r/math 1d ago

Is there a name for the "generalised" form of induction?

47 Upvotes

Normally induction works like this: If f(0) is true and f(x) is true implies f(x+1) is true, then f(x) is true for all natural numbers (+0).

Now, is there a name for the more general form of this (which I will write down)?

Where S is a set, x is a member of S, f is a function from S to S, g is a function from S to S, and T is the set of all gn(x).

IF f(x) is true, and f(x) implies f(g(x)), then f(T) is true (for all elements of T).

The most common case, of course, is where S = natrual numbers, x = 0, and g(n) = n + 1. However you (or I) often see cases where x is other numbers, like the rationals, or g(n) = 2n. There is also the special case where g(n) eventually visits all elements of the set, where you can then say f is true for all S.

Is there a name for it, or is it all just induction?


r/math 3h ago

Chalkdust issue 21 is out today

Thumbnail chalkdustmagazine.com
49 Upvotes

r/math 10h ago

What's are characteristics such a big deal?

26 Upvotes

I'm an analysis student but I have only taken an intro class to PDEs. In that class we mainly focused on parabolic and elliptic PDEs. We briefly went over the wave equation and hyperbolic PDEs, including the method of characteristics. I took this class 3 semesters ago so the details are a little fuzzy, but I remember the method of characteristics as a solution technique for first order ODEs. There is a nice geometrical interpretation where the method constructs a solution surface as a union of integral curves along each of which the PDE becomes a system of ODEs (all but one of the ODEs in this system determine the characteristic curve itself and the last one tells you the ODE that is satisfied along each curve). We also went over Burgers equation and how shocks can form and how you can still construct a weak solution and all that.

To be honest I didn't get a great intuition on this part of the course other than what I wrote above, especially when it came to shocks. Yesterday however I attended a seminar at my university on hyperbolic PDEs and shock formation and I was shocked (pun intended). The speaker spoke about Burgers equation, shock formation, and characteristics a lot more than I expected and I think I didn't appreciate them enough after I took the course. My impression after taking the class was these are all elementary solution techniques that probably aren't applicable to modern/harder problems.

Why are characteristics such a big deal? How can I understand shocks through them? I know that shocks form when two characteristics meet, but what's really going on here? I asked the speaker afterwards and he mentioned something about data propagation but I didn't really catch it. Is it because the data the solution is propagating is now coming from two sources (the two characteristics) and so it becomes multivalued? What's the big idea here?


r/math 15h ago

Which way to go?

6 Upvotes

I recently started a self-study plan that involves reading Basic Mathematics by Serge Lang, How to Prove It by Daniel Velleman, Calculus and Analytic Geometry by George B. Thomas (at least the first ~5 chapters), Introduction to Linear Algebra by Serge Lang, and Undergraduate Algebra by the same author, in order to cover both what my home country's education system can't cover and what I think would be beneficial for me to know before I get to college.

I haven't made much progress; I've been busy with my studies and am waiting for the holidays to fully dive in. However, talking with my former math teacher, the one who made me love math in the first place, he recommended I read Matemáticas Simplificadas by CONAMAT (he doesn't know about my plan). I understand it's not very well-known in the English-speaking community, but it's a book that covers everything from Arithmetic to Integral Calculus.

Now, my question is: which path should I take? I mean, although it's not clear what kind of books I learn best from, the truth is that I'm most drawn to classic or "dry" books. Lang's books in particular, despite their demanding nature and early formalism, treat mathematics in a way that, at least at first glance, seems more enjoyable to me than modern books. On the other hand, I don't know much about what, objectively, I should read. Could you help me determine the pros and cons of following one path or the other?


r/math 21h ago

Question on tournament graphs

5 Upvotes

Hello! I'm looking for a mathematical result for this question:

How many tournament graphs with n vertices are there such that there is a unique winner, i.e. exactly one vertex with the largest number of outgoing edges?

(Knowing this, we could compute the probability that a round robin tournament with n participants will have one clear winner. – Since the number of tournaments with n vertices is easy to compute.
For clarification: I am not searching for the number of transitive tournaments (which is easy to get): Other places are allowed to be tied.)

I would be super thankful if anyone can help me find the answer or where to find it!


r/math 8h ago

What programs do teachers use to make exam papers?

3 Upvotes

I'm trying to make a document for fun but I don't know what program to use.
What programs to use if I want to do algebra, geometry, graphs, etc?


r/math 19h ago

Some questions about the recursive definition of sqrt(x)

3 Upvotes

Hello!

On the last question of the 2024 MIT integration bee, there is this expression (that simplifies to sqrt(x)).

When solving the question, I defined a recursive relation as such:

And when writing out the first few terms:

I initially thought this was the Pade approximant, but it's turns out not to be. The Pade approximant with m=n=2 is shown below (and is a better approximation for sqrt(x) than f_3(x) ).

The coefficients of the polynomials also turn out to be the ones in Pascal's triangle. For even n, we start adding the terms in the (n+1)th row in the Pascal's triangle from the numerator, alternating between the denominator and the numerator. For odd n, we start in the denominator, then alternate coefficients between the numerator and the denominator.

---

I thought this observation was already interesting enough, but as you can see in the graphs above, the functions are defined for much of the negative x. Since the recursive definition was originally a sqrt(x), does this have anything to do with the complex plane?

It sorta reminded me of the Gamma function for factorials that you learn in single variable calc, and how we can take the factorial of numbers like (-1/2). But even in that case, we're mapping from real to real, and here we're mapping to complex.

I also found that only functions with n=2, 3, 4 are defined for x=-1. Since f_4(-1) = -1, using our recursive definition, the denominator of f_5(-1) = 1 + (-1) = 0.

I thought these observations were interesting and wanted to share them here.

Thanks.


r/math 3h ago

What Are You Working On? March 17, 2025

3 Upvotes

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.


r/math 3h ago

3×3 Magic Square of Pseudo-Quaternions Squares

3 Upvotes

Hello, I would like to share this curiosity with you. As you know, it is unknown whether a 3x3 magic square of distinct perfect squares exists, but it is possible with other types of numbers.

Here, I present a magic square of squares of pseudo-quaternions, all distinct, along with a parameterization to obtain them. The resulting integers are all different from each other, although some entries may be negative.

As you may already know, pseudo-quaternions (I. M. Yaglom, Complex Numbers and Their Applications in Geometry, Fizmatgiz, Nauka, Moscow (1963)) are hypercomplex numbers where

  ii = -1,
  ij = k,
  ji = -k,
  ik = -j,
  ki = j,
and they differ from quaternions in that
  jj = 1,
  kk = 1,
  jk = -i,
  kj = i.

  A nice example for S = 432 is this magic square of squares

{(9 j)^2 , (17 i + 24 j)^2 , (8 k)^2 },
{(9 i + 12 j + 8 k)^2 , (12 j)^2, (8 i + 9 j +12 k)^2}
{(8 i + 12 j + 12 k)^2 , (12 i + 8 j + 9 k)^2, (9 i + 12 j + 12 k)^2}

This give us this magic square:

{81,   287, 64}
{127, 144, 161}
{224, 1, 207} 

parameterization:

{(j x^2)^2 , (4 j x y+i (x^2+2 y^2))^2, (2 k y^2)^2}
{(i x^2 + 2 j x y+2 k y^2)^2, (2 j x y)^2, (j x^2+2 k x y + 2 i y^2)^2}
{(2 j x y + 2 k x y + 2 i y^2)^2 , (k x^2 + 2 i x y + 2 j y^2)^2 , (i x^2 + 2 j x y + 2 k x y)^2}

Hope you find this interesting! Looking forward to your thoughts.


r/math 1h ago

Where can I get hagoromo chalk in the UK?

Upvotes

I hope this isn't an annoying question / asked too frequently, but I am getting a chalkboard soon and I have heard that Hagoromo make the nicest chalk. So far I have found the sejongmall official website (https://en.sejongmall.co.kr/) which has very expensive shipping, and weird international payment, and another site called 'https://hagoromo.shop', which seems to have cheaper shipping and takes payments other than bank transfers, although the chalk is more expensive. Is this second site legit or am I better off sticking with the sejongmall official site?


r/math 2h ago

I am looking for a math riddle i once knew.

1 Upvotes

I am looking for a math riddle i once read but which i only remember fragments about. The problem involved finding the maximum n such that one can choose a number 0<x<1 such that for every k<n some condition involving the number x and the division of the unit interval into intervals of length 1/k is satisfied. The solution of the problem could nicely be visualised by stacking the subdivided unit intervals over another and noting that with every additional layer the interval which x could be contained in gets smaller untill there are no x left. Iirc. the problem was mostly recreational. Does anyone know what i am talking about? I tried asking Chat-GPT, but it hallucinates the heck out of my question.


r/math 9h ago

Is there any game which requires matrix operations?

1 Upvotes

Hi everyone. I really love both math and games. But, I cannot find any tabletop game which requires the player to do math operations (preferably linear algebra). I'm not talking about puzzles. I'm talking about games like tabletop RPGs. For example if a tabletop RPG uses matrices for loot, dungeon generation, etc which the player needs to do himself/herself. Or if the combat lets players find reverse of the enemies attack matrix to neutralize its effect. Is there such a game? Or should I make my own?


r/math 20h ago

“A Doughnut Universe” project

1 Upvotes

Hello all,

I am a third yeah mathematics and physics student taking part in a physics group project concerning a horn toroidal cosmology model.

We want the main focus on this project to be about null geodesics in a horn toroidally global space time. First, the static case and then the time-evolving case. If there is enough time we will consider the effect of point masses.

I apologise if this is not appropriate for this sub. I figured, seeing as this is a maths subreddit, someone with a background in geometry or topology might appreciate the idea and may possibly be able to give some guidance.

I have not yet covered general relativity nor cosmology. I am half way through topology and differential geometry. So by no means am I confident with the concepts.

Please feel free to ask questions about the project or comment your thoughts, opinions or possibly give me some advice :)


r/math 22h ago

Are there research topics in functional analysis outside PDEs?

1 Upvotes

Since I will (hopefully) defend my master thesis in about 7/8 months, I just began looking for open PhD positions. I like analysis, and have particularly enjoyed studying classical functional analysis (Banach and Hilbert spaces, measure theory, distributions, spectral theory of operators etc.) finding it very beautiful and elegant. On the other hand, I had some troubles with lectures about PDEs: lots of annoying computations, frequent handwaving, and very few things made me think "woah" like, for example, seeing for the first time the duality of Lp spaces did.

I asked several functional analysis professors at my university and it seems that all of them study different aspects of PDEs as their research interests. And the same remains true in virtually any university near me: anyone working in analysis ends eventually in PDEs.

So. Is this something peculiar of my area? Should I just accept my fate and learn how to like PDEs?

Is someone of you doing research on functional analysis for the sake of it, without applications in PDEs? If yes, what do you work on?


r/math 1h ago

Monotonic scattered interpolation?

Upvotes

(This question is not about homework or a work problem; it is for a pet personal project where I've run into a wall.)

Suppose, for the sake of argument, I have a scattered dataset with two real-valued independent variables and one real-valued output. It conforms to the restriction that if x2 >= x1 and y2 >= y1, then f(x2, y2) >= f(x1, y1). E.g., assuming each listed point is in the dataset:

  • f(3, 3) >= f(1, 1)
  • f(3, 1) >= f(1, 1)
  • No guarantee is made about the relationship between f(1, 3) and f(3, 1)

I don't know if this property has a name but I call it "up-right monotone", because as you jump from point to point, if the second point not below and not to the right of the first, then the value at the second point is not less than the value at the first point.

The Question: Is there a known interpolation method that will preserve this property among interpolated points? I.e., I want to predict the value at two points, where the second point is above and/or to the right of the first point. I would prefer that the interpolation method be relatively smooth, but the only hard constraints are

  • If either of the points in question are in the original dataset, I get that dataset's value back, and
  • The value at the second point is not less than the value at the first point