2 problems

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The multiplicity-two conjecture for the second Dirichlet eigenvalue on simply connected planar domains
Multiplicity-two conjecture.
Conjectureopeneigenvalue-multiplicities planar-domains spectral-geometryarXiv posted 2 days ago
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The conjectured formula for the first minimum zero eigenvalue multiplicity
Conjectured identity. For $w\geq 5$ and $n\geq 2w+1$ ,
Conjectureopeneigenvalue-multiplicities extremal-combinatorics johnson-schemearXiv posted 2 days ago