17 problems

Helix conjecture. The helix provides a solution of the variational shape problem by minimizing the total sliding. This conjecture proposes a distinguished minimal-sliding configura…

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 $f$,…

Higher-dimensional pluri-Lagrangian form conjecture. For $d>2$, integrable hierarchies of $d$-dimensional integrable partial differential equations can be described by pluri-Lagran…

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.

Let $Q=[-1,1]^2\subset\mathbb{R}^2$, and consider shear maps of the form … where $\sigma$ belongs to the specified subclass of $W^{1,2}(Q;\mathbb{R})$ and satisfies the uniform Lip…

Let $L_0$ be a prescribed constant, and consider the class of all smooth interpolating curves whose length is at most $L_0$. The bending energy is the energy being minimized among…

Automatic-adjustment conjecture. Independence with respect to the initial value of $\alpha$ means that $\alpha$ is automatically adjusted even if it undergoes strong variations.

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…

Higher-order non-existence conjecture. The same non-existence assertion holds for higher-order evolutionary systems with nondegenerate linearizations of their right-hand sides.

Let a linear scalar second-order ordinary differential equation be an r-CODE, and let a system of two linear second-order ordinary differential equations be obtainable from it usin…

Let a scalar linear second-order ordinary differential equation be complexified, and let the resulting system of two real second-order ordinary differential equations be linear and…