arXane › Mathematics › Mathematics
Wednesday, 4 November 2026: 3 new papers, 0 cross-lists and 2 replacements in General Mathematics (math.GM), in arXiv's order, with full abstracts. The same papers as arXiv's own listing, set as a newspaper; every title links to arXiv.
On a hereditarily precompact quasi-metric space, every bijective non-contractive self-map has the following structure: the quasi-metric splits as $d = \lambda^- + D_0$. The invariant limit register $\lambda^-$ is a quasi-pseudo-metric below $d$ for which the map is an isometry, with totally bounded symmetrization on every orbit, so that the map's rigid shadow is a rotation of a compact monothetic group; the asymmetric limit structure on it, a priori only paratopological, carries the compact topology. The defect $D_0$ lies in a band between a past fee difference and the fee of its source, the fee nondecreasing along orbits and vanishing in the deep past. The defect is zero precisely when the map is an isometry. Deep sources are thus almost rigid: in the deep past the map is an isometry to within any $\varepsilon$, against every target. Every such map that is not an isometry is a toll over its rigid shadow, levied on a forward-invariant zone that sheds into the past. Two further facts are forced: the orbit of the source of any strict pair is infinite and already strict, and the source pays its whole defect on its own backward returns, uniformly on a chain-closed set of displacements. The band is attained at both ends, and beyond it the shape of the defect is not forced: three spirals over the circle show that the toll can depend on the target, that a source carrying a fee can descend for free, and that an ascent can carry defect; no rate of shedding is forced either, and for zone tolls (two-valued descent tolls that are shift-monotone and subadditive along their own routes) the fee reaches zero already at a finite depth, by the ascending chain condition on submonoids of $\mathbb{N}$. The results are machine-checked in Lean, apart from a few statements listed at the end.
We study arc length on the first-quadrant unit curve $x^p+y^p=1$, $p>1$, when length is measured with respect to an $L^r$ norm. Denoting this length by $L(p,r)$, we prove the H\"older-duality identity $L(p,r)=L(r^*,p^*)$, where $p^*=p/(p-1)$ and $r^*=r/(r-1)$. The identity is interpreted as a special case of the symmetry of mixed area (equivalently, of the anisotropic perimeter pairing). For the distinguished dual metric $r=p^*$, the length element agrees with the areal angular element: $L(p,p^*)=\int_\gamma (x\,dy-y\,dx)=\frac1p B(1/p,1/p)$. Thus the familiar coincidence between arc length and angular parameter on the Euclidean unit circle is the self-dual case $p=2$ of a more general relation between $L^p$ geometry and its dual metric. As a consequence, the Euclidean arc length of the $L^p$ unit curve admits a dual trigonometric representation.
We enumerate dissections of a 30-60-90 triangle into smaller triangles similar to itself, counting unlabeled arrangements up to similarity and reflection. There are 18 four-piece, 121 five-piece, and 844 six-piece arrangements, corresponding to 11, 47, and 235 distinct multisets of piece areas. We present illustrated catalogs organized by these area multisets and distinguish layouts obtained by subdivision and local rotations or reflections from independent layouts. A separately implemented computational search reproduces every arrangement up to nine pieces, finding 460404 arrangements. Counts through nine pieces for five other classes of triangles are included for comparison.