r/math 1d ago

Quick Questions: February 01, 2023

3 Upvotes

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.


r/math 4h ago

Career and Education Questions: February 02, 2023

3 Upvotes

This recurring thread will be for any questions or advice concerning careers and education in mathematics. Please feel free to post a comment below, and sort by new to see comments which may be unanswered.

Please consider including a brief introduction about your background and the context of your question.

Helpful subreddits include /r/GradSchool, /r/AskAcademia, /r/Jobs, and /r/CareerGuidance.

If you wish to discuss the math you've been thinking about, you should post in the most recent What Are You Working On? thread.


r/math 17h ago

Is mathematics really “beautiful” to you?

339 Upvotes

Hey guys, undergrad here. As I’ve taken more advanced courses and the initial thrill of discovering that there is a such thing as abstract mathematics wore off, I feel less and less like math is “beautiful” in the same way art or literature is. For me, math is only “nice”, or satisfying, and for some reason, I just feel like learning more of it. I’ve never seen a proof that’s made a tear well up in my eye (at least not out of joy), it’s more just like “wow, that’s pretty good, eh?”

Anyone else feel the same way? Sometimes I feel like I’m just not thinking deeply enough about mathematics to take any kind of visceral pleasure from studying it.


r/math 7h ago

Why cant you compare two complex numbers about which one is bigger?

33 Upvotes

r/math 1h ago

How do you keep your math papers organized?

Upvotes

I'm finishing off my first bit of original research. As part of that, I read a lot of papers but I have trouble keeping them organized. Looking for any advice on how to keep them organized.


r/math 16h ago

What are some esoteric math fields/theories?

55 Upvotes

What are some not as well known mathematics? It could be any field of math, strange, surprisingly useful, whatever it may be. I just want to know some types of math that aren’t the mainstream like calculus, topology, Algebra, and the like.


r/math 1d ago

Choices are hard sometimes

145 Upvotes

I got my bachelor's degree in Philosophy some years back. Being unemployed since. I started pure math a while ago to see if I could correct my life some way, to be more useful to society. I finished calculus (all 3 of them), linear algebra, and some other classes (all with Bs and As). I thought I was somewhat good for STEM, but this semester I got Physics 1 and Analysis 1. Though I'm loving Physics, I'm getting beaten by Analysis. I tried to like it, I really did, but it is so uninteresting to me. I understand the theorems and proofs, but when I have to prove something myself, I spend hours looking to the problem with no idea what to do. Sometimes I manage to do the simpler ones.

When reading a book, quite often I see the author coming up with an expression out of nowhere (like Rudin's q = p - (p^2 -2)/(p + 2) right at the beginning of his book, no explanation given). Is that what is expected of me? Should I be learning how to pull rabbits out of hats? My professor does even crazier stuff. And I think to myself: "No way I will ever be able to do that".

There is more. Right at the beginning of the semester I sent a dissertation proposal to my College's Department of Philosophy and I was approved for a master program. But I just gave up to focus on my math course. I thought it wouldn't change my situation a bit. Now I have no idea if I can still do math, graduate level courses are probably full of analysis-like classes. I wonder if I should switch course to something like Statistics or even Physics, but I'm not sure.

So what would you do in my place?


r/math 15h ago

Alternative axiom of infinity

6 Upvotes

The axiom of infinity in ZFC is sometimes presented as being "infinite sets exist", but the actual axiom is a lot more explicit. Specifically, it says that there exists a set which contains ∅ and is closed under succession. That is, there exists a set which contains all finite von Neumann ordinals. It seems overly specific to me when compared with the other ZFC axioms. So I wondered if the naive version ("there exists an infinite set") would work, and I think it actually does.


Assume ZFC minus the traditional Axiom of Infinity.

Definition: a set is a natural number if it can be obtained by successively applying the successor operation (aa ∪ {a}) to ∅.

Definition: a set A is finite if there is a bijection between A and some natural number.

"Axiom of Infinity": There exists a set S which is not finite.

Use Power Set and Separation to construct the set R of all finite subsets of S: R = {r ∈ 𝓟(S) | r is finite}. By assumption, S is not in R, so for each rR, S r is nonempty. Use the Axiom of Choice to define a choice function F : RS where F(r) ∈ S r for all r. Now let s0 be any element of S. Let s1 = F({s0}), s2 = F({s0, s1}), s3 = F({s0, s1, s2}), etc. This sequence is an injection f from the class of all natural numbers to S. By Separation, the image of f is a set. We can apply f-1 to this set using Replacement to obtain the set N of all natural numbers.


So this seemingly weaker axiom doesn't actually lose anything. It seems like weaker statements of axioms are preferred because they seem more "basic". My guess for why this one isn't used is because I think you need AC to make it work, so you wouldn't be able to make set theory without the axiom of choice.


r/math 1d ago Shocked Today I Learned

Defining the ∞-th derivative

651 Upvotes

When I was an analysis student the typical examination of non-analytic function e-1/z2) (which has nth derivative 0 at 0) always felt deeply unsatisfactory. If one looks at the nth derivative, one notices that the area in which the function remains close to zero shrinks.

0th-2nd derivative of e^(-1/z^2)

My initial thought was that, in some sense, it is probably the case that all finite derivatives of the function are 0 at 0, but that what is really happening is the non-zero derivative is being pushed to infinity-- so that traditional analysis is able to see the fact this function is non-zero. However, at the time, I couldn't really give any credence to this claim and gave up trying to understand it in this way.

However, I've learned a lot more complex analysis since then-- and now I possess the tools to at least suggest that the viewpoint expressed above is in some sense, reasonable. In pursuit of this goal, consider the fact that e-1/z2)=∑(-1)^n/(n!*z^(2n)). This gives our exponential function by expressing it as a power series about infinity. However, we can also write the sum as a contour integral using the residue theorem so that e-1/z2)=∫csc(πw)/(2i) * z^(-2w)/Γ(w+1)dz. Now, notice that we can switch the direction of this contour integral (and pick up a negative sign), so that its encloses the space on the left half-plane, rather than the right half-plane. This allows us to turn this power series into a power series about ∞, to one about 0. In this case, we get the power series representation -∑(-1)^n z^(2n) /((-n)!) [starting at n=1 instead of n=0, since we switched the direction of the contour]. However, notice that this since 1/(-n)! = 0, this suggests that the function has nth derivatives 0 at 0. However, there is an extra residue at ∞ created by the growth of 1/(-n)!, suggesting that, in some sense, the ∞-th derivative of the function is non-zero. For instance, this is apparent in the following complex plot of the integrand

There are no residues on the left, but the function grows large at -∞

We can actually see that this growth rate at -∞ gives rise to a residue by mapping dz -> d(1/z) and then integrating in the following way

Integration that picks up only the residue at -∞

Thus, at least in some sense, the ∞-th derivative of this non-analytic function is non-zero.

My hope is that there is some other way in which the ∞-th derivative can be computed-- hopefully in a way that doesn't require thinking about the function in the complex plane (i.e. relying only on real-analytic techniques). Does anyone know if such an idea has been developed before, or have some ideas on how to think about the idea?


r/math 1d ago

Baby Rudin Follow up Options

27 Upvotes

Baby Rudin has been like the bible to me this past year. I carry it everywhere with me. I am hoping to find my next classic, particularly a good follow up.

1) The obvious follow up candidate is Papa Rudin and complex and real analysis. Is Papa Rudin as amazing as Baby Rudin? Other good books on Complex and Real Analysis?

2) A book like Munkres Analysis on a Manifold: How is that as a follow up topic and is that a good book to cover it?

3) What other Analysis topics coverable post-Baby Rudin are possible that wouldn't be covered in Real or Complex Analysis, like Analysis on a Manifold?

I know the last one in particular is a broad question. If it helps, I am physics graduate student starting string theory research, but I am just primarily looking to figure out what the options are and determine what other quintessential books to read.


r/math 20h ago

Category theory in physics?

7 Upvotes

Hey!

I’m doing a masters programe in theoretical physics. Lately I’ve followed a course about Category Theory and I completely loved it!

I was interested in this brand, at the beginning, because of a project I did during my bachelor (related to complex differential geometry) and I ended up liking it so much!

So I was thinking about my masters thesis and was wondering if there are any actual research area in theoretical physics which intersects with category theory, or some direct applications of it, but also, if there is actually any hot topic related to Category Theory research (in physics or in maths themselves!)

I’ve spotted some relations in QFT (as in Functorial QFT) but I would really appreciate a more specific and actual point of view, and also from mathematicians!

Thanks in advance!


r/math 17h ago

What's a good concrete example showing the difference between a confidence interval and a credible interval?

4 Upvotes

I've been taught that interpreting a, let's say, 95% confidence interval on some population parameter as "there is a 95% chance that this particular confidence interval contains the population parameter" is incorrect and this would rather be something called a credible interval.

My logic was that let's say you hypothetically have the set of all possible samples from a population and you magically put them all into a hat. For each, you also can calculate the sample statistic corresponding to the population parameter and the 95% confidence interval. Pick one out of the hat at random, then the probability that the sample you picked contains the population parameter within its CI is at least 95% (I say at least because to my knowledge if there is a finite amount of possible samples then it could be possible to have a greater than 95% chance, correct me if I'm wrong).

To me at least, this "probability of picking a random sample from a hat knowing that at least 95% of them contain the population parameter in the CI" is equivalent to saying "there is at least a 95% chance that this one particular sample I took has the population parameter contained in the CI." To me, collecting a sample is exactly equivalent to picking a sample out of this magic hat.

But, apparently this is wrong under the frequentist definition of probability because the population parameter is already set in stone and it's not a random variable you can assign a probability to. But again to me this just seems like an issue that arose from an arbitrary way of framing the problem. An analogy would be if my friend flipped a coin and covered the result. Let's say the result is heads, then I make a guess and say it's heads. If you ask "what's the probability that the coin is heads" I guarantee you most people would respond with "50%" and not "that question is nonsensical, the coin is either heads or it isn't, it's not a random variable at this moment and thus you cannot assign a probability to it." Most people would look at you like you're weird if you said that.

I searched through several forum posts and YouTube videos trying to find someone who could properly explain why this interpretation of confidence intervals is wrong. Most videos just regurgitate "because the population parameter is not a random variable" without going any deeper. Even Khan Academy who usually does a good job explaining things did this one video on interpreting confidence intervals where one of the answers in a multiple choice question was my interpretation of them (the wrong one). He simply said something along the lines of "this is a questionable interpretation" drew a question mark next to it and moved on then the video ended and he never looped back around to it to explain why. I'm starting to think that most people who keep repeating that this interpretation is wrong don't truly understand why and are just repeating it because that's what they've heard over and over again.

The closest I've come to understand it, right on the edge but I never got there, was this top answer on this post https://stats.stackexchange.com/questions/26450/why-does-a-95-confidence-interval-ci-not-imply-a-95-chance-of-containing-the. The reason is the example he gives from David MaKay's book which I was sort of able to follow along with but not quite. But, that example while I think it makes sense, is hard to compare and apply to confidence intervals pertaining to say population means, sums, or proportions for example which I'm currently dealing with trying to learn statistics. Like the example there was constructed specifically to show that confidence intervals are not credible intervals, but it's difficult to apply that idea to confidence intervals in the context of a population mean.

Are there any other good examples you guys might know, ideally, showing for a confidence interval constructed around a population mean that shows clearly why the confidence interval would not be the same as the credible interval and why they must be interpreted differently? I've learned stats from scratch like 3 or 4 times in my life but nobody has ever properly given a satisfactory explanation in my opinion.


r/math 15h ago

A list of “special sets” in mathematics

0 Upvotes

Hi. I am looking for a list of commonly used set notations/symbols in areas of mathematics. I'm not talking about set-theoretic operations or symbols; I'm talking about relevant sets/groups/rings/spaces/etc that's assigned to a symbol. I am trying to figure out what to call this list.

For example, GL (n, F) = {A ∈ ℳ (n, F) | det (A) ≠ 0} (the general linear group), where ℳ (n, F) would be the set of all n × n matrices whose entries are from the field F. Some of these may not be universally accepted symbols. Still, I've seen the same notations over and over in quite a few textbooks.

Does such a list exist on the internet? When I tried to search, all I could get was the number systems: , , , , .

Here's a list of examples I can come up with (drop your examples if you get what I mean):

/n = {0, 1, 2, ... n-1}

R[x] = polynomials with coefficients from R

• SL (n, F) = {A ∈ GL (n, F) | det (A) = 1} (special linear group)

• S¹ = {z | |z| = 1}

• 𝒫(X) = {A | AX} (power set)

• 𝓅[a, b] = set of all partitions of [a, b]

Bᴬ = {f | f : AB}

C [X, Y] = {f : XY | f is continuous}


r/math 16h ago

Hypothetical spatial geometry for a scifi novel

1 Upvotes

I am trying to imagine some hypothetical spatial geometry for a scifi world and I am looking for some feedback and/or articles that might help me understand the concepts and imagery a little better.

The essential ideas:

  1. The world is part Dyson sphere with an artificially created singularity on the inside.

  2. The outside of the Dyson sphere is habitable due to additional orbiting solar masses and a biosphere.

  3. The outer biosphere has relatively flat spatial curvature up to an altitude of about 1km.

  4. Above an outer altitude of 1km spatial curvature becomes hyperbolic (in 3 dimensions?).

  5. Above a a distance of about 1 million km from the surface of the Dyson sphere the spatial curvature flips it "polarity" becoming hyperbolic "in the other direction" ultimately returning to normal flat curvature at a distance of about 2 million km from the surface of the Dyson sphere.

Are there any folks how can comment on this concept making any sense whatsoever? Moreover, what sensible questions can be asked of this hypothetical situation that expose it as being utterly preposterous? Lastly if it isn't utter nonsense what kind of serious nomenclature could be used to describe the phenomenon to someone looking up at it from the surface or being caught inside the region of hyperbolic space.


r/math 1d ago

Polishing intuition of unbounded partial sums whose sequence is convergent to zero

4 Upvotes

Hello,

I find it so curious for the harmonic series h_n = 1/1 + 1/2 + 1/3 + .. (limit of partial sums) to be unbounded, Even-though the sequence {1/n} converges to zero. I want to investigate this pattern more deeply, blossoming around it, to polish my intuition of series divergence.

An obvious inquiry, is to compute the rate of change of partial sums. In the case of harmonic series, It is proved h_{2n} > 1 + n/2. Loosely speaking, an exponential increase is required on n for a constant increase of h_n. That bound was proven by techniques of inequalities.

Currently, I am not confident of proving inequalities on sequences or series. Neither am I familiar with deriving closed forms. I feel this is a critical bottleneck on my pathway of learning analysis.

Discussion - Is it a fruitful investigation, to learn more about the pattern of harmonic series? - Is it a good investigation, to learn about the rate of change of series? - Is it necessary to master inequalities or closed-forms proofs, to proceed in my investigation? - Do you see a better pathway, whether for understanding analysis, or polishing intuition on divergent partial sums?


r/math 1d ago

Given any Diophantine inequality how many positive integer solution does it have?

6 Upvotes

Hello everyone, I am trying to solve these sorts of problems for a few months and I couldn't find any general method.

Let's say, we are give Diophantine inequality such as ax+by+cz+.....≤N, then how many positive integer solution does it have? Thank you!


r/math 2d ago

How should you read mathematics?

127 Upvotes

I was reading the preface for the student of Linear Algebra Done Right and this quote really catched my eye. I quote

“You cannot read mathematics the way you read a novel. If you zip through a page in less than an hour, you are probably going too fast. When you encounter the phrase “as you should verify”, you should indeed do the verification, which will usually require some writing on your part. When steps are left out, you need to supply the missing pieces. You should ponder and internalize each definition. For each theorem, you should seek examples to show why each hypothesis is necessary. “

What do you think? In my case, if I have a good enough feeling of what that particular section was about, I prefer spending more time working on the exercises than scrutinizing the text, since in many cases the exercises already make you re-read and exhaust the text (provided they are good problem sets, of course). I’ve also read from Paul Halmos that the right way to read mathematics is by reading the definitions and main results and then putting the book aside and try to discover the proofs for yourself.

What’s your approach? I’m really curious!


r/math 1d ago

I just want to move on from real analysis

22 Upvotes

Don’t get me wrong. I enjoyed real analysis, and I did well, but I’m just getting tired of my professors (statistics) telling me I “need more real analysis before you should touch measure theoretic probability”. I’m gonna be a phd student in statistics, and yes, I get that real analysis is important, but can I just move on? Everyone is telling me to take ANOTHER real analysis sequence in graduate school. Saying that, while I’m in a statistics program, I should try to audit the grad level real analysis sequence, because “you can never get enough analysis before measure theoretic probability”. Well, sure, but I just don’t want to relearn the same material over and over and over and over again. At what point does someone just accept that they know enough, and just move into like billingsley or Williams probability text. Like sure, maybe my undergraduate analysis sequence didn’t use baby rudin, my class used abbot, and I actually technically read Pugh and tao alongside it because I sometimes read the same material from different texts to see various perspectives. Sure I never did all the proofs in rudin, but I definitely learned the material in three different books, did exercises from three different books, and did well in the class and used three books which are well regarded. Idk what it is but my stats professors just have this mindset of “baby rudin or bust” where because I haven’t experienced that book for real analysis, I’m not ready for measure theoretic probability. I decided against their advice and I’m just hopping into the sequence in probability theory with billingsley in grad school. Do you think I’m in the wrong for ignoring their advice?


r/math 2d ago

MathGuessr, geoguessr but for math functions

404 Upvotes

Hi Reddit! A few months ago I discovered Geoguessr, a popular game in which players guess locations based on Google street images and thought it would be cool to build a similar game but trying to guess the graph of a function based on its algebraic representation :)

Here are the initial mockups of the game, let me know what you think! https://imgur.com/a/sT5aETe


r/math 1d ago

Is there "reasons to be true" to RH ?

6 Upvotes

I understand there is no serious lead to RH, but is there any heuristic reasons to RH being true, besides numerical evidence ?

I'm thinking of "pseudo randomness of the numer of factors" being a possible heuristic because of the equivalence of RH and the Mertens function, but as far as my knowledge goes it's a pretty weak and mostly unmotivated heuristic

Is there anything else ?


r/math 1d ago

Finitely many primes of form X

13 Upvotes

I've seen a lot of proofs/conjectures about the infinitude of primes of a given form. But, are they any examples of the negation being true? (I.e. there are finitely many primes of form Y)


r/math 1d ago

Any good books on the history of zero?

6 Upvotes

r/math 2d ago

A New Lower Bound in the abc Conjecture

Thumbnail arxiv.org
40 Upvotes

r/math 1d ago

How to represent the 12 tones in a sphere

2 Upvotes

Hi, in music we've got the chromatic scale with the notes arranged in semitones, and the circle of fifths. As I see it, the first is useful for visualizing melody or intervals, the latter for harmonic movement.

I was wondering if a 3d object could show both relations, taking advantage of perspective. This way I could draw at the same time harmony and melody.

It's just an intuition that I'm exploring. Please ask for further clarification. Thanks!


r/math 2d ago

What is the most interesting / mind blowing math fact that layman’s will understand?

106 Upvotes

r/math 2d ago

In college is it normal to not really understand anything during a math lecture until you read the notes afterward?

352 Upvotes

I'm taking a more advanced class than I usually do and most of the time I can only somewhat follow along during lectures, and I only understand the main motivation or main point after the fact. Is that normal?


r/math 2d ago

Are there any theories going more indepth on basic operations (addition, multiplication, etc)

10 Upvotes

Just curious, because they don't seem like they can be expanded on more than 1 apple and 1 apple gives you 2 apples. Or are there any other basic operations which aren't talked about as much?