Search arXivSearch

arXiv · 2608.25169

ChatGPT solved the dynamic construction problem of Malfatti circles

Abstract

In this paper, we address the dynamic construction problem for Malfatti circles with respect to a fixed triangle, using the LC and CC modules. We show that the configuration of the Malfatti circles constitutes a stable fixed point and provide lower bounds for the main parameters of the LC and CC modules. We now remark that all mathematical proofs in this paper were generated by ChatGPT.

Explore related subjects

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Kazushi Ahara. 2026-08-25. ChatGPT solved the dynamic construction problem of Malfatti circles. https://arxiv.org/abs/2608.25169

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

Sobriety of Scott topologies under countability conditions

In this paper, we focus on the sobriety of the Scott topology in countable case. Specifically, we show that: (1) every meet-continuous core-compact countable dcpo is sober with respect to the Scott topology; (2)every countable locally compact dcpo is sober endowed with the Scott topology; (3) the lattice of all open sets for the rational numbers space Q equipped with the Scott topology is not sober.

math.GN

Exponentiable Objects and Function spaces in Lowen Fuzzy Topological Spaces

We study exponentiable objects and function spaces in the category of stratified Lowen fuzzy topological spaces over \(\I=[0,1]\). Using the Lowen fuzzy Sierpiński object \(\Sier\), which identifies \(τ_X\) with \(C(X,\Sier)\), we explicitly determine the largest splitting topology on this mapping set. Its open weights \(Φ:τ_X\to\I\) are precisely those satisfying Scott continuity and a finite-tier compatibility condition induced by finite powers of \(\Sier\). This yields an intrinsic characterization: \(X\) is exponentiable if and only if every \(μ\inτ_X\) satisfies \[ μ=\bigvee_{λ\triangleleftΦ} (\const{Φ(μ)}\wedgeλ), \qquad λ\triangleleftΦ \Longleftrightarrow \const{Φ(ν)}\wedgeλ\leqν \quad(ν\inτ_X). \] When this condition holds, \(Y^X\) has underlying set \(C(X,Y)\), with topology generated by \([Φ,v](f)=Φ(v\circ f)\). We also obtain a dual closed-set formulation and three applications. Exponentiability implies that \(τ_X\) is a continuous lattice, although the converse fails. Moreover, a classical space \(X\) is exponentiable exactly when its induced fuzzy space \(ωX\) is exponentiable in the entire stratified Lowen category. Finally, Lowen compact, strongly fuzzy compact, and \(N\)-compact Hausdorff spaces are exponentiable.

math.GN

Super calibers in topological spaces and topological hyperspaces

We study the notion of a super caliber of a topological space, which is closely related to the classical notion of caliber and has appeared in the literature under several different names. We collect and unify several known results and establish new results concerning the collections of super calibers of topological spaces and their hyperspaces. In particular, we investigate the relationship between the super calibers of a space $X$ and those of hyperspaces $\mathcal{H}(X)$ lying between $\mathrm{CL}(X)$ and $\mathcal{F}(X)$. For infinite metrizable spaces, we characterize several cases in which $\mathsf{SC}(X)$ and $\mathsf{SC}(\mathrm{CL}(X))$ differ and establish an independence result over \textsf{ZFC}; see Theorem~5.12.

math.GN