] | [
33
] | [
""
] | [
true
] | [
false
] | [
0.84
] | Note: I don't mean the total area that their union covers on the plane, I mean the sum of their areas taken as separate circles. Though, it would be interesting to see what changes between those two variants. I am thinking of trying to figure out how to code a tool to allow me to drag circles around and calculate areas of their intersections on the fly, so that I could explore this myself. If anyone else is interested in the problem (or if it's already a known thing that's been solved!), I'd love to see your thoughts - and perhaps some pictures. :) Oh, and another variant that might make it a bit harder would be to restrict it to only circles such that none of them strictly contains any of the others. It was pointed out that the N-1 circles constraint means they can't overlap. Dumbass that I am I totally failed to think about this fact. So scrap that constraint altogether. Just focus on the number of distinct internal regions of the unit circle into which the arcs passing through it from the other circles divide it. | Are the N-1 circles allowed to overlap? If they are then how do you define the equal parts when multiple circles overlap? | If only the area of the intersection region between the initial and each N-1 circle matters then it is a little simpler. So the area of the unit circle is just pi. Let A be the intersection area between it and one of the other circles. When pi -(N-1)A = A then you will have a solution. So A=pi/N. | If circles overlap, then they form too many regions, so either they are fully contained in each other or they have disjoint interior. Now I have a question about tangent circles. Suppose I place at (1/2, 0) and (-1/2,0) circles of radius 1/2. Would this circle necessarily be considered 4 parts or am I allowed to call the top and bottom pieces one part? | Indeed, but the question of course is, what is the optimal way to do this so that the total area of all the circles is minimized? I feel like something I'm saying is getting lost in translation here... | I didn’t realize it was to minimize the total area of the actual circles. My bad. That paper you posted is probably too advanced for this problem. I recommend checking out this webpage: http://mathworld.wolfram.com/Circle-CircleIntersection.html |
[
"How does mathematics and define characterize risk?"
] | [
"math"
] | [
"bxnyrl"
] | [
6
] | [
""
] | [
true
] | [
false
] | [
0.8
] | /I've been mulling over what I'm interested in for my PhD. Uncertainty quantification and risk analysis seems to be my cup of tea. Afterall, everyone wants to have some certainty in the future. I get the basics of how to quantify risk, like classical statistics, p-values, standard deviation and the like. But I'm sure there's more complicated risk measures for more complicated systems. For instance, I recall in my experience a research needing to characterize correlations in a graph network, and the pearson correlation not being enough to characterize the network. They needed a new statistical measure to characterize a better ranking system to see what most likely is the most influential factor. Are there "cutting edge" or more sophisticated ways of looking at risk? Also, recommendations for books or resources can help. Thanks! | Hi, I think a person could make a really valuable contribution to the theory. Starting with Bernoulli there was a realization that risk isn't only about expected wealth --- Bernoulli suggested that people act, or should act, to maximize the expected value of the logarithm of wealth. A modern example would be, an advisor shouldn't advise a retired person living on a pension to invest the whole pension on a bet with probability 50% of adding again an additional 1 dollar plus again the value of his pension and 50% of losing his whole pension, even though the expected wealth is larger. The expected value of log(wealth) is -\infty. Later analyses of Bernoulli's idea were that you should divide an asset into small pieces and invest them in statistically independent ways with as high expected gain as you can find, then the two concepts of expected gain coincide in the limit. A wonderful book by a non-mathematician named Lawrence Macdonald -- and he leaves the .pdf file of his book online for free somewhere -- explains how mathematicians would come down to him and challenge his work at Lehman brothers. He had bought distressed Delta shares after a colleague had told him that the value of the planes on the ground equates to 50c per share. The mathematicians were worried that there isn't enough statistical independence, but they allowed a large chunk of Lehman's investement to be junk property bonds, as long as they were in different areas of the country. In other words, although he doesn't say it explicitly, Macdonald thinks that a mathematically incorrect definition of risk caused the 2008 financial crash. There are similar stories about Kweku Adoboli and others, where the definition of risk becomes crucial. The reason I'm spending so much time about this is, people are trying to use a theory of risk to assess risks of allowing types of scientific, economic, biological development, and to try to evaluate things like environmental risks of eco-engineering projects, many of them well-intentioned, to try to ameliorate consequences of things like global warming. A group of more than 100 Nobel-prize-winning genetic engineers wrote a letter complaining that people mis-understand the risks of releasing GMO's. Their letter said that ignorant people without Nobel prizes incorrectly worry that there might be a risk somewhere, but that they are sure that the risk to Nature is nonexistent. Having mathematicians doing something in any way different than they were doing at Lehman might be a start. When you look at what mathematics does in understanding atoms and wave functions, cosmology, and coming up with an interpretation of things in the universe, it is clear that the types of calculations that led to particular small tragedies, like financial tragedies, can be blamed partly on mathematicians accepting a pragmatic here-and-now real numbers/linear regression interpretation of life, space, geometry, and meaning itself. A difficulty is, if you say "I want to be a mathematician who studies risk in a meaningful way, using the full range of understandings that mathematicians can have," what you are saying is, "I want someone to give me a salary to study risk in a meaningful way..." There are particular salary pipes sitting right there. Actually I don't know why there aren't more people like Chomsky who have a competent understanding of math/linguistics and are willing to make political statements. A big worry is that in some examples there seems to be almost no benefit to having a deep mathematical understanding. Riemann was a devout Christian, for example, saw his understanding of the relation between numbers and continuity as only a way of servicing his responsibility to God or Jesus or something like that. He was similar to the president's spokesperson, Sarah Sanders, replying to concerns about global warming, "I'll appeal to a higher authority for that, thank you," meaning, we should trust the Bible for an assessment of any risk that there might be a huge tragedy in the future. The bible says, there just will be, but not to worry about it because it will not be a problem for good people. Also, whereas micro-economics is a sensible subject, Macro-economics like Keynes work is being rejected with no comprehensible replacement having been found, mainly because it ignores things like environmental damage and environmental risk. There is a worry that any replacement would be a 'fringe' subject, or would be a 'corporate' project. There is an intuitive sense that mathematics can encompass meaning in ways that could help somehow, even though this has not turned out to be true yet except in irrelevant or distant things like calculating the colors of stars. | Maybe not entirely what you are looking for but in game theory we classify agents into three groups. Risk seeking risk avoiding and risk neutral. They correspond to concave convex and linear utility functions respectively. An example is a risk neutral agent will value a deterministic fifty cents the same as getting a dollar half the time and none otherwise. Whereas a risk seeking agent values the dollar more than twice that of the fifty cents and would prefer to gamble. | I never really understood why the word "risk" has become such a boner in the finance industry when it's essentially describing probability. You can call yourself an Uncertainty-Quantification-Accredited CFA Corporate Risk Management Specialist Executive all you want, but it doesn't change the fact that the entire financial industry is essentially predicated upon people giving you their money, and, likewise, borrowing money, on disadvantageous interest rates. Banks have all their little departments on different sectors and trading desks, but they really just diversify into everything at the end of the day. Nonetheless, the term "risk" basically encapsulates anything to do with probability, which in-turn encapsulates all of financial mathematics. So it's a broad topic. There are many inlets into the study of probability. It's expansive and non-generalizable, analogous to mathematical logic or number theory. Proper mathematical statistics and probability theory form the theoretical foundation. Probably the most fruitful developments with respect to finance are stochastic analysis (from which we can derive derivative assets pricing like Black-Scholes) and, more recently, machine learning - which is generating largely arbitrary statistical blueprints and running them through optimization. | Oh, hmm, interesting. | fortunes formula is pretty good https://www.amazon.com/Fortunes-Formula-Scientific-Betting-Casinos/dp/0809045990#reader_0809045990 also the theory that would not die https://www.amazon.com/Theory-That-Would-Not-Die/dp/0300188226 |
[
"Separable Equations"
] | [
"math"
] | [
"bxr2z5"
] | [
1
] | [
""
] | [
true
] | [
false
] | [
0.67
] | Hi guys, I'm currently taking Differential Equations this semester and I have a theory question that my professor wasn't able to answer. So far I've learned that some differential equations can be solved using the power series. We came across a separable differential equation problem in class and we were able to solve it using the geometric series instead of the power series. So now I'm curious if all separable DE's can be solved using the geometric series. I've googled this question and nothing has come up so I'm hoping that someone here can answer. Thanks guys! Happy finals season. | y=0 is geometric. | What do you mean? The solution to y'=y, y(0)=1 is y(t)=e , and I wouldn't call that the geometric series. | I teach ODE, I'm aware of power series solutions, e just can't reasonably be described as the geometric series | Lol, okay, fair enough. | How so? |
[
"What Are You Working On?"
] | [
"math"
] | [
"bxwg4s"
] | [
49
] | [
""
] | [
true
] | [
false
] | [
0.89
] | 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! | Said the algebraic geometer | Wrapped up my masters thesis today. Next step is to prepare the oral examination. | lmao imagine being this much of a nerd | Studying for final exams. ;-; | Continuing with Burago, Burago, Ivanov, but more importantly trying to find other ways to fill my time other than math. |
[
"What is/are your favorite math YouTube channels?"
] | [
"math"
] | [
"bxpkoy"
] | [
3
] | [
""
] | [
true
] | [
false
] | [
0.61
] | I need some recommendations. | I like Mathologer the most. I appreciate the no-frills approach and the clarity of explanation. Many people like 3Blue1Brown, and you probably will too. He's not for me (there's something about his choice of aesthetics that I find tedious), but I have respect for his work, and he's deservedly popular. Numberphile is of course well-known. I find that their quality varies. I can't quite forgive them for poisoning a crop of interested math students with the -1/12 nonsense; that was kind of irresponsible, and the right context should have been better explained. | Numberphile is of course well-known. I find that their quality varies. I can't quite forgive them for poisoning a crop of interested math students with the -1/12 nonsense; that was kind of irresponsible, and the right context should have been better explained. I like Numberphile and all but I feel like a more insidious problem with them is that they sort of embrace and perpetuate the ideas that: a) math is almost always fundamentally about numbers and b) math is sort of gleefully useless. Obviously the difficulty is that those go hand in hand with the kind of accessible math that's easy to popularize, so I really don't know what to do about it. | 3Blue1Brown is by far the best. | the IAS's YT channel is p lit, so is CIRM's. | In no particular order: |
[
"Does this argument convince you that ZFC is inconsistent? (the least ordinal containing all countable ordinals defined by a formula is an element of itself)"
] | [
"math"
] | [
"bxuyzm"
] | [
0
] | [
""
] | [
true
] | [
false
] | [
0.21
] | Here is a link to a longer argument, in the style of a rigorous proof. | Value is an ordinary English word. The discussion makes direct reference to the cardinal aleph_1. When you formalize that discussion into the formula 𝜓, the formula 𝜓 makes reference to aleph_1. More precisely, since 𝜓 is in the language L(∈), 𝜓 includes some description of aleph_1 expressed in this language. Then, when you interpret 𝜓 in the structure (aleph_1,∈), this description of aleph_1 ends up meaning something else (i.e. has a different value). Depending on the formalization, it might end up defining nothing, or it might define some countable ordinal, but it definitely doesn't still refer to aleph_1, because aleph_1 isn't an element of (aleph_1,∈). So the formula 𝜓 ends up defining a different element (or no element) in (aleph_1,∈). | No, it doesn't. You've shown that 𝜓 uniquely defines an element in (V,∈), but then you assume that it uniquely defines the same element in (aleph_1,∈), which of course it doesn't (since 𝜓 actually depends directly on the value of aleph_1 itself.) Taking the definitions you've written literally, I think 𝜓 will probably simply fail to define an element in (aleph_1,∈), but tweaking the definitions in reasonable ways, it might end up simply defining a smaller element in (aleph_1,∈) than it does in (V,∈). | I think 𝜓 will probably simply fail to define an element in (aleph_1,∈) It's more than this. The supremum of the ordinals that are first-order definable (without parameters) in the structure (aleph_1,∈) is definable in ZFC (in fact it is precisely omega ), but there's no way to even formalize this definition in (aleph_1,∈) itself. | By Tarski's undefinability theorem, first-order logical systems can't completely define truth. In particular, you can't actually construct the set of all P(x) such that P defines a countable ordinal. The only way you can do this is to work in the meta-theory, and talk about which countable ordinals are definable in a particular model. But in this case which ordinals are definable will depend on the model - it's even possible that of a model's countable ordinals are definable, since uncountability within the model does not imply uncountability in an "external" sense. | I'm a bot, , . Someone has linked to this thread from another place on reddit: /r/badmathematics ZFC is Inconsistent Info Contact |
[
"Intuition behind e^{ix} =cos(x)+isin(x)"
] | [
"math"
] | [
"bxsa6y"
] | [
315
] | [
""
] | [
true
] | [
false
] | [
0.94
] | I know this is true, and I understand the proof, but I am having a hard time understanding WHY it is true. Try as I might, I can't really find the connection between trig (ie circles and angles) and e (exponential growth and calc) | The whole purpose of e is to be a function that is its own derivative. So the derivative of e is ae by the chain rule. In particular, the derivative of e is ie . So when x travels along the real line, the point e will travel on the plane so that its velocity vector will be always equal to its position vector multiplied by i. Since multiplication by i is the same as turning a vector by 90 degrees, you can see why it makes a circle. | Since multiplication by i is the same as turning a vector by 90 degrees, you can see why it makes a circle. HOLY SHIT I GOT MIND BLOWN | it is truly an awesome intuition, isn't it? | Since we've established that e^(ia) is a circle, notice that at a=0, e^(ia)=1. Because that point is 1 away from the origin, the circle has a radius of 1, so we're talking about the unit circle. Recall that if 'a' is an angle, the x-coordinate for the point at that angle, by definition, is cos(a), and the y-coordinate is sin(a). Since a complex number can be written as x and y coordinates by x+iy, than we can write the unit circle as cos(a) + isin(a). | If you know some physics you might know the fact that when a object is traveling in a circular path, then the acceleration is pointed towards the center of rotation and the velocity is pointed tangentially to the circle and the position is pointed away from the path, meaning that they are all 90° rotations of each other. Now if you're willing to accept that also the converse is true (if something follows a path with these properties, then that path must be a circle), then the formula follows quite nicely. First we notice that the right hand side simply traces out a circle, so we simply have to show that the left hand side does as well. Now all we have to do is to simply differentiate, after all, the definition of exp(z) is still such that the derivative doesn't change, but because of the chain rule, the derivative becomes $ie (velocity) and the second derivative becomes $i -e (acceleration). If you now know anything about complex numbers you might know that multiplying by I corresponds to a 90° rotation which means that it satisfies our conditions and must therefore trace out a circle as well! Now I left out some details about such as the speed around the circle being the same as well as the radius but I think this gives a pretty good intuition. |
[
"Where would I be in university math?"
] | [
"math"
] | [
"bxm0om"
] | [
5
] | [
""
] | [
true
] | [
false
] | [
0.63
] | I'm in high school (in Canada) and our calculus teacher told us that we are ahead of the curriculum (doing more then needed) to help prepare for university ( I'm not complaining about this btw, I actually really like it). Just wondering where I would be in university math at the moment, as I plan to pursue a mathematics degree next year. We started with limits, optimization, and derivatives. We expanded on that with differential equations. We have done antiderivatives, and integrals by substitution, parts, and tabular integration. We are currently doing area between curves, and 3d shape areas on x/y axis (not using z-axis). Thanks in advance, I'd just like to know because our teacher said that we know most of what will be reviewed in our first year. | Depending on the school you're going to, this material might even be pre-university. See: the UK A-level curriculum. | I can only speak for the UK, but I’d say that most of what you’ve learned so far has been what I’d refer to as ‘mathematical methods’. These are the techniques used in applied mathematics. Pure mathematics at university is something quite different. To get you started, read about the irrationality of the square root of 2, the infinitude of primes, and conditional convergence. | You'd be in your first year. Depending on the school you went to, you'd be somewhere between late first semester calculus and mid second semester calculus. | It really depends. We did the same stuff at school but there is a huge difference between that and the same topics at University. One huge difference would be proofs: At school most technices are just used but at University you would have to proof that this always works. One example that you mentioned would be integration by substitution. While the method is quite simple the proof why this is always possible requires a lot of other theroems. And going just by topics you would be in the middle of calc 1 which you will probably take as a course during your first semester at least that's how it was at my University | Depends on the country. In the UK this (and more) is all covered in A-Level Maths/Further Maths, which are prerequisites for doing maths at uni. Can't really speak for Canada. |
[
"What are the biggest/your favorite black boxes?"
] | [
"math"
] | [
"bxmlby"
] | [
18
] | [
""
] | [
true
] | [
false
] | [
0.89
] | Mathematicians sometimes use theorems without ever working through the proof, and often never intend to do so. Sometimes theorems are used where we simply know a proof exists. The most interesting cases are ones frequently used in a serious manner (so NOT FLT to prove cbrt(3) irrational) where most people don't even cover the main ideas of the proof because it has a reputation for being long, tedious, and un-enlightening. This is sometimes referred to as "black boxing" the result. I was wondering what examples of this others might have. I'll start: If I'm off base with these, let me know! | The government is killing anyone who understands the proof of the Birkhoff ergodic theorem. Wake up sheeple. | Do people usually prove the Birkhoff ergodic theorem? I've seen the theorem covered once in a probability class, where it was proved, once in a dynamical systems class, where it was not proved, and once in a summer school on ergodic theory, where it was not proved. Even in the probability class where it was proved, the professor said in the middle of the proof "I don't understand this proof at all, by the way. I mean, locally I understand it, but globally, I have no idea." My advisor, who is the one who avoided proving it in his dynamical systems class, half-joked that at conferences he asks people if they actually understand the proof of the Birkhoff ergodic theorem. He said he has never found someone who does, although once someone said they knew someone who must actually understand it, but he later came to know that person had passed away. | Lol I had my exam in Ergodic Theory 2 weeks ago and my absolute mad lad of a professor decided to put several chunks of the proof as exam questions. Luckily it was optional, I don't think anyone chose to select it and answered something else. The "locally makes sense globally I have no idea" is probably what he had in mind since some of the individual steps are very acceptable, not that it would help me since I just """tactically""" avoided studying that proof :P | I think Sard's Theorem might count. My understanding is, at least, that in a typical differential topology class/book you quote it (and use it ) but don't prove it. The proof itself isn't bad. But it's just of a very different flavor than the rest of the subject, so going through it doesn't help prepare you for arguments to come. | The Atiyah-Singer index theorem of course, and the Hodge theorem if you're an algebraic geometer. |
[
"Simple Questions - June 07, 2019"
] | [
"math"
] | [
"bxwg62"
] | [
19
] | [
""
] | [
true
] | [
false
] | [
0.82
] | This recurring thread will be for questions that might not warrant their own thread. We would like to see more conceptual-based questions posted in this thread, rather than "what is the answer to this problem?". For example, here are some kinds of questions that we'd like to see in this thread: Can someone explain the concept of maпifolds to me? What are the applications of Represeпtation Theory? What's a good starter book for Numerical Aпalysis? What can I do to prepare for college/grad school/getting a job? Including a brief description of your mathematical background and the context for your question can help others give you an appropriate answer. For example consider which subject your question is related to, or the things you already know or have tried. | The important part of a natural transformation is that it commutes with the appropriate maps. If you aren't going to be using techniques of category theory, there isn't really a point in trying to show it really is a natural transformation. Just knowing it commutes is enough. | That isn't the typical way to do it, but it should work. The typical way is to say two cauchy sequences are equivalent if their difference converges to 0. Proving these give an ordered field is not hard at all. | How about (P,P implies Q) for the first and (R, R implies Q) for the second? | How about (P,P implies Q) for the first and (R, R implies Q) for the second? | How does one define the real numbers in terms of Cauchy sequences? In particular, how does one form the equivalence classes? In fact, it would be better if you could give me a hint instead of telling me the precise construction. I thought of one option: assert that any two Cauchy sequences (a_n), (b_n) are equivalent if their interleaving a_1,b_1,a_2,b_2,... is also a Cauchy sequence. And define operations on Cauchy sequences componentwise, so (a_n) - (b_n) would be a_1-b_1,a_2-b_2,.... Hopefully this gives a complete ordered field. But I haven't checked whether this makes sense yet. |
[
"Why is linear algebra useful?"
] | [
"math"
] | [
"bxkxws"
] | [
0
] | [
""
] | [
true
] | [
false
] | [
0.33
] | I'm teaching myself linear algebra and currently learning about bases and dimensions. I don't get how linear algebra is useful. It seems to be just a fancy way of doing systems of linear equations. What are some practical applications of linear algebra? | Even if it was only a fancy way to solve systems of linear equations it would still be the most applicable math in the world. | Linear algebra is quite useful for DEs, both for studying/solving systems of ODEs, studying PDEs, and for their numerics. | There's also that joke about representation theorists that they only really understand linear algebra so their whole field is reducing more complicated structures to linear ones. | Have you googled your question? It has been asked and answered a bazillion times here, on math.stackexchange, on quora, etc. Here is one such page: https://math.stackexchange.com/questions/1072459/what-are-some-applications-of-elementary-linear-algebra-outside-of-math | Spectral graph theory is close to my heart. |
[
"Stance on randomness: is there really a such thing?"
] | [
"math"
] | [
"bxjv1r"
] | [
0
] | [
""
] | [
true
] | [
false
] | [
0.33
] | I’m curious to see what stance people are taking on this topic. Is there anything that is ‘truly’ random. Not practically speaking, as in what we have the technological abilities to measure and perceive, but from a theoretical and perhaps philosophical stance. They say radioactive decay is a true random process, but I don’t know enough about that to comment. Is there anything on the macroscopic scale people claim is truly random? All good cases for randomness I have heard are quantum scale. | Look into Kolmogorov randomness . This gets into complexity theory, where if your sequence/image/whatever could very easily be produced by a computer program then that is one way of being "unrandom." One can find examples of random sequences here . | Something random on a macroscopic scale is the pixels on an lcd monitoring a radioactive decay experiment. | The Copenhagen Interpretation of Quantum Mechanics says that the universe is inherently random. Note: while Copenhagen is the most common interpretation, it is only by a plurality, not a majority. Other note: many interpretations besides the Copenhagen also insist on a fundamental randomness. Post-Einstein, the idea that the universe is random is probably more common than the idea that it is purely deterministic. | yes, by definition. | Last survey I saw had it in a distant second, but MWI also has a randomness baked in, so we're good. I would argue that the randomness of MWI is more interesting than Copenhagen as well. It's like... all humans exist, but my experience as human seems pretty much random. Only on a universal scale. |
[
"Where did I go wrong in my proof of the Goldbach Conjecture?"
] | [
"math"
] | [
"bxl7y4"
] | [
428
] | [
""
] | [
true
] | [
false
] | [
0.96
] | [deleted] | Haven't read it, yet, but I'm going to laugh myself to death if it turns out you did it and I start seeing citations of a Reddit post. | "Bad Proof" - Well I, for one, think anyone who at least attempts problems like this and puts 21 pages of work into it deserves applause | I didn't go through your argument in detail, but I can help you with how to find the error yourself. As you mentioned, since you didn't use any properties of the prime numbers, it's obviously wrong. So imagine following the proof with a different specific set of numbers for which the statement clearly fails. For example, remove 5 and 7 from the set of prime numbers, and then the corresponding Goldbach-type statement already fails for 8, for example. While personally don't have the patience to see where your proof falls apart, it shouldn't be too hard for you to do so, because although you have all these power series in your proof, to see the failure for 8, you can just write down the terms of these power series up to i=8. The mistake should become clear. | I can't verify everything before this, but 6.2.8 is certainly wrong; consider i=M-1, for which P )(0) = G )(0) = 0, and P+G = 0 != 1 = (M-i-1)!, since 0!=1. This eliminates a term in your final summation, and I expect that it accounts for the disparity. | for the life of me I can't figure out where I went wrong You didn't actually link a file, for one ;) |
[
"Is it possible to accurately rank the quality of math teachers/professors?"
] | [
"math"
] | [
"wmrl7h"
] | [
0
] | [
""
] | [
true
] | [
false
] | [
0.38
] | null | No. It's subjective. People learn differently, teach differently, etc. | Different people like different teaching styles, you can't have an objective ranking about that. Also I've had a couple students argue that teacher X or Y was "bad" because their exams were "too hard" or because "half the class failed". When I asked them if a good teacher would let more students pass even if they didn't have a good grasp on the subject, they couldn't answer that. | Most attempts to do this measure the improvement in the students’ scores on standardized tests. Even that’s controversial. | As a general statement about life: anything subjective is virtually impossible to assign a score or ranking to that can be agreed upon by all logical parties involved. | A good teacher can show the path, and instil in one the desire to follow that path. It is possible, but not probable, to quantify the results. Answer is yes. |
[
"What does 1/2000 mean?"
] | [
"math"
] | [
"wmmq9q"
] | [
0
] | [
"Removed - ask in Quick Questions thread"
] | [
true
] | [
false
] | [
0.42
] | null | I’m going to start with your second question: no. To understand why, think of a fair coin. Each toss gives you a 50/50 chance of getting “heads”. If you throw it twice, you don’t have 2/2 chances of getting heads, right? In fact you need to compute 1-“chances of not getting heads twice”: that is: 1 - 1/2*1/2 = 3/4 . So you have 75% chances of getting at least one heads in 2 throws. In your example: you would have 1- (1999/2000) chances of developing cancer. | No, it means if 2000 people had a CT scan, you would *expect* one of those to develop cancer from having had the scan later on. But obviously it's probability, none could, 5 could, can't say 'til you measure. | The second interpretation maybe incorrect. Two different persons having a CT scan can be assumed to be independent events. But the same person having a CT scan two times need not be independent at all. | As a possibly easier-to-understand reason for why "if I've had 2 ct scans, does that make my chances 2/2000" is "no" --- if you had 3000 CT scans, would your chances be 3000/2000? No; that doesn't even make sense as a probability (since probabilities are between 0 and 1). | This thought experiment probably happens every day, and based on how few people die of rabies, it’s clear most people choose correctly. Sadly though there are cases of people not wanting to get the vaccine and dying.. |
[
"Worst wikipedia sins"
] | [
"math"
] | [
"wmungz"
] | [
17
] | [
""
] | [
true
] | [
false
] | [
0.84
] | null | This is not a direct nor serious answer, but you may lament the revision to "Mapping cone (topology)" that occured at 20:40, 11 Jan 2019. | They might be referring to the picture that was there before that edit. | Not Wikipedia, but the illustration on MathWorld for Perko's pair has been wrong for years. See here , here , and here . | Shame it was edited, this is a masterpiece. | Another math world one rather than Wikipedia but every single sentence in the first paragraph of the Julia set article is wrong. https://mathworld.wolfram.com/JuliaSet.html |
[
"In your opinion, what invention/discovery contributed the most for math?"
] | [
"math"
] | [
"wn9nvh"
] | [
4
] | [
""
] | [
true
] | [
false
] | [
0.6
] | null | Numbers | Fire, because if human did not have efficient method of getting energy our brain might not have developed enough for high level reasoning, and it's unlikely dolphin would have taken up that mantle. | Bijections, but I'm biased. | An eraser. | For that matter, a few other important inventions/discoveries along this line include: I'm sure there are other such discoveries along these lines that primarily improve human life in general, and only end up improving our mathematical understanding as a side effect. |
[
"How exactly do you use higher math day to day so as to not forget the concepts if you don’t have it as your profession?"
] | [
"math"
] | [
"wmy4dm"
] | [
70
] | [
""
] | [
true
] | [
false
] | [
0.96
] | null | It's not my profession, so I let myself forget the concepts. If I ever need them, I'll review an old textbook. | Pretty much just read on a regular basis, take notes, and work exercises such that you can recall where you are. Get stuck as quickly as possible and give your mind time to process the material. | I think this is the best solution: learn to let it go and utilize the maths you already have to the max (if you really need it, forgetting things would be less of your concern). Even if it's arithmetic, there are ways to do it on the head more efficiently, or perhaps reduce hand movements on calculator. Another example, real analysis is basically estimation, so that soft skill can be used in daily activities. | Project Euler can certainly keep you sharp and keep you on top of certain number theory concepts. Lot of interesting discrete math problems on that website. | It's simple enough to do exercises from a book or two when you're done work. |
[
"good Mathematics wikis"
] | [
"math"
] | [
"wmyp3z"
] | [
3
] | [
""
] | [
true
] | [
false
] | [
0.71
] | I really like wikis as a medium for learning. I've use Wikipedia very extensively and i read multiple articles every day. It works great for most topics, but for learning mathematics I find it to be quite bad. The problems I have with Wikipedia are that: - Often, things are not nearly as general as they could be and as they are in my university lectures - The notation is often cumbersome and weird to me, proofs are harder to read than they should be. I'm looking for recommendations for wikis that are more intuitive and/or cover topics more formally and generally. I just completed my second semester of a pure math degree, so typical bachelor topics are most relevant to me. Thank you for your time :) | ncatlab is very general... but probably too general for a second year. Also, Wikipedia often does have the right level of generality -- you just need to go looking for the right page! What topics do you not think are covered in appropriate generality on Wikipedia? | ProofWiki : A compendium of proofs. So definitely ticks the "formal" box, but it's not expository like a textbook. | Wikis aren't really meant for learning but more as a reference/to look something up. I'd very much recommend looking for text books on the relevant topics, instead. | Thanks a lot for the recommendation :) I will definitely check it out. I was looking through my lecture notes, comparing then side-by-side with Wikipedia and think I may be wrong about the generality. I think I've had these problems with the German language version of the pages but I can't find these kinds of issues at all with the English ones (things like only considering |R or |R and |C, when a statement is valid for any field, only looking at |R instead of |R etc...) Sorry for that! If I do find something like that that bothers me in the future I hope I will remember this thread. | Canonical: https://encyclopediaofmath.org/wiki/Main_Page https://mathworld.wolfram.com/ |
[
"Category theory and logic"
] | [
"math"
] | [
"wmx2zv"
] | [
16
] | [
""
] | [
true
] | [
false
] | [
0.91
] | I recently had thoughts about category theory and logic. Suppose we have a set (a class ?) of all mathematical proposition, I think we can create a category out of this, by having arrows induced by implication, if I understand it correctly, products and coproducts are the categorical equivalent of the « and » and « or » connectors. Is there a deeper link around I do not see ? If so, what kind of implications are reflected by mono/epi-morphisms ? Do equivalence reflect as isomorphism ? What are the role of functors and higher construction ? Same for limits and representation of functors ? Finally, if a deep connection does exist, are there some reference books about it ? For context I’m a french master student in maths and if this is an interesting topic I’d love doing a master thesis about it | Yes, the mathematical object (or category in this case) is a boolean algebra (if you consider two equivalent statements to be equal). These objects are the dual of the so called Stone spaces: compact topological spaces made up of a basis of clopen sets. These spaces and their generalizations, type spaces are very important in mathematical logic, and they have found applications in real analysis (o - minimal theories) and algebraic geometry (stable theories). | It's a comonad but backwards | You might want to check out syntactic categories (unfortunately, that term also has a different meaning in linguistics, so you have to be a bit careful when searching) and the related classifying topos . | A tool to confuse Haskell programmers. | Theres something about this in elephant, I'll see if i can dig up the old email where i asked about it |
[
"What topics in maths should I look into if I want to analyze knotted surfaces of some curve in ℂ³?"
] | [
"math"
] | [
"wmuuyn"
] | [
15
] | [
""
] | [
true
] | [
false
] | [
0.95
] | Hi, Sorry for not being so precise in the title, I'm not sure exactly what I'm dealing with. I have this matrix H(k), where k is in traditionally in R³. By considering k in C³, I think the eigenvalues of this system ought to produce sheets or lines that become knotted up, and these correspond to physical processes. What tools might exist to analyze the topological invariants or knots that show up in this system? I think I could meaningfully reduce this to a 12-order characteristic polynomial with 3 degrees of freedom (Re(kx) ,Re(ky), Im(kx)) for some given energy slice, if that helps at all. Cheers | Every embedding S -> C is isotopic (via a real isotopy) to any other, so real knots probably aren't what you're looking for. You could look at holomorphic isotopy classes of hypersurfaces in C but that's a very murky subject. I have heard that for compact real surfaces in R morse theory can provide useful information about real isotopy classes, that might be a good place to start. Generally "knotting" really only happens in (real) codimension 2, that's the only setting where one thing 'winding around' another thing makes any sense. You should probably first guarantee that you know what you're looking for. There's no standard way that eigenvalues of a matrix would be related to knots, there has to be something else going on. Is H a linearization of a Hamiltonian operator? In that case you may be interested in Floer homology and properties of Lagrangian submanifolds. The surfaces themselves are always tori in this case. | It's a bit hard to answer without more details, what shape does H(k) have? Do we even how many eigenvalues it has? What sort of topological information would interest you? Anyway, I'd think I'd start by finding the intersection with various planes to try and get a picture of what's going on, and then trying to think of more concrete questions. More generally, the topic of algebraic geometry is essentially about the geometric shapes that are formed by zeroes of polynomials. | H(k) would be 12x12 in the simplest possible approximation, though I think a 20x20 H(k) would be the simplest model with all of the features I want. When certain, somewhat novel, perturbations are applied to H(k), it seems like certain eigenvalues are completely forbidden, even considering the complex k. I found a paper showing that these eigenvalue sheets become knotted with certain perturbations that are analogous to extending k -> C³, and I'm curious if there is some knot-driven phase transition that forbids these modes. Also curious about how people describe or characterize similar knotted objects. | I assume you talk about some self-adjoint Hamiltonian operator, right? Then your eigenvalues are all real, how can those form knots? Maybe you're asking about some preimage of those real paths, knotted in C - but then you'd have real codimention 1 which I don't think allows for any knots. Also, knots can remain knotted and still be continuously deformed. I don't see how knots would forbid certain eigenvalues. | For a real k vector, yes it is self-adjoint. Considering complex k, the Hamiltonian becomes non-hermitian and has evanescent states, which could host nontrivial topology. A similar system does. |
[
"A sequence I couldn't find anywhere"
] | [
"math"
] | [
"wmu44f"
] | [
6
] | [
""
] | [
true
] | [
false
] | [
0.75
] | I was playing around with sequences when I was board, and got the idea to construct a sequence that is defined as follows to my surprise and delight it repeats so I threw the terms into the OEIS to see what the sequence was and/or what applications it may have. However, it was not there. It sat in my head for a few days and so I started googling and still could not find a mention of this sequence. Next my question was what about other starting values, and so I checked all combinations of 1 - 9 in mod 10 as anything over 10 would just be redundant, and the ALL eventually repeated into a pattern at some point. Continuing on this generalization I (my computer this time) then tested all non-redundant inputs for every sequence like this up to 100 as the mod value, they still all repeated, however the sequences before repeating were longer when it was a prime number as the modulus. Dose anyone know why it is very stochastic in how long the sequences get, or have an idea where this may be applicable? | As to "Why has nobody thought of this before": There is a practically unlimited number of expressions you can make even with very few operations, and you can turn any expression into a sequence by simply applying it to several preceding elements to get the next. If you pick any moderately complex expression, chances are that nobody has tried investigating before. But this means about as much as saying that nobody has used the exact number 361,594,780 in a proof before. The possibilities that exist are countless, the possibilities that are useful for something other than its own merit are fairly limited. (I'm not saying that researching something for its own sake is a bad thing. I'm just saying that you shouldn't expect direct connections to some other popular concept from any possible example you can build.) As to "Why does the sequence eventually repeat": Notice two things: The sequence can only take on finitely many values. The two digits you multiply are between 0 to 9 (or 0 to m - 1), so every number of the sequence is going to be between 0 and 81 (or 0 and m²). Given two consecutive numbers in your sequence, the entire rest of the sequence is uniquely determined. This means that if some pair of consecutive numbers in your sequence occurs twice -- and some pair occur twice, because you only have finitely many pairs to choose from -- then the entire rest of the sequence will just keep repeating the run of the sequence between these two occurrences over and over. | Well we have 0 <= mod(x, 10) <= 9 , so 0 <= a(n) <= 81 for any n >= 2 . This, together with the fact that each a(n) only depends on the previous two values, guarantees that the sequence will repeat itself at some point. Think about all the triples (a(n - 2), a(n - 1), a(n)) . There can only be finitely many unique such triples. So by pigeonhole principle we will eventually have a(n) = a(m) and then the sequence repeats. | I guess I didn't phase my curiosity well, why is it that some values repeat after say 20 terms while others go well past 100 when doing for example mod 85 | why is it that some values repeat after say 20 terms while others go well past 100 when doing for example mod 85 because your sequence is related to the sequence of modulo Fibonacci numbers F_n mod m, and the periodicity of that sequence depends on the prime factorization of m. In particular, since the periodicity of F_n mod 5 is 20 and the periodicity of F_n mod 17 is 36, then I hypothesize that the periodicity of your sequence modulo 85 should be LCM(20, 36) = 180. There are actually a ton of open questions about computing the periodicity for these series modulo prime numbers. https://en.wikipedia.org/wiki/Pisano_period#:~:text=In%20number%20theory%2C%20the%20nth,Joseph%20Louis%20Lagrange%20in%201774 . https://sites.math.rutgers.edu/~zeilberg/essays683/renault | The fact that it repeats is actually a pretty good reason for why no one has studied it before as it kinda makes it less interesting in my opinion. Now, something that could be interesting is studying GENERAL recursive sequences over finite rings (such as the integers mod 10), but that's probably been done before |
[
"What are your favorite simple/straightforward proofs?"
] | [
"math"
] | [
"wmxa4i"
] | [
348
] | [
""
] | [
true
] | [
false
] | [
0.97
] | I am not a mathematician, just someone who likes math. So mine aren’t too advanced. Would love you hear your favorites and why! | Cantors diagonal argument for the uncountability of the reals | The proof that root 2 is irrational is the canonical proof “from the book” in my opinion. | Proof that an irrational raised to the power of another irrational can be rational: sqrt(2) is known to be irrational. Take sqrt(2) . It could either be rational or irrational and we don't know which. If it is rational then we have proved our statement. If it isn't, raise it to the power of sqrt(2). (sqrt(2) ) = sqrt(2) = sqrt(2) = 2 Either way, the statement that an irrational raised to the power of an irrational can be rational would be shown to be true. | Pi is transcendental. Wait... Is there a straightforward proof of this? I've only seen arguments that are very technical and seem like magic. | The proof that limits add. It’s a very straightforward demonstration of three indispensable techniques in analysis: min(delta1, delta2), epsilon/2, and the triangle inequality. |
[
"I finished Hartshorne… now what?"
] | [
"math"
] | [
"wmjnxw"
] | [
182
] | [
""
] | [
true
] | [
false
] | [
0.96
] | null | Algebraic geometry is huge. Hartshorne is a general introduction to algebraic geometry, now you have to go for something more specialized. What kind of algebraic geometry do you want to go into? For example, I'm doing a certain thing in arithmetic geometry. Then a good path is: 1. General introduction to Algebraic Geometry (like Hartshorne, Vakil, ...) 2. Introduction to Arithmetic Geometry (Liu's textbook is good) 3. Survey papers/very specialized textbooks/specialized learning seminars toward the specific topic you want to get into 4. Read research in the vicinity of the thing you care about You don't have to do all those steps very rigorously, but skipping too much will leave you lost! | If you finished reading all of Hartshorne before undergrad as you say, you're an incredibly exceptional student. I would have some faith. Many grad students at top programs find Hartshorne difficult, much less undergrads. | If you finished reading all of Hartshorne before undergrad as you say, you're an incredibly exceptional student. I would have some faith. Many grad students at top programs find Hartshorne difficult, much less undergrads. | a math degree is unironically better than a finance degree to get into finance/big banking job market. | a math degree is unironically better than a finance degree to get into finance/big banking job market. |
[
"Good at maths but suck at programming"
] | [
"math"
] | [
"wmkdy4"
] | [
97
] | [
""
] | [
true
] | [
false
] | [
0.93
] | I’ve been told that mathematical logic is applicable to programming, but I completely suck at it. What are your experiences of programming coming from a mathematical background? | Regardless of mathematical affinity, programming is a skill that requires you to practice. What kind of programming languages have you used? You'd probably do better in languages with a strong type system, plus ones like Rust and Haskell that have robust theoretical foundations. | With the exception of python, which IMO is a solid general-purpose scripting language, that list strikes me as a list of special-purpose tools. They definitely have their share of footguns and inelegant syntax. It's not surprising that a theoretical/math person would not particularly gel with them. When writing python, you can use type annotations + mypy to simulate a stricter type system. I find that "type-driven development" stimulates the mathy part of my brain. Perhaps that's not particularly useful advice if you don't have any experience with that approach, though. If you have the bandwidth, I think you'd feel like a stronger programmer after spending some time poking at some other stricter general-purpose languages, like C/C++, Haskell, or Rust. Each of these would be challenging to pick up, but would help you in different ways. | Do you like number theory? Check out 'project euler' for an endless supply of programing problems in number theory. My guess is you will do way better when you code about problems you care about. So ask yourself a computational problem which you can/cannot do it by hand and try coding it. There are obvious examples: implement the most naive factoring algorithm (which I'm sure you know), given two numbers find their gcd, i.e., implement the euclidean algorithm, print all prime numbers up to a million etc. once you are able to successfully program these small problems you will get some confidence.. | Honestly, I do not know where this notion comes from. First, there is not one math. There are many. Being good at differential geometry does not mean you are good at abstract algebra, neither of which translate into skills in harmonic analysis. And then programming... Programming is not a science. It's a craft. While algorithms are related to math.. kind of, that only makes up a small part of programming. Most of it is to know which data structure to use or how to group the operations together. To become a proficient programmer, you have to learn it like a craft, preferably as an apprentice from a master (or at least have someone criticizing your code), with lots and lots of repetition. No matter how good you are at any kind of math, there is no replacement for practicing programming and learning what works when. | I have used MATLAB, Maple, R, SQL and Python. |
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