- What are dimensions?by /u/senpaithesensei on May 6, 2021 at 3:44 am
So I just learned that circles have the largest ratio of area to perimeter in any 2 dimensional object and that made sense and i understood it. But does that mean that there has to be some type of rule in the 3rd dimension that has an effect on us like that? it might be a stupid question and i might not be explaining it right, but in my head its just like a video game and would another dimension just be another level? and if that question doesnt make sense, i do have another one i’ve been meaning to ask someone for a while. I’ve heard people say that ants can only see in two dimensions, i’ve just never understood why? and why can we know that but we dont know what the fourth dimension is? like i know we’re probably never going to know what the fourth dimension is like but do we at least know what’s stopping us or do we have no idea? submitted by /u/senpaithesensei [link] [comments]

- Real numbers vs natural numbersby /u/HelpfulPanda5 on May 6, 2021 at 1:09 am
Hi, I am not well-versed in maths, but I came across this page: https://www.science.org.au/curious/space-time/beyond-infinity which has a nifty little tool to help prove that there will always be a real number between 0 and 1 that will be unmapped with a natural number if you try to pair them up (if I understand it correctly). And this is meant to prove that the infinity of real numbers between 0 and 1 is bigger than the infinity of natural numbers. So I’m wondering, as far as mapping 1:1, why wouldn’t this work?: 1 = 0.1 2 = 0.2 3 = 0.3 … 9 = 0.9 10 = 0.01 11 = 0.11 and the number it told me would be unmapped: 11,112 = 0.21111 And so on.. I feel pretty confident that no matter what unique decimal you would provide, I would be able to give a unique natural number that corresponds to it. What am I misunderstanding? Thanks! submitted by /u/HelpfulPanda5 [link] [comments]

- What’s the formal way to say “numbers”?by /u/HXSC on May 6, 2021 at 12:48 am
This seems like a stupid and silly question, but what’s a formal way of saying “numbers”? By numbers, I loosely mean C, R, Q, Z, N, etc. … although I guess algebraically they are somewhat different (i.e. N is not a field) submitted by /u/HXSC [link] [comments]

- What is the topology of our universe?by /u/__-_—___ on May 6, 2021 at 12:30 am
A new paper (which is very readable) gives some interesting connections between assumptions made for cosmological models and a condition called global hyperbolicity. Almost all cosmological models are assumed to be globally hyperbolic. There’s not really any justification for this beyond “We know that the initial value problem version of Einstein’s field equations is well defined”. The paper shows that global hyperbolicity is a consequence of the assumption of the homogeneity of matter. The paper is very interesting, it touches on K-Theory, the classification of equivarient abstract bundles, harmonic analysis and has a bunch of interesting new concepts in Lorentzian geometry. If you are interested in cosmology and have a done a differential geometry course then the paper is a gateway drug to modern research as well as a plethora of topics of importance in modern topology and geometry. submitted by /u/__-_—___ [link] [comments]

- What is the limit of a transition semi group for a Markov jump process with a generator matrix Q?by /u/jilkz on May 5, 2021 at 10:46 pm
I know we can find stationary distributions for transition matrixes as well as conference to equilibrium using the limit theorem to find the limit but the theorem just gives us arbitrary values. How do you apply it for a specific generator matrix? submitted by /u/jilkz [link] [comments]

- Do you have a method to prevent forgetting already learned definitions and theorems?by /u/Windstro01 on May 5, 2021 at 8:25 pm
I understand that it is not necessary to keep everything in mind but rather develop your mathematical skills so you can re-derive things. That’s not always reliable. From time to time what happens is that I am not able to solve an exercise while trying everything and wasting multiple ours. In the end the exercise turned out to be fairly easy under consideration of a specific theorem or definition I didn’t thought of at all. Frankly that doesn’t happen too often. Usually I know in what direction an exercise goes so I at least know what to look up. But I’m kind of paranoid if an exercise takes too long, that I’m missing something I will never remember unless I know what it is. Do you have a system of (short) repetition of old topics to have everything you may need at hand? submitted by /u/Windstro01 [link] [comments]

- can two probabilities be equal but one still be more unlikely than the other?by /u/FazeDiogenes on May 5, 2021 at 6:34 pm
I’ve heard that since there are an infinite amount of points on a dart board, then the probability of hitting a perfect bullseye (or any specific point) is 0, since 1/x as x approaches infinity = 0, and there are infinite points and you’re focusing on hitting exactly one. So while it’s technically possible, there’s still a 0 probability of it occurring. But, intuitively speaking, hitting a perfect bullseye twice in a row is even more unlikely. And since your second throw is independent of your first, you can just square the probability of it occurring once to get the probability of it occurring twice in a row. So we’d get (1/x)2 as x approaches infinity. and both (1/x) and (1/x2) equal 0 as x approaches infinity, but for all values of x greater than 1, (1/x2 ) is smaller than (1/x). So would it be fair to say that while they have the same probability of occurring (0% chance of you hitting a perfect bullseyes and 0% chance of you hitting a perfect bullseye twice in a row), the likelihood of one is still lower than the other? submitted by /u/FazeDiogenes [link] [comments]

- Real numbers constants that were discovered to be rational.by /u/homestar_galloper on May 5, 2021 at 6:07 pm
Hey, as I’m sure you all know, there are many important real number constants that are not known to be rational or irrational. For example, the euler-mascheroni constant may be rational or irrational for all we know. But most of these number at least appear to be irrational on the surface, and typically end up being irrational if proven at all. I was just wondering if there are any real numbers that were discovered to be rational despite previously being believed to be irrational? submitted by /u/homestar_galloper [link] [comments]

- AOPS calculus book for AP calcby /u/InertiaOfGravity on May 5, 2021 at 5:59 pm
I intend to self study and take the AP calc AB/BC exam, and was looking for a book which is both interesting and would give me the requisite knowledge (or at least put me in a positon where obtaining the requisite knowledge for the is trivial). I’ve seen glowing recommendations for most AOPS books (both online and from friends) but I was significantly less able to find information specifically regarding AOPS calculus. Has anyone had any experience with this book? Thanks! submitted by /u/InertiaOfGravity [link] [comments]

- When self-learning math, what are your strategies if you keep getting stuck on a lot of the practice problems?by /u/gul_dukat_ on May 5, 2021 at 5:53 pm
I am currently trying to teach myself about manifolds, and have found that after about a month of working through one of the chapters, I am getting stuck on a lot of the practice problems at the end of the chapter. I don’t expect to automatically know how to solve everything, but what would you all do in this situation? Skip the problems you can’t solve and move ahead to the other ones? Reread the chapter? Any other strategies or suggestions to work through this? The last thing I want to do is google the proofs because that defeats the purpose of practice problems but at the same time I really do want to solve them and understand them. submitted by /u/gul_dukat_ [link] [comments]

- Is all math a generalization of logic and arithmetic?by /u/frankrot09 on May 5, 2021 at 4:28 pm
Historically, we start to look at math by developing arithmetic. This provides us with addition, subtraction, multiplication and division, and a concept of infinity. These operations allow us to define the integers and the rationals. Then, starting from multiplication of a number by itself we can define the square root, and from there we meet the irrationals. So now we have the entire real line. On the real line we can define calculus, with open sets, continuity etc. These concepts then inspire their generalization to topology. Also, the way addition and multiplication work on the integers inspire the concept of group, and more in general of algebraic structures. Elementary geometry also starts with the basic arithmetic operations, and then abstracts more and more using algebra as well. So what I am trying to pose as a discussion is if all math follows, directly or indirectly, from the rules of arithmetic and logic. In other words, we use these rules to build a more complex/abstract structure, and then we use the new structures to build even more complex/abstract ones. I can imagine a system which does not follow the rules of arithmetic. However, I guess (I might be wrong of course) that the systems we can imagine are systems that do or do not have some of the properties of arithmetic. For example, a group law can be commutative or not, but this distinction can be made because we know what commutativity means from arithmetic. It seems to me that we rely on concepts that come from arithmetic and then we try to see what happens if we modify/negate/generalize them, according to the rules of logic. I hope I explained myself appropriately, otherwise just let me know what is not clear. I am very curious to hear your opinions! submitted by /u/frankrot09 [link] [comments]

- Quick Questions: May 05, 2021by /u/inherentlyawesome on May 5, 2021 at 4:00 pm
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. submitted by /u/inherentlyawesome [link] [comments]

- Are there theorems that are true exclusively for non abelian groups? Ie they don’t hold for abelian groups but do hold for non abelian groups.by /u/h4dial on May 5, 2021 at 2:40 pm
I know that commutative groups have some extra exclusive properties, I was wondering if the same could be said for non commutative groups. Thanks. submitted by /u/h4dial [link] [comments]

- The mysterious connection between 5 and 15…by /u/Shadow_Reptile on May 5, 2021 at 2:23 pm
Hello. I am going to talk about a controversial topic, but please hear me out. I was working on the COLLATZ CONJECTURE. I began by defining five parameters: Even Probability – It is the ratio of even numbers in the collatz sequence of an integer to the total steps of the collatz sequence. Odd Probability – Same as Even Probability but for odd numbers Step-Integer Ratio – It is the total number of steps in the sequence divided by the integer whose sequence is being considered. Even-Integer Ratio – It is the number of even steps in the sequence divided by the integer whose sequence is being considered. Odd-Integer Ratio – Same as Even-Integer Ratio but for odd numbers. Eg: Consider 3. The sequence goes 3,10,5,16,8,4,2,1. (We stop once 1 is reached.) Even Probability = 5/8 = 0.625 Odd Probability = 3/8 = 0.375 Step-Integer Ratio = 8/3 = 2.66666… Even-Integer Ratio = 5/3 = 1.66666….. Odd-Integer Ratio = 3/3 = 1.0 What I found out was that all numbers until 10000 have a unique sequence, ie, no two integers have the same values for all five of these parameters, EXCEPT 5 AND 15. For some reason, they have the same values for all of them, ie, 0.6666…, 0.3333…, 1.2, 0.8, 0.4 respectively. Is there any link between 5 and 15 other than the fact that 5 is a factor of 15? submitted by /u/Shadow_Reptile [link] [comments]

- Could someone ELI5 how zero knowledge proofs work? Interactive ones?by /u/PsychologicalDrawer0 on May 5, 2021 at 2:00 pm
Hi, What I know/found out is that its basically proving to someone you know something, without actually letting him know how you know, reading mainly from Wikipedia, it explains it using the color blind friend analogy and other examples, the thing is, I haven’t really crossed passes with much proofs before, what I was thinking of my “zero knowledge interactive proof” was basically making a program, and whoever wants to check it works, can send me trial data, I process it, and then hand him back the results which he could easily check. I guess what I am essentially asking is, is this the most efficient way to go through it? is this even essentially a zero knowledge proof? TBH I don’t have that much of an idea of how Math proofs work, I mean like I studied proofs by contradiction, direct proof, proof by induction and so on ,very briefly though , but I guess, I don’t have to use those? Is proving something a science with rules that I have to follow, or is it more like guidelines that are preferred? Is proving something no matter which way enough? or do I have to do it using certain ways? I understand that these rules are there for a reason, and they have been built upon to improve more and more how we discover things, making things more organized, but wouldn’t my method, sort of, count as just a very abstract sort of direct proof? Or do I need to get into the complex Jargon and use graphs and so on? If it is of any help, I am mainly focusing on proofs regarding complexity classes, reduction from one problem in one class to another for example and so on. Thank you submitted by /u/PsychologicalDrawer0 [link] [comments]

- Inequality from Imperial’s Maths Comp solved in a Theorem Proverby /u/Soccolo on May 5, 2021 at 12:23 pm
submitted by /u/Soccolo [link] [comments]

- Image of unit ball by a compact operator.by /u/Frequent_Pharoh on May 5, 2021 at 9:08 am
A compact operator on a Hilbert space acts like an asymptotic contracting map and sends the closed unit ball into an “elliptic spiral”. I don’t see how, can someone provide a rigorous formulation of this? Thank you. submitted by /u/Frequent_Pharoh [link] [comments]

- Question on Squaring the Circleby /u/kronchkronch on May 5, 2021 at 8:17 am
I’m having trouble figuring out this question I thought of about a twist on squaring circles. Can anybody help me find out if T(x) has a global max (ignoring around 0) or if it has infinite ever growing maximums? This function finds when integer radius values make circles with square areas, peaks show close matches, but the catch is it divides by x, the radius, rewarding circles and corresponding squares that are not massive and near perfect matches in area. I’d love to help clarify any questions on how the function works and what it means intuitively; here is a little more detail. The x value is the radius of the circle. The y value is how close to square the area of that circle would be, but divided by the x value, the radius. That makes it so that numbers like 22 and 167 have higher peaks than some of the more ridiculously large radii you could choose for near perfect area matches. It rewards reasonably small numbers. But that begs the question, with this divide by x constraint, is there a non-trivial global max? And I honestly have no idea, or any idea where to begin. The functions and their plot submitted by /u/kronchkronch [link] [comments]

- Interesting questionby /u/FugBone on May 5, 2021 at 6:07 am
I seriously have no idea if it’s frowned upon to ask this here, but I saw a question on MathSE that some might enjoy a crack at, it’s about a fugitive that is 100 times faster than an infinite number of guards that are trying to “catch” him and the question is if the fugitive can stay away from the guards if they continuously pursue the spot that the fugitive is currently at: https://math.stackexchange.com/q/4127588/716998 I will try to keep this question updated (as in if the one on MathSE gets an answer, I will let you all know). If someone on this site gets a solid answer, I encourage you to post it on MathSE as well (I don’t want to take credit for anything, I’m just trying to get that question more attention) submitted by /u/FugBone [link] [comments]

- What’s your favorite math puzzle?by /u/Lucky_Motherducker on May 5, 2021 at 2:02 am
submitted by /u/Lucky_Motherducker [link] [comments]

- Finite abelian groups are just movements in a toroidal box.by /u/Electro_blob on May 4, 2021 at 10:08 pm
I’ve been really interested in group theory recently, but I’ve noticed the explanations for concepts rely on a web of obscure terminology. For example, I had to scour Wikipedia to finally figure out that a normal subgroup is just a subgroup where for every element n in the subgroup, x+n-x is also in the subgroup. As another example, groups are always described as the symmetries of an object, but for me this doesn’t help at all. I can’t “see” a symmetry. I can only see one element, or symmetry, of the object at a time. On the other hand, cyclic groups are easy to imagine, because they are just moving on a big loop, like a clock. On a clock you can add hours, and you can subtract hours. Each hour could be considered as an element. Adding by an element moves you a certain amount around the clock, and adding by its inverse moves you back. In fact, clock math is congruent to Z12. The cycle with 12 elements. Then I realized, that, in fact, every possible finite abelian group, by the fundamental theorem of abelian groups, is just a combination of cycles. For example, the Klein four-group Z2 x Z2, is just two cycles. Because these cycles share common factors, they can’t be represented as a larger cycle. Instead, they can be represented as the movement on a 2×2 grid that wraps around, like a little torus. One cycle in one dimension, one in the second. Similarly Z2 x Z3 can be represented as all the possible ways to move on a 2×3 grid, but because 2 and 3 are coprime, it can also be represented as 6 cycle, so it is still just Z6. Moving to larger example, you could have a 10 by 7 grid that loops around like an arcade game, and that would be Z70. You could also have a 10 x 10 x 10 grid, which would have to be a box, since all of those share common factors, the last 10 cycle would have to be pushed up into the third dimension. You could go to any dimension. For example, the simplest representation Z2^4 would be a 2 x 2 x 2 x 2 hyper-cube, and of course the edges wrap around, making it a 4d torus. Now, my biggest question is, what would non-abelian “spaces” look like. What kind of surface do they describe movements of. I’ve tried understanding this even with the simplest non-abelian group, S3, and so far I’ve gotten nowhere, but I know it can be done, even if the geometry of this space is abstract, because groups essentially share all the same axioms as a connected space. You can go to anywhere, but you can’t ever leave (closure.) You can always stay where you are (identity.) You can always return back where you came from (inverse.) The grouping of movements you take doesn’t matter, you’ll get to the same place no matter what order you calculate the direction you must head (associativity.) submitted by /u/Electro_blob [link] [comments]

- Do professional mathematicians remember all the basic math?by /u/IntertexualDialectic on May 4, 2021 at 8:08 pm
I was wondering whether people who are working in the research or industry as mathematicians remember basic algebra geometry and calculus from their school days. Do you remember the law of cosines or how to find roots of unity off-top or would you have to look it up? How well would you do on a multivariate calculus test if you had no time to prepare? Are mathematicians more specialized or does it require general mastery? submitted by /u/IntertexualDialectic [link] [comments]

- Students who wrote a thesis this semester (any level), what was your topic and how did it go?by /u/neutrinoprism on May 4, 2021 at 2:57 pm
Curious to hear about people’s experiences. submitted by /u/neutrinoprism [link] [comments]

- I am curious what your thoughts are on this? Someone asked if engineers remembered how to do some more advanced math after they graduated and it looks like every last one of them couldn’t.by /u/s_0_s_z on May 4, 2021 at 11:19 am
submitted by /u/s_0_s_z [link] [comments]

- Are there any mathematical discoveries which are absolutely useless?by /u/zippydazoop on May 4, 2021 at 10:40 am
I’ve heard a lot about theorems or discoveries that are used everyday, especially in the modern world, so I am wondering if the opposite also exists. Any there are such examples? submitted by /u/zippydazoop [link] [comments]

- Which was the first book that gave you the taste of real mathematics?by /u/dsengupta16 on May 4, 2021 at 7:22 am
The title says it all. Assuming that you are somewhat involved with the mathematical field. Which was the book which made you fall in love with math? The book that helped you make the transition from rote exercise solving to a real understanding about what math really is, so that you were able to solve some really interesting problem. From a high-school problem sheet junkie to a real appreciator of mathematics. You may list more than one such book also, if such is the situation for you. submitted by /u/dsengupta16 [link] [comments]

- What Are You Working On? May 03, 2021by /u/inherentlyawesome on May 3, 2021 at 4:00 pm
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: math-related arts and crafts, what you’ve been learning in class, books/papers you’re reading, preparing for a conference, giving a talk. 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 Career & Education Questions thread. submitted by /u/inherentlyawesome [link] [comments]