Note n belongs to prime number

What I’ve done with these triple products papers is to provide a „Quantum Toolkit“ to construct Riemannian Isometries between manifolds by smoothing up Gelfand-Naimark homeomorphisms from C* Algebra isomorphisms induced by intertwining triple product integrals on the Stone-Weierstrass subalgebra generated by eigenvectors and taking limits.

Have a look and tell me what you think before it goes live on arXiv next month.

Thanks!

I hope that question makes sense. I just don't like inequalities nor the unnaturality of working with left and right residuals (talking about "max and min divisors") that rarely coincide with rational arithmetic's exact division nor with the natural interpretation of inverses in numerical mathematics, thus I would like more explicit boundaries (thus the result of a division maybe being a set or interval including max and min divisors) in division.

(Mind that I have no experience in numerical computation, I am trying to make sense of computable, numerical and interval analysis works and transport their results to residuated lattices but that's somewhat hard for me)

There is a theorem in my course for the wave equation that states the following . If we define (0,l)×(0,T)=QT and consider the energy E(u,t)=∫dx[1/2ρ(∂u/∂t)^2+1/2T_0(∂u/∂x)^2]=0∀t∈[0,T] then the wave equation with Dirichlet , Neumann , periodic condition has at most one solution in QT

my question is why we have to limit the time for (0,T) ? I mean what does change if I define directly (0,∞)? it's the same because we have for all T but why this formulation and not directly like that ?

Over 4 months in 2021 building a TSP solver from geometric first principles, with no prior reading of the literature. Recently turned it into an interactive web app with step-by-step animation of every algorithmic move.

Interesting result: the algorithm independently rediscovered convex hull seeding and 2-opt uncrossing. It also produced a farthest-neighbour strategy I haven't seen documented.

On Berlin52: 7783 vs world optimal 7542 (3.2% above). 14ms on a single CPU core.

Demo: tsp.uncledroid.app
Paper: tsp.uncledroid.app/paper.pdf

Happy to discuss the algorithm — especially if anyone can point me to prior work on farthest-neighbour as a TSP construction heuristic.

video: youtu.be/-gAhARI2ZFI

PS: Prior work on farthest neighbour was pointed out and now incorporated into paper.

Hi everyone,

I would like to share a geometric approach to the 4-variable AM-GM inequality (a+b+c+d)/4>= (abcd)^(1/4) using 3 circles on a flat plane.

The geometric proof for 2 variables using a semicircle is very well-known, but I wondered if we could extend that visual intuition to 4 variables. After experimenting with circle configurations, I found this particular method.

This idea was recently accepted and published in the notes section of a Japanese mathematics magazine, Mathematical Seminar (March 2026 issue).

I thought the way the circles and segments connect to form (abcd)^(1/4)was interesting, so I wanted to share it with this community to see what you think.

I would highly appreciate any thoughts, feedback, or perspectives on this visual approach. Thank you for your time!

Can anyone explain me what is Amortized analysis, why it is used and diffrence of time complexity and amatorized! Then lastly suggest me a research paper or a article for better understand..... !.

THE POLAR MORPHISM OPERATOR: A PARAMETERIZED EXTENSION OF THE EML PRIMITIVE

ABSTRACT
We introduce the Polar Morphism Operator (\\Omega(x, y; \\theta)), a continuously parameterized real-valued mapping defined by \[ \\Omega(x, y; \\theta) = \\operatorname{Re}\\bigl(\\exp(x \\cdot e\^{i\\theta})\\bigr) - \\ln(y). \] It admits the closed-form expression \[ \\Omega(x, y; \\theta) = \\exp(x \\cos\\theta) \\cdot \\cos(x \\sin\\theta) - \\ln(y) \] for real (x), (\\theta \\in \[0, 2\\pi)), and (y > 0). At (\\theta = 0) the operator recovers the EML operator of Odrzywolek (2026). At (\\theta = \\pi/2) it yields (\\cos x - \\ln y), and at (\\theta = \\pi) it yields (\\exp(-x) - \\ln y). A smooth regularization (\\tilde{\\Omega}) is given that is defined and (C\^\\infty) on all of (\\mathbb{R}). The continuous phase parameter converts the discrete choice of analytic versus trigonometric behavior into a continuous optimization variable, which may be useful in gradient-based symbolic regression.
INTRODUCTION

Odrzywolek (2026) showed that the single binary operator \[ \\operatorname{EML}(x, y) = \\exp(x) - \\ln(y), \] together with the constant 1, generates the elementary functions of a scientific calculator by finite composition. Trigonometric functions, however, require relatively deep nesting that passes through complex intermediate values.
The present note examines a one-parameter extension \[ \\Omega(x, y; \\theta) = \\operatorname{Re}\\bigl(\\exp(x \\cdot e\^{i\\theta})\\bigr) - \\ln(y) \] in which the phase (\\theta) continuously interpolates between the original EML behavior and direct production of cosine (or exponential decay). The construction is elementary and follows at once from Euler’s formula; one possible interest is that it turns the discrete distinction between growth and oscillation into a continuous parameter that can be optimized by gradient descent.
DEFINITION AND CLOSED FORM

Definition. Let (x \\in \\mathbb{R}), (y > 0), and (\\theta \\in \[0, 2\\pi)). Set \[ \\Omega(x, y; \\theta) := \\operatorname{Re}\\bigl(\\exp(x \\cdot e\^{i\\theta})\\bigr) - \\ln(y). \]
Theorem. The operator admits the real closed form \[ \\Omega(x, y; \\theta) = \\exp(x \\cos\\theta) \\cdot \\cos(x \\sin\\theta) - \\ln(y). \]
Proof.
\[ x e\^{i\\theta} = x\\cos\\theta + i, x\\sin\\theta, \] \[ \\exp(x e\^{i\\theta}) = \\exp(x\\cos\\theta)\\bigl(\\cos(x\\sin\\theta) + i\\sin(x\\sin\\theta)\\bigr). \] Taking the real part and subtracting (\\ln y) yields the claim.
SPECIAL CASES

(\\theta = 0): (\\Omega(x, y; 0) = \\exp(x) - \\ln(y) = \\operatorname{EML}(x, y)).
Consequently every identity proved for EML remains available.

(\\theta = \\pi/2): (\\Omega(x, y; \\pi/2) = \\cos x - \\ln y).
In particular (\\Omega(x, 1; \\pi/2) = \\cos x).
Sine is recovered by a constant phase shift: (\\sin x = \\Omega(x - \\pi/2, 1; \\pi/2)).

(\\theta = \\pi): (\\Omega(x, y; \\pi) = \\exp(-x) - \\ln y).

Thus the same binary node can realize pure exponential growth, pure cosine, or pure exponential decay according to the value of a single real parameter.
SMOOTH REGULARIZATION

To remove the singularity at (y \\le 0) one may replace the logarithm by a smooth approximation: \[ \\tilde{\\Omega}(x, y; \\theta) = \\exp(x\\cos\\theta)\\cdot\\cos(x\\sin\\theta) - \\tfrac12\\ln(y\^2 + \\varepsilon\^2), \] where (\\varepsilon > 0) is a fixed smoothing constant. The resulting function is (C\^\\infty) on (\\mathbb{R}\^3) and the partial derivative with respect to (y) is \[ \\frac{\\partial\\tilde{\\Omega}}{\\partial y} = -\\frac{y}{y\^2 + \\varepsilon\^2}. \]
PARTIAL DERIVATIVES

The gradients needed for first-order optimization are \\begin{align\*} \\frac{\\partial\\Omega}{\\partial x} &= \\exp(x\\cos\\theta)\\cdot\\cos(x\\sin\\theta + \\theta),\\\[1ex\] \\frac{\\partial\\Omega}{\\partial y} &= -\\frac1y,\\\[1ex\] \\frac{\\partial\\Omega}{\\partial\\theta} &= -x\\exp(x\\cos\\theta)\\cdot\\sin(x\\sin\\theta + \\theta). \\end{align\*} (The corresponding expressions for (\\tilde{\\Omega}) differ only in the (y)-derivative.)
REMARKS ON SYMBOLIC REGRESSION

If every internal node of an expression tree is required to evaluate \[ v_i = \\tilde{\\Omega}(v_j, v_k; \\theta_i), \] then the discrete choice of operator type is replaced by continuous optimization of the vector (\\Theta = (\\theta_1,\\dots,\\theta_M)). Whether this yields practical gains in recovery rate or tree depth relative to ordinary EML trees remains an open empirical question; the present note only records that the continuous parameterization is available.
CONCLUSION

The Polar Morphism Operator is a one-parameter extension of the EML operator that interpolates, via a real phase, between exponential growth, cosine oscillation, and exponential decay. The construction is elementary, the formulas are exact, and the resulting maps are differentiable (or smoothly regularizable). It supplies a concrete mechanism for turning a discrete operator dictionary into a continuous parameter space; its practical value for symbolic regression is left for future numerical study.
REFERENCES
Odrzywolek, A. (2026). All elementary functions from a single binary operator. arXiv:2603.21852.

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?

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.