2 problems

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Conjecture on the optimal parameter domain for analytic response solutions
Optimal-domain conjecture. The optimal domain of $\varepsilon$ on which $U_{\varepsilon}$ can be considered as a function does not extend to the imaginary axis.
Conjectureopenfunctional-analysis quasi-periodic-systems response-solutionsarXiv posted 2 days ago
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Conjecture on nonexistence of regular response solutions at exceptional small parameters
Optimality conjecture. This domain is optimal: for these small parameter values and general forcing and nonlinearity, regular solutions do not exist.
Conjectureopendynamical-systems quasi-periodic-systems response-solutionsarXiv posted 2 days ago

Conjecture on the optimal parameter domain for analytic response solutions