Experimental detection of interactive phenomena and their analysis

Varkonyi and Domokos proved that homogeneous convex bodies with any prescribed numbers $S\ge1$ of stable and $U\ge1$ of unstable static equilibria exist, the case $S=U=1$ being the mono-monostatic Gomboc. We give a complete constructive answer for the simplest nondegenerate shapes: a homogeneous body whose boundary is the unit sphere perturbed radially by a single real spherical harmonic $Y_\ell^m$ of degree $\ell\ge2$ and order $1\le m\le\ell$ has exactly $S=U=m(\ell-m+1)$ stable and unstable equilibria, for every amplitude in the convex range when $m\ge2$, and for small amplitude when $m=1$. The reduction to the critical points of the perturbing harmonic is an identity when $m\ge2$: the symmetry of a single tesseral harmonic pins the centroid at the origin exactly, so the centroid-to-surface distance is a strictly increasing function of the harmonic. We count the critical points of $Y_\ell^m$ from its separable structure and verify the Poincare-Hopf balance in index form. Three consequences follow. (i) A single harmonic populates only the diagonal $S=U$ of the equilibrium-class lattice; no single harmonic of degree $\ge2$ is mono-monostatic. (ii) Any centrally symmetric (even-degree) perturbation has even $S$ and $U$ and cannot be mono-monostatic, a parity obstruction. (iii) Mono-monostaticity is therefore intrinsically multi-harmonic. A predict-then-confirm numerical study on eight single-harmonic bodies matches the formula exactly. For the multi-harmonic regime we give a certified equilibrium counter (interval arithmetic with Krawczyk uniqueness and Hessian classification) that, given the centroid, provably neither under- nor over-counts. It certifies that a member of a known analytic family, at an amplitude of our own choosing inside the strictly convex sub-regime, is mono-monostatic. For that body the centroid is exactly the origin, so its certificate is unconditional.

math.GM↗

arXiv2026

A Simple Proof of the Riemann Hypothesis

In this article, it is proved that the non-trivial zeros of the Riemann zeta function must lie on the critical line, known as the Riemann hypothesis.

math.GM↗

arXiv2026

Irreducibility of the Cuboid Polynomial $P_{a,u}(t)$ via a Rank-Zero Elliptic Curve

In this paper we study the even monic degree-8 cuboid polynomial $P_{a,u}(t)$ introduced by R. A. Sharipov in the first-cuboid specialization of his cuboid equations. For nonzero integers $a,u$ with $u^2\neq a^2$ we prove that $P_{a,u}(t)$ is irreducible in $\mathbb{Z}[t]$ (equivalently, in $\mathbb{Q}[t]$), thus confirming Sharipov's irreducibility conjecture in this two-parameter case. Over $K=\mathbb{Q}(\sqrt2)$ we have a factorization $P_{a,u}(t)=H_-(t)H_+(t)$ into two conjugate quartics. We show that reducibility of either $H_\pm$ would force the discriminant of a certain quadratic in $S=t^2$ to be a square in $K$, which in turn implies (via $τ=(au/Δ)^2$, where $Δ=u^2-a^2$) the existence of a rational point $(y,v)\in\mathcal{C}(\mathbb{Q})$ on the genus-one quartic $\mathcal{C}: v^2=16y^4+136y^2+1$ with $y^2=τ$. We give an explicit isomorphism $\overline{\mathcal{C}}\simeq E$ with the elliptic curve $E: Y^2=X(X-8)(X-9)$, which has conductor $48$ and Mordell-Weil rank $0$. Enumerating $E(\mathbb{Q})$ and tracing back to $\mathcal{C}(\mathbb{Q})$ rules out the only possible values $τ\in\{0,1/4\}$, and hence excludes any further factorization of $H_\pm$ in $K[t]$. A quadratic Galois descent then yields the irreducibility of $P_{a,u}(t)$ over $\mathbb{Q}$ and $\mathbb{Z}$.

math.GM↗