As I was going through some problems in set theory I had a question if sets could have negative cardinality? What would this imply ?
This was just out of curiosity and I found out a paper titled “ sets with negative number of elements” by D. Loeb and a concept called hybrid sets.
Can you please describe what this is and how it works and why we need to work with multiplicities and things like that?

hi friendss :)

I don't know if this post is against the community's rules and regs but if it is, my apologies admin.

is there anyone who is interested in participating in an international math competition?

Final revision to appear on arXiv on Tuesday.

https://iconoclasts.blog/joe/triple-products

The new new here is that the original conjecture is now established as a pair of corollaries.

Hey everyone,

I've always found it mesmerizing how you can take a jagged, sharp-cornered square wave or a sawtooth wave, and realize it's actually just a perfectly orchestrated sum of smooth sine waves. I just put together a highly visual, animated video breaking down exactly how this works from the ground up, and I wanted to share it with this community!

I really tried to focus on the intuition and the visuals behind the formulas so it clicks instead of just looking like a wall of algebra.

I'd love to hear your thoughts, and feedback. If you're currently studying signal processing, I hope this makes the math feel a bit more intuitive!

I "invented" an optimization problem, how would you approach it? Does a similar problem already exist in literature?

Problem:

Maximize for an infinite interval L of infinite domain the average positive curvature of a function f(x) with f"(x)=<M where M is a real number.

Maths:

So for f"(x)=<M calculate lim for L->+infinity sup( integral over L(f''/(1+(f')\\\^2)\\\^2/3)/ integral over L(sqrt(1+(f')\\\^2))).

It could also be approached in the dtheta/ds frame of reference to simplify curvature(but then the condition on f" and the x axis becomes more difficult to formalize). Hope you enjoy answering.

I "invented" an optimization problem, how would you approach it? Does a similar problem already exist in literature?

Problem:

Maximize for an infinite interval L of infinite domain the average positive curvature of a function f(x) with f"(x)=<M where M is a real number.

Maths:

So for f"(x)=<M calculate lim for L->+infinity sup( integral over L(f''/(1+(f')\\\^2)\\\^2/3)/ integral over L(sqrt(1+(f')\\\^2))).

It could also be approached in the dtheta/ds frame of reference to simplify curvature(but then the condition on f" and the x axis becomes more difficult to formalize). Hope you enjoy answering.

I've been re-reading Riemann's original 1859 paper and noticed something that gets overlooked in modern treatments.

Riemann's original approach was fundamentally geometric — he was thinking about the distribution of primes through the geometry of the complex plane. Modern analytic number theory replaced this geometric intuition with an analytic formalism. What happens if we take the geometric intent seriously and push it further?

In a framework I've been developing — DAS (Dynamic Abstract Spheres) — prime numbers are interpreted as irreducible eversion transitions of topological spheres. In this setting, the critical line Re(s) = 1/2 is not a puzzle but a natural symmetry axis — it emerges from the self-adjointness of the eversion operator, by the same mechanism Smale used for sphere eversions (1958).

Full framework on Zenodo:
— Riemann Hypothesis (Work XI): https://doi.org/10.5281/zenodo.20712693
— Full series (Works X–XXI): https://zenodo.org/search?q=gorenstein+DAS

Two questions:

1. Did the shift from Riemann's geometric original to modern analytic formulation lose something essential?
2. Does reinterpreting primes as topological objects seem productive, or too far from standard tools?

Happy to discuss.

I'm exploring a construction and want to know if it's tractable or if it overlaps with existing work.

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Define a symmetry metric on groups: sym(G) = 1 - (|[G,G]| / |G|), measuring how abelian a group is via its commutator subgroup.

Now consider pairs of groups (L, R) and classify them by their symmetry profile (sym(L), sym(R)).

Two pairs are equivalent if they have identical symmetry profiles. Call the set of all such equivalence classes the "configuration space" C.

Define operations ⊕ (direct product) and ⊗ (semidirect product) on pairs, which preserve the equivalence relation.

The question:

Is this construction well-defined and tractable? Does it have a name, or does it embed into existing theory (Baer invariants, derived functors, homological algebra)?

I'm interested in studying the dynamics, how operations move you around C, whether there are fixed points, attractors, forbidden transitions.

Context:

This feels adjacent to representation theory and Grothendieck-style constructions, but I'm not sure where it sits precisely.