6 problems

Topological one-dimensionality conjecture for regular resistance forms
arXiv@arxiv
Topological one-dimensionality conjecture. Any set that carries a regular resistance form is a topologically one-dimensional space when equipped with the associated resistance metr…
Conjectureopenanalysis-on-fractals resistance-forms topological-dimension
0
Applicability of the convergence estimates to the two-dimensional Sierpinski carpet
arXiv@arxiv
The two-dimensional Sierpinski carpet is a fractal space for which the paper's resistance-form and heat-kernel framework may require modifications. Applicability conjecture. The re…
Conjectureopenfractals heat-kernels resistance-forms
1
Hambly–Kumagai conjecture on the DHL resistance exponent
arXiv@arxiv
Let $R_n$ denote the effective resistance between $0$ and $1$ at level $n$ in the unit-resistance setting. Hambly–Kumagai's resistance-exponent conjecture. There exists $\alpha$ su…
Conjectureopenhierarchical-lattices random-walks resistance-forms
1
Resistance-form discretization conjecture for eigenmaps on the real line
arXiv@arxiv
Resistance-form discretization conjecture. Numerically accurate locally uniform approximations to the Laplacian eigenmap can be obtained by discretizing the left- and right-hand si…
Conjectureopeneigenmaps resistance-forms spectral-geometry
0
Approximation results for broader classes of compact resistance spaces
arXiv@arxiv
General approximation conjecture. Under suitable additional conditions, one can produce similar approximation results for p.c.f. self-similar sets with non-regular harmonic structu…
Conjectureopenfractal-geometry partial-differential-equations resistance-forms
0
The renormalization criterion for survival of the fractal resistance form
arXiv@arxiv
Renormalization criterion. The fractal part $\mathcal{E}_\mathcal{R}^\Sigma$ survives if and only if
Conjectureopenfractals resistance-forms spectral-asymptotics
1

6 problems

Topological one-dimensionality conjecture for regular resistance forms

Applicability of the convergence estimates to the two-dimensional Sierpinski carpet

Hambly–Kumagai conjecture on the DHL resistance exponent

Resistance-form discretization conjecture for eigenmaps on the real line

Approximation results for broader classes of compact resistance spaces

The renormalization criterion for survival of the fractal resistance form