9 problems

Let $k$ be a positive integer, and consider an additive $2k$th-order difference equation … Here $\mathbf{x}_{n}^{(m,l)}=(x_{n+m},\dots,x_{n+k})$ as defined in the source, and $f$,…

Symmetry-group correspondence conjecture. There is a one-to-one correspondence between symmetry groups acting on $(M,A,f)$ and stable solutions for $\psi$ in $(M,A,f)$.

Stable-symmetry characterization conjecture. Stable solutions are characterized by geometrical symmetries: there is a group of symmetries acting on the space that preserves the sol…

Probability interpretation conjecture. This quantity equals the real probability that the solution occurs.

Exterior-determination conjecture. The field $\psi$ is then determined precisely inside $U$.

Uniqueness and representation conjecture. The field $\psi$ is uniquely determined in $U$ by $f$, its boundary values, and those two equations. Moreover, $\psi$ can be expressed in…

Uniform finite-order stability conjecture. There exists a constant $K(\lambda,n,R)$ such that stability for the local length-minimization problem in every such $U$ can be determine…

Finite-variation stability conjecture. To determine whether the solution is stable over a compact domain, it suffices to examine derivatives of only finitely many orders.

Permanent tracking-control conjecture. Under certain conditions, still to be defined, the Euler–Lagrange equation applied to the variational model-free Lagrangian establishes the p…