I can algebraically explain how i^i is real. However, I am having trouble geometrically understanding this.

What does this mean in a coordinate system (if it has any meaning)?

I don't know if this is the right sub for such questions, but I want to get a cool but subtle math related tattoo. And I can't really find good inspirations — all I see are golden ration designs or some very symmetric but boring geometric designs.

I remember how I was fascinated by how Conway's Game of Life is used as a reduction to the halting problem, but I couldn't seem to come up with something that was satisfying to me.

I feel like it's so easy to mess up a math tattoo, I might as well not get one at all lol.

Note my question is the bottom paragraph, though the following is what motivates it.

Despite just how rich the universe of our axiomatic theories tend to be, some of the most continuously studied and celebrated mathematical structures remain those at the bottom of our universe, particularly those of ordered fields with cardinality less than or equal to the continuum. I'm trying to intuitively characterize what makes such sets in particular so appealing to us to study.

I argue there are three major non arbitrary extensions that characterize this: The naturals, the computables, and the dedekind completes.

The naturals: At the very bottom we have the natural numbers, which extends all that is finite. This is the smallest infinite set, with many group operations and other nice functions like addition and distance easy to define in full due to uniqueness of recursively defined functions on them. I don't think much needs to be said on why this set is so appealing.

The computable: The next very natural leap to me is ultimately what is computable, or more accurately numbers that's digits are recursively enumerable. However obviously, there is more structure than just being recursively enumerable. A computable real number is a (possibly infinite) computable sequence of digits 0 to n (depending on which n is the base of our system/the Turing machine alphabet), but we have more structure than this. We have an order structure, and we have distance between numbers, and furthermore the naturals are embedded into this structure such that distance and order between the natural numbers is consistent. While there may be other structures you can put on it, I'd say that is the minimal needed to characterize this set for me in a still natural way (the computable number line).

The last level is dedekind completeness of the second level.

Of these three, the leap from the second to the third is very natural, it's just filling the gaps. However the leap from the first to the second is far more arguable in terms of how many choices we had. This is because there are many extensions of the naturals in between them and the computables which yield the same third set under dedekind completeness. You could for instance, close the naturals under halving (dyadic rationals), or general multiplicative inverses (rationals), or under that and square roots (constructibles). If I had to guess, there's probably no "greatest" or "smallest" theory that's some closure of a function on the naturals while lying in between the naturals and computables and which yields the reals under dedekind completeness. However, that's precisely why I choose to fixate on the computables. Since there's so many options to choose from, the one that feels the least arbitrary is the greatest that's actually "knowable."

What I'm referring to by that is how at any point of time, what you can get from any logically valid combination of all the rigorous knowledge you have right now, is a recursively enumerable set. So in principle you cannot know what is not recursively enumerable, while (with enough time) you can read an encoding of what is recursively enumerable and enumerate it up to the nth element for any n (yes definable numbers are larger and still yield the reals under dedekind completeness while having finite descriptions, but I hypothesize that we can't actually perfectly define mathematical definability beyond "what's definable is a formula in a recursively enumerable theory", which is obviously not specific, so RE is the next best candidate to me).

Of course, this is still not satisfying since the computables actually need a lot more information to have the structure I mentioned than what we needed on naturals. Namely, we add a symbol "." (decimal point) and our order unlike the naturals is dense. As I mentioned, the definition of a distance function is fundamental to me, order is not enough. Even for the reals, just having a dedekind complete dense set may be order isomorphic to the real line, but just having that isn't enough to ensure we have the "length" structure that characterizes it (and is usually denoted by max(a, b) - min(a, b)). A part of me wonders if an algebraic characterization is therefore in some sense necessary to yield this, that you need a field baked in with at least two group operations (addition and multiplication). If that's the case, then it might be that closure under a second operation really is necessary on the path to the real number line (with lengths), thereby disproving my hypothesis.

A possible red herring that comes to mind is the Eudoxus reals, which is a way of constructing the reals directly from the integers without needing multiplication/closure under that (a similar construction can be used to go from naturals to the positive reals, which I want to highlight to show negativity is not an important property in constructing a number line, likely due to it having no relationship with density). While I'm still digesting the construction, it's by equivalence classes on functions on naturals, with addition ultimately being represented by pointwise addition on the functions. However, this field still has multiplication implicitly baked in, as function composition fulfills all the field properties of multiplication. This suggests to me that two group operations, one that distributes over the other, might really be necessary, though if so I would appreciate a proof or other resource to explore this connection.

Otherwise, my question more or less becomes what is the bare minimum needed to get some subset of positive computable numbers with the distance metric d(a, b) = max(a, b) - min(a, b) defined, such that when closed under dedekind completeness, it yields a dense dedekind complete order with the same metric. I also just wonder if the computable numbers themselves can be constructed as some minimal closure over the naturals with respect to inverses of some functions or class of functions.

I am studying Measure Theory from Capinski and Kopp's text, and my purpose of learning Measure Theory is given this previous post of mine for those who wish to know. So far, the theorems have been falling into two classes. The ones with ultra long proofs, and the ones with short (almost obvious type of) proofs and there are not many with "intermediate length" proofs :). Examples of ultra long proofs so far are -- Closure properties of Lebesgue measurable sets, and Fatou's Lemma. As far as I know, Caratheodory's theorem has an ultra long proof which many texts even omit (ie stated without proof).

Given that I am self-studying this material only to gain the background required for stoch. calculus (and stoch. control theory), and to learn rigorous statistics from books like the one by Jun Shao, is it necessary for me to be able to be able to write the entire proof without assistance?

So far, I have been easily able to understand proofs, even the long ones. But I can write the proofs correctly only for those that are not long. For instance, if we are given Fatou's Lemma, proving MCT or dominated convergence theorem are fairly easy. Honestly, it is not too difficult to independently write proof of Fatou's lemma either. Difficulty lies in remembering the sequence of main results to be proved, not the proofs themselves.

But for my reference, I just want to know the value addition to learning these "long proofs" especially given that my main interest lies in subjects that require results from measure theory. I'd appreciate your feedback regarding theorems with long proofs.

Hello all, I was wondering if there was any books or things in the literature that you could recommend that discuss differential equations that contain derivative terms in the argument of functions such as:

dy/dx + y = sin(dy/dx)

Are equations like the solvable or does it break some sort of differential equation rule I don’t know about ?

I read ONAG last year as an undergraduate, but I haven't really seen them mentioned anywhere. They seem to be a really cool extension of the real numbers. Why aren't they studied, or am I looking in the wrong places?

Out of all the classes I've taken, two have been conceptually impossible for me. Intro to ODEs, and Intro to PDEs. Number Theory I can handle fine. Linear Algebra was great and not too difficult for me to understand. And analysis isn't too bad. As soon as differentials are involved though, I'm cooked!

I feel kind of insecure because whenever I mention ODEs, people respond with "Oh, that course wasn't so bad".

To be fair, I took ODEs over the summer, and there were no lectures. But I still worked really hard, did tons of problems, and I feel like I don't understand anything.

What was your hardest class? Does anyone share my experience?

My former calculus teacher claims to have solved the Twin Primes Conjecture using the Chinese Remainder Theorem. His research background is in algebra. Is using an existing theorem a valid approach?

EDIT: After looking more into his background his dissertation was found:

McClendon, M. S. (2000). A non -strongly normal regular digital picture space (Order No. 9975272). Available from ProQuest Dissertations & Theses Global. (304673777). Retrieved from https://libproxy.uco.edu/login?url=https://www.proquest.com/dissertations-theses/non-strongly-normal-regular-digital-picture-space/docview/304673777/se-2

It seems to be related to topology, so I mean to clarify that his background may not just be "algebra"

https://preview.redd.it/htxdzv77lmge1.jpg?width=3024&format=pjpg&auto=webp&s=e85fa5f639f79ae07d3bfbdaed39635875b0d870

Let M be some smooth finite dimensional manifold (without boundary but I don't think this matters). Let U subset M be some open, connected subset.

Let p be in the interior of U and let q be on the boundary of U (the topological boundary of U as a subset of M).

Question 1:

Does there always exist a smooth path gamma:[0,1] --> M such that gamma(0)=p and gamma(1)=q and gamma(t) in U for all t<1?

Question 2: (A weaker requiremenr)

Does there always exist a smooth path gamma:[0,1] --> M such that gamma(0)=p and gamma(1)=q and such that there is a sequence t_n in (0,1) with t_n --> 1 and with gamma(t_n) in U for all n?

Ideally the paths gamma are also immersions, i.e. we don't ever have gamma'(t)=0.

Im fairly certain that alot of people can feel anxious when asking for help on a problem or understanding a concept, me included, so I wanna ask - how do you guys deal with it? Like, I just asked a question on math stackexchange a bit ago, and even though I dont think I said anything outrageous, I've still been having a near panic attack about it since then lol. Sometimes I'll feel so anxious/embarrassed about asking for help on something math related that I wont even message my friends about it, and I dont really know how to fix this.

Im sure that part of it is related to imposter syndrome, and I also have quite bad anxiety in general. However, I still think that most of it comes from the fact that alot of people in math communities (online especially) often act extremely arrogant and have this air of superiority, which makes it really discouraging to ask for help. Although I know they dont represent all mathematicians its still quite unfortunate :/ How does this affect u guys? What do you do about it?

I noticed something strange about this code which I sum up here.
First take digitsConstant, a small random semiprime… then use the following pseudocode :

2. Compute : bb=([[digitsConstant^0.5 ]]+1)^2 −digitsConstant
3. Find integers x and y such as (25^2 + x×digitsConstant)÷(y×67) = digitsConstant+bb
4. take z, an unknown variable, then expand ((67z + 25)^2+ x×digitsConstant)÷(y×67) and then take the last Integer part without a z called w. w will always be a perfect square.
5. w=sqrt(w)
6. Find a and b such as a == w (25 + w×b)
7. Solve 0=a^2 ×x^2 +(2a×b-x×digitsConstant)×z+(b^2 -67×y)
8. For each of the 2 possible integer solution, compute z mod digitsConstant.

The fact the result will be a modular square root is expected, but then why if the y computed at step 2 is a perfect square, z mod digitsConstant will always be the same as the integer square root of y and not the other possible modular square ? (that is, the trivial solution).

Hello all,

I ask as I was considering getting a copy and wanted to know what you thought of it and whether you’d be willing to post any pictures of the layout etc.

I can’t find any pages of it online, only a contents page and that’s about it.

Thanks

Good morning.

I know that is a rather naive and experimental question, but I'm not really a probability guy so I can't manage to think about this by myself and I can't find this being asked elsewhere.

I have been studying some papers from Eric Hehner where he defines a unified boolean + real algebra where positive infinity is boolean top/true and negative infinity is bottom/false. A common criticism of that approach is that you would lose the similarity of boolean values being defined as 0 and 1 and probability defined between 0 and 1. So I thought, if there is an isomorphism between the 0-1 continuum and the real line continuum, what if probability was defined over the entire real line?

Of course you could limit the real continuum at some arbitrary finite values and call those the top and bottom values, and I guess that would be the same as standard probability already is. But what if top and bottom really are positive and negative infinity (or the limit as x goes to + and - infinity, I don't know), no matter how big your probability is it would never be able to reach the top value (and no matter small the bottom), what would be the consequences of that? Would probability become a purely ordinal matter such as utility in Economics? (where it doesn't matter how much greater or smaller an utility measure is compared to another, only that it is greater or smaller). What would be the consequences of that?

I appreciate every and any response.

I came across this beautiful theorem and its proof and fell in love with it. That is why I am so very surprised to learn that IT HAS NO WIKI PAGE IN ENGLISH!!!!

Anyways, I think that this theorem is too beautiful to keep for myself, so I shall share it and its proof with this subreddit.

Notation:

[n]=the set of natural numbers up to n (with the convention that 0 is excluded)

P(X) = the powerset of X, set of all subsets of X.

X|n where X is a set of sets and k is a number = means all elements of X with size n

χ(G) where G a graph = the chromatic number. Least amount of colors needed to color G without neighboring vertices of the same color.

S^(n) = the n-dimensional topological sphere

H(x) where x is a point in S^(n) = the hemisphere of S^(n) polarized at x.

Theorem:

Let the Kneser graph G(n,k) be defined as P([2n+k])|n (the set of all n-length subsets of a 2n+k set) with disjoint subsets being connected by an edge.

The Kneser theorem conjectures that χG(n,k)=k+2.

This theorem itself may seem not that interesting, but first of all if that's what you think I seem you not worthy of living, and secondly, Greene's proof which I am about to present, is one of the most beautiful proofs I've ever seen!!!

Proof:

To show that χG(n,k) is k+2 we first must show a coloring of k+2. So let's take the given k sizes subsets and color them as follows:

We will assign a color to any number from 2n to 2n+k, and a collective color to the numbers from 1 to 2n-1. Now a subset is part of a certain color if it's maximal element is represented by that color. Let's make sure that connected vertices are really of different colors. Let's assume x,y are both in the color represented by the number A. Then x and y both contain A thus are not disjoint sets so are not connected by an edge. But if x,y are both part of the remainder 1 to 2n-1 set, then by Dirichlet principle the must have a joint element thus not be disjoint thus not be connected by an edge. So we know that G(n,k) is k+2 colorable. Since it's k+1 colors for numbers from 2n to 2n+k, and 1 color for the rest.

Now the more interesting part of the proof, proving that it is not k+1 colorable.

To do so, we shall do the bizarre thing of assigning each point in [2n+k] to a point on the topological k+1 sphere, S^(k+1). Let's call the points x(i). We can assume our points are in general position, scattered across the sphere and not lying all on one line.

Now let's assume the existence of a coloring C(i) of G(n,k) with size k+1. We can identify it with a coloring of the subsets of size n of the points x(i). Now let's define the following:

A(i) = {x€S^(k+1)|there exists an element of C(i) fully contained within H(x)}.

And let's define B=S^(k+1)/UA(i). In other words B is whatever the A sets don't cover.

Now A,B together cover the whole S^(k+1) sphere and are exactly k+2 sets. Note that A are open and B closed, so we can use the Borsuk-Ulam theorem to conclude that there exists one of the covering sets that has a pair of antipodal points. Let's call them {v,-v}.

Now there's two options. v€one of the A's or v€B.

Let's assume it's in one of the A's call it A(j). That means that both H(v) and H(-v) contain n-sets of the color C(j). But since H(v) and H(-v) are disjoint sets, also the n-sets contained in them are disjoint, but if they are disjoint, they are connected by an edge as defined in the Kneser graph. But they can't be connected by an edge if they are of the same color C(j). So that is a contradiction. From here we conclude that v can't be in an A set. So let's check if it's in B:

If v,-v€B, then they are not in any A, thus H(v) and H(-v) both dont contain n-sets of any color, since otherwise they would be in an A set. But if they don't contain n-sets of any color, and every n-sets has a color, then they don't contain n-sets. So H(v) and H(-v) both have by max n-1 elements. So that means the line S^(k+1)/H(v)UH(-v) contains at least 2n+k-2(n-1)=k+2 points. But that means that the points x are not in general position, because this line is a k+1 subspace of S^(k+1) so in general position it should have k+1 points.

Isn't this a beautiful connection of topology and graph theory?

My understanding has always been that we introduce the notion of manifolds as analogues of surfaces but in a way that removes the dependence on the ambient space. However, almost all examples you come across in the standard study of manifolds are embedded submanifolds of Euclidean space, making differentiation significantly easier. It’s also a well known theorem of manifolds that they can always be embedded in some Euclidean space of high enough dimension.

If we can always embed a manifold in some Euclidean space and doing so makes computations easier, what is the point in removing the dependence of the ambient space to begin with? Why remove any ambient space if you’re just going to put it in one to do computations?

In regard to this submission

https://arxiv.org/pdf/1603.08540

It's just basic undergrad calculus unless I am missing something? Are these topics not enforced for quality or accuracy? I don't see anything close to number theory here.

I'm working on improving my proof skills and studying for quals by doing a lot of exercises. One issue I'm constantly having is coming up with that first big idea from which everything follows. I usually have to look up this part of the proof and then from there I can fill in the details. After seeing the idea I can see why they did what they did, but I have a hard time coming up with that idea in the first place and I'd like to get better at that.

Here's a very simply example. I was working on proving that if f is in L^1 and g(x) is the integral of f from -infinity to x then g is continuous. I was able to do the trivial part which is reducing |g(x) - g(y)| to bounding int_x^y |f(t)| dt and from there I got stuck. Then I looked up some solutions and saw that you can define sets B_M = {x | |f(x)| > M} for some fixed number M and use the dominated convergence theorem to show that the limit of the integrals \int_(B_M) |f| -> 0 as M -> infinity. From there it's just the usual analysis estimates. In hindsight it makes perfect senes to do this but I don't think I would have thought of it on my own. Is this just something that comes with practice? How can I get better at this sort of thing?

https://preview.redd.it/d8hj6aye2fge1.png?width=1674&format=png&auto=webp&s=ee47db3a66d7310dad6e3fc24b3cbc95ce4c533e

https://preview.redd.it/5kz9xfye2fge1.png?width=1630&format=png&auto=webp&s=3165990ae0f4bf02fe861414bb1e5edbccef0879

https://preview.redd.it/w898feye2fge1.png?width=1630&format=png&auto=webp&s=3bfd52561eba2ab8815485c100f7d617db5c8f38

Hi r/math –

I have had a long-standing geometry question:

Why doesn't the Manhattan distance approximate the Euclidean distance as the width of the city-block goes to 0?

At the limit where the width of the city block goes to 0, the Manhattan path appears the same as the Euclidean path.🤔

As the only boy in my family, my parents often caught me teasing one of my four sisters. At the age of 9, my parents determined this teasing was coming from an excess of energy. Thus, my punishment was to run several laps around the block. We lived on the corner of the block, which consisted of 2 rows of 12 houses, the 2 rows separated by an alley.

□ □ □ □ □ □ □ □ □ □ □ □
□ □ □ □ □ □ □ □ □ □ □ □

__________________________■ <-- my house

At first, I would run. As time went on, I tried to cheat. When my dad found out, he would sit outside on our front lawn and make sure I fulfilled the requirement.

I still tried to cheat.

Going clockwise around the block, I would cut thru the alley on the leftmost side. I would continue thru the alley until I approached the rightmost part of the block, where I would loop around the last house.

At the time, I was studying geometry, and it didn't take me long to discover that cutting between houses wasn't actually shortening anything. Since my dad could observe both corners of the block (bottom-left and top-right), I could find no conceivable way to cheat.

My dad is a smart guy, but to this day, I don't think he realizes how airtight his punishment was.

</story time>

This (true) story led me down a rabbit-hole of trying to reconcile Euclidean and Manhattan distances, but 25 years later, and I'm still puzzled. Any help would be much appreciated!

Note:
The story is funny, but the real question is at the top / in the images.

Hi, I'm currently an undergraduate studying Mathematics. For reference, I have an Audhd diagnosis as well as a wide range of academic interests outside of maths that are more humanities/ologies orientated.

When I do it, I enjoy maths a lot, but I find it hard to motivate myself to do it, and while I understand the bigger picture of how mathematics is laced into all of my other interests, it can be hard to remind oneself of that all of the time.

Are there any books you'd recommend that might excite me to do maths, whether it might be more of fictional/story or more a slightly more technical book, the purpose not to teach me maths but to remind me why this is such an interesting and exciting subject. Thanks for any advice

Just like the title says. I have a hard time structuring the work I do on math problems and alot of the time I have a hard time finding stuff I wrote because it’s all just a jumbled mess. On tests I try to make it as readable and structured as possible, and it seems to work, but when I go back to problem solving it all becomes unstructured again. It’s fine because it’s only me reading it, but I would like to know, how do you structure your work?

This recurring thread is meant for users to share cool recently discovered facts, observations, proofs or concepts which that might not warrant their own threads. Please be encouraging and share as many details as possible as we would like this to be a good place for people to learn!

In your opinion who is the mathematician with the coolest Name, that makes you go "well that's kinda interesting" when you think about it?

Maybe because it actually uses math terminology, or it is just befitting to him or her as a human.

I don’t know if it’s because I have a hard time paying attention, or that my notes aren’t the best, or that maybe I’m not built for this path in pure mathematics, but I personally feel like I’m learning considerably less than I should be learning. Only when I put hours and hours of time into conceptually (not even understanding concepts through the proof) understanding the content can I say that I genuinely understand it. For example, in my probability theory course I have written down the Caratheodory theorem three times in my notes so far, but if you asked me why it’s so fundamental, I don’t know. With that being said, I want to know if others relate to this problem, or if it’s just me.

The international maths day is celebrated on the 14 of march, different schools in different part of the world have different activities organised specifically on this day. I am very curious what and how other schools in Asia and the Middle East do for this day.

I am currently working in Europe but I feel like this year I could implement something from another part of the world, new and interesting but also challenging.

Ive seen that a lot in mechanics, where polar and cartesian coordonates are just different representations of the same that they share magnitude and angle where components have multiplicative conversion.

What made me curious is: what if they share 2 of 3 attributes? Lets say mangitude and angle but dont have multiplicative but additive conversion? Or dont have magnutide but the rest is the same?

What does that mean logically in spatial language?

If all 3 are the same we can say that axies are identicall but in different reprrsentations. But if 2 are the same does that mean that there is different scaling (if magnitude is different), projection (if angle is different) and mapping (if axies are different)?

Hi, I was wondering if there are any other similar sites/projects that are similar to project Euler. In that problems can be solved or helped computationally, you can submit your answer to get checked and the problems encompass different math areas and levels of difficulties.

I would like to hear how other universities schedules are, in my third semester I take group theory, probability theory, ODE, metric and topological spaces, numerical analysis and a science course. I would like to hear how other universities organise their semesters and courses, I will add that in my university most students don't take all the courses above but it's recommended by the university